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System response to inputs
Control Engineering, 2nd Edition
W
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H
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Gupta

pg
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2


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
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Previous concept

Transient response
Steady state response
Test input signals
Laplace Transform

pg
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



0

0

f (t )   f (t )e  st   u (t )e  st dt


  1
...
4

Time response of a first order system
Consider a first order system with unity feedback as shown
...
5

Response of the first Order System with Unit Step input
C ( s)
1

R( s) sT  1
Therefore

C (S ) 

1
R( S )
sT  1

Since, input is the unit step function therefore

1
R( s) 
s
Substituting

Using partial fractions

1
C (S ) 
s( sT  1)
Ramp function?

pg
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6321 or 63
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2% of final or steady state value
...

Smaller the time constant,
faster is the system response
...
The figure shows exponential curves for two systems
having different time constants
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7

Exercise
The circuit below shows an operational amplifier integrator circuit;

Given that R=100 Ω, C=4
...
Determine the output of the given integrator
when subjected to a dc input voltage of 1
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Gupta

pg
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

b0n2
0 ( s)   2
 i ( s )
2 
 s  2n s  n 
For a unit step θi(s) = 1/s, then

b0
0 (s)  2
2
s

2

s



n
n s
2
n

pg
...

When ζ >1 then



2

 1 is a real number and the system is overdamped
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10

When ζ =1, the system is critically damped
...

Example
A system has the following relationship, in the s domain, between its output
θ0 and its input θi
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11

Solution

The roots of the s2 + 8s + 16 equation are thus m1 = m2 = -4
...
The system is thus critically damped
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12

roots
As you can see, there is a relation between the location of the roots and the
system response

pg
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T
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14

Exercise
Determine the characteristic equation for the system below

TF 

167
4s  6

Determine the characteristic equation of a closed-loop system, given the
following

G( s) 

1
s
H
(
s
)

0
...
15

Poles, Zeros and Stability
An important requirement for a control system is that it should be stable
...

For linear systems the stability requirement can be defined in terms of the poles
of the closed-loop transfer function
...

Pole is the value of s that makes (the value of) the transfer function approach
infinity (hence the term pole (rising to infinity))
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pg
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e
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Thus, for example, for every step input applied to a system
the output must be finite
...

Alternatively, a system can be defined as stable if when it is subject to an
impulse input the output dies away to zero as the time tends to infinity
...

If, however, the output does not die away to zero or increase to infinity but
tends to some finite but non-zero value then the system is said to be critically or
marginally stable
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17

Exercise
The characteristic equation of a system is given below
...

Determine the poles and zeros of the closed-loop system

0
...
18

Next lecture
Stability analysis
CAT2

pg

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