Notesale: Turn your study into money

Document Preview

Extracts from the notes are below, to see the PDF you'll receive please use the links above

---

MODULE-III
Introduction
A linear system designed to perform satisfactorily when excited by a standard test
signal, will exhibit satisfactory behavior under any circumstances
...
The stability of nonlinear systems is determined solely by the location of
the system poles & is independent entirely of whether or not the system is driven
...
In fact, the nonlinear system response may be highly sensitive to the input
amplitude
...
Further, the
nonlinear systems may exhibit limit cycle which are self sustained oscillations of fixed
frequency & amplitude
...
The amplitude of the
fundamental is usually the largest, but the harmonics may be of significant amplitude in
many situations
...

Jump Resonance
Consider the spring-mass-damper system as shown in Fig3
...
below
...
(3
...
3
...
2
...
3
...
3
...

Let us now assume that the restoring force of the spring is nonlinear,given by 𝐾1𝑥 +
𝐾2𝑥3
...
3
...
Now the system equation
becomes

𝑀𝑥 + 𝑓𝑥 + 𝐾1𝑥 + 𝐾2𝑥3 = 𝐹𝑐𝑜𝑠 𝑤𝑡 … … … … … …
...
2)
The frequency response curve for the hard spring(𝐾2 > 0) is shown in Fig3
...

For a hard spring, as the input frequency is gradually increased from zero, the measured
two response follows the curve through the A, B and C, but at C an increment in frequency
results in discontinuous jump down to the point D, after which with further increase in
frequency, the response curve follows through DE
...
This phenomenon which is peculiar to nonlinear systems
is known as jump resonance
...
3
...

Methods of Analysis
Nonlinear systems are difficult to analyse and arriving at general conclusions are tedious
...
It should be emphasised that very often the conclusions arrived at will be useful
for the system under specified conditions and do not always lead to generalisations
...


65

Linearization Techniques:
In reality all systems are nonlinear and linear systems are only approximations of the
nonlinear systems
...
Many techniques like perturbation method, series approximation
techniques, quasi-linearization techniques etc
...

Phase Plane Analysis:
This method is applicable to second order linear or nonlinear systems for the study of the
nature of phase trajectories near the equilibrium points
...
The periodic oscillations in nonlinear systems called limit cycle
can be identified with this method which helps in investigating the stability of the system
...
This method is useful for the study of
existence of limit cycles and determination of the amplitude, frequency and stability of
these limit cycles
...


Classification of Nonlinearities:
The nonlinearities are classified into
i) Inherent nonlinearities and
ii) Intentional nonlinearities
...
Examples
are saturation in magnetic circuits, dead zone, back lash in gears etc
...
Such nonlinearities introduced intentionally to
improve the system performance are known as intentional nonlinearities
...

Common Physical Non Linearities:
The common examples of physical nonlinearities are saturation, dead zone, coulomb
friction, stiction, backlash, different types of springs, different types of relays etc
...
All practical systems, when
driven by sufficiently large signals, exhibit the phenomenon of saturation due to limitations
of physical capabilities of their components
...

66

Fig
...
4 Piecewise linear approximation of saturation nonlinearity
Friction: Retarding frictional forces exist whenever mechanical surfaces come in sliding
contact
...
Vicous friction is thus linear in nature
...
One is the coulomb friction which
is constant retarding force & the other is the stiction which is the force required to initiate
motion
...


Fig
...
5 Characteristics of various types of friction

67

Dead zone: Some systems do not respond to very small input signals
...
This is called dead zone existing in a system
...


Fig
...
6 Dead-zone nonlinearity
Backlash: Another important nonlinearity commonly occurring in physical systems is
hysteresis in mechanical transmission such as gear trains and linkages
...
In
servo systems, the gear backlash may cause sustained oscillations or chattering
phenomenon and the system may even turn unstable for large backlash
...
7: (a) gear box having backlash (b) the teeth A of the driven gear located midway
between the teeth B1, B2 of the driven gear(c) gives the relationship between input and
output motions
...
This output motion corresponds to the segment mn of fig3
...
After the contact is
made the driven gear rotates counter clockwise through the same angle as the drive gear, if
the gear ratio is assumed to be unity
...
As the input
motion is reversed, the contact between the teeth A and B1 is lost and the driven gear
immediately becomes stationary based on the assumption that the load is friction controlled
with negligible inertia
...
7 (c) by the segment op
...
As the input motion is reversed the direction gear is again at standstill for the segment
qr and then follows the drive gear along rn
...
A relay
controlled system can be switched abruptly between several discrete states which are
usually off, full forward and full reverse
...
The characteristic of an ideal relay is as shown in figure
...
This dead zone is caused by the facts
that relay coil requires a finite amount of current to actuate the relay
...


