control chap4
TRANSCRIPT
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CONTROL SYSTEMS THEORY
Transient response
Chapter 4
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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
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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
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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)
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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
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Poles & zeros
Zero (o), Pole (x) Transfer function = Numerator
Denominator = Zeros
Poles
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Poles & zeros
Example : Given G(s), obtain the pole-zero plot of the system
Zero (o)Pole (x)
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Poles & zeros
Exercise : Obtain and plot the poles and zeros for the system given
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First order system
First order system with no zeros
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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
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First order system System response
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Second order system
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Second order system
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Second order system
Exercise : Is this system under/over/critically damped?
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Second order system Performance specifications
damping ratio
% Overshoot = cmax – cfinal x 100
cfinal
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Second order system Settling time, Ts
Peak time, Tp
nsT
4
a = 2ωn
21
n
pT
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Second order system
2nd order underdamped response
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Second order system
Second-order response as a function of damping ratio
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Second order system
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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)
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Second order system
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Exercise
Describe the damping of each system given the information below
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Solution
Find value of zeta
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2nd order general form
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Exercise
Given these 2nd order systems, find the value of and . Describe the damping
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Solution
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Example
Given
Find settling time, peak time, %OS Hint :
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Solution
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Block diagram: Analysis
Finding transient responseFor the system shown below, find the peak time, percent overshoot and settling time.
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Block diagram: Analysis
Answers:n=10
=0.25Tp=0.324
%OS=44.43Ts=1.6
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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
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Block diagram: Analysis and design
Solution:Closed-loop transfer function is
Kss
KsT
5)(
2
K
Kn
n
2
5
Thus,
and
52
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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
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Higher order systems
Systems with >2 poles and zeros can be approximated to 2nd order system with 2 dominant poles
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Higher order systems
Placement of third pole. Which most closely resembles a 2nd order system?
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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
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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
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Solution part (a)
ωn = 4 ζ = 0.375 Ts =4s Tp = 0.8472 s %OS = 28.06 %
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Solution part (b)
ωn = 0.2 ζ = 0.05 Ts =400s Tp = 15.73s %OS = 85.45 %
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Solution part (c)
ωn = 3240 ζ = 0.247 Ts =0.005 s Tp = 0.001 s %OS = 44.92 %