CONTROL SYSTEMS THEORY
Transient response
Chapter 4
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
Introduction In Chapter 1, we learned that the total
response of a system, c(t) is given by
In order to qualitatively examine and describe this output response, the poles and zeros method is used.
forced naturalc t c t c t
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)
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
Poles & zeros
Zero (o), Pole (x) Transfer function = Numerator
Denominator = Zeros
Poles
Poles & zeros
Example : Given G(s), obtain the pole-zero plot of the system
Zero (o)Pole (x)
Poles & zeros
Exercise : Obtain and plot the poles and zeros for the system given
First order system
First order system with no zeros
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
First order system System response
Second order system
Second order system
Second order system
Exercise : Is this system under/over/critically damped?
Second order system Performance specifications
damping ratio
% Overshoot = cmax – cfinal x 100
cfinal
Second order system Settling time, Ts
Peak time, Tp
nsT
4
a = 2ωn
21
n
pT
Second order system
2nd order underdamped response
Second order system
Second-order response as a function of damping ratio
Second order system
Second order system
Step responses of second-orderunder-damped systems as poles move:
a. with constant real partb. with constant imaginary partc. with constant damping ratio (constant on the diagonal)
Second order system
Exercise
Describe the damping of each system given the information below
Solution
Find value of zeta
2nd order general form
Exercise
Given these 2nd order systems, find the value of and . Describe the damping
Solution
Example
Given
Find settling time, peak time, %OS Hint :
Solution
Block diagram: Analysis
Finding transient responseFor the system shown below, find the peak time, percent overshoot and settling time.
Block diagram: Analysis
Answers:n=10
=0.25Tp=0.324
%OS=44.43Ts=1.6
Block diagram: Analysis and design
Gain design for transient responseDesign the value of gain, K, for the feedback control system of figure below so that the system will respond with a 10% overshoot
Block diagram: Analysis and design
Solution:Closed-loop transfer function is
Kss
KsT
5)(
2
K
Kn
n
2
5
Thus,
and
52
Block diagram: Analysis and design
Can be calculated using the %OS
= 0.591We substitute the value and calculate K, we getK=17.9
100/%ln
100/%ln22 OS
OS
Higher order systems
Systems with >2 poles and zeros can be approximated to 2nd order system with 2 dominant poles
Higher order systems
Placement of third pole. Which most closely resembles a 2nd order system?
Higher order systems Case I : Non-dominant pole is near
dominant second-order pair (=) Case II : Non-dominant pole is far from the
pair (>>) Case III : Non-dominant pole is at infinity
(=)
How far away is infinity? 5 times farther away to the LEFT from dominant poles
Exercises
Find , ωn, Ts, Tp and %OS
a)
b)
c)
T(s) = 0.04
s2 + 0.02s + 0.04
T(s) = 1.05 x 107
s2 + (1.6 x 103)s + (1.5 x 107)
T(s) = 16
s2 + 3s + 16
Solution part (a)
ωn = 4 ζ = 0.375 Ts =4s Tp = 0.8472 s %OS = 28.06 %
Solution part (b)
ωn = 0.2 ζ = 0.05 Ts =400s Tp = 15.73s %OS = 85.45 %
Solution part (c)
ωn = 3240 ζ = 0.247 Ts =0.005 s Tp = 0.001 s %OS = 44.92 %