Figure3
...
, are functions of more than one variable
...


Phase Plane Analysis
Introduction
Phase plane analysis is one of the earliest techniques developed for the study of second
order nonlinear system
...
The phase plane is thus a state
plane where the two state variables x1 and x2 are analysed which may be the output
variable y and its derivative 𝑦
...
The method is used for obtaining graphically a solution of the following
two simultaneous equations of an autonomous system
...
The state plane with coordinate axes x1 and x2 is called
the phase plane
...
The plot of the state trajectories or phase trajectories
of above said equation thus gives an idea of the solution of the state as time t evolves
without explicitly solving for the state
...
It can be extended to
cover other inputs as well such as ramp inputs, pulse inputs and impulse inputs
...
This will be true if 𝑓1 𝑥1, 𝑥2 and 𝑓2 𝑥1, 𝑥2 are
analytic
...
These points are called singular points
...
If the system is placed at such a point, it will continue to
lie there if left undisturbed
...
As time t increases, the phase portrait graphically shows
how the system moves in the entire state plane from the initial states in the different
70

regions
...
If the system has nonlinear elements which are piecewise linear, the complete state space can be divided into different regions and phase plane
trajectories constructed for each of the regions separately
...
9
𝜆1 Where c is
(a)
...
The trajectories become a set of parabola as shown in figure 3
...
In the original system of coordinates, these
trajectories appear to be skewed as shown in figure 3
...

If the eigen values are both positive, the nature of the trajectories does not change, except
that the trajectories diverge out from the equilibrium point as both z1(t) and z2(t) are
increasing exponentially
...
9
(d)
...


(c) Stable node in (X1,X2)-plane

71

(d)

Unstable node in (X1,X2)-plane
Fig
...
9
Saddle Point: Both eigen values are real,equal & negative of each other
...
10
...


Fig 3
...
A plot for negative
values of real part is a family of equiangular spirals
...
The
origin which is a singular point in this case is called a stable focus
...
When transformed
into the x1-x2 plane, the phase portrait in the above two cases is essentially spiralling in
nature, except that the spirals are now somewhat twisted in shape
...
11

Centre or Vortex Point:
Consider now the case of complex conjugate eigen values with zero real parts
...
, λ1, λ2 = ±jω
dy2
jwy1
−y1
=
=
dy1 −jwy2
y2

for which

y dy + y dy = 0
1

1

2

2

Integrating the above equation, we get 𝑦12 + 𝑦22 = 𝑅2 which is an equation to a circle of
radius R
...
The trajectories are thus
concentric circles in y1-y2 plane and ellipses in the x1-x2 plane as shown in figure
...


Fig
...
12 (a) Centre in (y1,y2)-plane (b) Centre in (X1,X2)-plane

Construction of Phase Trajectories:
Consider the homogenous second order system with differential equations
𝑑2𝑥
𝑑𝑥
𝑀 2 +𝑓
+ 𝐾𝑥 = 0
𝑑𝑡
𝑑𝑡
𝑥 + 2𝜉𝑤𝑛 𝑥 + 𝑤𝑛 2𝑥 = 0

73

where ζ and ωn are the damping factor and undamped natural frequency of the
system
...
The time response plots of
x1, x2 for various values of damping with initial conditions can be plotted
...
For example, if the spring force is
nonlinear say (k1x + k2x3) the state equation takes the form
𝑘1 𝑥1 = 𝑓𝑥2
𝑘2
𝑥 =− 𝑥 − 𝑥 − 𝑥 3
2
𝑀 1 𝑀 2 𝑀 1
Solving these equations by integration is no more an easy task
...
The coordinate plane
with axes that correspond to the dependent variable x1 and x2 is called phase-plane
...
A phase trajectory can be easily constructed by graphical techniques
...

From the above equation, the slope of the trajectory is given by
𝑑𝑥2 𝑓2 𝑥1, 𝑥2
=
=𝑀
𝑑𝑥1 𝑓1 𝑥1, 𝑥2
Therefore, the locus of constant slope of the trajectory is given by f2(x1,x2) = Mf1(x1,x2)
The above equation gives the equation to the family of isoclines
...
Knowing the value of
M on a given isoclines, it is easy to draw line segments on each of these isoclines
...
We can draw different lines in the x1-x2 plane for
different values of M; called isoclines
...
Different trajectories can be drawn from different initial conditions
...
A few typical trajectories are
shown in figure3
...


