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AC ELECTRIC MOTORS
CONTROL

AC ELECTRIC MOTORS
CONTROL
ADVANCED DESIGN TECHNIQUES
AND APPLICATIONS

n

The right of the author to be identified as the author of this work has been asserted in accordance with the Copyright,
Designs and Patents Act 1988
...
No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any
form or by any means, electronic, mechanical, photocopying, recording or otherwise, except as permitted by the UK
Copyright, Designs and Patents Act 1988, without the prior permission of the publisher
...
Some content that appears in print may not be
available in electronic books
...
All brand names and
product names used in this book are trade names, service marks, trademarks or registered trademarks of their
respective owners
...
This
publication is designed to provide accurate and authoritative information in regard to the subject matter covered
...
If professional advice
or other expert assistance is required, the services of a competent professional should be sought
...
and is used with permission
...
This book’s use or discussion of MATLAB? software or related
products does not constitute endorsement or sponsorship by The MathWorks of a particular pedagogical approach or
R
particular use of the MATLAB? software
...

pages cm
Includes bibliographical references and index
...
Electric motors, Alternating current–Automatic control
...
Giri, Fouad, editor of compilation
...
A33 2013
621
...
1
1
...
3

AC Motor Features
Control Issues
1
...
1
State-Feedback Speed Control
1
...
2
Adaptive Output-Feedback Speed Control
1
...
3
Fault Detection and Isolation, Fault-Tolerant Control
1
...
4
Speed Control with Optimized Flux
1
...
5
Power Factor Correction
Book Overview
1
...
1
Control Models for AC Motors
1
...
2
Observer Design Techniques for AC Motors
1
...
3
Control Design Techniques for Induction Motors
1
...
4
Control Design Techniques for Synchronous Motors
1
...
5
Industrial Applications of AC Motors Control
References

Part One
2

1
3
3
3
4
6
7
8
9
9
10
11
12
13

Control Models for AC Motors

Control Models for Induction Motors

17

Abderrahim El Fadili, Fouad Giri, and Abdelmounime El Magri

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4
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5

Introduction
Motor Representation and Estimation Issues
4
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1
Problem Statement
4
...
2
Short Literature Review
Some Observer Approaches
4
...
1
Estimation under known and constant speed and Parameters
4
...
2
Estimation under known Speed and Parameters
4
...
3
Estimation under unknown Speed and known Parameters
4
...
4
Estimation in the presence of unknown Speed and/or Parameters
Some Illustration Results
4
...
1
State and Parameter Estimation under known Speed
4
...
2
State and Speed Estimation under known Parameters
4
...
3
State, Parameter, and Speed Estimation
4
...
4
Estimation close to Unobservability
Conclusions
References

59
60
60
61
63
63
64
64
66
66
68
69
71
74
75
76

Contents

5

State Observers for Active Disturbance Rejection in
Induction Motor Control

vii

78

Hebertt Sira Ram´rez, Felipe Gonz´ lez Monta˜ ez, John Cort´ s Romero, and
ı
a
n
e
Alberto Luviano-Ju´ rez
a

5
...
2

5
...
A

6

Introduction
A Two-Stage ADR Controller Design for the Induction Motor
5
...
1
The Flux Simulator
5
...
2
Formulation of the Problem and Background Results
5
...
3
Assumptions
5
...
4
Problem Formulation
5
...
5
Control Strategy
5
...
6
Experimental Results
Field-Oriented ADR Armature Voltage Control
5
...
1
Control Decoupling Property of the Induction Motor System
5
...
2
Problem Formulation
5
...
3
Control Strategy
5
...
4
Experimental Results
Appendix
5
...
1
Generalities on Ultra-Models and Observer-Based Active
Disturbance Rejection Control
5
...
2
Assumptions
5
...
3
Observing the uncertain System through the Ultra-Model
5
...
4
The Observer-Based Active Disturbance Rejection Controller
References

78
80
80
81
81
81
82
86
90
91
92
92
95
99
99
99
101
102
103

Observers Design for Systems with Sampled Measurements, Application
to AC Motors

105

Vincent Van Assche Philippe Dorl´ ans Jean-Francois Massieu
e
¸
and Tarek Ahmed-Ali

6
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2
6
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4

6
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3
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7
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3

7
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5

Introduction
7
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1
Problem Statement
7
...
2
State of the Art and Objectives
SPMSM Modeling and its Observability
7
...
1
SPMSM Model
7
...
2
Quick Review on the Observability of SPMSM
Robust MRAS Observer
7
...
1
Reference Model
7
...
2
Adjustable Model
7
...
3
Adaptation Mechanism
7
...
4
Rotor Position Observer
Experimental Results
7
...
1
Nominal Conditions
7
...
2
Parameter Variation Effect
7
...
3
Load Torque Effect
Conclusions
References

123
123
124
125
125
125
125
125
127
128
129
129
130
132
133
133
134

Part Three Control Design Techniques for Induction Motors
8

High-Gain Observers in Robust Feedback Control of Induction Motors

139

Hassan K
...
Strangas

8
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4

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10
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5
11

Introduction
Induction Motor Modeling with Saturation Effect Inclusion
Controller Design
10
...
1 Control Objective
10
...
2 Rotor Flux Reference Optimization
10
...
3 Speed and Flux Control Design and Analysis
Simulation
Conclusions
References

188
190
194
194
194
197
202
205
205

Experimental Evaluation of Nonlinear Control Design Techniques for
Sensorless Induction Motor

207

Jes´ s De Le´ n, Alain Glumineau, Dramane Traore, and Robert Boisliveau
u
o

11
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11
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7
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7
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7
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8

Conclusions
References

231
231

12

Multiphase Induction Motor Control

233

Roberto Zanasi and Giovanni Azzone

12
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12
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4
12
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2
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3
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3
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4
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4
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1
13
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3

13
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5
14

Introduction
Modeling “AC/DC/AC Converter—Doubly-Fed Induction Motor”
Association
13
...
1 Doubly-Fed Induction Motor Model
13
...
2 Modeling of the System “DC/AC Inverter–DFIM”
13
...
3 AC/DC Rectifier Modeling
Controller Design
13
...
1 Control Objectives
13
...
2 Motor Speed and Stator Flux Norm Regulation
13
...
3 Power Factor Correction and DC Voltage Controller
Simulation Results
Conclusions
References

253
255
255
257
257
260
260
260
266
269
273
273

Fault Detection in Induction Motors

275

Alessandro Pilloni, Alessandro Pisano, Martin Riera-Guasp, Ruben Puche-Panadero,
and Manuel Pineda-Sanchez

14
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1
15
...
3
15
...
5

Introduction
PMSM Models and Problem Formulation
15
...
1 Problem Formulation
Controller Structure and Main Result
Unavailability of a Linearization-Based Design
Full Information Control
15
...
1 Port-Hamiltonian Model

313
314
316
316
318
319
319

xii

Contents

15
...
2 A Full-Information IDA-PBC
15
...
3 Certainty Equivalent Sensorless Controller
15
...
(2011)
15
...
1 Flux Observer and Stability Properties
15
...
2 Description of the Observer in Terms of ραβ
15
...
8 Proof of the Main Result
15
...
1 Currents and Speed Tracking Errors
15
...
2 Estimation Error for ραβ
15
...
3 Speed and Load Torque Estimation Errors
15
...
4 Proof of Proposition 15
...
1
15
...
9
...
9
...
10 Future Research
15
...
1
16
...
3
16
...
5
16
...
7
16
...
2012)
16
...
1 Response to Time-Varying Load Torque
16
...
2 Response to Parameter Uncertainties
Experimental Setup and Results (Bifaretti et al
...
1
17
...
3
17
...
5

Introduction
Preliminaries
17
...
1 PMSM Modeling
Control Design
17
...
1 A Robust Observer of Rotor Angular Position and Velocity for the
Tracking Problem
The Faulty Case
Simulation Tests
References

370
371
371
372
372
375
376
380

Contents

18

On Digitization of Variable Structure Control for Permanent Magnet
Synchronous Motors

xiii

381

Yong Feng, Xinghuo Yu, and Fengling Han

18
...
2
18
...
4
18
...
6

18
...
8

19

Introduction
Control System of PMSM
Dynamic Model of PMSM
PI Control of PMSM Servo System
High-Order Terminal Sliding-Mode Control of PMSM Servo System
18
...
1 Velocity Controller Design
18
...
2 q-Axis Current Controller Design
18
...
3 d-Axis Current Controller Design
18
...
4 Simulations
Sliding-Mode-Based Mechanical Resonance Suppressing Method
18
...
1 Load Speed Controller Design
18
...
2 d-Axis Current Controller Design
18
...
3 q-Axis Current Controller Design
18
...
4 Simulations
Digitization of TSM Controllers of PMSM Servo System
18
...
1 Backward Difference Discretization Method
18
...
2 Bilinear Transformation
Conclusions
References

381
382
383
384
385
386
386
387
387
388
390
391
391
392
393
393
393
396
396

Control of Interior Permanent Magnet Synchronous Machines

398

Faz Rahman and Rukmi Dutta

19
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2

19
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4

19
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6
19
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1
20
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3

20
...
5

Introduction
System Modeling
20
...
1 Three-Phases AC/DC Rectifier Modeling
20
...
2 Inverter-Motor Subsystem Modeling
Nonlinear Adaptive Controller Design
20
...
1 Control Objectives
20
...
2 Inverter-Motor Subsystem Control Design
20
...
3 Reactive Power and DC Voltage Controller
Simulation
20
...
1 Simulation and Implementation Considerations
20
...
2 Simulation Results
Conclusion
References

Part Five
21

429
431
431
433
435
435
436
443
446
446
448
450
450

Industrial Applications of AC Motors Control

AC Motor Control Applications in Vehicle Traction

455

Faz Rahman and Rukmi Dutta

21
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2

21
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4
21
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1
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1
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2
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2
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2
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4
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22
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22
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7
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2
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3
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7
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7
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Chattopadhyay

23
...
2

23
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4

23
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6

23
...
8
Index

Introduction
High-Power Semiconductor Devices
23
...
1 High-Power SCR
23
...
2 High-Power GTO
23
...
3 IGCT/GCT
23
...
4 IGBT
23
...
5 IEGT
High-Power Converters for AC Drives and Control Methods
23
...
1 Pulse Width Modulation for Converters
23
...
2 Control Methods of High-Power Converter-Fed Drives
Control of Induction Motor Drives
23
...
1 Induction Motor Drives with Scalar or Volts/Hz Control
23
...
2 Induction Motor Drives with Vector Control
23
...
3 Induction Motor Drives with Direct Torque Control (DTC)
Control of Synchronous Motor Drives
23
...
1 Synchronous Motor Drives with Scalar Control
23
...
2 Synchronous Motor Drives with Vector Control
Application Examples of Control of High-Power AC Drives
23
...
1 Steel Mills
23
...
2 Cement and Ore Grinding Mills
23
...
3 Ship Drive and Marine Electric Propulsion
23
...
4 Mine Hoists, Winders, and Draglines
23
...
5 Pumps, Fans and Compressors in the Industry
New Developments and Future Trends
Conclusions
References

509
510
511
511
513
514
514
515
516
516
517
517
527
531
534
534
537
539
539
544
544
546
547
548
548
549
553

List of Contributors
Haitham Abu-Rub
Department of Electrical & Computer Engineering, Texas A&M University at Qatar, Qatar
Tarek Ahmed-Ali
GREYC Lab, University of Caen Basse-Normandie, France
Vincent Van Assche
GREYC Lab, University of Caen Basse-Normandie, France
Giovanni Azzone
Dipartimento di Ingegneria “Enzo Ferrari”, Universit` di Modena e Reggio Emilia, Italy
a
Gildas Besancon
¸
Control System Department, GIPSA Lab, Grenoble Institute of Techology and Institut
Universitaire de France, France
Marc Bodson
Electrical and Computer Engineering, University of Utah, USA
Robert Boisliveau
Ecole Centrale de Nantes, LUNAM, France
Ajit K
...
Khalil
Department of Electrical and Computer Engineering, Michigan State University, USA
Zbigniew Krzeminski
Faculty of Electrical and Control Engineering, Gdansk University of Technology, Poland
Arkadiusz Lewicki
Faculty of Electrical and Control Engineering, Gdansk University of Technology, Poland
Xuefang Lin Shi
Ampere Lab, INSA Lyon, France
Alberto Luviano-Ju´ rez
a
UPIITA, IPN, M´ xico
e

List of Contributors

xix

Riccardo Marino
Dipartimento di Ingegneria Elettronica, Universit´ di Roma “Tor Vergata,” Italy
a
Jean-Francois Massieu
¸
GREYC Lab, University of Caen Basse-Normandie, France
˜
Felipe Gonz´ lez Montanez
a
Departamento de Energ´a, Universidad Aut´ noma Metropolitana, M´ xico
ı
o
e
Giuseppe Orlando
Dipartimento di Ingegneria dell’Informazione, Universit` Politecnica delle Marche, Italy
a
Romeo Ortega
LSS, SUPELEC, CNRS, France
Hamid Ouadi
FSAC, University of Casablanca, Morocco
Alessandro Pilloni
Deptartment of Electrical and Electronic Engineering (DIEE), University of Cagliari, Italy
Manuel Pineda-Sanchez
Deptartment of Electrical Engineering (DIE), Universidad Polit´ cnica de Valencia, Spain
e
Alessandro Pisano
Deptartment of Electrical and Electronic Engineering (DIEE), University of Cagliari, Italy
Ruben Puche-Panadero
Deptartment of Electrical Engineering (DIE), Universidad Polit´ cnica de Valencia, Spain
e
Faz Rahman
School of Electrical Engineering & Telecommunications, University of New South Wales,
Australia
Hebertt Sira Ram´rez
ı
Department of Electrical Engineering, CINVESTAV-IPN, M´ xico
e
Martin Riera-Guasp
Deptartment of Electrical Engineering (DIE), Universidad Polit´ cnica de Valencia, Spain
e
John Cort´ s Romero
e
Department of Electrical and Electronic Engineering, Universidad Nacional de Colombia,
Colombia
...
Strangas
Department of Electrical and Computer Engineering, Michigan State University, USA
Alexandru Ticlea
¸
Department of Control and System Engineering, Polytechnic University of Bucharest,
Romania

xx

List of Contributors

Patrizio Tomei
Dipartimento di Ingegneria Elettronica, Universit´ di Roma “Tor Vergata,” Italy
a
Dramane Traore
Ecole Centrale de Nantes, LUNAM, France
Cristiano Maria Verrelli
Dipartimento di Ingegneria Elettronica, Universit´ di Roma “Tor Vergata,” Italy
a
Xinghuo Yu
School of Electrical and Computer Engineering, RMIT University, Australia
Roberto Zanasi
Dipartimento di Ingegneria “Enzo Ferrari”, Universit` di Modena e Reggio Emilia, Italy
a

Preface
The last two decades have witnessed spectacular developments in the technologies of power
electronics and microprocessors
...
For this goal to be fully realized, one
should get as much benefit as possible from the considerable progress made in the field of
control theory
...
Of course,
not all nonlinear control methods apply to AC motors, but a significant fraction does
...
In this respect,
special focus is made on the topics of sensorless nonlinear observers, adaptive and robust
nonlinear controllers, output-feedback controllers, fault detection and isolation algorithms,
and fault-tolerant controllers
...
Most of the concepts and methods are presented by their own inventors
...
Although
it includes many aspects of the theory, it is nonetheless beneficial to practitioners who will be
able to use the methods without necessarily understanding every single detail of the theory
...
For students and
newcomers, the main prerequisites are undergraduate courses on linear and nonlinear system
control, on electric machines, and on power electronics
...

I am grateful to my colleagues from around the world who generously contributed to
this work, bringing together considerable knowledge from a wide range of aspects of the
research area
...
Marc Bodson and Vincent van Assche not only directly contributed by coauthoring chapters, but also helped in other ways, with Marc reading and correcting parts
of the manuscript, and Vincent retyping parts in Latex and compiling the whole manuscript
...
Special and warm thanks
go to Abderrahim El Fadili, Abdelmounim El Magri, and Hamid Ouadi, all three my former
PhD students and current collaborators
...
I am
particularly grateful to Abderrahim and Abdelmounim who kindly accepted to retype in Latex
some chapters initially written in Word by the authors
...

Fouad Giri
University Distinguished Professor
University of Caen Basse-Normandie
France

1
Introduction to AC Motor Control
Marc Bodson1 and Fouad Giri2
1
2

Electrical and Computer Engineering, University of Utah, USA
GREYC Lab, University of Caen Basse-Normandie, France

1
...
In this section, some basic facts are
recalled, focusing on the features that contribute to the success of AC motors in motion control
and to the continuing growth of their applications
...
There is a wide variety of such machines
that differ by their operating principles, physical characteristics, and power level
...

Induction motors exist in two main types, squirrel cage and wound rotor
...

The rotor coils are made accessible on the stator side through slip rings
...
The rotor bars are connected by circular conductors placed at the extremities
...

For both types of motors, the stator windings generate a rotating magnetic field when
supplied with polyphase AC
...
By Faraday’s
and Lenz’s laws, currents are induced in the rotor windings whenever the rotor speed differs
from the speed of the magnetic field produced by the stator
...
Under this
constraint, a change of rotational speed requires an adjustment of the stator electrical
frequency
...
Edited by Fouad Giri
...
Published 2013 by John Wiley & Sons, Ltd
...
1 AC motor control architecture

Synchronous motors also exist in two versions, namely, permanent-magnet and wound rotor
...
A motor torque
is developed due to the interaction between the stator rotating field and a rotor field generated
either by permanent magnets or by an injected rotor current
...
Until the development of modern power electronics,
there was no simple and effective way to vary the frequency of the motors’ supply voltages
...
Specifically, an AC motor is supplied with power through an association
of two power converters, a rectifier and an inverter (Figure 1
...
The former, also called AC/DC
converter, converts the AC power provided by the grid into DC power
...
The inverter, also referred to as DC/AC converter, transforms the DC
voltage into an AC voltage with a specified frequency
...
In this
respect, it is worth emphasizing the considerable progress made in computer technology, which
has resulted in fast multiprocessor computers and high-performance analog-digital interfaces
...

DC motors require schemes similar to Figure 1
...
However, ACs are produced in conductors through mechanical commutation, rather than electrical commutation
...
As a result, AC motors offer a higher power/mass ratio, relatively low cost, and
simple maintenance
...
For these reasons, AC drive systems
have already replaced DC drives in several industrial fields and this widespread proliferation

Introduction to AC Motor Control

3

is expected to continue
...

2
...

4
...


Transport: vehicle traction, marine propulsion
Milling in cement, steel, paper, and others industries
Pumping/compressing in oil and gas industry
Cranes and industrial vehicles
Domestic machines: lifts, washing machines, and others
...
2
1
...
1

Control Issues
State-Feedback Speed Control

The prime objective in AC motor control is to make the rotor turn at a desired speed despite
load variations
...
The desired speed, also called the speed reference signal, is often unknown a priori, making the control issue more difficult
...

In AC induction motors, the generation of a given torque necessitates a sufficient level of
rotor magnetization, that is, a sufficiently high flux magnitude in the rotor
...
These signals are binary signals commanding on and off conduction modes
...
Furthermore, the three-phase nature of the motor
means that the overall model is nonlinear, of high dimension, as well as controlled through
binary signals
...
g
...
The binary nature of the inverter signals is generally coped with by averaging
the signals over the PWM period and letting the control design be based on the corresponding
averaged two-coordinate model (see, e
...
, Sira-Ramirez and Silva-Ortigoza 2006)
...
1998; Isidori 1999; Sastry 1999; Vidyasagar 2002; Khalil 2003)
...
2
...
2
...
2 assumes that all controlled system
parameters are known
...
In particular, the stator and the rotor
resistances are sensitive to the magnitude of the currents, and thus undergo wide variations in
the presence of speed reference and load torque changes
...
g
...
To maintain the control

4

AC Electric Motors Control

Speed and flux
references
Speed-flux controller

2-3
Transformation

3-2
Transformation
AC
motor

AC
power
supply

Rectifier
AC/DC

Inverter
DC/AC

Load

Figure 1
...

Another limitation of the control strategy of Figure 1
...
However, reliable and cheap sensors are only available
for stator currents and voltages
...
Mechanical sensors (for speed
and, more rarely, torque measurements) are common, but also entail reliability issues and extra
maintenance costs due to physical contact with rotor
...

¸
Sensorless controllers involving online state estimation using observers are commonly referred
to as output-feedback controllers
...
3, combine
both features: parameter adaptation and sensorless output-feedback
...
2
...
Faults may originate from the failure of certain system components, for example,
sensors, inverter, rectifier, power supply, or even stator/rotor windings
...
Regardless, the controller designed on the basis of a faultless model may see its
performance deteriorate drastically, sometimes causing unsafe operation of the whole system
...
The development of FTC systems has been
an active research topic, especially over the past 15 years, and a review of relevant concepts
and methods can be found in Blanke et al
...
(2009)
...
3 AC motor modern control strategy: adaptive output-feedback speed control

approaches
...
The active
FTC approaches are those that dynamically react to fault occurrence by performing control
reconfiguration
...
Selecting online (within a set of predesigned laws) the control law that best fits the detected
fault type
...
Redesigning online the control law to adapt it to the detected faulty situation
...
The role of the
latter is twofold:
1
...

2
...

FDI techniques are broadly classified as information-based, model-based (MB), and artificialintelligence-based
...
(2010)
...
This is
illustrated in Figure 1
...


6

AC Electric Motors Control

Decision on FDI

Adaptive observer
Parameter
estimates

Reconfiguration

Speed torque and
flux estimates

Adaptive
speed-flux controller

2-3
Transformation

Speed and flux
references

3-2
Transformation
AC
motor

AC
power
supply

Rectifier
AC/DC

Inverter
DC/AC

Load

Figure 1
...
2
...
However, energetic efficiency is only maximal when the motor operating conditions, essentially determined by the load torque, remain close to the nominal conditions
...
Then, in presence of small
loads (compared to nominal load), maintaining the nominal flux entails a waste of energy and
a lower than optimal energetic efficiency
...
In general,
speed-control strategies involving constant flux references do not guarantee optimal machine
performance in the sense of maximal energetic efficiency and maximal torque
...
Thus, the flux reference must be state-dependent (Figure 1
...

Flux weakening is also used for both synchronous and induction machines to maximize the
torque at high speeds in the presence of voltage constraints
...
5 AC motor control strategy combining ftc and flux optimization

1
...
5

Power Factor Correction

The role of the rectifier in Figures 1
...
3, 1
...
5 is to convert the supply AC power
into DC power and transmit this power through a constant DC link voltage
...
Moreover, the rectifier-invertermotor set strongly interacts with the AC power supply net as the power flow is bidirectional:
the direction depends on the speed profile and on load variations
...
” This harmonic pollution has several damaging effects on
the quality of power distribution along the AC line, for example, electromagnetic compatibility
issues, voltage distortion, larger power losses, and so on
...
g
...
Of course, the harmonics and power factor correction
(PFC) can be improved using additional equipment and/or over-dimensioning the converter
...


8

AC Electric Motors Control

Decision on FDI

Adaptive observer
Parameter
estimates

Speed torque and
flux estimates
Flux
reference

Reconfiguration

Adaptive
Speed-flux controller

Flux
reference
optimizer

Speed
reference

PFC and
DC link voltage
controller

AC
power
supply

Rectifier
AC/DC

Park and park-inverse
transforamtion
AC
motor
Inverter
DC/AC

Load

Figure 1
...
Figure 1
...


1
...
The present book is not intended to be an encyclopedic survey of all existing solutions
for all types of AC machines
...
The focus is on
the most significant types of AC motors and a representative sample of application fields
...
In addition

Introduction to AC Motor Control

9

to the present introduction, the book includes 22 other chapters organized in five parts, briefly
overviewed in the following subsections
...
3
...
It consists of Chapters 2 and 3, respectively, which are devoted to induction motors
and synchronous motors
...
The corresponding three-phase models are established, and it is
shown how simpler two-phase models can be obtained using specific coordinate transformations
...
The established models will prove to be useful for control design in later
chapters
...
3
...
Chapter 4
is about the design and analysis of sensorless state observers for induction motors
...
The case of lack of speed
measurement and unknown electrical parameters is considered
...

The performance of the approach is evaluated, both by simulations and experiments
...
Generalized proportional integral (GPI) observers are proposed, in combination with
linear output feedback controllers, for the direct field-oriented control and the classical fieldoriented two-stage output feedback controller design
...
The high-gain GPI observers
estimate online the output phase variables and the lumped disturbance inputs affecting the
underlying linear dynamics
...

In Chapters 4 and 5, the nonlinear state observers are developed for induction motors
assuming a continuous-time implementation
...

Then, a classical solution consists of constructing a discrete-time approximation of the
(continuous-time) observer
...
Chapter 6 presents a different solution based on the hybrid continuous-discrete
estimation principle
...
They are formally shown to maintain satisfactory estimation
accuracy when applied to the (continuous-time) system model, and the theoretical result is
confirmed using experimental data
...
A
special emphasis is placed on the loss of observability arising at zero-speed without mechanical

10

AC Electric Motors Control

sensors
...

The obtained observer turns out to be robust against disturbances and avoids the chattering
phenomenon inherent to standard first order SMs
...
Experimental results are carried out to highlight
the technological interest of the proposed method
...


1
...
3

Control Design Techniques for Induction Motors

The third part of the book includes seven chapters dealing with control design techniques
for induction motors
...
The framework for
studying field orientation in the presence of model uncertainty is used to design and analyze
a nonlinear output feedback controller that requires only measurements of the rotor position
and stator currents
...
A high-gain observer is
used to estimate the rotor speed and acceleration from its position measurement
...
In this
case, a high-gain observer is used to estimate the speed from the field-oriented currents and
voltages
...
It addresses the
problem of adaptively controlling induction motors in order to achieve rotor speed and flux
magnitude tracking, all without resorting to mechanical sensors
...
Adaptive output-feedback controllers are developed and formally shown to solve the control problem
...
The specific observability and identifiability
conditions that allow for the exponential tracking and identification of the uncertain parameters are emphasized in terms of persistent excitation conditions
...

Chapter 10 is on nonlinear control of induction motors for speed regulation with maximal
energetic efficiency
...
The controller includes an optimal flux reference
generator (optimal in the sense of minimal stator current consumption) and a nonlinear regulator obtained using the backstepping design technique
...

Chapter 11 presents an experimental evaluation of two robust control design techniques
for induction motors, including a nonlinear backstepping technique with integral terms that
improve robustness against parametric uncertainties, and a high-order SM technique designed
for its intrinsic robustness quality
...

Chapter 12 is on multiphase induction motor control
...
Different field-oriented control strategies are
designed in the multiphase and compared by investigating the trade-off between the different
solutions that differ in required control degrees of freedom
...
A multiloop nonlinear controller is developed
using the backstepping design technique
...
The controller is formally shown to meet its objectives, that is,
accurate motor speed-reference tracking, tight regulation of the DC Link voltage, and PFC
...
Both MB and data-driven (DD) methods are
developed
...
DD methods, also referred to as “motor current signature analysis” (MCSA), are
designed using processing techniques based on FFT, Hilbert transforms, and more advanced
time/frequency combined analysis techniques
...
3
...
In Chapter 15, PMSMs are considered and the control problem is solved using a passivity-based
output-feedback control
...
It involves different observers that estimate the mechanical coordinates (speed and load
torque) and stator fluxes (which, in turn, are used to obtain information related to the mechanical position)
...

Chapter 16 is on adaptive output-feedback control of PMSMs
...
Satisfactory performance is obtained even in the presence of inaccurate motor
parameters, time-varying load torques, current sensing errors, and discrete-time controller
implementation
...
It proposes first a robust
speed observer making use of currents measurements only
...
Finally, a residual-based detection approach is discussed for sensor
faults affecting current measurements
...
SM controllers are
designed to control the motor position and velocity and currents
...
In addition, an SM-based mechanical resonance suppressing method is proposed
...
Finally, a high-order SM control is designed to guarantee the stability
of the system
...
The discretization behaviors of PMSM servo systems are analyzed,
which helps obtain approximate boundary conditions for the sampling period
...
The
control trajectories in terms of the current and voltage limit boundaries for optimum drive
response is discussed
...
Both open- and closed-loop operations are described
...

Chapter 20 is about nonlinear state-feedback controllers for three-phase wound-rotor
synchronous motors
...
To achieve
these objectives, an adaptive control strategy is developed, based on a nonlinear model
of the whole “converter-motor” set
...
The closedloop system stability and performance properties are formally analyzed using averaging
theory
...
3
...

Chapter 21 is on AC motor control applications in vehicle traction
...
The chapter reviews
recent trends of vehicle traction drive architectures in the marketplace and in the available literature
...
The
battery management system with temperature compensation for charging and discharging current limits and for monitoring the state of charge (SOC) are discussed
...
Examples of vehicle traction systems that use induction and IPM machines are
included, as well as other types of machines in commercially available traction applications
...
It illustrates how state observers can be useful in fault diagnostic for modern high-speed
electric traction applications
...
The analysis of speed and load torque signals makes it possible to assess the
state of speed sensor and torque transmission systems in electric traction vehicle
...
The proposed diagnostic system is applied
to a HST propelled by an induction squirrel-cage motor
...

Chapter 23 is on AC motor control applications in high-power industrial drives
...
The brief features of the
industrial AC drives developed by the leading manufacturers worldwide are presented together
with new developments and trends for the future
...
Springer
...
Springer
...
(2000) What is fault-tolerant control? IFAC Symposium on Fault Detection,
Supervision and Safety for Technical Process, pp
...

Blaschke F (1972) The principle of field orientation applied to the new trans-vector closed-loop control system for
rotating field machine
...

Chiasson JN (2005) Modeling and High Performance Control of Electric Machines
...

Hindmarsh, J (1985) Electrical Machines and their Applications 4th edn
...

Hwang I, Kim S, Kim Y, and Seah CE (2010) A survey of fault detection, isolation, and reconfiguration methods
...

Isermann R (1984) Process fault detection based on modeling and estimation methods: a survey
...

Isermann R (2005) Model-based fault-detection and diagnosis status and applications
...
Reviews Control, 29,
71–85
...
Springer
...
Prentice Hall, Upper Saddle River NJ
...
John Wiley & Sons
...
Springer, New York
...
Springer
...

Springer, Berlin
...
Clarendon Press, Oxford
...
Springer Verlag
...
Springer
...
Proceedings of Institution of
Mechanical Engineers, Part G: Journal of Aerospace Engineering, 219, 263–275
...
SIAM, PA
...
Automatica, 12, 601–611
...
Annual Reviews
in Control, 32, 229–252
...
Edited by Fouad Giri
...
Published 2013 by John Wiley & Sons, Ltd
...
1

Introduction

It is widely recognized that the induction motor has become one of the main actuators for
industrial use
...

It is largely agreed that these machines have promising perspectives in the industrial actuator
field
...

However, the problem of controlling induction motors is not a simple issue due to the
multivariable and highly nonlinear nature of these machines
...
The
problems of induction motor control and observation have been given a great deal of interest
over the last decade (see, e
...
, Lubineau et al
...
2006; Traore et al
...
2012a)
...

Control-oriented modeling of induction motors has first been accomplished by considering
simplified assumptions, for example, linear magnetic characteristic and constant (or slowly
varying) rotor speed (e
...
, Lubineau et al
...
Then, the obtained models turn out to be
linear and of quite limited use
...
g
...

2001)
...
Edited by Fouad Giri
...
Published 2013 by John Wiley & Sons, Ltd
...
That is, most control designs presented in this book are based on
the models that are by now standard
...
This will be accomplished by considering successively
the cases of triphase motors and doubly-fed induction motors (DFIM)
...
2 is devoted to a brief description of triphase
induction motor principles; the modeling of these motors is completed in Section 2
...
4; the modeling of the DFIM machine is presented in Section 2
...


2
...
In squirrel-cage induction motors (Figure 2
...
The rotor winding is made
of longitudinal bars in slots fixed just beneath the rotor outer surface
...
The rotor winding may be constituted
of individual bars and rings made of conducting material connected together, or it may be a
one-piece structure made by die casting together the rings and the bars
...
This provides the motor with robustness but makes its control
more complex
...
Connections from the coils are brought to
collector rings
...

Rather, it is completed through external circuit (resistors, converters, etc
...
2)
...
1 Squirrel-cage induction motor
...
(For a
color version of this figure, please see color plates
...
2 Wound rotor of the DFIM
...
ieee
...


For both types of machine, a three-phase equivalent circuit is associated to the stator and to
the rotor (Figure 2
...
By Faraday’s and Lenz’s Laws, the stator carrying a sinusoidal current
of pulsation ωs generates a rotating magnetic field
...
The induced currents tend to oppose the flux variation in the rotor coils resulting
in a mechanical torque applied on the rotor
...
The electromagnetic torque is
proportional to the pulsation ωr
...
This
is called synchronization
...
This difference is
called slip pulsation and constitutes an image of the torque
...
3 Induction motor structure equivalent

isc

20

2
...
3
...
Linearity: The fluxes and the corresponding induced currents are proportional, that is, all
self- and mutual inductances are constant
...
All iron losses are neglected
...
The machine air gap is constant, smooth, and symmetric
...
The stator and the rotor windings present a symmetrical structure providing the induction
machine with a three-phase equivalent circuit (equation 2
...

The machine triphase structure entails a sinusoidal spacial distribution of magnetomotive
force (MMF) in the air gap and three-phase currents in the stator and rotor currents whenever
the stator voltage is three-phase
...
3
...
g
...
The application
of the electromagnetic laws yields six voltage equations and six flux equations
...

dt
d
[vrabc ] = [Rr ][i rabc ] + [φrabc ]
...
1)
(2
...


(2
...


(2
...

φsc
⎡
⎤
φra
[φrabc ] = ⎣ φrb ⎦
...
5)

(2
...
The subscripts s and r refer to the stator and the
rotor, respectively
...


21

Control Models for Induction Motors

A direct consequence of the machine perfect symmetry is that all resistance and inductance
matrices are symmetric, that is,
⎡

⎡

los
[L os ] = ⎣ Mos
Mos

⎡

⎤
0
0 ⎦,
Rs

0
Rs
0

Rs
[Rs ] = ⎣ 0
0

Mos
los
Mos

0
Rr
0

Rr
[Rr ] = ⎣ 0
0
⎡

⎤
Mos
Mos ⎦ ,
los

lor
[L or ] = ⎣ Mor
Mor

⎤
0
0 ⎦,
Rr

⎤
Mor
Mor ⎦ ,
lor

Mor
lor
Mor

(2
...
8)

where Rs and Rr are the stator and the rotor resistances, los and lor are the self-inductances, Mos
is the mutual inductance between two stator phases, and Mor is the mutual inductance between
two rotor phases
...
3
...
Specifically, one has
⎡

cos( pθ )
?
?
⎢
[Mosr ] = Mo ⎣ cos pθ + 4π
3
?
?
cos pθ + 4π
3

?
cos pθ +

2π
3

?

cos( pθ )
?
?
cos pθ + 2π
3

?
cos pθ +
?
cos pθ +

?

4π ⎤
3
?
2π ⎥
,
3 ⎦

(2
...


Mechanical Equations
The rotor motion undergoes the following usual second order differential equation:
J

dωm
= −Fωm + Tem − TL − Td ,
dt

(2
...
J designates the inertia of the rotor-load set, and F the viscous
friction coefficient
...
Specifically, one has
Tem =

∂ Wmag
∂θ

with

Wmag =

?
1?
[i sabc ]T [φsabc ] + [i rabc ]T [φrabc ] ,
2

(2
...
The induction motor model, including the seven
equations (2
...
2), and (2
...
On the one hand, the system order is
relatively large and, one the other, the matrix (2
...
To overcome these difficulties, adequate coordinate transformations are available

22

AC Electric Motors Control

that reduce the system order and eliminate the dependence on θ
...


2
...
3

Park Transformations

The key idea is that the MMF, created by a physical three-phase system, can be equivalently
created by a fictive two-phase system involving two orthogonal windings (Figure 2
...

The three-phase current system (i a , i b , i c ) traversing n 1 turns and two-phase current system
(i d , i q ), traversing n 2 turns are said to be equivalent if they produce the same air-gap MMF
...


Similarly, the components of the MMF due to (i d , i q ) are the following:
εd = n 2 i d ,

εq = n 2 i q
...
4, the MMF due to (i a , i b , i c ) is represented by the vector ε , which is
?
a vector sum of the three MMF vectors (εa , εb , εc )
...
4 illustrates the projection of
? ? ?
the vector ε along two orthogonal axes referred to direct axis d and quadrature axis q
...
4 Triphase system (i a , i b , i c ) and its equivalent two-phase system (i d , i q )
...

− sin ψ − 4π
3
ε

(2
...
12) is clearly noninvertible as it involves a nonsquare matrix
...
The new variable is
defined to be proportional to the homopolar component of the triphase (εa , εb , εc )
...
To the
fictive MMF εo is associated a fictive current, denoted i o , referred to homopolar
...

Replacing in equation (2
...

3
ic
K

(2
...
For convenience, the homopolar axis is let to be orthogonal to the
plane dq
...
13), it remains to assign values to n 1 and K o
...

Park Transformation Preserving Amplitudes
The Park transformation goes back to 1929 (Blaschke 1972; (Vas 1990)
...
The spectacular advances
achieved in these fields have made it possible to implement real-time applications involving
the construction and manipulation of the Park transformation
...
13) letting the free parameters (i
...
, n 1 and K o ) be chosen to meet
n2
the following requirements:
1
...

2
...

The first requirement leads to the following double equality:
io =

1
n1
K o (i a + i b + i c )
...

n2
3

(2
...


(2
...
13) that
i d (t) =

n1 3
Im cos(ωt − ψ),
n2 2

i q (t) =

n1 3
I
n2 2 m

sin(ωt − ψ)
...
16)

Comparing equations (2
...
16) gives, using (2
...

2

(2
...
17), it follows from (2
...

3

1
2

(2
...

⎦

(2
...

⎦

(2
...
13), the free parameters, that is, the ratio n 1 /n 2 and K o , are chosen to
ensure power conservation when passing from the triphase system to the two-phase system
...
21)

va i a + vb i b + vc i c = vd i d + vq i q + vo i o
...
21) may be rewritten in the following more compact vector form:
[vabc ]T [i abc ] = [vdqo ]T [i dqo ]
...
22)

By (2
...
Then, it follows from
(2
...


(2
...
22) yields
[P(ψ)]T = [P(ψ)]−1

(2
...

2

Then, the direct Park matrix conserving the power is

[P(ψ)] =

?

⎡

cos(ψ)

2⎢
⎢ − sin(ψ)
3⎣
1
√

2

?
?
cos ψ − 2π
3
?
?
− sin ψ − 2π
3
1
√
2

?
? ⎤
cos ψ − 4π
3
?
?⎥
− sin ψ − 4π ⎥ ,
3 ⎦

(2
...

⎦

(2
...
Note that, the angle ψ = 0 is a free parameter in that transformation
...

⎦

(2
...
3
...

⎦

(2
...
1)–(2
...
10) get simplified by applying the Park transformation defined
by the matrix (2
...
Roughly, all mathematical relationships initially expressed in terms of
the triphase frame (a, b, c) are rewritten in terms of (d, q, o)
...
Then, the corresponding homopolar currents (i
...
the
components along the axis o) are null (Blaschke 1972; Vas 1990; Leonard 2001)
...
1), and (2
...
As mentioned
earlier, the angle ψ in (2
...
This entails
several variants of the two-coordinate frame (d, q)
...

• The rotating reference frame (d, q), linked to, for example, the rotor flux or the stator current
...
25), as follows:
• Set ψ = 0, for the transformation of the stator variables
...

The passage from the triphase frame (a, b, c) to the rotating frame (d, q) is accomplished
by choosing the transformation angle ψ as follows:
• Set ψ = θs , for the transformation of stator variables
...


27

Control Models for Induction Motors

Sa
θ

Ra
θr

d

θs
Rb

Sb

Sc

q

Rc

Figure 2
...
5):
?

?

vsd
vsq
vrd
vrq

?
?

= [P(θs )][vsabc ],
= [P(θr )][vrabc ],

?

?

i sd
i sq
i rd
i rq

?

?

= [P(θs )][i sabc ],
= [P(θr )][i rabc ],

?

?

φsd
φsq
φrd
φrq

?

?

= [P(θs )][φsabc ], (2
...
(2
...
29) and (2
...
1)
and (2
...
31)

+ ωs φsd ,

(2
...
33)

+ (ωs − pωm )φrd ,

(2
...

dt

Flux Equations in dq Coordinates
Similarly, the passage from the triphase frame (a, b, c) to the (d, q) frame necessitates the
following transformations of the fluxes:
[φsdq ] = [P(θs ][φsabc ]
...
35)
(2
...
3) and (2
...
35) and (2
...
29)
[φsdq ] = [P(θs ][L s ][P(θs ]−1 [i sdq ] + [P(θs ][Msr ][P(θr ]−1 [i rdq ]
...
37)

At the rotor, one has
[φrdq ] = [P(θr ][L r ][i rabc ] + [P(θr ][Msr ][i sabc ],
which implies, due to (2
...


(2
...
39)

with
L s = lso − Mos , cyclical stator inductance;

L r = lro − Mor , cyclical rotor inductance;
Msr =

3
Mo , mutual inductance between the stator and rotor windings
...
40)
(2
...
42)

29

Control Models for Induction Motors

Mechanical Equation in dq Coordinates
The mechanical equation of the induction motor describes the motion of the rotor carrying a
load
...

dt
J
J
J
J

(2
...
44)

where pm designates the mechanical power developed by the motor
...
45)

where the second equality is a direct consequence of the Park transformation preserving
power when passing from the three- to the two-coordinate frame
...
31),
(2
...
33), and (2
...
45), it follows that that the power pa can be
decomposed into three parts:
1
...


(2
...
The power related to the electromagnetic exchange with sources:
i sd

dφsq
dφrq
dφsd
dφrd
+ i sq
+ i rd
+ i rq

...
47)

3
...

dt
dt

(2
...
39), and replacing the rotor current components
(d, q) by their equivalent expressions, one obtains from (2
...

Lr
dt

(2
...
44) that the torque rewrites as follows:
Tem = p

Msr
(φrd i sq − φrq i sd )
...
50)

30

AC Electric Motors Control

The electromagnetic torque can also be expressed in the following two forms:
Tem = p(φsd i sq − φsq i sd ),

Tem = p Msr (φsd i rq − φsq i rd )
...
51)

Induction Motor Model in General dq Coordinate Frame
Eliminating the flux components (φsd , φsq ) and the current components (i rd , i rq ) in equations
(2
...
32), (2
...
34) and substituting equation (2
...
10), one gets
the model of the induction machine expressed in the rotating frame (d, q):
TL
Msr
Td
F
dωm
(φrd i sq − φrq i sd ) −
= p
−
− ωm ,
dt
J Lr
J
J
J
disd
Msr
Msr Rr
1
φrd + pωm
φrq +
vsd ,
= −γ i sd + ωs i sq +
dt
σ Ls L2
σ Ls Lr
σ Ls
r
disq
Msr
Msr Rr
1
φ − pωm
φrd +
vsq ,
= −γ i sq − ωs i sd +
2 rq
dt
σ Ls Lr
σ Ls Lr
σ Ls
dφrd
Rr Msr
Rr
i sd ,
= − φrd − (ωs − pωm )φrq +
dt
Lr
Lr
dφrq
Msr Rr
Rr
i sq ,
= − φrq + (ωs − pωm )φrd +
dt
Lr
Lr

(2
...
53)
(2
...
55)
(2
...

Lr

(2
...
58)

Induction Motor Model in Fixed αβ Coordinate Frame
The model of the induction motor in the fixed (α, β) frame is obtained by just letting ωs = 0
in the general model described by equation (2
...
Doing so, one gets the following model:
TL
Msr
Td
F
dωm
(φrα i sβ − φrβ i sα ) −
= p
−
− ωm ,
dt
J Lr
J
J
J

(2
...
60)

31

Control Models for Induction Motors

disβ
Msr Rr
Msr
1
φ − pωm
φrα +
vsβ ,
= −γ i sβ +
2 rβ
dt
σ Ls Lr
σ Ls Lr
σ Ls

(2
...
62)

dφrβ
Msr Rr
Rr
i sβ
...
63)

Induction Motor Model in Oriented d-q Reference Frame
The rotor-flux-oriented model is obtained by considering as rotating reference frame the one
whose d-axis coincides with the rotor flux ?r
...
An immediate consequence is that the model becomes a fourth order
(instead of fifth in the preceding model)
...
54) that the
pulsation ωs takes the following value:
ωs = pωm −

Msr Rr i sq

...
54) gives the-rotor-flux oriented model
TL
Msr
Td
F
dωm
?r i sq −
= p
−
− ωm ,
dt
J Lr
J
J
J
disd
Msr Rr
1
? +
vsd ,
= −γ i sd + ωs i sq +
2 r
dt
σ Ls Lr
σ Ls
disq
Msr
1
?r +
vsq ,
= −γ i sq − ωs i sd − pωm
dt
σ Ls Lr
σ Ls
Rr Msr
Rr
d?r
i sd
...
3
...
64)
(2
...
66)
(2
...

Considering the flux components, φsd and φsq , and the current components, i rd and i rq , as
state variables and assuming that magnetic circuit is linear, the two-phase model of the DFIM,
represented in a rotating reference frame (d, q), is as follows:
F
TL
Msr
Td
dωm
(φsq i rd − φsd i rq ) − ωm −
= p
− ,
dt
J Ls
J
J
J
dφsd
Msr
1
i rd + vsd ,
= − φsd + ωs φsq +
dt
τs
τs

(2
...
69)

32

AC Electric Motors Control

dφsq
Msr
1
i rq + vsq ,
= − φsq − ωs φsd +
dt
τs
τs
dird
γ2
= −γ1 i rd + (ωs − pωm )i rq + φsd − pωm γ2 φsq − γ2 vsd + γ3 vrd ,
dt
τs
dirq
γ2
= −γ1 i rq − (ωs − pωm )i rd + φsq + pωm γ2 φsd − γ2 vsq + γ3 vrq
...
70)
(2
...
72)

The parameters γ1 , γ2 , γ3 , σ , and τs are defined as follows:
γ1 =

2
Rr L 2 +Rs Msr
s
,
σ Lr L2
s

γ2 =

Msr
,
σ Ls Lr

γ3 =

1
,
σ Lr

σ =1−

2
Msr
,
Ls Lr

τs =

Ls

...
Then, the power exchanged between
the motor and the grid can be controlled by controlling the currents
...
Then, the model
(2
...
69), (2
...
71), and (2
...
73)

dφsd
Msr
1
i rd + Vs ,
= − φsd + ωs φsq +
dt
τs
τs

(2
...
75)

dird
γ2
= −γ1 i rd + (ωs − pωm )i rq + φsd − pωm γ2 φsq − γ2 Vs + γ3 vrd ,
dt
τs
dirq
γ2
= −γ1 i rq − (ωs − pωm )i rd + φsq + pωm γ2 φsd + γ3 vrq
...
4

(2
...
77)

Identification of Induction Motor Parameters

In this section, an experimental procedure is described to get estimates of the mechanical and
the electrical parameters of a squirrel-cage induction machine
...
2 kW power, whose characteristic are described in Table 2
...


2
...
1

Identification of Mechanical Parameters

Deceleration Test
The present experiment consists, in first running the motor, being loadless (TL = 0)
...
The time when the voltage is turned
off is considered as t = 0, for the present experiment, and the motor speed at that moment is

33

Control Models for Induction Motors
Table 2
...
2
380/420
4
...
81

kW
V
A
Hz
rad/min

denoted ωmo
...
52) simplifies to, for t > 0,
J

dωm
= −Fωm − Td
...
78)

Clearly, equation (2
...
Then, the least squares
J
J
estimator can be resorted to get estimates of these quantities using a sufficiently large sample
of measurements (t, ωmo ) (e
...
, Ioannou and Fidan 2006)
...
The key point is that the solution of the first order equation
(2
...
Specifically, one has
?
?
−t
Td
Td
e τm − ,
ωm (t) = ωmo +
F
F

(2
...
The expression
J
(2
...
Figure 2
...
1 is
submitted to the deceleration test
...
Writing equation (2
...

F
F

(2
...
81)

Subtracting side-to-side equation (2
...
81), gives
?
?
?
? −t1
−t1
Td
ωm2 − ωm1 = ωmo +
e τm e τm − 1
...
82)

Also, one immediately gets from (2
...
83)

ωm1 − ωmo = ωmo +

Td
F

??

?
−t1
e τm − 1
...
5 rad/s

140

ωm1 = 125
...
05 rad/s

80
60
40
20
0
0

5
t1=5 s

10
15
t2=10 s

20

25

Time (s)

30

35
40
ts=36 s

45

Figure 2
...
82) and (2
...
Solving that equation, one gets the following expression that determines
τm = F from available informations
J
τm = −

ln

?

t1
ωm 2−ωm 1
ωm 1−ωmo

(2
...


A second useful expression is immediately obtained from equation (2
...
e
...
Doing
so, one gets an equation where the only unknown quantity is Td
...

F
τm − 1
e

(2
...
6
...
2 concerning the quantities ωmo ,
ωm1 , ωm2 , t1 , and ts
...
2

The main results for the deceleration test

t1

ts

ωmo

ωm1

ωm2

5s

36 s

156, 5 rd/s

125, 82 rd/s

98
...
3 Measurements obtained during
a progressive startup test
Pas

Vas
21
...
9 W

Ias
1
...
84) and (2
...
17 s,

Td
F

= 149
...
86)

where τm = F
...
That is, a supplementary test is needed to determine the mechanical
parameters
...
Accordingly,
the stator voltage being initially null (stationary machine) is very slowly increased until the
motor only just starts moving
...
Then, it follows
from equation (2
...
44)
...
87)
Td = p
ωs
where Pas is the power absorbed at the moment when the motor only just begins turning and ωs
is the grid voltage frequency
...
Table 2
...
1
...
1, one readily gets ωs = 157 rd/s and from Table 2
...
9 W
...
87)
...
86), gives the numerical values of all mechanical parameters (see Table 2
...


2
...
2

Identification of Electrical Parameters

Traditionally, the steady-state model of a three-phase induction motor is represented by the
per phase equivalent circuit
...
7 (Leonard 2001)
...
4
J
0
...
0022 Nms/rd

0
...
7 Per phase equivalent circuit, viewed from the stator, of a three-phase induction motors
...
L s
is the magnetizing and stator inductance; Rc is the equivalent resistance for core loss; and s is the slip
...
The tests are quite similar to those performed on transformers
...
This experimental test
gives, for the induction motor characterized by Table 2
...
5 ?,

(2
...
Currents, voltages, and
powers are measured at the motor input
...
The no-load rotor current is then negligible and the rotor branch of the equivalent circuit
can also be negligible
...
8
...
8 Induction machine equivalent circuit in loadless test

37

Control Models for Induction Motors
Table 2
...
6 A

220 V

Pnl
140 W

1046
...
5 rad/s

In the loadless test, the losses are caused by the core, the stator copper, and the friction
...
89)

where the stator copper losses are given by
2
Pcop = 3Rs Inl
...
90)

2
Prot = Fωm nl + Td ωm nl ,

(2
...
92)

where Vnl , Inl , Pnl , Q nl , and ωm nl denote, respectively, the stator voltage, the stator current, the
absorbed active power, the absorbed reactive power, and the rotor speed
...
1, yield the numerical values of
Table 2
...

The stator inductance L s and the core resistance Rc are given by the following expressions:
Ls =

2
3Vnl
,
ωs Q nl

Rc =

2
3Vnl

...
93)

From equation (2
...
5, one gets the following values:
L s = 441
...
94)

Blocked Rotor Test
In this experiment the rotor is blocked, that is, prevented from turning
...
Again, all currents, voltages and powers are measured at the motor input
...
Then, the secondary impedance becomes much

38

AC Electric Motors Control

Rs

N

Ibl = In

R

Vbl

Figure 2
...
9
...

The three-phase reactive power Q bl
...

The line current Ibl
...


(2
...

N +L s

(2
...
1,
are described in Table 2
...


Table 2
...
5 V

Ibl

Pbl

Q bl

4
...
7

Electrical parameter estimates of the induction motor

Rs

Rr

Ls

Lr

Msr

Tr

σ

2
...
6 mH

441
...
6 mH

100
...
33

2
...
6 and using the approximation that L r = Msr , one gets from equations
(2
...
96) the electrical parameter estimates described in Table 2
...


2
...
First,
the triphase model is established applying electromagnetic and mechanical laws
...

However, it is hardly applicable in control design due to its high complexity
...
The simpler
(two-phase) models are still nonlinear but will prove to be tractable in control design
...
As a matter of fact, in real-life
machines, the magnetic circuit characteristic is nonlinear and linear approximations are only
accurate if the machine operation does not entail wide-range rotor flux variation
...
The point is that a constant-flux
operation cannot be optimal in the presence of varying load, when large speed variations are
needed
...
Models that account for the nonlinear nature of the
machine magnetic characteristic have been presented in Levi (1995), Novotnak et al
...
(2011), and El Fadili et al
...
An example of such models will be presented
in Chapter 10 of this book
...
Siemens Review, 93, 217–220
...
International Journal of Control, 74, 290–302
...
(2012a) Adaptive nonlinear control of induction motors through AC/DC/AC
converters
...

El Fadili A, Giri F, Magri A, et al
...
Speed regulation,
flux optimization and power factor correction
...

Ioannou PA and Fidan B (2006) Adaptive Control Tutorial
...

Leonard W (2001) Control of Electrical Drives
...

Levi E (1995) A unified approach to main flux saturation modeling in D-Q axis models of induction machines
...

Lubineau D, Dion JM, Dugard L, and Roye D (2000) Design of an advanced nonlinear controller for induction motor
and experimental validation on an industrial benchmark
...


40

AC Electric Motors Control

Montanan M, Peresada S, and Tilli A (2006) A speed-sensorless indirect field-oriented control for induction motors
based on high gain speed estimation
...

Novotnak RT, Chiasson J, and Bodson M (1999) High performance motion control of an induction motor with
magnetic saturation, IEEE Transactions on Control Systems Technology, 7, 315–327
...
International Journal of Modeling, Identification and Control
...

Traore D, Plestan F, Glumineau A, and De Leon J (2008) Sensorless induction motor: high order sliding mode
controller and adaptive interconnected observer
...

Vas P (1990) Vector Control of AC Machines
...


3
Control Models for
Synchronous Machines
Abdelmounime El Magri, Fouad Giri, and Abderrahim El Fadili
GREYC Lab, University of Caen Basse-Normandie, France

3
...
Due to these features, the synchronous machine is nowadays widely used in various fields, for example, electric traction, high-speed machining, automotive, robotics, watches,
computer peripherals, and energy production (Krause et al
...
Synchronous machines exist
in two main variants: wound-rotor synchronous machines (WRSM) and permanent-magnet
synchronous machines (PMSM)
...
Compared with induction motors, synchronous motors (especially PMSMs) are more
suitable for electric traction as due to their better mass/power ratio, their higher power level,
and their better efficiency
...
WRSMs are quite suitable for high-power
applications such as train driving (for instance, the French high-speed train “Atlantic TGV” is
equipped with such type of machines) (El Magri et al
...
They also feature three control
inputs (i
...
, stator current amplitude, flux angle, and filed current), which makes them, just as
their permanent-magnet cousins, able to ensure power optimization and high-power efficiency
...
In
the literature, three main modelling approaches have been developed for electrical machines
in general
...
1981;

AC Electric Motors Control: Advanced Design Techniques and Applications, First Edition
...

© 2013 John Wiley & Sons, Ltd
...


42

AC Electric Motors Control

Arkkio et al
...
This method provides a very accurate
description of the electromagnetic field distribution in the machine
...
The second modelling method,
referred to permeance network, consists of representing the machine magnetic circuit by an
equivalent circuit diagram (Hecquet and Brochet 1998; Srairi et al
...
The obtained models
are reasonably accurate but they suffer a large sensitivity to the air-gap permeance
...
The equations are made simpler by considering commonly
admitted assumptions, for example, sinusoidal induction in the air gap, linearity of the magnetic
circuit, negligence of iron losses, higher harmonics in slots, and spaces not account for
...
The models thus
obtained prove to be reasonably accurate and tractable for control design
...
2; in Section 3
...
4 and 3
...


3
...
It is
reversible in the sense that it can operate either as a current generator (alternator) or as a motor
...
1)
...

The windings are fixed in notches on the magnetic circuit
...
1 Electrical and magnetic structure of a wound-rotor synchronous machine with salient poles
(number of poles p = 2)

43

Control Models for Synchronous Machines

Stator winding
Air gap

Stator

Permanent-magnet
Rotor (the magnetic circuit)
Figure 3
...
The rotation speed of the turning field
is proportional to both the number of poles of the machine and the pulsation of the stator
currents (Krause et al
...
2005)
...
1),
or consisting of permanent magnets, in case of permanent-magnet machines (Figure 3
...
In
the first case, the rotor consists of poles and windings wounded around
...
In the second case, the MMF is generated by the permanent magnets
...
This interaction results
in an electromagnetic torque applied to the rotor entailing a rotation motion
...

This motivates the designation “synchronous machine” attributed to this type of machines
...
3
3
...
1

Preliminaries
Modeling Assumptions

The following assumptions will prove to be useful in simplifying the machine modelling
procedure, leading to tractable models:
A1: The induced electromagnetic force is assumed to be sinusoidal (with quasisinusoidally distributed stator winding)
...
Therefore, the
reluctance of the flux path is composed solely of the air gap and leakage reluctances
...

A3: The Foucault current and the hysteresis losses are insignificant
...
The
above assumptions are generally accepted as the resulting modelling errors are negligible in
normal operation modes
...
3
...


(3
...
2b)
(3
...
An inherent property to (balanced)
three-phase systems is that their homopolar components, that is, x 0 = x a + x b + x c , are null all
the time
...

It was already emphasized in Chapter 2 that, a three-phase system like equation (3
...
This frame is sometimes also
said to be stationary or stator-related
...
Accordingly,
one has
⎞
? ?
xa
⎝ x b ⎠ = C 32 x α
xβ
xc
⎛

(3
...
3)

where C 32 denotes the Concordia (3 × 2) matrix defined by
C 32

? ⎛ 1
2⎝
−1/2
=
3
−1/2

⎞
√0
3/2 ⎠
...


(3
...
3
...
5)

where, ρ is the angular position of the rotating reference frame (dq)
...
Note that P(ρ)T P(ρ) = I2
...
4), it turns out that the
passage from a stationary three-phase system (a, b, c) to the corresponding bi-phase system
expressed in the rotating dq-frame is performed as follows:
?

3
...

xc

(3
...
1)
...
3)
...
3 Electric structure of the wound-rotor synchronous machines

46

AC Electric Motors Control

magnetically coupled with each other
...
7)

and
vf = Rfif +

dφ f
,
dt

(3
...

The fluxes of phase windings a, b, c, and f can be expressed in terms of the self- and mutual
inductances
[φsabc ] = [L ss ][i sabc ] + [Ms f ]i f ,

(3
...
9b)

with,
?
[i sabc ] = i sa

?
[M f s ] = M f a
⎛

La
[L ss ] = ⎝ Mba
Mca

i sb

i sc

Mfb
Mab
Lb
Mcb

?T

;

Mfc

?
[φsabc ] = φsa

?T

;

⎞
Mac
Mbc ⎠ ;
Lc

φsb

φsc

[Ms f ] = [M f s ]T ;
⎛

Rs
[Rs ] = ⎝ 0
0

0
Rs
0

?T

;

⎞
0
0 ⎠
...
10a)
(3
...
10c)

In the above expressions, [i sabc ] and [φsabc ] are the stator current and flux, [M f s ] is the mutual
inductance between the rotor and the stator, [L ss ] is the stator inductance matrix, and [Rs ] is
the stator resistor
...
Accordingly, when the rotor makes a complete turn, the geometrical configuration
is repeated 2 p times with p being the number of pole pairs
...
The mutual inductance
between the stator and the rotor windings also has a varying period, equal to pθ
...


(3
...
11b)
(3
...
12a)

Mbc = Ms0 + L sv cos(2 pθ ),

(3
...


(3
...
13a)

Mb f = M f s cos( pθ − 2π/3),

(3
...
13c)

where L s0 , L sv , Ms0 , Ms f , and L f are nonzero real constants depending on the machine
structure
...
9a–b), it follows from equation (3
...
8) that the electrical equations
of the three-phase model (a, b, c) can be rewritten as follows:
[vsabc ] = [Rs ][i sabc ] +
vf = Rfif +

d
{[L ss ][i sabc ] + [Ms f ]i f },
dt

(3
...

dt

(3
...
16)

where TL denotes the load torque, F is the viscous friction coefficient, and J the global rotorload inertia
...
17)

where [L] includes all inductances and [i] all currents
...


(3
...
It immediately
follows that
?
?
?
?
d[Ms f ]
1
d[L ss ]
(3
...

Tem = [i sabc ]T
[i sabc ] + [i sabc ]T
2
dθ
dθ

48

AC Electric Motors Control

The stator and the rotor voltages equations (3
...
15) constitute, together with the
mechanical equation (3
...
From expressions (3
...
This makes this model difficult to be exploited in control design
...
This entails burdensome real-time implementations especially in transient
regimes
...
Analytically described by equation (3
...
14), (3
...
10b–c) to the corresponding rotating dq-frame two-phase model
...


3
...
1

Oriented dq-Frame Model of Salient Pole WRSM

As explained in Chapter 2, it is beneficial to let the dq-frame be rotating at the rotor speed
and be oriented along the the rotor flux d-axis so that the rotor flux q-component can be set to
zero, reducing the model size (Blaschke 1972)
...
6), on the (three-phase) current [i sabc ], the voltage
[vsabc ], and the flux [φsabc ], one obtains the two-phase system ([i sdq ], [vsdq ], and [φsdq ])
...
The obtained equations are listed in order
...
4 abc- and dq-coordinate frame in wound-rotor synchronous machines

49

Control Models for Synchronous Machines

Voltage Equations in dq Coordinates
[vsdq ] = [Rs ][i sdq ] +
vf = Rfif +

d[φsdq ]
+ pωQ[φsdq ],
dt

dφ f

...
20)
(3
...


(3
...
22b)
(3
...
23)

where L d = 3(L s0 + L sv )/2 is called d-axis inductance, L q = 3(L s0 − L sv )/2 is the q-axis
√
inductance, and M = 3/2M f s is the direct axis magnetizing inductance (mutual between
the stator and rotor windings)
...
20), (3
...
22), and (3
...
24a)

= −b1 i sq − b2 i sd ω − b3 i f ω + b4 vsq ,

(3
...
24c)

= −d1 R f i f + d2 i sd − d3 i sq ω − d4 vsd + d5 v f
...
24d)

The fourth order state-space representation (equations (3
...
This model is still nonlinear
but it will prove to useful for control design purpose
...
The model state vector includes
the stator currents (i sd , i sq ), the rotor excitation current i f , and the rotor speed ω
...
1) are constant and all its state variables
are measurable
...

The model coefficients are described in Table 3
...


50

AC Electric Motors Control

Table 3
...
24a–d)

a1 = p(L d − L q )
Rs
b1 =
Lq
Rs L f
c1 =
Ld L f − M2
Ld
d1 =
Ld L f − M2

a2 = p M
Ld
b2 = p
Lq
M
c2 =
Ld L f − M2
Rs M
d2 =
Ld L f − M2

1
Lq
Lf
c4 =
Ld L f − M2
M
d4 =
Ld L f − M2

M
Lq
pL f L q
c3 =
Ld L f − M2
pM Lq
d3 =
Ld L f − M2
b3 = p

b4 =

M
Ld L f − M2
Ld
d5 =
Ld L f − M2
c5 =

Rs , stator resistor; R f , rotor resistor; L d , L q , d- and q-axis stator inductances; L f , rotor inductance;
M, rotor and stator mutual inductance; p, number of pole pairs; F, combined rotor and load viscous
friction; J , combined rotor and load inertia; TL , machine load torque
...
5 Permanent-Magnet Synchronous Machine Modeling
3
...
1 PMSM Modeling in abc-Coordinates
As already mentioned, PMSMs differ from WRSMs in that the excitation is provided by
permanent magnets fixed on the rotor (Figure 3
...
Presently, the PMSM modelling is dealt
with following the same approach as for the WRSM, using the same assumptions, conventions,
and notations
...
Then, the voltages between the three phases is given by the expression
[vsabc ] = [Rs ][i sabc ] +

d[φsabc ]
dt

(3
...
In the rotor, a constant flux is

b-axis

β-axis

vsb
q-axis

d-axis

isb

f-axis
N
S φf

θi

sa

vsa
a-axis

α-axis

isc
c-axis

vsc

Figure 3
...
The distribution of the excitation field in the air gap and the MMFs
are assumed to be sinusoidal (by assumption A3)
...
26a)

φr b = φr cos( pθ − 2π/3),
φr c = φr cos( pθ + 2π/3),

(3
...
26c)

where φr is the amplitude of the flux produced by the permanent magnets, assumed to be
constant as the variation with temperature is insignificant
...
Specifically, one has
[φsabc ] = [L ss ][i sabc ] + [φr abc ]
...
27)

Furthermore, the fact that the rotor flux (PMSMs) is generated by permanent magnets, equation (3
...
Now, using the flux
expression (3
...
25) becomes
[vsabc ] = [Rs ][i sabc ] +

d
d
{[L ss ][i sabc ]} + ω [φr abc ],
dt
dθ

(3
...

dt
The WRSM mechanical equation (3
...
Doing so, one gets

Tem

1
= [i sabc ]T
2

?

?
?
?
d[L ss ]
d[φr abc ]
T
[i sabc ] + [i sabc ]

...
29)

For convenience, the usual motion equation is rewritten
Tem = TL + Fω + J

3
...
2

dω

...
30)

PMSM Model in the Rotating dq-Frame

Electric Equations
The PMSM model in the rotating dq-frame, linked to the rotor (Figure 3
...
28), by using the Concordia-Park transformation (3
...


52

AC Electric Motors Control

The dq-frame is made linked to the rotor by letting the angular position ρ (in equation (3
...
Doing so, we get
?
d ?
[L ss ]C 32 P(θe )[i sdq ]
dt
?
?
d
+ω
C 32 P(θe )[φr dq ]
...
31)

It can be checked that the matrix of inductances [L ss ] can be given in the following form:
[L ss ] = L sv I3 +

3
T
L sv C 32 P(θe )Q P(θe )T C 32
2

(3
...

1

T
Then, multiplying both sides of equation (3
...
32), it follows that:

?
?
T d
C 32 P(θe )[φr dq ]
[vsdq ] = Rs [i sdq ] + ω P(θe )T C 32
dθ
?
?
?
?
d
3
T
T
+P(θe )T
C 32 L sv I3 + L sv C 32 P(θe )Q P(θe )T C 32 C 32 P(θe )[i sdq ]
...
33)

The following properties are readily checked:
d
d
(•) = ω (•),
dt
dθ
d
(P(θe )) = p P(θe)Q ? ,
dθ
P(θe )Q ? P(θe )T = Q ? ,

(3
...
34b)
(3
...

0

Using equation (3
...
33) is rewritten as follows:
?
?
d[i sdq ]
3
?
[vsdq ] = Rs [i sdq ] + pωQ [φr dq ] + L sv I2 + L sv Q
2
dt
?
?
3
+ pω P(θe )T L sv P(θe )Q ? + L sv P(θe )Q ? Q [i sdq ]
2

(3
...
36)

where the second equality is obtained using the definitions of the the direct axis inductance,
L d = 3(L so + L sv )/2, of the quadratic inductance, L q = 3(L so − L sv )/2 and cylique inductance L cs = 3 L s0
...
36), the stator voltage equation (3
...

dt

(3
...
29) using
equation (3
...
34a–c)
...


(3
...
It
is also readily checked that

[i sdq ]T Q ? Q[i sdq ] = 2i sd i sq ;

L sv = 3(L d − L q )/4;

and

Q ? Q = Q Q ?T = I2
...
38) rewrites
Tem = TL + Fω + J

?
?
dω
= p L d − L q i sd i sq + [i sdq ]T Q ? [φr dq ]
...
39)

Recall that in the rotor-like dq-frame, the rotor flux is aligned with the d-axis
...
Then, the electromagnetic torque rewrites
Tem = TL + Fω + J

?
?
?
dω
= p L d − L q i sd i sq + p 3/2i sq φr
...
40)

From equations (3
...
40), replacing [L dq ] and Q ? by their expressions given by
equation (3
...
41a)
(3
...

= − i sd + p i sq ω +
dt
Ld
Ld
Ld

3
...
3

(3
...
41d)

PMSM Model in the Fixed Bi-Phase αβ-Frame

Electric Equations
The model of the machine in the fixed αβ-frame is obtained from equations (3
...
39)
by applying the Park-Concordia transformation defined in equations (3
...
3), (3
...
5),
and (3
...
Accordingly, one has
P(θe )T [vsαβ ] = Rs P(θe )T [i sαβ ] + [L dq ]

?
d ?
P(θe )T [i sαβ ]
dt

+ pωQ ? [L dq ]P(θe )T [i sαβ ] + pωQ ? P(θe )T [φr αβ ]
...
42)

Multiplying both sides of equation (3
...
43)

where we have used the fact that P(θe )P(θe )T = I2
...
34a–c), the
above equation becomes
d
[vsαβ ] = Rs [i sαβ ] + [L dq ] [i sαβ ]
dt
?
?
+ pω P(θe ) Q ? [L dq ] + [L dq ]Q ?T P(θe )T [i sαβ ] + pωQ ? [φr αβ ]
...
44)

?
?
It is easily checked that pω P(θe ) Q ? [L dq ] + [L dq ]Q ?T P(θe )T = pω(L d − L q )
...

dt

(3
...
2), (3
...
4), (3
...
6) to the torque expression
(3
...

2

(3
...

2

(3
...
Accordingly, the mechanical
equation rewrites as follows:
Tem = TL + Fω + J

dω
3
= p L sv [i sαβ ]T R(θe )[i sαβ ] + p(φr α i sβ − φrβ i sα )
dt
2

(3
...
26a–c) one obtains the corresponding two-phase system in the αβ-frame
...

dt

(3
...
49b)

Writing together equations (3
...
48), and (3
...

di sα
dt
di sβ
dt
dφr α
dt
dφrβ
dt
dω
dt

3
...
50a)
(3
...
50c)

= pωφr α ,

(3
...

3J
J
J
J

(3
...
The air gap being uniform (constant thickness) entails L sv = 3(L d − L q )/4 = 0
and
yielding L q = L d = L s
...
24a–d), (3
...
50) remain all valid
...

The Park-Concordia transformations have proved to be major instruments in passing from
the complex triphase machine model, expressed in the abc-frame, to the simpler fixed twophase αβ-coordinate model and the rotating dq-coordinate model
...
The real-time

56

AC Electric Motors Control

implementation of these models and the controllers based upon them has become possible in
recent years due to the advances made in digital computer technology
...
IEEE Transactions on Magnetics, 26, 551–554
...
I
...
IEEE Transactions on Energy Conversion, 14,
1465–1471
...
Siemens Reviews, 93, 217–220
...
IEEE
International Conference on Control Applications, pp
...

Hecquet M and Brochet P (1998) Simulations of synchronous machines using a electric-magnetic coupled network
model
...

Kothari DP and Nagrath IJ (2004) Electric Machines
...

Krause PC, Wasynczuk O, and Sudhoff SD (2002) Analysis of Electric Machinery and Drive Systems
...

Minnich SH, Csendes ZJ, Berkery J, and Tandon SC (1981) Load characteristics of synchronous generators by the
finite-element method
...

Multon B, Ben Ahmed H, et al
...
Techniques de
´
l’ing´ nieur
...

e
e
r
Srairi S, Djerdir A, and Miraoui A (2006) Use of permeance network method in the demagnetization phenomenon
modelling in a permanent magnet motor
...


Part Two
Observer Design
Techniques for
AC Motors

AC Electric Motors Control: Advanced Design Techniques and Applications, First Edition
...

© 2013 John Wiley & Sons, Ltd
...


4
State Observers for Estimation
Problems in Induction Motors
Gildas Besancon1 and Alexandru Ticlea2
¸
¸
1

Control System Department, GIPSA Lab, Grenoble Institute of Techology and Institut
Universitaire de France, France
2
Department of Control and System Engineering, Polytechnic University of Bucharest,
Romania

4
...
Those electrical dynamics generate the mechanical
torque and rotating speed of the machine
...
From this, a lot of efforts have been dedicated to methods of
reconstruction of the unmeasured variables, which amounts to a problem of observer design
...

In addition, the problem is made even more difficult by the fact that model parameters
themselves can change during operation, or just not be very well known
...

The main point of this chapter is to review those three items: (1) observability, (2) state
observer, state, and (3) parameter estimation, mostly in the spirit of former studies (Besancon
¸
2001; Besancon et al
...
Those problems are also put in the perspective of some of the main studies
that have been published in that respect over the last two decades
...
2 thus first recalls the problem(s) statement and some related references, while
Section 4
...
Section 4
...
Edited by Fouad Giri
...
Published 2013 by John Wiley & Sons, Ltd
...
5
...
2
4
...
1

Motor Representation and Estimation Issues
Problem Statement

The estimation issues and possible solutions will be settled for a model under the assumption
of symmetric operation with uniform values of parameters over each phase
...
Notations
i and φ will be used to respectively refer to currents and fluxes, and subscripts s and r will
respectively refer to the stator and the rotor
...

The corresponding state equations finally read as
d
i
dt s
d
φ
dt s
d
ω
dt m

? ?
= − σRr r +
L

Rs
σ Ls

= −Rs i s + u s ,

?

?
?
?
R
I + pωm J i s + σ L srL r I − p σ 1 s ωm J φs +
L

1
u ,
σ Ls s

(4
...

0

(4
...
Finally, in both electrical and mechanical parts, p represents the
number of pairs of poles
...
In general, such a measurement, usually denoted by y, will be a vector
at least corresponding to stator currents i s
...

In addition, the estimation problem can extend to the model parameters themselves, which
may not be accurately known, or vary significantly under operation (typically under heating
effect); a lot of work has been dedicated to the problem of rotor time constant estimation in
that respect for instance
...


4
...
2

Short Literature Review

Complex instrumentation placed on an induction motor would only cancel the major strength
of this device—simplicity
...
On the other hand, flux
information is crucial for the control of the drive, so flux estimation is an inescapable part
of any variable-speed control design
...
Notice then that “sensorless” in the context of induction motor usually means “speedsensorless
...
1) making the model linear time varying, which
therefore allow for a possible Kalman-like observer (see Section 4
...
However, under measured speed conditions, it is possible to build effective reduced-order observers for flux only,
which, as opposed to a Kalman solution, may employ a constant gain in the correction term
(Verghese and Sanders 1988)
...

When the rotor speed is not available through measurements, the observation problem
becomes significantly complex
...
(2000) and Ibarra Rojas
et al
...
The locus of the unobservability points is a straight line in
the speed-torque plane—called the unobservability line of the motor—that runs through the
origin of the plane and lies in the quadrants that correspond to a generator operating mode
of the motor
...
Based on this idea, some authors claim speed-sensorless estimation results with respect to low mechanical speed situations, which is actually a rather long
stretch from the precise unobservability conditions; in reality, at any constant speed, the synchronous speed is the one responsible for the presence of information exchange between rotor
and stator
...
Unfortunately, this is likely to be
the case during operation; in particular, the parameters that are the most susceptible to be
uncertain are the resistances, as they (significantly) change with the temperature
...
It is worth noticing at this point that it is impossible to identify all
five fundamental electrical parameters from the input-output data, even when using speed
measurements
...
2001)
...

The challenges raised by the induction motor observation problem motivated many research
groups into efforts to design effective solutions, from both pure observation and observation in
support of some particular control strategy perspectives
...
We shall therefore limit
ourselves here to mention the most relevant ideas that have been pursued in these efforts
...

Some factory values issued from equivalent circuit identification through some standard tests
(Chapman 2005) are normally available for the parameters of electric machines
...
g
...
1994),
linear least squares methods (e
...
, de Souza Ribeiro et al
...
2001) and nonlinear least squares methods (e
...
, Wang et al
...
2009) can also be applied, some without speed measurement, and with the advantage
that they can be adapted to online operation for critical parameter monitoring
...
Very popular in
the 1990s, the EKF also captured the attention of researchers in recent years (Alonge and
D’Ippolito 2010)
...
The unscented Kalman filter (UKF) adopts a superior technique to propagate
this quantity, which leads to improved accuracy (although some empirical tuning is involved)
...
(2006) and Jafarzadeh et al
...

