2. Introduction
Influence of Poles on Time Response
Transient Response of First-Order System
Transient Response of Second-Order System
3. The concept of poles and zeros, fundamental to the
analysis of and design of control system, simplifies
the evaluation of system response.
The poles of a transfer function are:
i. Values of the Laplace Transform variables s, that cause
the transfer function to become infinite.
ii. Any roots of the denominator of the transfer function
that are common to roots of the numerator.
The zeros of a transfer function are:
i. The values of the Laplace Transform variable s, that
cause the transfer function to become zero.
ii. Any roots of the numerator of the transfer function that
are common to roots of the denominator.
4. The output response of a system is a sum of
i. Forced response
ii. Natural response
a) System showing an input and an output
b) Pole-zero plot of the system
5. c) Evolution of a system response. Follow the
blue arrows to see the evolution of system
component generated by the pole or zero
6. a) First-order system
b) Pole plot of the
system
Effect of a real-axis pole upon transient response
7. General form:
Problem: Derive the transfer function for the following
circuit
1
)
(
)
(
)
(
s
K
s
R
s
C
s
G
1
1
)
(
RCs
s
G
8. Transient Response: Gradual change of output from
initial to the desired condition.
Block diagram representation:
By definition itself, the input to the system should be
a step function which is given by the following:
C(s)
R(s)
1
s
K
s
s
R
1
)
(
Where,
K : Gain
: Time constant
9. General form:
Output response:
1
)
(
)
(
)
(
s
K
s
R
s
C
s
G
1
1
1
)
(
s
B
s
A
s
K
s
s
C
t
e
B
A
t
c
)
(
)
(
)
(
)
( s
R
s
G
s
C
10. Problem: Find the forced and natural responses for the
following systems
12. Time constant,
◦ The time for e-at to decay 37% of its
initial value.
Rise time, tr
◦ The time for the waveform to go
from 0.1 to 0.9 of its final value.
Settling time, ts
◦ The time for the response to reach,
and stay within 2% of its final value.
a
1
a
tr
2
.
2
a
ts
4
13. Problem: For a system with the transfer function
shown below, find the relevant response specifications
i. Time constant,
ii. Settling time, ts
iii. Rise time, tr
50
50
)
(
s
s
G
14. General form:
Roots of denominator:
2
2
2
2 n
n
n
s
s
K
s
G
Where,
K : Gain
ς : Damping ratio
n : Undamped natural
frequency
0
2 2
2
n
ns
s
1
2
2
,
1
n
n
s
15. Natural frequency, n
◦ Frequency of oscillation of the system without damping.
Damping ratio, ς
◦ Quantity that compares the exponential decay frequency
of the envelope to the natural frequency.
(rad/s)
frequency
Natural
frequency
decay
l
Exponentia
16. Problem: Find the step response for the following
transfer function
Answer:
225
30
225
2
s
s
s
G
t
t
te
e
t
c 15
15
15
1
17. Problem: For each of the transfer function, find the
values of ς and n, as well as characterize the nature
of the response.
a)
b)
c)
d)
400
12
400
2
s
s
s
G
900
90
900
2
s
s
s
G
225
30
225
2
s
s
s
G
625
625
2
s
s
G
24. When 0 < ς < 1, the transfer function is given by the
following.
Pole position:
d
n
d
n
n
j
s
j
s
K
s
G
2 Where,
2
1
n
d
28. Rise time, Tr
◦ The time for the waveform to go from 0.1 to 0.9 of its
final value.
Peak time, Tp
◦ The time required to reach the first
or maximum peak.
Settling time, Ts
◦ The time required for the transient’s
damped oscillation to reach and stay
within ±2% of the steady-state value.
2
1
n
p
T
n
s
T
4
29. Percent overshoot, %OS
◦ The amount that the waveform overshoots the steady-
state, or final value at peak time, expressed as a
percentage of the steady-state value.
%
100
% )
1
/
( 2
e
OS
)
100
/
(%
ln
)
100
/
ln(%
2
2
OS
OS
31. Lines of constant peak time Tp, settling time Ts and
percent overshoot %OS
Ts2 < Ts1
Tp2 < Tp1
%OS1 < %OS2
32. Step responses of second-order underdamped
systems as poles move
a) With constant
real part
b) With constant
imaginary part
33. Step responses of second-order underdamped
systems as poles move
c) With constant damping
ratio
34. The steady state response is that part of the
output response where the output sognal
remails constant.
The parameter that is important in this is the
steady state error(Ess)
Error in general is the difference between the
input and the output. Steady state error is
error at t→∞
35. Static error coefficient
The response that remain after the transient
response has died out is called steady state
response
The steady state response is important to find
the accuracy of the output.
The difference between the steady state response
and desired response gives us the steady state
error.
Kp = positional error constant
Kv = velocity error constant
Ka = acceleration error constant
These error constant called as static error co
efficient. they have ability to minimize steady
state error.
STEADY STATE ERROR