Fig
...
13
The Procedure for construction of the phase trajectories can be summarised as below:
1
...
Determine the equation to the isoclines as

𝑑𝑥2 𝑓(𝑥1, 𝑥2)
=
=𝑀
𝑑𝑥1
𝑥2
3
...
On each of the isoclines, draw small line segments with a slope M
...
From an initial condition point, draw a trajectory following the line segments With slopes

M on each of the isoclines
...

With the help of this method, phase trajectory for any system with step or ramp or any time
varying input can be conveniently drawn
...

While applying the delta method, the above equation is first converted to the form

𝑥 + 𝑤𝑛 [𝑥 + 𝛿(𝑥, 𝑥 , 𝑡)] = 0
In general 𝛿(𝑥, 𝑥 , 𝑡) depends upon the variables 𝑥, 𝑥 𝑎𝑛𝑑 𝑡 but for short intervals the changes in
these variables are negligible
...

Let us choose the state variables as

𝑥1 = 𝑥 , 𝑥2 = 𝑥 𝑤𝑛 , giving the state equations
𝑥1 = 𝑤𝑛 𝑥2
𝑥2 = −𝑤𝑛 (𝑥1 + 𝛿)

Therefore, the slope equation over a short interval is given by

𝑑𝑥2 −𝑥1 + 𝛿
=
𝑑𝑥1
𝑥2
With δ known at any point P on the trajectory and assumed constant for a short interval, we can
draw a short segment of the trajectory by using the trajectory slope dx2/dx1 given in the above
equation
...


1
...

2
...

3
...

Example : For the system described by the equation given below, construct the trajectory
starting at the initial point (1, 0) using delta method
...
Therefore, the initial arc is centered at
point(0, 0)
...
By constructing the small arcs in this way
the arcs in this way the complete trajectory will be obtained as shown in figure3
...


Fig
...
14
Limit Cycles:
Limit cycles have a distinct geometric configuration in the phase plane portrait, namely, that
of an isolated closed path in the phase plane
...
A limit cycle represents a steady state oscillation, to which or from which all
trajectories nearby will converge or diverge
...
It should be pointed out that not all
closed curves in the phase plane are limit cycles
...
Closed curves of this kind are not limit cycles because none of these curves are
isolated from one another
...
On the other hand,
limit cycles are periodic motions exhibited only by nonlinear non conservative systems
...


In terms of the state variables = 𝑥1 𝑎𝑛𝑑 𝑥 = 𝑥2 , we obtained
𝑥1 = 𝑥2
𝑥2 = 𝜇(1 − 𝑥12)𝑥2 − 𝑥1
The figure shows the phase trajectories of the system for μ > 0 and μ < 0
...
On the other
hand, if initially x1(0) is small, the damping is negative, and hence the amplitude of x 1(t)
increases till the system state enters the limit cycle as shown by the inner trajectory
...
15
...
3
...
In this case, the system exhibits a sustained oscillation with constant
amplitude
...
The inside of the limit cycle is an unstable region in the
sense that trajectories diverge to the limit cycle, and the outside is a stable region in the sense that
trajectories converge to the limit cycle
...
In this
case, an unstable region surrounds a stable region
...
If a trajectory starts in the unstable region, it
diverges with time to infinity as shown in figure (ii)
...


78

Describing Function Method of Non Linear Control System
Describing function method is used for finding out the stability of a non linear system
...
This method is
basically an approximate extension of frequency response methods including Nyquist
stability criterion to non linear system
...
We can also called sinusoidal describing function
...

Let us discuss the basic concept of describing function of non linear control system
...


Let us assume that input x to the non linear element is sinusoidal, i
...
e
...
So in the describing function analysis, we assume that only the fundamental
harmonic component of the output
...
Most control systems are low pass filters, with the result that the higher
harmonics are very much attenuated compared with the fundamental harmonic component
...

We can write y1(t) in the form ,
Where by using phasor,

The coefficient A1 and B1 of the Fourier series are given by-

From definition of describing function we have,

80

Describing Function for Saturation Non Linearity

We have the characteristic curve for saturation as shown in the given figure3
...
3
...
Characteristic Curve for Saturation Non Linearity
...


On substituting the value of the output in the above equation and integrating the function
from 0 to 2π we have the value of the constant A1 as zero
...
17
...
3
...
Characteristic Curve for Ideal Relay Non Linearity
...


On substituting the value of the output in the above equation and integrating the function
from 0 to 2π we have the value of the constant A1 as zero
...
18
...
If X > &Delta, the relay produces
the output
...
3
...
Characteristic Curve for Real Relay Non Linearities
...