By exploiting the fact that the induction motor model is linear with respect to the current
and flux, one class of methods considers the speed as an unknown parameter (constant or time
varying) and focuses on building some speed-adaptive flux observers (Kubota et al
...

Efforts have also been made to include the adaptation of some parameters (resistances) along
with the adaptation of the speed
...
A somewhat related class of methods
uses reference models of flux or current dynamics in order to generate error signals that can
be used to adapt the value of the mechanical speed
...

Moving to nonlinear design techniques, the idea of adapting some parameters also appears
in the design of sliding-mode observers for induction motors (e
...
, Rao et al
...
For state
estimation only, nonlinear interconnected observers are also reported (Ghanes et al
...

Efforts were also aimed at building nonlinear controllers that include adaptation mechanisms
for the unmeasured states (speed, flux) and for some parameters (load torque, resistances)
...
For instance, speed-sensorless nonlinear
controllers with local convergence have been designed with torque adaptation in Montanari
et al
...
(2008)
...
This solution relies on the immersion of the induction
motor model into an affine structure with respect to the unknowns; a Kalman-like observer
can then be used for the observation of the system under this new (inherently redundant)
representation
...
Further details on the related observer
tools can be found in Besancon (2007) and references therein
...
3 Some Observer Approaches
4
...
1 Estimation under known and constant Speed and Parameters
In the case when the mechanical speed is known and assumed to be constant, or varying
in a slow enough way, as well as all parameters, the motor modeling can be reduced to the
current and flux dynamics, under the form of a classical linear time-invariant state-space
representation
˙
x(t) = Ax(t) + Bu(t),
y(t) = C x(t),

(4
...
1)
...


...


⎞

⎟
⎟
⎟ is the observ⎠

C An−1
ability matrix
...
4)

where the observer gain K is designed either by pole placement (as in Luenberger approach)
or by linear quadratic optimization (as in Kalman approach)
...
5)

for a positive tuning parameter λ
...
It can in general be approached via a so-called EKF technique, namely the application
of Kalman equations to a model linearized along the trajectories currently estimated by the
observer, but with the limitations of the underlying approximation that is made via this
linearization
...


4
...
2

Estimation under known Speed and Parameters

In the case when the speed variation may be significant, but remains measured, its measurement
can be injected in the observer so as to keep a linear structure for the estimation error; in this
situation, the speed variation is to satisfy some appropriate excitation condition so that the
convergence of an observer based on Kalman equations is still guaranteed
...
1) can again be reduced to state variables of i s and φs as in equations
(4
...

This means that the model structure is still linear, but now time varying, and consequently,
Kalman equations (e
...
, as in equation (4
...
6)

for some T0 , α > 0, any t ≥ t0 for some t0 ≥ 0, and ?ωm the transition matrix of system
˙
ζ (t) = A(ωm (t))ζ (t)
...
4) is again possible, where K is computed
as in Kalman equations (4
...


4
...
3

Estimation under unknown Speed and known Parameters

In the case when the speed is unknown and varying, then its dynamical equation is to be taken
into account in the model for observer design (as in model (4
...
In this situation, the model
becomes nonlinear, and the observer design is now subject to a nonlinear observability condition, corresponding to a rank condition based on the so-called observation space (Hermann
and Krener 1977)
...
7)

where x now gathers i s , φs , and ωm , while y = i s , and f, g, and h result from model (4
...

When applied to model (4
...
2000)
...
2004),
when the speed is not measured (so-called sensorless case), corresponding to the notion of
unobservability “at zero (low) speed” for instance, as it is commonly referred to in the literature
...
8)

with notations of equation (4
...

In practice, this problem may be overcome by fluctuations, for instance due to the noise
reinjected by feedback, or the inverter harmonic—omitted in the modeling considered here
...

1 The

Lie derivative of a function h : Rn → R along some f : Rn → Rn of components f i , is defined for any x ∈ Rn
n
? ∂h
of components xi , by L f h(x) =
f i (x) (Isidori, 1995)
...
3
...
1), and
gathering them into a vector θ , the model becomes
˙
x(t) = f un (x(t), θ ) + gun (x(t), θ )u(t),
y(t) = h(x(t)),

(4
...

This model can typically be rewritten in an extended form as
˙
x(t) = f un (x(t), θ ) + gun (x(t), θ )u(t),
˙
θ = 0,

(4
...

Alternatively, one can look for an appropriate transformation to rewrite this extended model
again in a linear-like structure as follows:
˙
X (t) = A(u(t), y(t))X (t) + B(u(t), y(t)),
y(t) = C X (t),

(4
...
6)
...
(2001), and that it appears that a set of four parameters,
¸
in bijection with
Rr
,
Lr

Rs ,

Ls,

σ

is indeed structurally identifiable, and that this result extends to the problem of simultaneous
state estimation
...
1) into the form (4
...

¸

4
...
The main idea here is to summarize some appropriate transformations, and show that the necessary excitation level required by the observer can be available
during normal operation within a closed-loop configuration that includes a voltage inverter
...
1 Setup for observer simulations
...
1 hereafter, which is
used as a test bench for our estimation method
...
1) with parameters taken from an experimental platform available at GIPSA-lab, which
is built around a two-pole induction motor with 7
...
The values of the electrical parameters that the observer will try to
estimate are as follows:
L s = 0
...
63 ?,

Lr
Rr

= 0
...
4 ?
...
091 H,

As far as the mechanical parameters are concerned, their values correspond to
Jm = 0
...
001 N s/rad,

and are supposed to be known at all times
...
g
...
The reference vector for the inverter is in its turn
generated by a torque and flux controller, designed by using available input-output (exact)
linearization techniques applied to induction motors (von Raumer et al
...
8) is provided by a (linear) PI speed regulator
...
In fact, for all the variables involved in the control law that are
not initially known, estimated values are used
...
01 rad/s for mechanical speed
...
In addition, both formulations
can be specialized to simultaneous state and parameter estimation, yielding adaptive observers
...
See Ticlea and Besancon (2008) for results with the continuous-time
¸
¸
adaptive observer, Ticlea and Besancon (2009) for the discrete-time exponential forgetting fac¸
¸
tor observer, and Ticlea and Besancon (2012) for the discrete-time adaptive observer version
...
1) and z for stator fluxes, each of them being a vector of dimension
2, with components yi and z i , i = 1, 2, respectively
...


(4
...

¸
¸
¸

4
...
1

State and Parameter Estimation under known Speed

In the case when the rotor speed is known, but electrical parameters are to be estimated together
with the state variables and the load torque, the following procedure can be used:
Set

Z 1 := θ5 z; Z 2 := θ4 z
...

An observer based on Kalman equations can then be designed
...
2, 4
...
4 where the speed tracking,
torque estimation, and flux and parameter estimation errors can be seen, respectively
...


69

State Observers for Estimation Problems in Induction Motors

50
τl
ˆ
τl

50
ωr
ωr∗

25

0

(Nm)

(rad/s)

75

0

2
...
5
(s)

5

(b) Load torque evolution
and estimation

Figure 4
...
4
...
One can indeed check here that
?
T
X 2 := y T

zT

ωm

ωm z T

2
z1

2
z2

z1 z2

Tl z T

now yields a state representation of the form

Tl

?T

˙
X 2 = A2 (u, y)X 2 + B2 u,
y = C2 X 2 ,

for new matrices A2 , B2 , and C 2
...
5 and the load torque evolution profile in Figure 4
...
Those figures
also show the actual speed against the reference and the estimated load torque against the
simulated one
...
7
...
5
0
−0
...
5
(s)

5

Figure 4
...
5

5

50

–40

0

2
...
5
(s)

5

–50

0

2
...
4 Relative parameter estimation errors under speed-sensed estimation
150
ωr
ωr∗

(rad/s)

100
50
0
–50

0

5

10
(s)

15

20

Figure 4
...
6 Speed-sensorless load torque estimation with known parameters

5

71

State Observers for Estimation Problems in Induction Motors

6
isα
isβ

(A)

3
0
–3
–6

0

5

10
(s)
(a) Stator current

15

20

0
...
2
0

−0
...
4

0

5

0

5

10
(s)
(b) Stator flux

15

20

10
15
(s)
(c) Mechanical speed

20

1

(rad/s)

0
−1
−2
−3
−4

Figure 4
...
4
...
8 Speed tracking under speed-sensorless control with unknown parameters

with
? T
Z := Z 1

T
Z2

θ2 z T

?T

,

?
ξ := ωm

2
θ4 z 1

2
θ4 z 2

θ4 z 1 z 2

θ2 ωm

θ4 ωm

?T

,

¯
and θ made of θ together with Rs θ2 , Tl θ2 , and Tl θ4
...

Some related simulation results can be found in Figures 4
...
9, 4
...
11
...
The speed reference profile is now presented in
Figure 4
...
The associated load torque is presented
in Figure 4
...
Figure 4
...
9 Speed-sensorless load torque estimation with unknown parameters

73

State Observers for Estimation Problems in Induction Motors

2
isα
isβ

(A)

1
0

–1
–2

0

5

10
(s)

15

20

(a) Stator current
2
φsα
φsβ

(Wb)

1
0

–1
–2

0

5

10
(s)

15

20

10
15
(s)
(c) Mechanical speed

20

(b) Stator flux
4

(rad/s)

2
0

–2
–4

0

5

Figure 4
...
11 presents the relative errors in the estimations
of the electrical parameters
...
One possible remedy is to slow down the update actions with
respect to the estimation of the parameters by employing an adaptive observer; see Ticlea and
¸
Besancon (2008) for an exploration of this idea
...
11 Relative parameter estimation errors under speed-sensorless estimation

4
...
4

Estimation close to Unobservability

As a last set of illustrative results, let us here consider the case of an operation close to this
unobservability line mentioned in Section 4
...
3, with the requirement that the parameters need
to be monitored (i
...
, the preceding working configuration)
...
5 rad/s under the action of a driving torque that determines zero
synchronous speed at steady state
...
4 Nm
...
12
...
12 Stator flux evolution under speed-sensorless control close to the unobservability line

75

State Observers for Estimation Problems in Induction Motors

0,4

4
Ls

Rs
0,2

0

0

–2

(%)

2

–0,2

–4

5

10

15

–0,4
4

4
σ

2

0

10

15

10
(s)

15

Lr
Rr

0

–2

(%)

2

5

–2

–4

5

10
(s)

15

–4

5

Figure 4
...
13
...
It is nevertheless an operation
close to the unobservability line and the results show that a sufficient excitation level can ensure
observer stability even under conditions in which, theoretically, the system is on the verge
of unobservability
...
This indicator can
be monitored in real time in order to detect potentially critical situations for the observer
...
5

Conclusions

In this chapter, various estimation problems related to the classical model of induction motors
have been reviewed, at the light of (nonlinear) observer approaches for possible solutions
...
The latter has also been
extended to the case of possible unknown parameters, and various simulation results have been
provided accordingly
...
Finally,

76

AC Electric Motors Control

it should be emphasized that similar approaches can also be of interest for other machines,
like synchronous ones for instance
...
IEEE/ASME Transactions on Mechatronics, 11(5), 634–643
...
International
Journal of Control, 75(10), 753–758
...
Proceedings of the 2010 IEEE Symposium on Industrial Electronics, pp
...

Alonge F, D’Ippolito F, and Raimondi FM (2001) Least squares and genetic algorithms for parameter identification
of induction motors
...

Besancon G (2001) On-line full state and parameter estimation in induction motors and application in control and
¸
monitoring
...

Besancon G and Ticlea A (2003) Simultaneous state and parameter estimation in asynchronous motors under sensorless
¸
¸
speed control
...

Besancon G, Besancon-Vod˘ A, and Bornard G (2001) A note on identifiability of induction motors
...

Besancon G (2007) Nonlinear Observers and Applications Number 363
...
Springer
...

Proceedings of the 10th IFAC Symposium on System Identification, Vol
...
553–558
...
(2000) Observability conditions of induction motors at low frequencies
...
2044–2049
...
McGraw-Hill, New York
...
IEEE Transactions
on Industry Applications, 36(3), 743–754
...
International Journal of Control, 83(3),
484–497
...
IEEE Transactions on Automatic
Control, 22(5), 728–740
...

e
Automatica, 40(6), 1079–1085
...
Springer-Verlag
...
IEEE Transactions on Industrial Electronics, 59(11), 4207–4216
...

IEEE Transactions on Industrial Applications, 29(2), 344–348
...
Automatica, 44(10), 2593–2599
...
International Journal of Adaptive
Control and Signal Processing, 14(2), 171–175
...
Automatica, 42(10), 1637–1650
...
IEEE Transactions on Industrial Electronics, 57(4), 1296–1308
...
4373–4378
...
International Journal of Control, 83(7), 1457–1464
...
Application aux syst` mes de puissance (In
¸
e
e
French)
...
Institut National Polytechnique de Grenoble
...
Proceedings of the 32nd Annual Conference of the IEEE Industrial
Electronics Society
...
European Journal of Control, 12(4), 400–412
...

¸
¸
Proceedings of the 17th IFAC World Congress, pp
...

Ticlea A and Besancon G (2009) State and parameter estimation via discrete-time exponential forgetting factor
¸
¸
observer
...
1370–1374
...
Proceedings of the 16th
¸
¸
IFAC Symposium on System Identification, pp
...

Verghese GC and Sanders SR (1988) Observers for flux estimation in induction machines
...

von Raumer T, Dion JM, Dugard L, and Thomas JL (1994) Applied nonlinear control of an induction motor using
digital signal processing
...

Wang K, Chiasson J, Bodso M, and Tolbert LM (2005) A nonlinear least-squares approach for identification of the
induction motor parameters
...

Zaky MS (2012) Stability analysis of speed and stator resistance estimators for sensorless induction motor drives
...

Zamora JL and Garc´a Cerrada A (2000) Online estimation of the stator parameters in an induction motor using only
ı
voltage and current measurements
...


5
State Observers for Active
Disturbance Rejection in
Induction Motor Control
Hebertt Sira Ram´rez1 , Felipe Gonz´ lez Monta˜ ez2 , John Cort´ s Romero3 ,
ı
a
n
e
and Alberto Luviano-Ju´ rez4
a
1

Department of Electrical Engineering, CINVESTAV-IPN, M´ xico
e
Departamento de Energ´a, Universidad Aut´ noma Metropolitana, M´ xico
...
1

Introduction

Asymptotic estimation of perturbation inputs, with the aim of exactly, or approximately,
canceling their influence on the system at the controller stage, has been treated in the existing
literature under several headings: disturbance accommodation control, active disturbance
rejection control (ADRC), and intelligent Proportional–Integral–Derivative (PID) control,
also called, model-free control
...
Roots of this line of work may be found in
Shipanov (1939)
...
D
...
Originally, the approach was cast in the context of linear systems (see Johnson 1971)
and included a known linear model for the exogenous disturbances
...
The ADRC method is represented by the
works of the late Prof
...
The emphasis of this work lies on nonlinear observerbased disturbance estimation, for the canceling, and efficient time derivative calculations,
AC Electric Motors Control: Advanced Design Techniques and Applications, First Edition
...

© 2013 John Wiley & Sons, Ltd
...


State Observers for Active Disturbance Rejection in Induction Motor Control

79

for the feedback (see, Han 2009)
...
Gao and his colleagues (see
Gao 2001; Sun and Gao 2005; Gao 2006)
...
This technique proposes controller design on the basis of
one-dimensional, or at most two-dimensional, local phenomenological models (called local
ultra-models) of the nonlinear plant
...

The theoretical support of this methodology stems from the differential algebraic viewpoint in
linear and nonlinear systems (see Fliess et al
...
In recent years, the authors of this chapter
have been involved in developing illustrative laboratory applications of the ADRC method
for nonlinear systems using only linear feedback and linear observers, known as generalized
proportional integral (GPI) observers, for simultaneous estimation of states and of nonlinear
disturbances
...
The method thus proposes
global ultra-models of the perturbed plant that require no online resettings
...
The method is circumscribed to either differentially flat
systems, or to minimum phase systems (see Luviano-Ju´ rez et al
...

a
ı
2009; Sira-Ram´rez et al
...
2012b; Sira-Ram´rez et al
...

ı
ı
ı
GPI observers, a dual counterpart of GPI controllers (see Fliess et al
...
The nonsliding version appears in Luviano-Ju´ rez et al
...
The linear GPI observer naturally incorporates a selfupdating polynomial model of the overall disturbance effects as well as iterated output error
integral injections aimed at attenuating the effects, on the estimation error dynamics, of lumped
exogenous and state-dependent perturbation input signals present in the input-output model
of the plant
...
2010a)
...
Two traditional control design techniques are cast into
this context
...
The fundamental advantage of this proposal lies
in the single-handed cancelation of the effects of time-varying torques, and of unmodeled
frictions and nonlinearities containing possibly uncertain parameters
...
(1998), Chiasson (2005), and Marino et al
...
The
chapter is organized as follows: Section 5
...
Section 5
...
Both
schemes present experimental results and comparisons with existing control strategies
...


5
...
1)
(5
...
3)

where ωm is the shaft’s angular velocity, i s is the complex armature current, φr is the complex
flux, vs is the complex input voltage, and the variable TL (t) is the unknown, time-varying,
load torque perturbation input
...
The following
jθψ
complex variable √
notation is used: i s = i sα + ji sβ ; φr = φrα + j φrβ = ψr e ; and vs = vsα +
jvsβ , where j = −1 is the imaginary unit, and z is the conjugate of z
...
2
...
For this reason, an observer is usually devised for the flux dynamics given in
equation (5
...
In this case, the observer is simply given by a replica of the system itself (see
Verghese and Sanders 1988; Martin and Rouchon 2000; Chiasson 2005)
...
4)

ˆ
the estimation error, defined as eφr := φr − φr , satisfies
Lr

deφr
= − (Rr − j pL r ωm ) eφr
dt

(5
...
Then, along solutions of
2
R
˙
equation (5
...
Hence, the origin of the complex simulation error space, eφr = 0, is a globally asymptotic exponential equilibrium point for equation
ˆ
(5
...
The flux simulator variable, φr , will be used, henceforth, in place of the actual flux
without further considerations
...


State Observers for Active Disturbance Rejection in Induction Motor Control

5
...
2

81

Formulation of the Problem and Background Results

The compact model in complex variables (5
...
2), and (5
...
6)

dis
Msr
= 2 (Rr − j pL r ωm ) ψr e jθψ − γ L s σ i s + vs
...
6) reveal an interesting indirect control decoupling property
...
Therefore, viewing the
complex current i s as auxiliary control inputs, both constitutive parts of the system can, in
principle, be controlled independently of each other
...


5
...
3

Assumptions

• It is assumed that only the shaft’s angular position θm is measured
...

• The gain parameters p Msr /J L r and 1/σ L s are assumed to be known
...

• The load torque TL (t) is assumed to be, time varying but unknown
...

The trajectory tracking problem will be formulated in terms of the angular velocity
...
This allows an alternative estimation of the angular velocity
...
2
...
1), (5
...
3)
...
These objectives are to be achieved regardless of the presence
of unknown but bounded perturbation inputs represented by (1) the load input torque, TL (t),
(2) unmodeled torques due to friction terms in the rotor shaft dynamics, and (3) the effects of all
additive, flux- and current-dependent, nonlinear terms acting on the (complex) stator current
dynamics
...
2
...
The first design stage controls the angular velocity of the motor shaft to
∗
track the reference signal ωm (t) by means of the stator currents i s taken as auxiliary control
inputs
...

As a result of the first design stage, a set of desirable current trajectories is synthesized
...
The second stage designs a feedback
controller to force, in a robust fashion, the actual currents to track the current references
obtained in the first stage
...
1)
...
7)

with ψr∗ being a constant desired flux magnitude reference value and vaux is a yet to be specified
auxiliary control input that does not affect the flux magnitude dynamics
...
Notice, that the partial
feedback equation (5
...

On the other hand, the partially controlled angular position dynamics satisfies the following
perturbed set of differential equations:
dθm
= ωm ,
dt
TL (t)
p Msr
dωm
vaux − Bωm −
=

...
8)

State Observers for Active Disturbance Rejection in Induction Motor Control

83

Considerations are shifted towards the following global ultra-model (see the Appendix for
definitions and properties):
dθm
= ωm ,
dt
dωm
p Msr
vaux + ξ1 (t)
...
9)

where ξ1 (t) = −Bωm (t) − TL (t)/J is regarded as a pure time-varying unknown but uniformly
absolutely bounded signal with a finite number of equally unknown but bounded time derivatives, directly related to the load torque and of the effects of the possibly unknown viscous
friction term
...
The control input vaux is the auxiliary control input appearing in
equation (5
...

Theorem 5
...
1 Consider the global ultra-model in equation (5
...
8)
...
The observer-based control input vaux ,
vaux =

?
?
?
J Lr ? ∗
∗
ˆ
ωm (t) − kωm ωm − ωm (t) − ξ1 (t) ,
˙
ˆ
p Msr

(5
...

ˆ
ˆ
ˆ
The variables ξ1 and ωm , are given, respectively, by ρ1θm and ωm , which, in turn, are
generated via the following linear GPI observer:
ˆ
d θm
ˆ
= ωm + λ(p+1)1 (θm − θm ),
ˆ
dt
p Msr
d ωm
ˆ
ˆ
vaux + ρ1θm + λp1 (θm − θm ),
=
dt
J Lr
ˆ
ρ1θm = ρ2θm + λ(p−1)1 (θm − θm ),
˙

...


...
, λ01 , and kωm > 0, so that the dominant
characteristic polynomials, respectively, governing the tracking error eωm = ω − ω∗ (t) and
ˆ ˜
˙
˜
˜
˜
the estimation errors eθm = θm − θm , eωm = eθm + λ(p+1)1 eθm
pc,ω (s) = s + kωm ,

po,θ (s) = s p+2 + λ(p+1)1 s p+1 + · · · + λ11 s + λ01 ,

(5
...
While
the position estimation error characteristic polynomial is Hurwitz, the angular velocity estimation error remains bounded
...
, eθm )T and the tracking error eωm
...


By an appropriate choice of the coefficients, λi ; i = 0, 1,
...
The estimation error is assured to be ultimately uniformly bounded by a small
disk around the origin of the estimation error state space as the observer gains are set to
produce eigenvalues sufficiently far at the left half of the complex plane (for more details, see
the Appendix at the end of this chapter
...

ˆ

(5
...
12) remains uniformly ultimately
∗
bounded
...


Inner Loop Controller Design Stage
Consider now the perturbed stator currents dynamics
dis
Msr
=
dt
σ Ls Lr

?

?
1
Rr
− j pωm φr − γ i s +
vs
...
7)
...
13)

State Observers for Active Disturbance Rejection in Induction Motor Control

85

with ξ2 (t) given by
ξ2 (t) =

Msr
σ Ls Lr

?

?
di∗ (t)
Rr
− j pωm φr − γ i s − s
...
In a more general context this condition has been found to be,
both, a necessary and sufficient condition for the existence of solutions of nonlinear differential
equations with state-dependent perturbation inputs (see Gliklikh 2006)
...
2
...


...


˙
ˆ
ϑ(q−1)eI = ϑqeI + λ12 (eis − eis ),
˙
ˆ
ϑqeI = λ02 (eis − eis ),
ˆ
ξ2 = ϑ1eI ,

(5
...
, λ02 } and kis are chosen so that the polynomials in the complex variables s,
pξ (s) = s q+1 + λq2 s q + λ(q−1)2 s q−1 + · · · + λ12 s + λ02 ,

pis (s) = s + kis ,

are Hurwitz polynomials with roots located sufficiently far from the imaginary axis in the
complex plane
...


A scheme of the proposed strategy is depicted in Figure 5
...


86

AC Electric Motors Control

∗
ωm

ωm
ˆ
ˆ
φr

i∗
s
Proportional
controller
outer loop

is

ˆ
ξ1

ˆ
φr

vs

Induction
motor

ωm
is

ˆ
ξ2

ωm
is

Proportional
controller
inner loop

GPI
observer

ˆ
ξ1
ωm
ˆ

First stage

is
i∗
s
vs

ωm
GPI
observer

ˆ
ξ2

is

ˆ
φr
Flux
reconstructor

Second stage

Figure 5
...
GPI, generalized proportional integral

5
...
6

Experimental Results

An induction motor coupled with a DC motor generating a time-varying load torque input,
was used for the experimental tests (see Figure 5
...
The induction motor, manufactured by
WEG, has the following nominal parameters: rated power, 0
...
2374 H, L r = 0
...
2505 H, Rs = 4
...
8807 ?
...
The torque
load was measured indirectly through the DC motor current, given that TL (t) = km i L , with
km = 1
...
The reference value for the flux magnitude was chosen so as to maximize
√
the induced torque when subject to nominal currents
...
5036
Wb, where i nom = 3A
...
125 ms
...
Analog data was channeled through
a National Instruments PCI-6025E data acquisition card
...


Figure 5
...
3 Block diagram of the control system
...
0 kHz
...
3
...
The characteristic polynomial associated with
the velocity control loop was set to be s + kωm , with kωm = 85 (the associated approximation
coefficient of Theorem 5
...
1, p, was 5)
...
The characteristic polynomial associated with
2
the velocity-control loop disturbance observer was specified by (s 2 + 2ζo,ω ωo,ω s + ωo,ω )3 (s +
po,ω ), with ζo,ω = 4
...
The characteristic polynomial associated with
2
the current-control loop disturbance observer was given by (s 2 + 2ζo,sc ωo,sc s + ωo,sc )3 (s +
po,sc ), with ζo,sc = 6
...
2
...

Figure 5
...
Figure
5
...

Figure 5
...
Finally, to illustrate the robustness of the strategy, a load torque was applied
via a DC motor, in generator configuration, whose armature current tracked a state variable of
a Chua’s circuit
...
The disturbance estimation, as
well as the applied load torque are shown in Figure 5
...

We carried out experiments under the same trajectory tracking task using two other effective
approaches
...
(2009), where a load torque estimator is proposed to solve the problem of controlling
the velocity of an induction motor in the presence of unknown mechanical torque inputs
...
4, δ = 0
...
The other methods used for comparisons included the classic
observer-based PI control scheme in a field-oriented control scheme, obtained from Bodson

88

AC Electric Motors Control

20

(s−1)

15
10
5
0

ωm(t )

−5
0

2

4

6

8

2

4

6

8

ω* (t )
m

10

12

14

16

18

10

12

14

16

18

m

eω (t ) (s−1)

0
...
05
0
−0
...
1
0

t (s)

Figure 5
...
(1995)
...

2
The closed-loop characteristic polynomial for the flux regulation was s 2 + 2ζ2 ω2 s + ω2 , with
ζ2 = 1 and ω2 = 8
...

Figure 5
...
However, in the flux regulation task, the passivity-based approach did not achieve
0
...
4
0
...
5 Flux magnitude regulation and control input

vsβ (t )
16

18

89

(A)

State Observers for Active Disturbance Rejection in Induction Motor Control

2
0
−2
−4

(A)

0

2

4

6

8

10

12

2
0
−2
−4
0

(V)

isα(t )

20
0
−20
−40
0

∗
isα(t )

14

16

i∗ (t)
sβ

isβ (t )
2

4

6

8

10

12

14

σLS ξ2α (t )
2

4

6

8

10

12

18

16

18

σLS ξ2β (t )

14

16

18

t (s)

Figure 5
...
Finally, since the observer-based PI control scheme is
not capable of estimating the disturbance input, the disturbance estimations were reported for
the passivity-based scheme and the GPI scheme (Figure 5
...
Both methodologies achieved
good performances in spite of abrupt load variations
...
6
(Nm)

0
...
2
0
−0
...
6
(Nm)

0
...
2
0
−0
...
7 Lumped mechanical disturbances estimation

16

18

90

AC Electric Motors Control

20

(s−1)

15
10
5
0
−5
0

ωm PAS(t)
2

4

6

8

10

ωm PI (t )
12

14

ωm GPI (t )
16

18

t (s)
0
...
6
0
...
2
ψ r PAS(t)

0
0

2

4

6

8

10

ψ r PI (t)

12

14

ψ r GPI (t )
16

18

t (s)

Figure 5
...
3

Field-Oriented ADR Armature Voltage Control

The classical indirect control decoupling property, exploited in the preceding paragraphs for
the two-stage controller design, is seen to be inherited in the form of a direct control decoupling
property, by the field-oriented voltage-control approach here explored
...

(1990) and Bodson et al
...


0
...
4
0
...
2
0

(TL)P AS (t)
2

4

6

8

10

12

14

16

18

(Nm)

0
...
4
0
...
2
0

(TL)G P I (t)
2

4

6

8

t (s)

10

12

14

16

18

Figure 5
...
3
...
15)

where
ξω (·) =

p Msr
Lr

ξψ (·) =

?

??

?
?
?
?
?
?
p
Rr
˙
− σ Im i s φr − pωm Re i s φr −
ωm ψr2 − B ωm − TL ,
˙
Lr
Ls Lr σ

2
2 2
4Rr
2Rr Msr
+
2
3
Lr
Ls σ Lr

+

?

?

ψr2 +

?

2
−6Rr Msr
2Rr γ Msr
−
2
Lr
Lr

? 2 2?
?
?
?
2Rr Msr 2
2Rr Msr p
ωm Im i s φr +
is
...
16)

As it can be easily seen, the nature of the nonlinear terms ξω (·) and ξψ (·) is quite involved
...
This procedure is, incidentally, in the
very same spirit of total ADR (see Gao 2006; Tian and Gao 2009) and nonlinear modelfree control (see Fliess et al
...
The armature input voltage vs can be directly used to
robustly control, in a naturally decoupled fashion, both the angular velocity of the motor
shaft and the squared magnitude of the flux via extended, second order, controlled equations
...
Denote
2
r
˙
˙
by ξω (t) the quantity ξω (ωm (t), ωm (t), ψr (t), dψdt(t) , TL (t), TL (t)) and by ξψ (t), the function
2

r
ξψ (ωm (t), ω˙m (t), ψr (t), dψdt(t) , TL (t))

?
?
p Msr
d 2 ωm
=
Im vs φr + ξω (t)
2
dt
J Lr Ls σ

?
d2 2
2Rr Msr ?
ψ =
Re vs φr + ξψ (t)
2 r
dt
Lr Ls σ

(5
...

θψ = pωm +
˙
dt
pψr2

(5
...
Note that ωm is just a state of the extended angular velocity
˙
dynamics written in equation (5
...


5
...
2

Problem Formulation

It is desired to have the rotor angular velocity ωm track, even if in an arbitrarily closed manner,
a given, desired angular velocity reference trajectory ω∗ (t), while the squared magnitude of
the complex flux is independently controlled, as closely as desired, towards a given constant
reference value, ψr∗ 2
...


5
...
3

Control Strategy

Suppose, for a moment, that the complex flux, φr , is perfectly known
...
19)

yielding the following set of control-decoupled linear disturbed systems,
?

?
p Msr
vb + ξω (t),
J Lr Ls σ
?
?
2Rr Msr
d2 2
ψ =
va + ξψ (t)
...
20)

Naturally, the lack of measurability of φr prompts us to rely on the asymptotic complex flux
ˆ
estimate, φr
...
359–360 in Ortega et al
...
For a justification in rather general
terms, see Hinkkanen (2004), and the many references therein
...
21)

are Hurwitz polynomials
...
, γ0 }, {πn+1 , πn ,
...
22)

are also Hurwitz polynomials, with roots located sufficiently far into the left half of the complex
(m)
(n)
plane
...

Theorem 5
...
1

The armature voltage field-oriented controller,
?
?
ˆ
φr
v, v = va + jvb ,
vs = vsα + jvsβ =
ˆ2
ψr
?
?
???
? 2
?
2
Lr Ls σ
ψ dψr
ψ
∗ 2
ˆ
?
+ κ0 ψr − (ψr )
,
va = −
ξψ + κ1
2Rr Msr
dt
?
?
?
?
J Lr Ls σ
ω
ω
?
¨
ω
˙
˙
ξω − ω∗ (t) + κ1 (?m − ω∗ (t)) + κ0 ωm − ω∗ (t) ,
vb = −
p Msr

(5
...


...

ψ
ψ ˆ 2
ψ
ψ
˙
ϑm−1 = ϑm + γ1 (ψr − ζ1 ),
ψ ˆ 2
ψ
˙ψ
ϑm = γ0 (ψr − ζ1 ),

(5
...
In practice, however, they are small and chosen within the range of 3–5
...
von
Neumann: “With four parameters I can fit an elephant, and with five I can make him wiggle his trunk!”

94

AC Electric Motors Control

ω
ω
?
?
and with ξω and ωm given, respectively, by the variables ϑ1 and ζ2 , generated by the following
˙
linear high-gain GPI observer:
ω
ω
ω
˙ω
ζ1 = ζ2 + πn+1 (ωm − ζ1 ),
?
?
p Msr
ω
ω
ω
˙ω
ζ2 = ϑ 1 +
vb + πn m (ωm − ζ1 ),
J Lr Ls σ

ω
ω
ω
˙ω
ϑ1 = ϑ2 + πn−1 (ωm − ζ1 ),

ω
ω
ω
˙ω
ϑ2 = ϑ3 + πn−2 (ωm − ζ1 ),


...


...
25)

globally ultimately drives the angular velocity ωm (t) and the squared magnitude of the
flux, ψr2 , towards vicinities of the reference trajectories ω∗ (t) and (ψr )2 , which can be uniformly made as small as desired, regardless of the nonlinearities present in the functions,
˙
˙
ξψ (ωm , φ, i s , φr , dψr2 /dt) and, ξω (ωm , ωm , φr , i s , TL , TL ), defined above
...
Likewise, let eω := ωm − ζ1 and eψ := ψr2 − ζ1 denote, respectively, the estimation
errors associated with the angular velocity and the squared flux magnitude
...

˜
eψ
ψ

ψ

(5
...
As for the control part, the closed-loop dynamics of the second
order tracking error dynamics for the angular velocity, ωm , and the closed-loop dynamics of
ˆ
the squared flux magnitude, ψr 2 (given that ψr → ψr , exponentially), are given by
ω
ω
ω˙
?
¨
˙
˜
eω + κ1 eω + κ0 eω = ξω (t) − ξω (t) + κ1 eω ,

ψ
ψ
ψ˙
?
˙
˜
¨
eψ + κ1 eψ + κ0 eψ = ξψ (t) − ξψ (t) + κ1 eψ
...
27)

?
˙
˜
The convergence of ξω (t) towards an arbitrarily small vicinity of ξω (t) and that of eω towards
˙
a small vicinity of zero establishes that the tracking errors eω and eω ultimately absolutely
ω
ω
converge towards a small as desired vicinity of the origin for gains, κ1 and κ0 , appropriately
chosen so that the roots of the dominant characteristic polynomial in the complex variable s,

State Observers for Active Disturbance Rejection in Induction Motor Control

95

ω
ω
pc,ω (s) = s 2 + κ1 s + κ0 are located sufficiently far into the left half of the complex plane
...

The “steady state” of the load torque perturbed complex flux argument dynamics, corresponding to the previously defined controller, is described by

?
Rr ? ∗
d
∗
∗
J ωm (t) + Bωm (t) + TL (t)
...
Moreover, it has been seen
that θψ has absolutely no influence on the closed-loop dynamics of ωm and ψr2
...
3
...
For the control scheme of Section 5
...
The angular velocity output reference trajectory ω∗ (t)
was defined as a series of ramps, which takes values of 0 to 35, 35 to −5, −5 to 15, 15 to
−5, and −5 to 20 rad − s−1 , during time intervals of 2
...
The characteristic polynomial
ω
ω
associated with the velocity control loop was set to be of the form: s 2 + κ1 s + κ0 , with
ω
ω
κ1 = 208 and κ0 = 6400
...
The characteristic polynomial associated
with the angular velocity control loop disturbance observer (setting p = 5) was chosen of the
2
form: (s 2 + 2ζo,ω ωo,ω s + ωo,ω )3 (s + po,ω ), with ζo,ω = 11, ωo,ω = 60, and po,ω = 60, while
the characteristic polynomial associated with the flux control loop disturbance observer (setting
2
m = 5) was set to be: (s 2 + 2ζψ,ω ωψ,ω s + ωψ,ω )3 (s + pψ,ω ), with ζψ,ω = 1, ωψ,ω = 200, and
pψ,ω = 200
...
10 depicts a rather accurate angular velocity tracking of the desired reference
trajectory
...
11 depicts the remarkable quality of the flux magnitude stabilization
...
12, the armature control input voltage and armature currents are shown in the
reference frame: α, β
...
13
...
14 shows the disturbance estimations, associated with the
flux and velocity control loops
...

The performance of our proposed control scheme was compared with other two armature
voltage field-oriented control strategies, differing only in the linear part of the design while
excluding the GPI observer
...
The experimental comparison results are depicted
in Figures 5
...
16, 5
...
18, where the GPI observer significantly improves the

96

AC Electric Motors Control

ωm (t)

20

(s−1)

30

*
ωm (t)
Detail at t = 1

10
0
−10

0

2

4

6

8

0

2

4

6

8

10

12

14

16

18

10

12

14

16

18

0

m

−1

eω (s )

1

−1
−2

t (s)

Figure 5
...
8

(Wb)

0
...
4

ψr

0
...
5

1

1
...
5

−3

(Wb)

eψ

r

0

−5

0

2

4

6

8

t (s)

10

12

14

16

Figure 5
...
12 Armature voltages and currents in the a, b reference frame

L

T (Nm)

1
...
5
0

0

5

10
t (s)

15

Figure 5
...
14 Estimated disturbance inputs, ξω (t), ξψ (t)

18

98

AC Electric Motors Control

ωm (t )

35

ωm (t)

GPI

ωm (t )

PID

ω*(t )

PD

30
25
20
Detail at t = 13

(s−1)

15
10
5
0
−5
−10

0

2

4

6

8

10

t (s)

12

14

16

18

Figure 5
...
5 (s)

0

2

4

6

8

t (s)

10

eω

mGPI

12

eω

mPID

14

mPD

16

18

Figure 5
...
6

(Wb)

0
...
2
Detail at t = 15
...
17 Performance comparison with PD and PID control schemes

18

State Observers for Active Disturbance Rejection in Induction Motor Control

150

PGPI (t )

PPID (t )

PPD (t )

99

Pm (t )

(W)

100

50

0
0

5

10
t (s)

15

Figure 5
...
The PD control scheme exhibits poor tracking results
...
However, its behavior in the presence of the applied torque input is slightly
affected by the time-varying disturbance
...
In Figure 5
...
In this case, even though all strategies had a
similar power consumption, the GPI control exhibited the lowest results
...
A

Appendix

5
...
1 Generalities on Ultra-Models and Observer-Based Active
Disturbance Rejection Control
Consider the following problem: it is desired to asymptotically stabilize to the origin the set
of phase variables of the scalar, nth order, nonlinear controlled system
˙
y (n) = φ(t, y, y ,
...
A
...


5
...
2

Assumptions

(n−1)
˙
• It is assumed that for every given set of initial conditions: Y0 = {y0 , y0 ,
...
A
...


100

AC Electric Motors Control

˙
• Assume that the scalar function φ(t, y, y ,
...
Notice that for a given
u(t), the solution y(t) of the above differential equation trivially satisfies: y (n) (t) =
˙
φ(t, y(t), y (t),
...

• It is assumed, based on the last fact, that as a time function, the first time derivatives
˙
φ ( j) (t, y, y ,
...

In other words, there exists constants K j such that,2
˙
sup |φ ( j) (t, y(t), y (t),
...
, m
...
A
...
2
...
A
...
, z 0 } = Y0 , is a global ultra-model of equation (5
...
1), if
for every u and for all t ≥ 0

˙
ξ (t) = φ(t, y(t), y (t),
...


(5
...
4)

˙
˙
Definition A
...
2 Two systems: y (n) = A(t, y, y ,
...
, z (n−1) )
with identical corresponding sets of initial conditions, Y0 and Z 0 , are trajectory equivalent if
y(t) = z(t) for all t ≥ 0
...
2
...
3), with initial conditions: Z 0 = Y0 , and satisfying equation
(A
...
1)
...
It follows that e0 = e0
= · · · = e0
Hence, e(t) = 0 for all t
...


2 This

= 0 for all j
...
However, in cases where the nonlinearity is known except for some of its parameters, as it is the case of our motor system, its validity can be assessed with
some work
...
A
...
3), and (5
...
4), are identical in the precise sense that their trajectories are the same, over any time interval, for a sheared control input function and with the
same initial conditions
...

Any pertinent consideration on the system (5
...
1) may be examined on the linear trajectory
equivalent system (5
...
3), viewed now without any ambiguity as
y (n) = μ(t, y)u + ξ (t),

(5
...
5)

which is devoid of the phase variables-dependent nonlinear structure
...


5
...
3

Observing the uncertain System through the Ultra-Model

˙
Setting y1 = y, y2 = y ,
...
, n − 1,

˙
yn = φ(t, y1 , y2 ,
...


(5
...
6)

Propose the following observer for the phase variables, {y1 , y2 ,
...
, yn , and complemented by m output estimation error iterated
integral injections, characterized by the variable, ζ1
...
, n − 1,

˙
ˆ
ˆ
y n = μ(t, y1 )u + ζ1 + λm (y1 − y1 ),

˙
ˆ
ζi = ζi+1 + λm−i (y1 − y1 ), i = 1,
...


(5
...
7)

˜
˜
˜
Let the estimation error, e y , in reference to the ultra-model system, be defined as e y = e1 :=
ˆ
ˆ
˜
ˆ
y1 − y1 = y − y1 with e2 = y2 − y2 , etc
...
, n − 1,

˙
˜
˜
en = ξ (t) − ζ1 − λm e1 ,

˙
˜
ζi = ζi+1 + λm−i e1 , i = 1,
...


(5
...
8)

˜
˜
It is not difficult to see that the estimation error, e y = e1 , satisfies, after elimination of all
variables ζ , the following n + mth order perturbed linear differential equation:
˙
˜y
˜
˜
˜y
e(n+m) + λn+m−1 e(n+m−1) + · · · + λ1 e y + λ0 e y = ξ (m) (t)
...
A
...
, n + m − 1, so that the characteristic polynomial in the complex variable s,
po (s) = s n+m + λn+m−1 s n+m−1 + · · · + λ1 s + λ0 ,

(5
...
10)

exhibits all its roots sufficiently far from the imaginary axis, in the left half of the complex
plane, then the trajectories for e y and for its time derivatives ultimately absolutely converge,
in an exponentially dominated manner, towards a small as desired vicinity of the origin
˜
˜y
˜ ˙
of the estimation error phase space, {e y , e y ,
...
The further away the roots are located in the left half of the complex plane the
smaller the vicinity of ultimate boundedness around the origin of the estimation error phase
˜
˜
space
...
, en+m )T denote the phase
variables of equation (5
...
9)
...
A
...
The Hurwitz character of A implies
that, given a positive definite matrix Q, there exists a positive definite matrix P, such that
−Q = A T P + P A
...
The Lyapunov function candidate, V (χ ) = 1 χ T Pχ ,
2
exhibits, along the solutions of the linear perturbed system, a time derivative of the form
˙
V (χ , t) = 1 χ T ( A T P + P A)χ + b T Pχ φ (m) (t)
...
Hence, all trajectories, x(t), starting
outside this sphere, defined in the estimation error phase space, converge towards its interior,
and all those trajectories starting inside S will never abandon it
...
From equation (5
...
8) it follows that
˙
˜
˜
ζ1 = ξ (t) − λm e1 − en
...
A
...
This fact demonstrates the self-updating character of the polynomial disturbance model, as an internal model in
the GPI observer
...
, m
...
, n,
reconstruct, in an arbitrarily close fashion, the time derivatives of y
...
A
...
A
...
, κn−1 }, chosen so that pc (s) = s n + κn−1 s n−1 + · · · + κ0
exhibits all its roots in the left half of the complex plane C
...


(5
...
13)

k=1

Theorem A
...
1 The disturbance rejection output feedback controller in equation (A
...
, y (n−1) ), provided the set of coefficients
{κ0 ,
...

˜y
Proof: According to the previous theorem, the term ξ (t) − ζ1 and the terms e(k) ,
k = 1, 2,
...
It follows that the right-hand side of the linear system (5
...
13) evolves, in an uniformly ultimately
bounded fashion, within a sufficiently small neighborhood of the origin of the output tracking
error phase space
...
, e(n−1) ), provided the roots of
pc (s) are located sufficiently to the left of the imaginary axis in C
...
Real life noises do not
preclude the application of high-gain observers, as it can be inferred from the experimental
results here presented
...
(2010b) for experimental details on other types of systems)
...
Control Systems Magazine, IEEE, 14(4), 25–33
...
IEEE Transactions on Industrial Electronics, 42(4), 337–343
...
Wiley-IEEE Press, Hoboken, NJ
...
36th IEEE Conference
on Decision and Control, Brighton, England, pp
...

Fliess M, Marquez R, Delaleau E, and Sira-Ram´rez H (2002) Correcteurs Proportionnels-Int` graux G´ n´ ralis´ s
...

Fliess M, Join C, and Sira-Ram´rez H (2008) Non-linear estimation is easy
...

Fliess M and Join C (2008) Intelligent PID controller
...
326–331
...
40th IEEE Conference on
Decision and Control 2001, 4578–4585
...
American
Control Conference, Minneapolis, MN
...
2399–2405
...
Abstract and Applied Analysis, 2006, 1–17
...
IEEE Transactions on Industrial Electronics, 56,
900–906
...
PhD Thesis
...

Johnson CD (1971) Accommodation of external disturbances in linear regulator and servomechanism problems
...

Johnson CD (2008) Real-time disturbance-observers; origin and evolution of the idea
...
40th
Southeastern Symposium on System Theory
...

Karagiannis D, Astolfi A, Ortega R, and Hilairet M (2009) A nonlinear tracking controller for voltage–fed induction
motors with uncertain load torque
...

Kim D, Ha I, and Ko M (1990) Control of induction motors via feedback linearization with input-output decoupling
...

Leonhard W (2001) Control of Electrical Drives Power Systems, edn
...

Luviano-Ju´ rez A, Cort´ s-Romero J, and Sira-Ram´rez H (2010) Synchronization of chaotic oscilators by means of
a
e
ı
proportional integral observers
...

Maggiore M, and Passino K (2005) Output feedback tracking: a separation principle approach
...

Marino R, Tomei P, and Verrelli C (2010) Induction Motor Control Design Advances in Industrial Control
...

Martin P and Rouchon P (2000) Two simple flux observers for induction motors
...
doi: 10
...
0
...

Ortega R, Lor´a-Perez J, Nicklasson P, and Sira-Ramirez H (1998) Passivity-Based Control of Euler-Lagrange
ı
Systems Mechanical, Electrical and Electromechanical Applications Communications and Control Engineering
...

Shipanov AG (1939) Theory and methods of designing automatic regulators
...

Sira-Ram´rez H, Luviano-Ju´ rez A, and Cort´ s-Romero J (2012) Flatness-based linear output feedback control for
ı
a
e
disturbance rejection and tracking tasks on a Chua’s Circuit
...
of Control, 85 (accepted for publication,
to appear)
...
International Journal of Control, 83(8), 1631–1640
...
American Control Conference 2009, Baltimore USA
...
Asian Journal of Control, (accepted for publication, to appear)
...
International Journal of Systems Science, 42(4), 621–631
...
Control Engineering Practice (accepted for publication, to appear)
...
International J
...

Sira-Ram´rez H, Ram´rez-Neria M, and Rodr´guez-Angeles A (2010b) On the linear control of nonlinear mechanical
ı
ı
ı
systems
...

Sun B, and Gao Z (2005) A DSP-based active disturbance rejection control design for a 1-kW H-bridge DC–DC
power converter
...

Tian G and Gao Z (2009) From Poncelet’s invariance principle to active disturbance rejection
...
ACC ’09
...
2451–2457, St
...

Trzynadlowski A (1994) The Field Orientation Principle in Control of Induction Motors
...

Verghese G and Sanders S (1988) Observers for flux estimation in induction machines
...


6
Observers Design for Systems
with Sampled Measurements,
Application to AC Motors
Vincent Van Assche, Philippe Dorl´ ans, Jean-Francois Massieu,
e
¸
and Tarek Ahmed-Ali
GREYC Lab, University of Caen Basse-Normandie, France

6
...
In the last decades, the design of nonlinear observers for
continuous systems with sampled measurements has received a great attention
...
For linear systems, it is usually possible to design observers by using the
discrete-time model of the continuous-time system
...
In this case, there
exist two main approaches for dealing with this problem
...

This approach provides a semi-global practical stability of the observation error
...

sc
The second one is based on a mixed continuous and discrete design
...
It consists of two steps
...
Edited by Fouad Giri
...
Published 2013 by John Wiley & Sons, Ltd
...
The exponential convergence of the observation error is then ensured under
some sufficient conditions on the sampling period through the stability analysis of impulsive
systems
...
(1992) use this approach to write a discrete-continuous version of the wellknown high-gain observer (Gauthier et al
...
Nadri and Hammouri (2003) have designed
observers for a MIMO class of state affine systems where the dynamical matrix depends on
the inputs when those inputs are regularly persistent
...
(2009)
...
(2002) used a similar method for a larger
class of systems and applied it to the observation of an emulsion copolymerization process
...
(2004)
and recently, Hammouri et al
...
Andrieu and Nadri (2010) extend the work from Zemouche
et al
...
Recently, a hybrid sampled-data observer
dedicated of a class of nonlinear systems has been presented by Karafyllis and Kravaris (2009)
...
This algorithm has been extended to some networked control systems by
Ahmed-Ali and Lamnabhi-Lagarrigue (2012) by using a Lyapunov Krasovskii approach
...
(2007b) by using a descriptor system approach and a
Lyapunov Krasovskii functional
...
This idea has also been used by Raff et al
...
(2011) for some classes of nonlinear systems with nonuniformly
sampled measurements
...
The first one is an extension to delayed
measurements of the continuous-discrete observer developed by Nadri and Hammouri (2003)
...
It can be viewed as an extension of the work by Ahmed-Ali et al
...

The chapter is structured as follows: first, two observer designs are presented, then the
application of those designs to the case of an induction motor without speed sensor is explained,
and eventually, some simulations results are given to illustrate this application
...
2

Nomenclature

The following notation will be used throughout this chapter
...
I p is the identity matrix of
0
dimension p × p and 0 p represents the zero matrix of dimensions p × p
...
The Euclidean norm of a vector a will be noted ?a? and the L2
norm of a matrix A will be noted ? A?
...


107

Observers Design for Systems with Sampled Measurements

6
...
3
...
Those designs can be applied to a larger class of nonlinear models
and are first presented within the framework of this class of systems
...

The value of the output of the system is known only at the sampling instants noted tk where
(tk )k∈N is an increasing sequence with limk→+∞ tk = +∞
...


(6
...
2)

y(tk ) = h(x(tk ))
...
3)

y(tk ) = C x(tk ),

with the following hypothesis:
H1: The state vector x ∈ Rn is composed of q subvectors, x 1 ,
...
4)

with ∀i = 1,
...

H2: A ∈ Rn×n is a block diagonal matrix
⎛

0
⎜
...

⎜
...

A=⎜
...

⎜
...


...


...


...


...


...


...
⎟

...
⎟
⎟
0⎟
⎟
⎟
Ip ⎠
0

(6
...

⎟
⎠

...
, xq , u)

(6
...
, x a,i , u) − φi (x b,1 ,
...
7)

i
i
where x a and x b are the vectors (x a,1 ,
...
, x b,i )? ,
respectively
...


6
...
2

Observer Design with a Time-Delay Approach

The first observer design we present handles the sample mechanism as a variable time-delay
...
(2011), based on the high-gain approach from Gauthier et al
...

The sampling mechanism is transformed into a time-variable delay by defining the following
delay function:
τ (t) = t − tk ,

∀t ∈ [tk , tk+1 ),

(6
...
3) is
˙
equivalent to
?
˙
x = Ax + φ(x, u),
(6
...

The following theorem applies to any variable time delay system of the form (6
...
In the
sequel, it will be applied to the particular case where the delay is of the form (6
...

Theorem 6
...
1

Consider the observer
?
?
˙ = A x + φ( x, u) − θ ?−1 S −1 C T C x(t − τ ) − y(t − τ ) ,
ˆ
ˆ
ˆ
ˆ
x

(6
...
11)

where θ is a positive constant satisfying θ > 1, S is a symmetric positive definite matrix
solution of the equation

109

Observers Design for Systems with Sampled Measurements

and ? is the following block-diagonal matrix:
?

?
1
1
1
? = Diag I p , I p , 2 I p ,
...

θ
θ
θ

(6
...


(6
...


(6
...
3) and (6
...


(6
...
16)

and the Newton-Leibniz formula
¯
¯
x(t − τ (t)) = x −

?

t

˙
¯
x(σ )dσ,

t−τ (t)

together with the property C? = C?−1 = C, one can rewrite equation (6
...


(6
...

(2008)
...

˙
¯

After some straightforward computations, the functional (6
...


(6
...
19)

t−T

with

I =

?

t

˙
¯
x(σ )dσ
...
20)

t−τ (t)

We will now bound the different terms of the right-hand side of this equation to express a
sufficient condition on T ensuring that this derivative is negative
...
11) in the first term of (5
...
21)

t−T

¯ ¯
where V = x ? S x
...
(1992) proved that
?? (φ(x, u) − φ(x, u))? ≤
ˆ

√

¯
qβ ? x?
...
22)

This implies the existence of a constant
k1 = 2

λmax (S) √
qβ,
λmin (S)

(6
...
24)

where λmax (S) and λmin (S) are the largest and smallest eigenvalues of S, respectively
...
21) becomes
? ?2
˙
˙
¯
¯
¯
¯
W ≤ −θ V + k1 V − θ x ? C ? C x + 2θ x ? C ? C I + T ? x ? −

Now, remark that

?

t
t−T

?
?
? x(σ )?2 dσ
...
25)

111

Observers Design for Systems with Sampled Measurements

therefore,
?
?2
˙
˙
¯
W ≤ −θ V + k1 V + θ I ? C ? C I + T ? x(t)? −

?

t
t−τ (t)

From equation (6
...

˙
¯

(6
...
Using equation (6
...

λmin (S)

(6
...
26) gives

t
t−τ (t)

?
?
?
? x(σ )?2 dσ
...
28)

so that equation (6
...

˙
¯

?
?
?
?? t
?
?
θ
3
? x(σ )?2 dσ
...
29)

(6
...
Hence, with this value of T , if θ − k1 > 0,
2
˙
then W is negative and equation (6
...

W
2

(6
...
32a)

θ > sup{1, 2k1 },
?
?
τ (t) ∈ 0, 2k12 θ ,

(6
...
33)

then
t→+∞

which proves the theorem
...
3
...
3), equation (6
...
Furthermore, equation (6
...
Obviously, this result can
be directly applied to the sampled measurements case, as in the following result
...
3
...
34)

for which the hypotheses H1–H5 apply
...


(6
...
36a)
(6
...
23) and (6
...
35) is a global
asymptotic observer for system (6
...


Observers Design for Systems with Sampled Measurements

113

Proof: This corollary is a straightforward application of Theorem 6
...
1: taking τ (t) = t − tk ,
with k such that tk ≤ t < tk+1 , system (6
...
3)
...
3
...

Remark 6
...
3 In the case where the measurements are acquired through a numerical network
inducing a further transmission delay δ, it is easy to adapt this scheme with the variable delay
τ = t − tk by taking k such that t ∈ [tk + δ, tk+1 + δ), that is, the total delay minimum value
equals δ and it is reset to this minimum value at each instant tk + δ, k ∈ R
...
36) are respected
...
3
...
Although this observer converges when the sampling
intervals are sufficiently small, its performance degrades rapidly as the sampling intervals
increase
...

In this case, the output delay δ, which comes in addition to the sampling of the measurements,
has to be explicitly taken into account
...
34)
but it is assumed that the measure y(tk ) is not available to the observer before instant tk + δ,
where δ represents the transmission delay
...
3
...
34) with hypotheses H1–H5
...
37)
⎪
⎩
w(tk + δ) = y(tk ),

is a global exponential observer
...
The observation equation itself is the classical
continuous-time high-gain observer equation, using the predictor output w instead of the
unavailable system output y
...

?

(6
...
39)

114

AC Electric Motors Control

The equations of the evolution of those errors are deduced from equations (6
...
37),
˙
˜
˜
ˆ
ˆ
x = A x + φ(x, u) − φ(x, u) − θ ?−1 S −1 C T (C x − w) ,

˙
˜
ˆ
w = C A x(t − δ) + Cφ( x(t − δ), u(t − δ)) − Cφ(x(t − δ), u(t − δ))
?

∀t ∈ [tk + δ, tk+1 + δ),

w (tk + δ) = 0
...
40)
(6
...
42)

˜
To prove the convergence of the error x towards 0, the following Lyapunov candidate is used,
based on the work from Hespanha et al
...
43)

¯
˜
where x = ? x, as in (6
...


Since, w (tk ) = 0 for all k ∈ N,
?

U (tk ) ≤ lim U (t)
...
44)

(6
...
Equation (6
...
46)

for some real ? > 0
...
45) and (6
...


6
...
4
...
The machine is controlled through the stator voltage

Observers Design for Systems with Sampled Measurements

115

vsα and vsβ and the stator current i sα and i sβ are the measured and sampled outputs, and
we note
?
? ?
?
i
vsα
and i = sα
...

The state of the system is a dimension 6 vector that will be subdivided into 3 subvectors of
dimension 2 for the design of the observer
⎞
i sα
⎜ i sβ ⎟ ⎛ ⎞
⎟
⎜
i,
⎜ φr α ⎟
⎟ = ⎝ ? ⎠,
⎜
x =⎜
⎟
⎜ φrβ ⎟
x3
⎝ ωm ⎠
TL
⎛

where the subvector ? =

?

φr α
φrβ

?

(6
...
The
of the third subvector x 3 are the rotor speed ωm and the load torque TL : x 3 =
TL
output vector y is measured at each sampling instant: y(tk ) = i(tk ) = C x(tk ), with k such that
?
?
tk ≤ t < tk+1 and C = I2 02 02
...
48)
pL m ?
⎪ ωm =
φr α i sβ − φrβ i sα − TL ,
⎪˙
⎪
J Lr
J
⎪
⎪
⎪ ˙
⎪ TL = u L ,
⎪
⎪
⎪
⎪
⎪
⎩
y = i(tk ), with k such that tk ≤ t < tk+1 ,

with

Rs
Rr L m 2
+
,
σ Ls
σ Ls Lr 2
Lm2
,
σ = 1−
Lr Ls
1
F(ωm ) = I2 − pωm J2 ,
τr
γ =

Lm
,
σ Lr Ls
Lr
τr =
,
R
?r
?
0
1
J2 =

...
1
...
1

AC machine parameters

Parameter

Definition

Ls
Lr
Lm
Rr
Rs
τr

Stator self-inductance
Rotor self-inductance
Mutual inductance between stator and rotor windings
Rotor resistance
Stator resistance
Rotor time constant

This model will also be noted in a shorter form,
?

˙
x = f (x, v) + h(u L ),

(6
...


To be able to design a high-gain observer as in Corollary 6
...
2, we follow the method
proposed by Dib et al
...
The following change of variable
is used:
⎛ ⎞
z1
(6
...
51)

z 2 = g2 (x) = k F(ωm )?,
?
?
k Lm
i + k F(ωm )?
z 3 = g3 (x) = pωm J2 −
τr
?
?
TL
k Lm
? J2 i +
J2 ?
...
52)

(6
...
54)

117

Observers Design for Systems with Sampled Measurements

is not constant almost everywhere (Dib et al
...
This implies that the AC Motor model
used is weakly observable in the sense of Hermann and Krener (1977)
...
48) becomes

where

⎧
⎪ z1
⎪˙
⎪
⎪
⎨ z2
˙
⎪˙
⎪ z3
⎪
⎪
⎩
y

= z 2 + ψ1 (z 1 , v),

= z 3 + ψ2 (z 1 , z 2 ),

(6
...

+
?+
∂i dt
∂?
∂ωm
∂ TL

This system can be rewritten
?

˙
z = Az + ψ(z, u),
y = C z(tk ),

(6
...

ψ3 (x, u L )
⎛

The AC machine model in z is of the form (6
...
3
...
3
...


6
...
2

Observer for AC Machine with Sampled and Held Measurements

The first observer, as written in Corollary 6
...
3, is a direct adaptation to the sampled measures
case of the one proposed by Dib et al
...
57)

118

AC Electric Motors Control

Applying the change of variables (6
...


(6
...
48) and (6
...
59)

ˆ
and, obviously, by replacing x with x,
∂g
ˆ
f (x, v) = Aˆ + ψ(ˆ , u)
...
60)

Multiplying equation (6
...
60), and remarking that
ˆ
ˆ
C x = C z , leads to
˙
ˆ
ˆ
z = Aˆ + ψ(ˆ , u) + θ ?−1 C T [C z (tk ) − y(tk )]
...
61)

This is indeed the form of the observer from Corollary 6
...
2
...
62)

provided that the gain θ and the sampling period Te are set such that conditions (6
...


6
...
3

Observer for the AC Machine with Predictor

The second observer design adds an output-predictor
⎧
? ?−1
⎪
⎪ x = f ( x, u) − θ ?−1 ∂g
˙
⎪
ˆ
ˆ
S −1 C T [C x(t − δ) − w] ,
⎪ ˆ
⎨
∂x
?
?
⎪ w = C f x(t − δ), v(t − δ) , t ∈ [tk + δ, tk+1 + δ),
ˆ
⎪ ˙
⎪
⎪
⎩
w(tk + δ) = y(tk )
...
63)

In the same way as in Section 6
...
2, we apply the change of variable (6
...
3
...
The first equation from (6
...
57) in Section 6
...
2 (except that w replaces y)
...
63), one can remark, from (6
...


(6
...
2

Simulation parameters

Parameter
Ls
Lr
Lm
Rr
Rs
Ts
T
θ

119

Value
140 mH
25 mH
50 mH
0
...
67 ?
2e-04 s
60 × Ts = 0
...

z

(6
...
63) can be rewritten in the form used in Corollary 6
...
3
⎧
−1 −1 T
˙
ˆ
z
z
⎪ z = Aˆ + ψ(ˆ , u) − θ ? S C [C z (t − δ) − w] ,
⎨ ˆ
z
w = C Aˆ (t − δ) + Cψ (ˆ (t − δ), u(t − δ)) , t ∈ [tk + δ, tk+1 + δ),
˙
z
⎪
⎩
w(tk + δ) = y(tk ),

(6
...


6
...
4

Simulation

Both observers introduced in Sections 6
...
2 and 6
...
3 have been applied to the same simulation
model of an AC machine of the form (6
...
The parameters values used
in the simulation are presented in Table 6
...
The physical values have been measured on a real
AC Motor bench at the GREYC Laboratory by Dorl´ ans and Massieu (2012)
...
1
...
1a
...
1c and 6
...
The effect on the motor speed is visible in the bumps seen on the curve of the
actual speed in Figure 6
...
For both observers, the speed estimation (Figures 6
...
1d)
and torque estimation (Figures 6
...
1e) are drawn
...
1b–c, with those parameters, the observer with sample and hold
equation (6
...
Note that we were able to make this observer converge by
reducing the sample interval Te
...
2, the observer with predictor
(6
...
1d,e
...


120

AC Electric Motors Control

125

120

115

110

105

100
3

4

5

6

7

8
9
Time in s

10

11

12

13

(a) Reference speed (square wave) and actual
speed of system (6
...
5

4

4
...
5

6

−150

6
...
57):
speed estimation

3

3
...
5

5
Time in s

5
...
5

(c) Observer with sample and hold (6
...
63):
speed estimation

12

13

−5

3

4

5

6

7

8
9
Time in s

10

11

(e) Observer with predictor (6
...
1 Simulation results

12

13

Observers Design for Systems with Sampled Measurements

6
...
The first design is a direct adaptation of the
continuous high gain observer proposed by Dib et al
...
The output is sampled and its
value is held, thus the correction term in the observer equation is constant between sampling
instants
...
At each sampling
instant, the predictor is reset to the actual value of the output
...

The Matlab/Simulink was used to test the implementation of those observer on a simulation
model of the induction machine
...
Simulations were done with different values
of sampling period and observer gain and the introduction of the predictor allows for larger
sampling intervals, as in the case shown on Figure 6
...

Further work involves the implementation of the observer with predictor on a real induction
machine
...


References
Ahmed-Ali T and Lamnabhi-Lagarrigue F (2012) High gain observer design for some networked control systems
...

Ahmed-Ali T, Cherrier E, and Lamnabhi-Lagarrigue F (2012) Cascade high predictors for a class of nonlinear systems
...

Ahmed-Ali T, Postoyan R, and Lamnabhi-Lagarrigue F (2009) Continuous-discrete adaptive observers for state affine
systems
...

Andrieu V and Nadri M (2010) Observer design fot Lipschitz systems with discrete-time measurements
...

Arcak M and Neˇ i´ D (2004) A framework for nonlinear sampled-data observer design via approximate discrete-time
sc
models and emulation
...

Astorga CM, Othman N, Othman S, et al
...
Control Engineering Practice, 10, 3–13
...
Systems &
Control Letters, 18, 295–299
...
(2011) High gain observer for sensorless induction motor
...

Dorl´ ans P and Massieu JF (2012) Plate-forme machine asynchrone et carte DSP DS1103
...

Fridman E, Dambrine M, and Yeganefar N (2008) On input-to-state stability of systems with time-delay: a matrix
inequalities approach
...

Gauthier J, Hammouri H, and Othman S (1992) A simple obsever for nonlinear systems: application to bioreactors
...

Hammouri H, Nadri M, and Mota R (2006) Constant gain observer for continuous-discrete time uniformly observable
systems Proceeding of the 45th IEEE on & Control, San Diego, CA, pp
...

Hermann R and Krener AJ (1977) Nonlinear controllability and observability
...

Hespanha J, Naghshtabrizi P, and Xu Y (2007b) A survey of recent results in networked control systems
...


122

AC Electric Motors Control

Jazwinski A (1970) Stochastic processes and Filtering Theory
...
Academic
Press
...
IEEE
Control, 54, 2169–2174
...
Applied Mathematics, 16, 967–974
...
European Journal of Control, 10, 252–263
...
ACC’08, American Control Conference, Seattle, WA
...
18th IFAC World Congress, Milan, Italy
...
Systems & Control Letters, 57(1), 18–27
...
1
7
...
1

Introduction
Problem Statement

Permanent-magnet synchronous motors (PMSM) drives are replacing classic DC and induction
motors drives, such as industrial robots and machine tools (Yang et al
...
Advantages of
PMSMs include high efficiency, compactness, high torque to inertia ratio, rapid dynamic
response, and simple modeling and control (Changsheng and Elbuluk 2001); (Peng et al
...
In the last years the price of rare-earth magnet material decreased significantly
...
2005)
...
The permanent magnets can be mounted on the surface of the rotor (surface permanent
magnet synchronous motor—SPMSM) or inside of the rotor (interior permanent-magnet synchronous motor—IPMSM)
...

This chapter is devoted to the first configuration, that is, SPMSM
...
The main drawback
of a SPMSM is the position sensor
...
Moreover, the sensor is vulnerable for electromagnetic noise in hostile

AC Electric Motors Control: Advanced Design Techniques and Applications, First Edition
...

© 2013 John Wiley & Sons, Ltd
...


124

AC Electric Motors Control

environments and has a limited temperature range
...
2009)
...

For these reasons, PMSM drive research have been concentrated on the elimination of
the mechanical sensors at the motor shaft (encoder, resolver, etc
...


7
...
2

State of the Art and Objectives

State of the Art
In order to avoid sensor position of SPMSM, several approaches for the so-called sensorless
control have attracted a great deal of attention recently (see, Bolognani et al
...

2007; Yu and Kaynak 2009; Arellano-Padilla et al
...
2010; Lee and Lee 2013)
...
The
first category is based on fundamental excitation methods, which are divided into two main
groups; nonadaptive (Benjak and Gerling 2010) or adaptive methods (Bolognani et al
...
The second category is
based on saliency and signal injection methods (Arellano-Padilla et al
...
The third one is
based on artificial intelligence methods (Gumus et al
...
2010)
...

To our best knowledge, the methods proposed in the literature to estimate the position and
speed of SPMSM motor are usually tested and evaluated experimentally at high speed whereas
as shown in Zaltni and Ghanes (2010) the main difficulties are primarily at very low speed of
the SPMSM unobservability
...
The observer is implemented and tested on experimental setup, with the aim
to compare its speed and position tracking capability on high speeds and low speeds where
particularly the motor state of SPMSM is unobservable
...
2, SPMSM model and a brief review of its observability analysis are presented
...
3, MRAS observer for SPMSM is described
...
4, experimental results of MRAS observer are illustrated on SPMSM high- and
low-speed benchmark
...
5, some conclusions are drawn
...
2

SPMSM Modeling and its Observability

7
...
1

SPMSM Model

In the (α − β) fixed reference frame, the dynamic model of the SPMSM reads (Zaltni and
Ghanes 2010; Lee and Lee 2013)
⎞ ⎛
˙
iα
⎜˙ ⎟ ⎜
⎝ iβ ⎠ = ⎝
⎛

ω
˙

R
− L ss i α +

φm
e
Ls α
φm
e
Ls β

⎞

⎛

1
Ls

⎟ ⎜
⎠+⎝ 0
fv
P
0
φ (− sin(θe )i α + cos(θe )i β ) − J ω
J m
R
− L ss i β +

0
1
Ls

0

⎞
uα
⎟⎜ ⎟
0 ⎠⎝ u β ⎠, (7
...
2)

refer respectively to BEMF and ω is the rotor speed; ωe = P
...


7
...
2

Quick Review on the Observability of SPMSM

The observability phenomena of SPMSM has been studied by several authors (see, for instance,
Junfeng et al
...
In Zaltni and Ghanes
(2010) we have presented sufficient and necessary conditions under which the PMSM is
observable and unobservable
...
The result is that the SPMSM observability cannot be established in the particular
case of zero speed and zero acceleration even if we use the higher derivatives of currents
...


7
...
1
...


7
...
1

Reference Model

The reference model consists of designing a second order sliding-mode observer (STO),
ˆ
ˆ
ˆ
which computes the reference BEMFs eα,β = [eα eβ ]T using only measured stator currents
and voltages
...


126

AC Electric Motors Control

uα, β

eˆα, β

Reference model
(STO observer)

iα, β

+
~
eα, β

Adjustable
model

ˆ
ωe

ε

–

Adaptation
mechanism
M

Figure 7
...
3)

ˆ
with e1 = x 1 − x 1 , λ, α > 0 are the observer parameters, v1 is the output of the differentiator, x 1 is the estimated variable and

sgn(e1 ), =
Let x = [i α

⎧
⎨

1,
−1,
⎩
∈ [−1

if
if
1],

e1 > 0;
e1 < 0;
if
e1 = 0
...
Consider only current dynamic equations of model (7
...

Ls

and c =

[x a

(7
...
5)

be the vector of unknown variables
...
4) becomes
?

˙
x 1 = ax 1 + x a + cu α ,

...
6)

127

Experimental Evaluation of Observer Design Technique for Synchronous Motor

Currents and voltages are assumed to be measurable
...
3) to system (7
...
7)

1
2

˙
ˆ
˜
ˆ
x 2 = x b + a x 2 + cu β + λ2 |e2 | sgn(e2 ),

˙
˜
x b = α2 sgn(e2 ),

ˆ
ˆ
where e1 = x 1 − x 1 , e2 = x 2 − x 2 , and λ1 , λ2 ,
Error dynamics of the observer are given by

α1 ,

α2 > 0
...
8)

1
2

˙
˜
ˆ
e2 = (x b − x b ) + a(x 2 − x 2 ) − λ2 |e2 | sgn(e2 ),

˙
eb = f 2 (x a ) − α2 sgn(e2 ),
with f 1 (x b ) = ωe x b and f 2 (x a ) = −ωe x a

Corollary 7
...
1 Taking into account the result proposed in Levant (1998) with respect to the
ˆ
STO (7
...
7)
...

The observer parameters α1 , α2 , λ1 , and λ2 have respectively the following expressions (Levant
1998):
?
2
+
+
,
α1 > f 1 and λ1 > ( f 1 + α1 )
α1 − f 1+
?
2
+
+
,
(7
...

˜
ˆ
˜
Having x a and x b we can easily deduce the estimated BEMFs eα,β from equation (7
...


7
...
2

Adjustable Model

The adjustable model is tunable by the estimated velocity and it computes the estimated
˜
˜
˜
BEMFs eα,β = [eα eβ ]T from the following equation:
˙
˜
ˆ
˜
ˆ ˜
eα,β = ωe J eα,β + M(eα,β − eα,β ),

(7
...

0

0
1

For convergence, a feedback loop is introduced and the feedback gain
M=m

?

1
0

?
0
,
1

where m is a positive constant
...
3
...


(7
...


(7
...

ˆ

(7
...

The BEMFs used in the reference model also satisfy the following equation
˙
ˆ
ˆ
eα,β = ωe J eα,β
...
14)

To show that the sliding mode can be enforced in the manifold S = 0, we need to show that
there exists M sufficiently high such that the manifold is attractive
˙
S S < 0
...
15)

After differentiating (7
...
10) and (7
...
16)

where f is a function of the reference and estimated BEMFs and speed
...
16) that sufficiently high K can be selected such that condition (7
...
Thus,
sliding mode is enforced in the manifold S and after sliding mode begins, we have S = 0
...
(1999) is used to find the equivalent
˙
control ωe,eq
...
The
expression of the equivalent control becomes
ωe,eq = ωe + m

˜
ε T J eα,β

...
17)

˜
From equation (7
...
The equivalent speed represents the low-frequency component
of the discontinuous term (7
...
Thus, while the high-frequency switching function is fed
into the observer, its low-frequency component can be obtained by low-pass filtering (LPF)
and represents the speed estimate
...
3
...
7) and (7
...

θ
ˆ
eβ

(7
...
To overcome this problem, an estimator/observer swapping system is proposed to ensure
position estimation in all speed range and to overcome position observability problems at very
low frequencies (see Section 7
...
2)
...
17)
...
Thus, since the speed is
always observable, there is no problem of observability using this position estimator
...
The position is equal to the
observer when the motor operate at high frequencies and swap to the estimator since the speed
become less than 5 rad/s
...
4

Experimental Results

The observer is tested on a significant benchmark where high- and very low-speed operation
modes of SPMSM are considered (Figure 7
...
It is tested in an open-loop control as showed
in Figure 7
...

The experimental testing is composed of an SPMSM with a 4096-pulse incremental encoder
to validate the observer
...
A
commercial 15 kW inverter is supplied with a DC voltage source Xantrex
...
A real-time controller board of dSPACE DS1104 and
interfaces are used to implement the speed, current control, and the proposed observer
...
The
experimental sampling time Te is equal to 200 μs
...
2 Block diagram scheme

The specifications and parameters of SPMSM and MRAS observer are listed in
Table 7
...


7
...
1

Nominal Conditions

Using the identified parameters of SPMSM (Table 7
...
It’s clear that due to experimental conditions, the identified parameters are not exactly
the real parameters of SPMSM
...
Furthermore, the identification methodology
introduces a certain uncertainty in its results
...

From now this case is called the nominal condition test
...
3
...
3 (dash-dot line)) coming from equation (7
...
1

Parameters specification of SPMSM and MRAS observer

SPMSM
parameters

value

MRAS observer
parameters

Pn
ωn
Un
In
Rs
Ls
P
φm
J
fv
Tl

value

1
...
8 A
2
...
15 mH
3
0
...
0249 kgm2
0
...
1 Nm

α1
α2
λ1
λ2
K
m

80 000
80 000
100
100
700
100

SPMSM, surface permanent-magnet synchronous motor; MRAS,
model reference adaptive system
...
3 (dashed line)) when the synchronous motor operates at high and very
low speeds
...
4
...
4 (solid line)) is obtained at high and very low frequencies
according to Section 7
...
4 and tracks well the measured position (Figure 7
...


350
ω*
300

ω
Observed

(rad/s)

250
200
150
100
50
0

0

1

2

3

4

5

Time (s)

Figure 7
...
4 Rotor position

However, at very low speeds (t = 4 s to t = 5 s) the estimated position (Figure 7
...

The applied stator voltages u α -u β and measured stator currents i α -i β are shown in Figure 7
...


