2. Objectives
To find time response from transfer
function
To describe quantitatively the
transient response of a 1st and 2nd
order system
To determine response of a control
system using poles and zeros
3. Introduction
In Chapter 1, we learned that the total
response of a system, c(t) is given by
c ( t ) = cforced ( t ) + cnatural ( t )
In order to qualitatively examine and
describe this output response, the poles and
zeros method is used.
4. Poles & zeros
The poles of a TF are the values of
the Laplace variable that cause the TF
to become infinite (denominator)
The zeros of a TF are the values of
the Laplace variable that cause the TF
to become zero (numerator)
5. Poles & zeros
Example : Given the TF of G(s), find
the poles and zeros
Solution :
G(s) = zero/pole
Pole at s=-5
Zero at s=-2
6. Poles & zeros
Zero (o), Pole (x)
Transfer function = Numerator
Denominator
= Zeros
Poles
7. Poles & zeros
Example : Given G(s), obtain the
pole-zero plot of the system
Zero (o)
Pole (x)
8. Poles & zeros
Exercise : Obtain and plot the poles
and zeros for the system given
10. First order system
Performance specifications:
Time constant, t
1/a, time taken for response to rise to 63%
of its final value
Rise time, Tr
time taken for response to go from 10% to
90% of its final value
Settling time, Ts
time for response to reach and stay within
5% of final value
20. Second order system
Step responses of second-order
under-damped systems as poles
move:
a. with constant real part
b. with constant imaginary part
c. with constant damping ratio
(constant on the diagonal)
31. Block diagram: Analysis and
design
Gain design for transient response
Design the value of gain, K, for the feedback
control system of figure below so that the
system will respond with a 10% overshoot
32. Block diagram: Analysis and
design
Solution:
Closed-loop transfer function is
2ζω n = 5
and
ωn = K
Thus,
5
ζ =
2 K
K
T (s) = 2
s + 5s + K
33. Block diagram: Analysis and
design
ζ Can be calculated using the %OS
ζ =
− ln ( %OS / 100 )
π 2 + ln 2 ( %OS / 100 )
ζ = 0.591
We substitute the value and calculate K, we get
K=17.9
34. Higher order systems
Systems with >2 poles and zeros can
be approximated to 2nd order system
with 2 dominant poles
35. Higher order systems
Placement of third pole. Which most closely
resembles a 2nd order system?
36. Higher order systems
Case I : Non-dominant pole is near
dominant second-order pair (α=ζω)
Case II : Non-dominant pole is far from the
pair (α>>ζω)
Case III : Non-dominant pole is at infinity
(α=∞)
How far away is infinity?
5 times farther away to the LEFT from dominant
poles
37. Exercises
Find ζ, ωn, Ts, Tp and %OS
a)
T(s) =
b) T(s) =
16
s2 + 3s + 16
0.04
s2 + 0.02s + 0.04
c) T(s) =
1.05 x 107
s2 + (1.6 x 103)s + (1.5 x 107)