85

On substituting the value of the output in the above equation and integrating the function
from 0 to 2π we have the value of the constant A1 as zero
...
19
...
3
...
Characteristic Curve of Backlash Non Linearity
...


On substituting the value of the output in the above equation and integrating the function
from zero to 2π we have the value of the constant A1 as

Similarly we can calculate the value of Fourier constant B for the given output and the value
of B1 can be calculated as

On substituting the value of the output in the above equation and integrating the function
from zero to pi we have the value of the constant B1 as

We can easily calculate the describing function of backlash from below equation

87

Liapunov’s Stability Analysis
Consider a dynamical system which satisfies
x = f(x, t); with initial condition 𝑥 𝑡𝑜 = 𝑥𝑜 ;

𝑥 ∈ 𝑅𝑛
...
3)

We will assume that f(x, t) satisfies the standard conditions for the existence and uniqueness
of solutions
...
A point 𝑥∗ ∈ 𝑅𝑛 is an equilibrium point of
equation (3
...

Intuitively and somewhat crudely speaking, we say an equilibrium point is locally stable if all
solutions which start near x* (meaning that the initial conditions are in a neighborhood of 𝑥∗
remain near 𝑥∗ for all time
...

We say somewhat crude because the time-varying nature of equation (3
...
Nonetheless, it is intuitive that a pendulum has a locally stable
equilibrium point when the pendulum is hanging straight down and an unstable equilibrium
point when it is pointing straight up
...
By shifting the origin of the system, we may assume that the
equilibrium point of interest occurs at x* = 0
...

3
...
3) is stable (in the sense of Lyapunov) at t = t0 if for any
𝜖 > 0 there exists a δ(t0, 𝜖 ) > 0 such that
𝑥(𝑡𝑜) < 𝛿 ⇒

𝑥(𝑡) < 𝜖 ,

∀𝑡 ≥ 𝑡𝑜 … … … … …
...
4)

Lyapunov stability is a very mild requirement on equilibrium points
...
Also,
stability is defined at a time instant t0
...
We insist that for a uniformly stable equilibrium
point x*, δ in the Definition 3
...
4) may hold for all
t0
...
2 Asymptotic stability
An equilibrium point x* = 0 of (3
...
x * = 0 is stable, and
2
...
e
...
(3
...

88

Uniform asymptotic stability requires:
1
...
x * = 0 is uniformly locally attractive; i
...
, there exists δ independent
of t0 for which equation (3
...
Further, it is required that the convergence in equation
(3
...

Finally, we say that an equilibrium point is unstable if it is not stable
...
In robotics, we are almost
always interested in uniformly asymptotically stable equilibria
...
Figure
below illustrates the difference between stability in the sense of Lyapunov and asymptotic
stability
...
1 and 3
...
We say an equilibrium point x* is globally stable if it is stable for all initial
conditions 𝑥0 ∈ 𝑅𝑛
...
We will concentrate on local stability theorems and indicate where it is
possible to extend the results to the global case
...
Thus, for time-invariant systems, stability implies uniform stability
and asymptotic stability implies uniform asymptotic stability
...
20 Phase portraits for stable and unstable equilibrium points
...

1
...

2
...

3
...

4
...

Theorm-1
Consider the system
𝑥 =𝑓 𝑥 ; 𝑓 0 =0
Suppose there exists a scalar function v(x) which for some real number ∈> 0 satisfies the
following properties for all x in the region 𝑥(𝑡) < 𝜖
(a) V(x)>0; 𝑥 ≠ 0 that is v(x) is positive definite scalar function
...
e dv/dt is negative semi definite scalar function)
𝑑𝑡

Then the system is stable at the origin
Theorem-2
If the property of (d) of theorem-1 is replaced with (d) 𝑑𝑣 < 0 , 𝑥 ≠ 0 (i
...

It is intuitively obvious since continuous v function>0 except at x=0, satisfies the condition
dv/dt <0, we except that x will eventually approach the origin
...

Theorem-3
If all the conditions of theorem-2 hold and in addition
...


90

Instability
It may be noted that instability in a nonlinear system can be established by direct recourse to
the instability theorem of the direct method
...

Direct Method of Liapunov & the Linear System:
In case of linear systems, the direct method of liapunov provides a simple approach to
stability analysis
...

However, the study of linear systems using the direct method is quite useful because it
extends our thinking to nonlinear systems
...
(3
...
(3
...
(3
...
Consider thescalar
function
...
(3
...
7) we get

Since Q is positive definite, V(x) is negative definite
...