7
...
2

Parameter Variation Effect

The robustness to internal disturbance of the proposed observer is tested by variation of
+100% of stator resistance Rs
...
5 Stator voltages and currents

133

Experimental Evaluation of Observer Design Technique for Synchronous Motor

(rad/s)

300

Observed
ω

200
100
0

0

1

2

3

4

5

(Degree)

200

6

7

8

Observed
θ

100
0

−100
−200

0

1

2

3

4
Time (s)

5

6

7

8

Figure 7
...
7
...
Compared to the results of nominal condition (Figures 7
...
4), it can be remarked that the proposed observer is very robust to this variation
...


7
...
3

Load Torque Effect

The robustness to external disturbance of the proposed observer is tested by applying to the
motor a constant load torque Tl at high and low speeds
...
The experimental results (Figure 7
...
7
...

However, at very low speeds the observed speed (Figure 7
...


7
...
This observer is designed to achieve sensor-less
control scheme for SPMSM
...
To overcome the problem of observability at very low speed,
the rotor position has been obtained using an estimator/observer swapping system when the
speed becomes less than 5 rad/s
...


134

(rad/s)

AC Electric Motors Control

300
280
260
240
220
200

ω*
ω

Observed

−0
...
5

1

(rad/s)

20

1
...
5

3

3
...
7 Rotor speed with +100% of Rs and load torque

References
Arellano-Padilla J, Gerada C, Asher G, and Sumner M (2010) Inductance characteristics of PMSMs and their impact
on saliency-based sensorless control
...

Bolognani S, Zigliotto M, and Zordan M (2001) Extended-range PMSM sensorless speed drive based on stochastic
filtering
...

Benjak O and Gerling D (2010) Review of position estimation methods for IPMSM drives without a position sensor
...
International Conference on Electrical Machines - ICEM
...
IEEE IAS Annual Meeting, 2, 1273–1278
...
IEEE Transaction on Industry Applications, 45, 582–590
...
International Aegean Conference on Electrical Machines and Power
Electronics
...
IEEE-ICECE, 78–81
...
(2004) A new control method for permanent magnet synchronous machines
with observer
...

Lee H and Lee J (2013) Design of an iterative sliding mode observer for sensorless PMSM control
...

Levant A (1998) Robust exact differentiation via sliding mode technique
...

Montesinos D, Galceran S, Sudria A, and Gomis O (2005) Low cost sensorless control of permanent magnet motors,
an overview and evaluation
...

Peng H, Lei H, and Chang-yun M (2010) Research on speed sensorless backstepping control of permanent magnet
synchronous motor
...

Utkin U, Guldner J, and Shi J (1999) Sliding mode control in electromechanical systems, 1st edn
...


Experimental Evaluation of Observer Design Technique for Synchronous Motor

135

Vaclavec and Blaha P (2008) A new control method for permanent magnet synchronous machines with observer
...
265–270
...
(2009) An Adaptive robust nonlinear motion controller combined with disturbance
observer
...

Yu X and Kaynak O (2009) Sliding-mode control with soft computing: a survey
...

Zaltni D and Ghanes M (2010) Synchronous motor observability study and an improved zero-speed position estimation
design
...


Part Three
Control Design
Techniques for
Induction Motors

AC Electric Motors Control: Advanced Design Techniques and Applications, First Edition
...

© 2013 John Wiley & Sons, Ltd
...


8
High-Gain Observers in Robust
Feedback Control of Induction
Motors
Hassan K
...
Strangas
Department of Electrical and Computer Engineering, Michigan State University, USA

8
...
In this approach, no speed measurement is used, but instead either reliable position
measurement by optical encoders or resolvers or no position measurement at all is utilized
...

Classic field orientation requires the transformation of variables, to either the stator, or, most
commonly, the rotor flux frame of reference
...
This makes all the new variables
available for feedback
...
The
technique, known as high-gain observers, works for a wide class of nonlinear systems and
guarantees that the output feedback controller recovers the performance of the state feedback
controller when the observer gain is sufficiently high
...
First, to design a state feedback controller that
stabilizes the system and meets other design specifications
...


AC Electric Motors Control: Advanced Design Techniques and Applications, First Edition
...

© 2013 John Wiley & Sons, Ltd
...


140

AC Electric Motors Control

Uncertainty in the rotor resistance and the load torque is allowed
...
The flux controller
is designed using either the traditional method with two PI controllers or a continuous approximation of variable structure control
...
In the current-control scheme, a PI controller is used to regulate the current to a
desired value that is designed using sliding-mode control
...
This
change of variables, which is dependent on the uncertain quantities, results in a state equation
where the uncertain terms satisfy the matching condition
...
The control design is a continuous approximation of sliding-mode control
...
We also show that in the special case when the speed reference and load
torque are constant, the speed tracking error tends to zero asymptotically as time approaches
infinity
...
We derive
a third order nonlinear model that captures the essence of the speed control problem
...

We show how the model can be used to investigate the design of a feedback controller with
integral action
...
2

Field Orientation

The squirrel-cage induction machine can be described by the fundamental equations that result
from electromagnetic considerations
...
We start with these equations
describing a linear model, neglecting saturation and nonuniform current distribution in the
rotor bars
...
The angle θ is the relative position of the rotor (axis of phase a with respect to the
stator),
dλr
= 0,
dt
dλs
R s is +
= us ,
dt

Rr ir +

(8
...
2)

with the flux linkages λr and λs in their corresponding natural stator and rotor frames of
reference,
?
?
λr = M (1 + σr )ir + is e−i pθ ,
?
?
λs = M (1 + σs )is + ir ei pθ
...
3)
(8
...
5)
(8
...
The torque, resulting from the interaction of
currents and fluxes, can be written in a variety of ways
...
7)

1
d 2θ
= ω = [Td − TL ] ,
˙
2
dt
J

(8
...

This system of equations can be rewritten with all variables transformed in any common
frame of reference
...
(2010)
...

In the reference system of the stator, and using matrix notation, the equations become
˙
θ = ω,

ω=
˙

T
−μλr

(8
...
10)

Ei s − TL /J,

˙
λr = (−αr I + pωE)λr + αr Mi s ,

(8
...
12)

where αr = Rr /L r , αs = Rs /L s , β = M/σ L r L s , γ = 1/σ L s , η = 1/σ , μ = p M/J L r ,
σ = 1 − M 2 /L s L r , and
I =

?

1
0

?
0
,
1

E=

?

0
1

?
−1

...


(8
...


(8
...

cos ρ

(8
...
16)

In particular,
? ?
? ?
id
i
= F(ρ) a ,
iq
ib

?

?
? ?
ud
u
= F(ρ) a ,
uq
ub

?

?
? ?
λ
λd
= F(ρ) a
...
17)

The torque equation in the rotor flux frame of reference becomes
Td =

pM
λd i q ,
Lr

(8
...
11),
while the angle of this flux, ρ, can be calculated from the stator currents and speed
˙
λd = −αr λd + αr Mi d ,
αr Mi q
ρ =ω+
˙

...
19)
(8
...
18) and (8
...
19)
and (8
...
Finally, the stator current satisfies
2
˙
i d = p ωi q + αr βλd − (αs η + αr β M)i d + αr Mi q /λd + γ u d ,

˙
i q = − p ωi d − p ωβλd − (αs η + αr β M)i q − αr Mi d i q /λd + γ u q
...
21)
(8
...

Of those, errors in the rotor resistance, Rr , and the corresponding rotor time constant, 1/αr ,
cause inaccuracies in the estimates of both flux magnitude and angle
...
11) requires the speed ω
...
It is estimated from the position θ when
a mechanical sensor is used or from the stator current and voltage when θ is not measured
...
23)

where αr 0 is a nominal value of αr and ω0 is a replacement for ω, with the most common
choices are a speed estimate ω or the speed reference ω∗
...
23), are different from the
ones obtained when αr = αr 0 and ω0 = ω
...

λd

(8
...
25)

The stator current and voltage transformed to this estimated frame of reference are given
by equation (8
...
To calculate the developed torque in this
estimated frame of reference we start again from equation (8
...

Lr

(8
...
27)

and transforming the variables for the torque to the estimated frame of reference
?

ia
ib

?

?

?
id
= F(−ρ)
,
iq

and since λq = 0, we obtain

?

λa
λb

?

= F(−ρ)

?
??

λa
λb

λd
0

?

?

?

?
λd
= F(−ρ)
,
λq

(8
...
29)

e
+ d
eq

??

,

giving the torque equation using variables in the estimated frame of reference
Td =

?
pM ?
i q λd + (i q ed − i d eq )
...
30)

The stator currents i d and i q satisfy the equations

2
˙
i d = p ω0 i q + αr βλd − (αs η + αr β M)i d + αr 0 Mi q /λd + γ u d + αr βed + p βωeq , (8
...
32)

while the rotor flux errors ed and eq satisfy the equations
˙
ed = −αr ed + ( p ω0 − p ω + αr 0 Mi q /λd )eq + (αr − αr 0 )(Mi d − λd ),

˙
eq = −( p ω0 − p ω + αr 0 Mi q /λd )ed − αr eq + (αr − αr 0 )Mi q + p(ω − ω0 )λd
...
33)
(8
...
3

AC Electric Motors Control

High-Gain Observers

To motivate the design of high-gain observers, consider the second order nonlinear system
˙
x1 = x2 ,

˙
x 2 = φ(x, u, w, d),
y = x1 ,

(8
...
36)
(8
...
The function φ is
locally Lipschitz in (x, u) and continuous in (d, w)
...
Suppose the state feedback control u = γ (x, w)
stabilizes the origin x = 0 of the closed-loop system
˙
x1 = x2 ,

˙
x 2 = φ(x, γ (x, w), w, d),

(8
...
39)

uniformly in (w, d), where γ (x, w) is locally Lipschitz in x and continuous in w
...
40)
(8
...
The estimation error
? ? ?
?
˜
ˆ
x1
x1 − x1
˜
x=
=
˜
ˆ
x2
x2 − x2
satisfies the equation
˙
˜
˜
˜
x 1 = −h 1 x 1 + x 2 ,

˙
˜
˜
˜
x 2 = −h 2 x 1 + δ(x, x, w, d),

(8
...
43)

where
ˆ ˆ
˜
ˆ
ˆ
δ(x, x, w, d) = φ(x, γ ( x, w), w, d) − φ( x, γ ( x, w), w)
...
In the presence
˜
of δ, we design h 1 and h 2 with the additional goal of rejecting the effect of δ on x
...
While this is not possible, we can make supω∈R ?G o ( jω)?
arbitrarily small by choosing h 2 ? h 1 ? 1
...
44)

for some positive constants α1 , α2 , and ε, with ε ? 1, it can be shown that
ε
G o (s) =
(εs)2 + α1 εs + α2

?

?
ε

...
The disturbance rejection property of the high-gain observer can
also be seen in the time domain by using the scaled estimation errors
ξ1 =

˜
x1
,
ε

˜
ξ2 = x 2 ,

(8
...


(8
...
47)

This equation shows that reducing ε diminishes the effect of δ
...
The initial condition ξ1 (0) could be O(1/ε) when
ˆ
x 1 (0) ?= x 1 (0)
...
46) could contain a term of
the form (1/ε)e−at/ε for some a > 0
...
In fact, the function (1/ε)e−at/ε approaches an impulse function as ε tends to
zero
...
5)
...

ˆ
This property can be always achieved by saturating u and/or x outside compact sets of interest
...
Then we determine bounds M1 and M2 such that |x 1 (t)| < M1 and |x 2 (t)| < M2
ˆ
ˆ
over ?
...
The global
ˆ
ˆ
boundedness of γ and φ in x provides a buffer that protects the plant from peaking because
ˆ
during the peaking period the control γ (x, w) saturates
...
After the peaking period, the estimation

146

AC Electric Motors Control

ˆ
error becomes of the order O(ε) and the feedback control γ ( x, w) becomes O(ε) close to
γ (x, w)
...
Thus, the performance achieved under state feedback can be recovered by
output feedback by choosing ε small enough
...
For the purpose of feedback control, we act as if both x 1 and
x 2 were measured
...

The high-gain observer design can be extended to higher order systems of the form
˙
xi = xi+1 ,

for 1 ≤ i ≤ r − 1;

˙
xr = φ(x, z, u, w, d);

˙
z = ψ(x, z, u, w, d);

(8
...
49)
(8
...
51)

where x and z constitute the state vector
...
The vector z represents
the state of the internal dynamics, which are not estimated by the high-gain observer
...
52)
(8
...
In feedback control, we first design a state feedback controller as if
ˆ
the vector x was measured; then we replace x by its estimate x provided by the high-gain
observer
...


8
...
4
...

ω = μ i q λd + (i q ed − i d eq ) −
˙
J

(8
...
55)

High-Gain Observers in Robust Feedback Control of Induction Motors

147

A high-gain speed observer is taken as
˙
ˆ
ˆ
θ = ω + (α1 /ε)(θ − θ ),
ˆ

˙
ˆ
ω = μi q λd + (α2 /ε 2 )(θ − θ ),
ˆ

(8
...
57)

where α1 and α2 are positive constants that assign the roots of s 2 + α1 s + α2 at desired
locations in the left-half plane and ε is a small positive parameter, which is tuned to achieve
the desired performance
...
4
...
6, the feedback controller uses the speed ω and the
acceleration a = ω
...
58)

˙
a = F(ω, ω0 , λd , i d , i q ) + μγ λd u q + δ2 (·),

(8
...
59)

where F is given by
?
?
F = μλd − p βωλd − p ω0 i d − (αr 0 + αs η + αr 0 β M)i q ,

and δ2 is an uncertain function whose components are proportional to ed , eq , or (αr − αr 0 )
...
61)

˙
ˆ
ˆ
a = F(ω, ω0 , λd , i d , i q ) + μγ λd u q + (α3 /ε 3 )(θ − θ ),
ˆ

(8
...
62)

where the constants α1 , α2 and α3 assign the roots of s 3 + α1 s 2 + α2 s + α3 in the left-half
plane and ε > 0 is chosen sufficiently small
...
23) with ω0 = ω, in which case ω0 in (8
...


8
...
3

Speed Estimation without a Mechanical Sensor

Information about ω is contained in the derivative of i q , as seen from the term pωβλd on the
right-hand side of (8
...
Rewrite equation (8
...
64)
(8
...


We view i q as the measured output and use it, together with the models (8
...
65),
to build an observer to estimate ω
...
3 can be designed if
the uncertain terms appear only on the right-hand side of (8
...
The presence of δ3 in (8
...
The change of variables
δ3
p β λd
?
?
λ d + ed
1
=
[(αr − αr 0 )β Mi q − αr βeq ]
ω+
λd
p β λd

?=ω−

(8
...
64) and (8
...
67)
(8
...
The change of variables (8
...
We use the high-gain observer
?α ?
1
˙
ˆ
ˆ
ˆ
i q = − p βλd ? − f 1 (λd , i d , i q , u q , ω0 ) +
(i q − i q ),
ε
?
?
α2
˙ = μi λ −
ˆ
ˆ
?
(i q − i q ),
q d
ε 2 p βλd

(8
...
70)

where ε is a small positive parameter and α1 and α2 are positive constants that assign the roots
of s 2 + α1 s + α2 = 0 at desired locations in the left-half plane
...

εξ
p βλd

(8
...
72)

For small ε, the closed-loop system will be a singularly perturbed one, with ξ1 and ξ2 as the
fast variables
...
1999), the stability
c
of the fast dynamics is determined by the matrix
⎡
⎢
⎣

−α1

− p βλd

α2
p βλd

0

⎤
⎥
⎦

in which λd > 0 is treated as a constant
...
From the high-gain observer theory (Atassi and
Khalil 1999), we know that if the control input u s is bounded uniformly in ε, then the estimation
ˆ
error ? − ? will be O(ε) after a short transient period [0, T (ε)], where limε→0 T (ε) = 0
...
Hence, we can design the
feedback controller with feedback signals from λd , i d , i q , and the virtual speed ?, defined by
ˆ
(8
...
In implementation, ? is replaced ?, which is provided by the speed observers (8
...
70)
...


8
...
This is achieved by the design of a state feedback control law for u d using the
second order model
˙
λd = −αr 0 λd + αr 0 Mi d ,
2
˙
i d = p ω0 i q + αr βλd − (αs η + αr β M)i d + αr 0 Mi q /λd + γ u d

+ αr βed + p βωeq
...
73)

(8
...
The
traditional approach uses two PI controllers (Leonhard 1996), which are derived as follows
...
73) and design the PI flux controller
def
˙
˜
φλ = λd − λ∗ = λd ,

˜
i D = −K f (τ f λd + φλ ),

(8
...
Then, we design the PI current controller
def
˜
˙
φd = i d − i D = i d ,

˜
u d = −K d (τd i d + φd ),

(8
...
74) to regulate i d to i D
...
The
design should ensure that λd starts at a positive value and approaches λ∗ monotonically so that
λd is always positive
...
23), which is at our disposal
...
As an example, we describe
a continuously implemented sliding-mode controller
...
77)

we can rewrite equations (8
...
74) as
˙
λd = −αr 0 λd + αr 0 λ∗ + αr 0 s1 ,
˙
s1 = M[ p ω0 i q + αr βλd − (αs η + αr β M)i d +

2
αr 0 Mi q /λd

˙
+ αr βed + p βωeq ] − λ∗
...
78)
+ γ ud
(8
...

The choice
vd = −k1 sat(s1 /μ1 ),
with
k1 ≥ k0 + |αr βed + p βωeq − (αr − αr o )β Mi d |,
for some positive constants k1 and k0 and a small positive constant μ1 , ensures that
˙
s1 s1 ≤ −k0 M|s1 |,

for |s1 | ≥ μ1 ,

which shows that, within finite time, s1 and |λd − λ∗ | will be of the order O(μ1 )
...


High-Gain Observers in Robust Feedback Control of Induction Motors

8
...
We describe two designs
...
The second design is a
voltage-control scheme where the voltage u q is treated as the control input
...
23) is implemented using
the speed estimate ω provided by the high-gain observer of Section 8
...
1
...
Therefore, the current i q is designed using the model
˙
θ = ω,
?
?
ω = μ i q λd + (i q ed − i d eq ) − TL /J,
˙

(8
...
83)

(8
...
82)

where equations (8
...
83) are obtained from equations (8
...
34), respectively,
by setting ω0 = ω
...


Assuming that the flux λd has been regulated to a constant value λ∗ , the term (Mi d − λd ) on
the right-hand side of equation (8
...
80), (8
...
82),
and (8
...
84)
∗

∗

∗

˙
x 2 = μ(λ + ed )i q − μλ eq /M − ω − TL /J,
˙
∗

˙
ed = −αr ed + αr 0 Mi q eq /λ ,

˙
eq = −αr 0 Mi q ed /λ∗ − αr eq + (αr − αr 0 )Mi q
...
85)
(8
...
87)

The function
2
2
V = ed + eq

satisfies the equation
˙
V = −2αr V + 2(αr − αr 0 )Mi q eq ,
which shows that for any bounded i q the flux errors ed and√q will be bounded and, after some
e
finite time, their ultimate bounds will be proportional to |αr − αr 0 |
...
84) and (8
...
Consider the continuously
implemented sliding-mode control
i q = Iq sat(s2 /μ2 ),
where
s2 = ax 1 + x 2 ,

a > 0,

Iq is the maximum value of i q , and μ2 is a small positive constant
...
If the inequalities
λ∗ + ed ≥ k 1

and

Iq ≥ k 2 +

ax 2 − μλ∗ eq /M − ω∗ − Y L /J
μ(λ∗ + ed )

are satisfied in ? for some positive constants k1 and k2 , then it a positively invariant set,
˙
˙
because on the boundaries |x 1 | = b/a, |s2 | = b and V = c2 , we have x 1 x 1 ≤ 0, s2 s2 ≤ 0, and
˙
V ≤ 0, respectively
...
Furthermore, if ω∗ and TL are constant and
(αr − αr 0 ) is sufficiently small, it can be shown that the trajectories converge to an equilibrium
point where ω = ω∗
...
32)
...
88)

˙
x2 = x3 ,

(8
...
90)

∗

˙
ed = −αr ed + αr 0 Mi q eq /λ ,

˙
eq = −αr 0 Mi q ed /λ∗ − αr eq + (αr − αr 0 )Mi q ,

(8
...
92)

153

High-Gain Observers in Robust Feedback Control of Induction Motors

where δ6 is an uncertain term whose components are proportional to ed , eq , or (αr − αr 0 )
...
Thus, the design of
u q is a robust control problem for the equations (8
...
89), and (8
...
Using the highgain observer of Section 8
...
2 to estimate ω and ω, and relying on the performance recovery
˙
property of high-gain observers, we can proceed to design a state feedback control in terms of
x 1 , x 2 , and x 3
...
The details can be found in Khalil et al
...


8
...

This time, however, we do not have measurements of θ and we use the high-gain observer of
Section 8
...
3 to estimate the speed
...
23) is implemented with ω0 = ω∗
and we assume that flux λd has been regulated to a constant value λ∗
...
32) as
˙
i q = −(αs η + αr β M)i q − αr 0 i q + γ u q + d2 ,

(8
...
For any current command i Q , we can design a state feedback
controller for u q , with sufficiently large gains, to regulate i q to i Q
...


(8
...
Thus, the motor dynamics can be described by
the third order model
˙
ed = −αr ed + ( pω∗ − pω + αr 0 Mi Q /λ∗ )eq ,
∗

∗

(8
...
96)
(8
...
98)

where ? is viewed as the measured output and a = (η − η)/( p βλ∗ )
...
(2009) it
ˆ
is shown how to apply the singular perturbation theory (Kokotovi´ et al
...
95), (8
...
97), and (8
...

The model (8
...
96), (8
...
98) enables us to design the current i Q as a feedback
function of ? to regulate ω to ω∗ , and perform rigorous analysis of the nonlinear closed-loop

154

AC Electric Motors Control

system
...
Under the condition
? = ω∗ , the equilibrium equations are
¯
¯
¯
¯
0 = −αr ed + ( pω∗ − p ω + αr 0 M i Q /λ∗ )eq ,

(8
...
100)

¯
¯
¯
¯
0 = −( pω∗ − p ω + αr 0 M i Q /λ∗ )ed − αr eq

¯
¯
¯
0 = μ[i Q (λ + ed ) − eq λ /M] − TL /J,
? ∗
?
¯
¯
αr eq
λ + ed
¯
ω∗ =
+ ai Q
...
101)
(8
...
99) and (8
...
102), it can be shown that

?

− pω +
˜

¯ ??
¯ ?
(αr − αr 0 )M i Q
αr 0 M i Q
pω +
˜
ωc = 0,
λ∗
λ∗

(8
...

∗
λ

Assuming that ωc ?= 0, the equation has two solutions:
ω=
˜

¯
−(αr − αr 0 )M i Q
∗
pλ

or

ω=
˜

¯
αr 0 M i Q

...
The equilibrium point corresponding to
this solution is
¯
¯
ed = eq = 0,
∗
bω + TL /J
¯
,
iQ =
μλ∗
ω = ω∗ −
¯

¯
(αr − αr 0 )M i Q

...
104)

(8
...
95), (8
...
97), and (8
...
105) to
obtain the linear model
¯
˙
x = Ax + B(i Q − i Q ),

¯
? − ω∗ = C x + D(i Q − i Q ),

155

High-Gain Observers in Robust Feedback Control of Induction Motors

where
⎡

⎢
⎢
⎢
A = ⎢−
⎢
⎣

¯
αr M i Q
λ∗

−αr
¯
αr M i Q
λ∗

¯
μi Q

0

⎤

⎥
⎥
⎥
pλ∗ ⎥ ,
⎥
⎦
−0

−αr
−

⎡

0

μλ∗
M

⎤

⎥
⎢
⎥
⎢
B = ⎢ (αr − αr 0 )M ⎥ ,
⎥
⎢
⎦
⎣
μλ∗
C=

?

ω
¯
λ∗

−

αr
pλ∗

?
1 ,

D=

(αr − αr 0 )M
,
pλ∗

with the transfer function
G(s) = C(s I − A)−1 B + D =

n(s)
,
d(s)

in which
?
¯ ?
ωc αr M i Q
n(s) = μλ∗ s 2 + αr s +
λ∗

?
?
(αr − αr 0 )M
× 1+
s ,
μpλ∗ 2
?
?
?
¯ ?2
αr M i Q
2
d(s) = s (s + αr ) +
λ∗
pμλ∗ 2
+
M

?

s + αr −

¯2
αr M 2 i Q
λ∗ 2

?


...
When ωc i Q = 0, G(s) has
a zero at the origin
...

¯
This follows from the theory of servomechanisms (Davison 1976)
...
It is possible to design a
controller with integral action to stabilize the system, but such a controller cannot be a simple
PI controller
...
For a PI controller, the root locus will always have a branch that

156

AC Electric Motors Control

¯
lies entirely on the positive real axis
...
In
this case the transfer function G(s) is minimum phase and we can design a PI controller with
high-gain feedback, of the form
˙
φω = ? − ω ∗ ,

i Q = −K [(? − ω∗ )φω ],

to stabilize the closed-loop system and achieve good tracking properties
...

¯
¯
The condition ωc i Q = 0 is satisfied if i Q = 0 or ωc = 0
...
It is well known in the induction motor literature that operating the motor
at zero (or low) frequency is challenging, and that a design for such case will have to exploit
secondary phenomena of the machine, which are not conveyed in our model (see, e
...
, Ferrah
¯
et al
...
The case i Q = 0 indicates that the motor is running at constant speed without
producing electromagnetic torque, which is unrealistic because balancing the motor’s friction
alone would require production of electromagnetic torque
...
8

Simulation and Experimental Results

In Khalil and Strangas (1996), Aloliwi et al
...
(1997), the speed
tracking problem is discussed and a controller is developed using rotor position sensors and
under uncertainty of the stator and rotor resistance and the load torque
...
(1996) and Aloliwi et al
...
(1997) presents
experimental results
...
In
the experimental results of Aloliwi et al
...
The actual and estimated speed and flux for rotor resistance at 200% of its nominal value
showed good performance
...
(1999), a torque controller is presented
...
The speed of the rotor is estimated from
the stator currents using a high-gain observer
...
This was demonstrated for speed reversal where the load was only inertia
and for torque command reversal
...

In Khalil et al
...
The discussion is centered on the development of a high-gain speed
observer to estimate the speed from field-oriented currents and voltages
...
It allows for

High-Gain Observers in Robust Feedback Control of Induction Motors

157

errors in both rotor and stator resistance
...
These were compared for 10% increase in Rs and
¯
Rr
...


8
...
The work has
focused on field-oriented control of induction motors, but the same tools can be applied to
other machines and different control strategies
...


References
Aloliwi B, Khalil HK, and Strangas EG (1997) Robust speed control of induction motors
...

Aloliwi B, Khalil HK, and Strangas EG (2000) Robust speed control of induction motors: application to a benchmark
example
...

Atassi AN and Khalil HK (1999) A separation principle for the stabilization of a class of nonlinear systems
...

Davison EJ (1976) The robust control of a servomechanism problem for linear time-invariant multivariable systems
...
Automat
...
, AC-21(1), 25–34
...
Prentice Hall, Upper Saddle River, NJ
...

IEEE Transactions on Automatic Control, 41, 1216–1220
...
IEEE Transactions on Control Systems Technology, 17, pp
...

Ferrah A, Bradley KJ, and Asher GM (1992) Sensorless speed detection of inverter fed induction motors using slot
harmonics and fast fourier transform
...

Kokotovi´ PV, Khalil HK, and O’Reilly J (1999) Singular Perturbations Methods in Control: Analysis and Design
...

Leonhard W (1996) Control of Electrical Drives 2nd edn
...

Novotony DW, Lipo TA, and Jahns TM (2010) Introduction to Electric Machines and Drives
...

Krishnan R (2001) Electric Motor Drives
...

Strangas EG, Khalil HK, Aloliwi B, et al
...
Robust tracking controllers for induction motors without rotor
position sensor: analysis and experimental results
...


9
Adaptive Output Feedback Control
of Induction Motors
Riccardo Marino, Patrizio Tomei, and Cristiano Maria Verrelli
Dipartimento di Ingegneria Elettronica, Universit´ di Roma “Tor Vergata”, Italy
a

9
...
The availability of low-cost powerful digital signal processors and
significant advances on power electronics allow for the design of complex induction motor
(IM) controllers with the aim of achieving high performance on speed tracking and power
efficiency
...
On the other hand, speed sensors may fail or be eliminated on purpose to
increase reliability and noise immunity as well as to reduce cost and maintenance: in this case,
the estimation and tracking control problem is called “sensorless” since only stator currents are
assumed to be measured and available for feedback
...

The sensorless estimation and tracking control problem with no use of nonrobust openloop integration of flux dynamics (or equivalently rotor flux measurements) has been recently

AC Electric Motors Control: Advanced Design Techniques and Applications, First Edition
...

© 2013 John Wiley & Sons, Ltd
...


Adaptive Output Feedback Control of Induction Motors

159

addressed
...
(2009), Lin
et al
...
(2005, 2008), Montanari et al
...
2012), the problem of designing an estimation and tracking control algorithm and
e
of proving its closed-loop stability for sensorless IMs with uncertainties in load torque and
stator and rotor resistances still remains open to the best of our knowledge
...
(2006) (see also Mitronikas et al
...
2007; Jadot et al
...
2010), errors in estimating the stator resistance may lead to steady-state rotor speed and
flux modulus tracking errors and even to instability, especially at low speeds
...
(2003), Fattah and Loparo (2001), Feemster et al
...
(2009), Karagiannis et al
...
(1999), Peresada and
Tonielli (2000), Peresada et al
...
(1997) do not solve, via a
priori verifiable persistency of excitation conditions, the critical case of output tracking in the
presence of uncertain load torque, rotor and stator resistances
...
(1999), in the presence, however, of not a
priori verifiable persistency of excitation conditions; (2) only a qualitative sensitivity analysis
of the persistency of excitation conditions is provided by Jadot et al
...

The aim of this chapter is to show that, under specific observability and identifiability
conditions, solutions to the above sensorless and output feedback estimation and tracking
control problems exist
...
In particular, owing
to the use of a sufficiently slow adaptation for the stator resistance estimate (see Montanari
and Tilli (2006) and Jadot et al
...
Exponential rotor speed and flux modulus tracking is thus,
in both cases, achieved along with exponential estimation of the unmeasured state variables
and uncertain parameters
...


9
...
2010)
TL
dωm
= μ(φr a i sb − φr b i sa ) −
,
dt
J
dφr a
= −αφr a − ωm φr b + α L m i sa ,
dt
dφr b
= −αφr b + ωm φr a + α L m i sb ,
dt
?R
?
1
di sa
s
=−
+ βα L m i sa + νsa + βαφr a + βωm φr b ,
dt
σ
σ
?R
?
1
di sb
s
=−
+ βα L m i sb + νsb + βαφr b − βωm φr a ,
dt
σ
σ

(9
...
The model
parameters are: load torque TL = TLn + θ , where θ ∈ [−θm , θm ] denotes the constant uncertain variation from the constant nominal value TLn (TL is typically uncertain since it depends
on applications); (known) motor moment of inertia J ; rotor and stator windings resistances
Rr and Rs and (known) inductances L r and L s ; and (known) mutual inductance L m
...

Besides the load torque TL , the parameters α and Rs are also assumed to be uncertain taking
into account resistance variations during operations due to motor heating
...
(1999), an angle ε0 (t), whose dynamics dεdt(t) = ω0 (t) will be later
defined (ε0 (0) is an arbitrary initial condition), we can equivalently consider the vectors
the
[φr d , φrq ]T , [i sd , i sq ]T , and [νsd , νsq ]T , which are obtained by multiplying ? corresponding
?
cos ε0 sin ε0
T
T
T
(a, b) vectors [φr a , φr b ] , [i sa , i sb ] , and [νsa , νsb ] by the matrix R(ε0 ) =

...

Using the state coordinates (ωm , φr d , φrq , i sd and i sq ) and the control variables (νsd and νsq )
the motor dynamics (9
...

σ
σ
=−

s

(9
...
Following the
field-oriented control strategy in Blaschke (1972), our goal is to design dynamic sensorless
and output feedback compensators of the form

dε0 (t)
= ω0 (t),
dt
? ?
?
cos ε0 (t)
νsa (t)
=
νsb (t)
sin ε0 (t)

(9
...
4)

and
?
?
lim φr d (t) − φ ∗ (t) = 0,
t→∞
?
?
lim φrq (t) = 0,
t→∞

(9
...
6)

which imply that
lim

t→∞

??

?
2
2
φr a (t) + φr b (t) − φ ∗ (t) = 0
...
5) and (9
...


9
...
The simplest feedback action is to add PI controls on the direct and quadrature
stator voltage vector components on the basis of the direct and quadrature stator currents errors

162

AC Electric Motors Control

(see, Marino et al
...
Field orientation is not attained in the presence of parameter
errors (especially in load torque and rotor resistance) and steady-state errors may appear:
they cannot be arbitrarily reduced by increasing the PI gains in the current loop
...
Field orientation and consequently vanishing speed tracking
errors can be achieved by online estimation of the critical parameters load torque and rotor
resistance: this estimation is inherently linked with the estimation of rotor speed and fluxes and
of stator resistance
...
2010, and references therein)
...
In this
section we present a sensorless solution to the estimation and tracking control problem for IMs
with uncertain load torque and rotor and stator resistances, which relies on: (1) persistency of
excitation conditions, which may be interpreted in terms of rotor speed and flux observability
and rotor resistance identifiability and involve the rotor speed and flux modulus reference
signals only; (2) conditions for the identifiability in first approximation of the stator resistance
at steady state
...
3
...
7)

∗
∗
in which: the reference signals i sd and i sq for i sd and i sq and the speed ω0 of the (d, q) rotating
frame, which, as in field-oriented control, are responsible for rotor speed and flux modulus

Adaptive Output Feedback Control of Induction Motors

163

tracking, are chosen as
˙
φ∗
φ∗
+
,
Lm
αLm
ˆ
?
ˆ
1 ?
TLn
sat(θ )
∗
=
ˆ
+
+ ωm ,
˙∗
−kω (ωm − ωm ) +
μφ ∗
J
J

∗
i sd =
∗
i sq

ε0 = ω 0 = ω m +
˙
ˆ

∗
α L m i sq
ˆ

φ∗

(9
...
7) and (9
...
9)
βφ ∗
?R
?
ˆs
1
˙
∗
ˆ
ˆ
ˆ
ˆ
i sd = −
ˆ
+ β α L m i sd + νsd + ω0 i sq − ωm (i sq − z q ) − α(i sd − z d ) + ki (i sd − i sd ),
ˆ
σ
σ
ˆ
Rs
1
α
ˆ
˙
ˆ
ˆ
ˆ
z d = − i sd + νsd + ω0 z q + (i sd − i sd ),
σ
σ
γ1
ˆ
Rs
1
ω∗
˙
ˆ
ˆ
ˆ
z q = − i sq + νsq − ω0 z d + m (i sd − i sd ),
σ
σ
γ1
?
?
˙
β φ∗
˙
ˆ
ˆ
(i sd − i sd ), α ,
α = Proj −
ˆ
γ2 α
ˆ

α(0) ∈ [αm , α M ], 0 < αm − εα ,
ˆ
?
? ∗
ˆ
˙ = −ε (ωm i sq + αi sd ) (i − i ) ,
ˆs
ˆsd
R
R
sd
σ γ1

ˆ
Rs (0) ∈ [Rsm , Rs M ],

164

AC Electric Motors Control

where Proj[ζ, α] is the projection algorithm (see, Marino et al
...

(α M + εα )2 − α M 2

The load torque uncertainty saturated estimate appearing in equations (9
...
9) is defined
as

ˆ
sat(θ ) =
l0 =
l2 =

⎧
ˆ
⎨θ ,
?
⎩

3
ˆi
i=0 li θ ,

θm + ε,

2
θm (θm +
ε2

ε)

ε + 3θm
,
ε2

,

ˆ
if 0 ≤ θ ≤ θm ;
ˆ
if θm < θ < θm + ε;
ˆ ≥ θm + ε;
if θ

2
−2θm ε − 3θm + ε 2
,
ε2
1
l3 = − 2 ,
ε

l1 =

in which sat(x) is a class C 1 odd function that is linear in the closed set [−θm , θm ] and satisfies
|sat(x)| ≤ θm + ε for all x ∈ ?
...
7),
(9
...
9) depends on: the available i sa and i sb measurements; the smooth bounded
∗
reference signals (ωm and φ ∗ ) and their bounded first and second order time derivatives; the
known motor parameters J, L r , L s , and L m ; the known bounds θm , αm , α M , Rsm , and Rs M ;
the positive control parameters kω , ke , k, ki , λ1 ?= λ2 ?= λ3 , γ1 , γ2 , εα , ε R , and ε
...
3
...
(2008), introduce the angle ε0 that satisfies

?
α L m ? TL
∗
∗
+ ωm (t) ,
˙∗
ε˙0 (t) = ωm (t) +
μφ ∗2 (t) J

∗
ε0 (0) = ε0 (0),

depending on the uncertain parameters α and TL ; define the tracking and estimation
∗
∗ ˜
∗ ˜
˜
errors: ωm = ωm − ωm , φr d = φr d − φ ∗ , φrq = φrq , ed = i sd − i sd , eq = i sq − i sq , i sd = i sd
˜

165

Adaptive Output Feedback Control of Induction Motors

ˆ ˜
ˆ
ˆ
ˆ
˜
ˆ ˜
−i sd , i sq = i sq − i sq , eω = ωm − ωm , eφd = φr d − φr d , eφq = φrq − φrq , θ = θ − θ , α = α
ˆ
ˆ
˜ s = Rs − Rs
...

min{λ1 , λ2 , λ3 } + inf
t≥0 φ ∗ (t)

Assume, as in Marino et al
...
10)

which may be physically interpreted in terms of motor observability and rotor resistance
identifiability (see Marino et al
...

⎣
⎦
˙
β φ ∗ (t)
− α
∗
∗
Note that ε0 (and consequently (9
...


Remark 9
...
1 The proposed estimation and tracking control algorithm is obtained by
ˆ
replacing the uncertain stator resistance Rs by its estimate Rs in the estimation and tracking
control law presented in Marino et al
...


ˆ
ˆ
The fluxes are estimated through the estimates z d and z q of the auxiliary variables z d =
i sd + βφr d and z q = i sq + βφrq whose dynamics
Rs
1
i sd + νsd + ω0 z q ,
σ
σ
Rs
1
˙
z q = − i sq + νsq − ω0 z d
σ
σ

˙
zd = −

depend on neither the unmeasured rotor speed ωm nor the uncertain parameter α
...
The
closed-loop error system can be written as
˜
˙
y = Q 1 (y, t) + Q 2 (y, t) Rs

...

˜ s ]?
?[y, R

Since, according to Marino et al
...
e
...

As in Jadot et al
...
In
set (containing the origin), a steady-state solution h( R
particular, let Br (0) be the closed ball centered at the origin with sufficiently small radius r
˜
and assume that for all ( Rs , t) ∈ Br (0) × [0, +∞) the following condition holds:
˜
A) there exists a smooth solution h( Rs , t) to the nonlinear partial differential equation
˜
∂h( Rs , t)
˜
˜
˜
= Q 1 (h( Rs , t), t) + Q 2 (h( Rs , t), t) Rs
∂t
˜
with h = [h ωm , h φr d , h φrq , h ed , h eq , h e1 , h e2 , h e3 , h i˜sd , h za , h zb , h α ]T (h( Rs , t) being
˜
˜
˜
˜
˜
˜
bounded on Br (0) × [0, +∞) along with its first order partial derivatives) and
satisfying
h(0, t) = 0,

∀ t ≥ 0
...
To this purpose, we first recall that βeφd
˜
ˆ
˜
z d , and βeφq = z q − z q = z q , so that
?

˜
za
˜
zb

?

=

?

∗
cos ε0
∗
sin ε0

∗
− sin ε0
∗
cos ε0

??

?
˜
zd
,
˜
zq

˜
˜
and we then consider the dynamics of z a and z b
...

˜
= pa (y, t) − A za (y, t) Rs − Bza (y, t) p(y),
˜
1
Rs
∗
∗
∗ ˜
∗
∗
˙
˜
˜ ˜
ˆ
[i sd sin ε0 + i sq cos ε0 ]
z b = −ω0 z a − [α sin ε0 + ωm cos ε0 ]i sd −
γ1
σ

...


From the (˜ a , z b )-dynamics we obtain at steady state
z ˜
˜

...
∂h zb ( Rs , t)
˜
˜
f b ( Rs , t) =
∂t
˜
˜
˜
= pb (h( Rs , t), t) − A zb (h( Rs , t), t) Rs
˜
˜
−Bzb (h( Rs , t), t) p(h( Rs , t))
...
11)

Even though the signals Bza and Bzb are not measurable owing to their dependence on the
∗
unmeasurable ε0 , the variable (we omit, for the sake of brevity, the dependence on y and t)
?

?
∗
αi sd + ωm i sq
ˆ
˜
sπ = [ A za Bza + A zb Bzb ] p =
i sd
σ γ1
˜
is available for feedback and shows the following useful property: at steady state for Rs ∈ Br (0)
˜
˜ s , h( Rs , t), and t), it satisfies
(we omit, for the sake of brevity, the dependence on R
˜
sπ = − A za f a − A2 Rs + A za pa
za

˜
− A zb f b − A2 Rs + A zb pb ,
zb

(9
...
13)

where f¯a,0 = f¯a (0, t), and f¯b,0 = f¯b (0, t)
...
14)

with
2
2
i qr
i dr
f¯a,0
f¯b,0
∗
∗
∗
∗
+ 2 +
[i dr cos ε0 − i qr sin ε0 ] +
[i dr sin ε0 + i qr cos ε0 ],
2
σ
σ
σ
σ
˙
φ∗
φ∗
=
+
,
Lm
αLm
?
1 ? TL
=
+ ωm
...
In
analytical terms, according to (9
...

The remainder of the section is devoted to show that, if B is satisfied, designing the stator
resistance estimation law as
˙
ˆ
R s (t) = −bε R sπ (y(t), t)
solves the problem stated in the previous section
...

2

2

i
i
Remark 9
...
2 If f¯a,0 = f¯b,0 = 0, then λ = σdr + σqr , whose positiveness is clearly related
2
2
to stator resistance identifiability conditions
...
(2010) for the case of constant rotor
speed and flux modulus and zero load torque, which led to the stator resistance estimator

˜
˜
˜
˜
that, by Hadamard’s Lemma in Arnold (1992), h( Rs , t) = h ∗ ( Rs , t) Rs with h ∗ ( Rs , t) a suitable bounded
function on Br (0) × [0, +∞)
...

σ sd
Define
˜
y∗ (t) = h( Rs (t), t),
˜
so that we can write for Rs ∈ Br (0)
˙
˜
˜
˜
˙
y∗ (t) = gt ( Rs (t), t) + g R ( Rs (t), t) R s (t)

˙
˜
˜
˜
˜
˜
= Q 1 (h( Rs (t), t), t) + Q 2 (h( Rs (t), t), t) Rs (t) + g R ( Rs (t), t) R s (t)

˜
˜
= A y (t)y∗ (t) + B y (y∗ (t), t)y∗ (t) + A R (t) Rs (t) + B R (y∗ (t), t) Rs (t)
˜
+ε R g R ( Rs (t), t)[ A za (y(t), t)Bza (y(t), t)
+ A zb (y(t), t)Bzb (y(t), t)] p(y(t)),

with
˜
∂h( Rs , t)
˜
gt ( Rs , t) =
,
∂t
˜
∂h( Rs , t)
˜
g R ( Rs , t) =

...


(9
...
To this purpose, from equation (9
...

˜
˜
= − A za (y(t), t) Rs (t) − f a ( Rs (t), t) + Na (y(t), y∗ (t), t),

˜
˜
m b (y(t), y∗ (t), t) = − A zb (y(t), t) Rs (t) − f b ( Rs (t), t)

˜
+ pb (y∗ (t), t) − [ A zb (y∗ (t), t) − A zb (y(t), t)] Rs (t)

−[Bzb (y∗ (t), t) − Bzb (y(t), t)] p(y∗ (t))

...


(9
...
Thus we have
˜ s ∈ Br (0))
(R
˙
˜
R s (t) = ε R [ A za (y(t), t)m a (y(t), y∗ (t), t)
+ A zb (y(t), t)m b (y(t), y∗ (t), t)]

+ε R [ A za (y(t), t)Bza (y(t), t)

+ A zb (y(t), t)Bzb (y(t), t)]ξi˜sd (t)
...
17)

˜
Recalling that for Rs ∈ Br (0) (a1 and a2 are suitable positive reals that do not depend on ε R
and t)
˜
?y∗ ? ≤ a1 | Rs |,
? ∂h( R , t) ?
˜s ?
?
˜
?g R ( Rs , t)? = ?
? ≤ a2 ,
˜
∂ Rs
?y? ≤ ?y∗ ? + ?ξ ?,

(9
...
12), (9
...
14), (9
...
16), (9
...
18)) with state variables (ξ and Rs ) is locally exponentially
2
stable provided that a sufficiently small design parameter ε R is chosen
...
Therefore exponential convergence to zero of all
tracking and estimation errors is achieved3
...

3 Note that, since the origin of the closed-loop error system is locally exponentially stable, sufficiently small initial
˜
tracking and estimation errors guarantee that Rs (t) ∈ Br (0) for all t ≥ 0
...
3
...
Assume that: (1) the motor is controlled, under persistency of excitation
(9
...
(2008), which is not adaptive with respect to
the stator resistance; (2) the stator resistance is slightly different from its nominal value used
˜
by the controller so that a sufficiently small stator resistance estimation error Rs appears
...

The gist of the control and estimation design can be alternatively explained in the following
simplified terms: if the controller in Marino et al
...

Remark 9
...
4 The ninth order estimation and tracking control algorithm (9
...
8), and
(9
...
The parameters kω , ke , ki , (λ1 , λ2 , λ3 ) directly affect the dynamics of the
˜
˜ ˜ ˜
tracking and estimation errors ωm , (ed , eq ), i sd , ( x 1 , x 2 , x 3 ), respectively, while the parameter
˜
˜
γ1 determines the influence of the estimation error i sd on the dynamics of the error variables
−1
ˆ
ˆ
(˜ a , z b ); the parameter γ2 and ε R are the adaptation gains for α and Rs , respectively, while
z ˜
the parameters k, εα and ε characterize the robustifying terms in νsd and νsq , the projection
algorithm Proj[·, ·], and the saturation function sat(·), respectively
...
3
...
(2008), the rotor flux modulus reference sig˙
¨∗
nal is required to be time varying: if ?[φ ∗ (t), ωm (t)]? = 0 for all t ≥ 0, then all the
˜
˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜
˜
˜
points (ωm , φr d , φrq , ed , eq , x 1 , x 2 , x 3 , i sd , z a , z b , α, Rs ) = (−G α, 0, 0, 0, 0, 0, G ∗ α, 0, 0,
˜ ˜
βφ

Lr
˙∗
0, 0, α, 0), with G = φ ∗2 (TL + J ωm ), are equilibrium points for the closed-loop system so
˜
that, when both rotor speed and flux modulus reference signals are constant, local exponential
rotor speed tracking may not be guaranteed by the sensorless estimation and tracking control
algorithm (9
...
8), and (9
...
This is strictly related to the fact that when rotor speed and
(nonzero) flux modulus are constant so that L m i sa − φr a = −cφr b , L m i sb − φr b = cφr a with
T Lr
c = (φ 2 L+φ 2 ) model (9
...
1)
...
10) to be satisfied (see Marino et al
...
A sufficient condition for inequality (9
...
Note that the
∗
smaller the ωm is, the larger t p results: roughly speaking, the convergence to zero of the
tracking and estimation errors is faster at higher speeds
...
3
...
7), (9
...
9) for
sensorless IMs exhibits the following interesting similarities with the estimation and tracking control algorithm for sensorless permanent-magnet synchronous motors (PMSMs) rigorously derived in Tomei and Verrelli (2011): (1) both estimation and tracking control algorithms are straightforward modifications of the field-oriented controls for IMs and PMSMs,
which incorporate closed-loop observers for the unmeasured variables (rotor speed ωm , rotor
fluxes φr a and φr b for IMs—rotor speed ωm , sine and cosine functions of electrical angle
sin(θe ) and cos(θe ) for PMSMs) and the uncertain parameters (load torque TL and motor
resistances Rr and Rs for IMs—load torque TL for PMSMs); (2) both estimation and tracking control algorithms use estimates of stator currents (as auxiliary variables) and estimates
of advantageous variables whose dynamics do not depend on unmeasured variables and
uncertain parameters4 ; (3) the closed-loop observers incorporated by both the estimation
and tracking control algorithms rely on persistency of excitation conditions, which can be
physically interpreted in terms of motor observability and parameter identifiability and only
restrict the family of reference signals5
...

5 The role played by ε ∗ and φ ∗ for the IMs case is played by ω ∗ for the PMSMs case
...
4 Nonlinear Estimation and Tracking Control for the Output
Feedback Case
The intuitive steps of the estimation and tracking control design presented in the previous
section allow for a straightforward simplification when the rotor speed is measured
...


9
...
1

Estimation and Tracking Control Algorithm

When rotor speed is measured and available for feedback, an estimation and tracking control
design similar to the one presented in the previous section leads to the following slight
modifications to the adaptive observer (9
...
19)

α(0) ∈ [αm , α M ], 0 < αm − εα ,
ˆ
?
? (ω∗ i + αi )
∗
(−ωm i sd + αi sq )
ˆ sd
ˆ
˙
m sq
ˆ
ˆ
ˆ
R s = −ε R
(i sd − i sd ) +
(i sq − i sq ) ,
σ γ1
σ γ1

ˆ
Rs (0) ∈ [Rsm , Rs M ]
...
4
...

This simplified algorithm (locally) extends the result presented in Marino et al
...
(2009),
the following, a priori verifiable, persistency of excitation conditions depending on reference
signals only:
P) there exist two positive reals t p and k p such that
?

t+t p
t

¯
¯
? T (τ )?(τ )dτ ≥ k p I3 ,

∀t ≥0

(9
...


In particular, in contrast to the sensorless case, constant rotor speed and flux modulus references
and nonzero load torques suffice to satisfy equation (9
...
On the other hand,
∗
ε˙0
equation (9
...

Remark 9
...
1 Even though the structure of both estimation and tracking control algorithms
(sensorless case and output feedback case) is similar7 , the observation and adaptation strategies are substantially different: while in the sensorless case the information about (ωm , TL ) and
(φr d , φrq , α) is taken from i sq and i sd respectively, in the output feedback case the information
about TL is directly extracted from ωm and the remaining information about (φr d , φrq , α) is
taken from (i sd , i sq )
...
5

Simulation Results

We perform simulations for the two controllers designed in the previous sections
...
The simulations are carried
out with reference to the three-phase single pole pair 0
...
(2010) and whose parameters are:
J = 0
...
3 ?, Rr = 3
...
365 H, L r = 0
...
34 H
6 Note

that the simplified controller is apparently much simpler than the one presented in Marino et al
...

7 Note that, even though the measured speed ω is available, no modification of i ∗ and ω , as well as of ν
m
0
sd and
sq
νsq , is introduced: from a practical point of view, this allows the filtered estimate ωm to appear in crucial parts of the
ˆ
controller instead of the possibly noisy signal ωm
...
4

100

1
...
8

60

(Wb)

(rad/s)

120

40

–20

0
...
4
6
...
6

5

2

6
5
...
6

1
0

0
...
4

20
0

5
...
2

0

5

10

(s)

15

Figure 9
...
8 of Marino et al
...
All initial conditions of the motor and of the estimation and tracking control algorithms
ˆ
are set to zero except for φr a (0) = 0
...
1 s−1 (the value of the parameter α is
ˆ
8
...
39 ?
...
5
...
7), (9
...
9) by
simulation with parameters (the values are in SI units) kω = 120, ke = 900, k = 0
...
5, α M = 13
...
9, and ε R = 0
...
The references for rotor speed and flux modulus along with the applied
torque and the stator resistance are reported in Figure 9
...
The rotor flux modulus reference
is time varying according to the persistency of excitation condition (9
...
The rotor speed
reference goes from 0 to 100 rad/s and back to zero, which is a critical theoretical situation
for stator resistance uncertainty
...
3 ?
to 6
...
2 ? as rotor speed decreases
...
104 Nm) is applied at t = 0
...
Figures 9
...
3 show the time histories of the rotor speed and flux modulus and the
corresponding tracking errors, Figures 9
...
5, and 9
...
2 Sensorless case: rotor speed and corresponding tracking error

Rotor flux modulus

1
...
5
0

5

10

(s)

15

20

25

20

25

Rotor flux modulus tracking error

(Wb)

0
...
4
0
...
2
0
...
1
–0
...
3 Sensorless case: rotor flux modulus and corresponding tracking error

179

Adaptive Output Feedback Control of Induction Motors

Load torque estimate

6
5
(Nm)

4
3
2
1
0

0

5

10

15

20

25

20

25

Load torque estimation error

1
...
5
0

–0
...
4 Sensorless case: load torque estimate and estimation error

α estimate

12

(1/s)

10
8
6
4
2

0

5

10

15

20

25

20

25

α estimation error

6

(1/s)

4
2
0
–2
–4

0

5

10

15
(s)

Figure 9
...
5

(Ω)

6
5
...
4
0
...
2
0
...
1
–0
...
3
–0
...
6 Sensorless case: stator resistance estimate and estimation error

vector are reported in Figure 9
...
Rather satisfactory dynamic performances, in the presence
of parameter uncertainties, are achieved by only measuring the stator currents
...
5
...
Since no constraint concerning the time-varying nature of the rotor flux
modulus reference is required, we are able to illustrate the benefits of choosing a definitely
constant rotor flux modulus reference signal that asymptotically minimizes the power losses
depending on the uncertain parameters estimates
...
8
...
21)

minimizing the power losses at steady state (see, Vedagarbha et al
...
In particular, after a flux excitation phase (of
8 As

in the previous subsection the uncertainty θ is −12% of the load torque nominal value TLn
...
7 Sensorless case: stator voltage vector (a, b) components
Rotor speed reference

120
100
60

(Wb)

(rad/s)

80
40
20
0
–20

0

5

10

15

20

25

Applied load torque

6

Rotor flux modulus reference

0

5

10

15

20

25

20

25

Stator resistance

6
...
4

5

6
...
6
1
...
2
1
0
...
6
0
...
2
0

3

6
5
...
6

1
0

5
...
2

0

5

10

15
(s)

Figure 9
...
4

(rad/s)

0
...
2
0
...
1

0

5

10

15
(s)

Figure 9
...
The saturation function
sat(l1 ,l2 ) (x) is a continuous odd function, which is linear in the closed set [l1 , l2 ] and satisfies
sat(l1 ,l2 ) (x) = l1 for all x ≤ l1 and sat(l1 ,l2 ) (x) = l2 for all x ≥ l2
...
9 and 9
...
11, 9
...
13 show the convergence to zero of the uncertain parameters estimation errors, while the absorbed power and the power losses are reported in Figure 9
...

Satisfactory dynamic performances are achieved in the presence of parameter uncertainties,
while stator currents and voltages are within physical limits: transient tracking and estimation behaviors are, as expected, largely improved when the rotor speed signal is available for
feedback (compare Figures 9
...
10 with Figures 9
...
3)
...
16 Wb) of the
rotor flux modulus reference9 instead of suitably adjusting it via the converging estimates of
9 Note

that 1
...


183

Adaptive Output Feedback Control of Induction Motors

Rotor flux modulus

2
1
...
5
0

0

5

10

15

20

25

20

25

(s)
Rotor flux modulus tracking error

0
...
1
(Wb)

0
...
05
–0
...
10 Output feedback case: rotor flux modulus and corresponding tracking error

Load torque estimate

6

(Nm)

5
4
3
2
1

(Nm)

0

0
...
1
0
–0
...
2
–0
...
4
–0
...
11 Output feedback case: load torque estimate and estimation error

184

AC Electric Motors Control

α estimate

12
11
(1/s)

10
9
8
7
6

0

5

10

15

20

25

20

25

α estimation error

2

(1/s)

1
0
–1
–2
–3

0

5

10

15
(s)

Figure 9
...
5

(Ω)

6
5
...
4
0
...
2
0
...
1
–0
...
3

0

0

5

10

15
(s)

Figure 9
...
14 Output feedback case: absorbed power and power losses

Absorbed power
1000

(W)

800
600
400
200
0

0

5

10

15

20

25

15

20

25

Power losses
500

(W)

400
300
200
100
0

0

5

10
(s)

Figure 9
...
15 (recall Figure 9
...


9
...
1) with uncertain load torque and resistances has
been demonstrated
...
Specific
contributions of the chapter are: (1) a proof for the the exponential estimation of both rotor
and stator resistances in the presence of uncertain load torque for IMs with no speed sensors;
(2) a novel estimation and tracking control algorithm when the rotor speed is measured;
(3) a clear interpretation, in terms of parameter identifiability and power loss minimization, of
the theoretical potentialities of feeding back speed measurements in the presented scenario of
uncertain load torque and motor resistances
...
Springer, Berlin
...

IEEE Transactions on Control Systems Technology, 11, 248–252
...
Siemens-Review, 39, 217–220
...
IEEE Transactions on Automatic Control, 46, 1979–1983
...

International Journal of Systems Science, 31, 1195–1208
...
IEEE Transactions on Industrial Electronics, 47, 842–853
...
IEEE Transactions on Power Electronics, 25, 1173–1183
...

IEEE Transactions on Control Systems Technology, 17, 646–657
...
IEEE Transactions on Control Systems Technology, 17, 608–619
...
IEEE Transactions on Industrial
Electronics, 54, 167–176
...
Prentice-Hall, Upper Saddle River, NJ
...
IEEE Transactions on Control Systems Technology, 17, 327–339
...
McGraw-Hill, New York
...
Springer-Verlag, Berlin
...
International Journal of Adaptive Control and Signal Processing, 14, 109–140
...
IEEE Transactions on Automatic Control, 44, 967–983
...
Automatica, 40, 1071–1077
...
Automatica,
41, 1071–1077
...
Automatica, 44, 2593–2599
...
Springer, London
...
IEEE Transactions on Industrial Electronics, 48, 1148–1157
...
IEEE Transactions on Control Systems Technology, 15, 1049–1064
...
Automatica, 42, 1637–1650
...
32nd IEEE Annual Conference on Industrial Electronics, IECON, pp
...

Peresada S and Tonielli A (2000) High-performance robust speed-flux tracking controller for induction motor
...

Peresada S, Tonielli A, and Morici R (1999) High-performance indirect field-oriented output-feedback control of
induction motors
...

Tomei P and Verrelli CM (2011) Observer-based speed tracking control for sensorless permanent magnet synchronous
motors with unknown load torque
...

Traor´ D, De Leon J, and Glumineau A (2012) Adaptive interconnected observer-based backstepping control design
e
for sensorless induction motor
...

Vedagarbha P, Dawson DM, and Burg T (1997) Rotor velocity/flux control of induction motors with improved
efficiency
...

Zaky MS (2012) Stability analysis of speed and stator resistance estimators for sensorless induction motor drives
...


10
Nonlinear Control for Speed
Regulation of Induction Motor
with Optimal Energetic Efficiency
Abderrahim El Fadili1 , Abdelmounime El Magri1 , Hamid Ouadi2 ,
and Fouad Giri1
1
2

GREYC Lab, University of Caen Basse-Normandie, France
FSAC, University of Casablanca, Morocco

10
...
However,
most works have focused on the speed/flux/torque regulation supposing the machine magnetic
circuit to be linear
...
Several studies have dealt with
speed/flux regulation (with constant flux reference (CFR)) following several control strategies
ranging from simple techniques, for example, field-oriented control (Montanan et al
...
g
...
2006; Singh et al
...
2008)
...
This model assumes
a linear representation of the magnetic circuit and this assumption is not true in real-life
machines
...
1)
...
Edited by Fouad Giri
...
Published 2013 by John Wiley & Sons, Ltd
...
1 Control strategy involving constant flux reference (the controller is obtained from the
standard model)

to its nominal value generally located at the elbow of the machine magnetic characteristic
(Moreno-Eguilaz and Peracaula 1999; Canudas et al
...
Doing so, energetic efficiency is
only maximal when the machine operates all the time in the neighborhood of its nominal point
...

Indeed, in presence of small loads, the operation point is below the nominal value causing
useless energy stored in stator inductances, which reduces the machine efficiency
...
Then, the standard model is no longer valid and, consequently, the
control performances are no longer guaranteed
...
2)
...
(1999), Elfadili et al
...
(2010), El Fadili et al
...
The
proposed controllers include flux reference optimizers, the design of which relies on a machine
model that takes into account the nonlinearity of the magnetic characteristic
...
2 Control strategy involving state-dependent optimal flux (SDOF) reference

190

AC Electric Motors Control

The control strategy presented in this chapter enjoys the following features:
System modeling: The induction machine is represented by an experimentally
validated model that accounts for the nonlinearity of the magnetic characteristic
(Ouadi et al
...

Control design: A multivariable controller is built up on the basis of the preceding
model, using a nonlinear design technique
...
(2) An online flux reference optimizer that provides the
preceding speed/flux regulator with the optimal flux reference (OFR) trajectory
...
The flux optimizer design relies on
the above motor model and expresses in function of state variables, especially the
stator currents
...

The chapter is organized as follows: in Section 10
...
3; the control performances are illustrated through numerical simulations in
Section 10
...


10
...
1, the achievement of speed regulation and flux optimization
in presence of wide-range load variation necessitates that the control design is based on a
model that takes into consideration the nonlinear nature of the machine magnetic circuit
...
2009)
...

Alternative modeling methods that provide control-oriented models are still very few, for
example, Novotnak et al
...
(2011)
...
(2011)
...
The magnetic fluxes for the stator and rotor
phases are given by Leonard (2001)
φs = φls + φμ = ls i s + φμ ,

φr = φlr + φμr = lr ir + φμr ,

(10
...
2)

Nonlinear Control for Speed Regulation of Induction Motor with Optimal Energetic Efficiency

191

where φμ and φμr designate the magnetizing air-gap fluxes along one phase of the stator and
rotor, respectively; φls and φlr are the leakage fluxes at the stator and the rotor, respectively
...
1)-(10
...
This is valid in real-life machines because the large leakage flux
is circulating in air, not in iron
...
1) and (10
...


(10
...
4)

The contribution of the stator and the rotor to the air-gap flux generation is expressed in term
of the magnetizing current, denoted by i μ (Ouadi et al
...
Then, the (d, q) components of the system are given by the expressions
i μd = i sd + ki rd ,

i μq = i sq + ki rq
...
5)

For squirrel-cage motors, the transformation ratio can be chosen equal to unity, that is,
k = 1
...
On the other
hand, the norms ?μ and Iμ are related to each other by the magnetic characteristic ?μ = λ(Iμ ),
depicted by Figure 10
...
Using these observations, it follows that the instantaneous values
φμ and i μ are related by
φμ =

?μ
iμ
...
6)

The forthcoming development involves the static magnetizing parameter L m , defined as follows
(Ouadi et al
...

Iμ
λ (?μ )

(10
...
In
Ouadi et al
...

μ
μ
Lm

(10
...
3b illustrates the polynomial approximation for a (loadless) 7
...
In view of equation (10
...


(10
...
3), (10
...
5), and (10
...


(10
...
Considering the first option, amounts to let lr = 0 and ls = lseq
...
10) becomes
L s = lseq + L m ,

Lr = Lm ,

φr = φμ ,

(10
...

Induction Motor Model
Using the previous inductance expressions, it is shown in Ouadi et al
...

= − φrβ − pωm φr α +
dt
Lr
Lr

(10
...
13)
(10
...
15)
(10
...
11), the above model simplifies to
TL
p
F
dωm
= (φr α i sβ − φrβ i sα ) −
− ωm ,
dt
J
J
J
Rr
1
1
di sα
κ(?r )φr α + p
ωm φrβ +
vsα ,
= −γ i sα +
dt
lseq
lseq
lseq
Rr
1
1
di sβ
κ(?r )φrβ − p
ωm φr α +
vsβ ,
= −γ i sβ +
dt
lseq
lseq
lseq
dφr α
= −Rr κ(?r )φr α + pωm φrβ + Rr i sα ,
dt
dφrβ
= −Rr κ(?r )φrβ − pωm φr α + Rr i sβ ,
dt

(10
...
18)
(10
...
20)
(10
...
In this case, the parameter σ = 1 − L s m r becomes σ = L s −L m = L s
...
22)
(10
...
24)
(10
...
26)

where the various notations are defined as follows:
• κ(?r ) is the only varying parameter depending on the machine magnetic state as shown by
Figure 10
...

(2011)
2
n
κ(?r ) = q0 + q1 ?r + q2 ?r + · · · + qn ?r ,

(10
...
3a
...
1
0
...
6

L (H)

0
...
8

0
...
06
0
...
02

0

5
10
15
Magnetic current (A)

20

(a) Rotor flux norm Φr versus magnetic current Iμ

0

0
...
4
0
...
8
Rotor flux norm (Wb)

(b) Characteristic (Lm , Φr )

Figure 10
...
2011) for a 7
...
The crosses (+++) indicate experimental points, the solid line the corresponding polynomial
interpolation

194

AC Electric Motors Control

• ?r denotes the amplitude of the (instantaneous) rotor flux, denoted as φr
...
28)
?r = φr α + φrβ

where (φr α , φrβ ) (resp
...
d, q) components
...
(i sd , i sq )) are the α, β (resp
...

• (vsα , vsβ ), (resp
...
d, q) coordinate (obtained
after Park transformation) of the three-phase stator voltages
...

• Rs and Rr are the stator and rotor resistances
...

• p is the number of pole pairs
...

• γ = Rsl+Rr
seq
Numerical values of the model parameters are given by Table 10
...
5 kW
...
3 Controller Design
10
...
1 Control Objective
The aim is to develop a controller able to achieve the two following objectives:
1
...

2
...
) has yet to be determined so that ?r = ?r entails the minimization of the
stator current consumed by the machine
...
3
...
1)
...
Then, for coherency, the flux
reference signal is generally given a constant value coinciding with the machine nominal flux
value (Ouadi and Giri 2002; Jasinski et al
...
2006; Singh et al
...
2008)
...
In El Fadili et al
...

(2010), new controllers have been proposed that include flux reference optimizers
...
The approach of El
Fadili et al
...

The first step is to find a relationship between the rotor flux norm and the stator current norm
...
Then, it makes sense for simplicity to conduct the
present development within the oriented dq-coordinates frame
...
That is,
the dq-coordinate model described by equations (10
...
23), (10
...
25), and (10
...
It turns out that, the machine electromagnetic torque Tem is simply expressed
as follows:
Tem = pφrd i sq = p?r i sq
...
29)

On the other hand, equation (10
...
30)

yielding the steady-state current i sd ,
i sd = κ(?r )?r
...
31)

In turn, the stator current norm expression simplifies to
Is =

?
2
2
i sd + i sq
...
32)

Then, using equations (10
...
30), (10
...
32), one gets the following expression
of the electromagnetic torque:
Tem = p?r

?

Is2 − (κ(?r )?r )2
...
33)

Figure 10
...
It is clearly seen that, to a given torque it corresponds
a multitude of operation points differing by the value of the flux ?r and the current Is
...
1, with
• a flux ?r = 0
...
5 A;
• a flux ?r = 0
...
21 Wb and a current Is = 16 A
...
Let Tei (i = 1,
...
It is
readily seen from Figure 10
...
That is, a set of couples (?ri , Isi ) can be obtained using

196

AC Electric Motors Control

Electromagnetic torque (Nm)

60
Is = 16A

50
Is = 13A

40
Is = 10A

30

I = 8
...
1

0
...
3
0
...
5
Rotor flux norm (Wb)

0
...
7

0
...
4 Electromagnetic torque Tem in function of rotor flux φr , for different stator currents
norm Is

equation (10
...
) such that ?ri = F(Isi )
...


(10
...
The optimizer associated to the machine defined by the numerical characteristics
of Table 10
...
5
...

Remark 10
...
1 The polynomial interpolation yielding the function F(
...


Rotor flux reference (Wb)

0
...
6
0
...
2
0
−0
...
5 Optimal current-flux characteristic obtained, for the induction machine with physical
characteristics of Table 10
...
3
...
17), (10
...
19), (10
...
21)
...
These requirements can always be met by filtering the
∗
original reference through second order linear filters
...
34)
...
e
...

The speed/flux controller design will now be performed in two steps using the backstepping
technique (Krstic et al
...
First, introduce the tracking errors
∗
z 1 = ωm − ωm ,

z2 =

∗2
?r

−

2
(φr α

(10
...
36)

∗
from the polynomial interpolation of the experimental points (?∗ej , I Tej )
...
It follows from equation (10
...
20), and (10
...

˙
z 1 = ωm −
˙∗

(10
...
38)

p
In equations (10
...
38), the quantities J (φr α i sβ − φrβ i sα ) and Rr (i sα φr α + i sβ φrβ )
stand up as virtual control signals
...

2 1

(10
...
39) can be made negative
definite function of (z 1 , z 2 ), that is,
2
2
˙
V1 = −c1 z 1 − c2 z 2 ,

(10
...
41)

and
˙∗ ∗
ν1 = c2 z 2 + 2?r ?r + 2Rr κ(?r )?r ,
where c1 and c2 are any positive design parameters
...
42)

198

AC Electric Motors Control

p
As the quantities J (φr α i sβ − φrβ i sα ) and Rr (i sα φr α + i sβ φrβ ) are not the actual control
signals, they cannot be let equal to μ1 and ν1 , respectively
...


z 3 = μ1 −

(10
...
44)

Then, using the notations (10
...
42), (10
...
44), the dynamics of the errors z 1
and z 2 , initially described by equations (10
...
38), can be rewritten as follows:
∗
˙
z 1 = ωm − (μ1 − z 1 ) + TL + Fωm ,

˙
z 1 = −c1 z 1 + z 3 ,
˙
z 2 = −c2 z 2 + z 4
...
45)
(10
...


(10
...
To this end, it must be made clear how
these errors depend on the actual control signals (vsα and vsβ )
...
43) that
˙
z 3 = μ1 −
˙

p ˙
˙
˙
˙
(φr α i sβ + φr α i sβ − φrβ i sα − φrβ i sα )
...
48)

Using equations (10
...
19), (10
...
21), and (10
...
48)
˙
p
F ∗
TL
˙
˙
˙
¨∗
+ (ωm − z 1 ) − ((−Rr κ(?r )φr α + pωm φrβ + Rr i sα )i sβ
˙
z 3 = c1 z 1 + ωm +
J
J
J
?
?
1
1
1
+ φr α −γ i sβ +
Rr κ(?r )φrβ − p
ωm φr α +
vsβ
lseq
lseq
lseq
− (−Rr κ(?r )φrβ − pωm φr α + Rr i sβ )i sα
?
?
Rr
1
1
− φrβ −γ i sα +
κ(?r )φr α + p
ωm φrβ +
vsα )
...
49)

For convenience, the above equation is given the following compact form:
˙
z 3 = μ2 +

p 1
(φr α vsβ − φrβ vsα ),
J lseq

(10
...

lseq
lseq

(10
...
44) that z 4 undergoes the following differential equation:
˙
˙
˙
˙
˙
˙
z 4 = ν1 − Rr (i sα φr α + i sα φr α + i sβ φrβ + i sβ φrβ )
...
52)

Using equations (10
...
18), (10
...
20), (10
...
42), it follows from equation (10
...
53)

where the derivative κ(?r ) is obtained from (10
...

=
d?r dt
d?r ?r
?r

(10
...
53) is given the following compact form:
˙
z 4 = ν2 −

Rr
(vsα φr α + vsβ φrβ ),
lseq

(10
...


(10
...
45), (10
...
50),
and (10
...

2 1 2 2 2 3 2 4

(10
...


(10
...
47), (10
...
55), equation (10
...

+ z 3 μ2 +
J lseq

(10
...
59) and rearranging
terms, yields

?
?
2
2
2
2
˙2 = −c1 z 1 − c2 z 2 − c3 z 3 − c4 z 4 + z 4 ν2 + z 2 + c4 z 4 − Rr (vsα φr α + vsβ φrβ )
V
lseq
?
?
p 1
(φr α vsβ − φrβ vsα ) ,
(10
...
Equation (10
...
60))
...
61)

p
φr α ,
J lseq
Rr
λ3 = −
φrβ
...
62)

with
p
φrβ ,
J lseq
Rr
φr α ,
λ2 = −
lseq

λ0 = −

λ1 =

201

Nonlinear Control for Speed Regulation of Induction Motor with Optimal Energetic Efficiency

Note that inversed matrix in equation (10
...

The properties of the speed/flux regulator thus designed are described in the following
theorem:

Theorem 10
...
2

(Speed/flux regulator)
...
17–10
...
61) and the flux optimizer (10
...

Then, one has the following properties:
1
...
63)

˙
z 3 = −c3 z 3 − z 1 ,

(10
...
64)

˙
z 4 = −c4 z 4 − z 2
...
66)

2
...
Consequently, all errors vanish exponentially
2
2
2
2
fast, whatever their initial conditions
...
Equations (10
...
64) are immediately obtained from equation (10
...
46)
...
65) is obtained by substituting the control law (10
...
50)
...
66) is obtained by substituting the
control law (10
...
55)
...

Part 2
...
57) that V2 is a Lyapunov
function of the error system (10
...
64), (10
...
66)
...
61) to (vsα , vsβ ) on the right-hand side of equation (10
...


(10
...
But asymptotic stability implies exponential stability due to
system linearity (Khalil 2003)
...
3
...

Remark 10
...
3

˙∗
¨∗
The derivatives ?r and ?r are obtained using the relation
?r = F(Is ) = h 0 + h 1 Is + h 2 Is2 + · · · + h n Isn
...
68)

202

AC Electric Motors Control

Specifically, one has:
d F(Is ) d Is
˙∗
?r =
d Is dt
?˙
?
˙
d F(Is ) i sα i sα + i sβ i sβ
=

...

−
d Is
Is
dIs
Is3
Note that the derivatives
equation (10
...


10
...
69)

(10
...
The comparison is performed using a 7
...
1
...
6, 10
...
8, and 10
...
The applied load torque TL (Figure 10
...
7) are chosen so that the induction machine works in
two very different zones of its magnetic characteristic
...
61); and (2) the flux reference optimizer
(10
...
The following values of the controller design parameters proved to be suitable:
c1 = 100, c2 = 400, c3 = 500, and c4 = 1000
...
56 Wb
...
The performances of both controllers are illustrated by
Figures 10
...
8, and 10
...
It is seen in Figure 10
...
Figure 10
...
1

Numerical values of considered motor characteristics

Characteristic

Symbol

Value

Unity

Nominal power
Nominal voltage
Nominal flux
Stator resistance
Rotor resistance
Inertia moment
Friction coefficient
Number of pole pairs
Leakage equivalent inductance

PN
Usn
?r n
Rs
Rr
J
F
p
lseq

7
...
56
0
...
52
0
...
001
2
7

kW
V
Wb
?
?
Kgm2
Nm s rad−1
mH

Nonlinear Control for Speed Regulation of Induction Motor with Optimal Energetic Efficiency

203

Load torque (Nm)

50
40
30
20
10
0
0

10

20

30

Time (s)

Figure 10
...
7 Rotor speed ωm (rad/s) response: the identical speed responses obtained by the OFR
controller and the constant flux controller

0
...
6

Rotor flux norm (Wb)

Rotor flux norm (Wb)

0
...
4
Φ*
r

0
...
5
0
...
3

Φ

r

0
...
1
0

0

10

20
Time (s)

(b) CFR controller

Figure 10
...
Dashed, rotor flux reference; solid, measured flux response

30

204

AC Electric Motors Control

Stator current (A)

20
15
10
CFR
OFR

5
0

5

10

15
20
Time (s)

25

30

(a) Absorbed stator current (A) (Solid: constant flux
controller, Dashed : flux-optimize-based controller)

6000
Reactive power (Var)

Active power (W)

8000
CFR
OFR

6000
4000
2000

5000
4000
3000
2000
1000

0
0

10

20

CFR
OFR

0

30

0

10

Time (s)

(b) Active power P (W)

30

(c) Reactive power Q (Var)

0
...
8

η

power factor

0
...
7

CFR
OFR

0
...
5

20
Time (s)

0
...
4
0
...
9 Supremacy of the OFR controller (containing flux reference optimizer) over the CFR
regulator (involving constant flux reference)

Nonlinear Control for Speed Regulation of Induction Motor with Optimal Energetic Efficiency

205

CFR controller)
...

Figures 10
...

