Modern Control System
EKT 308
Time Response
Transient and Steady-State
Response Analysis
Typical Test Signals
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Step functions (*)
Ramp functions (*)
Impulse functions (*)
Parabolic functions
Sinusoidal functions (Later in frequency analysis)
White noise
Test signals
Transient and Steady-State Response
Time response has two parts:
1) Transient Response: System response from initial state to
final state
2) Steady-state Response: System response when t
approaches infinity.
c(t )  Ctr (t )  Css (t )
First Order Systems
R(s)
+
-
E(s)
1
Ts  1
C ( s)
1

R( s) Ts  1
C(s)
Unit step response of first order systems
C ( s)  G ( s) R( s)
1
C ( s) 
Lu (t )
Ts  1
1 1 A
B

 
Ts  1 s s Ts  1
1
T
1
1
 
 
s Ts  1 s s  (1 / T )
c(t )  1  e

t
T
for t  0
From Eq (1),
At t  0, c(t )  0
At t  , c(t )  1
(1)
c(t )  1  e t / T
Slope,
dc(t )
 1 t / T
 0 e
dt t 0
T
t 0
1

T
Unit ramp response of first order systems
C ( s)  G ( s) R( s)
1
C (s) 
Lr (t )
Ts  1
1 1
1 T
T2

 2 
2
Ts  1 s
s
s Ts  1
c(t )  t  T  Te t / T for t  0
t
Error signal e(t )  r (t )  c(t )  T (1  e t / T )
At t  , e()  T
Unit impulse response of first order systems
C ( s)  G ( s) R( s)
1
C (s) 
L (t )
Ts  1
1
1
1

.1 
Ts  1
T s 1/ T
1 t / T
c(t )  e
for t  0
T
1 t /T
c(t )  e
T
Example of first order system
Second Order System
Let us start with an example
Second Order System (Contd…)
Transfer function of a second order system (K=1)
…. (1)
From equation (1), we can write
n2
Y ( s)  2
R( s)
2
s  2n s  n
For unit step
n2
1
Y (s)  2
s  2n s  n2 s
 Y ( s ) 
 s  2n
( s  n )  n
2


y (t )  1 
y (t )  1 
1
n
2
 1  
2
1
 1  
2
 1  
2
2
1

s


1/ 2
 2   2 1   2 2  e  nt sin( 
n
n
 n



e  nt sin( n 1   2 t   )
 1   t   )
2
Assuming   1 - 
y (t )  1 
1

e
 n t
2
sin( n  t   )
n 
1 1 - 
where,   tan
 tan
n

1
tan  
1-  2

sec   1 
2
  cos 1 
1  2

2

1
2
2
Signal flow and block diagram of the system
Transient response
Unit Impulse response of a second order system

Y ( s)  2
R( s)
2
s  2n s  n
2
n

 2
.1
2
s  2n s  n
2
n
n  t
y (t ) 
e
sin( n  t ).

n
Unit Impulse response (contd…)