In order to show that the result is also necessary, suppose that the system is asymptotically
stable and P is negative definite, consider the scalar function
V 𝐗 = 𝐗𝐓𝐏𝐗
...
8)
Therefore
V X = − 𝐗 𝐓𝐏𝐗 + 𝐗𝐓𝐏𝐗
= 𝐗𝐓𝐐𝐗
>0
There is contradiction since V(x) given by eqn
...
8) satisfies instability theorem
...
(3
...

Methods of constructing Liapunov functions for Non linear Systems
As has been said earlier ,the liapunov theorems give only sufficient conditions on system
stability and furthermore there is no unique way of constructing a liapunov function except in
the case of linear systems where a liapunov function can always be constructed and both
necessary and sufficient conditions Established
...
Since this treatise is meant as a first
exposure of the student to the liapunov direct method, only two of the relatively simpler
techniques of constructing a liapunov‟s function would be advanced here
...
(3
...

Now
V = 𝐟 𝐓𝐏𝐟 + 𝐟𝐓𝐏𝐟
...
10)
f =
∂f1 ∂f1
∂x1 ∂x2
∂f2
∂f2
J = ∂x1 ∂x2
⋮
⋮
∂fn
∂x1

∂fn
∂x2

……
……

……
...
10), we have
V = 𝐟𝐓𝐉𝐓𝐏𝐟 + 𝐟𝐓𝐏𝐉𝐟
= 𝐟𝐓(𝐉𝐓𝐏 + 𝐏𝐉)𝐟
Let
𝐐 = 𝐉𝐓𝐏 + 𝐏𝐉
Since V is positive definite, for the system to be asymptotically stable, Q should be negative
definite
...


93

POPOV CRITERION

94

95

96

MODEL QUESTIONS
Module-3
Short Questions each carrying Two marks
...


2
...

4
...
Extend this
concept to show that a ferro resonant circuit can be used to stabilize wide
fluctuations in supply voltage of a
...
mains in a CVT(constant voltage
transformer)
...

Bring out the differences between Liapunov‟s stability criterion and Popov‟s
stability criterion
...

5
...

[5]
97

(b) Draw the phase trajectory for the system described by the following differential
equation
𝑑𝑋
𝑑2𝑋
+ 0
...

[5]
𝑑𝑡

6
...
Distinguish between the concepts of stability, asymptotic stability and global
stability
...
Write short notes on
[3
...
(a) The origin is an equilibrium point for the pair of equations
𝑋1 = 𝑎𝑋1 + 𝑏𝑋2
𝑋2 = 𝑐𝑋1 + 𝑑𝑋2
Using Liapunov‟s theory find sufficient conditions on a, b, c and d such that the
origin is asymptotically stable
...
707

Draw the phase plane trajectory when the initial conditions are 𝑥 0 = 𝜋 , 𝑥 0 = 0
...
Compute x vs
...
1 sec
...
Determine the amplitude and frequency of oscillation of the limit cycle of the system
shown in Figure below
...

[16]

11
...
[4]

98

12
...

[14]

13
...


[3]

14
...
Distinguish between the concepts of stability, asymptotic stability & global stability
...
(a) What are singular points in a phase plane? Explain the following types of
singularity with sketches:

[9]

Stable node, unstable node, saddle point, stable focus, unstable focus, vortex
...
Derive the formula used
...
(a) Evaluate the describing function of the non linear element shown in figure below
...

Making use of the describing function analysis
...

[10]

18
...
1 𝑥 )𝑥 = 0
Using delta method obtain the first five points in the phase plane for initial condition
X(0)==1
...
6
19
...

20
...

[5]
(b) 𝑥1 =𝑥2
𝑥2 =-0
...
5
Find the nature of singular points lying between 𝑥1 = 00 to 1800
[10]
21
...
Obtain the phase trajectory of the system for the initial condition e(0)=0
...
does the system has a limit cycle? If so determine its amplitude and time period
...
Explain the phenomena of jump resonance in a non-linear system
...
Sometimes non-linear elements are intentionally introduced into control system
...

[4]
100

24
...
Plot the trajectory passing through((𝑋1 =
2, 𝑋2 = 0) without any approximation
...

[12+3]
25
...

[8]
i) ideal on off relay
ii)ideal saturation
(b) Derive a Liapunov function for the defined by
[8]
𝑥1 =𝑥2
𝑥2 = −3𝑥1 2 − 3 𝑥2
Also check the stability of the system
...
(a) Determine the singular points in the phase plane and sketch the plane trajectories for a
system of characteristics equation
𝑑2 𝑥(𝑡)
+ 8𝑥 𝑡 − 4𝑥2 𝑡 = 0
[8]
𝑑𝑡2

(b) A system described by the system shown in fig below

Will there be a limit cycle? If so determine its amplitude and frequency

---