First, let us focus on Figure 10
...
It is seen that the OFR controller requires a smaller current
than the constant flux controller
...
The reduction of the
absorbed stator current entails the reduction of the consumed reactive power Q (Figure 10
...

The absorbed active power P, which is mainly determined by the speed and load torque,
remains quasi the same with both regulators (Figure 10
...
Let us evaluate the consequence
of these observation in terms of energetic efficiency and of power factor correction
...
71)

where the active and reactive powers are given by the following expressions, respectively:
P = vsα i sα + vsβ i sβ ,

Q = vsβ i sα − vsα i sβ
...
72)

Figures 10
...


10
...
A speed-flux controller described by equation (10
...
17), (10
...
19), (10
...
21)
...
The latter performs
an online tuning of the flux reference so that the absorbed stator current is minimized
...
3
...
That is, the
tracking objective is perfectly achieved both for the rotor speed and flux
...


References
Abdel Fattah HA and Loparo KA (2003) Passivity-based torque and flux tracking for induction motors with magnetic
saturation
...

Abdelaziz M and Ghedjati K (2007) Control by feedback linearization of the torque and the flux of the induction
motor
...
VII, pp
...

Canudas de Wit C and Ramirez J (2000) Optimization, Discretization, Observers and Control of Inductions Motors
...


206

AC Electric Motors Control

El Fadili A, Giri F, Magri A, et al
...
Speed regulation,
flux optimization and power factor correction
...

El Fadili A, Giri F, Ouadi H, et al
...
European Control Conference, Budapest, Vol
...
2542–2547
...
Electric Power Systems
Research, 34, 205–210
...
International Journal for Computation and Mathematics in Electrical and
Electronic Engineering, 25, 235–242
...
Prentice Hall, Upper Saddle River NJ
...
John Wiley & Sons
...

Levi E (1995) A unified approach to main flux saturation modelling in D-Q axis models of induction machines
...

Montanan M, Peresada S, and Tilli A (2006) A speed-sensorless indirect field-oriented control for induction motors
based on high gain speed estimation
...

Moreno-Eguilaz JM and Peracaula J (1999) Efficiency optimization for induction motor drives: past, present and
future
...
187–191
...
IEEE Transactions on Control Systems Technology, 7, 315–327
...
Mediterranean conference on control and
Automation-MED 2002, Lisbon Portugal
...
Control
Engineering Practice, 18, 55–66
...
International Journal of Modelling, Identification and Control, 14, 27–36
...
Electric Power Systems
Research, 79, 1054–1061
...
Power India Conference, pp
...

Thiringer T (1996) Measurements and modelling of low-frequency disturbances in induction machines
...

Traore D, Plestan F, Glumineau A, and De Leon J (2008) Sensorless induction motor: high order sliding mode
controller and adaptive interconnected observer
...


11
Experimental Evaluation of
Nonlinear Control Design
Techniques for Sensorless
Induction Motor
Jes´ s De Le´ n1 , Alain Glumineau2 , Dramane Traore2 ,
u
o
and Robert Boisliveau2
1
2

FIME, Universidad Autonoma de Nuevo Leon, France
Ecole Centrale de Nantes, LUNAM, France

11
...
Unlike DC motors, the IM is difficult
to control due to the coupling nonlinear dynamics
...

To facilitate the design and implementation of the IM controller, it is necessary to introduce
sensors to measure the electrical currents, the rotor position, and velocity
...
A reduction in the number of sensors reduces the cost and the maintenance of the
overall control system and becomes attractive economically
...
For these reasons the IM drive without mechanical sensor has had a considerable
interest in the last years
...

Several efforts have focused on sensorless schemes in order to solve the IM control problem, taking into account the robustness with respect to parametric uncertainties and their
AC Electric Motors Control: Advanced Design Techniques and Applications, First Edition
...

© 2013 John Wiley & Sons, Ltd
...


208

AC Electric Motors Control

performance under different operation conditions
...
For instance, in
Ibarra-Rojas et al
...

In this chapter, a comparative experimental study between nonlinear robust sensorless IM
controllers, taking into account different operation conditions and under parametric uncertainties, is presented
...
2012) and (b) a high-order sliding-mode control
(HOSMC) (Traore et al
...
These control schemes are designed to improve the performance of the sensorless IM, at different operation conditions, in particular at low frequencies
and in presence of unknown load torque
...

Regarding the high-order sliding-mode (HOSM) speed-flux controller, a sliding manifold
is designed in order to ensure finite-time convergence of sliding variable and its high-order
time derivatives to zero in spite of uncertainties and disturbances
...

Furthermore, experimental results comparing the performance of both control schemes are
obtained on the framework of a specific sensorless IM benchmark (Benchmark 2005)
...
Section 11
...
The robust integral backstepping is developed in
Section 11
...
The HOSMC is presented in Section 11
...
To implement the proposed controller
and to estimate the nonmeasured variables in Section 11
...
Furthermore, experimental results are given and discussed
related with the performance of the control schemes in Section 11
...
Finally, conclusions are
drawn in Section 11
...


11
...
1)

where i sd , i sq , φr d , φrq , vsd , vsq , ωm , Tl , and ωs , respectively, denote the stator currents, the
rotor fluxes, the stator voltage inputs, the angular speed, the load torque, and the stator
frequency
...
The parameters a = Rr /L r , b =
L 2 R +L 2 R
L m /σ L s L r , c = Fv /J , γ = r σsL s L m r , σ = 1 − (L 2 /L s L r ), m = pL m /J L r , m 1 = 1/σ L s ,
2
m
r

Experimental Evaluation of Nonlinear Control Design Techniques for Sensorless IM

209

L2 R

m
and γ1 = σ L s Lr2
...

The following assumptions are introduced:

1
...
The load torque is unknown and is considered as a disturbance modeled by a piecewise
function
...
The stator resistance is considered as a bounded parameter slowly varying with the temperature
...
The other parameters are constant and given by offline identification with bounded uncertainties
...
2
...

The IM observation problem can be established as follows: to estimate the speed and flux,
and, moreover, to identify the load torque and the stator resistance simultaneously, from the
measurement of the stator currents and the stator voltages under different operation conditions
(at low and high speed: Ghanes et al
...


11
...
Denote ωm and φ ∗ as
?

2
2
the smooth bounded reference signals of the speed ωm and the rotor flux modulus φr d + φrq ,
?
2
2
respectively
...


11
...
1

Controller Design using an Integral Backstepping Method

Following the classical backstepping method (Krstic et al
...
2012), the controller design is
done in two steps:
1
...

2
...

sq
sd

210

AC Electric Motors Control

It is well known that the control performances of IM is still affected by the uncertainties
of the plant, such as mechanical parameter uncertainties, external load disturbance, no ideal
field orientation in a transient state, and unmodeled dynamics in practical applications
...
This design is detailed in the
following text
...
2)

where ??ωm (m, c, Tl ) and ??φr d (a, L m ) are the parametric uncertain terms satisfying |??σ | ≤
J
ησ , for σ = ωm , φr d
...
Consequently, references i∗ and i∗ can then be considered as virtual inputs
sq
sd
of the reduced model (11
...

To solve speed and flux tracking problem, let us define the tracking errors as
?
zσ = σ ∗ − σ + K σ

?

t
0

(σ ∗ − σ )dt,

for

σ = ωm , φr d ,

where a supplementary integral term is introduced with respect to the classical backstepping
algorithm
...
2), it follows that the dynamics of
z ωm and z φr d are expressed as
?

∗
?
∗
˙
z ωm = ω˙m ∗ − mφr d i sq + cωm + Tl + K ωm (ωm − ωm ) − ??ωm ,
J
∗
∗
?
∗
˙
˙
z φr d = φr d + aφr d − a L m i sd + K φ (φr d − φr d ) − ??φr d
...
3)

2
Choosing the following candidate Lyapunov function Vzωm = 1 z ωm and by taking the deriva2
tive along the trajectories of equation (11
...

V m
m
J
∗
Following the backstepping methodology, by choosing the virtual control inputs i sq as
∗
i sq =

1
mφr d

[ω˙m ∗ + cωm +

Tl
J

?
∗
+ K z ωm + K ωm (ωm − ωm )],

(11
...


2
Following the same procedure as above, consider the Lyapunov function Vzφr d = 1 z φr d
...

∗
Choosing the virtual control inputs i sd as follows:
∗
i sd =

1
˙∗
[φr d
aLm

?
∗
+ aφr d + K φ z φr d + K φ (φr d − φr d )]
...
5)

Finally, we get
2
˙
Vzφr d = −K φ z φr d − z φr d ??φr d ,

K φr d > 0
...
1), one introduces the following
tracking errors:
?t ∗
?
?
??
∗
(11
...
K iq and K id are positive constants
...

2
2

(11
...
3) and by replacing
equation (11
...
8)
?
?? ∗
+z iq K iq (i sq − i sq ) − z ωm ??ωm
...
1), dt = baφrq − bpωm φr d − γ i sq − ωs i sd + m 1 vsq and by choosing
the control vsq as
?
∗ ?
di sq
1
(11
...
10)

212

AC Electric Motors Control

??
??
where K iq and K iq are positive constants with K iq > K iq and K 1 = min{K ωm , K idq −
??
??
K iq , K iq }
...
3) and replacing
equation (11
...

(11
...
1),
the control vsd
vsd

di sd
dt

= baφr d + bpωm φrq − γ i sd + ωs i sq + m 1 vsd and by working out

?
∗ ?
di sd
1
=
,
K id z id − baφr d − bpωm φr d + γ i sd − ωs i sq +
m1
dt

(11
...
11) can be rewritten as
2
??
2
?? ?2
˙
Vzid = −K φ z φr d − {K id − K id }z id − K id z id − z φr d ??φr d
= −K 2 Vzid − z φr d ??φr d

(11
...


Proposition 11
...
1 Consider the reduced order model of IM drive system represented by
∗
∗
equation (11
...
Then, system (11
...
4), (11
...
9), and (11
...

Proof:

Consider the following candidate Lyapunov function
Vc = Vziq + Vzid
=

(11
...

2
2
2
2
2
2

Taking its time derivative and replacing the suitable terms, after straightforward computations,
one has
˙
Vc = −κ Vc + ϒ,
where

κ = min{K ωm −

1
,
2?2
ξ1

Kφ −

1
,
2?2
ξ2

??
??
??
K iq , K iq − K iq , K id , K id − K id , K id }

?2 ηω
ξ1 2 m
2

?2 ηφ
ξ2 2
rd
2

11
...
15)
and

ϒ=

High-Order Sliding-Mode Control

+
, ∀ ? ∈ (0, 1) i = 1, 2
...
1990)
...


The success of sliding-mode (SM) control for electric drives is mainly due to its disturbance
rejection property, strong robustness and simple implementation, as shown by the large number

Experimental Evaluation of Nonlinear Control Design Techniques for Sensorless IM

213

of papers on sensorless IM drive (see e
...
, Zhang et al
...
2005), that use the
standard approach of SM control
...
However, one of the disadvantages associated
with SM control is the chattering phenomenon that may occur on the neighbor of the sliding
surface
...
In order to overcome this drawback and to improve the controller performances,
an approach called HOSM algorithm has been proposed to keep the main advantages of the
standard SM control, the chattering effect being attenuated and high-order precision provided
(Levant 2001)
...

(2008), Levant (2001), and Laghrouche et al
...
In the sequel, the HOSM speed-flux
control is based on Plestan et al
...
(2008) where it is proposed an
easy implementation, an a priori well-known convergence time with robustness with respect
to uncertainties and disturbance
...
Define σφ and σωm , the sliding variables as σφ = φr d − φr d and
∗
σωm = ωm − ωm
...
26) and (11
...

Furthermore, to reduce or eliminate the chattering effect and to improve the robustness of
the controller, third order HOSM controllers are designed for the two outputs, which means
(3)
(3)
˙
that the discontinuous term is applied to σφ and σωm through u
...

(11
...

0
mm 1 φr d

(11
...
18)
(11
...
20)

N
N
N
such that ϕα1om , ϕα2om , and ϕβ om are the well-known nominal terms whereas ?ϕα1 , ?ϕα2 ,
and ?ϕβ contain all the uncertainties due to parameters variations and disturbance
...
The control input u reads as (note that matrix ϕβ om is
1
invertible on the work domain (φr d ?= 0))

?

vsd
vsq

?

=

N −1
ϕβ om

? ?
−

N
ϕα1om
N
ϕα2om

?

+

?

νsd
νsq

??


...
21)

From equations (11
...
17), (11
...
19), (11
...
21), switching variables
dynamics read as
?

(2)
φr d

(2)
ωm

?T

?
= ?α + ?β νsd

νsq

?T


...
22)

N
N
N
ϕα1om , ϕα1om , and ϕβ om are bounded C 1 -functions in the operation domain D of IM, which
implies that ?α and ?β are uncertain bounded C 1 -functions
...

∗
νsq
˙
???? νsq
ωm (3)
ϕ2
??
?
ϕ1

The control law synthesis is made in two steps: the design of the switching variable and the
discontinuous input
...
4
...
(2008), the switching vector reads as
(2)
(2)
• For t ≤ t F
...


(2)
2
(2)
2
˙
˙
• For t > t F
...


Experimental Evaluation of Nonlinear Control Design Techniques for Sensorless IM

215

with F being a 2r × 2r -dimensional stable matrix (strictly negative eigenvalues), T being a
2r × 1-dimensional vector and

with c := φ, ωm
...
4
...
23)

Discontinuous Input

The control discontinuous input reads as
?

νsd
˙

νsq
˙

?T

?
= −αφ
...
sign(Sωm )

From equation (11
...


?T


...
24)

(11
...
(2008), it yields that there exist
gains αφ and αωm such that
˙
Sφ Sφ ≤ −ηφ |Sφ |,

˙
Sωm Sωm ≤ −ηωm |Sωm ,

to obtain the convergence to the sliding surfaces
...
5

Adaptive Interconnected Observers Design

In this section, an observer is designed to estimate the unmeasurable variables of the IM
...
One the
most important difficulties is the observability problem at low speed
...
In Canudas et al
...

(2004), Ghanes et al
...
Nevertheless, in the literature, some
sensorless algorithms are tested and evaluated at high and low speed (Ghanes et al
...

Then, taking into account the observation problem, an adaptive interconnected observer is
designed for the sensorless IM to estimate speed, flux, load torque, and stator resistance
...
1) and from the previous assumptions, the model can be
extended by the equations
˙
Tl = 0,

˙
Rs = 0
...
26) and (11
...

?
˙
?1 : X 1 = A1 (X 2 , y)X 1 + g1 (u, y, X 2 , X 1 ) + ?Tl ,

?
˙
?2 : X 2 = A2 (X 1 )X 2 + g2 (u, y, X 1 , X 2 )
...
26)
(11
...

Remark 11
...
1
values

Furthermore, the IM physical operation domain D is defined by the set of

D = {X ∈ R 6 | |φr d | ≤ ?d max , |φrq | ≤ ?q max , |i sd | ≤ Id max , |i sq | ≤ Iq max ,
max
|ωm | ≤ ωm , |Rs | ≤ Rs max },
max
where X = (φr d , φrq , i sd , i sq , ωm , Tl , Rs ), and ?d max , ?q max , Id max , Iq max , ωm , Tl max , and
max
are respectively the actual maximum values for the fluxes, currents, speed, torque
Rs
load, and stator resistance determined from the motor specification sheet
...
Moreover, the bounds of the motor parameters uncertainties are
known
...
26) and (11
...
The terms g1 (u, y, X 2 , X 1 ) and g2 (u, y, X 2 , X 1 ) are globally Lipschitz with respect
to X 2 , X 1 and uniformly with respect to (u, y), as long as the IM state remains in D
...
26) and (11
...
28) and (11
...

⎪ ?1
⎪X
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪˙
⎨ ˆ
Tl
O1 :
⎪ ˙
⎪
⎪ S1
⎪
⎪
⎪
⎪ ˙
⎪ S3
⎪
⎪
⎪
⎪
⎩ ˙
?

ˆ
= A1 ( ?2 , y) ?1 + g1 (u, y, ?2 , ?1 ) + ?Tl
X
X
X X
? T
?
−1
−1
T
ˆ
ˆ
+ ? ?S3 ?T + ?S1 C 1 (y1 − y1 ) + K C 2 (y2 − y2 ),

−1
T
ˆ
ˆ
ˆ
= ? S3 ?T C 1 (y1 − y1 ) + B1 ( ?2 )(y2 − y2 ) + B2 ( ?2 )(y1 − y1 ),
X
X
T
T
= −θ1 S1 − A1 ( ?2 , y)S1 − S1 A1 ( ?2 , y) + C 1 C 1 ,
X
X

(11
...

⎨? = A ( ? ) ? + g (u, y, ? , ? ) + S −1 C T (y − y ),
ˆ2
X2
X1 X2
2 X1 X2
2
2
2
2
O2 :
⎩ S = −θ S − A T ( ? )S − S A ( ? ) + C T C ,
˙2
2 2
2 2 X1
2 X1 2
2 2

(11
...
θ1 , θ2 , and θ3 are positive constants, S1 and S2 are
ˆ
ˆ
X
X
symmetric positive definite matrices, with S3 (0) > 0, B1 ( ?2 ) = km φr d , B2 ( ?2 ) = −km φrq ,
T
T
? = diag(1, 1, α), K C 2 = (−kc1 , −kc1 , 0) , where k, kc1 , kc2 , α, and ? are positive constants
...
28) and S2 C 2
is the gain of observer (11
...

Following Remark 11
...
1, one now assumes that all parameters of IM are uncertain and
bounded with well-known values
...
26) and (11
...
30)
(11
...

Let us define the estimation errors as
?
X
?1 = X 1 − ?1 ,

?2 = X 2 − ?2 ,
X

ˆ
?3 = Tl − Tl
...
30), (11
...
28), (11
...
32)

+g2 (u, y, X 1 , X 2 ) − g2 (u, y, ?1 , ?2 ) + ?g2 (u, y, X 1 , X 2 ),
X X
?
?
?
?
−1
−1
T
T
?3 = − ? S3 ?T C 1 C 1 ? + ?2 ? ?3 − ? S3 ?T C 1 C 1 + ?2 ?1 − ??1 ?2 ,
˙

T
where ?1 = B1 ( ?2 )C 2 , ?2 = B2 ( ?2 )C 1 , and K ? = K C 2 C 2
...
Taking
the time derivative of Vo and by using equation (11
...
33)

where μ is a constant associated with the nonlinear terms that are Lipschitz and ψ > 0 (see
Traore et al
...
Then, using practical stability results, it
follows that the observation error is uniformly practically stable
...
5
...
The d-axis angle
θm can be computed from the stator frequency (ωs ):
Lm
˙
i sq
...
34)

ˆ
The estimation θm of the d-axis angle is then computed by
ˆ
(i sq − i sq )
Lm
˙
ˆ
ˆ
i −
k ωs
...
35)

Then, the error dynamic of the estimation is given by
?θm = p?ωm −
˙

a L m i sq
k ωs
? +
? ,
ˆr d φr d β1 φr d isq
ˆ
φr d φ

(11
...
The gain kωs can
be tuned to ensure the convergence to zero for the nominal case or to a small ball for the
uncertain case for which the radius can be balanced by the gain tuning
...
5
...
2004)
...
Thus, it is necessary to use a notion of stability
that is more suitable than asymptotic stability, this is the practical stability (Lakshmikantham
et al
...

Then, using Lyapunov-like arguments, an analysis of stability has been presented in Traore
et al
...


11
...
1
...
2
...
5 kW, nominal angular speed
1430 rpm, number of pole pairs 2, nominal voltage 220 V, nominal current 7
...

The identified parameters values are Rs = 1
...
099 H, Rr = 0
...
0111 Nms2 /rad, L s = 0
...
0018 Nms/rad, L r = 0
...


219

Experimental Evaluation of Nonlinear Control Design Techniques for Sensorless IM

Figure 11
...
(For a color version of this figure, please see color plates
...
Three-phase inverter operated with a symmetrical pulse-width modulation (PWM) technique with 5 kHz switching frequency
...
A permanent-magnet synchronous motor controlled by an industrial drive to provide a
desired load torque
...
Inv
...
Inv
...
Obs
Inverter

∗

Ti
Tl

Induction motor

Figure 11
...
(For a color version of this figure, please
see color plates
...
6
(Wb)

7

c

0
...
2
0

0

1

2

3

4

5
6
Time (s)

Figure 11
...
a, reference speed ωm (rad/s); b, load torque disturbance Tl (Nm);
c, reference flux φr∗d (Wb)

3
...

The dSPACE board performs data acquisition (two stator currents, DC-link voltage, load
torque and rotor speed, by means of a 512 ppr incremental encoder (only for monitoring
purposes), computes the control algorithm and generates the PWM signals for the inverter
...
Matlab/Simulink,
2
...

Benchmark trajectories
The sensorless trajectories of the benchmark are such that (see Figure 11
...
5 to 2
...
This first step allows
to test the performances and the robustness of the controller without mechanical sensors at
low speed but under observable conditions
...
Again the load torque is applied from
5 s
...

Then, the motor is driven to reach a negative constant low-speed value from 7 to 9 s
...
This last step allows to illustrate the
IM unobservability phenomena (from t = 7 s to t = 9 s)
...

All control and observer algorithms are computed with a sampling rate of 200 μs
...


221

Experimental Evaluation of Nonlinear Control Design Techniques for Sensorless IM

For the observer design, the gains are chosen as follows: α = 50, ? = 10, k = 0
...
5, θ1 = 5000, θ2 = 7000, and θ3 = 10−9
...
28) and (11
...
Stator resistance observer is initialized as Rs0 = 1
...
The initial value of φr d in the observer is φr d0 =
0
...

Now, experimental results for the two proposed-observer schemes are given for nominal
conditions and robustness test cases, then compared
...
6
...

Nominal case
The experimental responses obtained by considering the nominal case with identified parameters are shown on Figure 11
...
Notice the good performance of the proposed scheme that
maintains the speed close to the desired reference even though the presence of disturbance
(load torque)
...

Note that, for experimental test, “nominal case” means with the use of the identified parameters (thus with already errors on parameters)
...
4d)
converges to the measured load torque (Figure 11
...
6

1

2

3

4

5

6

2

3

0
...
59

8

9

10

11

7

8

9

10

11

7

8

9

10

11

d

c
1

7

6

f

0
...
58
0

1

2

3

4

5

6

Figure 11
...
6
0
...
59
0
...
58

(Ω)

10
8
6
4
2
0
–2

(Wb)

0
0

1

2

3

4

5

6

c

0

1

2

3

4

7

8

9

10

11

7

8

9

10

11

7

8

9

10

11

7

8

9

10

11

d

5

6
e

f
0

1

2

3

1
...
86
1
...
82
1
...
5 Integral backstepping control (IBC) +50% for Rr ; a, c, measured speed and load torque;
e, reference flux; b, d, f, g, observed speed, load torque, flux(φr d ), and stator resistance

very low frequency (conditions of unobservability between 7 and 9 s)
...

The following cases are experimented under parameters variations for robustness tests:
+50% rotor resistance variation (Rr )
In Figures 11
...
6 the results for a +50% variation of Rr that are included in the
observer-controller parameters are shown
...
4)
...
5b) converges to the measured one (Figure 11
...
5e and f) and a good estimation of the load torque
(Figures 11
...
6d),
• the estimation of the stator resistance is always good (Figure 11
...

+10% rotor inductance variation (L r )
Figures 11
...
8 display the good performances for the case of a +10% rotor inductance
variation applied at the same time for the observer and the controller parameters
...
04

c
2

3

4

5

d

0
...
02
–0
...
6 Integral backstepping control (IBC) +50% for Rr ; a, b, speed and load torque estimation
errors; c, d, flux tracking error (φr d ) and flux angle error (φrq ) du flux

(rad/s)

100

b

50
a

0
1

2

3

4

(Wb)

10
8
6
4
2
0
–2

0
...
595
0
...
585
0
...
88
1
...
84
1
...
8
1
...
7 Integral backstepping control (IBC), +10% of L r ; a, c, measured speed and load torque;
e, reference flux; b, d, f, g, observed speed, load torque, flux(φr d ), and stator resistance

224

AC Electric Motors Control

(rad/s)

5

a

0
–5
0

1

2

3

4

(Nm)

6

7

8

9

10

11

5

6

7

8

9

10

11

4

5

6

7

8

9

10

11

7

8

9

10

11

b

2
0
–2

(Wb)

5

4

4

6
4
2
0
–2
–4

0

1

× 10–3

2

3

c

0

1

2

3

(Wb)

0
...
05

0

1

2

3

4

5

6

Time (s)

Figure 11
...
The flux angle is well oriented Figure 11
...

+10% stator inductance variation (L s )
A robustness test is given by a +10% variation on the stator inductance for the controller
and the observer parameters (Figures 11
...
10)
...
9c) to the measured load torque (Figure 11
...
In
addition, there are not many oscillations of the IM speed for the high speed (Figure 11
...
10d)
...
9a–f for t = 2
...
A
good stator resistance estimation is obtained (Figure 11
...
It is clear that the IBC has good
robutness performances when associated to the adaptive interconnected observer
...


11
...
2

High-Order Sliding-Mode Control and Adaptive Observer

To optimize the behavior and the performances of the motor, two parameters tuning have
been chosen: the first one to induce the reaching of the motor flux, the second one to reject

225

Experimental Evaluation of Nonlinear Control Design Techniques for Sensorless IM

(rad/s)

100
b

a

50
0

(Wb)

0
...
595
0
...
585
0
...
88
1
...
84
1
...
8
1
...
9 Integral backstepping control (IBC), +10% of L s ; a, c, measured speed and load torque;
e, reference flux; b, d, f, g, observed speed, torque, flux (φr d ), and stator resistance

(rad/s)

5

a

0
–5
0

1

2

3

4

5

6

7

8

9

10

11

6

7

8

9

10

11

6

7

8

9

10

11

7

8

9

10

11

(Nm)

2
0

b

–2
–4

(Wb)

(Wb)

0
6
4
2
0
–2
–4

0
...
01
0
–0
...
02
–0
...
10 Integral backstepping control (IBC), +10% of L s ; a, b, speed and load torque estimation
errors; c, d, flux tracking error (φr d ) and flux angle error (φrq )

226

AC Electric Motors Control

disturbance (such as load torque) and to ensure high-level accuracy for the trajectory tracking
...
3 s and
• t ≤ 5 s
...
35, ωnφ = 316 rad/s, αφ = 6
...
56, ωnωm = 32 rad/s, αωm =
8
...
ζφ = 0
...
104 , ζωm = 0
...
106
Nominal Case

(rad/s)

100
50
0

(Nm)

The experimental results of the nominal case with identified parameters (except stator resistance) are shown in Figure 11
...
These figures show the good performance of the complete
system observer-controller in trajectory tracking and disturbance rejection
...
11b) converges to the measured speed (Figure 11
...
It is the same conclusion for estimated flux (Figure 11
...
11e)
...
11d) converges
to the measured load torque (Figure 11
...
Nevertheless, it shows a
small static error when the motor speed increases (between 4 and 6 s)
...
01

5

0

1

0

0

6

7

8

9

10

11

7

8

9

10

11

d

3

4

5

6

2

3

4

5

6

7

8

9

10

11

1

2

3

4

5

6

7

8

9

10

11

1

2

3

5

6

7

8

9

10

11

5

6

7

8

9

10

11

5
6
Time (s)

7

8

9

10

11

0
...
9
1
...
8

4
c

0

0
...
01
0

1

2

3

4

Figure 11
...
11h and j at time 1
...
11h, j at time 2
...
In Figure 11
...
This test shows the capability of the proposed controller to guarantee
flux and speed tracking of slowly varying speed reference with excitation frequency close to
zero (between 7 and 9 s)
...
12 (evaluation of robustness with respect to inductances
variations has also been successfully made)
...
It shows a static error when the motor is under unobservable condition (between
7 and 9 s) (Figure 11
...
The static error transitory increases when the load torque is
applied at time 1
...
12h, j)
...
The results of these tests are shown on Figure 11
...
The rotor inductance

10
5
0

a
0

1

2

(Wb)

(rad/s)

5
0
–5

(Nm)

(Ω)

1

2

0

1

2

0
...
9
1
...
8

2
0
–2

3

4

5

e

6

7

8

9

10

11

d

c
0

0
...
01

0

1

2

3

4

5
6
Time (s)

j

0

–0
...
12 HOSM, rotor resistance variation (+50%); a, b, estimated and measured speeds; c, d,
measured and estimated load torques; e, f, reference and estimated fluxes; g, estimated stator resistance;
h, speed error; i, torque error; j, flux error versus time

228

(Nm)

(rad/s)

AC Electric Motors Control

100
50
0
10
5
0

1

2

(rad/s)

5
0
–5

(Nm)

(Ω)

(Wb)

0

1
...
85
1
...
6
0
...
01

1

2

3

4

5
6
Time (s)

7

8

9

10

11

0

–0
...
13 HOSM, rotor inductance variation (+10%); a, b, estimated and measured speeds; c, d,
measured and estimated load torques; e, f, reference and estimated fluxes; g, estimated stator resistance;
h, speed error; i, torque error; j, flux error versus time

does not affect the performances of both observer-controller schemes
...
13a, b) at time 5 s
...
The results of these tests are shown on Figure 11
...
By analyzing
this figure, we can see that the stator inductance variation does not affect the performances of
both observer-controller schemes
...
14a and b) at time 5 s
...
7

Robust Nonlinear Controllers Comparison

In this section, we compare the performance of the two proposed control approaches
...


229

(rad/s)

100
50
0

(Nm)

Experimental Evaluation of Nonlinear Control Design Techniques for Sensorless IM

10
5
0

b
a
0

1

2

3

4

5

6

7

8

9

10

11

6

7

8

9

10

11

d

c

0
...
9
1
...
8

(Nm)

(Ω)

(Wb)

0

1

2
e

3

4

5

0

1

2

3

4
g

5

6

7

8

9

10

11

0

1

2

3

4

5

6

7

8

9

10

11

5

6

7

8

9

10

11

5

6

7

8

9

10

11

5
6
Time (s)

7

8

9

10

11

0
...
01

f

h
0

1

2

3

4
i

0

1

2

3

4
j

0

–0
...
14 HOSM; stator inductance variation (+10%); a, b, estimated and measured speeds; c, d,
measured and estimated load torques; e, f, reference and estimated fluxes; g, estimated stator resistance;
h, speed error; i, torque error; j, flux error versus time

The HOSM and the IBCs are implemented and validated experimentally on the same IM
setup and use the same electromechanical model for the control design
...
7
...
The HOSM control introduced in Section 11
...
Convergence properties: Finite-time convergence is achieved
...

2
...

Limitations
1
...

2
...

3
...


230

11
...
2

AC Electric Motors Control

Integral Backstepping Control

The proposed IBC design is a robust recursive design methodology
...

Advantages
1
...

2
...

3
...

4
...

5
...

6
...

Limitations
The backstepping methodology (with integral term or not) needs the knowledge of a sufficiently
precise model
...
7
...
The same hardware and software are used to implement the different control laws on the
same setup
...
The transient performance is improved by using IBC
...
Even if attenuated with respect to SM of order one, the HOSM controller leads to significative high-frequency content in the control signals
...
The behavior of the IBC is more desirable since the chattering voltages and currents can
damage the motor and hence shorten the longevity
...
The price to pay for the elimination of the chattering is an important computation effort to
achieve the tracking performance
...
1, a computation time comparison is reported for the two controllers and the
observer presented in this paper
...
1

Time computation comparison

Controller

Control time (μs)

Observation time (μs)

10
11
12
35

20a
20a
34b
34b

PI
SM order 1
Backstepping
HOSM

SM, sliding mode; HOSM, high-order sliding mode
...


Experimental Evaluation of Nonlinear Control Design Techniques for Sensorless IM

231

given: an FOC-PI controller (Traore et al
...
2007a) associated to an interconnected oberver
...


11
...

The major contributions of this study are summarized as follows:
1
...

2
...
The advantages and limitations are
analyzed in terms of: (a) practical implementation, (b) robustness under uncertainties,
(c) computational effort, (d) simplicity to tune, (e) transient performances, (f) structural
properties
...
The implementation of the proposed control schemes on experimental setup with a significant sensorless control benchmark, has been presented
...


This research has been partially supported by CONACYT Ciencia Basica No
...
CONACYT Estancias Sabaticas
...


References
Benchmark (2005) Lunam Universit´ , Ecole Centrale de Nantes, IRCcyN www2
...
ec-nantes
...
Siemens Review, 93, 217–220
...
(2000) Observability conditions of induction motors at low frequencies IEEE
Conference on Decision and Control, Sydney, Australia
...
Proceedings of the 45th IEEE CDC, Manchester Grand Hyatt Hotel San Diego USA,
December 13–15, pp
...

Ghanes M, De Leon J, and Glumineau A (2006b) Observability study and observer-based interconnected form for
sensorless induction motor
...
1240–1245
...
Proceedings of the IEEE, 90, 1359–1394
...

Automatica, 40, 1079–1085
...
IEEE Press Series, Power
Engineering
...

Krstic M, Kanellakopoulos I, and Kolotovic P (1995) Nonlinear and Adaptive Control Design
...


232

AC Electric Motors Control

Laghrouche S, Plestan F, and Glumineau A (2006) Practical Higher Order Sliding Mode Control: Optimal Control
Based Approach and Application to Electromechanical Systems
...
334, pp
...
Springer-Verlag
...
World Scientific
Publishing
...

Lascu C, Boldea I, and Blaabbjerg F (2005) Very-low-speed variable-structure control of sensorless induction machine
drives without signal injection
...

Levant A (2001) Universal siso sliding-mode controllers with finite-time convergence
...

Montanari M and Tilli A (2006) Sensorless control of induction motors based on high-gain speed estimation and
on-line stator resistance adaptation
...

Plestan F, Glumineau A, and Laghrouche S (2008) A new algorithm for high order sliding mode control
...

Traore D, De Leon J, and Glumineau A (2012) Adaptive interconnected observer-based backstepping control design
for sensorless induction motor
...

Traore D, De Leon J, Glumineau A, and Loron L (2007a) Sliding mode controller design for sensorless induction
motor: experimental test on low frequencies benchmark
...

Traore D, De Leon J, Glumineau A, and Loron L (2007b) Speed sensorless field-oriented control of induction motor
with interconnected observers: experimental tests on low frequencies benchmark
...

Traore D, Plestan F, Glumineau A, and De Leon J (2008) Sensorless induction motor: high-order sliding-mode
controller and adaptive interconnected observer
...

Utkin V (1993) Sliding mode control design principles and applications to electrical drives
...

Zhang Y, Changxi J, and Utkin V (2000) Sensorless sliding-mode control of induction motors
...


12
Multiphase Induction
Motor Control
Roberto Zanasi and Giovanni Azzone
Dipartimento di Ingegneria “Enzo Ferrari”, Universit` di Modena e Reggio Emilia, Italy
a

12
...
2007), especially in the last
two decades when the multiphase machines started to be considered as potential alternative to
the conventional three-phase machines
...
Combining all these enhancements with the intrinsic
induction motors robustness and versatility, it is quite clear that their use in industrial applications has seen a substantial growth worldwide
...
Moreover, the
advantages of the odd order harmonic injection, existing in concentrated-winding multiphase
machines (see Toliyat et al
...

In literature, different field-oriented control strategies addressed to induction motors with
a specific number of stator and rotor phases have been discussed, especially for five-phase
machines (see, for instance, Xu et al
...
2006; Duran et al
...

This chapter presents a new complex dynamic model of a multiphase induction motor
considering an arbitrary number of stator and rotor phases and including the odd order
harmonic injection
...
Edited by Fouad Giri
...
Published 2013 by John Wiley & Sons, Ltd
...
1 POG main blocks

control (IRFOC) is extended to the multiphase general case
...


12
...
The POG block schemes are normal block diagrams
combined with a particular modular structure essentially based on the use of the two blocks
shown in Figure 12
...
e
...
); the connection block redistributes
the power within the system without storing nor dissipating energy (i
...
, any type of gear
reduction, transformers, etc
...
In the vectorial case, G(s) and K are
matrices: G(s) is always a square matrix of positive real transfer functions; matrix K can also
be rectangular, time varying, and function of other state variables
...
The connection block transforms the power variables imposing
the constraint x∗ y1 = x∗ y2
...
1) has the physical meaning of the power flowing through that
particular section
...
This correspondence is shown in Figure 12
...
The
energy matrix L is always symmetric and positive definite: L = L∗ > 0
...
The dynamic equations L˙ = −Az+Bu and y = B z of the “reduced” system can be
obtained from the original one using a “congruent” transformation x = Tz (matrix T can also
˙
be complex and/or rectangular) where L = T∗ LT, A = T∗ AT − T∗ LT, and B = T∗ B
...
The POG
schemes maintain their physical meaning even when the dynamic system is described using
complex variables
...
1)

(x = Tz)

1
s

y

B∗

˙
Lz = −Az + Bu
x

(12
...
2 POG block scheme of a generic dynamic system

An example of POG modeling of an electrical RC circuit is shown in Figure 12
...
The
C and R physical elements are described by two POG elaboration blocks
...
The
summation elements present in the elaboration blocks are a mathematical description of the
current and voltage Kirchhoff’s laws applied to the considered electrical system
...
2
...

? Ri, j ? = ⎢
...

⎦,

...
⎥,

...


...

⎣
...


...

1:n
1:n
1:m
Rn
Rn1 Rn2 · · · Rnm
i
?
|[ Ri ]| = R1

R2

···

1:n

Kirchhoff’s
current law

Rn

?T

R

I1

,

j
??
?? ?
? R j ? = R1
1:m

I2

···

?
Rm
...
3 POG modeling of an electrical RC circuit

2

3

236

AC Electric Motors Control

The symbol δ(n)|m denote the following function:
k
δ(n)|m =
k

?

1,
0,

if n ∈ [k, k ± m, k ± 2m,
...
The symbol Im denotes an identity matrix of order m
...
3

Multiphase Induction Motor Complex Dynamic Modeling

The basic structure of a multiphase star-connected induction motor is shown in Figure 12
...

The electrical and mechanical parameters of the motor are shown in Table 12
...
All the
electrical parameters of the motor have been obtained connecting in series the p polar couples
of the motor
...
3
...
+ Ism s = 0 ,

mr
?
h=1

Ir h = Ir 1 + Ir 2 +
...
(12
...
4 Structure of a multiphase induction motor

237

Multiphase Induction Motor Control
Table 12
...
This can be mathematically
expressed defining the self- and mutual inductance coefficients L si , L ri , Msi j , and Mri j as
follows:
⎧
m s −2
?
⎪
⎪
⎨ Msi j = Ms0
a s cos(n (i − j )γs )
n

⎪
⎪
⎩

n=1:2

,

L s0 = L s − Ms0 > 0

⎧
m r −2
?
⎪
⎪
⎨ ri j = Mr 0
M
a r cos(n (i − j)γr )
n

⎪
⎪
⎩

n=1:2

(12
...
, m s } for the stator and i, j ∈ {1, 2,
...
Parameters an
r
and an are the stator and rotor Fourier coefficients
...


H3: The rotor phases are short-circuited: Vrr = Vr 0 (see Figure 12
...

?m s
H4: The input stator voltages Vi are supposed to be balanced: i=0 Vi = 0
...
3
...
4):
⎤
Vs1
⎢ Vs2 ⎥
⎥
⎢
t
Vs = ⎢
...
⎦

...
⎥,

...


Ism s

⎤
Vr 1
⎢ Vr 2 ⎥
⎥
⎢
t
Vr = ⎢
...
⎦

...
⎥

...


Ir m r

238

AC Electric Motors Control

where Vsi = Vi − Vs0 for i ∈ {1, 2,
...
, m r }
...
(2009) and Zanasi and
Azzone (2010), one obtains the following dynamic equations of the multiphase induction
motors:

d
dt

??

?? t ? ?
?? t ? ? t ?
?t
Re + t Fe t Ke
Le 0
Ie
Ie
Ve

...
5)

The structures of the energy matrix t L(t q), the dissipating matrix t R, and the energy redistribution matrix t W are the following:

⎡

⎤
t
?
?t
Ls
MT (θm ) 0
sr
t
t
t
⎣ t Msr (θm )
⎦ = Le 0 ,
Lr
0
L( q) =
0 Jm
0
0
Jm
t

⎤ ⎡
⎤
? t
?
R s Im s 0
Rs 0 0
0
Re 0
t
R = ⎣ 0 t Rr 0 ⎦ = ⎣ 0 Rr Im r 0 ⎦ =
,
0 bm
0 0 bm
0
0
bm
⎡

t

t

⎡

0

⎢
˙
W = ⎢ − 1 t Msr
⎣
2
t

˙ sr
− 1 t MT
2
0

T M
− 1 t Ir ∂ ∂θmsr − 1 t IT
2
2 s

∂ t MT
sr
∂θm

1
2
1
2

∂ t MT t
sr
Ir
∂θm
t
∂ Msr t
Is
∂θm

0

⎤

⎥
⎥
...
4) of Hypothesis H2, the self- and mutual inductance matrices t Ls , t Lr , and t Msr are

239

Multiphase Induction Motor Control

supposed to have the following structure:
i

t

Ls = L s0 Im s

j
?? m −2
??
s
?
? ?
?
?
s
+ Ms0 ?
an cos(n (i − j)γs ) ?,
?
?
n=1:2
1:m s

1:m s

i

t

Lr = L r 0 Im r

j
?? m −2
??
r
?
? ?
?
?
r
+ Mr 0 ?
an cos(n (i − j)γr ) ?,
?
?
n=1:2
1:m r

1:m r

i

j
??m −2
??
sr
?
? ?
?
?
sr
t
Msr (θ ) = Msr 0 ?
an cos(n(θ + iγr − jγs )) ?,
?
?
n=1:2
0:m r −1

0:m s −1

where m sr = min{m s , m r }, L s0 = L s − Ms0 , and L r 0 = L r − Mr 0 , that means the stator and
s
r
sr
rotor phases have a concentrated-winding structure
...


Once the motor dynamic equations have been obtained, a state space transformation has to be
˜
performed in order to project them onto a new rotating reference frame
...
6), the following transformation matrix
t

Tω ∈ C(m s +m r +1)×(m s +m r )/2+2

?


...
6)

240

AC Electric Motors Control

can be defined as
⎡

⎤
˜
?
?t
TωN (m s , θs )
0
0
TωN 0
t
t ˜
Tω = ⎣
0
TωN (m r , θ p ) 0 ⎦ =
0 1
0
0
1
t

⎡

⎤⎡
⎤
˜
0
0
N ms 0 0
Tω (m s , θs )
t ˜
0
=⎣
Tω (m r , θ p ) 0 ⎦ ⎣ 0 Nm r 0 ⎦
0 0 1
0
0
1
=

(12
...
8)

(12
...
The complex matrix t Tω , whose columns are orthogonal vectors, is used
˙
˙
to perform a “pseudo” state space transformation t q = t Tω ω q from the original external
¯
reference frame ?t to a new complex rotating one ?ω
...

+
¯e
0 Jm ωm
− ω K∗
bm
ωm
−τe
˙
? ?? ? ? ?? ?
?
??
? ? ?? ? ? ?? ?
ω
ω
ω
ω
ω
˙
L
R + ωW
V
q
¨
q
?ω

(12
...
7)):
⎡

⎡ω

⎤
? ω ?
Is
¯
Ie
ω
t ∗ t
¯
⎣ ω Ir ⎦ =
˙
˙
q = Tω q =
,
ωm
ωm

⎤
¯
?ω ?
Vs
¯
Ve
ω
t ∗ t
¯
⎣ ω Vr ⎦ =
V = Tω V =
−τe
−τe

ω¯

while the transformed matrices ω L, ω R, and ω W are obtained using matrix t Tω (see equation
(12
...


Note that the transformation matrix t Tω does not change the diagonal structure of the transformed matrix ω R with respect to t R, it only reduces its dimension:

ω

R=

t ∗ t t
Tω R Tω

⎡

¯
Rs
⎣ 0
=
0
ω

⎤ ⎡
⎤
?
?ω
Rs I m s −1
0
0
0 0
¯
2
Re 0
ω¯
⎦=⎣
⎦=
m r −1
Rr 0
0
Rr I
0

...


Note that ω p is the slip velocity of the motor
...

1:2:m r −2 1:2:m s −2

The transformed complex conjugate stator and rotor torque vectors are defined as
ω

p ω ¯∗
¯s
K∗ = − j Msr e Ir km r asr ,
2

ω

p
¯∗
¯s
Kr = j Msr e ω I∗ km s aT
...
The components
and ω Vsm s in (12
...
11)

⎤

mr
1 ?
Ir m s = √
Iri ,
m r i=1

ω

ω

Ism s ,

mr
1 ?
Vsm s = √
Vsi
...
3)), it results
that ω Ism s = 0 and ω Ir m r = 0
...
The vectors ω Is , ω Ir , and ω Vs in (12
...

1:2:m s −2

¯
Finally, the mechanical torque τm in the reference frame ?ω can be expressed as follows:
¯e ¯
τm = Re ( K∗ ω Ie ) = Re
ω

=

?
?

ω

? ω ??
¯
Is
ω¯
Ir
?
??
¯
? ω Is
¯s
j ω I∗ km s aT
ω¯
sr
Ir

¯s
K∗

ω

?
?
p
¯∗
Msr e Re − j ω Ir km r asr
2

= p Msr e

m sr −2
?
k=1:2

¯∗
Kr

?

sr
k ak (Idr k Iqsk − Idsk Iqr k )
...
12)

1 3
A POG graphical representation of system (12
...
5
...
Function “Re(·)” denotes the “complex to
3 4
real conversion” of the input
...
5,
represents the electrical part of the system: note that this part is described only by complex
matrices and complex variables
...
Section ?–? represents
the energy and power conversion (without accumulation or dissipation) between the electrical
and mechanical parts
...
5 POG graphical representation of a multiphase induction motor in the frame ?ω

7

243

Multiphase Induction Motor Control

¯ ¯
It can be easily proved that in (12
...

+
¯e
0 Jm ωm
− ω K ∗ bm
ωm
−τe
˙
? ?? ? ? ?? ?
?
??
? ? ?? ? ? ?? ?
ω
ω
ω
ω
ω
˙
L
R+ ω W
V
q
¨
q

?ω

(12
...
4 Multiphase Indirect Field-Oriented Control with
Harmonic Injection
The field-oriented control in the multiphase case can be now investigated starting from equation (12
...
Let us now consider the case m s = m r = m sr and let Pπ denote the following
permutation matrix:
h
??
??
Pπ = ? eh em s +h ?,

(12
...
Applying the transformation ω Ie = Pπ ω Iek to the electrical part of system
¯
(12
...
15)

T ¯
¯
¯
¯
¯
¯
where matrices and vectors ω Lek = PT ω Le Pπ , ω Rek = PT ω Re Pπ , ω ?ek = Pπ ω ?e Pπ ,
π
π
ω¯
T ¯
T ¯
¯
¯e
¯e
Iek = Pπ ω Ie , ω Vek = Pπ ω Ve , and ω K∗k = ω K∗ Pπ are expressed as follows:

ω

k
??
??
? L sek Msr ek ?
?
¯
L ek = ?
? Msr ek L r ek ?,
1:2:m sr −2

k

ω

??
??
? R 0 ?
?
¯
R ek = ? s
? 0 Rr ?,
1:2:m sr −2

ω

??
¯e
K∗k = ? j

k
??ω ??
? I¯sk ?
Iek = ? ω ¯ ? ,
? Ir k ?

ω¯

p
2

k Msr ek ω I¯r∗k

ω

1:2:m sr −2

1:2:m sr −2

k

ω

k
??ω ??
? Vsk ?
¯
?
¯
V ek = ?
? 0 ?,

??
??
? j ωs k L sek j ωs k Msr ek ?
?
¯
? ek = ?
? j ω p k Msr ek j ω p k L r ek ?,
−j

p
2

k

??
∗
k Msr ek ω I¯sk ?,

1:2:m sr −2

1:2:m sr −2

s
r
sr
with L sek = L s0 + m2s ak Ms0 , L r ek = L r 0 + m2r ak Mr 0 , and Msr ek = Msr e ak
...
15) it
clearly follows that a multiphase induction motor with an odd order harmonic injection can

244

AC Electric Motors Control

¯
be mathematically described by m sr−1 sets of decoupled equations in the frame ?ωk : in other
2
m sr−1
words, the system can be seen as the set of 2 three-phase induction motors whose dynamic
equations are the projection of system (12
...
In addition, the overall mechanical torque of the motor τm is the
sum of the torques generated by the m sr−1 internal decoupled motors:
2
τm =

m sr −2
?

k=1:2
⎛

⎜
= Re ⎜
⎝

τm k =

m sr −2
?

¯e ¯
Re ( ω K∗k ω Iek )

k=1:2

k
??
p
ω ¯∗ ?
j k Msr ek Isk ?
2

m sr −2 ?
? ?
k=1:2

= p Msr e

p
?
? − j k Msr ek ω I¯r∗k
2

m sr −2
?
k=1:2

⎞
k
??ω ??
? I¯sk ? ⎟
?ω
?⎟
? I¯r k ? ⎠

1:2:m sr −2 1:2:m −2
sr

sr
k ak (Idr k Iqsk − Idsk Iqr k )
...
12)
...
For this purpose the multiphase IRFOC is considered
...
Let ω ?ek = ω Lek ω Iek denote the flux vector in the frame ?ωk
...
15) one obtains the following reordered motor dynamic equations as function
of the flux:
k
k
??ω ??
??
??
? Vsk ?
?
?
¯
? = ? Rs 0 ?
?
? 0 ?
? 0 Rr ?

1:2:m sr −2

where

k
k
??ω ?? ??
??
?
? I¯sk ? ? j ωs k 0
?
?ω
? +?
? I¯r k ? ? 0 j ω p k ?

1:2:m sr −2 1:2:m sr −2

ω

k
k
??ω
?? ?? ˙ ??
? ?sk ? ? ω ?sk ?
¯
¯ ?
?ω
? ?
? ?r k ? + ? ω ? ?
...


Imposing the field orientation condition, that is the rotor flux is supposed to be aligned with
the direct axis ensuring null quadrature components of the rotor flux, one can write the IRFOC
¯
main equations in the frame ?ωk as follows:
τm =
?r =

m sr −2
?
k=1:2

m sr −2
?
k=1:2

ω pk = k

τm k = p
?dr k =

Iqsk
,
Tr k Idsk

m sr −2
?
k=1:2

m sr −2
?

k

Msr ek
?dr k Iqsk ,
L r ek

Msr ek Idsk ,

(12
...
17)

k=1:2

k ∈ {1 : 2 : m sr − 2},

(12
...
Note that each subspace is characterized by the respective ω pk slip velocity
...
16), (12
...
18)
...
4
...
The IRFOC equations can be obtained
from equations (12
...
17), and (12
...

Tr 3 Ids3

(12
...
20)
(12
...
In this case it is clear that the field-oriented
control of the two subspaces is totally decoupled: the stator direct-current components Ids1
and Ids3 directly control the rotor direct flux components ?dr 1 and ?dr 3 , while the stator
quadrature current components Iqs1 and Iqs3 independently control the torque components τ1
and τ3 , assuming that the rotor flux components remain constant
...


(12
...
15 in order to avoid saturation effects ensuring a
more flattened flux (see Xu et al
...
Substituting the expressions of the direct and
quadrature third harmonic stator current components Ids3 and Iqs3 defined in equation (12
...
19), (12
...
21), one obtains the following direct and quadrature
fundamental stator current components Ids1 and Iqs1 and slip velocity:

Iqs1 =

?

Msr e1
Msr e1 + k3 Msr e3

?

τm
?
?,
2
p ?r Msr e
Msr e3
1
2
+ 3 k3
L r e1
L r e3 Msr e1
?r
,
+ k3 Msr e3

Ids1 =

Msr e1

ω p1 =

Iqs1

...
23)

(12
...
25)

246

ref
τm

PIω

equation
(12
...
24)

ref
Ids1

K

ref
ref
[Ids1 , Ids3 ]

PId1,3

ref
ref
[Vqs1 , Vqs3 ]

Σt

Field
weakening
[Iqs1 , Iqs3 ]

Σt
t

[Ids1 , Ids3 ]

Is

Five-phase induction motor

ref
ωm

AC Electric Motors Control

¯
Σωk

θs
equation
(12
...
6 IRFOC of a five-phase induction motor: fundamental plus third harmonic injection

These variables are used to implement the IRFOC control of a five-phase induction motor
as shown in Figure 12
...
The inner
PIdk and PIqk controllers respectively regulate the stator direct-current components Ids1 and
Ids3 , directly controlling the rotor flux ?r , and the stator quadrature current components Iqs1
and Iqs3 , directly controlling the mechanical torque τm
...
Note that K = [1 k3 ] according to equation (12
...

An alternative strategy that can be considered is to control the first subspace only, that
ref
corresponds to the fundamental harmonic, and generate the voltage references Vds1 and
ref
Vqs1 , while the third harmonic injection contributions can be obtained using an appropriate
ref

ref

scaling coefficient k V 3 = f (k3 ) in order to calculate the voltage references Vds3 = k V 3 Vds1
ref
ref
and Vqs3 = k V 3 Vqs1 that act in the third subspace
...
6) is
more flexible because a custom control for each harmonic subspace can be designed, but its
implementation has a higher cost in terms of number of used controllers and tuning
...
7) has a simpler structure but it is limited to
the first subspace only, so the control is focused only on the subspace of the fundamental
...


247

Multiphase Induction Motor Control

ref
τm

ref
Iqs1

Lre1
p Msre1

ref
Vqs1

PIq

Φdr1
kV 3

Iqs1

1
Msre1

ref
Ids1

¯
Σωk

ref
Vqs3
t

Vs

ref
Vds1

PId
kV 3

Ids1

ref
Vds3

Σt

Figure 12
...
6 for the remaining scheme)

12
...
2

Five-Phase IRFOC Simulation Results

The simulation results presented in this section have been obtained in Matlab/Simulink by
implementing the complex and reduced order model of a five-phase induction motor considering the third harmonic injection and using the IRFOC strategy described in Figure 12
...
The
electrical and mechanical parameters of the considered motor are listed in Table 12
...
Note
that this motor is supposed to have a concentrated winding phase structure in order to take
into account the third harmonic injection
...
2 Electrical and mechanical parameters of the considered
five-phase induction motor
Stator parameters
ms
Ls
Rs
Ms0

Rotor parameters

5
115
...
7 ?
114 mH

mr
Lr
Rr
Mr 0

5
43
...
85 ?
42
...
002 Kg m2
0
...
2 mH
50 Hz
1390 rpm
0
...
06

t

Msr (1, :) (H)

0
...
02
0
−0
...
04
−0
...
1

0
...
3

0
...
5

0
...
7

0
...
9

1

Figure 12
...
)

where k indicates the injected harmonic order and h the number of stator phases
...
8
0

...
2
These coefficient matrices define the shape of the mutual inductance t Msr
...
8
...
9: the dashed line represents the trapezoidal
ref
profile reference ωm (see Figure 12
...
The tracking error is quite small and it increases only during the rising and
falling edges
...
5

1

1
...
5

4

Angular velocity ωm —falling ramp

200

ωm (rad/s)

ωm (rad/s)

200
150
100
50
0

2
...
5

1

Time(s)

1
...
4

2
...
8

3

Time(s)

3
...
4

Figure 12
...
5

1

1
...
5

3

3
...
5

4

Stator quadrature current component Iqs1

Iqs1 (A)

10
5
0
0

0
...
5

2
Time (s)

2
...
10 Controlled stator current components Ids1 and Iqs1 in the fundamental subspace: actual
values (solid) and reference values (dashed)

and the third harmonic subspaces: in each subspace the stator direct and quadrature current
components are controlled, as depicted in Figures 12
...
11, in order to perform the flux
and torque control
...
12 and 12
...
4
0
...
2
0
...
5

1

1
...
5

3

3
...
5

4

Stator quadrature current component Iqs3

Iqs3 (A)

1
...
5
0
0

0
...
5

2

2
...
11 Controlled stator current components Ids3 and Iqs3 in the third harmonic subspace: actual
values (solid) and reference values (dashed)

250

AC Electric Motors Control

Voltages t V
s
(V)

100
50

tV

s

0
−50
−100
0
...
8

1
...
4

1
...
45

1
...
47

1
...
49

1
...
51

1
...
53

1
...
55

Time (s)

Figure 12
...
)

contributions
...
14: for t ∈ [0, 0
...
5, 4] the torque τm is null because no load
torque τe is applied and no velocity is tracked, while for t ∈ [0
...
5] and t ∈ [2
...
5] the
torque τm evolves according to the load torque profile τe and to the velocity rising and falling
transients (see Figure 12
...


Currents tIs

0

t

Is (A)

5

−5
0
...
8

1

1
...
2

1
...
5
0
−0
...
45

1
...
47

1
...
49

1
...
51

1
...
53

1
...
55

Time (s)

Figure 12
...
)

251

Multiphase Induction Motor Control

Mechanical torque τm
τm (Nm)

2
1
...
5
0
0

0
...
5

2

2
...
5

4

3

3
...
5

0
0

0
...
5

2

2
...
14 Mechanical torque τm and load torque τe

12
...
First of all the motor dynamic equations in the complex rotating
reference frame have been obtained considering a generic number of stator and rotor phases and
the corresponding odd order harmonic injection
...
Finally, a five-phase induction motor has been considered and the control decoupling
between the fundamental and the third harmonic subspaces has been presented
...


References
Duran MJ, Salas F, and Arahal MR (2008) Bifurcation analysis of five-phase induction motor drives with third
harmonic injection
...

Jones M and Levi E (2002) A literature survey of state-of-the-art in multiphase AC drives
...
505–510
...
Springer-Verlag, 3rd edn, Berlin
...

Levi E, Bojoi R, Profumo F, et al
...
IET Electric
Power Applications, 1(4) 489–516
...
32nd Annual
Conference of the IEEE Industrial Electronics Society, IECON, Paris, pp
...

Pereira LA, Scharlau CC, Pereira LFA, and Haffner JF (2006) General model of a five-phase induction machine
allowing for harmonics in the air gap field
...

Toliyat HA, Lipo TA, and White JC (1991) Analysis of a concentrated winding induction machine for adjustable
speed drive applications
...
Motor analysis
...

Toliyat HA, Lipo TA, and White JC (1991) Analysis of a concentrated winding induction machine for adjustable speed
drive applications
...
Motor design and performance
...


252

AC Electric Motors Control

Vas P (1990) Vector Control of AC Machines
...
ISBN 0-19-859370-8
...
IEEE APEC, 1, pp
...

Xu H, Toliyat HA, and Petersen LJ (2002) Five-phase induction motor drives with DSP-based control system
...

Zanasi R (1991) Power Oriented modeling of dynamical system for simulation
...
2, pp
...

Zanasi R, Grossi F, and Azzone G (2009) The POG technique for modeling multiphase asynchronous motors
...
14–17
...
Proceding of Vehicle
Power and Propulsion Conference, VPPC, Lille, France, pp
...

Zanasi R and Azzone G (2010) Complex dynamic model of a multiphase asynchronous motor
...

Zanasi R and Azzone G (2011) Field oriented control of a multiphase asynchronous motor with harmonic injection
...


13
Backstepping Controller for
DFIM with Bidirectional
AC/DC/AC Converter
Abderrahim El Fadili, Vincent Van Assche,
Abdelmounime El Magri, and Fouad Giri
GREYC Lab, University of Caen Basse-Normandie, France

13
...
2006)
...

It has proved to be quite suitable both as a motor in various applications (Metwally et al
...
2004; Bonnet et al
...
2007; Vidal et al
...
2009; Abo-Khalil 2012; Song et al
...
It also turns out to
be a possible alternative to the synchronous machine in high-power applications, for example,
railway traction, marine propulsion, metallurgy, rolling mills, or hydroelectric stations, and in
very low-speed applications, for example, coiler-uncoiler
...
The first option is one where the
DFIM is (doubly) acted upon, that is, both from the stator and the rotor (Figure 13
...
This
configuration necessitates two converters, powering respectively the stator and the rotor
...
Edited by Fouad Giri
...
Published 2013 by John Wiley & Sons, Ltd
...
1 DFIM supplied by tow inverters in stator and rotor

control inputs, yielding four freedom degrees (Ourici 2012)
...
2)
...
2010)
...

Indeed, as the power supplied from the rotor (slip power) is proportional to the slip, only a
small fraction of the overall system power can be handled by means of the rotor-side power

AC grid

AC
DC

DC
AC

DFIM
3~
Figure 13
...
Moreover, in varying-speed drive applications, the slip power regenerated by the
rotor-side converter, during motor operation stages, is released to the line grid resulting in
highly efficient energy conversion (Peresadaa et al
...

The problem of controlling DFIMs has been dealt with following different approaches
...
(2000), a field-oriented controller without position sensor has been
proposed for DFIMs in motor applications with one converter on the rotor side whereas the
stator side is connected to the network
...
(2002) and Khojet El Khil
et al
...
In Gritli et al
...
Other control strategies have been proposed including direct torque
control (Bonnet et al
...
2008), output-feedback control
(Peresadaa et al
...
2007)
...
The DFIM stator windings are directly connected
to the line grid, while the rotor windings are controlled via a bidirectional power converter
...
A multiloop nonlinear adaptive
controller is designed, on the basis of the DFIM nonlinear model, using the backstepping
technique
...

The chapter is organized as follows: in Section 13
...
3; the control performances are illustrated through
numerical simulations in Section 13
...


13
...
3
...
The rectifier is an AC/DC converter operating, just
as the DC/AC inverter, according to the well-known PWM principle
...
2
...
It was obtained by operating the (energy preserving) Park
transformation on the three-phase electrical quantities
...

Considering as state variables the flux components (φsd and φsq ) and the current components
(ir d and irq ), the DFIM two-phase model has been shown to be defined by the following statespace representation:
TL
Msr
F
dωm
(φsq ir d − φsd irq ) −
= p
− ωm ,
dt
J Ls
J
J

(13
...
3 Controlled system including a DFIM and associated AC/DC/AC converters

dφsd
Msr
1
ir d + Vs ,
= − φsd + ωs φsq +
dt
τs
τs

(13
...
3)

γ2
dir d
= −γ1 ir d + (ωs − pωm )irq + φsd − pωm γ2 φsq − γ2 Vs + γ3 vr d ,
dt
τs

(13
...
5)

where φsd and φsq are stator flux dq-components; ir d and irq are rotor current dq-components;
Vs is the stator voltage norm; vr d and vrq are rotor voltage dq-components; ωm is the motor
speed; ωs is the dq-frame speed; Rs and Rr are stator and rotor resistances; L s and L r are
stator and rotor self-inductances; Msr is the mutual inductance between the stator and the
rotor; F, J , and TL are friction coefficient, rotor inertia, and load torque, respectively; and p
is the number of pole pairs and rotor windings
...

σ Lr

257

Backstepping Controller for DFIM with Bidirectional AC/DC/AC Converter

13
...
2

Modeling of the System “DC/AC Inverter–DFIM”

The inverter is featured by the fact that the rotor voltage d and q components are independently
controlled
...
g
...
1998):
vr d = vdc u 1 ,

vrq = vdc u 2 ,

i in = u 1 ir d + u 2 irq ,

(13
...
Specifically, the former are obtained by operating the Park transformation on the
latter and averaging the result over the PWM periods; i in designates the input current inverter;
vdc denotes the voltage in capacitor C; and s1 , s2 , and s3 are binary input signals, defined as
follows:
si =

?

1,
0,

if
if

Si
Si

?

On and Si Off;
?
Off and Si On;

i = 1, 2, 3
...
7)

Now, let us define the averaged state variables:
ωm = x 1 , φ sd = x 2 , φ sq = x 3 , i r d = x 4 , i rq = x 5 , v dc = x 6 , vr d = u 1 x 6 , v rq = u 2 x 6
...
Then,
it is proved in many places that instantaneous “DFIM-inverter” representation (equations
(13
...
2), (13
...
4), and (13
...
6):
d x1
dt
d x2
dt
d x3
dt
d x4
dt
d x5
dt

13
...
3

Msr
TL
F
(x 3 x 4 − x 2 x 5 ) −
x1 + p
,
J
J Ls
J
Msr
1
x 4 + Vs ,
− x 2 + ωs x 3 +
τs
τs
Msr
1
x5 ,
− x 3 − ωs x 2 +
τs
τs
γ2
−γ1 x 4 + (ωs − px 1 )x 5 + x 2 − pγ2 x 1 x 3 − γ2 Vs + γ3 x 6 u 1 ,
τs
γ2
−γ1 x 5 − (ωs − px 1 )x 4 + x 3 + pγ2 x 1 x 2 + γ3 x 6 u 2
...
8)

=

(13
...
10)
(13
...
12)

AC/DC Rectifier Modeling

The AC/DC rectifier, connected to the triphase power grid, is depicted in Figure 13
...
It
consists of six (semiconductor) insulated gate bipolar transistors (IGBTs) and antiparallel
diodes, allowing for bidirectional current flow mode, arranged in three legs denoted 1, 2,
and 3
...
On a given leg, only one
switch is conducting at a time
...
4 AC/DC converter power with triphase input

It is readily checked, applying Kirchhoff’s laws, that the rectifier is described by the following set of differential equations:
Lo

d[ir e ]123
= [vs ]123 − vdc [k]123 ,
dt

(13
...
14)

T
i in = [k]123 [ir e ]123 ,

(13
...


Backstepping Controller for DFIM with Bidirectional AC/DC/AC Converter

259

Specifically, one has
ki =

?

1,
0,

if
if

Ki
Ki

?

On and K i Off;
?
Off and K i On;

i = 1, 2, 3
...
16)

For control synthesis purpose, the Park transformation is applied to the triphase representation
(13
...
14), with the d-axis being linked to the stator voltage
...
17)
(13
...
19)

where ir ed and ir eq denotes the rectifier-side grid current dq-coordinates; and u 3 and u 4
represent the averaged d- and q-axis components of the triphase duty ratio system (k1 , k2 , k3 )
...

= −ωs x 7 −
dt
Lo

(13
...
21)
(13
...
For future referencing, the complete model is
rewritten as
d x1
dt
d x2
dt
d x3
dt
d x4
dt
d x5
dt

Msr
TL
F
(x 3 x 4 − x 2 x 5 ) −
x1 + p
,
J
J Ls
J
Msr
1
x 4 + Vs ,
− x 2 + ωs x 3 +
τs
τs
Msr
1
x5 ,
− x 3 − ωs x 2 +
τs
τs
γ2
−γ1 x 4 + (ωs − px 1 )x 5 + x 2 − pγ2 x 1 x 3 − γ2 Vs + γ3 x 6 u 1 ,
τs
γ2
−γ1 x 5 − (ωs − px 1 )x 4 + x 3 + pγ2 x 1 x 2 + γ3 x 6 u 2 ,
τs

=−

(13
...
24)

=
=
=

(13
...
26)
(13
...

= −ωs x 7 −
dt
Lo

13
...
3
...
28)
(13
...
30)

Controller Design
Control Objectives

There are two operational control objectives:
1
...

2
...

The flexibility offered by the available four control inputs, that is, u 1 , u 2 , u 3 , and u 4 , makes
possible to add two more control objectives
...
Controlling the continuous voltage vdc making it track a given reference signal x 6 = vdc
...

?
2
2
4
...


13
...
2

Motor Speed and Stator Flux Norm Regulation

The problem of controlling the rotor speed and stator flux norm is presently addressed for
the DFIM described by equations (13
...
24), (13
...
26), and (13
...
The speed
∗
∗
reference x 1 = ωm is any bounded and derivable function of time and its two first derivatives
are available and bounded
...
The stator
flux reference ?∗ is set to its nominal value
...
1995)
...

s

(13
...
32)

Backstepping Controller for DFIM with Bidirectional AC/DC/AC Converter

261

It follows from (13
...
24), and (13
...
33)

˙
˙
˙
˙
z 2 = 2?∗ ?∗ − 2(x 2 x 2 + x 3 x 3 )
s s
˙
= 2?∗ ?∗ +
s s

2
2Msr
(x 2 2 + x 3 2 ) −
(x 2 x 4 + x 3 x 5 ) − 2x 2 Vs
...
34)

Msr
In equations (13
...
34), the quantities p J L s (x 3 x 4 − x 2 x 5 ) and 2Msr (x 2 x 4 + x 3 x 5 ) stand
τs
up as virtual control signals
...
33)
Msr
and (13
...

s s
τs
def

˙∗
μ 1 = c1 z 1 + x 1 +

(13
...
36)

Presently, the load torque TL is not assumed to be known
...
35)
...
37)

ˆ
where c1 and c2 are any positive design parameters and TL is the estimate of TL (yet to be
determined)
...
Nevertheless, we retain the
expressions of μ1 and ν1 as first stabilizing functions and introduce the new errors
Msr
(x 3 x 4 − x 2 x 5 ),
J Ls
2Msr
z 4 = ν1 −
(x 2 x 4 + x 3 x 5 )
...
38)
(13
...
37), (13
...
39), the dynamics of the errors z 1 and z 2 ,
initially described by (13
...
34), can be rewritten as follows:
˙
z 1 = −c1 z 1 + z 3 +
˙
z 2 = −c2 z 2 + z 4 ,

˜
TL
,
J

(13
...
41)

262

AC Electric Motors Control

where,
˜
ˆ
TL = TL − TL

(13
...
To this end, we should make clear how these
errors depend on the actual control signals (u 1 , u 2 )
...
38)
that

˙
z 3 = μ1 − p
˙

Msr
˙
˙
˙
˙
( x 3 x 4 + x 3 x 4 − x 2 x 5 − x 2 x 5 )
...
43)

Assume that the load torque TL is constant or slowly time varying and using equations (13
...
24), (13
...
26), (13
...
42) and (13
...
43)
?

F
˙
z 3 = μ 2 + c1 −
J

? ˜
˙
˜
Msr
TL
TL
γ3 x 6 (x 3 u 1 − x 2 u 2 ),
−
−p
J
J
J Ls

(13
...
(13
...
39) that z 4 undergoes the following differential
equation:

˙
˙
z 4 = ν1 −

2Msr
˙
˙
˙
˙
( x 2 x 4 + x 2 x 4 + x 3 x 5 + x 3 x 5 )
...
46)

Using equation (13
...
24), (13
...
26), (13
...
36), it follows from (13
...
47)

Backstepping Controller for DFIM with Bidirectional AC/DC/AC Converter

263

with
¨
˙s
ν2 = c2 (−c2 z 2 + z 4 ) + 2(?∗ )2 + 2?∗ ?∗
s s
?
?
?
?
Msr 3
4
1 2
+2
+ γ1 (x 2 x 4 + x 3 x 5 ) +
− ?s + Vs x 2
τs
τs
τs
τs
?
?
Msr
1
x 4 + Vs
−2Vs − x 2 + ωs x 3 +
τs
τs
?
?
γ2 2
Msr Msr 2
2
(x + x 5 ) + ?s + px 1 (x 3 x 4 − x 2 x 5 ) + x 4 Vs − γ2 x 2 Vs
...
48)
−2
τs
τs 4
τs
To analyze the error system composed of (13
...
41), (13
...
47), let us consider
the following augmented Lyapunov function candidate:
V =

˜2
1 2 1 2 1 2 1 2 1 TL
z1 + z2 + z3 + z4 +

...
49)

Its time derivative along the trajectory of the state vector (z 1 , z 2 , z 3 , z 4 ) is
˙ ˜
˜
T L TL
˙
˙
˙
˙
˙
V = z1 z1 + z2 z2 + z3 z3 + z4 z4 +

...
50)

Using equations (13
...
41), (13
...
47), equation (13
...
51)

2
2
2
2
adding c3 z 3 − c3 z 3 + c4 z 4 − c4 z 4 to the right-hand side of equation (13
...

J
J

(13
...
52) immediately suggests the following parameter adaptation law:
?
?
F
˙
˜
z3 − z1,
T L = − c1 −
J

(13
...
42), yields
?
?
F
˙
ˆ
T L = c1 −
z3 + z1
...
54)

˙
˜
Substituting the parameter adaptation law (13
...
52), gives
2
2
2
2
˙
V = −c1 z 1 − c2 z 2 − c3 z 3 − c4 z 4
?
?
?
?
??
?
?
1
F
1
Msr
+z 3 μ2 + c3 +
γ3 x 6 (x 3 u 1 − x 2 u 2 )
c1 −
z3 + 1 +
z1 − p
J
J
J
J Ls
?
?
2Msr
γ3 x 6 (x 2 u 1 + x 3 u 2 ) ,
(13
...
Equation (13
...
55)) are set to zero
...
56)

with
Msr
γ3 x 6 x 3 ,
J Ls
2Msr
λ2 =
γ3 x 6 x 2 ,
τs
λ0 = p

?

λ1 = − p
λ3 =

Msr
γ3 x 6 x 2 ,
J Ls

2Msr
γ3 x 6 x 3
...
57)

?
λ1
is nonsingular as its determinant, D = λ0 λ3 −
λ3
?
M2
2 2
2
2
2
λ2 λ4 = 2 p J L ssrτs γ32 x 6 (x 2 + x 3 ), stay away from zero because the flux ?s = x 2 + x 3 never
vanishes in practice due to the presence of the remnant flux
...
56) to u 1 and u 2 on the right-hand side of (13
...


(13
...
58) is a negative definite function of the state vector
(z 1 , z 2 , z 3 , z 4 ), the latter are globally asymptotically vanishing (Khalil 2003)
...
3
...
Consider the closed-loop system composed of:

1
...
23), (13
...
25),
(13
...
27),
2
...
56), and
3
...
54)
...
The closed-loop error system undergoes, in the (z 1 , z 2 , z 3 , z 4 , TL ) coordinates, the following
equations:
˙
z 1 = −c1 z 1 + z 3 +
˙
z 2 = −c2 z 2 + z 4 ,

˜
TL
,
J

(13
...
60)

?

? ˜
F TL
˙
z 3 = −c3 z 3 − z 1 + c1 −
,
J
J
˙
z 4 = −c4 z 4 − z 2 ,
?
?
F
˙
˜
T L = − c1 −
z3 − z1
...
61)
(13
...
63)

2
2
2
...


˜2
TL
J

and the errors (z 1 , z 2 , z 3 , z 4 ) are exponentially vanishing, whatever their

Proof: Equations (13
...
60) are immediately obtained from (13
...
41)
...
61) is obtained by substituting the control law (13
...
54) to (u 1 and u 2 ) on the right-hand side of (13
...
Equation (13
...
56) to u 1 and u 2 on the right-hand side of (13
...
Equation (13
...
53)
...
On the other hand, it is readily seen
˜
T2
2
2
2
2
from (13
...
53) that V = 1 z 1 + 1 z 2 + 1 z 3 + 1 z 4 + 1 JL is a (radially unbounded)
2
2
2
2
2
˙
Lyapunov function of the error system (13
...
60), (13
...
62)
...
Besides, asymptotic stability implies exponential stability due to system linearity (Khalil 2003)
...
3
...


266

AC Electric Motors Control

Remark 13
...
2 Note that the exponential nature of stability guarantees stability robustness
with respect to modeling and measurements errors (Khalil 2003)
...
3
...
Accordingly, one seeks
a regulator that makes the q-components of the input current, that is, i gq = i sq + i r eq , track a
null reference signal i gq
...

∗
∗
As the reference signal i gq = 0, it follows that the tracking error z 5 = i gq − i gq undergoes
the following equation:
z 5 = −i sq − x 8
...
64)

x 3 = L s i sq + Msr x 5 ,

(13
...
64) becomes
z5 = −

x3
Msr
+
x5 − x8
...
66)

In view of equations (13
...
27), and (13
...

τs

(13
...
5z 5
...
68)

Backstepping Controller for DFIM with Bidirectional AC/DC/AC Converter

267

with c5 > 0 is a design parameter and
h 1 (x) =

Lo
Ls
+

?

−

Msr
1
x 3 − ωs x 2 +
x5
τs
τs

L o Msr
Ls

?

?

γ1 x 5 + (ωs − px 1 )x 4 −

?
γ2
x 3 − pγ2 x 1 x 2 − γ3 x 6 u 2
...
69)

DC Link Voltage Regulation
The aim is now to design a control law u 3 so that the rectifier output voltage x 6 = v dc is steered
∗
∗
∗
to a given reference value x 6 = vdc
...
The point is that u 3 acts also on the current
x 7
...
This double requirement is to be met using only one
control input, namely u 3
...
Accordingly, an
∗
inner loop is first designed so that the current x 7 is enforced to track a reference signal x 7 , by
acting on the only available control signal u 3
...

Inner Loop Design for Current x7
Introduce the current tracking error z 7 :
∗
z 7 = x7 − x7
...
70)

The z 7 -dynamics are described by the following equation:
˙
˙∗
z 7 = x 7 − ωs x 8 −

Vs
x6 u 3
+

...
71)

To get a stabilizing control signal for this first order system, consider the following quadratic
Lyapunov function:
V7 =

1 2
z
...
72)

˙
It is easily checked that the time derivative V7 can be made negative definite in the state z 7 by
letting the quantity x 6 u 3 be chosen as follows:
∗
˙∗
x 6 u 3 = −c7 L o x 7 + c7 L o x 7 − L o x 7 + L o ωs x 8 + Vs ,

with c7 > 0 is a design parameter
...
73)

268

AC Electric Motors Control

Outer Loop Design for DC-Voltage x6
2
Now, let us introduce the squared voltage variable y = x 6
...
28), it follows
that that y undergoes the following equations:
˙
y=

2
(x 7 x 6 u 3 + x 8 x 6 u 4 − x 6 i in )
C
2
∗
˙∗
= − (c7 L o x 7 x 7 + L o x 7 x 7 + c5 L o x 8 z 5 ) + h 2 (x),
C

(13
...
75)

where the second line in (13
...
73)
...
e
...
It is given the nominal value of the rotor voltage amplitude
...
74) that the tracking error z 6 = y ∗ − y undergoes the following equation:
˙
˙
z6 = y∗ +

2
∗
˙∗
(c7 L o x 7 x 7 + L o x 7 x 7 + c5 L o x 8 z 5 ) − h 2 (x)
...
76)

To get a stabilizing control law for the system (13
...

2 6

(13
...
76) yields
˙
˙
V6 = z 6 z 6
...
78)

∗
This suggests for x 7 the following control law:
∗
˙∗
x 7 = −c7 x 7 − c5 z 5

?
x8
C ?
˙
−c6 z 6 − y ∗ + h 2 (x) ,
+
x7
2L o x 7

(13
...
Indeed, substituting x 7 to (13
...


Proposition 13
...
3 Consider the control system consisting of the subsystem (13
...
29),
and (13
...
68), (13
...
79)
...

⎥⎢ ⎥ + ⎢
⎣ z7 ⎦ ⎣ 0
⎦ ⎣ z7 ⎦ ⎣
⎦
0 −c7 0
0
˙
c6 C
x8
C
∗
∗
∗
˙
(h 2 (x) − y )
−c5 x7 − 2L o x7 0 −c7
˙
x7
x7
2L o x 7
⎡

(13
...


Proof: Equation (13
...
68), (13
...
79) to x 6 u 3 , x 6 u 4 , and x 7 on the right-hand side of equations (13
...
78)
...
80) is globally
exponentially stable
...
3
...


13
...
3, including the control laws
(13
...
68), (13
...
79) and the parameter adaptive law (13
...
5
...
5 mF; modulation frequency 10 KHz
...
5 kW power, the remaining characteristics are summarized in Table 13
...

The simulation protocol is described by Figures 13
...
7, which show that, the reference signals and the machine load are profiled so that the machine is enforced to operate,
successively, both at high and low speeds
...

∗
The DC-link voltage reference is set to the constant value vdc = 220 V
...
7 wb)
...
1 Numerical values of the considered doubly-fed induction motor
characteristics
Characteristic
Nominal power
Nominal stator voltage
Nominal stator current
Nominal flux
Stator resistance
Stator inductance
Nominal rotor voltage
Nominal rotor current
Rotor resistance
Leakage inductance
Rotor inductance
Inertia moment
Friction coefficient
Number of pole pairs

Symbol

Value

Unity

PN
Usn
Isn
?sn
Rs
Ls
Ur n
Ir n
Rr
Msr
Lr
J
F
p

1
...
3
0
...
75
0
...
5
1
...
195
0
...
35
0
...
5 Control system including AC/DC/AC converters and a doubly-fed induction motor

Sensors

DFIM
3~

123/dq

ir1

ire3

igq

N

Lo

Lo

ire2

x7
x8
Vs
ws
igq

Lo

ire1

ig3

ig2

ig1

vs1 vs2 vs3

123/dq

is3
is2
is1

Sensors

271

Backstepping Controller for DFIM with Bidirectional AC/DC/AC Converter

10

Load torque (Nm)

8
6
4
2
0
0

2

4
Time (s)

6

8

Figure 13
...
7 Speed ωm (rad/s)

Stator flux norm (Wb)

0
...
6
0
...
4

s

Φ

s

0
...
2
0
...
8 Stator flux norm (Wb)

8

272

AC Electric Motors Control

DC−link voltage (V)

200
V dc

150

*

V dc

100
50
0

0

2

4
Time (s)

6

8

Figure 13
...
5
1
0
...
5
−1

0

2

4
Time (s)

6

8

Grid current ig1 and grid voltage vs1

Figure 13
...
7

0
...
72

ig1
4
...
76
Time (s)

4
...
8

Figure 13
...
The experimental setup is simulated in the Matlab/Simulink environment
with a computation step of 5 μs
...
In the light of the closed-loop responses (see Figures 13
...
9, 13
...
11), it is seen that the multiloop nonlinear adaptive controller meets all its objectives
and enjoys quite satisfactory transient performances
...
5

Conclusions

In this chapter, the problem of controlling systems including DFIMs and the associated AC/DC
rectifier and DC/AC inverter, has been addressed
...
23), (13
...
25), (13
...
27),
(13
...
29), and (13
...
Based on such a model, the nonlinear controller, defined by
(13
...
68), (13
...
79), and (13
...
The controller, depicted by (Figure 13
...
It is formally shown that this controller makes the motor velocity track well
is reference trajectory and ensures a unity power factor, despite the external load torque
uncertainty and variations
...
The control system performances are formally analyzed and numerically confirmed
using simulation
...


References
Abo-Khalil AG (2012) Synchronization of DFIG output voltage to utility grid in wind power system
...

Bonnet F, Vidal PE, and Pietrzak-David M (2007) Dual direct torque control of doubly fed induction machine
...

Boukhezzar B and Siguerdidjane H (2009) Nonlinear control with wind estimation of a DFIG variable speed wind
turbine for power capture optimization
...

Gritli Y, Stefanib A, Rossib C, et al
...
Electric Power Systems Research, 81,7 51–766
...
IEEE Proceeding-Electrical Power Application, 147, 241–250
...
Prentice Hall, Upper Saddle River, NJ
...
Mathematics and Computers in Simulation, 71, 360–368
...
John Wiley & Sons
...
Energy Conversion and Management, 43, 3–13
...
IEEE Conference on
Industrial Electronic Conference, pp
...

Ourici A (2012) Double flux orientation control for a doubly fed induction machine
...

Peresadaa S, Tillib A and Toniellib A (2004) Power control of a doubly fed induction machine via output feedback
...

Poitiers F, Bouaouiche T, and Machmoum M (2009) Advanced control of a doubly-fed induction generator for wind
energy conversion
...


274

AC Electric Motors Control

Salloum G, Mbayed R, Pietrzak-David M, and De Fornel B (2007) Loop shaping H∞ control for doubly fed induction
motor
...
pp
...

Song Z, Shi T, Xia C, and Chen W (2012) A novel adaptive control scheme for dynamic performance improvement
of DFIG-Based wind turbines
...

Verma V, Maiti S and Chakraborty C (2010) Grid-connected vector-controlled slip-ring induction machine drive with
out speed sensor
...

Vidal P
...
Electrical
Engineering, Archiv fur Electrotechnik, Elsevier-Verlag
...

Xiying D and Jian W (2010) A new control strategy of doubly fed induction machine for hybrid electric vehicle
...


14
Fault Detection in Induction
Motors
Alessandro Pilloni,1 Alessandro Pisano,1 Martin Riera-Guasp,2
Ruben Puche-Panadero,2 and Manuel Pineda-Sanchez2
1
2

Department of Electrical and Electronic Engineering (DIEE), University of Cagliari, Italy
Department of Electrical Engineering (DIE), Universidad Polit´ cnica de Valencia, Spain
e

14
...
Remarkably, methods
for IM fault detection in both steady-state and transient operating conditions will be
outlined
...
2 is a short introduction in
which the more common faults of IM are concisely described as well as their causes,
consequences, and symptoms
...
3 introduces an example of model-based approach
for fault detection and isolation (FDI) in IMs based on dynamical observers
...
4,
14
...
6, 14
...
8, and 14
...
Sections
14
...
5 are dedicated to signal analysis techniques valid for steady-state operation;
these methods, commonly designated as motor current signature analysis (MCSA) include
the conventional Fourier transform (FT) approaches, and the more recently developed
Hilbert-transform (HT)-based approaches
...
Thus, in Sections 14
...
7, 14
...
9,

AC Electric Motors Control: Advanced Design Techniques and Applications, First Edition
...

© 2013 John Wiley & Sons, Ltd
...


276

AC Electric Motors Control
Table 14
...


state-of-the-art approaches based on time-frequency analysis tools, which enable for diagnostic in transient conditions are introduced
...
6, the continuous wavelet transform (CWT), in Section 14
...
8, and the instantaneous frequency (IF)
approach, in Section 14
...


14
...
Although its design takes
into account the possibility of the most typical faults, such as overvoltage and overcurrent, it is
impossible to warrant that the electrical machine hold out the normal status during all its life
...
A fault in the machine causes a nonprogrammed stop
in the process, with serious economic consequences
...
1 summarizes the percentage
of fault occurrences in the electrical machines
...
(2003)
...
), the element of the machine (stator or rotor) where the
failure occurs, and so on
...
2
...

Their origin, causes, and possible methods of detection are outlined in this chapter
...
The most common types of
failures are related to failures in the windings insulation, and the different possible failures,
along with the corresponding effects on the machine’s operation, are listed as follows:
• Short-circuits between adjacent turns
...

• Short-circuits between coils of the same phase
...


277

Fault Detection in Induction Motors

• Short-circuits between phases
...

• Short-circuits between one phase and earth
...

• Open circuit of a given phase
...

These faults can cause high temperatures in the coils or stator core, stator core degradation,
deterioration of clamp short-circuit end-rings, oil, moisture and dirt contamination, imbalances
in supply or electric shocks, and leaks in refrigeration systems
...
2005):
?
?
(1 − s)
f1
f sc = k ± n ·
p

k = 1, 3, 5,
...


(14
...

Rotor Part Broken Bar Damages
The most common electrical rotor failures, in the case of squirrel-cage IMs, are breakages in
the rotor-cage winding
...
1988; El Hachemi Benbouzid 2000; Puche-Panadero et al
...

The characteristics frequencies of these fault-related harmonic components f br b are given by
f br b = ((1 − s) k/ p ± s)) f 1

k/ p = 1, 3, 5
...
2)

The left sideband harmonic (LSH) is obtained by substituting k/ p = 1 in (14
...
2001), with frequencies given by
f br b = (1 ± 2 k s) f 1

k = 1, 2, 3,
...
3)

Although broken rotor bars do not immediately cause the motor to fail, they can be a
serious problem with several secondary effects (i
...
, overheating, bar hitting, damaging of
motor insulation and consequent winding failure, etc
...


14
...
2

Mechanical Faults

Almost always, mechanical failures occur in the rotor, as it is the moving part, and are identified
as rotor’s imbalances, misalignment, bearing failures, gear failures, and eccentricities in all
their variants
...
Because of this there are two types of imbalances:
1
...
This failure is detectable with the rotor stopped
...
Dynamic imbalance, which is produced by a nonhomogeneous longitudinal distribution
of weights in the rotor
...
When the coupling is not perfect, fault-related
components appear in the motor current at frequencies given in Cabanas and Melero (1998)
...
The rotor may be positioned slightly off-center in
the stator bore
...
2004)
...
Static eccentricity can be produced by a stator ovality, or by a misalignment of the
mounted bearings or the bearing plates
...
It is characterized by a
displacement of the axis of rotation, which can be caused
...
The
nonuniform air gap gives rise to a radial force of electromagnetic origin, called UMP
...
4)

f1,

where k is a positive constant, N is the number of the rotor slots, and υ is the harmonic
order
...
Dynamic eccentricity corresponds to the case where the rotation axis of the rotor does
not coincide with its geometric center
...
For each air gap position, the
radial length of the air gap varies with time, sinusoidally modulated, and this results in an
asymmetric magnetic field
...


(14
...
Mixed eccentricity is the combination of static and dynamic eccentricity
...
(2009), and Puche-Panadero et al
...


= f s ± k fr ,

(14
...
Levels of air-gap eccentricity should not exceed a maximum of 10% in three-phase
IMs to avoid catastrophic damage
...
Axial eccentricity appears when the eccentricity varies along the axis of the rotor
...


Gear Failures
The use of the gears in the electrical machines is due to the use of different speed and
torque reference in different industrial applications
...

Of course, these external elements of the machine have influence in the electrical machine
behaviour
...


Damaged Bearings
Electrical machines have two bearings where the rotor shaft end is held
...
A bearing is a rolling element with extremely small tolerances to avoid any internal
displacement, except the rotational one
...
The following equation shows the characteristic frequency if there is a fault
in the inner or outer raceway (Bl¨ dt et al
...
7)
(14
...
1)
...


(14
...
1 Geometry of a rolling-bearing

14
...
3
...
2003)
...
Ideally, each residual should selectively react to a specific fault only to allow fault
isolation
...
2003)
...
2000; Simani et al
...
2008)
...
The method is based on
a mathematical model of the motor under the considered faulty conditions, which represents
the faults by means of suitably located fault injection signals
...
Noticeably, to overcome the uncertainty
in the load torque, and to cope with, at the same time, the unknown fault injection signals, an
unknown-input observation approach, robust to the presence of certain unmeasurable exogenous input terms, is taken
...
2004; Fridman et al
...
The convergence
property of the proposed algorithm will be supported by a Lyapunov-based stability proof
...
The standard model describing the nominal (i
...
,
healthy) dynamics of the three-phase induction (wound-rotor or squirrel-cage) machine in
the fixed (α, β) stator reference frame is expressed by the next fifth order nonlinear system
(Krause and Thomas 1965; Marino et al
...
10)

281

Fault Detection in Induction Motors
Table 14
...

Lr

(14
...
10) and (14
...
2
...
3
...
2, regarding fault patterns in the considered
fault scenarios, we can devise that inserting additional exogenous voltages f sα and f sβ in the
stator current equation of (14
...

Therefore, a mathematical model representing a faulty IM can take the following form:
⎧
˙
⎪ x1
⎪
⎪x
⎪ ˙2
⎨
˙
x3
⎪
⎪ x4
⎪˙
⎪
⎩
˙
x5

= a1 (x 3 x 4 − x 2 x 5 ) − a2 x 1 + a3 TL ,
= b1 x 4 − b2 x 2 + b3 x 1 x 3 + b4 (u sα + f sα ),
= b1 x 5 − b2 x 3 − b3 x 1 x 2 + b4 (u sβ + f sβ ),
= c1 x 2 − c2 x 4 − n p x 1 x 5 ,
= c1 x 3 − c2 x 5 + n p x 1 x 4 ,

(14
...

The load torque TL being generally not available for measurements in applications, it is
treated as an “unknown input” within the observer design problem using the model (14
...

It is concluded from the inspection of (14
...
6) that an accurate estimation of the
speed x 1 is a prerequisite for making a reliable diagnosis by spectral methods
...
3
...
2003), shall be presented hereinafter
...
This is, of course, not possible in practice, as an accurate and complete mathematical
description of a process is never available
...
2003)
...
Particularly, we exploit the desirable feature of SMOs of providing, under certain
conditions, the capability of reconstructing the unknown inputs acting on the observed system
...

We will then devise a scheme that can reconstruct both the unknown exogenous faults
signals f sα and f sβ in (14
...

The detection of faults will be achieved by a nonconventional, threshold-based residual
evaluation procedure applied to the reconstructed fault signals
...


14
...
4

Residual Generation and Evaluation

The aim of residual generation is to reconstruct fault symptoms using available inputs and
outputs from the monitored system in order to obtain information about the occurrence of
fault
...
13)
⎪˙
⎪ x 4 = c1 x 2 − c2 x 4 − n p x 1 x 5 ,
ˆ
ˆ
⎪ˆ
⎪
⎩˙
ˆ
ˆ
ˆ
x 5 = c1 x 3 − c2 x 5 + n p x 1 x 4
...
13) is a replica of the faulty motor model (14
...
Shaft
speed x 1 and stator currents x 2 and x 3 are supposed to be available for measurements, whereas
ˆ
xi (i = 1,
...


283

Fault Detection in Induction Motors

Denote the observation error variables as
ˆ
ei (t) = xi (t) − xi (t)

i = 1,
...
14)

and note that the error variables e1 , e2 , and e3 are accessible for measurements, while the flux
errors e4 and e5 are unknown
...

Assumption 14
...
1 There exist a priori known constant Fs and FL such that, at any t ≥ 0,
the time derivatives of the unknown inputs f sα , f sβ , and TL satisfy the next inequalities
?
?
?
?d
? f s (t)?
? dt i ?

i=α,β

?
?
?d
?
? TL (t)? ≤ FL
...
15)

The observer injection terms are built according to the next relations
νi (t) = νi1 (t) + νi2 (t),

i = 1, 2, 3,

(14
...
17)

where (ki , wi ), i = 1, 2, 3, are proper tuning constants
...

Theorem 14
...
2 Consider the faulty IM model (14
...
3
...
Then, the observer (14
...
16), and (14
...
18)
(14
...
20)
∗

t≥T ,

where ξ1 (t), ξ2 (t), and ξ3 (t) are exponentially vanishing signals
...
(2012)
...
21)
(14
...
21) and (14
...
Signal ν1 represent an asymptotically exact estimation of the unknown load torque
TL , which may be useful in direct torque control (DTC) applications (see La et al
...
The suggested observer also provides an exponentially converging
estimate of the rotor flux components, as well as a consequence of the fact that e4 and e5 are
exponentially vanishing
...
21) and (14
...
The instantaneous power of
the two residuals are summed up to build a scalar measure of fault occurrence
2
2
r (t) = ν2 (t) + ν3 (t)
...
23)

The two considered faults will be referred to as BBF (broken bar fault) and EF (eccentricity
fault), respectively
...
2003):
?

if r (t) ≤ ?,
if r (t) > ?,

then
then

machine is healthy;
BBF or EF is active;

(14
...
However, the above fault detection logic suffers
of the oscillating nature of ν2 (t) and ν3 (t), and it may be highly sensitive to the errors in
the reconstruction of the fault injection signals due to the instantaneous dependence on ν2 (t)
and ν3 (t)
...
Then the next signal is constructed

E(t) =

??

t

r (τ )dτ ,
t−?T

t ≥ ?T,

(14
...


(14
...


14
...
5

Experimental Results

In order to support the treatment presented in the previous sections we are going to present
some experimental result validating the suggested procedure
...
2 Experimental setup (left) and a drilled rotor-cage bar (right)

commercial drives intentionally damaged in order to reproduce a BBF and the two considered
types of eccentricity
...
2 (left plot) depicts the structure of the experimental setup, where a DC motor is
mechanically coupled to the rotating shaft of the IMs under test in order to apply a constant load
torque
...
3 shows a picture of the two motors
...
Several commercial squirrel-cage
IM drives, manufactured by Siemens, in healthy and (intentionally induced) faulty condition
have been tested
...

BBF has been realized by drilling a single rotor bar (see right plot of Figure 14
...
We have tested respectively a
machine with 0
...
07 mm of
dynamic eccentricity
...
3 summarizes the experimental condition tests carried on the load conditions in
terms of slip’s percentage, the measured speeds, and the theoretical sideband frequencies
associated with (see equations (14
...
6))
...
After few
trial and error tests, devoted to guarantee an accurate convergence to zero of the measurable

Figure 14
...
3 Experimental working condition for the broken bar and
eccentricity tests and fundamental fault sideband frequencies
Test

Speed
(rpm)

Slip
(%)

2 Slip f s
(Hz)

fr
(Hz)

f brb
(Hz)

f ecc
(Hz)

Broken bar

1452

3
...
21

–

46
...
21

–

Static eccentricity
Mixed eccentricity

2908
2906

3
...
15

−
–

48
...
43

−
–

98
...
43

estimation errors e1 , e2 , and e3 in healthy operating condition, the observer gains in all tests
have been set as follow
w1 = 14
,
k1 = 80

w2 = 22
,
k2 = 200

w3 = 22
,
k3 = 200

(14
...
The observer scheme is digitally
implemented by means of Euler discretization with sampling time Ts = 10−4 s
...

Hereafter the effectiveness of proposed UIO (14
...
12), will be investigated by experiments with real data
...
4 (left) refers to data from a motor undergoing the BBF, and compares the spectra
of the measured machine α current and that of the corresponding injection signal ν2
...
Figure 14
...
The conclusion is, however, the same as before
...
25) and (14
...
5 shows the E(t) residual profiles obtained using measurements from an healthy and a
faulty motor in the two cases of BBFs and EFs
...
78
Y : 0
...
4 46
...
8 47
Frequency (Hz)

X : 98
...
8

–30

20 40 60 80
Frequency (Hz)

–35
97
...
5 99 99
...
4 Comparison between the normalized spectra of the faulty motor stator currents and that
of the observer injection signal ν2 for the broken bar test (left) and for the eccentricity test (right)

287

Fault Detection in Induction Motors

Comparison between current residual
for healthy and faulty machine

Comparison between current’s residuals
for healthy and faulty machine
600

8000

E healthy test
E faulty test
Fault threshold

400

6000
5000

200
0

Healthy test
Static eccentricity test
Mixed eccentricity test
Fault threshold

7000

4000

0

2

4

6
8
Time (s)

10

12

3000

2

2

4

6
Time (s)

8

10

12

0

2

4

6
Time (s)

8

10

12

2

1
...
5

1

1

0
...
5

0

0

2

4

6
8
Time (s)

10

12

0

Figure 14
...
25) is activated after 9 s, to let the machine reach
the sinusoidal steady state during which the the receding horizon mean (14
...
26)
...
It is apparent that
the faulty conditions are diagnosed almost instantaneously after that the algorithm (14
...
26) is activated
...
25) has been chosen as 0
...
The choice of ?T turned out
to be not critical, provided it is not chosen too short to avoid permanently unsteady profiles of
E(t)
...


14
...
This technology was developed in the 1980s (Kliman et al
...
Initially, the signals chosen for carrying out the diagnosis were mainly vibration-related
mechanical ones, but quickly the interest turned on the stator currents, originating a family
of methodologies designated as MCSA methods, which has been continuously improved
since then
...
The basis of the MCSA approaches based on
FFT is very simple
...
, I N in the discrete time domain,
were N = f s T is the number of samples
...
6 FFT spectrum of a simulated IM in healthy condition (left), with a broken rotor bar (center)
and with a mixed eccentricity fault (right)

which the signal is decomposed into N /2 sinusoidal components with frequencies below half
the sampling frequency and a frequency resolution ? f = 1/T
...
2, different faults originate or substantially increase specific components of the signal,
whose frequencies depend on the speed s, as it is shown in equations (14
...
2), (14
...
4), (14
...
6), (14
...
8), and (14
...
Diagnostic is based on the identification into the
current spectrum of these fault-related components
...
6 shows three
spectra corresponding to a simulated IM under different fault conditions: in healthy state (left),
with a broken bar (center), and with a forced mixed eccentricity (right)
...

MCSA is based on FFT, which is the most used diagnostic approach in the industrial
environment
...
The
energy of the main frequency spreads over the other frequencies and can hide the sideband
components
...
(2004), Douglas et al
...
(2006), and Jung et al
...
Solutions such as the use of a Hanning
window and the Barlett periodogram have been proposed in Didier et al
...
2005)
...
It is inversely proportional to the time of
measurement T
...
Diverse techniques
that can improve frequency resolution without increasing the measurement time have been
presented in (Aiello et al
...
(2005), Bellini et al
...

2007)
...
These
changing frequencies can invalidate an MCSA-based diagnosed process, because they produce a typical smearing effect on the current spectrum (Thomson and Fenger 2001), which
make it difficult to identify the fault components (Ian and Wendell 2007)
...

Some of the proposed solutions to the aforementioned problems are conflicting: long time
periods are needed to achieve sufficient spectral resolution, but load conditions are likely to
vary during this time, thus generating distorted data
...


14
...
5
...
The HT of a real signal x(t), as the phase current, is used to emphasize
its local properties
...

t −τ

(14
...
29)

where
?1/2
?
,
a(t) = x 2 (t) + y 2 (t)

θ (t) = arctan (x(t)/y(t)) ,

(14
...
Three key properties of the HT and the
?
AS that give a more physical insight about it are:
• The HT of a trigonometric function x(t) is a version of itself with a π/2 phase shift: sins
are transformed to cosines and vice versa
...

• The AS has a one-sided FT, that is, its components at negative frequencies are 0
...

• All the low frequencies of the original signal are in the amplitude , and the high frequencies
in the phase of the AS
...
(2004) and Puche-Panadero et al
...

Analytic Signal of the Current in a Healthy Machine
The steady-state phase current in an ideal machine, running at constant speed, is purely
sinusoidal:
?
? j ωt
e + e− jωt

...
31)
i(t) = Im cos(ωt) = Im
2
The FT of i(t) shows two distinct components, at frequencies f = ω/2π and f = −ω/2π ,
with an amplitude of Im /2
...

i

(14
...
32) will show a single spike corresponding to the positive frequency
f = ω/2π , with an amplitude double respect to the original signal spectrum component
(14
...
The modulus of the AS reflects how the energy of i(t) varies with time, and contains
its low-frequency components
...
32), it has a constant value
of Im , indicating that the energy of the phase current does not vary with time
...
33)

where β denotes the modulation depth (modulation index) and ω0 = 2π f 0
...
31) in (14
...


(14
...
35)

which shows the presence of the two sideband frequencies characteristic of the fault
...
36)

291

Fault Detection in Induction Motors

which can be expressed as
H T (i b (t)) = Im sin(ωt) [1 + β cos(ω0 t)]
...
37)

The AS is constructed by combining (14
...
37) (imaginary part):
?b (t) = [1 + β cos(ω0 t)] Im (cos(ωt) + j sin(ωt)) = [1 + β cos(ω0 t)] Im e jωt
...
38)
i
In this case, and in contrast to the case of a healthy machine (14
...


14
...
2

Experimental Results

The proposed method has been applied to the analysis of a commercial 1
...
Tests
were carried out in two different conditions: healthy state and faulty condition in which a
single bar was broken by drilling
...
Additional test were performed with the faulty machine under three different loads
(low, medium, and full) to assess the performance of the proposed method over the full range
of motor load
...
Table 14
...

Two types of spectra are presented for the sake of comparison, the FFT of the phase current
(classical method), and the FFT of IHilbert , the alternating component of the modulus of IHilbert
...
7 (left) presents the spectrum of the line current
...
16%) and medium load (s = 2
...
In the low-load test (s = 0
...
4

Experimental tests of the healthy and faulty motor and theoretical fault sidebands

Motor

Load

Speed
(rpm)

Slip
(%)

2s f 1
(Hz)

Healthy

a) No

1497
...
19

0
...
8

0
...
20

c) Low

1492
...
61

0
...
2

2
...
96

e) Full

1407
...
16

6
...
86
50
...
44
50
...
04
52
...
84
56
...
44 Hz
42

44

46

50
...
99 Hz
48 50 52 54
Frequency (Hz)

56

58

60

43
...
12
Hz
44

46

48 50 52 54
Frequency (Hz)

56

58

60

0

2

× 10–3

4
6
Frequency (Hz)

8

10

4
6
Frequency (Hz)

8

10

4
6
Frequency (Hz)

8

10

8

10

2
1
0

0
...
62 Hz

1
0

2

× 10–3

2

0

2

× 10–3

2
...
07 Hz

–50

IHilbert(pu)

IPhase (dB)

0
(d)

1

3
IHilbert(pu)

IPhase (dB)

0
(c)

2

3
IHilbert(pu)

IPhase (dB)

0
(b)

× 10–3

0

2

× 10–3

4
6
Frequency (Hz)

2

6
...
7 Spectrum of the phase current i(t) (left) and spectrum of the modulus of IHilbert (right)
in five experimental tests
...


components are hardly detected, and in the case of absolute no-load (s = 0
...
In this last case the
classical MCSA method is unable to correctly diagnose the fault
...
Its spectrums, in the
five cases presented in Table 14
...
7 (right)
...


14
...
6
...
, s N ) onto n + 1
wavelet signals: an approximation signal an (t) and n detail signals d j (t) with j varying from
1 to n:
s(t) = an + dn + · · · + d1
...
39)

293

Fault Detection in Induction Motors

The parameter n is an integer known as number of decomposition levels and can be set
freely
...
Conceptually, the detail signal d1 is calculated as

d1 (t) =

?
i

βi1 · ψi1 (t),

(14
...
The detail signal d j is
calculated in a similar way, but using as a base the wavelet with level j, which is a scaled and
time-expanded version of the mother wavelet

d j (t) =

?
i

j

j

βi · ψi (t)
...
41)

The approximation signal an is obtained similarly, but using the scaling function φ n and
j
scaling coefficients α n , instead of the wavelet function and coefficients
j
an (t) =

?
i

α n · φ n (t)
...
42)

Each mother wavelet is associated with a family of scaling functions, which are perfectly
determined once the mother wavelet is selected
...
1998) or sub-band coding algorithm; the
approximation signal behaves as a low-pass filter, whereas each wavelet signal behaves as a
passband filter, extracting the time evolution of the components of the original signal included
within its corresponding frequency band
...
8 shows the sub-band coding algorithm
regarding the coefficients of the transform at the different levels according to the description
by Polikar et al
...
In this figure the length of those coefficients and frequency content at
each level is shown considering an original signal with n = 1024 samples and sampling rate
of f s samples/s
...


Figure 14
...
5

Frequency bands for the wavelet signals

Level

Frequency band
f s = 5000 samp/s

Frequency band
f s = 2000 samp/s

d1
d2
d3
d4
d5
d6
a6

1250–2500 Hz
625–1250 Hz
312
...
25–312
...
12–156
...
06–78
...
6 Hz

500–1000 Hz
250–500 Hz
125–250 Hz
62
...
25–62
...
625–31
...
625 Hz

Source: Riera et al
...


The main idea that underlies the application of the DWT is the dyadic band-pass filtering
process carried out by this transformation
...
5)
...
More concretely,
if f s (samp/s) is the sampling rate used for capturing s(t), then the detail d j contains the
information concerning the signal components with frequencies included in the interval:
?
?
f (d j ) ∈ 2−( j+1) f s , 2− j f s Hz

(14
...
44)

For instance, Table 14
...
2008b) gives the frequency bands obtained
for two different sampling frequencies ( f s = 5000 and 2000 samp/s) and n = 6 levels of
decomposition
...
9
...
2006)
...
9 Filtering process performed by the DWT

295

Fault Detection in Induction Motors

the signal is close to the limit of a band
...
g
...

Due to the automatic filtering performed by the wavelet transform, the tool provides a
very attractive flexibility for the simultaneous analysis of the transient evolution of rather
different frequency components present in the same signal
...
In addition, the DWT is available in
standard commercial software packages, so no special or complex algorithm is required for
its application
...
6
...
10 shows the sampled start-up current (signal s, at the top) and the signals resulting
from the DWT with n = 6 (a6 , d6 ,
...
This current was sampled at f s = 5000 samp/s
...
This is because, for the
sampling frequency used, the frequency band corresponding to this signal is 39
...
12
Hz (see Table 14
...

• The approximation a6 does not show any relevant pattern, once the initial oscillations due
to electromagnetic transient and border effects are extinguished
...
06 Hz) within the signal
...
At t ≈ 1
...
012 Hz and the component
penetrates within the detail d5
...
2 s, crossing successively the frequency bands of d4 ([156
...
5
Hz), and d3 (312
...
The pattern described above fits well the evolution during the start-up of the
current principal slot harmonic (IPSH) of the machine, the frequency of which, as function
of slip (Nandi et al
...
45)

At the beginning of start-up s ≈ 1 and f PSH ≈ 50 Hz
...
Fourier analysis of the
stationary portion of d2 confirms the previous interpretation, showing a predominant component of 640 Hz
...

• The detail d1 includes the high-frequency components of the signal with frequencies in the
interval (1250–2500 Hz); nothing relevant is observed in this detail signal
...
3
a6 0
–0
...
3
d5 0
–0
...
3
d4 0
–0
...
3
d3 0
–0
...
3
d2 0
–0
...
3
d1 0
–0
...
3
0–39
...
3
10
39
...
12 Hz d
0
6
–10
0
...
12–156
...
3
0
...
3–312
...
3
0
...
5–625 Hz
d3 0
–0
...
3
625–1250 Hz
d2 0
–0
...
3
1250–2500 Hz
d1 0
–0
...


9

10

1250–2500 Hz

625–1250 Hz

312
...
3–312
...
12–156
...
06–78
...
06 Hz

Figure 14
...
Diagnostic based on left sideband
extraction

s

Decomposition at level 6 :
s = a6 + d 6 + d 5 + d 4 + d 3 + d 2 + d 1
...
2

a9

0

d9

–0
...
5

1
Time (s)

1
...
02
0
–0
...
08
0
–0
...
2
0
–0
...
5
0
–0
...


0–4
...
88–9
...
76–19
...
53–39
...
06–78
...
11 Theoretical evolution of the LSH component of the start-up current of a cage motor with
one broken bar (left) and DWT of the start-up current of a machine with a broken bar (right), showing
the characteristic pattern of low-frequency wavelets signals

14
...
3 Application of the DWT to the Analysis of the Start-up Current of a
Motor with a Broken Bar in the Rotor
The previous test was repeated, but using a machine in which a rotor bar was artificially broken
...
10 (left) shows the DWT of the start-up current for this case
...
10 (left) and 14
...
(2008a), is caused by the left sideband component; its amplitude increases
substantially when a rotor asymmetry is present and its frequency evolves during almost the
whole start-up within the frequency band of a6 ; the similitude between the waveform of the
approximation a6 and the theoretical evolution of the left sideband component during the
start-up, deduced in Riera-Guasp et al
...
11 (left), should
be highlighted
...
An alternative way for detecting a rotor asymmetry is shown in Figure
14
...
In this way, the
evolution of the sideband along the start-up is spread across four consecutive wavelet signals
(d7 , d8 , d9 , and a9 ), with frequency bands covering from near the main frequency to zero Hz
...
This also constitutes a reliable signature
for the left sideband identification
...
6
...
This was achieved by sanding down the inner and

298

AC Electric Motors Control

0
...
05

0

0–34
...
05
10

d5

34
...
4 Hz

0
–10
0
...
7 Hz

–0
...
7–69
...
05

0

–0
...
4–138
...
05

69
...
9 Hz
0
...
5

2

3
2
...
5

4

4
...
5

Figure 14
...
Conventional Fourier analysis shows steady-state mixed eccentricity-related components
f 1 ± fr increased by a factor of 3 after this process, growing their amplitude from 0
...
6%
...
12 shows respectively the DWT of the start-up current before (left) and after modifying the bearings (right); only the significant wavelet signals
for this diagnosis are shown, that is, the signals containing the related eccentricity components
( f 1 ± fr ) during the start-up
...
These changes are produced by the components f 1 ± fr , which eccentricity introduces into the current of the faulty machine, that are included within the detail d5
at the beginning of start-up; as the rotor speed increases their frequencies change, reaching the
limits of the band of d5 when the start-up is almost finished
...

Subsequently, the diagnosis based on the DWT not only detects the fault through the increase
of the amplitudes of the signals, like the steady-state-based analyses, but also through the
characteristic pattern due to the progressive increment in this amplitudes during the transient
...


14
...
7
...
11 (left), or the EF pattern (see Figure 14
...
A solution
to this problem is to assemble a unique matrix, with each row containing the time evolution of
the signal in a single frequency band
...
13 for the case of the simulated LSHst presented in Figure 14
...

The image displayed in Figure 14
...
8125
0
...
625

0
...
14

31
...
12
0
...
5

0
...
06
0
...
02
0

0
...
4

0
...
8

1
1
...
4

1
...
8

2

0

Figure 14
...
The frequency pattern is superimposed in dashed line

amplitude variation
...
All these drawbacks can be
overcome with the use of the CWT
...
The most natural way to obtain this representation (Cohen
1989) is to define a family of scaled and translated functions
1
ψa,b (t) = √ ψ
a

?

?
t −b
,
a

a > 0, b ∈ R,

(14
...
The CWT of a function f ∈ L 2 (R) is defined as
?
?
1
CWT f (a, b) = f, ψa,b = √
a

?

f (t) ψ

?

?
t −b
dt
...
47)

From a practical point of view, the CWT can only be computed on a discrete grid of points
(an , bn )n∈Z
...
N − 1, where f s is the sampling frequency and N is the total number of sampled values
...
47) becomes
?
?
N −1 ?
1 ? k+1
t − n?t
i(k?t) · ψ
dt ,
CWT i (m, n) = √
m
m k=0 k

with

?

m = 1
...
N − 1
...
48)

300

AC Electric Motors Control

The analyzing mother wavelet used in this paper is the nth derivative of a Gaussian (DOG)
function, given by (14
...
A value n = 8 has been selected in this work (gaus8 wavelet): it is
a symmetric wavelet, it is infinitely differentiable in the time domain, and has eight vanishing
moments, the minimum value proposed by Douglas et al
...

ψn (t) = ?

(−1)n

2

d n (e−t )
dt n
2(n−1/2) ·?(n + 1/2)

(14
...
14 (left), which offers a clear visual insight of the timefrequency evolution of the LSHst
...
The use of the complex CWT (CCWT) solves
this problem, as it is shown in the next section
...
7
...
49) but using complex valued wavelets, instead of
real ones, so that the modulus of the coefficients can be computed:
ψn (t) = ?

(−1)n

2

d n (e− jt · e−t )

...
50)

The CCWT has been applied to the LSHst, and the absolute value of the resultant coefficients
is shown in Figure 14
...
The advantage of using the CCWT to represent the timefrequency evolution of the LSHst is that the modulus of the CCWT tracks the evolution of its
envelope, instead of its instantaneous value
...
14 (right), which facilitates the work of automatic recognition systems
...
7
...
15)
...
8 Wigner-Ville Distribution Approach
14
...
1 Basis for the Application of the WVD to Diagnostic
of Electrical Machines
The distribution of the energy of a signal x(t) over the two description variables time and
frequency, that is, its energy density function, Px (t, ν), can be obtained with the use of a

0

0
...
750)

600

Analyzed signal (length = 2000)

1600

1600

1800

1800

2000

2000

–0
...
1

61
...
6
55
...
2
49
45
42
...
4
36
...
8
26
...
4
20
...
8
10
...
4
4
...
1

0

0
...
14 CWT (left) and complex CWT (right) of the LSH component of the start-up current of a cage motor with one broken bar
...
1

0

0
...
1

Amplitude (A)

Scale

Amplitude (A)

Amplitude (A)
Scale
Amplitude (A)

302

AC Electric Motors Control

16

Frequency (Hz)

17
20
23
28
35
46
69
137
0
...
5

2

Time (s)

Figure 14
...
The broken bar faults pattern has been superimposed in dashed line

time-frequency distribution (Hlawatsch and Boudreaux-Bartels 1992), as proposed recently
for motor fault detection in Bl¨ dt et al
...
(2008)
...
The class of energy time-frequency distributions
verifying this property, the Cohens class, has the following general expression (Cohen 1989):
Px (t, ν) =

+∞
?
? ?
τ ? − j 2π ντ
τ? ?
e j2π ξ (s−t) f (ξ, τ )x ∗ s −
dξ dsdτ
...
51)
x s+
e
2
2
−∞

Individual distributions (Spectrogram, Wigner-Ville, Choi-Williams, etc
...
51)
...

+∞
?
?
τ ? − j2π ντ
τ? ?
x∗ t −
dτ
...
52)

−∞

This expression shows that the WVD of a function is obtained as the FFT with respect the
variable τ (delay) of the convolution of the signal with a translations in time and frequency of
itself
...
8
...
16 shows the result of applying the WVD to a monocomponent signal such as
the simulated LSH of Figure 14
...
In this spectrogram a clear “V” pattern is shown
...
2

0
...
6

0
...
2
Time (s)

1
...
6

1
...
16 Wigner-Ville distribution of the LSH during a start-up

interpretation of this pattern is that the frequency of the analyzed signal first decreases with
time, from 50 to 0 Hz (at t ≈ 0
...
85 s)
...
This behavior
matches perfectly the evolution of the frequency and amplitude of the LSH during a start-up
(see Figure 14
...
(2008b); therefore, the detection of this “V”
pattern into a start-up current enables diagnostic of a rotor breakage
...
16
...
In real multicomponent signals, these artifacts
can completely hide the searched patterns
...


14
...
3

Application of the WVD to Multicomponent Signals

Another alternative for minimizing the undesirable effects of the cross-terms consists of
applying some kind of pretreatment to the signal before computing the WVD, in order to isolate
the component of interest into a frequency band in which it is predominant
...
This technique is applied to analyze the startup current of Figure 14
...
The WVD is applied to the approximation a6 , which
contains the components of the current belonging the interval [0, 39
...
The resulting

304

AC Electric Motors Control

Real Ecc 37%, log
...
1%

0
...
25

40
Frequency (Hz)

50

30
20
10
0

EC-175
EC-150

0
...
1

EC-100
EC-75

0
...
17 Wigner-Ville distribution of the start-up current of a machine with a broken bar (left) and
of a machine with 37% eccentricity (right)

energy distribution is shown in Figure 14
...

A more general and systematic methodology for the pretreatment of the signal before
applying the WVD is proposed in Climente-Alarcon et al
...
Then the HT is applied to the filtered signal to obtain an AS, whose
spectrum does not contain negative frequencies
...
In this way, the WVD is applied to a signal
in which the more important components not related with the fault have been suppressed, and
thus, the cross-terms are strongly reduced; as a result, the computed energy distribution enables
appreciation of simultaneously several fault components evolving through a wide frequency
band
...
17 (right) (Climente-Alarcon et al
...

The spectrogram shows the evolution of the couple of main eccentricity components ( f 1 ± fr )
designed as EC-25 clearly and EC-75 in Figure 14
...
The trajectory in the t– f plane
of these components can be followed during the full start-up transient, reproducing the same
conceptual pattern appearing in Figure 14
...

Also, the evolution of other second order eccentricity-related components (designed as EC100, EC-125, EC-50, EC-175 in Figure 14
...


14
...
9
...


305

Fault Detection in Induction Motors

For instance, and considering only the most important components, it has been demonstrated
that:
• Rotor asymmetry produces a substantial increase in the amplitude of the lower and upper
sideband component of stator currents, whose frequencies are given by
f bS (s) = |(1 ± 2 · s) f 1 | ,

(14
...

• A mixed eccentricity induces stator current harmonics at specific frequencies
...
6), at the frequency given by
?
?
?
?
? f 1 − f 1 (1 − s)? ,
f ecc (s) = ?
?
p

(14
...

Conventional MCSA uses equations (14
...
54) as formulas that give the frequencies
at the current spectrum where the fault components appear when a fault happens
...
It is remarkable that these trajectories are straight
lines, with a specific slope and offset, different for every fault, irrespective of the way in which
speed varies (increasing, decreasing, oscillating) and the machine characteristics (rated power,
rated voltage), thus, constituting very reliable patterns for diagnostic purposes
...
18 gives the theoretical evolution of the IF of the fault components
related to a rotor asymmetry (left) and a mixed eccentricity (right)
...
53) and (14
...


14
...
2

Calculating the IF of a Monocomponent Signal

The IF of a monocomponent signal sm (t) can be calculated using different mathematical tools,
as for instance the HT and the WVD, among other
...


1 d (arg(x a ))
1 dφ
=
,
2π
dt
2π dt

(14
...
2

0
...
6

0
...
2

Slip

0
...
6

0
...
18 Characteristic slip-frequency pattern of the IF of the LSH of a machine with broken bars
(left) and with eccentricity (right) during a start-up transient

• From the WVD, the IF of sm (t) can also be calculated as the first conditional moment of
frequency for a given time of the WVD of the signal; that is, as the average of the frequencies
existing in the time-frequency plane for a given time (Cohen 1989):
?
ν Px (t, ν)dν
1 dφa
= νi (t)
...
56)

It can be demonstrated that both definitions are equivalent
...
9
...
This problem
has been reported extensively in the technical literature: the IF of a multicomponent signal can
exhibit large fluctuations, and it can extend beyond the band defined by any of the individual
components (Nho and Loughlin 1999)
...
Figure 14
...
10 (right)
...
Also, in this case, the DWT has been used for isolating the LSH; actually this
graph shows the calculated IF (blue dots) of the approximation a6 of the tested current, since
this wavelet signal contains the components of the start-up current below the fundamental
frequency; therefore, if the LSH is present in the start-up current, it will be prominent in this
approximation
...


307

Fault Detection in Induction Motors

IF as the derivative of the phase of the AS

Frequency (Hz)

50
40
30
20
10
0
1

0
...
6
0
...
2

0

Figure 14
...
IEEE Transactions on Instrumentation and Measurement, 54 (5),
1811–1819
...
IEEE Transactions on Industry
Applications, 42 (4), 990–996
...
(2009) Transient detection of eccentricity-related components
in induction motors through the hilbert-huang transform
...

Bellini A, Filippetti F, Franceschini G, et al
...
IEEE Transactions on Industry Applications, 37 (5), 1248–1255
...
IEEE Transactions on Industry Applications, 42 (1), 69–78
...
W Zhang), InTech, pp
...

Burrus C, Gopinath R, and Guo H (1998) Introduction to wavelets and wavelet transforms: a primer
...

Cabanas M and Melero M (1998) T´ cnicas para el mantenimiento y diagn´ stico de m´ quinas el´ ctricas rotativas
...

Cizek V (1970) Discrete Hilbert transform
...

Climente-Alarcon V, Antonino-Daviu J, Riera-Guasp M, et al
...
Mechanical Systems and Signal Processing,
25 (2), 667–679
...
Proceedings of the IEEE, 77 (7), 941–981
...
IEEE Transactions on Industry Applications, 42 (1), 79–88
...
IEEE Transactions on Energy Conversion, 20 (1), 135–141
...
Industry Applications Magazine,
IEEE, 10 (5), 12–21
...
Automatica,
36 (4), 541–553
...

IEEE Transactions on Industrial Electronics, 47 (5), 984–993
...
IEEE Transactions on Industrial Electronics, 51 (1), 107–121
...
International Journal of control, 77 (7), 622–629
...
46th IEEE
Conference on Decision and Control, 2007, pp
...

Hlawatsch F and Boudreaux-Bartels G (1992) Linear and quadratic time-frequency signal representations
...

Ian MC and Wendell R (2007) Using current signature analysis technology to reliably detect cage winding defects in
Transaction on Industry Application squirrel-cage induction motors
...

Jung JH, Lee JJ, and Kwon BH (2006) Online diagnosis of induction motors using MCSA
...

Kia S, Henao H, and Capolino GA (2007) A high-resolution frequency estimation method for three-phase induction
machine fault detection
...

Kliman G, Koegl R, Stein J, et al
...

IEEE Transactions on Energy Conversion, 3 (4), 873–879
...
IEEE Transactions on Industry Applications, 40 (4), 1101–1106
...
Power Apparatus and Systems, IEEE
Transactions on, 84 (11), 1038–1053
...
IECON 2000
...
2, pp
...

Liu Z, Zhang X, Yin X, and Zhang Z (2004) Rotor cage fault diagnosis in induction motors based on spectral analysis
of current hilbert modulus
...
IEEE, pp
...

Marino R, Peresada S, and Valigi P (1993) Adaptive input-output linearizing control of induction motors
...

Nandi S, Ahmed S, and Toliyat H (2001) Detection of rotor slot and other eccentricity related harmonics in a three
phase induction motor with different rotor cages
...

Nandi S, Toliyat H, and Li X (2005) Condition monitoring and fault diagnosis of electrical motors—a review
...

Nho W, and Loughlin P (1999) When is instantaneous frequency the average frequency at each time? Signal Processing
Letters, IEEE, 6 (4), 78–80
...

12th International Workshop on Variable Structure Systems (VSS), pp
...

Pisano A, Davila A, Fridman L, and Usai E (2008) Cascade control of PM DC drives via second-order sliding-mode
technique
...

Polikar R, Udpa L, Udpa S, and Taylor T (1998) Frequency invariant classification of ultrasonic weld inspection
signals
...

Puche-Panadero R, Pineda-Sanchez M, Riera-Guasp M, et al
...
IEEE Transactions on Energy
Conversion, 24 (1), 52–59
...
(2008) Nonstationary motor fault detection using recent quadratic time
frequency representations
...

Riera-Guasp M, Antonino-Daviu JA, Pineda-Sanchez M, et al
...
IEEE Transactions on Industrial
Electronics, 55 (12), 4167–4180
...
IEEE Transactions on Industry Applications,
44 (3), 716–726
...
Recherche, 67, 02
...
Electric Power Systems Research
...
Industry Applications
Magazine, IEEE, 7 (4), 26–34
...
Cables et transmission, 2 (1), 61–74
...
(2005) Improvement of frequency resolution for three-phase induction
machine fault diagnosis
...
20–25
...

International Aegean Conference on Electrical Machines and Power Electronics, 2007
...
403–408
...
Edited by Fouad Giri
...
Published 2013 by John Wiley & Sons, Ltd
...
1

Introduction

Sensorless control of electrical machines is a topic that imposes the challenging problem of
eliminating the use of sensors for mechanical variables (position and speed) for controller
design purposes (Rajashekara et al
...
Its solution is both important from the applications
perspective (due to its economic impact) and quite attractive from the control theory approach
(for the mathematical complexity that it exhibits)
...
1998; Ortega et al
...

2003; Nam 2010), the sensorless control problem is currently recognized as a longstanding
essentially open problem
...
For solving it, three variables must be estimated out of the
measurement of the electrical coordinates: (1) rotor position, (2) rotor speed, and (3) load
torque—the latter assumed constant
...
g
...
(2006) and Fabio et al
...

Many results are also available for the (practically unrealistic) cases of known initial position
(Tomei and Verrelli 2008; Ezzat et al
...
2010a), or
the (theoretically unjustifiable) assumption of bounded trajectories (Ezzat et al
...
An
approximate stability analysis of the scheme proposed in Matsui (1996) is carried out in Nahid
et al
...
In Marino et al
...
Edited by Fouad Giri
...
Published 2013 by John Wiley & Sons, Ltd
...
A key difference of the latter machine from the PMSM is
the availability of flux measurements that considerably simplifies the observation problem
...
2008, Ezzat
et al
...
(2011)
...

2006), hence it is adopted in this paper
...
(2011) is considered,
which has been successfully evaluated in an experimental setting, combining it with an ad hoc
linear speed estimator and a standard field-oriented controller (Lee et al
...

The main objective of this paper is to prove that direct application of two well-established
design methodologies—immersion and invariance (I&I) (Astolfi et al
...

(2011) to design an asymptotically stable sensorless controller
...
(2011) and Shah et al
...
(2007) and Petrovic et al
...
To the best of our knowledge, this is the first time
a complete theoretical analysis of a sensorless controller is done—under reasonable practical
and theoretical assumptions
...
The models of the PMSM and the
problem formulation are given in Section 15
...
The controller structure and the main result
are presented in Section 15
...
In Section 15
...
A full information IDA-PBC is given in Section 15
...
Section 15
...
(2011), while in Section 15
...
The proof of the main result is given in Section 15
...
Some
simulation and experimental results are given in Section 15
...
(2010)
...
10
...
2

PMSM Models and Problem Formulation

The classical fixed-frame αβ model of the unsaturated nonsalient PMSM is given by Chiasson
(2005) and Krause (1986):
?
?
di αβ
cos(θ )
+ vαβ ,
= −Ri αβ − n p ω?J
L
sin(θ )
dt
?
?
cos(θ )
?
− τL ,
J ω = n p ?i αβ J
˙
sin(θ )
˙
θ = n p ω,

(15
...

To design the observer it is convenient to embed the dynamics (15
...
2)

?
J ω = n p ?i αβ J ραβ − τ L ,
˙

(15
...
4)

ραβ = n p ωJ ραβ ,
˙
where the vector
ραβ :=

?

ρα
ρβ

?

=

?

cos(θ )
sin(θ )

?

(15
...
Notice that, if ραβ is known, θ can be easily reconstructed inverting the trigonometric
functions
...
1) can be written in rotating dq-coordinates by means of the transformation

eJ θ =

?

cos(θ )
sin(θ )

− sin(θ )
cos(θ )

?

= ρα I 2 + ρ β J ,

(15
...
7)

where the rotated signals

i=
are defined
...
8)

316

AC Electric Motors Control

Remark 15
...
1 The main advantage of the dq model is that it transforms the periodic orbits
associated to the constant speed operation of the αβ model of the PMSM into equilibrium
points
...
5
...

Remark 15
...
2 The industry standard field oriented control (Nam 2010) is designed for this
model, hence the need to reconstruct θ
...


15
...
1

Problem Formulation

The main contribution of the paper is the solution of the following Sensorless control problem
...
2), (15
...
4) with some desired constant speed ω∗ ?= 0,
under the following conditions:
A1 The only variables available for measurement are i αβ
...

A3 The parameters R, L , ? and J are known
...

Remark 15
...
3 Even though we have restricted ourselves to the case of constant desired
speed and constant load torque, it is clear that the controller, being exponentially stable hence
robust, will be able to track (slowly) time-varying references and reject changes in the load
torque
...
9 show that the
proposed controller yields a good performance even in the face of fast changes in the speed
reference and the load torque
...
g
...
2008; Ezzat et al
...
2011), that the rank condition for observability is violated when the motor is at standstill
...


15
...
The proposed controller is a fourth order certainty equivalent
version of a full-information globally asymptotically stabilizing controller, which is a static
state-feedback IDA-PBC of the form vαβ = q(ραβ , ω, τ L , i αβ )
...

The dynamics of the controller is, then, due to the I&I observer, which generates the estimates
ˆ
ˆ
that we denote ραβ , ω, and τ L , respectively
...
7) yields a seventh order closed loop system
...
To simplify the notation, all these errors are
lumped into a seventh dimensional vector denoted χ , and defined as1
⎡

⎤
χ1
⎤
⎡
⎢ χ2 ⎥
L(i − i ∗ )
⎢ ⎥
⎢ χ3 ⎥
⎢
J (ω − ω∗ ) ? ⎥
⎥
⎢ ⎥
⎢
?
ˆ
χ = ⎢ χ4 ⎥ := ⎢ e−J θ ραβ − ραβ ⎥
...
9)

Notice that the errors in both, the currents and the vector ραβ , are defined in the dq coordinates
...
8
...

Proposition 15
...
1

There exists a fourth order observer-based speed regulator of the form

⎡

˙
ψ = g(ψ, i αβ , vαβ ),
⎤

ραβ
ˆ
⎥
⎢
⎣ ω ⎦ = h(ψ, i αβ , vαβ ),
ˆ
τL
ˆ

ˆ
ˆ ˆ
vαβ = q(ραβ , ω, τ L , i αβ ),

where ψ ∈ R4 and g, h, q are suitably defined functions that solve the sensorless control
problem2
...
10)

with zero (locally) exponentially stable equilibrium
...


1 The constants L and J are introduced because—consistent with the Hamiltonian formulation—the IDA-PBC is
derived with the motor dynamics represented using the energy variables, flux, and momenta
...
18), the position observer (15
...
25), and the speed-load
torque observer (15
...


318

15
...
This question is particularly
relevant in our case since, as explained below, the stability analysis of the proposed controller
relies on the linearization of the closed loop
...
7), with measurable
output signals the currents i αβ
...
Now, as the measurable signal is i αβ , invoking (15
...


The linearization of (15
...


⎡

1
I
L 2

⎤

⎢
⎥
B := ⎣ 0 ⎦ ,
0

Although, apparently, this is an innocuous linear time-varying system for which an observerbased controller could be designed, there are several aspects that stymies this task
...

On the other hand, the position θ ∗ (t) is also unknown, due to its dependence on θ (0)—see
the remark below
...
On top of that, the
∗
“output” δ y is known up to the bias term eJ θ (t) i ∗
...


319

Sensorless Speed Control of PMSM

Remark 15
...
1 It is sometimes argued that the motor operation often starts at a known
rotor position, hence θ ∗ (t) can be computed
...


15
...
(2007) and Petrovic et al
...
This scheme
serves as a basis for our certainty equivalent design
...
5
...
2002; Ortega and Garcia–Canseco 2004)
it is convenient to write the system dynamics in port-Hamiltonian form (van der Schaft 2000),
thus we define the state vector as
x=

?

x 12
x3

?

=

?

?
Li
,
Jω

(15
...
12)


...
7) can be written in the form
˙
x = F(x)∇ H (x) +

?

v
−τ L

?

(15
...


˙
Notice that H = −R|i|2 + v ? i − τ L ω, which is the power balance equation for the motor
...
13) is given by
?

∗
x ∗ ∈ R3 | x 2 =

?
L
τL ,
n p?

320

AC Electric Motors Control

∗
∗
with arbitrary x 1 and x 3
...
See the remark below
...

This is achieved by modifying the interconnection and damping matrices, endowing the closed
loop with the port-Hamiltonian form

˙
x = Fd (x)∇ Hd (x),

(15
...
This ensures stability of the equilibrium x ∗ with Lyapunov function
Hd (x)
...
g
...
8, van der Schaft 2000),
the equilibrium is shown to be asymptotically stable
...
5
...
5
...
13) with a desired equilibrium point
⎡

⎤

0

⎥
⎢ L
x ∗ = ⎢ n p ? τL ⎥
...
15)

J ω∗

The full-information control

v
where d :=
stable
...
16)

with r > 0 a damping injection term, renders x ∗ globally asymptotically

Define the desired closed-loop energy function as the quadratic in the errors form
Hd (χ13 ) =

1 ?
χ Qχ13 ,
2 13

with
χ13 =

?

χ12
χ3

?

=

?

∗
x 12 − x 12
∗
x3 − x3

?

,

where Q is as in (15
...

In order to achieve the required matching between the right-hand sides of equations (15
...
14), it is considered that matrix Fd (x) is partitioned in an appropriate way, with

321

Sensorless Speed Control of PMSM

∗
∗
elements given by Fi j (x), that x 1 = 0, and the definition of x 2 in (15
...
Thus, the third row
of this marching equation, which actually states the only constraint to be solved since the first
and second components can be easily satisfied with a suitable selection of the control inputs
v1 and v2 , can be written in an equivalent way as

n p?
1
1
1
χ2 = F31 χ1 + F32 χ2 + F33 χ3
...


The nonpositivity condition on the symmetric part of Fd (x) suggests to define F23 = −F32 =
−n p ?
...
A solution to this equation is obtained by selecting
Ln
F21 = − J p x 3 , F22 = −r and
v2 = d x 2 +

r ∗ n p? ∗
x +
x ,
L 2
J 3

∗
∗
which, upon replacement of the definitions of x 2 and x 3 , yields the expression given in the
proposition
...

J 2

Finally, the closed-loop system takes the desired port-Hamiltonian form (15
...
Asymptotic stability follows, verifying that
r
˙
Hd = − 2 |χ12 |2
L
and that |χ12 |2 is a detectable output for the closed-loop system (15
...


(15
...
5
...
16) with
F
vαβI = eJ θ v F I = (ρα I2 + ρβ J )v F I
...

r
ˆ
n p ?ω∗ + n p ? τ L

(15
...

ˆ

15
...
(2011)

In this section, the observer presented in Ortega et al
...
Also, an alternative representation of the
observer, which is instrumental for the speed-load torque observer given in the next section,
is presented
...
To facilitate the
reference to Ortega et al
...
In particular, we
˜
ˆ
ˆ
define the observation error λ := λ − λ, with λ the stator flux and λ its estimate
...
6
...


(15
...
2) can be equivalently written as
˙
λ = −Ri αβ + vαβ
...
20)

This representation of the electrical dynamics of the PMSM is used in Ortega et al
...
To explain this observer, we make the important observations that
˙
λ is measurable, and that the vector function
η(λ) := λ − Li αβ

(15
...


(15
...
23)

In Ortega et al
...
It is also proven
˜
that the dynamics of the observation error λ is described by the second order nonautonomous
equation
˙
˜
˜
˜
˜
λ = −γ [|λ|2 + 2?λ? ραβ (t)][λ + ?ραβ (t)],

(15
...
This means, that all trajectories of (15
...

P2: Exponential stability under persistent excitation: The zero equilibrium of
equation (15
...


P3: Constant non-zero speed: If the speed is constant and satisfies
|ω| >

1
γ ?2 ,
4

then the origin is the unique equilibrium of (15
...


15
...
2

Description of the Observer in Terms of ραβ

Instrumental for the development of the position and load-torque observer, as well as for the
analysis of the closed-loop system, is the representation of the previous flux observer, and its
estimation error, in terms of ραβ
...


324

AC Electric Motors Control

Proposition 15
...
1

From equation (15
...
23) define the estimate
ραβ =
ˆ

?
1 ?ˆ
λ − Li αβ
?

(15
...
The observer (15
...
26)

while the estimation error ραβ satisfies
˜

?
??
?
˙
˜
˜
˜?
ρ αβ = −γ ?2 |ραβ |2 + 2ραβ ραβ ραβ + ραβ
...
27)

˜
First, notice that λ = ?ραβ , which replaced in equation (15
...
27)
...
27) yields (15
...

ˆ

15
...
2007)
...
3), as well as the representation of the flux observer (15
...
26), are used
...

˜
S3: A globally exponentially convergent I&I observer of ω and τ L is designed
neglecting the perturbation in the system4
...
3) and the position observer (15
...

ˆ
ˆ
ˆ
ˆ
˜

(15
...
8
...
29)

where, for completeness, the last (trivial) equation has been added
...
7
...
29) and the speed and load torque observer

˙
ξ = A33 ξ +

? a2

2
− n p a1

?

J
n p a1 a2
?
? ? ?
? ?
a1
ω
ˆ
ρβ
ˆ
=ξ+
A
,
−a2
τL
ˆ
ρα
ˆ

?

ρβ
ˆ
A
ρα
ˆ

?

⎡

⎤
n p? ?
ˆ
i αβ J ραβ ⎦
+⎣ J
,
0
(15
...


(15
...

lim e ?
t→∞
τ L (t) − τ L ?
ˆ
αt

(15
...
30) is a globally exponentially convergent speed and load torque observer
for the unperturbed system (15
...

Proof: Following the I&I procedure (Astolfi et al
...
As is well known, to achieve the latter objective a partial differential equation (PDE)
should be solved
...
29), we propose the manifold
M := {(ξ, ω, ραβ ) : ξ −
ˆ

?

ω
τL

?

+ ζ (ραβ ) = 0} ⊂ R5 ,
ˆ

(15
...

ˆ
5 As

explained below, the operator A(z), which is widely used in the drives community, is “essentially” equal to
arctan(z), and is introduced to avoid singularities and jumps
...
34)

the norm of which determines the distance of the state to the manifold M, is such that:
• χ67 (0) = 0 ⇒ χ67 (t) = 0, for all t ≥ 0 (invariance);
• χ67 (t) asymptotically (exponentially) converges to zero (attractivity)
...

ˆ
τL
To obtain the dynamics of χ67 , differentiate (15
...
29), yielding

Notice that, if χ67 (t) → 0, an asymptotic estimate of

?

?

˙
ˆ
ˆ
ˆ
χ67 = ξ − ∇ζ [γ ? |ραβ | − 1 ραβ − n p ωJ ραβ ] +
˙
2

2

?τ

L

J

−

n p? ?
i J ραβ
ˆ
J αβ

0

?


...
Towards this end, notice that selecting ξ as
?

?

˙
ξ = A33 (ξ + ζ ) + γ ? |ραβ | − 1 ∇ζ ραβ +
ˆ
ˆ
2

2

? n p?
J

?
i αβ J ραβ
ˆ

0

?

(15
...


Consequently, if we can solve the PDE
∇ζ J ραβ =
ˆ

?

?
a1
,
−a2

(15
...
34), one gets
χ67 = A33 χ67 ,
˙

(15
...
The PDE (15
...

−a2
ρα
ˆ

(15
...

ˆ?

Consequently, ∇ζ ραβ = 0, and the second right-hand term in (15
...
The proof is
ˆ
completed noting that
A33

?

a1
−a2

?

?
2
− n p a1
,
n p a1 a2

? a2
J

=

replacing the function arctan by the operator A in equation (15
...


Remark 15
...
2 If the arctan function is used instead of the operator A in order to recover
the estimate ραβ , some Dirac delta functions might appear in the speed estimation and the
ˆ
error dynamics
...
Then, in view of (15
...
In
ˆ
this scenario, the arctan jumps instantaneously from the value π to the value −π inducing a
2
2
train of Dirac delta functions, δT (t), in the derivative of arctan
...


As illustrated in the simulations of Section 15
...

Remark 15
...
3 Proposition 15
...
1 refers to the unperturbed dynamics (15
...
Some simple calculations show that if this term is not zero the
˜
error dynamic of χ67 takes the form
χ67 =
˙

?

−n p a1
n p a2

−1
J
0

?

n pω
χ67 −
|ραβ |2
ˆ

?

a1
−a2

?

ραβ ραβ +
ˆ? ˜

? n p?
J

?
i αβ J ραβ
˜

0

?


...
39)

In the next section, the effect of the additional terms on the overall dynamics is analyzed
...


328

15
...
7), the output-feedback controller (15
...
23) and (15
...
30) are studied
...
9), which yields a set of nonlinear differential equations of the form (15
...
For ease of reference, these equations are
sequentially derived for χ13 , χ45 , and χ67
...
Towards this end, the equations are written in the
form
χ = Aχ + ?(χ ),
˙

(15
...
9) and (15
...
The proof of the claim of asymptotic stability of
Proposition 15
...
1, follows showing that A is a Hurwitz matrix
...
8
...
8
...
7) in closed loop with the output-feedback
controller (15
...
The first three components, χ13 , of the error vector χ —defined in (15
...
41)

where
A11 = Fd (x ∗ )Q,
⎡
∗
− JL τ L x 3
?
⎢
⎢ ∗
∗
A12 = ⎢ d x 2 + n pJ? x 3 + n r? τ L
p
⎣
0
⎡ L
∗⎤
− ? τ L − JL x 3
?
⎢
⎥
r
⎥,
A13 = ⎢ 0
⎣
n ? ⎦
0

∗
−d x 2 −

n p? ∗
x3
J

−

∗
− JL τ L x 3
?

0

r
τ
n p? L

⎤

⎥
⎥
⎥,
⎦
(15
...
17) and (15
...
Moreover, the matrix A11 is Hurwitz
...
18) can be written as
?
?
vαβ = ρα I2 + ρβ J v + eJ θ v,
˜
˜
ˆ
ˆ

which, in dq coordinates, that is, considering v = e−J θ vαβ , takes the form
v = v + [v1 I2 + v2 J ] χ45 ,
ˆ
ˆ
ˆ

(15
...

˜
ˆ
ˆ
ˆ
On the other hand, some simple calculations show that

v = vFI +
ˆ

?

L
− ? τL

∗
− JL x 3
?
r
n p?

0

?

χ67 −

?

Lχ7
?

?1
J

χ3 + χ6
0

??

,

with the full-information control v F I given by (15
...
Using the definition of χ3 , the latter
can be decomposed as

v

FI

⎡

= d x 12 + ⎣

∗
− JL τ L x 3
?

n p? ∗
x3
J

+

r
τ
n p? L

⎤

⎦+

?

− JL τ L χ3
?
0

?


...
Using all the expressions above to
define v, and replacing in equation (15
...
41) and (15
...

To prove that the matrix A11 is Hurwitz we use equations (15
...
12) to evaluate
⎡

r
−L

⎢ n ∗
Fd (x ∗ )Q = ⎢ − Jp x 3
⎣
0

np ∗
x
J 3
r
−L

n p?
L

0
−

n p?
J

0

⎤

⎥
⎥
...


330

AC Electric Motors Control

15
...
2

Estimation Error for ραβ

Lemma 15
...
2 Consider the mechanical equation of the PMSM model (15
...
23)
...
9)—satisfy the following differential equation:
χ45 = A22 χ45 + ?45 (χ ),
˙

(15
...
45)

∗
and ?45 (χ ) is such that ∇?45 (0) = 0
...


Computing the time derivative of χ45 = e−J θ ραβ yields
˜

Proof:

χ45 = −
˙

np
˙
x 3 J χ45 + e−J θ ρ αβ
...
27), and using the facts that |ραβ | = |χ45 | and that e−J θ ραβ = e1 , it
˜
is possible to write
?
?
?
˙
ρ αβ = −γ ?2 |χ45 |2 + 2χ45 e1 eJ θ (χ45 + e1 )
...

∗
The proof that, for all x 3 ?= 0, A22 is Hurwitz follows trivially computing the characteristic
polynomial
...
8
...
8
...
3) and
(15
...
23)
...
9)—satisfy the following differential equation:
χ67 = A32 χ45 + A33 χ67 + ?67 (χ ),
˙

(15
...
31), and ?67 (χ ) is such that ∇?67 (0) = 0
...
7
...
39)
...
47)

where x 12 , defined in (15
...
On the other hand,
after some lengthy but straightforward computations, the second right-hand term of equation
(15
...
48)

where ?(|χ |2 ) contains term of order higher or equal to |χ |2
...
47) and (15
...
39)
...
8
...
3
...
8
...
8
...
8
...
40) where
⎡

A11

⎢
A=⎣ 0
0

A12
A22
A32

A13

⎡

⎤

?13 (χ )

⎤

⎢
⎥
?(χ ) = ⎣ ?45 (χ ) ⎦
...
For, we notice that A is
similar to a block triangular Hurwitz matrix
...


15
...
For the simulations, the considered motor parameters were L = 0
...
225 ?, ? = 0
...
012 kgm2 , which correspond to an experimental setup located in the Laboratoire de Genie Electrique de Paris, where the experiments
were carried out
...
9
...
Finally,
we carried out a third set of simulations to compare the performance of the proposed scheme
with one proposed in the drives community, namely the one reported in Nam (2010)
...

ı
In order to evaluate the scheme under stringent conditions, the motor was at standstill at
the beginning of the simulations
...
To
avoid singularities, the initial conditions of the position observer were set as ρα (0) = ? and
ˆ
ρβ (0) = 0
...
5 s and from t = 5 s to the end of the
ˆ
experiment, a 1 Nm load torque was applied
...
In both cases, the selection was taken
to obtain a better response of the closed–loop system under parametric uncertainty operation
...

In Figure 15
...
At the top of this figure, both the actual and the desired speeds are shown
...
Moreover, when the speed reference is time varying
7 The

complete evaluation procedure can be consulted in http://www2
...
ec-nantes
...


333

Sensorless Speed Control of PMSM

Speed (rpm)

150
ω
ˆ
ω∗
ω

100

50

0
0

1

2

3

4

5
Time (s)

6

7

8

9

Torque (Nm)

1
...
5
0
−0
...
1 Reference and actual speed (top) and load torque and its estimate (bottom) in nominal
operation

the speed error still remains within reasonable values
...
In Figure 15
...
In both cases their
magnitudes are negligible, even in the presence of changes in the load torque perturbation and
under time-varying speed references
...
In Figure 15
...

To illustrate the controller robustness against parametric uncertainty, in Figure 15
...
5 and
Figure 15
...
It is important to mention that these parameter variations correspond
to the maximum uncertainty that the controller can manage without going to instability
...
2010), the observer of Ortega et al
...
To compare the performance
of our new speed and load torque observer and the proposed IDA-PBC, we show in Figure 15
...
It is clear from the figure
that our scheme outperforms the one in Nam (2010), both in speed regulation as well as load
torque estimation
...
5
0
−0
...
5
0

τL error

5

0

−5

0

Voltage (V)

Voltage (V)

Current (A)

Current (A)

Figure 15
...
3 Stator currents and voltages in nominal operation

335

Sensorless Speed Control of PMSM

2

ρ error

0
−2
−4
−6

0

1

2

3

4

5
Time (s)

6

7

8

9

10

0

1

2

3

4

5
Time (s)

6

7

8

9

10

0

1

2

3

4

5
Time (s)

6

7

8

9

10

2

ω error

0
−2
−4
−6
−8

τL error

5

0

−5

Figure 15
...
5
0
−0
...
5 Observer and speed tracking errors with a 50% increase of the stator inductance

2

ρ error

0
−2
−4
−6

0

1

2

3

4

5
Time (s)

6

7

8

9

10

0

1

2

3

4

5
Time (s)

6

7

8

9

10

0

1

2

3

4

5
Time (s)

6

7

8

9

10

Error (rpm) ω error

2
1
0
−1
−2

τL error

5
0
−5

Speed (rpm)

Figure 15
...
7 Comparative behavior of the proposed scheme (denoted TAC) with the one reported in
Nam (2010) (denoted NAM)

337

Sensorless Speed Control of PMSM

650

ω∗
ω

600

ω
ˆ

Speed (rpm)

550
500
450
400
350
300
250
0

1

2

3

4

5
Time (s)

6

7

8

9

4

10

τL
ˆ

Torque (Nm)

3
2
1
0
−1

0

1

2

3

4

5
Time (s)

6

7

8

9

10

Figure 15
...
9
...
Unfortunately,
at the moment of writing this paper the evaluation of the full-information IDA-PBC and the
observers was carried out in a separated way, that is, it was not possible to present the outputfeedback operation
...
8 for a
positive speed reference, while in Figures 15
...
10, the operation for a speed reference
that crosses through zero is shown
...
10

Future Research

? ?
ρ
ˆ
From a theoretical viewpoint the need to include the operator A ρβ to avoid the presence
ˆα
of spikes may seem unsatisfactory
...
Given the theoretical complexity of the problem, we
tend to believe that the problem does not admit a “smooth” solution
...

Another research line that we are currently pursuing is the establishment of a nonconservative estimate of the region of attraction of the equilibrium point
...
9 Measured and observed speed (top) and estimated load torque (bottom) in the experimental
rig

Current (A)

5

Iα
Iβ

0

−5

0

1

2

3

4

5
Time (s)

6

7

8

80

9

Vα
Vβ

60
Voltage (V)

40
20
0
−20
−40
−60
−80

0

1

2

3

4

5

6

7

8

Time (s)

Figure 15
...
This
research is, obviously, related with the analysis of the full-fledged nonlinear dynamics, that
seems a formidable task
...
To enhance robustness it would be interesting to incorporate an adaptation
algorithm, but this task is far from trivial given the nonlinearly parameterized nature of the
problem
...

Some preliminary experimental results, which have confirmed the remarkable properties of
the observers, have been reported here
...

Part of the work of Gerardo Espinosa–P´ rez was developed during a sabbatical leave at LSS–
e
SUPELEC supported by SUPELEC Foundation
...
The work of Dhruv Shah was supported by the Indo-French project No
...
The authors want to thank Alain Glumineau and Robert
Boisliveau (IRCCyN, France) for the computational code to generate the operator A
...
A

Appendix

In computer programming languages the single argument arctan(u) function is computed in
such a way that its output value e is wrapped in the set (−π, π ]
...
,−π ) then it is assigned the value −π (resp
...
From a mathematical perspective, the result is an operator, denoted as A(u),
that has as input the argument of the arctan function and as output a continuous variable that
corresponds to the unwrapped version of the original output of the arctan function
...
The code for doing this considers two consecutive values of e at two consecutive
sampling times, kT and (k + 1)T , and compute its difference di f = e[kT ] − e[(k + 1)T ] in
order to know if there has been a jump from π to −π or viceversa
...
if di f < −2π then n = n + 1;
2
...
otherwise the value of n is not changed
...


340

AC Electric Motors Control

References
Akrad A, Hilairet M, Ortega R, and Diallo D (2007) Interconnection and damping assignment approach for reliable
pm synchronous motor control
...

Astolfi A, Karagiannis D, and Ortega R (2007) Nonlinear and Adaptive Control with Applications
...
Springer-Verlag, Berlin
...
John Wiley & Sons
...

Ezzat M, de Leon J, Gonzalez N, and Glumineau A (2010a) Observer–controller scheme using high order sliding
mode techniques for sensorless speed control of permanent magnet synchronous motor
...

Ezzat M, Glumineau A, and Plestan F (2010b) Sensorless speed control of a permanent magnet synchronous motor:
high order sliding mode controller and sliding mode control observer
...

Fabio G, Miceli R, Rando C, and Ricco–Galluzzo G (2010) Back EMF sensorless-control algorithm for high-dynamic
performance PMSM
...

Ichikawa S, Tomita M, Doki S, and Okuma S (2006) Sensorless control of PMSM using on-line parameter identification
based on system’s identification theory
...

Khorrami F, Krishnamurthy P, and Melkote H (2003) Modeling and Adaptive Nonlinear Control of Electric Motors
...

Krause PC (1986) Analysis of Electric Machinery
...

Lee J, Hong J, Nam K (2010) Sensorless control of surface-mount permanent magnet synchronous motors based on
a nonlinear observer
...

Marino R, Tomei P, and Verrelli CM (2008) Adaptive field-oriented control of synchronous motors with damping
windings
...

Matsui N (1996) Sensorless PM brushless DC motor drives
...

Nahid Mobarakeh B, Meibody–Tabar F, and Sargos F (2001) Robustness study of a model–based technique for
mechanical sensorless control of PMSM
...

Nam K (2010) AC Motor Control and Electric Vehicle Applications
...

Ortega R, van der Schaft AJ, Maschke BM, and Escobar G (2002) Interconnection and damping assignment passivitybased control of port-controlled Hamiltonian systems
...

Ortega R and Garcia-Canseco E (2004) Interconnection and damping assignment passivity-based control: a survey
...

Ortega R, Loria A, Nicklasson PJ, and Sira-Ram´rez H (1998) Passivity–based Control of Euler–Lagrange Systems
...

Ortega R, Praly L, Astolfi A, et al
...
IEEE Transaction on Control Systems Technology, 19–2, 284–296
...
IEEE Transaction of Control System Technology, 9–6, 811–820
...
IEEE Press
...
48th IEEE Conference
on Decision and Control, Shangai, China
...
International Journal of Adaptive Control and Signal Processing, 22–3,
266–288
...
Springer-Verlag, Berlin
...
2009 International Conference
on Signals, Circuits and Systems, Jerba, Tunisia
...
1

Introduction

Two major classes of permanent-magnet AC machines exist: (1) the trapezoidally excited
machines (typically known as brushless DC motors), which are specifically designed to
develop nearly constant output torque when they are excited with six-step switched current trapezoidal waveforms; (2) the sinusoidally excited machines in which the windings are
typically distributed over multiple slots in order to approximate a sinusoidal distribution
...
Due to their excellent
serviceability and durability, high efficiency and power density, as well as high torque to
inertia ratio and absence of external rotor excitation and rotor windings, the (sinusoidally
excited) permanent-magnet motors are used in practical applications such as printers, tape
drives, hard drives in PCs, process control systems, home appliances, and have been gradually replacing DC motors in a wide range of drive applications such as machine tools and
industrial robots
...

When the motor mechanical variables (rotor position or speed) are available from measurements, high closed-loop performances can be achieved in rotor position or speed tracking
applications for permanent-magnet synchronous motors (PMSMs) even in the presence of
uncertain model parameters (see, for instance, Zribi and Chiasson 1991; Bodson et al
...
Edited by Fouad Giri
...
Published 2013 by John Wiley & Sons, Ltd
...
1995, 2012; Dawson et al
...
2003;
Chiasson 2005; Bifaretti et al
...
The speed
or the position tracking control of PMSMs requires the knowledge of rotor shaft position
and speed signals in order to control the stator current vector suitably and then achieve high
performance (Zribi and Chiasson 1991; Marino et al
...
When all machine parameters (in
addition to load torque) are exactly known, the classical control is a nonlinear state-feedback
algorithm which is designed after performing the direct-quadrature dq coordinate transformation for the two-phases currents and voltages: it may be viewed as a feedback linearizing
control law (Zribi and Chiasson 1991); experimental results reported in Bodson et al
...
A full state-feedback nonlinear
adaptive control is presented in Marino et al
...
The
extension to the case in which the rotor speed measurement is not required can be found in Di
Gennaro (2000)
...
Even though relevant contributions
concerning this problem have been presented and experimentally validated in the literature
(see De Angelo et al
...
2007; Rashed et al
...

2008; Bisheimer et al
...
2012), a rigorous solution to such a problem
(i
...
, a solution guaranteed by a closed loop stability proof) turned to be rather difficult to
be derived since: motor dynamics are nonlinear and multivariable; measured outputs (stator
currents) do not coincide with one of the controlled outputs (rotor speed) that are required to
track smooth bounded reference signals; and the load torque depends on applications and is
typically an uncertain model parameter
...
(2012)
...
The control algorithm in Tomei
and Verrelli (2011) is then reported
...
(2011) and experimentally validated in Lee et al
...
The overall
closed-loop stability analysis shows that local exponential speed tracking is guaranteed under
a persistency of excitation condition that only restricts the family of speed reference signals
1 For instance, the installation

of position (and speed) sensors in applications involving vacuum pumps is troublesome
due to the difficulties of extending the motor shaft out of the motor housing, while in some crane and elevator
applications the large distance between the motor and the inverter causes high sensor signal attenuation and noise
interference
...


Adaptive Output-Feedback Control of Permanent-Magnet Synchronous Motors

343

and admits a clear physical interpretation in terms of motor observability
...
(2012) illustrate how the reported theoretical analysis
provides actually effective tools for identifying conditions in which satisfactory performances
can be obtained in practice
...
2

Dynamic Model and Problem Statement

Assuming linear magnetic materials, nonlinear flux density distribution due to the air-gap
geometry only, and negligible magnetic hysteresis and Foucault currents, the dynamics of a
PMSM with no saliency and a sinusoidal flux density distribution in a fixed reference frame
attached to the stator are given by the well-known fourth order model2 (see, for instance,
Marino et al
...
(1998); Khorrami et al
...
1)

KM
νsb
Rs
di sb
ωm cos( pθr ) +
,
= − i sb −
dt
Ls
Ls
Ls
in which θr is the rotor angle, ωm is the rotor speed, i sa and i sb are the stator currents
θr , ωm , i sa and i sb constitute the state variables, νsa and νsb are the stator voltages (which
constitute the control inputs); the output to be controlled is the rotor speed ωm
...
The
load torque TL , which depends on applications, is assumed to be an unknown constant model
parameter
...
While J, Rs , L s , and K M are positive parameters, F is assumed to be
nonnegative, so that the main stability result of this paper is not based on the positiveness of F,
which gives some advantage for the design in the speed-sensorless scenario
...

= R(θr )
,
wb

2 Model (16
...


344

AC Electric Motors Control

then the dynamics (16
...

= − i sq − pωm i sd −
dt
Ls
Ls
Ls

(16
...
2) is suitable for control design since the rotor speed dynamics are influenced
by the stator current vector q-component i sq only: adaptive backstepping techniques may be
successfully applied provided that cos( pθr ), sin( pθr ) are available for feedback along with the
rotor speed ωm
...

∗
In the following, we denote by ωm (t) the known smooth bounded reference signal with
∗
∗
¨∗
known bounded time derivatives ωm (t) and ωm (t) for the rotor speed ωm (t), and by i sd (t) the
˙
∗
di sd (t)
known smooth bounded reference signal with known bounded time derivative dt for the
∗
current i sd (t), which may be simply chosen as i sd = 0 (see Marino et al
...

Remark 16
...
1 There are cases, as in the master–slave synchronization problems, in which
the reference signals foreknowledge is typically not available so that reference signal time
derivatives cannot be directly compensated by feed-forward actions
...


16
...
3)

? T
ˆL
KM ?
F

...

ˆsa ) + K M ψθ c (i sb − i sb ) ,
ˆ
ψθ s (i sa − i
= γT −
Ls
Ls

?
p ?ˆ
?
ξa − L s i sa ,
ψθ c = cos( pθr ) =
KM

?
?
2
˙ = −R i + ν + γ ?ξ − L i ? K M − ?ξ − L i ?2 − ?ξ − L i ?2 ,
ˆ
ˆa
ˆa
ˆb
ξa
s sa
sa
θ
s sa
s sa
s sb
p2
?
p ?ˆ
?
ξb − L s i sb ,
(16
...

ξ b = −Rs i sb + νsb + γθ ξb − L s i sb
p2

346

AC Electric Motors Control

It consists of: (1) a stator current-control loop (containing feedforward actions and suitable
∗
∗
∗
stabilizing terms) asymptotically forcing (i sd , i sq ) to (i sd , i sq ), where i sq , designed according
to the field-oriented control strategy, is responsible for the rotor speed tracking; (2) a sixth
ˆ ˆ ˆ ˆ
order closed-loop observer (which includes the auxiliary variables i sa , i sb , ξa , ξb ) providing
estimates for the unmeasured cos( pθr ), sin( pθr ), ωm , and the unknown TL
...
3) and (16
...
5)

in terms of a positive scalar T p ∈ R+
...
3
...

sin( pθr ) =
KM

Adaptive Output-Feedback Control of Permanent-Magnet Synchronous Motors

347

Remark 16
...
2 The algorithm (16
...
4) is constituted by: (1) the second order
observer (16
...
(2011) and experimentally validated in Lee et al
...
3) designed in Tomei and Verrelli (2008) which relies on
the estimates of rotor speed and load torque
...
3) and
(16
...


16
...
e
...
4))
of known dynamics from known initial conditions
...
3) and (16
...

sin( pθr
?
?
Theorem 16
...
1 Assume that cos( pθr (t0 )) = 1 and that cos( pθr )(t0 ) = 1, sin( pθr )(t0 ) = 0
...
3) and (16
...
1), guarantees boundedness of ωm , i sa , i sb , ωm , i sa , i sb ,
ˆ
and TL and uniform local asymptotic stability of the closed-loop system equilibrium point
∗
∗
∗
ˆ
ˆ
ˆ
ˆ
(ωm − ωm , i sd − i sd , i sq − i sq , i sa − i sa , i sb − i sb , ωm − ωm , TL − TL ) = 0 with domain of
attraction

where

?
?
B = [ξ1 , ξ2 ,
...

Proof: Assume that cos( pθr (t0 )) = 1
...
1),
cos[ pθr (t)] = ψθ c (t),
sin[ pθr (t)] = ψθ s (t)
...

J
γ + r FλJ 2 L s
Ls

˙
ωm = −
˜
˙
˜
i sd =
˙
˜
i sq =
˙
˜
i sa =
˙
˜
i sb =
˙
eω =
˙
˜
TL =

(16
...

Ls
J
J
4J 3 r
The last four equations in (16
...
7)

349

Adaptive Output-Feedback Control of Permanent-Magnet Synchronous Motors

˜ ˜
˜
with x = [i sa , i sb ]T , z = eω , w = TL , and
A=

?

B(t) =

?

D(t) =
H (t) =

R
− L ss − ke

0

0
R
− L ss − ke

− K M sin( pθr (t))
Ls
KM
Ls

cos( pθr (t))

2γ 2 + r FλJ 2 ?
γ 2 λ + r Fλ2 J 2

?

?

?

,

,

KM
Ls

?
sin( pθr (t)), − K M cos( pθr (t)) ,
Ls

?
0
− p ωm (t)
ˆ

...
5)
so that system (16
...

The second and third equations in (16
...
Since (i sd (t), i sq (t)) are bounded on [t0 , +∞) and, for any initial condition
˜sa (t0 ), i sb (t0 ), eω (t0 ), TL (t0 )]T in A, eω (t) tends exponentially to zero, the system above
˜
˜
[i
˜ ˜
complies with the hypotheses of Lemma III
...
Recalling
exponentially to zero for any initial condition [i
˜
the first equation in (16
...
6) with domain of attraction B
...
4
...
5) are satisfied, then,
˜
˜
˜
˜
for any initial condition [i sa (t0 ), i sb (t0 ), eω (t0 ), TL (t0 )]T in A: (1) since ωm tends uniformly
˜ ˜
asymptotically to zero, uniform asymptotic speed tracking is achieved; (2) since (i sd , i sq )
tend exponentially to zero, stator currents tend exponentially to the corresponding reference signals; (3) since eω tends exponentially to zero, rotor speed is exponentially esti˜
mated; (4) since TL and eω tend exponentially to zero, load torque TL is exponentially
estimated
...
4
...

Proposition 16
...
3 “s-alignment” procedure: Let u cs ∈ R+ be a positive real scalar and
suppose that no-load torque TL is applied to the motor
...
8)

substituted in model (16
...

Rs

the stator current vector b-component regulation error,

?
KM ?
u cs
˜
cos( pθr ) ,
− i sa sin( pθr ) + i sb cos( pθr ) +
J
Rs
KM
ωm sin( pθr ),
Ls
KM
ωm cos( pθr ),
Ls

Adaptive Output-Feedback Control of Permanent-Magnet Synchronous Motors

351

whose equilibrium points are

with θes =

π 3π
, ,
...

2 p

Consider the function

?
? K u ?
1? J
M cs
˜2
ωm 2 + i sa 2 + i sb −
sin( pθr ) − 2 ,
2 Ls
pL s Rs

which is always positive
...

U s = − ωm −
Ls
Ls
˙
Since Us (t) ≤ 0 [recall that F ≥ 0], by virtue of LaSalle theorem (see Khalil 2002), we can
˙
establish that every solution is attracted into the largest invariant subset Ms of the set Us = 0,
˜
consisting of the equilibrium points (θr , ωm , i sa , i sb ) = (θes , 0, 0, 0)
...
4
...
Let u cs ∈ R+ and u cc ∈ R+ be positive real
scalars
...
??
with δs ? R+
...
9)

substituted in model (16
...

Rs

352

AC Electric Motors Control

˜
Proof: Denoting by i sa = i sa −
the closed-loop system is

u cc
Rs

the stator current vector a-component regulation error,

˙
θr = ωm ,
ωm = −
˙

?
F
KM ? ˜
u cc
sin( pθr ) + i sb cos( pθr ) ,
ωm +
− i sa sin( pθr ) −
J
J
Rs

Rs ˜
KM
˙
˜
i sa = − i sa +
ωm sin( pθr ),
Ls
Ls
di sb
KM
Rs
ωm cos( pθr ),
= − i sb −
dt
Ls
Ls
whose equilibrium points are
˜
(θr , ωm , i sa , i sb ) = (θec , 0, 0, 0),
with θec = 0, π , 2π ,
...
Consider the function
p
p
p
Uc =

?
? K u ?
1? J
M cc
˜2
ωm 2 + i sa + i sb 2 −
cos( pθr ) − 2 ,
2 Ls
pL s Rs

which is always positive
...

U c = − ωm −
Ls
Ls
˙
Since Uc (t) ≤ 0 [recall that F ≥ 0], by virtue of LaSalle theorem (see Khalil 2002), we can
˙
establish that every solution is attracted into the largest invariant subset Mc of the set Uc = 0,
˜sa , i sb ) = (θec , 0, 0, 0)
...


Adaptive Output-Feedback Control of Permanent-Magnet Synchronous Motors

353

Choosing a sufficiently small ?z s (t0 )? such that
pL s Rs
|ϑs (z s (t0 ))| ≤ δs ,
K M u cc
suffices to guarantee
lim cos( pθr (t)) = 1
...
4
...
5
...
9) does not guarantee
lim cos( pθr (t)) = 1

t→+∞

for any initial condition, since for instance
?

u
θec =

?
π 3π
π
,
,
...


16
...
For the sake of simplicity,
we set t0 = 0
...

ˆ
ˆ
ˆ
i sq (t), i sa (t) − i
∗
Theorem 16
...
1 Assume that the rotor speed reference signal ωm (t) is persistently exciting,
that is, there exist positive reals T and c p such that the persistency of excitation condition

P:

?

t+T
t

∗
ωm (τ )2 dτ ≥ c p ,

∀t ≥0

is satisfied
...
1), (16
...
4), boundedness of
ˆ
ˆ
ˆ
?
?
ˆ
(ωm (t), i sa (t), i sb (t), cos( pθr )(t), sin( pθr )(t), ωm (t), i sa (t), i sb (t), TL (t)) on [0, +∞) is guaranteed along with exponential convergence to zero of ??e (t)?, provided that ??e (0)? is
sufficiently small
...
2010)
2
2
2?
KM ?
?
?
cos( pθr )2 + sin( pθr )2 − cos( pθr ) − sin( pθr )
p2
? 2
?
?2 ?
?
KM ? ˆ
ˆb − L s i sb 2 ,
=
− ξa − L s i sa − ξ
p2

F =

ˆ ˆ
which is available for feedback3
...
Let us introduce the tracking and estimation errors the
˜
˜ ˜
same, excepting for ξa , ξb , of the previous section with z and w directly in place of eω and TL :
∗
ωm = ωm − ωm ,
˜
∗
˜
i sd = i sd − i sd ,

ˆ
˜
i sa = i sa − i sa ,

z = ωm − ωm ,
ˆ

∗
˜
i sq = i sq − i sq ,

˜
ˆ
i sb = i sb − i sb ,

ˆ
ˆ
w = TL − TL − F[ωm − ωm ],

˜
ˆ
ξa = ξ a − ξa ,

˜
ˆ
ξb = ξb − ξb ,

and the vector
˜ ˜
q = [ωm , i sd , i sq , i sa , i sb , z, w, ξa , ξb ]T ,
˜ ˜ ˜ ˜ ˜
so that the error system can be written as
?
?
F
KM
w
˜
(ωm + z) − kω satκ (ωm + z) + 0,
˜
μ− ,
J
J
J
? R
?
− L ss − ki
0,
μ=
˙
μ + h i (q, t),
Rs
0
− L s − ki

˙
ωm = −
˜

˙
x = Ax + B(t)z + H (t)x + h x (q, t),
w
˙
z = D(t)x + + h z (q, t),
J
F
λJ γ
w =− w+
˙
D(t)x + h w (q, t),
J
2γ 2 + r FλJ 2
2K M
˙
˜
ξ =
p
3 Recall

?

ϕa (t)
ϕb (t)

−ϕb (t)
ϕa (t)

that sin( pθr )2 + cos( pθr )2 = 1
...
10)

(16
...


If we choose the yet to be designed functions ϕa and ϕb as
ϕa = −

γθ K M ?
γθ K M
˜
cos( pθr ) = −
cos( pθr ) + γθ ξa ,
p
p

ϕb = −

γθ K M ?
γθ K M
˜
sin( pθr ) = −
sin( pθr ) + γθ ξb ,
p
p

˜
then the ξ -subsystem (16
...
12)

356

AC Electric Motors Control

so that, by introducing the nonsingular change of variables
?T
?
˜
η = ηd , ηq = R(θr )ξ ,

the dynamics (16
...


(16
...
4
...

Remark 16
...
2 The exponential stability of the origin of the closed-loop error system
allows for establishing certain robustness properties: ultimate boundedness of the solutions
of the closed-loop error system when perturbed by modeling errors, aging, uncertainties, and
disturbances, which exist in any realistic application, is guaranteed, according to Lemma
9
...

Remark 16
...
3 Since ωm exponentially tends to zero, exponential rotor speed tracking
˜
˜ ˜
is achieved; since (i sd , i sq ) exponentially tend to zero, stator currents exponentially tend
to the corresponding reference signals; since z exponentially tends to zero, rotor speed is
exponentially estimated; since w and z exponentially tend to zero, the load torque TL is
exponentially estimated; since ηd and ηq exponentially tend to zero, cos( pθr ) and sin( pθr ) are
exponentially estimated
...
5
...

It is related to motor observability (see Basic et al
...
1)
?

νsa νsb
(θr , ωm , i sa , i sb ) = θ∗ , 0,
,
Rs Rs

?

,

with θ∗ satisfying
KM
[−νsa sin( pθ∗ ) + νsb cos( pθ∗ )] = TL ,
Rs
4 Recall

˜
that ?η? = ?ξ ?
...
In particular, when ωm (t) ≡ 0
for all t ≥ 0 (so that θr (t) ≡ θc for all t ≥ 0 with θc a constant value), all the points
2K M
˜ ˜
(cos( pθc ), sin( pθc ))
(ξa , ξ b ) =
p
2

2

?
?
˜
for which cos( pθr ) + sin( pθr ) = 1, are equilibrium points for the ξ -subsystem (16
...

˜
˜
Remark 16
...
5 Note that the origin ξ = 0 is an equilibrium point for the ξ -subsystem so
?r )(0) = 0 and sin( pθr (0)) − sin( pθr )(0) = 0, then cos( pθr (t)) −
?
that if cos( pθr (0)) − cos( pθ
?
?
cos( pθr )(t) ≡ 0 and sin( pθr (t)) − sin( pθr )(t) ≡ 0 for all t ≥ 0
...
4
...


16
...
2012)

The proposed control algorithms (16
...
4) have been tested by simulations with control
parameters (the values are in SI units) ki = 20, kω = 100, κ = 9, ke = 12000, γθ = 180,
γω = 10, and γT = 9 for the (nonsalient pole surface) PMSM with sinusoidal flux distribution
in Marino, Peresada, & Tomei (1995), whose parameters are:
viscous friction coefficient F = 0 kg m2 s−1
number of pole pairs p = 6

total rotor-load inertia J = 0
...
006 H

torque constant K M = 2 N m A−1
...
The necessary first and second order derivatives
have been obtained from the state space realization of the filter
...


16
...
1

Response to Time-Varying Load Torque

A time-varying load torque (whose profile includes regenerating mode and ramp-wise variations) has been applied at t = 0
...
The rotor speed reference and the applied torque
are reported in Figure 16
...
The initial conditions for the controller have been set to zero
ˆ
ˆ
excepting for ξa (0) = KpM cos( pθu ), ξb (0) = KpM sin( pθu ) with θu = θr (0) + π/576 (obtained
by using the previously described finite-time alignment procedures)
...
2, 16
...
4

358

AC Electric Motors Control

Rotor speed reference
60

(rad/s)

40
20
0
0

0
...
2

0
...
4

0
...
6

0
...
8

0
...
7

0
...
9

1

Applied load torque
2
...
5
1
0
...
0
–1

0

0
...
2

0
...
4

0
...
6

Figure 16
...
1

0
...
3

0
...
5

0
...
7

0
...
9

1

0
...
8

0
...
1

0
...
3

0
...
5
(s)

0
...
2 Rotor speed and load torque estimate

359

Adaptive Output-Feedback Control of Permanent-Magnet Synchronous Motors

5

Estimation error for cos(pθ)

×10–3

0
–5

–10

0

0
...
2

0
...
4

0
...
6

0
...
8

0
...
7

0
...
9

1

Estimation error for sin(pθ)
0
...
03
0
...
01
0
–0
...
1

0
...
3

0
...
5
(s)

0
...
3 Estimation errors: cos( pθr (t)) − cos( pθr )(t); sin( pθr (t)) − sin( pθr )(t)
Rotor speed tracking error

1

1
...
5

–0
...
4

0
...
5

–1
–1
...
2

0
...
6

0
...
5

1

Stator current vector d-component

5

0
...
2

0
...
6

0
...
2

3

0
...
1

0

–0
...
3

2

0

0
...
4

0
...
8

1

–1

0

0
...
4

0
...
8

1

(s)

Figure 16
...
2

0
...
6

0
...
2

0
...
6

0
...
2

0
...
6

0
...
2

0
...
6

0
...
5 (a, b)-components of stator current and voltage vectors

and 16
...
Similar
results (see Figures 16
...
7) can be achieved for different controller initial conditions
ˆ
leading to larger initial estimation errors in sin( pθr ) and cos( pθr ) (ξa (0) = KpM cos( pθu ),
ˆ
ξb (0) = KpM sin( pθu ) with θu = θr (0) + π/414 and θu = θr (0) + π/306, respectively) larger
transient output tracking/regulation errors are accordingly obtained
...
6
...
05 A); (2) variations in stator resistance Rs (a 3%
increase at t = 0
...
018
kg m2 is different from the nominal value 0
...
The load torque
(2 Nm, the rated value) has been applied at t = 0
...
The rotor speed reference (including
low speed operation) and the applied torque are reported in Figure 16
...
The controller initial
conditions are the same of the previous subsection
...
9,
16
...
11, and 16
...
5
...


361

Adaptive Output-Feedback Control of Permanent-Magnet Synchronous Motors

Rotor speed tracking error

Rotor speed tracking error

2

0

(Nm)

(rad/s)

1

0

–1
0
0
...
8
0
...
2
0
...
01

0
...
6
0
...
4
1
Estimation error for sin(pθ)
0
...
04
0
...
02
0
...
01
0
...
6
0
...
4
0
1
Stator current vector q-component
5
(A)

(A)

–0
...
8
0
...
2
0
...
5

–2
0

0

–0
...
2

0
...
4

0
...
2

0
...
4

0
...
6 A (θu = θr (0) + π/414)
...
2

0
...
4

0
...
01

Rotor speed tracking error

2
(Nm)

(rad/s)

2

0
...
6

0
...
8

1

0
...
01

0
...
02
0

0
...
4

0
...
05
0

Stator current vector d-component

0
–1
0

5

(A)

(A)

0
...
2

0
...
6

0
...
8

1

0
...
6

0
...
8

1

Stator current vector q-component

0
–5
0

0
...
6

0
...
8

1

(s)

Figure 16
...
Rotor speed tracking error; load torque estimation error;
?
?
cos( pθr (t)) − cos( pθr )(t); sin( pθr (t)) − sin( pθr )(t); stator current vector (d, q)-components

362

AC Electric Motors Control

Rotor speed reference

60
50
(rad/s)

40
30
20
10
0
–10

0

0
...
2

0
...
4

0
...
6

0
...
8

0
...
7

0
...
9

1

Applied load torque
2
...
5
1
0
...
5

0

0
...
2

0
...
4

0
...
6

Figure 16
...
1

0
...
3

0
...
5

0
...
7

0
...
9

1

0
...
8

0
...
1

0
...
3

0
...
5
(s)

0
...
9 Rotor speed and load torque estimate

363

Adaptive Output-Feedback Control of Permanent-Magnet Synchronous Motors

Estimation error for cos(pθ)

0
...
02
0
...
01

0

0
...
2

0
...
4

0
...
5

0
...
8

0
...
7

0
...
9

1

Estimation error for sin(pθ)

0
...
08
0
...
04
0
...
1

0
...
3

0
...
5
(s)

0
...
10 Estimation errors: cos( pθr (t)) − cos( pθr )(t); sin( pθr (t)) − sin( pθr )(t)

Rotor speed tracking error

Load torque estimation error

3

5

1

(Nm)

(rad/s)

2

0

0

–5

–1
–2

0

0
...
4

0
...
8

1

Stator current vector d-component
1
...
2

0
...
6

0
...
5
–0
...
5
0

0
...
4 0
...
8

1

–5

0

0
...
4

0
...
8

1

(s)

Figure 16
...
2

0
...
6

0
...
2

0
...
6

0
...
2

0
...
6

0
...
2

0
...
6

0
...
12 (a, b)-components of stator current and voltage vectors

16
...
2012)

A Tetra 56SR1
...
Its main specifications are: stall torque of 1
...
9 Ar ms
...
The PMSM is fed by a three-phase bridge using 70 V DC bus, while
the DC motor is fed by a H -bridge using 25 V DC
...
The
experimental tests have been performed applying a 16 kHz switching frequency for the power
MOSFET (IXYS FMM50-025TF) used for both the three-phase and the H -bridges
...
13 and 16
...
The sensorless control algorithms (16
...
4) are executed with
a sampling interval Ts = 62
...
At the beginning
of each sampling interval, the phase currents values, provided by two Hall effect current
sensors, are acquired
...
14 in which the main
mechanical and electronic subsystems are highlighted by dashed boxes
...
(2003) (with a 20 Hz cut-off frequency),
the rotor speed measurements required to evaluate the rotor speed tracking performance
...
1) are zero (the motor is aligned and at rest); a zero

365

Adaptive Output-Feedback Control of Permanent-Magnet Synchronous Motors

Three-phase
grid

PMSM
iu

Variac

Controller

iv

VDC
70V
Three-phase
rectifier

C
2
...
13 Functional block diagram
∗
reference signal i sd for the stator current vector d-component i sd is chosen (field orientation);
all the initial conditions for the sensorless controls (16
...
4) are set to zero except
?
cos( pθr )(0) = 1; the control parameters (all values are in SI units) are: ki = 700, kω = 1800,
ke = 3000, γθ = 1000, γω = 100, γT = 0
...
The nominal parameter values
directly provided by the manufacturer, that is
...
8 · 10−5 kg m2 , K M = 0
...
48 Ω, L s = 1
...
14 Experimental prototype

366

AC Electric Motors Control

0
...
2

(A)

0
–0
...
4
–0
...
8
–1

0

1

2

3

4

5

6

7

8

9

(s)

Figure 16
...
A timevarying load torque (negative for t ≤ 4
...
5 s) is provided by the DC
motor and applied even when the rotor speed reference is zero (see Figure 16
...
Figure 16
...
17 and 16
...
According to Figure 16
...
5% and 2% maximum speed tracking errors are obtained when the rotor speed
reference is imposed to 52 rad/s and 105 rad/s, respectively (better performance are reasonably
achieved at higher speeds according to P); even in the case of a negative load torque (braking
120
100

(rad/s)

80
60
40
20
0
–20

0

1

2

3

4

5

6

7

8

9

(s)
∗
Figure 16
...
5
0

–0
...
5
0

–0
...
17 Stator current vector (a, b)-components
100
(V)

50
0
–50
–100

0

1

2

3

4

0

1

2

3

5

4

6

7

8

9

5

6

7

8

9

100
(V)

50
0
–50
–100

(s)

Figure 16
...
5% maximum speed
tracking error is obtained
...
8

Conclusions

According to the recent advances in Tomei and Verrelli (2008, 2011), and Bifaretti et al
...
3) and (16
...
1): it relies on the theoretical result (not based on motor friction) stated in Theorem 16
...
1
...
Satisfactory performances are obtained in practice in conditions (persistently exciting rotor speed reference signals, relatively small initial tracking and estimation
errors, relatively accurate knowledge of motor parameters, relatively accurate measurements

368

AC Electric Motors Control

of available variables, relatively fast algorithm execution), which can be inferred by deeply
analyzing the reported theoretical analysis
...
4)—instead of the nonrobust open-loop estimators
introduced in Tomei and Verrelli (2008) for theoretical purposes (see Theorem 16
...
1)-turns to
be crucial: it is theoretically related to the exponential stability of the origin of the closed-loop
error system, which allows for establishing certain robustness properties
...
IEEE Transactions on Automatic Control, 55(1), 212–217
...
Automatika, 1-2, 67–74
...
IEEE Transactions on Industrial Electronics, 58(10), 4654–4663
...
Control Engineering Practice, 20(7), 714–724
...
Automatica, 47(1), 227–234
...
(2010) Full speed range permanent magnet synchronous motor
control without mechanical sensors
...

Bodson M, Chiasson JN, Novotnak RT, and Rekowski RB (1993) High-performance nonlinear feedback control of a
permanent magnet stepper motor
...

Chan TF, Wang W, Borsje P, et al
...
IET Electric Power Applications, 2(2), 88–98
...
Wiley-IEEE Press, Hoboken, NJ
...
Marcel Dekker, New York
...
(2006) Mechanical sensorless speed control of permanent-magnet AC motors
driving an unknwon load
...

Di Gennaro S (2000) Adaptive output feedback control of synchronous motors
...

Hinkkanen M, Tuovinen T, Harnefors L, and Luomi J (2012) A combined position and stator-resistance observer for
salient PMSM drives: design and stability analysis
...

Khalil H (2002) Nonlinear Systems, Prentice Hall, Upper Saddle River, NJ
...

Lee J, Hong J, Nam K, et al
...
IEEE Transactions on Power Electronics, 25(2), 290–297
...
Automatica,
31(11), 1595–1604
...
IEEE Transactions on Automatic Control, 40(7), 1300–1304
...
European Journal of Control, 14(3), 177–195
...
International Journal of Robust and Nonlinear Control, 22, 645–675
...
IEEE Transactions on Industrial
Electronics, 43(2), 485–494
...
(2011) Estimation of rotor position and speed of permanent magnet synchronous
motors with guaranteed stability
...


Adaptive Output-Feedback Control of Permanent-Magnet Synchronous Motors

369

Park RH (1929) Two-reaction theory of synchronous machines – generalized method of analysis – part I
...

Rashed M, MacConnell PF
...
IEEE Transactions on Industrial Electronics, 54(3), 1664–1675
...
International Journal of Adaptive Control and Signal Processing, 22(3),
266–288
...
IEEE Transactions on Automatic Control, 56(6), 1484–1488
...
Applied
Mathematics and Computation, 217(9), 4502–4515
...
International Journal of Control, 84, 1024–1030
...
Nonlinear
Analysis: Real World Applications, 13(1), 395–409
...
IEEE Transactions
on Automatic Control, 36(5), 620–625
...
1

Introduction

Permanent-magnet synchronous motor (PMSM) drives are currently extensively used in many
industrial applications, where continuity of operation and reliability are major features to
be pursued
...
As a consequence, fast and
accurate diagnosis of failures in the drive system is important for preventing major damages
to the motor and, mostly, to guarantee continuity of operation and minimize eventual machine
downtime for maintenance
...
2006; Kim
et al
...

The most common electrical faults, however, are known as stator winding fault, and are
caused by the breakdown of insulation as a result of the voltage, current, and thermal stress
AC Electric Motors Control: Advanced Design Techniques and Applications, First Edition
...

© 2013 John Wiley & Sons, Ltd
...


Robust Fault Detection for a Permanent-Magnet Synchronous Motor

371

acted on the stator winding (Kim et al
...
In this case, the early detection of the fault
is important to avoid the propagation to more stator turns, in order to reduce repair costs
and maintenance time
...
2008), negative sequence current (Quiroga et al
...
2006), experimental methods using current monitoring (Kim et al
...
2005), and AI tools (Awadallah and Morcos
2003)
...
According to this approach, a model
is needed of the unsupervised process, in order to build a residual function able to signal any
difference between nominal and faulty status
...

It follows that any model-based fault detection technique should be made robust with respect
to structured uncertainties, to avoid generation of false alarms
...
2000) for the generation of residuals, a nonlinear
control technique being claimed for
...
2008) where second order sliding modes are used for control and observation, under
the assumption of stable zero dynamics for ensuring the existence of a standard sliding
surface, and assuming the availability of the whole state vector
...
A further paper deserving attention because model-based
fault detection is considered is Liu and Carter (2005), but the quasi-dynamic approach is
adopted there and nonlinearity and coupling affects are basically neglected
...
Convergence of such
observer is ensured by the enforcement of a sliding motion on given surfaces, based on
motor currents, which are of course affected by all faults influencing stator currents and can
easily detect them
...
Note that observer and control design are not performed in the
(d, q) reference frame but in the (α, β) frame, since any fault affecting the angular position
measurement could invalidate the whole model and the consequent design
...
2 Preliminaries
17
...
1 PMSM Modeling
In the (α, β) reference frame, the electrical equations of motion of a PMSM can be written as
⎧
⎪ di α
⎪
⎨
dt
⎪ di β
⎪
⎩
dt

λ0
1
R
= − i α + ωe sin(θe ) + vα ,
L
L
L
λ0
1
R
= − i β − ωe cos(θe ) + vβ ,
L
L
L

(17
...

The electrical torque Te is given by
Te = K t (i β cos(θe ) − i α sin(θe )),

(17
...

2
For the electrical angular position/speed and the mechanical angular position/speed, the
following relations hold:
Nr =

ωe
θe
= ,
ωr
θr

(17
...
In the following it will
be assumed that Nr = 1, therefore ωe = ωr
...
3 Control Design
17
...
1 A Robust Observer of Rotor Angular Position and Velocity for the
Tracking Problem
In standard drives, rotor position is given by encoder measurements, and rotor speed is usually
estimated as the incremental ratio of encoder positions over one sampling period
...

The mechanical motion equation is described by
J ωr + Bωr = Te − τ,
˙
˙
θr = ωr ,

(17
...
5)

where J is the total mechanical inertia of the the PMSM
...
It is likely to introduce the following assumption
...
3
...
4) are uncertain, with bounded

¯
J = J + ?J ;
|? J | ≤ ρ J ;

¯
B = B + ?B;
|?B| ≤ ρ B ;

τ
¯
τ
= + ?τ ;
¯
J
J
|?τ | ≤ ρτ ;

(17
...
7)

Robust Fault Detection for a Permanent-Magnet Synchronous Motor

373

The tracking problem is considered here, that is, the variable ωe is required to track a known
reference ω∗
...
8)

ωe
...

Theorem 17
...
2 With reference to the plant (17
...
4), (17
...
8) is able
to robustly guarantee the asymptotic vanishing of the observation error and of the tracking
error for a suitable choice of the control variables vα , vβ and of the auxiliary inputs να , νβ
...


(17
...
10)

It is straightforward to verify that a sliding motion can be enforced on each surface by the
following auxiliary inputs:
? M
?
ˆ
ˆ
να = −R(i α − i α ) + ωe + |ωe | λ0 sign(sα ),
? M
?
ˆ
νβ = −R(i β − i β ) + ωe + |ωe | λ0 sign(sβ ),
ˆ

(17
...
12)

where it has been taken into account that physical constraints of the device are such that a
M
maximum achievable velocity ωe and a maximum admissible nominal torque τ M exist
...
9) and (17
...

ˆ
dt
dt
L
This implying that
ˆ
ωe = ωe ,
that is, that the observation error vanishes
...
13)
(17
...
15)

whose derivative is
˙
˙
sω = ω ∗ +

¯
Kt
τ
¯
B
ˆ
ˆ
ω −
ˆ
(i cos(θe ) − i α sin(θe )) + + λ1 (ω∗ − ωe )
...
16)

∗
According to the previous expression, one can consider two reference currents: i β =
∗
∗
∗
∗
ˆ
ˆ
I (t) cos(θe ) and i α = −I (t) sin(θe ), whose tracking will be ensured later and where I (t) is
∗
∗
ˆ
ˆ
to be determined
...
17)

and choosing I ∗ (t) as follows:
¯
Kt ∗
B
τ
¯
˙
ˆ
I (t) = ω∗ + ωe + + λ1 (ω∗ − ωe ),
ˆ
¯
¯
¯
J
J
J

(17
...

∗
∗
The imposition of the conditions i α = i α and i β = i β can be easily carried out by considering
the following Lyapunov function:
W =

1
1
∗
∗
(i α − i α )2 + (i β − i β )2 ,
2
2

(17
...

ˆ
ˆ
ˆ
+ (i β − I (t) cos(θ
L
L
L
Splitting the previous inequality into the following ones:
?
?
λ0
R
1
ˆ
ˆ
ˆ
ˆ
(i α + I (t) sin(θe )) − i α + ωe sin(θe ) + vα + I˙(t) sin(θe ) + I (t)ωe cos(θe ) < 0,
L
L
L

(17
...
21)

375

Robust Fault Detection for a Permanent-Magnet Synchronous Motor

ˆ
and using the fact that both θe and ωe tend to θe and ωe , respectively, possibly within a finite
ˆ
time (as previously proved), it turns out immediately that a possible solution is the following:
?
?
1
M
˙(t) sin(θe ) − I (t)ωe cos(θe ) + R i α − ζ ωe λ0 + η sign(i α + I (t) sin(θe )),
ˆ
ˆ
ˆ
ˆ
vα = − I
L
L
L
?

?

(17
...
23)

with ζ > 1, η > 0
...
4

The Faulty Case

Variations of stator currents with respect to the healthy behavior may arise due to several
reasons, such as interturn short-circuit, inherent motor instrumentation asymmetries, load
variations, and unbalanced supply voltages (Quiroga et al
...
Such conditions can be
reflected in negative sequence components, whose spectral behavior at high frequency has been
largely studied in order to discriminate, not always easily, between fault occurrence and load
fluctuation (Quiroga et al
...
This section is devoted to show that the introduction of the
previously described observer allows to easily detect variations of stator currents with respect
to the healthy behavior, regardless of the presence of load variations within a prescribed range
...
24)

¯
i β = i β + f β (t),

where f α (t) and f β (t) represent (possibly time-dependent) perturbations affecting the stator
currents acting from a given time on
f α (t) = f 1 (t)δ−1 (t − T f α );

f β (t) = f 2 (t)δ−1 (t − T f β );

| f α (t)| ≤ ρ f α ;

| f β (t)| ≤ ρ fβ ;

f 1 (t) and f 2 (t) being unknown but bounded functions, and T f α and T fβ representing the times
of occurrence of the faults
...
11) and (17
...
9) and
ˆ
(17
...
Therefore ωe deviates
from the actual ωe
...
18) deviates from the variable corresponding to the vanishing of the surface
sω (17
...
19) is no longer ensured
...
4
...
1) fed by the controllers (17
...
23),
whenever it holds W (τ ) ?= 0 for any time τ ≥ t¯ then a fault has occurred at a time t ≤ τ , t¯
being the reaching time for the surface W = 0, W being given by (17
...

Proof: Recalling the proof of the previous result, it is verified that, whenever the auxiliary
inputs (17
...
12) are unable to enforce sliding motions on surfaces (17
...
10),
then ωe deviates from the actual ωe and the described control policy cannot ensure the vanishing
ˆ
of W
...
Note that load variations can be excluded since they are allowed as long as they
are below the admissible threshold τ M
...
5

Simulation Tests

Simulations have been performed using technical data of the Technosoft MBE
...
E500
PMSM, as a preliminary step before experimental implementation
...
1
...
1, 17
...
3, 17
...
5
...
1 t)
...
01 K g, with a 20% variation
...

Boundary layers have been used to avoid chattering
...
1

Technosoft MBE
...
E500 PMSM parameters

Coil-dependent parameters
Phase-phase resistance
Phase-phase inductance
Back-EMF constant
Torque constant
Pole pairs

?
mH
V/1000 rpm
mNm/A
–

8
...
13
3
...
8
1

V
V
mA
rpm
mA
mNm
rpm
mNm

36
58
73
...
01

0
...
03

0
...
05

Figure 17
...
01

0
...
03

Figure 17
...
15)

0
...
05

378

AC Electric Motors Control

I(t) (continuous line) and detection signal (dashed line)
6

4

2

0

−2

−4

−6

0

0
...
02

0
...
04

0
...
3 Variable I (t) (17
...
01

0
...
03

0
...
4 Sliding surfaces sα (continuous line), sβ (dashed line)

0
...
01

0
...
03

0
...
05

Time (s)

Figure 17
...
19)

Moreover, a time-varying fault has been considered to affect current i α additively as
follows:
?
1?
t−0
...
001 )
2

for t > 0
...

Figure 17
...
Notice the presence of an initial transient of duration of nearly ts = 0
...
This fact is also particularly evident in Figure 17
...
15)
...
3 reports the variable I (t) (17
...
0263 s, which
signals the detection of the time-varying fault
...
Finally, Figure 17
...
5 shows the Lyapunov function (17
...


380

AC Electric Motors Control

References
Awadallah M and Morcos M (2003) Application of AI tools in fault diagnosis of electrical machines and drives-an
overview
...

Awadallah M, Morcos M, Gopalakrishnan S, and Nehl T (2005) A neuro-fuzzy approach to automatic diagnosis and
location of stator inter-turn faults in CSI-fed PM brushless DC motors
...

Cusido J, Romeral L, Ortega J, et al
...
IEEE Transactions on Industrial Electronics, 55(2), 633–643
...
3rd IEEE Conference on Industrial Electronics and Applications, pp
...

Kim KH, Choi DU, Gu BG, and Jung IS (2010) Fault model and performance evaluation of an inverter fed permanent
magnet synchronous motor under winding shorted turn and inverter switch open
...

Liu L and Carter DA (2005) On-line identification and robust fault diagnosis for nonlinear pmsm drives
...
2023–2027
...
) (2000) Issues of Fault Diagnosis for Dynamic Systems
...

Quiroga J, Liu L, and Carter DA (2008) Fuzzy logic based fault detection of PMSM stator winding short and long
fluctuations using negative sequence analysis
...
4262–4267
...
(2006) Broken bearings and eccentricity fault detection for a permanent magnet
synchronous motor
...
964–969
...
1

Introduction

AC Motors have been widely used in various applications, such as factory automation, household electrical appliances, computers, CNC (Computer Numerical Control) machine tools,
industrial robots, high-speed aerospace drives and high-technology tools used for outer space
in the past decades (Jang et al
...
2006; Feng et al
...
There are two main
types of AC motors depending on the principle of operation, the induction motor (IM) and the
permanent magnet synchronous motor (PMSM)
...
The rotor of PMSMs turns
at the same speed as the rotating synchronous magnetic field of the stator which is generated
by the stator currents
...
The difference is called the slip, which
generates the torque of the motor (Trzynadlowski 2001)
...


AC Electric Motors Control: Advanced Design Techniques and Applications, First Edition
...

© 2013 John Wiley & Sons, Ltd
...


382

AC Electric Motors Control

PMSMs, on the other hand, can be used in specialized high-performance applications
requiring smooth and fast operation, and demanding low torque ripple because PMSMs offer
the advantages of relatively high power density, high efficiency, low rotor inertia, fast dynamics,
good compatibility, reduced rotor losses, efficient heat dissipation structure, reduced motor
size, the elimination of brushes, ease of control, and minimal maintenance requirements
...
Since PMSMs are mainly used in
specialized high-performance applications, this chapter focuses on the control of PMSM
servo systems, which play an important role in achieving the objective of high precision,
high speed, high efficiency, and high reliability in many practical applications
...
The control
of PMSM servo systems aims at the goal of good performances of servo systems subjected
to parameter variations and external load disturbances
...
2000)
...
1999)
...
2

Control System of PMSM

Vector control is an important technique used for the control of AC motors
...
A PMSM employing vector control can be used for high-performance
servo applications
...
1
...
An encoder, a tachometer, or a resolver can
be used to measure the speed of the rotor
...
Generally, the measurement of both the position and speed of the rotor can be done
using an encoder in practical applications
...
The design process of the three
controllers is from the inner loop (current loop) to outer loop (position loop)
...
The parameters of the current controller mainly depend on the measureable parameters of
the motor stator, such as the inductance and resistor
...

Secondly, the speed controller is designed based on the current closed-loop control system
...
Compared to the current controller and the position controller, the design of the
velocity controller is more challenging, since mechanical resonances, backlash, and friction
in servo systems mainly influence the performance and stability of the velocity loop
...
1 A typical structure of PMSM servo systems

the velocity loop to eliminate a single frequency or narrow band of frequencies, as shown in
Figure 18
...


18
...

(b) The eddy current and magnetic hysteresis loss are negligible
...

With the above assumptions and taking the rotor coordinates (d–q axes) of the motor as the
reference coordinates, the dynamic model for the PMSM can be expressed in the state-space
by both an electrical model (Su et al
...
2011):
?

˙
i d = − Rs i d + pωi q +
L
˙
i q = − pωi d −

Rs
i
L q

−

ud
,
L
pψ f
L

ω+

uq
,
L

(18
...
2005; Feng et al
...
2)

where u d and u q are the d–q axes stator voltages, i d and i q the d–q axes stator currents, L the
d–q axes inductance, Rs the stator winding resistance, ψ f the rotor flux, p the number of pole
pairs, TM the driving torque of the motor (Nm), TL the load torque of the motor (Nm), J the
moment of inertia of the motor rotor and the load (kgm2 ), B the viscous damping coefficient,
ω the angular speed of the motor (rad/s), and θ the angular position of the motor (rad)
...
2) can be expressed by the following equation (Zhu et al
...
3)

where L d and L q denote the d- and q-axis inductances, respectively
...
1) and (18
...
Therefore, equation (18
...
4)

where kq = (3 p/2)ψ f denotes an electrical-mechanical energy conversion constant, or simply
a torque constant, which can be determined by using the experimental data
...
4) that the driving torque TM is only proportional to i q and kq is a constant,
which means that TM can be controlled only by the q-axis current i q
...
4

PI Control of PMSM Servo System

PMSM servo systems generally utilize PI controllers in industrial applications
...
2
...
Three phase currents of the motor stator
(i a , i b , i c ) are transformed into the direct and quadrature components i d and i q in two rotor
coordinates (d–q axes)
...
On the other hand, i q is the component of the stator flux that
is producing the driving torque
...
2 PID control of PMSM servo system

ib
ic

385

On Digitization of Variable Structure Control for PMSMs

e

Iqm

iq

PI
controller

i qr

I qm

Figure 18
...
2, the anti-reset windup method (Shin 1998) can be used
...
3
...
5)
i q = k p eω + eω − (i q − i q ) ,
τi
kc
s
∗
r
where, kc > 0 is the compensation coefficient, both i q and i q are depicted in Figure 18
...

Equation (18
...

τi kc s + 1
kc s + 1 q

(18
...
5 High-Order Terminal Sliding-Mode Control of PMSM
Servo System
In order to improve the performances of PMSM servo systems, such as the precision, response
speed, and robustness, high-order terminal sliding-mode (TSM) control strategy is employed
to control PMSM servo systems (Feng et al
...
The structure of the
system is shown in Figure 18
...
2008)
...
4 High-order TSM control of PMSM servo system

ib
ic

386

18
...
1

AC Electric Motors Control

Velocity Controller Design

The design of the velocity controller can be described step by step as follows:
1
...
7)

where eω = ω∗ − ω denotes the motor speed error, ω∗ the desired motor speed
...
The TSM control strategies are designed as follows:
∗
i q = i qeq + i qn ,
?
?
J
B
ω ∗ + γω eω q ω / p ω +
˙
,
i qeq =
kq
kq

(18
...
9)

˙
i qn + Tω i qn = vω ,

(18
...
11)

where k T ω , Tω , and kd L are constants, which satisfy the following condition:
?∗ ?
?i ? < k T ω ,
qn
Tω
? ∗?
?T ? < kd L kq
...
12)
(18
...
The error speed of the motor eω = ω∗ − ω will converge to
zero in finite time
...
8)–(18
...
Although a switching function exists in equation (18
...
8) is softened to be smooth, hence there is no
chattering phenomenon in the TSM controller
...
5
...
A TSM manifold is chosen as follows:
q / pq

˙
sq = eq + γq eqq

(18
...

2
...
15)
q / pq

˙∗
u qeq = L i q + L pωi d + Rs i q + pψ f ω + Lγq eqq
˙
u qn + Tq u qn = vqn ,

vqn = (k Tq + ηq )sgn(sq ),

,

(18
...
17)
(18
...

Tq

(18
...


18
...
3

d-Axis Current Controller Design

1
...
20)

∗
where, γd > 0, pd and qd are odds, 1 < pd /qd < 2; ed = i d − i d = −i d is the current error;
∗
i d = 0 is the desired current;
2
...
21)

u d = u deq + u dn ,
u deq = Rs i d − L pωi q +
˙
u dn + Td u dn = vdn ,

q /p
Lγd edd d ,

vdn = (k Td + ηd )sgn(sd ),

(18
...
23)
(18
...

Td

(18
...


18
...
4

Simulations

Some simulations are carried out for evaluating the TSM control strategies of PMSM servo
systems
...
5 kW, n N = 1000 rpm, I N = 3
...
875 ?, L = 33 mH, J = 0
...
002, ψ f = 0
...
The
q-axis current limit is Iq max = 4 A
...
5 and 18
...
The motor speed responses for the different perturbation
of the inertia, ?J = ±0
...
5
...
2ψ f , are displayed in Figure 18
...
It can
be seen from these simulation results that the robustness of the TSM controller is better than
that of the PI controllers
...
5J

900

ΔJ = 0

800

ΔJ = –0
...
2

0
...
5J

900

ΔJ = 0

800

ΔJ = –0
...
6

0
...
2

0
...
6

t (s)
(a)

0
...
5 Motor speed response when J has a perturbation: (a) PI control; (b) high-order TSM
control

18
...
It is usually caused by a combination
of high servo gains and a flexible coupling between the motor and load, such as drive shafts,
belts, gears, harmonic speed changer, and so on
...
The phenomenon of mechanical resonance can deteriorate the system stability,
damage motors and transmission mechanisms, and reduce the system reliability
...


1050

1000

1000

950

Δψf = –0
...
2ψf

800

0

0
...
2ψf

950

Δψf = 0

900

0
...
2ψf

850
0
...
8

1

800

0

0
...
4

0
...
8

1

t (s)
(b)

Figure 18
...
Three kinds of methods have been proposed to suppress the
phenomenon of mechanical resonance
...
1
...
The notch
filter is utilized to reduce the effects of resonance in Schmidt and Rehm (1999)
...
If the mechanical parameters such as load inertia
or spring constant changes, the notch filter may be ineffective
...
1995)
...
This method is robust, but the system response becomes slow to the input
signal
...
The third method is to use the observer of the motor acceleration or
load torque (Ji and Sul 1995; Valenzuela et al
...
However, they still have disadvantages
such as high dependency on system models
...
2007; Zheng and Feng 2008)
...
A full-order state observer is applied to estimate
the load speed and the shaft torsion angle which cannot be measured directly in applications
...

Compared to the acceleration feedback and the notch filter method, the TSM controller can
suppress mechanical resonance more effectively, increase convergence speed of the system,
and reduce the overshoot
...
Based on the vector control principle, high-order TSM controllers are designed
for two stator currents i d , i q , and the load speed ω L , respectively, as shown in Figure 18
...

ˆ
ˆ
?θ is the estimate of the shaft torsion angle ?θ = θ M − θ L , and ω L is the estimate of ω L
...
7 PMSM servo system with mechanical resonance suppressing

390

AC Electric Motors Control

full-order state observer is utilized to obtain the two variables, for they are difficult to measure
directly in applications but the stator currents, the angular position and angular velocity of the
motor are measurable
...
6
...
e
...

L

(18
...

JL

(18
...
28)

˙p
lω = sω + γω sω ω /qω ,

(18
...
In order to eliminate
chattering, a third-order sliding mode controller is designed step by step as followings (Zheng
et al
...
The LSM manifold and the TSM manifold are chosen as equations (18
...
29),
respectively;
2
...
30)

where
i qeq

J M JL
=
bs pφ f

?

ω∗
¨L

+

?

2
bs
Ks
−
J p JL
JL

?

K s bs
?θ
(ω M − ω L ) +
J p JL

?

(18
...
32)

?
?
? b
b
¨ ?
˙
where kω > ?− J s2 TL + (−βω J s2 + J1L )TL + βω TL ?, ηω > 0, 1/J p = 1/JM + 1/JL
...


391

On Digitization of Variable Structure Control for PMSMs

18
...
2

d-Axis Current Controller Design

Suppose that ed is the error between the given value and the actual value of the d-axis current
i
...

∗
ed = i d − i d
...
33)

Then, the d-axis current error system is
˙
ed = − pω M i q +

Rs
ud
id −
...
34)

In order to realize fast convergence and better tracking precision, a nonsingular terminal sliding
mode manifold is designed as follows:
p /qd

˙
sd = ed + γd ed d

,

(18
...
The d-axis current controller can be
designed as following steps:
(a) An NTSM manifold is chosen as equation (18
...

(b) The control law is designed as follows:
(18
...
q
˙
[qd /(γd pd )ed d d + kd sgn(sd )]dτ,
u dn = L

(18
...
38)

0

where, kd > 0
...
34) can converge to zero in finite
time
...
6
...
e
...


(18
...

L
L
L

(18
...
41)

392

AC Electric Motors Control

where, γq > 0, pq , qq are all odds, 1 < pq /qq < 2
...
41);
(b) The control law is designed as follows:
(18
...
q
˙
u qn = L
[qq /(γq pq )eq q q + kq sgn(sq )]dτ,

(18
...
44)

0

where kq > 0
...
40) can converge to zero in finite
time
...
6
...
The parameters of the controllers are designed as βω = 0
...
008,
?
pω = 7, qω = 5, kω + ηω = 500 and kω = 200
...
8a
...
8b
...
The effect of suppressing mechanical resonance using the acceleration
feedback is better than the notch filter
...
The speed response time of the TSM control is similar to the notch filter
...
2

0
...
6

0
...
2

0
...
6

0
...
8 Step responses of the motor speed and load speed: (a) motor speed response; (b) load
speed response

393

On Digitization of Variable Structure Control for PMSMs

18
...
It means that all controllers for a PMSM servo system are executed by a program
...

There are many existing numerical approximation methods to transform continuous-time system representations to discrete-time and vice versa, such as standard z-transform, backward
difference, forward difference, bilinear transform, pole-zero mapping, impulse invariance,
Simpson’s rule, and matched z-transform, (Phillips and Nagle 1995)
...
Here, we use
backward difference and bilinear transform discretization method to digitize the continuous
TSM controllers in previous sections
...
45)

where u l (t) and u i (t) are two parts of the TSM control law
...
7
...
From equations (18
...
46)

(k−1)T
0

u i (t)dt
...
47)

?

(18
...
47) from equation (18
...


Therefore, the digitization of TSM controllers of PMSM servo system can be obtained using
the backward difference method as follows from equation (18
...


18
...
2

(18
...
It is the most commonly used method for converting controllers
from the continuous-time domain into the discrete time domain
...
48):
u(k) = u(k − 1) + u l (k) − u l (k − 1) +

T
(u i (k) + u i (k − 1))
...
50)

It should be mentioned that discretization of sliding mode control may result in periodic,
near-periodic, and irregular orbits
...
2012) that the reason of emergence
of these behaviors is due to the fact that the sampling period must satisfy certain boundary
conditions between the system parameters and the sampling period
...
For the purpose of the simplicity, the following two assumptions are
made:
(a) The LSM manifold is utilized to replace the TSM manifold in (18
...
51)
T

˙
where eω = ω∗ − ω denotes the speed error, ω∗ the desired speed, x = [eω eω ]
...
2) is time invariant, that is,
˙
TL = 0
...
52)

From equations (18
...
4), it can be obtained:
˙
eω = ω ∗ − ω = ω ∗ −
˙
˙
˙

pn ψ f ∗ B
TL
i + ω+
J q
J
J

(18
...
54)

B
ω + v
...
55)

kq
TL
v+

...
56)

∗
the control i q is designed as follows:

∗
iq = −

Then equation (18
...
56) with the time gives
¨
eω = −

˙
kq
kq
TL
v+
˙
= − v
...
57)

395

On Digitization of Variable Structure Control for PMSMs

The system (18
...
58)

˙
eω ]T , u = −kq /J v, A and b are as follows:
˙

where x = [ eω

?

0
A=
0

?
1
,
0

? ?
0
b=

...
51), it can be assumed c = [ c1 1 ]T
...
The control signal u in
system (18
...
59)

u = u eq + u n ,

where u eq = −(c T b)−1 c T Ax = −c T Ax and u n = −α(c T b)−1 sgn(g(x)) = −αsgn(g(x))
...
58) is converted into
the discrete form firstly:
x(k + 1) = e

Ah

x(k) +

?

h
0

e Ah dτ bu k ,

(18
...
System (18
...
61)

x 1 (k + 1) = x 1 (k) + vx 2 (k) − γ1 αsk ,

(18
...
System (18
...


(18
...
62) and (18
...
The behaviors of
the system can be described using the following theorem:
Theorem 18
...
1
of Lyapunov if

(Yu and Chen 2003): The system (18
...
63) is stable in the sense

− 1 < d < 1,
Furthermore, the system state is bounded by
|x 1 (∞)| ≤ γ1 α +

?
?
? υγ2 ?
?
?
? 1 − |d| ? < |γ1
...
64)

|γ2 |α

...
65)

396

AC Electric Motors Control

For analyzing the discretization behaviors, some strict assumptions have been made and the
linearized models of the PMSM systems have been obtained
...

But at least near the desirable operation points, the linearized models would help obtain
approximate boundary conditions for the sampling period
...


18
...
The typical
structure of the systems consists of the position, velocity, and current control loops
...
The
digitization of TSM controllers of PMSM servo system has been discussed for the purpose of
practical implementation
...


References
Baik C, Kim KH, and Youn MJ (2000) Robust nonlinear speed control of PM synchronous motor using boundary
layer integral sliding mode control technique
...

Edwards C and Spurgeon SK (1998) Sliding Mode Control Theory and Applications
...

Ellis G and Lorenz RD (2000) Resonant load control methods for industrial servo drives
...
1438–1445
...
Automatica,
38, 2159–2167
...
International Journal of Robotic and Automation, 19, 91–102
...
International Journal of Control, 80, 856–862
...
IEEE Transactions on Industrial Electronics, 56, 3424–3431
...

Jang JH, Sul SK, Ha JI et al
...
IEEE Transactions on Industrial Applications, 39, 1031–1039
...
IEEE Transactions on Industrial Electronics, 42, 564–571
...
(1999) Acceleration feedback control strategy for improving riding quality of
elevator system
...

1375–1379
...
IEEE Transactions on Education, 52, 157–168
...


On Digitization of Variable Structure Control for PMSMs

397

Rahman MA and Hoque MA (1998) Online adaptive artificial neural network based vector control of permanent
magnet synchronous motors
...

Shin HB (1998) New antiwindup PI controller for variable-speed motor drives
...

Schmidt PB and Lorenz RD (1992) Design principles and implementation of acceleration feedback to improve
performance of DC drives
...

Schmidt P and Rehm T (1999) Notch filter tuning for resonant frequency reduction in dual inertia systems
...
1730–1734
...
IEEE Transactions on Industrial Electronics, 52, 814–823
...

Utkin V, Guldner J, and Shi J (1999) Sliding Mode Control in Electromechanical Systems, (1st edn), Taylor & Francis,
London
...
IEEE Transactions on Industrial Applications, 41, 1458–1466
...
IEEE
Transactions on Automatic Control, 48, 1642–1664,
Yu X, Wang B and LX (2012) Computer-controlled variable structure systems: the state of the art
...

Zhang Y, Akujuobi CM, Ali WH et al
...
IEEE
Transactions on Industrial Electronics, 53, 1198–1208
...
Proceedings of the 33rd Annual Conference of IEEE Industrial Electronics Society, pp
...

Zheng J and Feng Y (2008) High-order terminal sliding mode based mechanical resonance suppressing method in
servo system
...
1–6
...
Proceedings IEEE International Workshop on Variable Structure Systems
(VSS), Antalya, Turkey, pp
...

Zhu G, Dessaint L-A, Akhrif O, and Kaddouri A (2000) Speed tracking control of a permanent-magnet synchronous
motor with state and load torque observer
...


19
Control of Interior Permanent
Magnet Synchronous Machines
Faz Rahman and Rukmi Dutta
School of Electrical Engineering and Telecommunications,
University of New South Wales, Australia

19
...
The availability of Neodymium in the earth’s crust is as plentiful as lead (though quite localized, such as in China, as of now)
...
The Sumitomo (later
Hitachi) production process offers the sintered variety with remanent flux density (Br ) as
high as 1
...
8
...
Both versions
also suffer from high temperature dependence of its flux density, becoming more prone to
demagnetization as the temperature of the magnet rises towards 100◦ C
...
The world market for the sintered variety is now 10 times larger than the
bonded variety, largely because of the more compact machines that become possible with the
sintered material
...
1
...
Edited by Fouad Giri
...
Published 2013 by John Wiley & Sons, Ltd
...
25
1
...
0

0
...
2

AlNi Co 9

ic
m

Ce

ra

dF
N

Nd

Fe

B

eB

@

@

25

10

o

0

C

o

C

0
...
5
0
...
1 Demagnetization characteristics of several PM materials

interest to machine designers and users
...
1, showing the knee of the B H characteristic encroaching into the second
quadrant at this temperature
...
This implies
some permanent demagnetization of the magnets, which must be avoided
...
A sufficient margin at the operating
point for preventing demagnetization at the maximum working MMF of the stator and at the
highest operating temperature and also achieving high energy product B Hmax , given by the
shaded area in Figure 19
...

Proper consideration of these issues leads to an IPMSM that yield high power density, high
efficiency and wide field weakening/constant-power speed range (CPSR)
...
The IPMSM, as opposed to the surface magnet SM, has the
magnets buried inside the rotor iron in slots
...
This also allows the q-axis inductance (L q ) of the
machine to be larger than the d-axis inductance (L d ), resulting in the exploitation
of the inherent reluctance torque capability of such a machine, while simultaneously weakening the magnet flux by armature reaction, allowing operation of the machine above the
base speed with constant power
...

Figures 19
...
The stators of these machines all have sine distributed
winding, as with other conventional AC machines
...
2 Cross-sections of some well-known IPM rotor structures: (a) Radially magnetized;
(b) tangentially magnetized; (c) inset magnet; (d) multi-layer magnets; and (e) V-shaped magnet

Cross-sections of some well-known IPM rotor structures are indicated in Figure 19
...

Depending on the direction of flux crossing in the air gap, the IPM rotors can be broadly
categorized as axially laminated, transversally magnetized, and radially magnetized
...
1995) can produce very large ratio of L q to L d , but
mechanical construction is expensive and complex
...
In the tangentially magnetized IPM rotor, the magnet poles
are in the form of spokes
...
Because of this reversal of inductance values, the reluctance torque
does not enhance the torque contributed by the magnets, as it does in a conventional IPM
machine (Bianchi and Bolognani 1999)
...
However, some recent studies are showing its potential in wind power generation
(Haraguchi et al
...


401

Control of Interior PMSM
Table 19
...
8 (?)
0
...
0448 (Henry)
0
...
7 (Nm)
1 (kW)

Since the IPM machine is a new emerging technology, none of the configurations yet has
been standardized
...
2011)
...
The
flux barriers/guards made of nonmagnetic material at the either sides of a magnet pole prevent
magnetic short circuit of the adjacent opposite poles
...
The conventional radial flux design has modest L q
d
ratio and a narrow constant power speed range
...
2011)
...
2004)
...
The required magnet volume in this
type of IPM rotor is larger than for other types
...
2d can
L
produce relatively large L q ratio, hence larger reluctance torque than any other types (Honda
d
et al
...
However, the design is relatively complex and mechanically less robust
...
With this design very good flux-concentration is possible by
optimizing the magnet pole pitch angle (Dutta and Rahman 2008)
...

The typical machine parameters of a radially magnetized IPM machine is shown in Table
19
...

A typical torque-speed characteristic of a radially magnetized IPM machine is shown in
Figure 19
...
Figure 19
...


19
...
Equations (19
...
3) are for d- and q-axes voltages,

AC Electric Motors Control

1
...
0
0
...
8
0
...
6
0
...
4
0
...
2
0
...
5

1 1
...
5

120

0

Power (kW)

300

Torque

Torque (Nm)

Power and torque in pu

402

40

0
0

2000 4000 6000 8000 10000 12000 14000

Speed (rpm)

(a)

(b)

Figure 19
...
3) for d- and q-axes flux linkages and equation (19
...
The circuit representation of the machine in the rotor dq frame, as indicated
in Figure 19
...
1):
dθ
di d
dλd
− λq
= Rd + L d
− ω L q iq ,
dt
dt
dt

(19
...
2)

vd = R d +
vq = Rq +

λd = L d i d + λ f
T =

λd = L q i q ,

?
?
?
3p ?
3p
(λd i q − λq i d ) =
λ f iq + L d − L q iq id
...
4 Circuit representation of an IPMSM in rotor d- and q-axes

(19
...
4)

403

Control of Interior PMSM

Figure 19
...
3) and (19
...
In order to understand the torque-speed envelope of the
machine and some of its control restrictions, it is useful to represent the machine in terms of
its steady-state equivalent circuit, in which the dq stator windings are stationary, but produces
their MMFs along the rotor dq axes
...

Figure 19
...
V0 is the rated phase voltage
...
θ and δ are the power factor and load angles, respectively
...
2
...
5), in which the first term, the
so-called reluctance torque, is due to the PM excitation and the second term is because of the
difference in the d- and q-axes inductances of the machine
...
6:
T =

3p
ωs

?

EfV
V2
sin δ +
Xd
2

?

Xd − Xq
Xd Xq

?

?
sin 2δ
...
5)

It is clear from equation (19
...
For operation at higher speeds, the inverter output voltage

404

AC Electric Motors Control

2
...
5
1
0
...
Deg
...
6 Developed torque components of an IPMSM

remains fixed at this value corresponding to the DC link voltage, while the speed is increased
by increasing the supply frequency ωs (= 2π f s )
...
The phasor diagrams of Figure 19
...
These
figures also indicate that operation at higher than base speed can be arranged by controlling
the phase angle of the input current phasor from the excitation voltage E f , when a current
source inverter drive is used
...
Equation
(19
...
It
other words, the increased reluctance torque afforded by increased −Id may allow reduction
in Iq , helping the drive to operate within its maximum current limit while at the same time
maximizing the total developing torque
...
Operation above this speed may not be required
...
4) that for an IPM machine, a given torque can be obtained by many
combinations of i d and i q
...
1986)
...
In other words, such applications require constantpower operation over an extended speed range
...
The CPSR is covered by weakening
the total air-gap field by armature reaction
...
The −ve i d helps the machine
to maintain torque so that a wide speed range is covered at constant power, by exploiting its

405

Control of Interior PMSM

reluctance torque capability
...


19
...
2 Optimum Control Trajectories for IPM Synchronous Machines
in the Rotor Reference Frame
The torque reference, T ∗ , which is determined by an outer speed loop, must be translated into
∗
∗
i q and i d references, so as to achieve the maximum efficiency and minimum inverter capacity
for both constant torque and constant power (field-weakening) speed ranges
...
4))
...
In equation (19
...
7
...

dβ
2
2

(19
...


Figure 19
...
7)

406

AC Electric Motors Control

Equation (19
...
i q can be determined by the outer speed loop (Sebastiangordon and Slemon 1987)
...

MTPA control was initially proposed by Jahns et al
...
1986) in order to achieve
high-efficiency operation of IPM motors by improving torque production in the constant torque
region
...
It was shown that
the MTPA trajectory is tangent to the q-axis at the origin of the rotor flux reference frame
axes and asymptotes to a 45◦ trajectory on a normalized i dn , i qn plane
...
This reflects the hybrid nature of the torque production
...
1994)
...
1990, Morimoto et al
...
These constraints can be expressed as
Is =
Vs =

?

?

2
2
i d + i q ≤ Ism ,

(19
...
9)

where Ism and Vsm are the available maximum current and voltage of the inverter/motor
...
9) may be expressed in terms of i d and i q instead of vd and vq
...
1) as follows:
?

vd
vq

?

?

R −ωL q
=
ωL d R

??

id
iq

?

?

0
+
ωλ f

?


...
10)

From equation (19
...
9),
Vs =

?

(R i d − ωL q i q )2 + (R i q + ωL d i d + ωλ f )2 ≤ Vsm
...
11)

Equation (19
...
12) if the stator resistance is neglected:
?

L q iq

?2

?2
?
+ L d id + λ f ≤

?

Vsm
ω

?2


...
12)

Equation (19
...
12) gives ellipses located at (0, − L d ) that become smaller as speed ω increases,

407

Control of Interior PMSM

30
0, –

q-Axis current (A)

20

λf
Current limit circle

Ld

ω1

10 ω2

0

ω3

–10

–20
Voltage limit ellipses
–30

–30

–20

ω1 < ω2 < ω3

–10
0
10
d-Axis currecnt (A)

20

30

Figure 19
...
8
...
7), (19
...
12),
respectively
...
8
...
Otherwise,
if the current vector lies outside the circle, the motor current will exceed its limitation
...

It is seen that the voltage limit trajectory is an ellipse which contracts when the rotor speed
increases
...
The d-axis current represented by this point is the so-called characteristic
current, Ich , of the machine
...


(19
...
If it is outside, the maximum −ve i d satisfying
the current limit of Equation (19
...
A special case applies when |Ich | = Ism , in which case the machine produces high
torque as it approaches infinite speed by exploiting the reluctance torque to the fullest extent
(the optimum field weakening condition) (Jahns 1987)
...
7), (19
...
12), respectively
...


19
...
3
...
9
...
The i d reference is
obtained from equation (19
...
The intersection of the trajectory at point A with the current
limit circle is found from
?
λ2f
λf
2
2
+ Ism − i d A
−
id A =
2(L q − L d )
4(L q − L d )2
λf
=
−
4(L q − L d )
iq A =

?

?

λ2f
16(L q − L d )2

+

2
Ism
,
2

2
2
Ism − i d A
...
14)

The intersection point A of the MTPA and the current limit trajectory occurs for a speed ωb
at which the machine develops its maximum torque
...
15)

?
?2
...
5

VL 1500 rev/min

A

iq (A)

VL @ 2200

Current limit

B
2
C

VL @ 2400
0
...
5

VL max
...
5

O
0

0
...
5

2

id (A)

Figure 19
...
i d A and i q A are
the current limits that must be placed on the d- and q-axes current controllers when operation
is along this trajectory during transient operation
...
3
...
11)
...
9, the voltage
limit trajectory for the base speed of 1500 rpm is indicated
...
14), along the intersections of
current and voltage limit trajectories for each speed
...
16)

where, vdo = −ωL q i q , vqo = ωλ f + ωL d i d , and Vom = Vsm − R Ism
...
12) with
Vsm replaced by Vom in order to include the effect of stator resistance drop
...

ω2

(19
...
17), the terminal voltage is always
kept within Vsm in the steady state
...
These limit values are given by

i dv = −

λ f Ld
1? 2 2
λ f L d − ab,
+
a
a

i qv =

?

V2

2
2
Ism − i dv ,

(19
...
19)

2
2
2
where a = L 2 − L q , b = Ism L q + λ2f − ωsm
...
9 for the crossover speed (ωc ) of
2400 rpm when the load is zero (or i d = i q = 0)
...
For example, when
the motor runs at 2200 rpm, the corresponding voltage limit trajectory is BCO in Figure 19
...

If the machine is heavily loaded, the i d − i q trajectory is along BC
...


410

AC Electric Motors Control
Speed
controller

ω* +
ω

*
iq

−

ω ≤ ωc?

Yes

*
iq

id from eqn 19
...
16
No

Voc ≤ V am ?

Yes
*
id = id

*
id = id

Field-weakening
control

+

Gcq(s)

−

No

id from eqn 19
...
10 Trajectory selection and control: (a) Trajectory calculation, (b) dq current control with
decoupling

19
...
3

Implementation Issues of Current Vector Controlled IPMSM Drive

As mentioned earlier, the speed controller output determines the q-axis current reference
∗
∗
i d
...
10a and b
...
10b can
be further decoupled using the back-emf and voltage drops in the dq axes as indicated in
this figure
...
10b
...
In the flux-weakening operation, the stator voltage is kept equal to the
∗
∗
maximum stator voltage Vsm
...
As a result, the dq current controllers can become
saturated and the current control performance may become poor
...

(1986), and suggested by Morimoto (Morimoto et al
...
The voltage compensation in
which vqo is determined from equation (19
...
11, can be used to
prevent the current controls from saturating
...
12
...
11 Voltage compensation scheme to prevent current controller saturation
∗
position feedback obtained from a high-resolution mechanical sensor
...
1) and (19
...

The limit values for the i d and i q current references are determined by equations (19
...
19), respectively, according to the trajectory selected
...
13a–d illustrate a few

iq*

Speed
controller

ω* +
−

T*

Current
reference
generator

ω

dq Current
controllers
*
va

vq*

+
−

v*
d

id* +

dqDecoupling,
voltage
compensation
–1
and dq

*
vb

vc*

SM
3- phase
PWM
inverter

−

ω

ia

iq
id

dq

ib
ic

d/dt

Figure 19
...
13 Trajectory following of a current vector controlled IPMSM drive
...
Precise trajectory following with
smooth transition between the two trajectories, using the controls indicated in Figures 19
...
11, and 19
...


19
...
The demanded torque reference from the outer speed control loop
primarily defines the i q current reference
...
14 Block diagram of a DTC drive in the stator reference frame

in order to exploit the machine’s reluctance torque and field-weakening capabilities
...

Mechanical sensorless drive systems have drawn industrial interest for a variety of reasons
(Corley and Lorenz 1998)
...
A
more direct torque and flux control approach than the current vector control of the previous
section is the so-called direct torque control (DTC), in which the developed torque T and the
stator flux linkage λs are controlled using estimations of these quantities from the machine
model
...
This approach is also
inherently mechanical sensorless
...
2004)
...
The control of torque T and λs is via
application of stator voltage vectors relative to the estimated rotor angular position or load
angle δ of the machine (Zhong et al
...
1998; Sayeef and Rahman 2009)
...
14 in which subscripts α and
β refer to the quantities in the stator α and β axes (α-axis being the magnetic axis of the
stator winding A)
...
This figure also indicates
the hysteresis torque and flux comparators, the discrete outputs of which, in conjunction of
the six sectoral information (θ ) of the rotor flux λ f , selects the required machine voltage
vectors for optimum torque and flux linkage response
...
2 and 19
...
1997; Rahman et al
...


414

AC Electric Motors Control

Figure 19
...
4
...
15, is based on applying the six to
eight voltage vectors available from the inverter and according to the errors in torque T and
stator flux linkage λs to the machine
...
14 are used,
full voltage vectors indicated in Figure 19
...
Additionally,
the current and voltage limits described in Section 19
...
The stator flux
linkage in the stator reference is obtained from a simple integration of the back EMF, as given
by equation (19
...
16 Stator flux trajectory with voltage of a two-level inverter

?


...
20)

415

Control of Interior PMSM

Figure 19
...
16 (left)
...
17
...
21), when the stator flux linkage is held constant:

T =
where i y =

1
2L d L q

?

3p
λs i y ,
2

(19
...

4L d L q

(19
...
23)

q
where a = L qs−L d , so that the limiting load angle δm is not exceeded
...
18
...
3:0
...
9

40

δm

35

Torque (Nm)

30
25
20
15
10
5
0

0

20

40

60

80

100

120

140

160

δ (°)
Figure 19
...
4
...


(19
...
25)
(19
...
7), the flux linkage and load angle trajectories
are given by

λs =

?

λ2f

−

?

?
?
?
L2
d
2 2
+ L q − L d λ f id + L 2 + L q id ,
d
Lq − Ld

δ = tan−1

?

?
?
λf
Lq
2
+ L d id id −
id
...
27)

(19
...
5
100
80

1
...
5
0

60
40
20

0

0
...
4

0
...
8
1
λs, (Wb)

1
...
4

0

0
...
4

0
...
8
λs, (Wb)

1

Figure 19
...
19
...
It is seen from Figure
19
...
As is also seen
from this figure, the load angle δ will not exceed δm with the MTPA control when the torque
is limited below the maximum torque the motor can produce
...
24)–(19
...
Provided the torque is known,
maximum torque per ampere (MTPA) control is achieved if the amplitude or the angle of the
stator flux is determined from the MTPA trajectory of Figure 19
...
For direct torque control, it is obvious that the torque and the amplitude of the
stator flux linkage, rather than its angle, should be controlled (Zhong et al
...
When the
torque and λs are controlled in this way, the angle δ will be automatically controlled and for
the prototype motor it will not exceed δm
...
e
...

The current and voltage constraints are rewritten below in equations (19
...
30):
|i d | =

?

2
2
Ism − i q ,

(19
...


(19
...
24), (19
...
29), the current limit trajectory in the T –λs plane can
be plotted as shown in Figure 19
...
Because the stator voltage is proportional to the product

418

AC Electric Motors Control

1500 rpm
(Base speed)

Torque (Nm)

2

2400 rpm

1
...
2

0
...
6

0
...
2

Current limit

40

0
...
5

1
...
2

λs (Wb)

0
...
6

0
...
20 (a) MTPA, current limit and voltage limit trajectories with DTC; (b) requirement for
limiting the minimum λs

of the rotational speed and the amplitude of the stator flux linkage, if the stator resistance is
neglected,
Vs = ωr λs ,

(19
...
In the steady state, the rotational
speeds of the stator flux linkage and the rotor magnet flux linkage are the same and equation
(19
...
32)

where ωr , ωb , and ωc are the rotor speed, base speed, and crossover speed, respectively
...
ωc is the crossover speed for which the unloaded motor develops
the rated phase voltage Vsm
...
1) and (19
...
From
equation (19
...
20a
...
The current and voltage limit trajectories in the T –λs plane
are as shown in Figure 19
...
If these trajectories are plotted in the δ–λs plane, it is revealed
that with very low λs (deep flux weakening), the limiting δm angle may be exceeded when an
IPMSM is operated at its current and voltage limits, as indicated in Figure 19
...

The current limit is satisfied if T and λs are controlled below the current limit trajectory
...
20a, which corresponds to the operating point with the maximum torque and
current
...
Below base speed, the voltage

Control of Interior PMSM

419

limit trajectories are to the right of the MTPA OA in Figure 19
...
Therefore, there is no
requirement here to control the amplitude of the stator flux linkage λs to satisfy the voltage
limit
...
For operation below the base speed, the MTPA should be selected
...
However,
for the operation between the base and crossover speeds, the control mode is determined by
load torque
...
20a
represents the voltage limit corresponding to the operation with the rotor speed between ωb
and ωc , there is an intersection of this line and the maximum torque-per-ampere trajectory, and
at this point the torque is TB
...
Otherwise, if the actual torque is smaller than TB , MTPA is selected even though
the rotor speed is above the base speed
...
4
...
21a
...
For field-weakening
operation, the amplitude of the stator flux linkage is simply determined by the inverse of the
speed
...
21b
...
21 (a) Block diagram of the speed loop for trajectory control in DTC; (b) flow chart for the
control mode selection

420

4
3

AC Electric Motors Control

1
...
5

1
Current
control

0

δrated

0

–1

–δ
rated

–0
...
05

57°

–1

–3
0

0
...
1
Time (s)
(a)

0
...
2

–1
...
5

–1

–0
...
5
λα (Wb)

1

1
...
22 (a) Torque control dynamics under DTC and current control; (b) δ angle limitation during
operation with MTPA

For applying DTC with trajectory control, it is necessary to check, if δ is smaller than δm
...

The field-weakening operation starts from point A in Figure 19
...
20b
...
After point C, at which δ is equal
to δm , δ should be limited at the value of δm
...
22, in which the amplitude of the stator flux linkage is already
known, with δm obtained from Figure 19
...

Figure 19
...
The dark trace represents torque response in the current controlled
system, while the faint trace indicates the torque response under DTC
...
22b, clearly showing the limit values
of δ during acceleration and deceleration along the MTPA trajectory
...
23a–e show
a few more dynamic responses for torque, fluxes and trajectory following, with MTPA and
field weakening controls
...
13
...
5

Sensorless DTC with Closed-Loop Flux Estimation

It is be noted from Figure 19
...
20 that the stator flux linkage, which is
used also to produce the feedback signal for torque, may be obtained from a simple, open-loop
integration of the machine back EMF
...
1

2
...
5
1
0
...
1
0
...
3

–0
...
1

0
...
1
0
...
5

MTPA

2

MTPA

Field
weakening

1
...
75
0
...
65
0
...
55
0
...
45
0
...
35
0
...
25
–0
...
5
0

0

0
...
2
Time (s)
(c)

0
...
3

0
...
4

0

0
...
4
0
...
8

1

1
0
...
6
0
...
2
–0
...
6
–0
...
5

0
0
...
23 DTC drive over MTPA and FW speed range: (a) speed response; (b) torque response; (c)
stator flux linkage response; (d) trajectory following; and (e) stator flux locus
...
Integration of signals that may have offset in measurements also causes unacceptable
errors at low speed
...
The errors in
flux estimation at low speed also causes unacceptable error in the estimation of speed that is
required for closing the speed loop in a speed control system
...
23 also
show the high torque and flux ripples when DTC is realized with cost-effective controllers
of limited sampling speed (about 75 μs)
...
14 for drives covering full rated torque cannot generally be guaranteed below 10%
of the base speed
...
2004)
...
14 are used to produce the stator voltage references vα and
vβ or the space vector modulation (SVM) in several ways (Xu and Rahman 2007; Boldea
et al
...
A sliding-mode (SM; variable-structure) controller is suitable for the nonlinear
IPMSM
...
Design of such a controller and the proof of its stability have been fully described
in references (Xu and Rahman 2007; Sayeef and Rahman 2009)
...

SM closed-loop flux observers based on forcing the errors in estimated currents from the
actual measured currents have delivered more accurate flux linkage estimation than is possible
with integrating the bemf for flux estimation
...
2003; Boldea et al
...
2008) with online compensations for switch voltage drops, inverter dead-time and stator

422

AC Electric Motors Control

PI
ω* +

SVM
T*

*
να

SM direct
torque
controller

–
λ s*
ˆ
T

ˆ
ω

Inverter

*
νβ

IPMSM

Vdc

ˆ
λs
νcomp Switch voltage
and dead-time
compensation
iα

SMO flux, torque,
speed and stator
resistance
observer and SKF

iα
iβ

iβ

(a)

150

6

100
0
20

0
...
5

Torque (Nm)

8

2

0
–20
0
7

4
2
0
0

0
...
5

0
...
5

2

0
...
5

1

1
...
5

3

1
...
5

3

2

6
5
4
0
0
...
56
0
...
5

1
Time (s)

(b)

1
...
55
0
...
45
0
...
24 (a) Sliding-mode DTC drive with closed-loop estimators for λs and speed; (b) steady-state
performance with full load torque 150 rev/min and (c) at 95 rev/min
...
Full
analysis and design of these SM controllers and flux observers are included in Xu and Rahman
(2007) and Sayeef and Rahman (2009) and will not be covered here
...
24a shows
the block diagram of the SM DTC drive with closed-loop SM observer for flux for a 4-pole
IPMSM with rated load torque applied
...
It
may be noted that persistent operation at very low speeds and at zero speed, with variation
of load torque from no load to full rated torque, is still not achievable with these methods
...


423

Control of Interior PMSM

Figure 19
...
2 and 19
...
25
...
Current
vector control of Section 19
...
3 is then used for indirect torque and flux control
(Corley and Lorenz 1998; Bolognani et al
...
2002)
...


19
...
Several high frequency signal injection techniques for current
vector controlled IPMSM drives have been reported in recent literature (Jung-Ik and SeungKi 1999; Spiteri et al
...
2003)
...
The following section will describe the dq
frame rotating injection that has been adopted industrially
...

We assume that a carrier frequency voltage given by equation (19
...

?

vdc
vqc

?

= Vc cos(ωc t)

? ?
1

...
33)

424

AC Electric Motors Control

The model of the machine to the carrier frequency voltage is then given by
?

vαc
vβc

?

d
≈
dt

?

L 0 + L 1 cos(2θr e )
−L 1 sin(2θr e )

−L 1 sin(2θr e )

L 0 − L 1 cos(2θr e )

??

i αc
i βc

?

,

(19
...
35)

L −L

where L 0 = q 2 d and L 1 = q 2 d are the average inductance and the amplitude of the
V
V
spatial modulation of the inductance, respectively, and Ic0 = ωc L 2L 0 2 and Ic1 = ωc L 2L 1 2
...
35) into the estimated dq axes, the HF component in the estimated
q-axis is
?
?
ˆ
ˆ
i qc = Ic1 cos(ωc t) sin 2(θr e − θr e )
...
36)

By using a band-pass, demodulation, and low-pass filters as indicated in Figure 19
...
26b
...
2010), the HF injection scheme loses its merit when the
drive is operated at high speeds where the separation between the fundamental AC frequency

( )

ˆ qc
i

ˆ
ε ≈ 2Kε sin 2θre
BPF

LPF

sin(ωct)
(a)

θm +
–

~

θm

2Kε

ε

PI
LPF

ˆ
ωm 1
s

ˆ
θm

(b)

Figure 19
...
LPF, low-pass filtering

425

Control of Interior PMSM

W1 1

W2

–500

–100

W1

0 +100

W2

+500

Rev/min

(a)
iabc

ω*

+

+–

T*

–

eT

+

–

Variable
structure
controller

ˆ
T

PI

ˆ
ω

eλ

Vhf , f c

Torque
estimator

λs
Flux and
speed
handover

ˆ
ω SMO
ˆ
λ SMO

HF signal injection

λ

*

Vα*
Vβ

*

SVM and
FVD, DT
compensation
Vdc

IPM
SM

Inverter
ia

ib

SM flux and
speed observer
id dq

λ sc

Current model
flux estimator

iq

ab

θ

ˆ
ωshf

HF signal
injection-based
position estimator

(b)
Figure 19
...
Clearly, the HF injection is best applied at very
low speed and it should be removed when the drive reaches a reasonable speed
...
The transfer
algorithm may consist of a linear transition weighting of 0 − 1 between speeds of ±100 and
±500 rev/min, as indicated in Figure 19
...
The flux and speed
estimations are mainly supplied by the HF injection scheme (i
...
, W1 = 1) from 0 to ±100
rev/min, linearly reducing to zero at ±500 rev/min while these are mainly supplied by the
SM observer (i
...
, W2 = 1) from and ±500 rev/min to top speed, linearly reducing to zero at
±100 rev/min
...
27b shows the combined HF and SM flux observers with SM DTC
...
28 shows the steady-state (at zero speed) and dynamic performance of the controller
of Figure 19
...
Figure 19
...
The HF injection scheme fully supplies the position, speed,
flux linkage and torque feedback signals during this test
...
28b shows these signals

AC Electric Motors Control

0
–200
0

10

15

20

25

30

–1000

0

5

5

10

15

10

15

20

20

25

25

30

30

2

2
...
5

4

4
...
5

3

3
...
5

2

2
...
5

4

4
...
6

Act Spd (rpm)

0

Torque (Nm)

0
–200
8
6
4
2
0
–2

0

2
...
5

4

4
...
5

3

3
...
5

1000

200

0
...
6
0
...
28 Performance of the combined SM DTC and HF injection scheme of Figure 19
...
[experimental results]

when the machine is driven with speed reference reversing between ±1000 rev/min
...


19
...
It starts with some description of the magnetic circuits of the machine
followed by a description of its steady-state torque-speed characteristics in a stator quadrature
frame and its dynamic model
...
The above
models are used in determining the references for its inner torque control
...
There are no
issues with operating this machine under such control to cover a wide speed range from zero
to full field weakening speed range
...
Sensorless control
techniques, which employ the direct torque control and high-frequency injection approaches,
are described in detail
...
The high-frequency signal injection scheme which estimates the stator
flux with adequate accuracy by exploiting the saliency of such machines for operation at very
low and zero speed is then described
...


References
Bianchi N and Bolognani S (1999) Performance analysis of An IPM Motor with segmented rotor for flux-weakening
application
...

Boldea I, Pitic CI, Lascu C, et al
...
IEEE Transactions on Power Electronics, 21(3), 711–
719
...
IEEE
Transactions on Power Electronics, 23, 2612–2618
...
IEEE Transactions on Industrial Electronics, 46, 184–191
...
IEEE Transactions on Industry Applications, 39(5), 1372–1378
...
IEEE Transactions on Industrial Electronics, 50, 288–295
...
IEEE Transactions on Industry Applications, 43(4), 784–789
...
IEEE Trans on Energy Conversion, 23(1), 25–33
...
IEEE Transactions on Industrial Electronics, 57(4), 1270–1278
...
IEEE Transactions on Energy Conversion, 25(1), 25–33
...
IEEE Transactions on Power Electronics, 18(6), 1376–1383
...

In Electrical Machines and Systems ICEMS 2009, 1–6
...
Electric Power Applications, IEE Proceedings, vol
...

Honda Y, Higaki T, Morimoto S, and Takeda Y (1986) Interior permanent-magnet synchronous motors for adjustablespeed drives
...

Jahns TM (1987) Flux-weakening regime operation of an interior permanent magnet synchronous motor drive
...
on Industry Applications, 23, 398–407
...
IEEE Transactions on Industry Applications, 22, 738–747
...
IEEE Transaction on Industry Applications, 35(1)
...
Transactions of the Institute of
Electrical Engineers of Japan, 126, 473–479
...
Conference REC IEEE-IAS Annual Meeting, 86–91
...
IEEE Transactions on Industry Applications, 40(5), 1369–1378
...
University Press, Oxford
...
IEEE Transaction on Industry Applications, 26, 866–871
...
IEEE Transactions on Industry Applications, 30, 920–926
...
IEEE Transactions on Industrial Electronics, 55, 570–576
...
IEEE Transactions on Industry Applications, 34(6), 1246–1253
...
In SAE
World Congress, Detroit, Michigan
...
Journal of Power Electronics (JPE),
9(3), 438–446
...
IEEE
Transactions on Industry Applications, IA-23, 327–333
...

IEEE Transaction on Industry Applications, 31, 358–367
...
Power Electronics Machines and Drives Conference Publication (487) IEE2002
...
IEEE Transactions
on Power Electronics, 19(2), 346–354
...
IEEE Transactions on Magnetics, 47(10), 3606–3609
...
IEEE Transactions on Industrial Electronics, 54(5), 2398–2406
...
IEEE Transactions on Power Electronics, 12(3), 528–536
...
1

Introduction

Compared to others, synchronous motors present several benefits
...

It is well known that for WRSMs (and generally for all AC machines), the speed variation can
only be performed by changing the stator frequency
...

In the case of AC supply, the (three-phase) net is connected to the three-phase DC/AC inverter
through a transformer and an AC/DC rectifier
...

The control problem at hand is to design controllers able to ensure speed regulation for the
system including the AC/DC converter, the DC/AC inverter and the WRSM (Figure 20
...

As the rotor is wound, a variable DC voltage source is needed
...
1
...
Then, undesirable current harmonics are
likely to be generated in the AC line
...
Edited by Fouad Giri
...
Published 2013 by John Wiley & Sons, Ltd
...
1 Structure of the AC/DC/AC-WRSM system

voltage distortion in the AC supply line and cause electromagnetic compatibility problems
...
The last objective is referred to
power factor correction (PFC) (Singh et al
...

In most works dealing with WRSM control, the control design is simplified by only
focusing on the subsystem “DC/AC inverter—Machine
...
The simpler problem has been dealt with using several control strategies ranging from simple techniques, for example, field-oriented control (Saleh et al
...
1998),
direct torque control (Pyrhonen et al
...
2006), or nonlinear adaptive control (Tomei and Vorelli 2008)
...
First, the
controller design relies on the assumption that the DC voltage (provided by the AC/DC
rectifier) is perfectly regulated
...
” The second drawback lies in the entire negligence of the PFC
requirement
...
(2010a) and El Magri et al
...

A nonlinear multiloop control strategy has been developed and shown to meet speed regulation
and PFC
...
It
turns out that the controlled “AC/DC/AC-WRSM” system is directly supplied by a three-phase
power grid
...
First, a
three-variable loop is designed that (i) makes the speed motor track its varying reference value;
(ii) regulates the d-component of stator current to zero optimizing thus the absorbed stator
current; and (iii) enforces the excitation current (and consequently the flux in the excitation

431

Nonlinear State-Feedback Control

winding) to track a given reference signal
...
Finally, a reactive power loop is designed that controls the
reactive power delivered to the supply grid
...
1995)
...
This parameter uncertainty is coped with by equipping the controller
with a parameter adaptation capability
...
These theoretical results are
obtained making judicious use of adequate control theory tools (Khalil 2003)
...
e
...
2; the controller
design and the closed-loop system analysis are presented in Section 20
...
4
...
2

System Modeling

The controlled system is illustrated by Figure 20
...
It consists of three subsystems: an AC/DC
boost rectifier, a combination “inverter-synchronous motor,” and a DC/DC converter (excitation circuit)
...


20
...
1

Three-Phases AC/DC Rectifier Modeling

The considered rectifier is a three-phase DC/AC converter of boost type (Figure 20
...
It
involves six semiconductors (insulated gate bipolar transistors (IGBTs) with antiparallel diodes
for bidirectional current flow mode) displayed in three legs a, b, and c
...
Applying Kirchhoff’s laws, this subsystem is shown to be described by the following
AC/DC
converter
Ka
AC Grid

R
S
T

L0

egc egb ega

Kb

is
Kc

iga
igb
igc

ir

V
dc

2C
'
Ka

'
Kb

'
Kc

Figure 20
...
1a)
(20
...
1c)

where [i gabc ] = [ i ga i gb i gc ]T is the three-phase input currents in the electric grid, [egabc ] =
[ ega egb egc ]T is the sinusoidal three-phase net voltages (with known constant frequency
ωg ), vdc denotes the voltage in capacitor 2C, ir designates the output current converter and
ki (i = a, b, c) is a binary control input determining the switch position
...
2)
ki =
(i = a, b, c)
...
1a) for the synthesis of control laws, the Park
transformation is resorted to project the triphase electrical quantities onto the two-coordinate
dq-frame
...
3a)
(20
...
3c)

where (E gd , E gq ), (i gd , i gq ), and (u gd , u gq ) denote, respectively, the network voltage and
current and input control of the rectifier in dq-coordinate
...

The instantaneous power absorbed by the AC/DC converter is given by the well-known
expression PLoad = [egabc ]T [i gabc ] = E gd i gd + E gq i gq
...
Using the power conservation principle, that is,
POut ≈ PLoad , one gets ir vdc = E gd i gd + E gq i gq
...
3a-c), this rewrites:
2vdc

?
dvdc
1 ?
1
= E gd i gd +
E gq i gq − vdc ir ,
dt
C
C
di gd
1
1
E gd + ωg i gq −
u gd vdc ,
=
dt
L0
L0
di gq
1
1
E gq − ωg i gd −
u gq vdc
...
4a)
(20
...
4c)

433

Nonlinear State-Feedback Control

DC/AC
converter

S4
S1

S2

S3

Vdc

WRSM

S1'

'
S2

'
S3

Figure 20
...
2
...
3
...
The whole system
modeling is generally accomplished in the rotating dq-coordinate frame
...
It is
shown in El Magri et al
...
(2010b) that the WRSM model, expressed
in the dq-coordinates, can be given the following state space form:
dω
dt
di sq
dt
di sd
dt
di f
dt

=−

a2
1
F
a1
ω + i sd i sq + i f i sq − TL ,
J
J
J
J

(20
...
5b)

= −c1 i sd + c2 R f i f + c3 i sq ω + c4 vsd − c5 v f ,

(20
...
5d)

where (vsd ,vsq ) denotes the averaged stator voltage in dq-coordinates, v f denotes the averaged
rotor excitation voltage
...
1
...
To this end, these voltages are expressed in function of the corresponding
control action (Michael et al
...

In turn, the averaged current writes:
i r = u 1 i sq + u 2 i sd + u 3 i f ,
where u 1 = u sq ; u 2 = u sd represent the averaged d- and q-axis components of the three-phase
duty ratio system (s1 , s2 , s3 ); u 3 = u f is the averaged duty ratio of s4
...
1

Notations used in the WRSM model (20
...


inputs si (i = 1
...


(i = 1, 2, 3)

(20
...
6b)

Now, let us introduce the state variables,
x 1 = ω, x 2 = i sq , x 3 = i sd , x 4 = i f , x 5 = v 2 , x 6 = i gd , and x 7 = i gq ,
dc
and the inputs,
u 4 = u gd , u 5 = u gq
...
Wherever it comes in, the averaging is intended over the PWM cutting
periods
...
For
convenience, the whole model is rewritten here for future referencing:
d x1
dt
d x2
dt
d x3
dt
d x4
dt
d x5
dt

=−

a1
a2
1
F
x 1 + x 2 x 3 + x 2 x 4 − TL ,
J
J
J
J

(20
...
7b)

= −c1 x 3 + c2 R f x 4 + c3 x 1 x 2 + c4 u 2 vdc − c5 u 3 vdc ,

(20
...
7d)

=

?
1 ?
1
E gd x 6 +
E gq x 7 − vdc i r ,
C
C

(20
...
3

1
u 4 vdc ,
L0
1
u 5 vdc
...
7f)
(20
...
3
...
7a–d), a controller has to be developed in order to achieve the
following control objectives:
CO 1: The machine speed ω must track, as closely as possible, a given reference
signal ωref
...

As there are five control inputs at hand, namely, u 1 , u 2 , u 3 , u 4 , and u 5 , we seek three additional
control objectives, namely:
CO 3: Regulate the current i sd to a reference value i sdref , preferably equal to
zero in order to guarantee the absence of d-axis stator current, implying thus no
reluctance torque
...

In particular, no direct magnetization/demagnetization along the d-axis is needed,
that is, only the field winding contributes to producing the flux along this direction
(Muhammad and Rashid 2001)
...

CO 5: Control the continuous voltage vdc making it track a given reference signal
vdcref
...

The above control objectives must be achieved despite the fact that the load torque TL , the
inertia J , the viscous friction F, and the rotor resistor R f are presently allowed to be unknown
and changing
...
The adaptive controller design will be performed using the backstepping
technique (Krstic et al
...
This technique has already used in previous chapters (e
...
, 10
and 13)
...
Given the system complexity and the control objective
multiplicity, the resulting adaptive controller will consist of several loops the design of which
will be organized in two major stages
...


436

20
...
2

AC Electric Motors Control

Inverter-Motor Subsystem Control Design

The control inputs that act on the inverter-motor subsystem are (u 1 , u 2 , u 3 )
...
These objectives will be separately investigated
in three separate subsections, where the control law of each input (u 1 , u 2 , u 3 ) is established
...
Based on equations (20
...
1995), so that the motor speed ω tracks well any reference ωref
...
7a–b) is of relative degree 2, the design towards that equation is performed in
two steps
...

=

(20
...
7a), this error undergoes the following equation:
˙
z1 = −

F
1
TL
˙∗
x 1 + (a1 x 3 + a2 x 4 ) x 2 −
− x1
...
9)

In (20
...
Following the backstepping design technique, the following Lyapunov function
candidate is considered:
2
V1 = 0
...


(20
...
9) yields
?
?
∗
˙1 = z 1 z 1 = z 1 − F x 1 + 1 α − T L − x 1
...
11)

This suggests the following control law for the (virtual control) α:
˙∗
α ∗ = −k1 J z 1 + F x 1 + TL + J x 1

(20
...
Indeed, substituting α ∗ to α gives V1 = −k1 z 1 that is negative
definite in z 1
...
Then, these must be replaced in
ˆ ˆ
ˆ
equation (20
...

Doing so, one gets the following stabilizing function:

ˆ
ˆ
ˆ
ˆ ˙∗
α ∗ = −k1 J z 1 + F x 1 + TL + J x 1
...
13)

437

Nonlinear State-Feedback Control

As α = (a1 x 3 + a2 x 4 ) x 2 is just a virtual control input, one cannot set α = α ∗
...


(20
...
13)–(20
...
9) that the z 1 -dynamics undergo the
following equation:
˙
z 1 = −k1 z 1 +

˜
˜
˜
? TL
z2
J ?
F
˙∗
+
k1 z 1 − x 1 −
−
x1 ,
J
J
J
J

(20
...

Step 2: Now, the aim is to make the couple of errors (z 1 , z 2 ) vanish asymptotically
...
14):
˙∗
˙
˙
˙
˙
ˆ
z 2 = (a1 x 3 + a2 x 4 ) x 2 + (a1 x 3 + a2 x 4 ) x 2 − α
...
16)

Using equations (20
...
15) and the subsystem (20
...
16) yields the following
z 2 -dynamics:

with

˙
z 2 = α1 u 1 vdc + α2 u 2 vdc + α3 u 3 vdc
?
?
?
?
ˆ
ˆ 2
¨∗
˙∗
+β (x 1
...
17)

β (xi=1
...
For convenience, the error equations (20
...
17) are put together:
˙
z 1 = −k1 z 1 +

˜
˜
˜
? TL
z2
J ?
F
˙∗
+
k1 z 1 − x 1 −
−
x1 ,
J
J
J
J

(20
...
18b)

J

γ = β (x 1
...
19)

Note that the actual control inputs (u 1 , u 2 , u 3 ) have come out for the first time in the second
differential equation (20
...

To determine a stabilizing control law for (20
...

2
2
2J
2J
2J L

(20
...
18a), one gets from (20
...


˙
˙
˙
V2 = z 1 z 1 + z 2 z 2 +
=

˜
J
J

˙˜
J+

˜
F
J

˙
˜
F+

˙ˆ?
J

(20
...
21) can be canceled by using the following
adaptive laws:
?
?
?
?
˙ˆ
2
ˆ
ˆ
˙∗
˙∗
J = z 1 k1 z 1 − x 1 − k1 z 1 + ( J k1 − F) k1 z 1 − x 1 z 2 ,

˙
2
ˆ
ˆ
ˆ
F = −x 1 z 1 − z 2 − ( J k1 − F)x 1 z 2 ,

˙
ˆ
ˆ
ˆ
T L = −z 1 − ( J k1 − F)z 2
...
22a)
(20
...
22c)

Substituting (20
...
21), one gets
1
F 2
2
˜
˙
z + (a1 c2 − a2 d1 ) x 2 x 4 z 2 R f
...
23)

This suggests for the control variable γ the following choice:
γ = −k2 z 2 ,

(20
...
Indeed, substituting equation (20
...
23) yields
(20
...

V
J
J

(20
...
24) in (20
...

J

(20
...
27)

J

Finally, combining (20
...
19) gives a first equation involving the three actual control
inputs (u 1 , u 2 , u 3 ):
?
?
ˆ
˙∗
(α1 u 1 + α2 u 2 + α2 u 2 )vdc = −k2 z 2 − β(x 1
...
28)
?
? ˙
?
˙ˆ ?
˙
ˆ
ˆ
ˆ 2
¨∗
˙∗
+ J k1 z 1 + x 1 − k1 z 2 − J k1 z 1 − x 1 + T L + F x 1
...
To harmonize notation throughout this
∗
section, the corresponding tracking error is denoted z 3 = x 3 − x 3 = x 3
...
7c) that z 3 undergoes the differential equation z 3 = −c1 z 3 + v with,
v = c2 R f x 4 + c3 x 1 x 2 + c4 u 2 vdc − c5 u 3 vdc
...
29)

˙
Clearly, v acts as a virtual input in the differential equation z 3 = −c1 z 3 + v
...
The point is that the
parameter R f is unknown making this resistor not accessible to measurements
...
30)

ˆ
˙
where R f is an estimate (yet to be found) of R f
...
31)

440

AC Electric Motors Control

˜
ˆ
where R f = R f − R f
...

ˆ

(20
...
30) to v in (20
...


(20
...
33) cannot be performed before a variation
ˆ
˜
law is associated to R f = R f − R f (or, equivalently, to R f )
...
In other words,
the update law for R f must selected bearing in mind both control objectives CO 3 and CO 4,
which concern the currents i sd and i f , respectively
...
30) and (20
...
Combining these, one gets
a second equation in the three control inputs (u 1 , u 2 , u 3 ), that is,
ˆ
− c4 u 2 vdc + c5 u 3 vdc = R f c2 x 4 + c3 x 1 x 2 − c1 z 3 + k3 z 3
...
34)

Excitation Current Control Design
According to the control objective CO 4, the excitation current x 4 = i f must track its reference
∗
signal x 4 = i fref , generally equal to the nominal value of that current
...
7d) that this error undergoes the following equation:
˙∗
˙
z 4 = −d1 R f x 4 + w − x 4 ,

(20
...


(20
...
35) as a virtual control input
...
35) is a first-order
system, it can be asymptotically stabilized using a simple proportional control law, that is,
˙∗
w = d1 R f x 4 + x 4 − k4 z 4 , with k4 is any positive real parameter
...
35) gives z 4 = −k4 z 4 which clearly is globally asymptotically stable
because k4 is positive
...
Then, the above
ˆ
control law is replaced by its equivalence form, simply obtained by substituting R f to R f :
ˆ
˙∗
w = R f d1 x 4 + x 4 − k 4 z 4
...
37)

Combining (20
...
35), one obtains the following differential equation describing the
closed-loop for the current i f :
˜
˙
z 4 = −k4 z 4 − d1 R f x 4
...
38)

441

Nonlinear State-Feedback Control

Combining equations (20
...
37), one gets a third equation in the control inputs (u 1 ,
u 2 , u 3 ):
ˆ
˙∗
d4 u 2 vdc − d5 u 3 vdc = − R f d1 x 4 − d3 x 1 x 2 + d2 x 3 + k4 z 4 − x 4
...
39)

Control Laws and Rotor Resistor Update Law
Solving equations (20
...
39) with respect to (u 2 , u 3 ), one gets
?

u2
u3

?

=

1
vdc

?

c4 −c5
−d4 d5

?−1 ?

?
ˆ
c1 z 3 − k 3 z 3 − R f c2 x 4 − c3 x 1 x 2

...
40)
ˆ
˙∗
R f d1 x 4 − k 4 z 4 − d2 x 3 + d3 x 1 x 2 + x 4

Now that (u 2 , u 3 ) are available, one can obtain u 1 from (20
...
4 ) − F(k1 z 1 − x 1 )
˙ˆ
˙
˙
ˆ
ˆ
˙
− J (k z − x ∗ ) + T + F x }
...
41)

1

ˆ
ˆ ˆ ˆ
The above control laws involve the parameter estimates J , TL , F, and R f
...
22a–c)
...
To this end, consider the following augmented Lyapunov function:
V = V2 +

?
1? 2
2
˜f
z3 + z4 + R2
...
42)

Using equations (20
...
33), and (20
...
42) gives
˙
˙
˜
˜ ˙
˙
˙
V = V2 + z 3 z 3 + z 4 z 4 + R f R f
? 2 1
?
2
2
2
= −k1 z 1 − k2 + F z 2 + J z 1 z 2 − k3 z 3 − k4 z 4
J
?
˙ ?
˜
˜
+ R f c2 x 4 z 3 − d1 x 4 z 4 + (a1 c2 − a2 d1 ) x 2 x 4 z 2 − R f
...
43)

˜
The last line on the right side of equation (20
...


(20
...
3
...
40) is actually invertible because its determinant
c4 d5 − c5 d4 , equal to L sd L 1−M 2 , is nonzero
...
In this regard, recall that vdc is the DC voltage at the
output of the AC/DC rectifier
...
That is, the (theoretical)
singularity risk may only occur in transient periods
...
Practically, this situation is
mainly faced at start-up stages
...
In practice, it is also usual to substitute in (20
...

Theorem 20
...
2 (inverter-motor control performances)
...
7a-d), and the adaptive
regulator (20
...
41)
...

(1) The error vector Z 1 = [z 1 z 2 z 3 z 4 ]T undergoes the following equation:
˙
˜
Z 1 (t) = B1 Z 1 (t) + η(Z 1 (t), ?),

(20
...

⎟
⎟
⎟
⎟
⎠

(2) Let the design parameters (k1 , k2 , k3 , k4 ) be any positive real numbers such that δ > 0,
F
J
def
˜
with δ = min(k1 , (k2 + )) −
...

?
? T
˜ T is defined by equations
Proof: The system with the augmented state vector Z 1 ?T
(20
...
44), and (20
...
46)
(20
...
22a–c) and
(20
...
To analyze the nonautonomous system (20
...
42) and its time-derivative (20
...
Substituting the right side of (20
...
42),
one obtains the time derivative of Lyapunov function V :
?
?
F
1
2
2
2
2
˙
V = −k1 z 1 − k2 +
z 2 + z 1 z 2 − k3 z 3 − k4 z 4
...
48)

443

Nonlinear State-Feedback Control
2
2
Using the well known inequality z 1 z 2 ≤ 1 (z 1 + z 2 ) in (20
...

V ≤ −δ z 1 + z 2 − k3 z 3 − k4 z 4 with δ = min k1 , k2 +
J
2

(20
...

˜
This immediately implies that V , and so z i (i = 1
...
Furthermore,
applying Lasalle’s invariant principle, it follows from equation (20
...


20
...
3

Reactive Power and DC Voltage Controller

In this subsection, we seek the realization of the control objectives (CO 2) and (CO 5)
...
The objective (CO 5) entails the regulation of the continuous voltage vdc making
it track a given reference signal vdcref
...
The DC voltage regulation loop is designed first
...

DC voltage Loop
Based on equations (20
...
As the subsystem (20
...

Step 1: Let z 5 denote the squared DC-link voltage tracking error:
∗
z 5 = x5 − x5
...
50)

In view of (20
...
7 , z 1
...
51)

where
β1 =

1
vdc
(u 1 x 2 + u 2 x 3 + u 3 x 4 )
...
51), the quantity ρ = C E gd x 6 stands up as a (virtual) control input for the
z 5 -dynamics because the actual control input u 4 acts on z 5 indirectly through ρ
...

2
Deriving V5 along the trajectory of (20
...

V5 = z 5 z 5 = −z 5 −
C

(20
...


(20
...
Indeed, substituting ρ ∗ to ρ = C E gd x 6 gives V5 = −k5 z 5
which clearly is negative definite in z 5
...
Nevertheless, the above expression of ρ ∗ is retained and a new error is introduced:

z6 = ρ − ρ ∗
...
54)

Using (20
...
51) that the z 5 -dynamics undergoes the following equation:
˙
z 5 = −k5 z 5 + z 6
...
55)

Step 2: Now, the aim is to make the couple of errors (z 5 ,z 6 ) vanish asymptotically
...
54), that is,
˙
z6 =

E gd
˙
˙
˙
¨∗
x 6 + k5 z 5 + β1 (x, z) − z 5
...
56)

Using equations (20
...
7e–f) in (20
...
57)

with
˙
˙
¨∗
β2 (xi , z i ) = k5 z 5 + β1 (x, z) − x 5 +

2
E gd

C L0

−

E gd
ωg x 7
...
58)

To determine a stabilizing control law for (20
...
Using (20
...
55), one gets from the expression of
2
2
V6 that
?
?
2
˙6 = −k5 z 5 + z 6 z 5 + β2 (x, z) + E gd u 4 vdc
...
59)

This suggests for the control variable u 4 the following choice:
u4 = −

C L0
(k6 z 6 + z 5 + β2 (x, z)) ,
E gd vdc

(20
...
Indeed, substituting (20
...
59) yields
2
2
˙
V6 = −k5 z 5 − k6 z 6 < 0
...
61)

445

Nonlinear State-Feedback Control

Now, substituting (20
...
57) one obtains the DC voltage closed-loop control system:
˙
z 5 = −k5 z 5 + z 6 ,

(20
...


(20
...
In this respect, note that the electrical reactive power injected in the grid is given
g
by Q g = E gd x 7 − E gq x 6
...
It follows from (20
...
63)

?
?
˙g
β3 (x 6 , vdc ) = −ωg E gd x 6 + E gq x 7 − Q ∗
...
63) is a first order, it can be (globally asymptotically) stabilized using a
simple proportional control law:
?

E gq u 4 − E gd u 5

? vdc
= −k7 z 7 − β3 (x 6 , vdc )
L0

with

k7 > 0
...
64)

Then the control law u 5 is given by
u5 =

?

?
E gq
L0
(k7 z 7 + β3 (x 6 , vdc )) +
u4
...
65)

It can be easily checked that the dynamic of z 7 undergoes the following equation:
˙
z 7 = −k7 z 7
...
66)

The DC voltage and reactive power regulators, defined by (20
...
65), are analyzed
in the following theorem
...
3
...
7e–g) and
the control laws (20
...
65)
...
67)
⎞
0
0 ⎠
...


z 7 ]T converges

z6

Proof: Equation (20
...
62) and (20
...
It is clear
that the matrix B2 is Hurwitz, implying that the closed loop system (20
...
This completes the proof of theorem 20
...
3
...
3
...
This is a direct result of Theorem 20
...
2
...
3
...


20
...
4
...
4 is simulated using the Matlab/Simulink
(V
...
The ODE14x (extrapolation) solver is selected

Ifref

+

if

-

regulator

if

u3

DC/AC

converter

converter
u5

Qg
Qgref

+

WRSM

DC/AC
converter

u4

u2

Qg

isd

controller

u1

regulator

-

isdref

Vdc

regulator

Vdcref

+ -

Vcd

+ -

isd
Speed
controller

-

Speed
measurements

AC-Grid

AC/DC

+ Speed
reference

Figure 20
...
2

System characteristics

Characteristics

Symbol

Supply network
Three-phase voltages
Network frequency
AC/DC/AC converters
Inductor
Capacitor
Modulation frequency
Synchronous motor
Nominal power
Stator resistor
d-axis stator inductance
q-axis stator inductance
Number of pole pairs
Combined rotor and load inertia
Combined rotor and load viscous friction

Value

Unity

fg

380–220
50

V
Hz

L0
C

10
...
00
10
...
00
1
...
40
37
...
22
0
...
The controlled part is a system including the associated three-phases
AC/DC/AC power converters and the wound rotor synchronous motor with the numerical
values of Table 20
...
The adaptive nonlinear feedback controller, including the control laws
(20
...
41), (20
...
65), is also implemented using Matlab/Simulink resources
...

For convenience, let us recapitulate the underlying control and update laws generating the
five control actions (u 1 , u 2 , u 3 , u 4 , u 5) :

?

ˆ
˙∗
u 3 vdc − k2 z 2 − β(x 1
...

ˆ
Rf
2 4 3
1 4 4
1 2
2 1
2 4 2

As a matter of fact, the control performances depend, among others, on the numerical values
given to the controller parameters, that is, ki (i = 1
...
The point is that there is no systematic
way, especially in nonlinear control, to make suitable choices for these values
...
Doing so, the following
(nonunique) suitable numerical values are obtained:
k1 = 9; k2 = 100; k3 = 20; k4 = 20; k5 = 10; k6 = 900; k7 = 9
...
According to the control design (Section 20
...
5 A,
vdcref = 600 V and Q gr e f = 0 V A R
...
e
...


20
...
2

Simulation Results

Presently, both the rotor speed reference ωref and load torque TL are filtered step-like signals (Figure 20
...
First, Figure 20
...
1
0
...
05
−0
...
5
4
3
...
5
2

2

4

0

2

F (Nm/rad/s)
6

0

2

4

Time (s)

6

(c) Control signals u1, u2

10

8

10

8

, u3

10

x 10−3

10
5

0

2

4

6

8

10

6

8

10

6

8

10

Time(s)
0
...
5
4
3
...
5
2

Time (s)

8

(b) d -axis stator current isd (A) and Excitation current if (A)

J (Nm/rad/s2 )

isd (A)

0
...
05
0
−0
...
1
4

10

Time (s)

(a) Rotor speed ω(rad/s) and Load torque TL(Nm)

2

8

6

4

Time (s)

0

6

Time (s)

0

30
20
10

0

2

4

0

2

4

Time(s)

Time(s)

(d) inertia J (Nm/rad/s2), viscous friction F (Nm/rad/s)
and rotor resistor Rf (Ω) variations

Figure 20
...
41) and (20
...


449

Nonlinear State-Feedback Control

40
Qg (VAR)

600

0

–20
400

ig1 (A) & e (V)
g1

Vdc (V)

500

20

300
200

0

2

4
6
Time (s)

8

0

4
6
Time (s)

2
...
1
Time (s)

8

10

10
0
−10
2

10

(a) DC-link voltage Vdc (V)

2

2
...
2

(b) Reactive power Qg and unitary power factor
checking (PFC)
1

80
U4

60
40
ig1 (A)

20

0

0
−20

0

2

4

0

2

4

0
...
5

−80
−100

0
...
6 Tracking performances of the controller defined by (20
...
65) in response to the
varying speed reference and load torque
...
Filtering is resorted to
make these references derivable and their derivatives available because these are needed in the
controller
...
5d
...
0096Nm/rd/s, between t = 9 s and t = 9
...
Similarly, a variation of 100% (resp
...
the rotor resistor) between t = 8 s and t = 8
...
t = 7 s and 7
...
All parameter variations are entirely ignored in the controller which solely relies
on the nominal values
...
5 and
20
...

Curves (a) and (b) show that the machine speed, x 1 = ω, the d-component of the stator
current, x 3 = i sd and the excitation current x 4 = i f , all perfectly converge to their respective
references; confirming thus theorem (20
...
2)
...
3s for all variables
...
5c
...

Figure 20
...

Figures 20
...
6c show respectively the reactive power (Q g ) extracted at the threephases AC-grid and the corresponding current in phase a (i ga )
...
Figure 20
...
5a)
...
6b)
...
This is particularly demonstrated by Figure 20
...


20
...
The controlled system is
connected with a three-phase supply grid
...
The system dynamics have been described by the averaged seventh order nonlinear statespace model (20
...
Based on this model, the multiloop nonlinear controller defined by the
control laws (20
...
40), (20
...
65) is developed using the backstepping design
technique
...
This theoretical result is confirmed
by simulation
...
Control Engineering Practice, 18, 540–553
...
IFAC Symposium on Nonlinear Control Systems, pp
...

El Magri A, Giri F, Abouloifa A, and Haloua M (2006) Nonlinear control of wound-rotor synchronous-motor
...
3110–3115
...
(2009) Nonlinear control of association including synchronous motors and
AC/DC/AC converters: a formal analysis of speed regulation and power factor correction
...
3470–3475
...
International Journal of Adaptive Control and Signal Processing, 26, 10
...
2271
...
Prentice Hall
...
John Wiley & Sons,
Inc
...
International Conference on Power Electronics and Variable-Speed Drives, pp
...

Michael J, Ryan D, and Rik W (1998) Modeling of sinewave inverters: A geometric approach
...

Muhammad H and Rashid (2001) Power electronics handbook
...


Nonlinear State-Feedback Control

451

Pyrhonen O, Niemela M, Pyrhonen J, and Kaukonen J (1998) Excitation control of DTC controlled salient pole
synchronous motor in field weakening range
...
294–298
...
IEEE Transactions on Energy Conversion, 19, 95–101
...
Power India Conference, p
...

Tomei P and Verrelli CM (2008) A nonlinear adaptive speed tracking control for sensorless permanent magnet step
motors with unknown load torque
...


Part Five
Industrial
Applications of AC
Motors Control

AC Electric Motors Control: Advanced Design Techniques and Applications, First Edition
...

© 2013 John Wiley & Sons, Ltd
...


21
AC Motor Control Applications
in Vehicle Traction
Faz Rahman and Rukmi Dutta
School of Electrical Engineering & Telecommunications, University of New South Wales,
Australia

21
...
2011)
...

The impetus for the current trend towards HEVs and EVs stems not only from considerations
of environmental issues and fuel saving but also from adding enhanced functionalities and
features that can be derived from electric traction (Hori et al
...
At the heart of a modern
EV or HEV is an electric motor that can deliver bidirectional torque quickly and precisely
over a wide speed range on demand set by a driver or the vehicle control system
...
Ability to produce high torque at low speed, including zero speed, for operation without a
clutch
...

2
...
The base speed is the speed up to which the motor is capable of
producing its maximum or rated torque
...
Operation of the motor and its drive circuits over the whole speed range with high efficiency,
high reliability and low maintenance
...
Four-quadrant operation with regenerative braking, returning the overhauling energy into
a storage device
...
Edited by Fouad Giri
...
Published 2013 by John Wiley & Sons, Ltd
...
Recently developed switched
reluctance motors are also showing signs of comparable performance, notwithstanding the
poorer volt-ampere capacity of its power converter, compared to the other AC machines
...
These
make possible many new functionalities to be included in EVs (Hori et al
...
Although
various configurations of HEVs and EVs have been tried over more than a century, HEVs
became commercially viable since 1997 when Toyota introduced the Prius model
...
Toyota has now sold more than 2
...
Almost all major
vehicle makers now have an HEV product line
...
GM Volt, introduced in 2011, uses this strategy
...

Even Boeing has declared its intention of using electric propulsion in subsonic aircraft in order
to shorten the take-off distance and to reduce noise
...
These
combine an ICE with an electric motor for traction, and for regenerative braking in which some
of the energy of the overhauling vehicle are returned to a battery for later use in traction and
start-stop duty
...
The latter two modes of operation immediately improve the fuel
efficiency of the vehicle
...

The development and adoption of commercial electric and plug-in electric vehicles have lagged
behind HEVs mainly owing to lack of cost- and space-effective battery capacity
...
Figure 21
...
1b
...
1 (a) Configuration Toyota parallel HEV and (b) its planetary gear system

Driveline
shaft

457

AC Motor Control Applications in Vehicle Traction

(a)

(b)

Figure 21
...
(For a
color version of this figure, please see color plates
...
A separate smaller capacity electric
machine, operating as generator, may also be used for charging the battery when the ICE
drives the traction load
...
Once the acceleration is completed,
that is, during steady-speed running, the ICE drives the vehicle
...
Re-use of the regenerative braking energy and use
of the ICE for nearly constant speed operation are primarily responsible for the fuel saving,
while faster vehicle dynamics are afforded by a combination of electric traction, ICE and
the additional battery charging machine, if used
...
2a–b depict the Toyota and Ford
HEVs
...
3, in which an electric motor of lower capacity than the Toyota system is located
on the driveshaft between the ICE and the transmission
...

Figure 21
...
In this scheme, an
ICE/generator charges a battery that supplies the traction motor
...
This scheme is easily amenable to plug-in EV with drive range
extension via the ICE
...
5 depicts an Audi retro-fit approach to HEVs in which the front-wheel ICE drive
is kept unchanged except for some reduction of the ICE capacity, while adding an independent
electric traction to rear wheels
...

The advent of pure electric vehicles of configuration indicated in Figure 21
...
Some of the drive power electronics in
this scheme may be shared between plug-in charging and traction, as indicated in this figure
...
3 (a) Configuration of Honda Insight parallel hybrid and (b) schematic of Honda Insight
drivetrain
...
)

in this category
...

This chapter primarily aims to describe machine and control techniques that are applied in
control of traction of modern automotive vehicles
...
The focus is on the control of the traction motor

D
F
ICE

G

Legend
ICE
G
C
S
C
M
DF

C

S

C

M

Signification
Internal combustion engine
Generator
Charger
Electricity storage devices
Converter
Electric motor
Differential

Figure 21
...
5 (a) Schematic of an Audi experimental HEV, and (b) photo of rear electric drivetrain from
PCIM2010, Nuremberg
...
)

Figure 21
...
7
...
Elaborate descriptions of the power converters
and the energy management of the storage devices such as batteries, ultra-caps and fuel cells
are excluded
...
7 Schematic of a plug-in EV system

460

AC Electric Motors Control

21
...
1 Electromechanical Requirements for Traction Drives
in the Steady-State
Torque-speed Requirements for Vehicle and Motor
Before an electric motor can be contemplated for electric traction, the short-time and steadystate torque-speed requirements for the vehicle which the traction motor must match needs to
be considered
...

dt 2

(21
...
2)

where V and Vair are the vehicle and tail wind velocities, A is the frontal area of the vehicle,
ρ is the density of air and C d is it’s drag coefficient
...
3)

where Rof is the rolling resistance coefficient in kg/kg and m vf is the vehicle kerb mass (kg)
referred to the front wheel, g is the acceleration due to gravity and α is the road gradient:
Frr = rear wheel rolling resistance force = Ror m vr g cos α N,

(21
...
5)

where the grade angle α should be taken as negative when the vehicle is moving down grade
...
This
difference in speeds is normally expressed as a ratio, called the slip-speed ratio λ, which is
the ratio of the slip speed to vehicle speed
...
The longitudinal aerodynamic friction coefficient indicated in
equation (21
...
8 (Miller 2003)
...
1) must take into account the effective gear ratios and their transfer efficiencies,
so that the reflected traction force on the motor and ICE shaft can be determined
...
Assuming

Coefficient of friction, μ

AC Motor Control Applications in Vehicle Traction

461

Longitudinal
driving force

Slip-speed ratio, λ

Figure 21
...
The tractive force Ftrac in equation (21
...
Tractive forces along the pitch (lateral) and yaw (vertical,
along gravity) axes, which also govern the dynamics of the vehicle along these axes are in
general of secondary consideration (Miller 2003)
...
The speed and acceleration of the
vehicle along the longitudinal direction is governed by input signals from the acceleration
and brake pedals indicated in Figure 21
...
Signals from both pedals stand for the desired
torque, accelerating or braking
...
The maximum traction force is required at low speed, from start of motion, when the
? 2?
acceleration d x2 is the highest
...
1)
...
Over this speed range, the first term in equation (21
...

Because production of maximum torque is desired in this mode, the air-gap flux should
be maintained constant; that is, the machine should be driven with its maximum torque
per ampere (MTPA) characteristic
...
The maintenance of flux to a constant level during this period of operation usually
requires decoupled vector control of the machine
...

3
...
It
may be noted that the maximum short time torque/current rating of an electric motor is
about twice its continuously rated torque/current limit from power loss and heat dissipation

462

AC Electric Motors Control

T (Nm)

Constant
torque

Constant
power

Tmax
Continuous
operation

Trated
Short time
operation

Prated
Tmin

ωb

(rad/s)

ω max ≈ 5 × ω b

Figure 21
...
The torque-speed envelops of machines for traction loads generally have
a large constant-power speed range as indicated in Figure 21
...
Most electrical machines
generally have a CPSR, except machines with permanent magnets at the surface of the air
gap
...
9 indicates the rated toque-speed envelope in the steady
state, indicating constant torque and constant power speed ranges
...
The base speed
ωb is also defined as the speed at which the machine develops its continuous rated torque
and power with rated inverter output voltage, the available DC link supply voltage and rated
magnetic field in the air gap
...

4
...
In order to maintain steady speed beyond the base speed, the torque demand is
much lower than the maximum torque mentioned above
...

5
...
Thus a maximum speed, ωmax , is defined beyond which the motor cannot meet the
torque demand
...
Note that in the speed range
of wb to wmax , the maximum power which the machine can develop remains fixed because
of the voltage and current rating limitations of the machine and the inverter
...
It has been
found that for clutch less (i
...
, without variable gearbox) operation over the desired vehicle
speed range, the required CPSR ≈ 5
...
Such constant power operation above the base speed
is normally achieved by weakening the air-gap field of the machine linearly with speed,

463

AC Motor Control Applications in Vehicle Traction

so that operation along the full CPSR may take place without exceeding the voltage and
current limits of the machine
...
It should be noted here that machines with such wide CPSR have to be
carefully designed in order to cover this speed range
...
In order to have adequate dynamic performance for longitudinal traction, and for pitch
and yaw motion controls, the motor drive system needs to employ high-performance controllers
...
Sensorless control
techniques, such as direct torque control (DTC), are also capable of achieving the required
response requirements, with the advantage of not requiring the delicate electromechanical position and speed sensor
...
For
mechanical sensorless drives, this is still a significant challenge
...
1
...
6)

where J is the total equivalent rotating moment of inertia, ωb is the base speed and Tm is the
rated maximum torque which the motor develops
...
6),
J

??

ωb
0

dω
+
PR /ω

?

ωrv
ωb

dω
PR /ω

?

(21
...


From equation (21
...

2TF b

(21
...
8) becomes
ωb
?

1+

ωmax
2
CPSR2 ωrv

?

=

2
J ωb
2TF

?

1+

2
ωb
2
ωrv

?


...
9)

464

AC Electric Motors Control

If ωb = 0, so that the entire speed range is covered with constant power and CPSR = ∞, the
required power rating of the machine is minimum, with a value
PR =

2
J ωb
Watts
...
10)

If ωb = ωmax , so that the entire speed range is covered with constant torque and CPSR = 1,
the required power rating of the machine is given by
PR =

2
J ωb
Watts,
TF

(21
...
10)
...
25PR min
...
12)

Thus, a high CPSR reduces the required power rating of the traction motor, based on the
assumptions taken for this simple analysis
...

CPSR > 5 may lead to diminishing returns because of the increase of Faero with vehicle speed
...
2

Machines and Associated Control for Traction Applications

Traction machines which have been adopted for automotive vehicles in recent years include
induction machines (IMs), interior permanent magnet synchronous machines (IPMSM), and
switched reluctance machines (SRM)
...

Table (21
...

The present scarcity of permanent magnet materials like Neodymium and associated materials
to enhance its temperature withstand capability has led to renewed development efforts in
SRMs that require no PM material (Rahman and Schultz 2002; Takano et al
...
Promising
developments in enhancing the torque density and efficiency of SRMs were reported by
Rahman and Schultz (2002)
...
(2010) compares very
favourably with a similar sized Toyota IPMSM machine used in its current Prius HEVs
...

It may be noted that an SRM based HEV was used in a showcase GM Commodore vehicle in
Australia as far back as in 2000 (Lovatt and Dunlop 2002)
...
Also because of this MTPA rather
than rated flux operation is preferred in the low-speed region
...

The IPMSMs typically offer efficiencies higher than 92% over the full CPSR (Dutta and
Rahman 2008; Olszewski 2008; Reddy et al
...
The IMs offer lower efficiencies and
larger volume because of the extra electric loading in the stator in order to establish the
rotor flux (Gosden et al
...
Figures 21
...
1

Traction motor, ICE and battery data of a few recent HEV and EVs

Vehicle

Traction machines used

ECE data

Battery

Honda Insight
hybrid, 2010
Ford Fusion hybrid

Brushless PMDC, 9
...
3 L petrol

100
...
75 Ah

2
...
8 L petrol, 73 kW
@ 5200 rev/min

201
...
5 L V6 petrol

Inverter DC link
voltage = 600 V
37 kW, NiMH

1
...
4 L Petrol

Li-Polymer

Induction machine, 215 kW

–

PMSM, 47 kW

–

Li-Ion, 375 V, 53 kW;
range: 365 km
16 kWh, Li-ion, range:
160 km
22 kWh Li-ion, range:
185 km
32 kWh, Li-ion, range:
151 km
23 kWh, Li-ion, range:
122 km

Renault Fluence
EV
BMW Active EV

PMSM, 7 kW

–

PMSM, 125 kW

–

Ford Focus BEV

PMSM, 107 kW

–

traction motors superimposed on their limiting T − ω characteristics
...


21
...
1

Induction Machines

This machine has the advantage of being low-cost and simple construction and wide availability
...
Unlike PM machines, the position of the rotor flux vector (or phasor) cannot be
directly measured for an IM
...
2008)
...
By fully decoupling the d- and q-axes equations of the machine, the slip

466

AC Electric Motors Control

300

180
140

90
200
88
150

86

100

75

120
100

84

50

Torque (Nm)

Torque (Nm)

250

70

160

92

82

80

80
82
84
86

60

88

40

90

20
0

500 1000 1500 2000 2500 3000 3500 4000 4500 5000 5500 6000

84
84
80

1

Speed (rpm)

3
2
Speed (rpm × 1000)

(a)

5

4

(b)

450
400
Torque (Nm)

350

91%
92%

85%

300

93%

250

94%

200

90%

150

95%

100

96%
97%

50
0

0

1000 2000

4000 5000

3000

6000 7000 8000

Rotational speed (rpm)

(c)

Figure 21
...
1994), and (c) an SRM (Takano et al
...
(For a color version of this figure, please see color plates
...
The IM machine
dynamics in this reference frame, the slip speed, rotor flux and torque equations are given as
⎡

vds

⎤

⎡

Rs + pL s

⎥ ⎢
⎢
⎢ vqs ⎥ ⎢ ωe L s
⎥ ⎢
⎢
⎥=⎢
⎢
⎢ vdr ⎥ ⎢ pL m
⎦ ⎣
⎣
vqr
(ωe − ωr )L m

−ωe L s

pL m

Rs + pL s

ωe L m

pL m

(ωe − ωr )L r

−(ωe − ωr )

Rr + pL r

ωe − ωr = ωsl =

L m Rr
i qs ,
λr L r

−ωe L m

⎤⎡

i ds

⎤

⎥⎢ ⎥
⎥ ⎢ i qs ⎥
⎥⎢ ⎥
⎥⎢ ⎥,
−(ωe − ωr )L r ⎥ ⎢ i dr ⎥
⎦⎣ ⎦
Rr + pL r
i qr
pL m

(21
...
14)

467

AC Motor Control Applications in Vehicle Traction

L r dλr
+ λr = L m i ds ,
Rr dt
3P L m
λr i qs
...
15)
(21
...
11
...
16), and in turn defines the slip reference from equation (21
...
Below
the base speed ωb , the rotor flux remains at the rated value defined by equation (21
...
At higher speed, the rotor flux reference, λr , is reduced in order
to maintain the motor/inverter voltage at the rated value
...
The slip speed ωsl cannot increase beyond a certain limit from power loss consideration,
and in any case, the limited slip mode of operation in the high-field weakening range is not
useful for traction applications
...
12
...

These can be overcome in a number of ways
...
Neural and observer methods that do not require a temperature sensor are also
effective (Telford et al
...
2005)
...
11 The RFOC control structure for induction machines

E

468

AC Electric Motors Control

T (Nm)

ω sl max
ω sl

Speed (rev/min)
Figure 21
...
11 (West and Lorenz 2009)
...
Space vector modulators, replacing the
hysteresis controller that introduces variable switching frequency, have also been used
...
The torque and flux control responses of the DTC scheme of Figure 21
...
13 DTC scheme for an induction motor drive

–

is β

–

469

AC Motor Control Applications in Vehicle Traction

be just as fast as, if not higher, than the RFOC scheme of Figure 21
...
The elimination of
the position sensor, considering the harsh environment in which a traction drive is expected
to operate in, combined with the fast torque and flux responses of the DTC drive, makes
this scheme a strong contender for traction drives
...
17), in the stator orthogonal xy
reference frame:
? t
?
?
¯
¯
¯
vs − Rs i s dt + λso ,
¯
(21
...
18)

where
σ =1−

L2
m

...
19)

Also,
¯
¯
¯
λs = L s i s + L m i r ,

¯
¯
¯
λr = L m i s + L r ir ,
?
Lr ? ¯
¯
¯
λs − σ L s i s = λrx + jλry ,
λr =
Lm
?
?
λr y
θr = tan−1

...
20)
(21
...
22)
(21
...

dt

(21
...
This implies, from equation (21
...
When the inverter applies full voltage vectors to the motor without significant delay,
and the delays in the controllers are also negligible, the torque and flux responses under DTC
can be as fast as, or even faster than, RFOC
...
Typically, sampling frequency of the order
of about 25 kHz is called for
...
14a and b compare the dynamic torque responses of
an IM under RFOC and DTC
...
17) and
(21
...

The parameters used in the computation of torque and rotor flux using equation (21
...
194 0
...
198

0
...
202 0
...
206

–8
0
...
694

0
...
698

Time (s)

(a)

0
...
702

Time (s)

(b)

Figure 21
...
22) also have to be compensated
...

Furthermore, the inductances L m amd L r may both suffer from magnetic saturation when the
applied voltage and load are high, so these must also be adjusted with the help of look-up
¯
¯
tables using i s and ir or by using some observer techniques
...
17) for estimating the stator flux is not very accurate
at low speed
...

Figure 21
...

The DTC scheme for vehicles also entails operation of the machine with the rated rotor
flux up to the base speed ωb , followed by linear reduction in rotor flux with speed, up to
the maximum speed ωmax
...

When operation at very low speed is required, for instance, when the vehicle is operated at
crawling speed for parking, other means of improving the very-low speed performance with
stability is required
...
Traction control for vehicles must have full
zero-speed performance with guaranteed stability
...
Industrial adoption
this HF injection technique is still awaited
...
2

0

0
...
4

0
...
8

8

1

Torque (Nm)

6
4

Current mode estimator

2
0
–2
–0
...
2

0
...
6

0
...
1
SMO

Current mode estimator

0
...
2

0

0
...
4
Time (s)

0
...
8

1

Figure 21
...
2
...
1
...

Although the IPM synchronous machines in early HEVs did not deliver a sufficiently high
CPSR (it was not perhaps needed because the motor was required only to operate below the base
speed for acceleration duty only), however, traction machines for EVs must have a CPSR close
to five or higher
...
PM machine structures are still evolving vigorously
...
16, has
been adopted in the HEVs from Toyota and a number of other vehicle manufactures and
in a few recently commercialised EVs
...
17 and 21
...


472

AC Electric Motors Control

400

SUV-’05

Torque (Nm)

300

120

Power

200

80
Torque

100
0

0

2000

Power (kW)

Prius-’00

160

40

0
4000 6000 8000 10 000 12 000 14 000
Speed (rpm)

Figure 21
...
(For a color version of this figure, please see color plates
...
17 The electromagnetic design of the SPM machine with fractional-slot winding developed
at the University of Wisconsin-Madison (Reddy et al
...
(For a color version of this figure, please
see color plates
...
18 The electromagnetic design of the SPM machine with fractional-slot winding developed
at the University of New South Wales (Dutta et al
...
(For a color version of this figure, please see
color plates
...

(1998)
...
These techniques have been reported further
in Foo et al
...
(2010)
...


21
...
3

Switched Reluctance Machines

The SRM, having no PM excitation in the rotor and a doubly salient structure as indicated
in Figure 21
...
These qualities have provided the incentive
for using this type of machine for vehicle traction, leading to recent developments of SRMs
which are claimed to have power densities and efficiencies comparable to IPMSMs (Rahman
and Schultz 2002; Takano et al
...


θ = 0°

θ = –15°

θ = +15°

A

A
D

D

B

T

C

C

D

B
A

θ = 0°

T

B

C

C

B

D
A

Figure 21
...
(For a color version of this figure, please see color plates
...
20 (a) Torque production mechanism of phase A of an SRM; (b) Torque versus current
versus displacement characteristic of an SRM

The control of torque of an SRM is via control of amplitude, duration and angular position
of rectangular phase current pulses relative to the rotor position
...
Placement of a phase current
pulse over the displacement angle where the rate of change of co-energy of the winding with
respect to the rotor angular position is the most negative than for all other windings maximizes
the reverse (regenerative) developed torque, as indicated in Figure 21
...
Each phase of the
machine contributes to the developed torque according to the representation of Figure 21
...
A fast response torque controller based
on the flux-linkage versus current versus angular position characteristics of Figure 21
...
21
...
The reference and estimated torque of each phase
of four-phase SRM is indicated in Figure 21
...
22b shows the actual torque
response when the machine accelerated and decelerated
...
21 requires a high-resolution position sensor
...
In this scheme, direct control of torque via

475

AC Motor Control Applications in Vehicle Traction
Asymmetric halfbridge converter

Commutation

T*

Torque
distribution
and
commutation
sequence controller

θ

TA TB TC TD

i*
A

i*
B
*
iC
*
iD

PWM
current
controllers
i A iB

v*
A

v*
B

T1

iA

Phase
A

v*
C

iB

T2

v*
D

iC
iD

SRM

iC iD

E

Torque calculator
using
look-up table: T(i, θ)

iA
iB

iC
iD

θ

Figure 21
...
This approach is similar to the DTC schemes that have
been used with induction and IPM synchronous machines, except that phase commutation
signals must be derived from a position sensor
...
23
...
23 is depicted in Figure 21
...
The dynamic response of torque
obtained from this scheme is indicated in Figure 21
...


21
...
26a–b, in order to optimize the selection of motor and DC supply voltage levels
...
The reverse arrangement is also possible
...
26a and b respectively, and the three-phase inverter for the
traction motor allow some flexibility when ultra-capacitors are used for rapid acceleration and
deceleration
...
Transfer of energy between the two storage devices is arranged
through outer voltage and inner current controls of each converter
...
Control system
designs for these two storage systems will not be discussed here
...
04

0
...
03

0
...
02
0
...
04

0
...
03

0
...
05

0
...
05

0
...
02

0
...
5 s 0
...
5 Nm/DIV

Time (0
...
5 s 0
...
22 Reference and estimated torque for each phase of an SRM and total estimated torque during acceleration
and deceleration

2
Phase A
1
0
–1
0
...
01
2
Phase C
1
0
–1
0
0
...
01
Total torque (Nm)

Phase torque look-up table
i = 0, 0
...
, 6
...
5

3

ij

2
1
...
5

1

λj

1

0
...
5
0

T (i, λ)

i

2
...
05

0
...
15
Flux linkage (Wb)

0
...
05

0
...
15

Flux linkage (Wb)

0
...
2

0
...
23 Torque versus current versus flux linkage of an SRM
Asymmetric half
bridge converter

Commutation

TA TB TC TD

iA

v*
A

*
TA

v*
B
v*
C

Direct
torque
controller

*
TB
*
TC
*
TD

TA TB

Phase
A

T1

iB


...
24 Direct torque control structure for an SRM
Experimental results
Reading floppy disk drive

Reference

Estimated
0
...
5 s 0
...
5 s 0
...
2 s/DIV)

Figure 21
...
26 Bidirectional (a) buck converter/inverter and (b) boost converter/inverter, with ultracapacitor for traction drives

21
...
Slip-speed ratio and torque control for the desired longitudinal and lateral motion
...
Control of regenerative braking
...
Sensorless control
...
4
...
The slow response of an ICE-driven vehicle stems from the requirements
for adjustment of air valve, fuel and oil pressure
...
In contrast, the AC electric motors when driven with RFOC and
DTC have delays less than 10 ms
...

This has significant impact on the control of vehicle dynamics and stability, as well as longer
driving range per battery charge because low-drag tyres can be used (Fujii and Fujimoto
2007)
...
These forces depend significantly on the slip-speed
ratio, λ, as defined in terms of the difference between the wheel and vehicle speeds, vw and v,
respectively, as defined in equation (21
...

v

(21
...
1998)
...
27, for a certain road surface
condition
...


(21
...
It may be noted that the lateral force diminishes
rapidly with increase in the slip-speed ratio
...
The torque control system for the
vehicle primarily receives its torque reference from the angle of the acceleration and brake
pedals (with brake pedal over-riding the two references); however, a slip-speed controller,
as indicated in Figure 21
...
27 Longitudinal and lateral force coefficients versus slip-speed ratio

480

AC Electric Motors Control

μopt

Torque
reference
generator

Torque-controlled
system

Driven
wheel

Look-up table
for friction
coefficient
Vehicle acceleration

Figure 21
...
With estimated vehicle speed, the slip-speed is also estimated and controlled
in a closed loop for the maximum adhesion to be achieved
...

1998; Fujii and Fujimoto 2007)
...
It suffices to mention here that with very fast control
of torque developed by an electric motor, the requirement for optimum slip speed control
becomes much more important than with an ICE driven vehicle
...
29

21
...
2

Control of Regenerative Braking

In a vehicle, the reference for braking torque is derived from the brake pedal angle
...
When the brake pedal is depressed hard, mechanical friction braking may
also be applied simultaneously
...
When regenerative braking is employed,
the braking current charges the ultra-capacitor across the DC link and the battery system taking
into account the voltage rise of the storage systems, state of charge, current limits and battery
temperature
...
The d-axis current must also be appropriately adjusted with
speed as the machine slows down, in order to ensure that the machine operates with rated back
EMF and MTPA characteristic when slowing down in the constant power speed range
...
30
...
The

481

AC Motor Control Applications in Vehicle Traction

16
Wheel speed

(km/h)

12
8
4

Vehicle speed

0
1000

1500

0
...
3

2000

2500 (ms)

Slip ratio

0
...
1
0
–0
...
29 Experimental results of optimum slip-speed ratio control (Fujii and Fujimoto 2007)

dynamics of the ultra-cap is assumed to be much faster than the battery, so the battery model
is ignored
...

2

i1

iC
A

RDC

(21
...
30 Representation of the DC link capacitor charging circuit

482

AC Electric Motors Control

For the DC side of the inverter,
i 1 = i DC − i C − i R = i DC − C

dvDC
vDC

...
28)

From (21
...
28),
vDC

?
v2
dvDC
PDC
3 ?
+ DC =
−
vqs i qs + vds i ds
...
29)

By ignoring the small power loss in the stator resistance, compared to the power from the
machine back EMF,
vDC

vDC

v2
dvDC
Lm
PDC
3
λdr i qs for an IM,
+ DC =
−
ωe
dt
RDC C
C
2C
Lr

?
?
?
v2
dvDC
PDC
3 ?
+ DC =
−
ωe {λf i q + L d − L q i d i q } for an IPMSM
...
30)

(21
...
30) and (21
...

+ 2vDC = 2RDC PDC − ωe
dt
2 Lr

(21
...
32) can be drawn as in Figure 21
...
With the fast current control and
complete decoupling of RFOC, the above control block diagram simplifies to Figure 21
...

It may be noted that with a large input capacitor C and fast current controls in the d–q axes,
the DC bus voltage control is independent of machine dynamics (IM and IPMSM)
...
32
shows experimental data of the DC bus voltage dynamics when an IM is operated with fast
acceleration and deceleration into and out of field weakening range
...


Dynamic Model of DC-Link Voltage Control for SRM
Regenerative control of the switched reluctance generator entails control of the switches for
each phase winding while the flux linkage is decreasing, as opposed to increasing that is
required for motoring
...

∂i ∂t
∂θ ∂t

(21
...
During this period, switches
∂θ
T1 and T2 in the asymmetric half-bridge converter indicated in Figure 21
...
31 Block diagram representation DC link capacitor voltage controller

increasing the flux in the winding
...
Figures 21
...
The duty cycle of the switches is adjusted
in order to control the regenerative current, and hence, the charging rate of the DC capacitor
...
32 Capacitor voltage during acceleration and regenerative braking of an IM

484

AC Electric Motors Control

Figure 21
...
As with IM and IPMSM drives, it
can be shown that when phase current control is much faster than the capacitor voltage dynamics, the capacitor voltage control with regenerative braking is essentially decoupled from the
current controls
...
34 shows the dynamic response of DC link voltage change when
an SRM is operated as a generator charging the DC link capacitor and supplying a DC load
that changes abruptly to zero
...
This is regarded as an
essential requirement for vehicles
...
The latter
technique relies on some saliency in the rotor magnetic circuit that is conveniently offered
by IPMSMs
...
Mechanical sensorless methods are currently regarded as possible

485

AC Motor Control Applications in Vehicle Traction

Iload(A)

Reading floopy disk drive

0
...
5 s 0
...
5 s 0
...
34 DC link capacitor voltage during abrupt change of regenerative braking

standby technique of controlling a vehicle when signals from the shaft-mounted mechanical
sensor become unavailable
...
1998; Foo et al
...
Methods for IMs are
available in reference by Holtz (2006)
...
5

Conclusions

This chapter has brought out the characteristics required for electric machines for traction
applications
...
Together
with Chapter 19, the coverage of induction, permanent magnet synchronous and SRM, which
are the three types of machines currently used in vehicles, has described the high-performance
control, both mechanical sensor based and sensorless
...
Apart from the control of torque,
stator and rotor flux linkages in the stator or rotor reference frames, the tractive forces on tyres
can be controlled more effectively with electric traction, leading to enhanced energy efficiency
and stability
...


486

AC Electric Motors Control

References
Casadei D, Milanesi F, Serra G et al
...
Proceedings of the 18 ICEM, pp
...

Dutta R and Rahman MF (2008) Design and analysis of an interior permanent magnet (IPM) machine with very wide
constant power operation range
...

Dutta R, Chong L, Rahman M-F (2011) Analysis of CPSR in motoring and generating modes of an IPM motor
...

Foo G, Sayeef SM and Rahman MF (2010) Low speed and standstill operation of a sensorless direct torque and flux
controlled IPM synchronous motor drive
...

Fujii K and Fujimoto H (2007) Traction control based on slip ratio estimation without detecting vehicle speed for
electric vehicle
...
688–693
...
Proceedings of IEE Power Electronics and Variable Speed Drives, No
...
710–715
...
IEEE Transactions on
Industrial Electronics, 53, 7–30
...
IEEE Transaction on Industry Applications, 34(5), 1131–1998
...
Journal on Power Electronics, 11(4), 418–423
...
63–68
...
IEEE Transactions on Energy Conversion, 20(4), 771–777
...
International Conference on Power Electronics, Machines and Drives, Bath UK, pp
...

Miller J (2003) Propulsion Systems for Hybrid Electric Vehicles
...
31
...
IEEE Transactions on
Industry Applications, 33(2), 333–341
...
Oak Ridge National Laboratory
Report FY 2008, US DOE, FreedomCAR and Vehicle Technologies, April 2008
...
IEEE Transaction on Industry Application 38(6), 1500–1507
...
IEEE Transactions on Industry Applications, 34(6), 1246–1253
...
IEEE Transactions on Industry Applications,
30(4), 897–904
...
(2011) Comparison of interior and surface permanent magnet machines
equipped with fractional-slot windings for hybrid vehicles
...
2252–2259
...
(2010) Torque density and Efficiency Improvements of a Switched Reluctance
Motor without Rare-Earth Material for Hybrid Vehicles
...
2653–2659
...
IEEE Transactions on Industrial Electronics, 50, 253–261
...
IEEE Transactions on Industry Applications, 45(2), 729–736
...
IEEE Transactions on Industrial
Electronics, 57(11), 3715–3723
...
1

Introduction

In the modern electric traction drive systems particular attention is being paid to improve their
reliability by numerous diagnostic systems (Ohnishi et al
...

2006; Kia et al
...
Such solutions of the faults detection, which are utilized, for example,
in high speed railway systems (Madej 2000; Kadowaki et al
...
If the failure mode is detected, then presence of the serious faults is possible
...

Mostly, diagnostic systems of the mechanical parts are based on analysis of the signals of the
measured torque
...
Unfortunately, all torque oscillation
sensors are sensitive to disturbances and troublesome in practical use
...

The use of the advanced microprocessor techniques makes it possible to realize sensorless
diagnostic systems in such applications
...
The existing sensors are being used for other purposes, for
AC Electric Motors Control: Advanced Design Techniques and Applications, First Edition
...

© 2013 John Wiley & Sons, Ltd
...


488

AC Electric Motors Control

example, motor control
...
2009)
...
Previously, the diagnosis of such system was based on
motor and gear vibration measurement (Kowalski 2005)
...
2009; Guzinski et al
...

In the transmission diagnostic, an analysis of mutual shaft positions of the gear or of drive
mechanical oscillations makes it possible for early fault detection, for example, gear damages (Hedayati et al
...
2009a)
...
After an initial period of operation, the problems with the transmission can appear as a result of wearing and material consumption
...
In case of transmission
faults, the train might be stopped or may work with some mechanical oscillations and noises
...
This is the reason of the necessity to introduce the proposed
diagnostic system
...

Numerous traction drives are equipped with different speed sensors used for traction motor
control
...
If there is only one drive in the train, the whole vehicle is stopped,
which blocks the track
...

To solve this problem, the motor speed calculations or estimation instead of speed measurement could be implemented
...
A lot of speed observer systems
are presented in the literature (Luenberger 1971; Rajashekara et al
...

2000; Orlowska 2003; Depenbrock and Evers 2006; Holtz 2006; Orlowska and Szabat 2007;
Krzeminski et al
...
Most popular observers are based on Luenberger theory (Luenberger
1971), Kalman filters (Magureanu et al
...
The differences between such algorithms depend on speed calculation accuracy in
steady state and in transients and on algorithms implementation and tuning complication
as well
...
The presented solution is small part of developed bigger
high-speed train (HST) predictive maintenance system
...


22
...

The analyzed multiple unit, presented in Figure 22
...


Induction Motor Control Application

489

Figure 22
...
2008; Guzinski et al
...
Nowadays, these motor are
replaced by more reliable and cheaper induction motors or permanent magnet synchronous
motors
...
In each bogie, one or two
motors may exist
...
In the presented HST
solution, each of the driving cars is propelled by four 1
...
Each of these traction motors is coupled with individual
driving axles through complex torque transmission system—Figure 22
...

The torque transmission unit consists of two-toothed gears and coupling systems to ensure
a compensation of the position change between motor shaft and drive axle
...
Therefore,
a sliding axle and some cardans are installed between gears shafts
...
This sensor is for motor control
...

The whole model of the HST drive contains four main parts: traction motor, transmission,
converter, and controller
...
The dedicated simulation programs in C language and in Matlab were
prepared
...


Figure 22
...
1

Per unit system definitions

Base value
√
Vb = 3Un
√
Ib = 3In

Voltage

ω0 = 2π f n

Electrical speed

Zb =

Un
In

Ub I b p
ω0
b
?b = U0
ω
ωb = ωp0
L b = ?bb
I
Tb
Jb = ωb ω0

Tb =

τ = ω0 t

22
...
1

Meaning

Current
Impedance
Torque
Flux
Mechanical speed
Inductance
Inertia
Time

Induction Motor

For train propulsion, high-power induction motor is used
...
2[MW], Un = 810[V], n n = 269[rpm], In = 586[A], f n = 133[Hz], ηn = 96[%],
cos ϕn = 0
...
When used in HST gear and wheels, the motor nominal and
maximum speeds are related to the train speeds of the values about 225 km/h and 320 km/h,
respectively—depending on train wheels wear
...
The x y coordinates
me rotating with arbitrary speed ωa
...
Whole system variables
were recalculated into per unit system as shown in Table 22
...

Equations of the used motor model are as follows
2
Lm
Rr L m
Lr
Rs L r + Rr L 2
di sx
m
i sx +
φr x + ωa i sy + ωm
φr y +
vsx ,
=−
dτ
L r wσ
L r wσ
wσ
wσ

(22
...
2)

Lm
Rr
dφr x
i sx ,
= − φr x + (ωa − ωm )φr y + Rr
dτ
Lr
Lr
dφr y
Lm
Rr
i sy ,
= − φr y − (ωa − ωm )φr x + Rr
dτ
Lr
Lr
?
?
dωm
1
Lm
(φr x i sy − φr y i sx ) − TS1 ,
=−
dτ
JM L r

(22
...
4)
(22
...
2
Parameter
Rs
Rr
Lm
Ls
Lr

Parameters of the motor equivalent circuit
Value

Description

0
...
u
0
...
u
...
56150 p
...

4
...
u
...
68320 p
...


Stator resistance
Rotor resistance
Mutual inductance
Stator inductance
Rotor inductance

where Rr , Rs , L r , L s , L m are motor equivalent circuit parameters presented in Table 22
...
1 kg m2 ), and wσ is the coefficient defined
as wσ = L r L s − L 2
...
1)–(22
...


22
...
2

Torque Transmission System

For simulation purposes HST transmission system was reduced to a two-mass system which
is presented in Figure 22
...

The integral property of the traction vehicle is a macroslip / microslip effect
...
Approximated
relation between the adhesion force and the slip velocity is presented in Figure 22
...
2007)
...

In the investigated HST the stable area is limited to the slip up to 2%
...
In the investigated
HST each driving axle is propelled by one electric motor which is connected to the separate
inverter
...
Adhesion
limit for HSTs imposed by European Technical Specification Interoperability is 0
...

In the considered two-mass model, the macroslip phenomena was omitted according to
assumption that superior adhesion control is implemented in the system
...


JM

JL

ωm ϕm Tem
Motor

Ts1

ni

Transmission

Ts2

TL

ωL ϕL

Load

Figure 22
...
4 Relation between adhesion force μ(vs ) and slip velocity vs of the coach

The two-mass model of the mechanical system is as follows:
dωm
dt
dω L
JL
dt
dϕr
dt
dϕ L
dt
TS1

JM

= Tem − TS1 ,

(22
...
7)

= ωm ,

(22
...
9)

= K 2 (θm − n i θ L ) + H2 (ωm − n i ω L ) ,

TS2 = n i |TS1 |,

(22
...
11)

where ω L is the mechanical speed of the load, θm is the angle position of the motor rotor, θ L is
the angle position of the load, Te is the motor electromagnetic torque, TL is the traction load
torque, TS1 is the transmitted torque on gear input = motor load torque, TS2 is the transmitted
torque on gear output, K 2 is the stiffness function, H2 is the damping coefficient (170 Ns/m),
JL is the load inertia (3000 kgm2 ), and n i is the gear ratio (1
...

Function K 2 depends on gear wheel tooth stiffness (Muller 1979):
K 2 = K S + K D sin(zϕm ),

(22
...
5 × 105 N/m), K D is the stiffness maximum value
(5
...

In (22
...
It is a result of skew teeth
in the modeled gear
...


493

Induction Motor Control Application

The misalignment in the transmission system was modeled as an additional load torque
component Tw :
JM

dωm
= Tem − TS1 − Tw
...
13)

The additional torque component Tw is sinusoidal value with frequency equal to the motor
shaft rotation:
Tw = Twav (1 + sin(ϕm )) ,

(22
...

In (22
...
15)

where T f is the viscous friction load torque component
...
16)

where F is the viscous friction coefficient (0
...


22
...
3

High-Power Electronic Converter

The HST traction system has double way supply: ac supply 25 kV 50 Hz and dc supply 1
...

The train high-speed range is obtained only with the ac high-voltage traction mode
...
The transformer
output ac supply is converted into dc by full controlled transistorized rectifier, filtered in the
intermediate dc link and converted to ac motor supply by a three-phase voltage inverter with
insulated gate bipolar transistor (IGBT) transistors (Figure 22
...
In the simulation program,
the voltage inverter was modeled with ideal switches controlled with a pulse width modulation
method
...


Figure 22
...
2
...
The control
structure used in the presented work is illustrated in Figure 22
...
In the traction application
of the field-oriented method, torque mode of operation is used and thus as such the speed
controller was not used
...
The traction system of the HST is working also in the field
weakening region that is used to obtain high speed of the train
...
3 Estimation Methods
22
...
1 Speed Observer
Speed computation in real time is presented in many publications (Luenberger 1971;
Rajashekara et al
...
2000; Krzeminski 2001; Orlowska
2003; Depenbrock and Evers 2006; Holtz 2006; Krzeminski et al
...
Most popular solutions are based on motor models using the concept of Luenberger state observer (Luenberger
1971)
...

The use of state observer makes it possible to simultaneously compute the angular speed and
magnetic flux (Krzeminski 2001)
...

For diagnostic purposes, the observer bandwidth should satisfy the interesting diagnostic
bandwidth
...
The bandwidth of the chosen observer is significantly higher than
other comparable observers (Bogalecka and Kolodziejek 2008)
...
These signals are signals of inverter output currents and signals of

com

Pl

–
com

com

isd

Pl

com dec

vsq

vsd

vsq
Decoupling

com

Tem

Commanded
DQ currents
control

com

isq

com

vα

com dec

vsd

ψr
isd

d-q

isq

α-β

vdc
PWM

Inverter
U

com

vβ

V

θ

vdc
Calculation
block

isU

M

isV
ωrm
T

Figure 22
...
7 Input/output block of the induction motor observer

the PWM commanded voltages (Figure 22
...
The equations of the speed observer in stationary
αβ coordinates are as follows (Guzinski et al
...
17)

com
ˆ
ˆ
ˆ
ˆ
a1 i sβ + a2 φrβ − a3 ξα + a4 vsβ + k1 (i sβ − i sβ ),

(22
...
19)

ˆ
ˆ
ˆ
ˆ
a5 φrβ + a6 i sβ + ξα − k2 Sb φrβ − k3 φr α (Sb − SbF ),

(22
...
21)

ˆ
ˆ
ˆ ˆ
ˆ ˆ
a5 ξβ + a6 ωm i sβ − ωm ξα + k1 (i sα − i sα ),

(22
...
23)

ˆ ˆ
ˆ ˆ
φr α ξα + φrβ ξβ
,
2 + φ2
φr α
rβ

(22
...

With observer calculation, the motor electromagnetic torque could be also identified:
ˆ
ˆ ˆ
ˆ ˆ
Tem = φr α i sβ − φrβ i sα
...
25)

The block structure of the observer is presented in Figure 22
...
The speed observer (22
...
24) is based on the Luenberger observer theory (Luenberger 1971) in addition using motor
electromotive forces as additional state variables
...
Such
approach was also successfully implemented for the synchronous permanent magnet motor
(Zawirski and Urbanski 2000)
...
8 Speed observer structure

In the observers equations (22
...
24), in contrary to Krzeminski (2008), the component
related to motor speed derivative was omitted assuming that for small step of the observer
calculation this component is close to zero for high-inertia train system
...
Practically that process is based mainly on the experience of the designer
...
With the modern simulation tools and
high-speed computers, this approach seems to be more precisely performed than using formal
mathematical computation
...
1997)
...
At the beginning, the observer gains were tuned randomly in wide range of the
field of research
...
That way is very close to the designer experimental
tuning
...
3
...
For the diagnostic of traction torque transmission system using
calculation-based methods the analysis of the motor supply current or motor load toque
signals is used (Tondos 1993; Begg et al
...
Especially, an analysis of the motor load torque looks promising
(Hedayati et al
...

The simplest method for load torque calculation is to use the mechanical equations:
dωm
ˆ
ˆ
,
TS1 = Tem − JM
dτ
ˆ
where TS1 is the estimated motor load torque
...
26)

497

Induction Motor Control Application

The load torque calculation method (22
...

This results from its sensitivity to any inaccuracy and the simplifications in the mechanical
equations
...

Instead of simple estimation (22
...
Some
of the load torque observers are presented in the literature, for example, the concept of the
observers for systems with unknown and inaccessible inputs or the concept of full order
Luenberger-based systems (Brdys and Du 1991)
...

In the presented application, different methods for the load torque calculations were previously tested (Guzinski et al
...
The best results were obtained with an observer based
on the Gopinanth’s method (Ohnishi et al
...
2007)
...
27)

(22
...
25)
...
27)–(22
...


22
...
An example of the results obtained using
speed observer operation are depicted in Figure 22
...
In Figure 22
...

For improving simulation time a smaller inertia of the whole train was assumed
...

The error between the measured and estimated speeds is also shown, and it is observed that
this error is significantly small and it is less than 2%
...
Also in this case, the operation of the proposed
observer is correct as illustrated in Figure 22
...

In Figure 22
...
The results show correct performance except at initial time, in
which load observer started computation
...
5

Experimental Test Bench

In the second step of verification of the speed and load torque observers the experiments were
done
...
u
...
02
− ωrm
–0
...
5

Ts 1
0
1
...
5

Tem
0
1
ψr
0

0

6

12

18

24

(s)

Figure 22
...
In the test, bench singular train drive with 1
...
The traction motor,
power converter, and control system are the same as in real train
...
The differences are in the gear, which has different
gear ratio and different gear wheels’ teeth number
...
The simplified structure of the test bench is presented in Figure 22
...
On the test
bench, series of tests were performed to investigate observer action at different motor speeds
and load torque levels in steady states and in transient conditions
...
u
...
5

ΔTs1
–0
...
2

0
...
6

0
...
10 Load torque observer simulation results

(s)

499

Induction Motor Control Application

25 kV 50 Hz
AC traction

AC genarator
Grid
inverter

Machine
inverter

Figure 22
...
The DSP control board has several
programmable analog outputs where some internal control variables are sent
...
The chosen signals in analog form are sent through wiring system to the general
purpose data acquisition board (DAB) connected to PC computer through USB interface
...
The block structure of the data acquisition system is presented in Figure 22
...
Due
to limited time access to the test bench, a two-step, procedure was used in the experiments
...

The next tests were done: starting of the train and work with constant speed with different
load levels and motor speed, breaking of the train with different load levels and work with
different constant motor speed for 0%, 25%, 50%, 75%, 100% load, and so on
...
13
...
The
experimental data were used as inputs for observers S-function
...
12 Structure of the test bench, data acquisition system

Figure 22
...
3
ωr

(p
...
)

0
...
1

0
...
14 Speed observer experimental results—the train start to 50 km/h

22
...
14
and 22
...
In the test of Figure 22
...
5%
...
15, an increase in computation error with train speed is noticed
...
This error is caused by a distortion in motor stator
current
...
The changes in PWM are done
to fully utilize the inverter dc supply voltage in the motor high-speed range
...
16
...
Comparable experimental investigation should be realized for correctly
working torque transmission system and for faulty system for different fault types
...
5
ωm

0
...
0
1
...
u
...
0

(p
...
)

1
...
5

0
...
15 Experimental results—the train start up to 320 km/h

502

AC Electric Motors Control

2298
...
7
A

motor phase current 1

–2298
...
16 Motor supply phase current at motor speed related to train speed of 320 km/h

healthy system only
...

The realized tests have shown torque observer’s acceptable precision
...
The vibration
amplitudes that result from such frequency were out of measurement bandwidth of the existing
instrumentation system of the torque sensor
...

The examples of the experimental results of the load torque observer are presented in
Figures 22
...
18
...
17 and 22
...
17 Identification of the gear meshing frequency for motor speed related to 50 km/h train
velocity
...
)

503

Induction Motor Control Application

Ts1

Frequency (Hz)

Load
(%)

Figure 22
...
150 km/h
train velocity
...
)

speeds—related to next train speeds: 50 km/h and 150 km/h
...
Fast Fourier transform (FFT) harmonic analysis
was then realized for the load torque and next the amplitudes levels related to meshing
frequencies were plotted
...

Higher precision of computation was observed for lower speed 50km/h which is, similarly
to the case of speed observer, caused by distorting current waveform with decreasing inverter
transistors switching frequency at higher speeds
...

Nevertheless, torque observer is applicable for diagnosis purposes, because it is not required
that it works all the time but, for example, at lower speeds only
...
Assembly defects of transmission system are characterized by violent failures, which could be detected during initial test for train travels at lower
speeds
...
7

Diagnosis System Principles

Diagnosis system is currently intended to evaluate diagnostic criterion
...
The main parts of our diagnostic system are speed and load torque

504

AC Electric Motors Control

observers
...

This aims for proper fault detection appearing and to the prediction of future faults before
their occurring
...
7
...
29)

|ωm − ωm | > E mlimit
ˆ

where E mlimit is the error limit between the measured and the computed speed
...
This value was selected to be higher than the maximum observed
speed observer computational error, a 2% value
...
19)
...
19, the speed sensor fault appeared at instant 5 s
...
The control system detected fault and a
diagnostic indicator dωm has been changed from 0 to 1
...
Simultaneously, the whole system was switched to
the speed sensorless operation mode without any noticeable disturbances
...

The other fault of the speed sensor is also possible—in the speed scanning buffer the last
proper value of the measured speed could be memorized
...
In such case, the whole system still works

(p
...
) 2
com
Tem
0
2
Tem
0
1
ωm
0
0
...
02
1
ψr
0
1
dωm
0

0

2
...
5

10

(s)

Figure 22
...
20 Load torque harmonics analysis

properly until the moment when the torque commanded signal is set or when the system starts
losing its operating point
...

It is assumed that to differentiate between the sensor failure and motor failure a different
motor stator and rotor diagnostic system has to be used
...


22
...
2

Diagnosis of Traction Torque Transmission

Vibrations of the torque transmission unit are caused by, for example nonexact meshing in the
gear, eccentricity of toothed wheel, and variable teeth stiffness (Guzinski et al
...
Such
vibration causes a creation of harmonics with frequencies of torque transmission elements
rotation and of meshing frequencies
...
20)
...
21
...
21 is divided into two parts: “online” and “off-line
...
An “off-line” subsystem is used to predict slowly increasing faults
...
More advanced signal spectrum computation or wavelet analysis could be also employed
in future development
...
8

Summary and Perspectives

The provided simulation and experimental investigations have proved an applicability of
presented observers for diagnostic purposes
...


506

AC Electric Motors Control

Figure 22
...
The designed DSP system will first operate in an open-loop mode to collect speed
and load torque values calculated online by observer systems
...
On the basis of that data, the diagnostic criteria will be elaborated
and verified
...
By this way, DSP system will obtain
the essential information needed to analyze the measured values and command values for the
control system
...

Thus, in this way it is possible to prepare essential data base of the system faults
...

The presented diagnostic system concept is general in nature and is not limited to only
HST induction motor drives
...

Nowadays diagnostic systems are also under implementation for tramway vehicles and also
for the newest traction drives with permanent magnet synchronous motors
...
Mechanical System Modeling for Failure Diagnosis and Prognosis, Maintenance and Reliability
Conference (MARCON’99) Gatlinburg, Tennessee, USA, pp
...

Blaschke F (1971) Das Prinzip der feldorientierung, die grundlage fur transvector-regelung von drehfeld-maschine
...

Bogalecka E and Kolodziejek P (2008) Frequency characteristics of induction machine speed observers
...
125–126
...
A and Du T (1991) Algorithms for joint state and parameter estimation in induction motor drive systems
...
II, pp
...


Induction Motor Control Application

507

Burton B, Kamran F, Harley RG, et al
...
Transaction on Industry Applications, 33, 697–704
...
Proceedings of
IEEE/ IAS Annual Meeting, Conference, Tampa, Florida, USA, pp
...

Depenbrock M and Evers C (2006) Model-based speed identification for induction machines in the whole operating
range
...

Guzinski J, Diguet M, Krzeminski Z, et al
...

13th International Power Electronics and Motion Control Conference (EPE-PEMC 2008), Pozna´ n, Poland, pp
...

Guzinski J, Diguet M, Krzeminski Z, et al
...
IEEE Transaction on Industrial Electronics, 56, 248–256
...
9th International Scientific Conference Modern Electric Traction (MET’2009), Gda´ sk –
n
Krynica Morska, Poland, pp
...

Guzinski J, Krzeminski Z, Lewicki A, and Abu-Rub H (2009c) State observers for diagnostic applications in modern
electric traction
...
Gdansk University of Technology,
Faculty of Electrical and Control Engineering, Gdansk
...
1-8
...
IEEE Transaction on
Industrial Electronics, 53, 7–30
...
Transaction on Industrial
Electronics, 53, 1842–1852
...
Transacton on Industrial Electronics, 54, 2295–2304
...
(2007) Antislip readhesion control based on speed-sensorless vector control
and disturbance observer for electric commuter train—series 205-5000 of the East Japan Railway Company
...

Kowalski CT (2005) Monitoring and diagnostic of the induction motor faults using neural networks (in polish—
Monitorowanie i diagnostyka uszkodzen silnik´ w indukcyjnych z wykorzystaniem sieci neuronowych)
...

Krzeminski Z (2000) Sensorless control of the induction motor based on new observer
...

Krzeminski Z (2001) Digital control of the asynchronous machines (in polish—Cyfrowe sterowania maszynami
asynchronicznymi)
...

Krzeminski Z (2008) Observer of induction motor speed based on exact disturbance model, in International Conference
(EPE-PEMC 2008), Poznan, Poland, pp
...

Kucharski T (2002) Mechanical vibration measuring system (in polish—System pomiaru drgan mechanicznych)
...

Luenberger DG (1971) An introduction to observers
...

Madej J (2000) Traction torque transmission mechanics (in polish—Teoria ruchu pojazdow szynowych)
...

Magureanu R, Ilas C, Bostan V, Cuibus M, and Radut V (2000) Luenberger, Kalman, neural observers and fuzzy controllers for speed induction motor control
...
XLVI (L), pp
...

Muller L (1979) Toothed gears—design (in polish—Przekładnie zebate projektowanie)
...

Ohnishi K, Shibata M, and Murakami T (1996) Motion control for advanced mechatronics
...

Orlowska-Kowalska T (2003) Sensorleess drives with induction motors, sensorless control of the induction motor
˚
based on new observer
...

Orlowska-Kowalska T and Szabat K (2007) Neural-network application for mechanical variables estimation of a
two-mass drive system
...


508

AC Electric Motors Control

Pajchrowski T and Urbanski K (2001) DSP application to robust speed control of PMSM by means of artificial neural
network (in Polish—Zastosowanie DSP do odpornej regulacji predkosci SSMT z wykorzystaniem sztucznych sieci
neuronowych)
...
II, pp
...

Rajashekara K, Kawamura A, and Matsuse K (1996) Sensorless control of ac motor drives
...
IEEE Press
...
European Power Electronics Conference (EPE1993), Brighton, United Kingdom, pp
...

Vedmar L and Andersson A (2003) A method to determine dynamic loads on spur gear teeth and on bearings
...

Wilk (2009) Winding internal faults diagnosis of the traction transformer based on analysis of energy dissipated per
period
...
72–78
...
9th International
Conference on Power Electronics and Motion Control (EPE-PEMC 2000), Koszyce, Slovakia, vol
...
239–243
...
Chattopadhyay
Electrical Engineering Department, Bengal Engineering & Science University,
India

23
...
2–13
...
The upper limit is
decided by the requirements of the applications rather than by the technology of the converters
and machines (Stemmler 1994)
...
g
...
g
...
The limits for the dc motors such as
cost, size, commutator problems and inability to operate satisfactorily in a dirty and explosive
environment have called for ac motor drives for high power applications
...
The machine may be excited by a voltage source inverter
(VSI) or a current source inverter (CSI)
...
The operation near unity power factor reduces
armature copper loss and permits inverter size reduction with simplicity of commutation (load
commutation) with thyristors as switches
...

The recent trends in high-power ac drives are to use pulse width modulated (PWM) VSI
or CSI with self-commutated devices like insulated gate bipolar transistors (IGBTs), gate turn

AC Electric Motors Control: Advanced Design Techniques and Applications, First Edition
...

© 2013 John Wiley & Sons, Ltd
...


510

AC Electric Motors Control

off thyristors (GTOs), integrated gate commutated thyristors (IGCTs) and injection enhanced
gate transistors (IEGTs) for efficient VVVF control with harmonic reduction
...
3/4
...
7/1
...
5 kV, 5
...
5 MVA to
about 30 MVA (Chattopadhyay 2010)
...
The converters for such drives meeting the high performance requirements must:
• generate smoothly variable frequency and voltage;
• produce nearly sinusoidal current waveforms throughout the operating range to avoid undesirable torque oscillations;
• permit highly dynamic control both in forward and reverse motoring and braking applications;
• provide as nearly as possible or even better performance than that of the dual converter-fed
dc drives as regards cost, service reliability and harmonic effects on the system
...

Thus, rapid and remarkable progress has been made over the years in the ac drive technology
used in the high-power drives and their control
...
1 shows a block diagram of a typical
high power ac drive system for a mill with its various components
...
The brief features of the industrial ac
drives developed by the leading manufacturers worldwide are also provided as well as new
developments and possible future trends
...
2

High-Power Semiconductor Devices

Rapid advances in industrial ac drives and power conversion systems have been possible
due to continuous and astonishing development of the rating and performance of the power
semiconductor devices over the last 50 years
...
1 Block diagram of a typical drive system
...
The voltage and current ratings
of these devices as commercially available today for high power converters are shown in Figure
23
...
Some typical high-power devices are shown in Figures 23
...
4 (Sato
and Yamamoto 2001; Ichikawa et al
...


23
...
1

High-Power SCR

Figure 23
...
5 kA SCR that is a high-power press-pack thyristor-based device
with three terminals: gate, anode, and cathode
...
The
turn-on time is 14 μs and turn-off time is 1200 μs
...

This device blocks voltage in both forward and reverse directions
...


23
...
2

High-Power GTO

The GTO is a self-commutated thyristor-based device that can be turned off by a negative gate
current
...
3b shows a 6 kV, 6 kA press-pack GTO (high-power GTOs being developed
by Japanese since 1980s), which is turned on by a pulse of positive gate current and turned
off by a negative gate current pulse
...
5 kA
(Mitsubishi)

SCR

10
6
...
6 kA
(Eupec) (Toshiba)

8

7
...
65 kA
(Eupec)
6
...
5 kA
(Mitsubishi)

6 kV/3 kA
(ABB)

6
...
2 kA
(ABB)

6 kV/6 kA
(Mitsubishi)

6
GTO/GCT
IEGT

4

0

4
...
5 kA
(Toshiba, press pack)

4
...
9 kA
(Mitsubishi)

2

2
...
8 kA
(Fuji, press pack)

3
...
2 kA
(Eupac)
(Toshiba, press park)
0

1

4
...
7 kV/3
...
2 Voltage and current ratings of high-power semiconductor devices
...
5-kA SCR

(b) 6-kV, 6-kA GTO

(c) 6-kV, 6-kA IGCT/GCT

Figure 23
...
Reprinted with
permission from IEEE (Chattopadhyay 2010)

AC Motor Control Applications in High-Power Industrial Drives

(a) 3
...
5 kV HVIGBT

(c) IEGT with gate driver

Figure 23
...
Reprinted with
permission from IEEE (Chattopadhyay 2010)

to turn off
...
The
typical turn on time is 2
...
The on-state voltage drop is typically
4
...
The GTO switching frequency is lower than that of IGBTs and IGCTs (to be described
later)
...
The GTO can be fabricated with asymmetrical structures suitable for VSIs
or symmetrical structures suitable for CSIs
...
2
...
However, the current pulse should be very narrow with
low energy for fast turn off
...
3c shows an ABB press-pack type 6
...
The IGCTs have replaced the GTOs for the medium-voltage drives over the past
few years due to their special features like snubberless operation and low switching loss
...
The rate of the gate current change
at turn-off is normally greater than 3000 A/μs compared to around 40 A/μs for GTO
...
Though the IGCT does not

514

AC Electric Motors Control

require a turn-off snubber, it requires a simple turn-on snubber or a clamping circuit since the
di/dt capability of the device at turn-on is around 1000 A/μs only
...
4 V for a GTO at 4000 A
...
IGCTs
have a higher switching frequency (typically 1
...
5 kHz)
...
3c, marketed by ABB),
symmetrical SGCTs (suitable for CSI) are available from Mitsubishi for smaller ratings
...
g
...


23
...
4

IGBT

After completely dominating the low-voltage converters, IGBTs are increasingly used for
medium-voltage converters
...
The ratings of these devices have reached as
high as 6
...
6 kA or 3
...
5 kV/1
...
It can be turned on with a 15 V gate voltage
and turned off when the gate voltage is zero or negative
...
4a and b
...
The main advantages of IGBT are simple gate driver, snubberless operation,
high-switching speed, modular design, and controllability of switching behavior providing
reliable short-circuit protection
...
The device has only forward blocking capability and can be used in a VSI
with a feedback diode
...

High-voltage IGBTs have a higher voltage drop (e
...
, 4
...
3 kV/1
...
IGBT devices can be available in intelligent
power module (IPM or HVIPM in Figure 23
...


23
...
5

IEGT

IEGT is basically an advanced high-voltage high-power IGBT with special gate construction
commercially developed by Toshiba in 1999 (Ichikawa et al
...
It is designed in such a
way that large numbers of electrons accumulate at its electrodes and it exhibits low on-state
voltage (compared to IGBTs and GTOs of the same rating)
...
4c shows a 4
...
1
kA (turn-off current 5
...
It can be turned on by the gate voltage
of +15 V and turned off by that of −15 V
...
Another advantage over the IGCT is
the power required to turn the device on and off
...
5 (Tessendorf and Hosoda 2004)
shows the comparison of typical gate trigger pulses required for equivalent power devices
...
The on-state voltage drop across this device is of the order of 3
...
In the IEGT-based system, neither turn-on
nor turn-off snubber is required for each IEGT as in the case of GTO
...
5 A

...
5 A

4000 A

Figure 23
...
Reprinted with permission from
IEEE (Chattopadhyay 2010)

leg needs simple and efficient clamp circuits to eliminate the snubbers
...
5%, which is 2% more than that of an equivalent GTO-based
system thus saving a lot of energy
...
3

High-Power Converters for AC Drives and Control Methods

Figure 23
...

The direct topology connects the load directly to the source through power semiconductors
and a suitable control logic, while the indirect topology transfers the power in two stages,
rectification and inversion
...
6 Classification of converters for high-power drives

High power
two-Level VSI

516

AC Electric Motors Control

in high-power applications that uses an array of naturally commutated power semiconductor
devices such as thyristors, to connect directly the power supply to the machine, converting a
three-phase ac voltage with fixed magnitude and frequency to a three-phase ac voltage with
variable magnitude and variable frequency (VVVF)
...
Only very
recently they have been used for medium-voltage, high-power drives with multilevel operation
(Yamamoto et al
...
Indirect dc-link inverters may be current source (CSI) or voltage source
(VSI) type depending on the dc-link energy-storage component which can be a capacitor that
provides a stiff dc voltage in voltage source drives or an inductor that smoothes the dc current
in current source drives
...
Inverter
originally developed as the neutral-point clamped (NPC) three-level inverter in 1981 (Nabae
et al
...
Other topologies of the multilevel inverters that have been commercialised are
flying capacitor (Rodriguez et al
...
2007) for
medium-voltage drives up to about 40 MVA
...
3
...
These
modulation techniques have become mature technology and implemented in power converters for high-performance drives as commercial products (Rodriguez et al
...
Out of
these schemes, SVM is an advanced digital modulation technique preferred over the SPWM
technique as it provides better utilisation of the dc bus voltage and lower harmonics
...
The concept of rotating space vectors are involved
here and it needs a microcomputer or digital signal processor (DSP) (Bose 2002) for its
implementation
...
3
...
To obtain high performance, closed loop control is preferred, while the
open-loop control is popular for pump, fan- and compressor-type drives, because this control
is simple and does not involve any complex feedback signal measurement or estimation as
needed for closed loop control
...
DTC is an advanced
scalar control with performance comparable with the VC method
...

Scalar control, unlike VC, means control of the magnitude of a variable, whereas in the latter,
both magnitude as well as phase of the space vector variables are controlled
...
4
23
...
1

Control of Induction Motor Drives
Induction Motor Drives with Scalar or Volts/Hz Control

The simplest type of scalar control is open-loop Volts/Hz (V / f ) control with low performance compared to closed-loop VC
...


PWM Two-Level VSI Induction Motor Drive
The well-known two-level VSI as shown in Figure 23
...
To increase the converter voltage, a series connection of
these switches is applied
...

A simple and most common SPWM method of 2-level voltage control is shown in Figure
23
...
An isosceles triangle carrier is compared with the sine wave reference signal and the
crossover points determine the points of switching
...
7 (a) Two-level PWM Rectifier-PWM-VSI Inverter, (b) Two-level carrier-based sinusoidal
PWM
...
8 Constant V / f or scalar control of a rectifier-inverter induction motor drive with slip
regulation

is synchronised with the signal and an integral ratio (multiple of 3) is maintained to improve
harmonic content
...

The classical flux regulation control scheme for an SCR-rectifier-inverter fed drive with
simple constant V / f ratio or scalar control for the constant torque region with slip regulation
is shown in Figure 23
...
The slip frequency ωsl which is proportional to torque
is regulated by the speed loop error
...
The voltage control signal Ve is generated from ωe through a function generator
so as to maintain the airgap flux nearly constant
...
The drive accelerates with the clamped value of slip corresponding to the maximum
torque and then settles down to a value as dictated by the load torque
...
Instead of regulating the slip, it can be maintained
constant and the speed loop error may control the dc-link voltage
...

The open-loop scalar control is popular in the industry when a small drift in speed and air-gap
flux due to fluctuation are of no significance
...


PWM Three-Level VSI-Induction Motor Drive
Figure 23
...
In two-level inverters, the output voltages consist of

519

AC Motor Control Applications in High-Power Industrial Drives

S1U

S1V

S1W

S2U

S2V

S2W

Vd/2

S3U

U

S3V

V

S3W

W

Vd/2
S4U

S4V

S4W

Vd /2

0

−Vd /2
Figure 23
...
For high-power high-speed drives, three-level inverters have been preferred as
they can be operated with twice the rated voltage without any series connection and therefore
with twice the rated power with significantly improved output voltage waveform when compared to a two-level inverter
...
Here, the connection in each phase may be represented by a threepoint changeover switch, the output of which can be connected to the positive pole, zero or
the negative pole of the dc supply
...
Till 1993, the rated power and frequency
of GTO-VSIs were limited to about 2 MW/60 Hz for two-level VSIs and 4 MW/130 Hz for
three-level VSIs but now with IGCTs (developed by ABB in 1996) and IEGTs (developed by

520

AC Electric Motors Control

Rectifier block

Inverter block

SM

Gate control

Gate control

PWM control

PWM control

PFC
power factor control
DC voltage command

Vector control
DC voltage
Speed command

Figure 23
...
Siemens has introduced SIMOVERT-ML drive with three-level converters of
MW range with VC for application to synchronous or induction motor drives
...
, Japan, also developed three-level GTO-based 6
...
The synchronous motor is often the most costeffective solution for applications with a wide field weakening range and for high surge
load requirements
...
Figure 23
...
1996) shows a three-level GTO converter-inverter system used for steel
rolling mills with VC (to be discussed later)
...


Current Source Inverter-Induction Motor (CSI-IM) Drive
A schematic diagram of a dual PWM CSI-fed drive using GTOs is shown in Figure 23
...
The system is a dual of the PWM-VSI rectifier-inverter system in Figure
23
...
The PWM rectifier provides sinusoidal input current at unity power factor
...
12
(Chattopadhyay 2002)
...
The freewheeling diodes,
typical of VSI, are absent in CSI as when supplied by a current source, current in any half-leg
of the inverter cannot change in polarity and can only flow through the power switches
...
11 CSI-PWM rectifier PWM-inverter fed induction motor drive

(pre-computed for given parameters of the machine) to maintain constant air-gap flux (as in
the V / f control of the VSI-fed drive) is shown in Figure 23
...
The
full four-quadrant capability of the drive can be obtained
...
For large drives, 6 or 12 pulse converter
Id

Ld
Q1

D1

Q3

D3

Q5

Ia

D5

Ib

a

Three-phase
AC 50 Hz

L

Vb

Ic

b

Vc

c
D4

Q4

D6

Q6

D2

Q2

Figure 23
...
13 CSI-drive with slip-frequency control

bridges are used requiring 36 or 72 thyristors in total, respectively
...
The circuit
operates in phase control line commutation mode and the firing angles are modulated to
synthesise a mean sine wave voltage
...
14 (Chattopadhyay 2010)
with voltage and current waveforms for the non-circulating current one
...
14 (a) Non-circulating current-type cycloconverter, (b) circulating current-type cycloconverter, (c) voltage and current waveforms for (a)
...
Due to line commutation, its output frequency is
limited to, typically, 1/3 to 1/2 of the line frequency and is suitable for low-speed high-power
drives, with easy four quadrant operation
...

The output voltage of a cycloconverter contains a complex harmonic pattern given by
k1 n f i ± k2 f 0 , where f i is the input frequency, f o is the output frequency, n is the pulse number
and k1 , k2 are integers
...
Because
of the phase control principle, the cycloconverter presents lagging reactive power at the input
irrespective of the power factor of the load and various schemes to improve this power factor
have been developed including fast current control loop or trapezoidal modulation
...
The
cycloconverter-fed induction/synchronous motor drives have been used with scalar control for
low-speed multi-motor-driven steel mill roller tables and with VC, in cement mills and rolling
mill drives as discussed later
...
With the initial progress made by Venturini (1980), it has received considerable
attention in recent years as it provides a good alternative to the double-sided PWM voltage
source rectifier-inverters having the advantages of being a single-stage converter with only
nine switches for three-phase to three phase conversion and inherent bi-directional power
flow, sinusoidal input/output waveforms with moderate switching frequency, possibility of a
compact design due to absence of dc link reactive components and controllable input power
factor independent of the output load current
...
866), complex control,
commutation and protection strategy and above all the non-availability of a fully controlled
bi-directional high-frequency switch integrated in a silicon chip
...
15a that uses nine
bi-directional switches so arranged that any of three input phases can be connected to any
output phase as shown in the switching matrix in Figure 23
...
Thus, the voltage at any
input terminal may be made to appear at any output terminal or terminals while the current
in any phase of the load may be drawn from any phase or phases of the input supply
...
New perspective configuration of the bi-directional switch is to use two RB-IGBTs with
reverse blocking capability in anti-parallel eliminating the diodes reducing the conducting
losses in the converter significantly
...
The switches
should be controlled in such a way that, at any time, one and only one of the three switches

524

AC Electric Motors Control

VAD

VBD
0

iA

A

iB

B

Matrix converter
SAa

SAb

Bidirectional switches
SAc

SBa

SBc

SCa

iC

SBb
SCb

SCc

ia

VCD

ib

ic

C

Three-phase input Input filter

a
Three-phase
inductive
load

b
Van

c
Vbn

Vcn

M
(a)
SAa

VA0
SAb

Van

SBb

SAc SCa
VB0

SBb

SCb
VC0

Vbn

SBc
Vcn

SCc
(b)

Figure 23
...

The control methods adopted so far for the MC are quite complex and are subjects of
continuing research (Zhang et al
...
Out of the several methods proposed for independent
control of the output voltages and input currents, two methods are of wide use: (i) the Venturini
method based on a mathematical approach of transfer function analysis and (ii) the SVM
approach (as has been standardised now in the case of PWM control of the dc link inverter)
...
(1998)
...
16 Multi-level MC: (a) configuration, (b) switch symbol, (c) switch realisation (Erickson
and Al-Naseem 2001)

A vector controlled high-performance MC-induction motor is described in Ishii et al
...

A multilevel MC with four-quadrant dc link H-bridge switching cells suitable as shown in
Figure 23
...
The use of four transistors in the switch cell of Figure 23
...
With dc capacitor, the switch cell
is capable of producing instantaneous voltages +V , 0, −V
...
2011) utilises a series connected multilevel topology shown in Figure 23
...
Connecting three cells in series, each designed for 635V
yields a line-neutral voltage of 1905 V, corresponding to a line-line voltage of 3300 V
...
3 kV, 3 MVA skin-pass mills (Yamamoto et al
...
Compared to PWM converter-inverter system, the MC scheme has higher reliability,
improved efficiency from 92
...
9% and weight about 62% (Yamamoto et al
...


Slip-Power Controlled Induction Motor Drive
With wound rotor IM, while the stator is connected to the ac system, the rotor side slip power
can be controlled by a converter cascade, either a rectifier-inverter or a cycloconverter, via
slip rings as described in (Akagi 1998)
...
Figure 23
...
17 Schematic of multicell MC with three cells in series in each phase (Yamamoto et al
...
(a) One cell, (b) Single-phase and line-to-line voltage

Power system
60 Hz 500 kV

18 kV

Collector ring
Cycloconverter
72 MVA

Rotor
(cylindrical)
Generator/motor
400 MW
Stator
Pump turbine
(Francis type)

Figure 23
...

The armature terminals, rated at 18 kV, of the 20-pole induction machine are connected to
a 500 kV utility grid through a step-up transformer
...
With
a synchronous speed of 360 rpm the speed can be controlled from 330 rpm to 390 rpm
...


23
...
2

Induction Motor Drives with Vector Control

The vector or FOC, introduced in the beginning of 1970s (Blaschke 1972), has revolutionised
the control of high-performance ac drives when, with this control, an induction motor drive
can be operated like a separately excited dc motor drive
...
In a separately excited dc machine, use of power electronic converters with
current feedback provides a direct control of the magnitude of the armature current and in
proportion, the torque
...
In addition, unlike the dc machine,
where the orientation of the field flux and the armature MMF is fixed by commutator and
brushes, ac machine requires external control to fix this orientation without which the space
angle between various fields vary with load (and during transients), giving rise to oscillatory
dynamic response
...
The technique can be applied to
either induction or synchronous motor fed from either CSI/ CRPWM (current regulated PWM
VSI) / VSI or cycloconverter (Chattopadhyay 1997a)
...
In VC, the currents i ds and i qs , the d-axis
and q-axis components, respectively, of the stator current in synchronously rotating reference
frame are analogous to the field current I f and to the armature current Ia of the dc machine
e e
and therefore the torque can be expressed as Te = K t ?m i qs = K t? I f Ia = K t?? i qs i ds
...
For normal operation as in the dc machine, the
e
e
current i ds remains constant and the torque is varied by varying the i qs component
...
The direct
method is based on the measurement or computation of the magnitude as well as the position
of the flux vector and the indirect method uses a slip relation to compute θe as a sum of θr and
θsl (corresponds to ωr and ωsl , respectively)
...
19 shows the block diagram of a direct VC scheme for a Current-regulated PWM
(CRPWM) inverter-fed induction motor drive (Chattopadhyay 1997a)
...
c

IM
TG
Ia
Ib
Ic

Comp &
switching

-

-

-

-

Va Vb Vc

PI2

Set flux

3/2 Trans

ia* ib*

+-

idse*
PI1

Set speed
+

iqs

Vector
rotator

e*

ic*

ids

sin θe

Ψdr

ψr =
cosθ e =

2

s
ψdr
s
ψdr

vqss

s'

Flux
observer

C
Ψqrs '

cos θ e

-

3/2 Trans
iqss

s

+ ψqr

sinθ e =

,

ψr

vdss

ωr

2

s
ψqr

ψr

Figure 23
...
A flux feedback loop provides precision flux control
...

s
s
The air-gap fluxes ?dm and ?qm can be measured directly by search coils /Hall probes or
estimated (observed) from stator voltage and current signals
...

Indirect Vector Control (Flux Feed-Forward)
An alternative to direct measurement or estimation of the flux position for application of VC
is to employ a slip relation derived from rotor voltage equations in a synchronously rotating
reference system with rotor flux entirely in the d-axis (Bose 2002),
Lm
ωsl = ? e
?r

?

?
Rr
?
Lr

?

e
i qs

(23
...
&
control
6

CRPWM

θe*

Ia Ib Ic
D

ωsl*
+

ωe*

+
N
Slip calculator ωr

IM

TG

Figure 23
...


(23
...
20 shows the block diagram of the indirect vector-controlled induction motor
e∗
e∗
drive (Chattopadhyay 1997a)
...

e∗
e∗
i qs is controlled according to the desired torque and constant rotor flux
...
Indirect VC, also known as flux feedforward control, has the
limitation in the slip calculation that depends on the commanded machine parameters that
may differ from the actual values during the running condition of the drive
...
A universal field
oriented controller applicable to both direct and indirect field orientation was reported in
DeDoncker and Novotny (1994)
...
21 for both indirect and direct VC
...
22)
...
00

Set speed
Actual speed

500
Speed(r/min)

Speed(r/min)

600

0
–100
–500
–1000

3
...
00

4
...
00

5
...
0

6
...
5

4
...
5

5
...
5

6
...
00

5
...
00

5
...
50

6
...
0

5
...
0
Speed(r/min)

15
...
0

Speed(r/min)

15
...
0
–5
...
0
–5
...
0

–10
...
0
3
...
50

4
...
50

5
...
50

6
...
0
3
...
50

4
...
50

Time(s)

Time(s)

(c)

0
...
4

0
...
0

0
...
0

0
...
2
0
...
0

3
...
5

4
...
50

Time(s)

5
...
5

6
...
3
0
...
1
0
...
00

3
...
00

4
...
21 Simulation results showing (a) speed, (b) torque and (c) rotor flux of a vector-controlled
induction motor drive for speed reversal (+600 to −600 rpm) (i) direct VC and (ii) indirect VC
(Chattopadhyay 1997a)

problems, besides the need for shaft extension and mounting arrangement
...
1996; Bose 2002) such as slip calculation, direct synthesis from machine
state equations, model referencing adaptive systems, speed adaptive flux observer, extended
Kalman filter and slot harmonics
...
22 Block diagram of a sensor-less VC system

machine parameters
...
Attempts
have been made to inject auxiliary signal at a carrier frequency from the stator side for a machine
designed with saliency and processing the response but with limited success (Holtz 2002)
...
with a wide range of speed control with speed control accuracy of ±0
...

Two commonly used methods for flux estimation by sensing the machine terminal voltages
and currents are the voltage model and current model as described in Bose (2002)
...
A hybrid model (Jansen and Lorenz 1992) is possible where the
voltage model is effective at higher speed ranges but transitions smoothly to the current model
at lower speed ranges
...
A speed-sensorless vector-controlled inverter equipped with automeasuring of the parameters is reported in Ohmori et al
...
A DSP-based speed adaptive
flux observer is described in Kubota et al
...


23
...
3

Induction Motor Drives with Direct Torque Control (DTC)

A new concept to control the torque and flux in induction motor drives, popularly known as
DTC, which is basically a performance-enhanced scalar control was developed in the late
eighties and commercialised in the late nineties by the ABB with IGCT inverters
...
The main variable to be
controlled in the DTC scheme is ?s that can be directly controlled by the stator voltage vs
(neglecting stator resistance)
...
23 DTC control of induction motor (Wu 2006)
...
The scheme is shown
in Figure 23
...
The machine voltages and currents are
sensed to estimate the torque and the flux vector that gives information about the angle θs in
one of the 60◦ sectors as shown in Figure 23
...
The vector ?s rotates in a circular
orbit within a hysteresis band covering six sectors as shown
...
24 (Bose 2006) shows
the six active voltage vectors and two zero vectors of the two-level inverter (relevant to the
Space Vector PWM control) controlled by the voltage switch logic unit (SLU) of Figure 23
...

If a voltage vector is applied for time ?t, the corresponding flux vector increment is given
by the relation ?ψ s = V s ?t
...
The flux is initially established at zero frequency in the radial

qs

ΔΨ3

V4 ·Δt4

V3 ·Δt3

E

B

S(4)

C
S(2)
S(1)

a
π
3

S(5)

Ψs

Ψs

ωe

ΔΨ5
V4 (011)

ΔΨ6

V1 (100)

ds

Ψ

ΔΨ1

ΔΨ4

V3 ·Δt1

Ψs A

ΔΨ2

V2 (110)

V4 ·Δt2

D

S(3)

V3 (010)

V0 (000)
V7 (111)
V5 (001)

S(6)

V6 (101)

2HB

Figure 23
...
24a
...
The motor state calculations are updated in
each sampling period ?t (e
...
25 μs) in the software model by a DSP
...

The delays associated with the PWM stage are replaced by an optimal switching flux vector
selection table SLU (can be realised by an ASIC hardware or through a DSP software) or
a look-up table which selects the most appropriate voltage vector to satisfy the flux and the
torque demands
...
However, as the feedback signals are estimated from the
machine terminals, the low speed limitation and the parameter variation problems are similar
to those of the stator flux oriented direct VC
...
Recently, a number of solutions of the inherent problems have been developed with
the use of improved switching logic, discrete SVM techniques, three-level inverters, adaptive
hysteresis-band control (Okumas and Aktas 2007) and introduction of fuzzy and neuro-fuzzy
techniques involving more computer power
...
25 (Malik and Khage 1998) shows a
circuit configuration of a compact 5 MW three-level IGCT converter motor system ACS 1000
with front-end rectifier and DTC control where the static speed control error is in the range
of 0
...
Typical torque response of a DTC drive is <10 ms compared with 10–20 ms for a
vector-controlled drive and >100 ms for an open-loop PWM drive
...

ABB has supplied IGCT-based ACS 6000 (3–27 MVA) with front-end controlled rectifier
(Active Rectifier Unit) designed to meet specific challenges faced by plate mills and reversing
cold mills-one in the new 5 m wide plate mill in China in 2005 where 10 MW synchronous
motors are used
...

motor

a

b

Figure 23
...

Reprinted with permission from IEEE (Chattopadhyay 2010)

534

23
...
5
...
One is
the true synchronous mode in which the machine is controlled by inverter or cycloconverter
through an independent oscillator just like the V / f control of an induction motor drive
...
26) where the inverter or cycloconverter firing signals are derived from a rotor
shaft position sensor; dc-CLM, when supplied from an inverter and ac-CLM when supplied
from a cycloconverter
...
Here, the frequency is slaved to the speed and not vice versa
...
Any slow down of the motor, no matter sudden, causes a corresponding drop
in frequency
...


LCI Synchronous Motor Drive
An important feature of the synchronous machine is that it can be operated at leading power
factor and when supplied by a CSI, load commutation can be used
...
27 shows the
power circuit for such a drive
...
Special starting arrangements with forced commutation through fourth leg or ‘current
pulsing’ are to be made at starting /low speed running
...
2000) with unaided
start-up capability, having the field winding in the dc link (Figure 23
...
LCI-fed drives in CLM mode
are widely used in high-power drives such as pumps, compressors, pumped storage hydroand gas turbine start-up applications besides continuous rolling mills and traction drives
...


+

Frequency
converter

SM

Shaft
position
sensor

Gating
circuit
DC CLM

Figure 23
...
27 LCI-fed synchronous motor drive

power rating for this drives has gone up to 100 MW for a NASA wind tunnel drive with a
single synchronous motor as discussed later
...
29) is best suited (Das and Chattopadhyay 1996)
...
1983; Nakano et al
...
1986);

Th1
Three
phase
AC IN

Th5

Th3

C

φa

F
i
e
l
d

Synchronous
machine

φb
φc
Th4

Th2

Th6

ω

Position
encoder

Inverter
controller

Figure 23
...
2000)

536

AC Electric Motors Control

TAE: Torque angle estimator
AR: Angle resolver
FCG: Field current generator
Cycloconverter
∗
Va

Function
generator

ω∗
r

PI1

δ

+

ωr

β

Limiter
r∗

r∗

Field

Phase
delay

Arm
...
29 Cycloconverter synchronous motor drive with scalar control (ac-CLM) (Das and Chattopadhyay 1996)

• Drives for mine hoists with high power ratings (Madiseti and Ramlu 1986);
• Icebreakers and other ships equipped with diesel generator-fed cycloconverter-fed SM with
power rating up to about 20 MW rating per unit (Hill et al
...

The cycloconverter is normally operated with line commutation but can have load commutation if the output frequency approaches or exceeds the line frequency
...
The method of driving the motor from a cycloconverter
with the transvector control principle involving FOC was patented and used by Siemens
in Germany, for years, in very high capacity cement Mills (>8 MW) and also for rolling
mills rated above 3 MW since 1978
...
The control concept
and relevant vector diagrams for field oriented operation is detailed in Salzmann (1978) and
Sugi et al
...


537

AC Motor Control Applications in High-Power Industrial Drives

23
...
2

Synchronous Motor Drives with Vector Control

The VC of SM is different from that of induction motors primarily due to the fact that, in
the latter, the magnetising current can be supplied from the field side independently of the
armature current and the space position of the field is located by the position of the rotor
...
Therefore, the indirect or feed-forward type of VC as used
extensively for the induction motor does not seem obvious for a synchronous machine
...
The response of the field current is sluggish because of the large time
constant and as a result, the response of a self-controlled synchronous machine is slow
...


Vector Control of a Cycloconverter-Fed Synchronous Motor Drive
Figure 23
...
2005) shows the vector diagram
of the synchronous machine (as preferred for a high-capacity steel mill), used to develop
FOC required to adjust speed and torque, where es is the air-gap emf, i q (torque-producing
component) is the quadrature axis component of the current i s , i d (flux-producing component)
is the direct axis component of current i s , ? the magnetic flux, i μ is the magnetizing current,
ϕ L is the load angle, ϕs is the flux axis angle and λ is the rotor axis angle
...
These two components can be independently controlled with
VC
...
31 shows the simplified block diagram (Rodriguez et al
...
The speed controller delivers
∗
the reference value of the torque-producing current i q , while the flux controller delivers the
∗
reference value of the field-controlling current i d
...
30 Vector diagram of the synchronous machine
...
31 Block Diagram of the speed and torque control for the the mill with cycloconverter
(Siemens)
...
The position of the flux is used to
transform from d − q to α − β reference axis in block 2
...
M2 in block 6 is the current model that uses
the current components in field coordinates (i d , i q ) to determine the flux position with respect
to the rotor axis
...
The current model is useful during low-operating speeds as needed
at starting and positioning of the mill when the machine voltage terminals are very noisy for
using voltage model
...
Nakano et al
...
Here, the flux linkage
was kept constant by feeding part of the field current to the armature windings transiently and
the power factor could be controlled to unity
...
It is briefly
described in the following sub-section
...
32 shows the block diagram of an observer-based stator flux oriented vector controlled six-pulse cycloconverter drive as applicable to rolling mills (Das and Chattopadhyay
1997) and the corresponding phasor diagram
...
1972; Trantner and Wick 1988) as well as the Japanese one (Nakano et al
...

The implementation aims at a control that maintains a spatial orthogonality between the flux
vector ?s and the armature current vector i a as shown in the space phasor diagram
...
The stator flux is estimated by a closed loop reduced order observer
...
The magnetisation current along the flux axis i m
is obtained from a flux controller (PI) C2
...
The steady state displacement angle is decided by the displacement angle controller and
the power factor can be maintained at unity
...
C3 is the field current controller (PI) that generates the control
voltage for triggering the field converter
...
The observer and the
control circuit design aspects together with the PC-based implementation are detailed in (Das
and Chattopadhyay 1997)
...
It is easier for digital computer implementation as it does not
contain any derivative term
...

Figure 23
...


23
...
6
...
Since 1970s, AC
motor drives having either IM or SM fed from either direct ac/ac cycloconverters or ac/dc/ac
link inverters have replaced the earlier Thyristor-Leonard DC motor drives
...
Synchronous motor drives

540

AC Electric Motors Control

x
F lu

r
To

e
qu

ax

is

b

Va
ia

θM

im

=0

ν

ia

is

ψS

im

α

δ β

ax

Rotor a

ωr

xis
θr

i fd

a

Rotor

A dc Machine Analogy

ia
α

Fi

c

el

d

ψ

(a) Phasor diagram

6PCC: 6-Pulse cycloconverter & control
VR: Vector rotator

Three-phase ac
re
K PC

ψs
C1

ωr∗

+

–

Te∗

∗
isa

∗
iST

÷

VR

i∗

ωr

SM

+

–
C5
C6

ν∗
sa
ν∗
sb

6PCC

ν∗
sc

+

Displacement angle control

∗
i 'SM

θr

Current regulator
δ

C2

∗
ψs

i'm

–

ψs

δ

ψs
Eca
VR

OBSERVER

C3

∗
ifd

isα
isβ

cos δ

Ecb
Ece

FLUX

ifd cos δ

+

–

ωr ψs

+
Eco
K PC

∗
isc

cos θM

0
e, M

∗
isb

C4

θr

ωr

vs α
vs β
ifd
vfd
Field

a

b

c

SM

÷
θr , ωr

+

e, T

–
ifd

encoder

(b) Block diagram

Figure 23
...
Reprinted with permission from IEEE (Das and Chattopadhyay 1997)

541

AC Motor Control Applications in High-Power Industrial Drives

1
...

current –0
(pu)

...



...

Time (s)
(a)

...
00v 2 2
...



...
10V

V 2(1) – 0
...
:0 V

(b)

Figure 23
...
The entire experiment lasts 5 seconds
...


have the advantages over the induction motor drives in that these can be operated in near
unity or even leading power factor with excitation control, reducing armature copper loss and
permitting simplicity of commutation with thyristors or SCRs (silicon-controlled rectifiers)
as switches (as in a LCI-fed drive) and it runs at a precisely set speed independent of load
and voltage fluctuations
...

With the introduction of FOC, a high performance 4 MW Blooming Mill with a
cycloconverter-synchronous motor drive having a speed of 60–120 rpm was commissioned
by Siemens in 1981 together with a 4 MW roughing stand of a strip mill (Chattopadhyay

542

AC Electric Motors Control

2010)
...
Cold rolling
mills such as tandem mills require high dynamic response, accurate speed and torque control
of main and auxiliary drives while hot rolling mills, such as roughers, and hot strip mills
require good torque control and momentary overloadability; all such performance criteria are
met by these drives
...
Hitachi and
Mitsubishi of Japan reported the development of high-performance three-level GTO-based
6
...
One such configuration developed by Mitsubishi is shown
earlier (Figure 23
...
Regenerative snubber circuit developed to have high efficiency and
a SVM method to minimise harmonic distortion are discussed in Okayama et al
...

Hitachi (Tobise et al
...
4 MW
and 2 MW IGBT-based three-level inverters for steel rolling mills at the same time
...
These converters compete with cyclocoverters in the capacity region of 10 MVA or less
...
10 were introduced in many steel
plants up to 3 MW of medium capacity, for example 1
...
(1996), where three inverters were driven by a common converter
...
Mitsubishi MELVEC 2000 N three-level IGBT inverter
(1
...

Figure 23
...
2008) shows the application of DTC controlled IGCT-based
ACS 6000 as supplied by ABB to a cold reversing mill and plate mill, respectively
...
5 kHz), higher efficiency (98%) and higher input power factor (0
...
The DTC method employed allows accurate control of both rotor
speed and torque without pulse encoder feedback from the motor shaft
...
2000, 2004; Suzuki et al
...
10,
resulting in higher efficiency and less size of the equipments
...
5 kV, 5
...
2000)
...
2005), for a hot strip mill of Hunan Valin Liangang Steel
Co
...
A new method of PWM control named as fixed pulse
pattern PWM to reduce the harmonics in the source input currents without increasing the
switching frequency for use with these inverters has been reported in Tsukakoshi et al
...

GE-Toshiba has developed 6–26 MVA Dura-bilt5 MV drives with IEGT-based NPC inverters
(GE Toshiba 2003)
...
35 shows one phase leg of a three-level 10 MVA IEGT Inverter
with its packaging unit (Tessendorf et al
...


(a)

Motor

Motor

Reversing cold Mill

Gear
box

ACS 6000

Top motor

(b)

Bottom
motor

Extension
Joint
shaft
spindles

Hot
material

Work
rolls

Figure 23
...
(For a color version of this figure, please see color plates
...
35 One phase leg of a three-level 10 MVA IEGT inverter with packaging unit for a rolling
mill (Tessendorf et al
...
Reprinted with permission from IEEE (Chattopadhyay 2010) (For a color
version of this figure, please see color plates
...
6
...
The world’s first gearless Tube/Ball mill drive with motor rating of 8700 hp (6400 kW)
at 15 rpm (44 poles, 5
...
The
electrical aspects of the first large gearless ball mill installed at St
...
(1975) with a motor rating 8750 hp (6500
kW) at 14
...
84 Hz)
...
The
motor thus cannot fall out of step and the characteristics are similar to dc machine
...
First gearless drive with the ring motor
(rotor of the synchronous machine wrapped directly round the mill cylinder) and FOC is
reported by Siemens in 1978 (Salzmann 1978) and later an improved version in Trantner and
Wick (1988)
...
5
...
ABB developed world’s largest gearless ball mill drive in cement rated
15 000 hp (11 200 kW) installed in the United States for a mineral grinding process (Errath
1996)
...
36
...
The flange is bolted on to the mill drum
...

2005)
...
5 MW for a copper mine is reported in (Pontt et al
...

2005; Bose 2011b)
...


23
...
3

Ship Drive and Marine Electric Propulsion

Electric propulsion is now well-established in large ship drives and in the merchant marine,
particularly, in cruise liners, icebreakers, shuttle tankers, and so on, as well as in warships
...
36 View of a wrap-around or ring motor for cement or ore grinding mill

motors to facilitate variable motor speed and thrust from fixed or controllable pitch propellers
...
The feasibility of a practical marine MC for electrical
propulsion system has been studied recently (Bucknall and Ciaramella 2010)
...
For example, US Coast Guard
Icebreaker Healy is equipped with 2 × 11
...
6 MW 12-pulse ALSTOM Alspa CL9000
Cycloconverters capable of providing 175% full load trorque for 30 s at zero speed (English
2001; Radan 2004)
...
A vector-controlled cycloconverter-fed drive designed for
icebreaker to deliver 16 000 hp to the twin propeller shafts of a Canadian Coast Guard
icebreaker is reported in (Hill et al
...

LCI-fed synchronous motor drives (also known as Synchroconverter-CSI drives) are ideally
suited to normal high-power ship propulsion applications such as the cruise liners, for example,
RCI Cruise Liner INFINITY with two 19 MW Mermaid podded propulsers which use 2 × 7
MW, 0–118/135 rpm motors with 2 × 12-pulse synchroconverters (English 2001; Radan 2004)
...
1999), where a
24-pulse SCR-rectifier-inverter system serves as a frequency converter to convert a voltage of
14–25
...
6 kV, 60 Hz for the ship’s
main distribution system
...
6 kV have been used in drill ships
...
37 Schematic diagram of a IPS drive system of a ship (Crane and McCoy 1999)

Africa is fitted with 7 medium voltage ALSTOM VDM5000 IGBT variable speed thrustor
drives up to 4
...
World’s first electric warship-UK’s “daring class” Type45 Destroyer, in service from 2007 is fitted with two 15-phase 20 MW, 4
...
An integrated
power system for all electric ship in a full-scale main propulsion drive for US navy (Crane and
McCoy 1999) consists of a main propulsion 19 MW induction motor drive system
...
37) consists of three 6-pulse rectifier bridges, three 6 kV dc links and 15
IGBT-based H bridges feeding a 15-phase induction motor
...
6
...

Torques of about six times the rated torque at low speed are possible and with digital monitoring
the winding cycle can be optimised, smooth and accurate
...
g 2
...
7 Hz, 65
...
The Pyhasalmi mine hoist is the first in the world to

AC Motor Control Applications in High-Power Industrial Drives

547

use ABB’s state-of-the-art ACS6000SD with DTC control using IGCTs to power the16-pole
2
...
This technology offers several advantages
over the alternative systems like cycloconverter and PWM converter in terms of footprint, high
reliability, high torque control over the entire speed range, unity power factor and lower energy
consumption
...
5-m-deep Majaling coal mine
in China equipped with one 3 500 kW synchronous motor and another 435 kW, having Simatic
programmable logic controllers for the automation system
...
7 MW for hoist and dragline for use in Zhungeer coal line in China
...
6
...
AC
drive applications in this market, sometimes referred as HVAC (Heating, Ventilation and Air
Conditioning) have been developing over the last 35 years
...
Later in 1980s, adjustable-speed LCI-fed synchronous motor
drive system with constant V/Hz control for pump and compressors were discussed in Weiss
(1983)
...
2010)
...
Largest variable-speed synchronous motor fan drive is
for the NASA 100 MW wind tunnel consisting of a converter with two independent channels,
resulting in a total of four identical six-pulse thyristor bridges and a six-phase synchronous
motor having two sets of stator windings with 30◦ electrical phase shift between them (Bhatia
et al
...
The efficiency of the LCI drive is very high (99%) and is very important in
high-power drives in energy saving
...

TMEIC (Toshiba-Mitsubishi Electric Industrial Systems Co) has developed a 30 MVA IGCT
controlled five-level VSI-fed synchronous motor (25 MW, 7
...
2009)
...
The five-level inverter output voltage and current are
much more sinusoidal and of higher magnitude compared to three-level inverter
...
2 kV
30 MVA converter can be applied in parallel up to four sets for a maximum capacity of 120
MVA using balancing reactors
...
0 kV five-level
IEGT Inverter for the LNG Industry with efficiency more than 99% (Tsukakoshi et al
...

Commercial drives developed by Siemens (e
...
Siemens Sinamics GM150 converter with
IGBTs) and ABB (e
...
ACS 1000, ACS 5000 & ACS 6000 with IGCTs and DTC) are widely
used for high-power pump and compressor applications
...
7

AC Electric Motors Control

New Developments and Future Trends

With the continued development of power semiconductor devices, multi-level inverters, control
and estimation technologies, variable speed high power ac drives have gone through a dynamic
evolution and poised for new developments in future
...
However,
the challenges are the SiC device fabrication processes which are expected to advance in
the next few years
...
2009) and direct torque control of IM (Kowalski et al
...
The
extremely fast FPGA computation time allows higher throughput and parallel architecture
to overcome the typical bottlenecks of DSP sequential algorithms
• Adaptive, optimal and intelligent control based on fuzzy logic (FL) and neural network
are emerging technologies (Bose 2012) that, when commercialised, will have dominant
impact on high power drives in future
...
Optimal control may be model-based predictive control where a performance parameter like
response time, efficiency, or energy consumption is optimised
...
2009)
...
Fuzzy logic has been used in online search-based flux programming efficiency optimisation control of indirect vector-controlled induction motor drive
(Sousa et al
...
Neural network applications have been proposed in motor drives
and power electronics as discussed in Bose (2007)
...
2004; Delgado et al
...
The concept here is that the drive will continue to
operate at a minimum level of performance as per system requirements after sustaining
a fault
...
8

Conclusions

A comprehensive but brief state-of-the-art review of the development of AC motor control in
industrial high-power drives involving high-power semiconductor devices, power converter
topologies, induction and SM, advanced control strategies used and their implementation, along
with their application examples is presented in this chapter
...
Scalar control
includes V/Hz control and DTC control as used extensively in high power industrial drives
...
Recent
improvement in MC, for medium-voltage highpower drives is also reported
...

At the end, new technology developments and future trends in this field have been indicated
...


References
Akagi H (1998) The state-of-the-art of power electronics in Japan
...

Allan JA, Weyth WA, Hwerzog GW, and Young JAL (1975) Electrical aspects of the 8750 hp gearless ball mill drive
at st
...
IEEE Transactions on Industry Applications, IA-11(6), 681–686
...
Siemens Review, 39, 220–223
...
(1999) Adjustable speed drive using a single 135,000 hp synchronous motor
...

Blaschke F (1972) The principle of field orientation as applied to the new transvector closed loop control system for
rotating field machines
...

Bose BK (1982) Adjustable speed ac drives-a technology status review
...

Bose BK (2002) Modern Power Electronics and AC Drives
...
(Asia)
...
Academic, Burlington, MA
...

IEEE Transactions on Industrial Electronics, 14–33
...
IEEE Power
Electronics Society Newsletter, 26–30
...
IEEE
Industrial Electronics Magazine, 12–22
...
IEEE Power
Electronics Newsletter, 35–38
...
IEEE Trans on Power Electron, 25(6), 1497–1508
...
International Mining, 38–44
...
Sadhana (Journ
...
of
Science), 22(6), 797–820
...
Special Issue of Journal of
Indian Institute of Science, 77(5), 397–419
...
Proceedings of Indian
International Conference on Power Electronics (IICPE), 1–12
...
IEEE Industrial Electronics Magazine,
4(4), 30–42
...
Proceedings of 9th International Conference on Electro Machanical Drives, 312–316
...
Proceedings of IEEE Industry
Applications Society Conference, vol
...

Das SP and Chattopadhyay AK (1996) Comparison of simulation and test results for an ac commutatorless motor
drive
...

Das SP and Chattopadhyay AK (1997) Observer-based stator flux oriented vector control of a cycloconverter-fed
synchronous motor drive
...

DeDoncker RW and Novotny DW (1994) The universal field oriented controller
...


550

AC Electric Motors Control

Delgado DVC, Trejo DRE, and Palacious E (2008) Fault tolerant control in variable speed drives: A survey
...

English P (2001) The evolution of marine and drilling drives in today’s market
...

Erickson EW and Al-Naseem OA (2001) A new family of matrix converters
...

2, 1515–1520
...

GE Toshiba (2003) A new family of MV drives for a new converter DURABUILT-5i MV Product Brochure, March
...
Wiley-Interscience,
New York
...

IEEE Transactions on Industry Applications, IA-23(6), 1030–1041
...
IEEE Industry Applications Magazine, 16(2), 22–30
...
IEEE-PEDS 2005 Conference
REC, pp
...

Ichihara K, Katayama S, Kokubu T, et al
...
IEEE
Transactions on Industry Applications, IA-22(2), 345–351
...
(2000) New advanced high voltage inverter employing IEGTs
...
994–999
...

Proceedings of 19th Annual IEEE Applied Power Electronics Conference (IEEE-APEC), 1663–1668
...
(2000) A vector-controlled high performance matrix converter induction motor
drive
...

Jansen P and Lorenz RD (1992) A physically insightful approach to the design and accuracy assessment of the flux
observers for field oriented induction machine drives
...

J
...
Proceedings of IEEE, 90(6), 1359–1394
...
(2009) Model predictive control – a simple and powerful method to control power
converters
...

Kowalski CT, Lis J, and Kowalski TO (2007) FPGA implementation of dtc control method for the induction motor
drive
...

Kubota H, Matsuse K, and Nakano T (1993) DSP-based speed adaptive flux observer of induction motor
...

Madiseti MK and Ramlu MA (1986) Trends in electronic control of mine hoists
...

Malik S and Khage D (1998) ACS 1000–world’s first standard ac drive for medium voltage applications
...

Masuda H and Toyoda M (2000) Variable speed drive systems for steel plants
...

Monmassom E and Christea M (2007) FPGA design methodology for industrial control systems-a review
...

Nabae A, Takahasi T, and Akagi H (1981) A new neutral point clamped pwm inverter
...

Nakano T, Oswawa H, and Endoh K (1984) A high performance cycloconverter-fed synchronous motor drive system
...

Ohmori Y, Nakanishi T, and Kobayashi H (1995) A speed sensorless spatial vector controlled inverter adding an
auto-measuring function
...
3, 452–457
...
(1996) Large capacity high performance 3-level gto inverter system for steel
main rolling mill drives
...

Okumas HI and Aktas M (2007) Direct torque control of induction machine drives using adaptive hysteresis band for
constant switching frequency
...

Pontt J, Rodriguez J, Valderrama W, et al
...
Proceedings of ISIE, 369–374
...


AC Motor Control Applications in High-Power Industrial Drives

551

Rajashekara K, Kawamura A, and Matsuse K (1996) Sensorless Control of AC Motor Drives: Speed and Position
Sensorless Operation
...

Richlen D (1971) Gearless drive for a cement mill
...

Rodriguez J, Benett S, Wu B, et al
...
IEEE Transactions on Industry Electronics, 54(6), 2930–2943
...
(2005) Technical evaluation and practical experience of high power grinding
mill drives in mining applications
...

Rossman WC and Ellis RG (1998) Retrofit of 22 pipeline pumping stations with (3000) hp motors and variable
frequency drives
...

Salzmann T (1978) Cycloconverter and automatic control of ring motors driving tube mill
...

Sato K and Yamamoto M (2001) The present state-of-the-art in high power semiconductor devices
...

Sengupta S, Mukherjee K, Bhattacharya TK, and Chattopadhyay AK (2000) Performance of an SCR-inverter based
commutatorless series motor with load commutation and un-aided start-up capability
...

Siemens (2011) Gearlessly Into the Mine Gearless drives, July, 26–27
...
industry
...
com/
...
/26_MM0711_GearlessMM0711_technology
...

Sinard S, Begunenane R, and Maillox JG (2009) Performance evaluation of rotor flux oriented control on FPGA for
advanced ac drives
...

Singh R and Pecht M (2008) Commercial impact of silicon carbide-opportunities and challenges in realizing the full
potential of sic power devices
...

Sousa GCD, Bose BK, and Cleland JG (1995) Fuzzy logic based on-line efficiency optimization control of an indirect
vector controlled induction motor drive
...

Stemmler H (1994) High power industrial drives
...

Sugi K, Naito Y, Kurusowa R, et al
...
Proceedings of IPEC, 744–755
...

Proceedings of Association of Iron and Steel Engineers (AISE), 1–7
...
Proceedings of AISTechnology, pp
...

Tessendorf R, Kodama S, and Hosada H (2008) Large PWM inverters for rolling mills
...

Timpe W (1982) Cycloconverter drives for rolling mills
...

Tobise M, Onda K, Sugiyama S, and Yamashina M (1996) Inverter-fed ac drive system for steel rolling mills
...

Trantner J and Wick A (1988) Dc link converter and cycloconverter-fed ac motors: The concept and properties of large
variable speed drives
...
(Energy and Automation Special Issue on Large Electric Motor AC Variable
Speed Drives), 1, 16–31
...
IEEE Energy Conversion Congress and Exposition (ECCE2009), 3485–
3492
...
(2010) Introduction of a large scale high efficiency 5-level iegt
inverter for oil and gas industry
...

Tsukakoshi M, Mukunoki M, and Nakamura R (2005) High performance IEGT inverter for main drives in the steel
industry
...
S15-2, 724–729
...
Proceedings
of POWERCON’80, vol
...
1–E3
...

Wechko BA, Lipo TA, Jahns TM, and Schulz SE (2004) Fault tolerant three phase ac motor drive topologies: A
comparison of features, cost and limitations
...

Weiss HW (1974) Adjustable speed ac drive systems for pump and compressor applications
...

Weiss HW (1983) Power transmission to synchronous machines for adjustable-speed pump and compressor drive
systems
...

Wu B (2006) High Power Converters and AC Drives, IEEE Press/Wiley-Inter-science 18
...
IEEE Transactions on Industril and Gen Appl
...

Yamamoto E, Hara H, Uchino T, et al
...
IEEE Industrial
Electronics Magazine, 4–12
...

Millenium Steel, 168–170
...
IEE Proceedings of Elecric Power Applications, 145(4), 284–294
...
Edited by Fouad Giri
...
Published 2013 by John Wiley & Sons, Ltd
...
1 Squirrel-cage induction motor
...

(See page 18)
...
1 Experimental setup
...


AC Electric Motors Control: Advanced Design Techniques and Applications, First Edition
...

© 2013 John Wiley & Sons, Ltd
...


~
ωs

~
ωs Estimate stator
pulsation

Integrator

Conc
...
Park
...
νsd

Inverter

3

2

2

θ
ˆm

2

Isd control

i∗
sd
isd Flux control
i∗
sq

νsq

Isq control

Park
2 2

isq

isq
i∗
sq
φ∗
rd
ˆ
φrd
ωm

Speed
control

ω
ˆm

Control laws
Obs1

Concordia
3 2

isd
isq
Obs2
Interconn
...
2 Experimental setup and observer-control scheme
...


Mutual inductance: first + third harmonic
0
...
04
0
...
02
−0
...
06
0

0
...
2

0
...
4

0
...
6

Rotor revolutions

0
...
8

0
...
8 Mutual inductance between the first stator phase and the rotor phases
...


Voltages t V
s
(V)

100
50

tV

s

0
−50
−100
0
...
8

1
...
4

1
...
45

1
...
47

1
...
49

1
...
51

1
...
53

1
...
55

Time (s)

Figure 12
...
(See
page 250)
...
6

0
...
4

1
...
6

Currents tI s —zoom

t

Is (A)

1
0
...
5
−1
1
...
46

1
...
48

1
...
5

1
...
52

1
...
54

1
...
13 Stator currents in the original reference frame ?t during the velocity rising ramp
...


(a)

(b)

Figure 21
...

(See page 457)
...
3 (a) Configuration of Honda Insight parallel hybrid and (b) schematic of Honda Insight
drivetrain
...

Inverters

Wheel motors

M
DF
ICE
M

(a)

(b)

Figure 21
...
(See page 459)
...
10 Torque-speed characteristics and efficiency maps of (a) the IMPMSM in Toyota Camri
Hybrid (Olszewski 2008), (b) an induction machine (Gosden et al
...

2010)
...

400

Torque (Nm)

SUV-’05

300

Power

200

80
Torque

100
0

120

0

2000

Power (kW)

Prius-’00

160

40

0
4000 6000 8000 10 000 12 000 14 000
Speed (rpm)

Figure 21
...
(See page 472)
...
17 The electromagnetic design of the SPM machine with fractional-slot winding developed
at the University of Wisconsin-Madison (Reddy et al
...
(See page 472)
...
18 The electromagnetic design of the SPM machine with fractional-slot winding developed
at the University of New South Wales (Dutta et al
...
(See page 473)
...
19 Cross section of an 8/6 SRM
...


Figure 22
...
(See page 502)
...
18 Identification of the gear meshing frequency for motor speed related to ca
...
(See page 503)
...
34 (a) Cold reversing mill with ACS6000 and (b) plate mill with ACS 6000 ABB Reprinted
with permission from IEEE (Chattopadhyay 2010)
...


Clamp
Snubber

Neutral

Output

Figure 23
...
2008) Reprinted with permission from IEEE (Chattopadhyay 2010)
...




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