Course roadmap
ME451: Control Systems
Modeling
Analysis
Laplace transform
Lecture 15
Time response of 2nd-order systems
Transfer function
Models for systems
• electrical
• mechanical
• electromechanical
Block diagrams
Linearization
Dr. Jongeun Choi
Department of Mechanical Engineering
Michigan State University
Design
Time response
• Transient
• Steady state
Design specs
Root locus
Frequency response
• Bode plot
Frequency domain
PID & LeadLead-lag
Stability
• RouthRouth-Hurwitz
• Nyquist
Design examples
(Matlab simulations &) laboratories
1
Performance measures (review)
ƒ Transient response
2
Second-order systems
ƒ A standard form of the second-order system
(Today’
(Today’s lecture)
ƒ Peak value
ƒ Peak time
ƒ Percent overshoot
ƒ Delay time
ƒ Rise time
Next, we will connect
these measures
with ss-domain.
ƒ Settling time
Amplifier
ƒ Steady state response
ƒ Steady state error
ƒ DC motor position control example
Motor
ClosedClosed-loop TF
(Done)
3
4
Step response for 2nd-order system
ƒ Input a unit step function to a 2nd-order system.
What is the output?
u(t)
u(t)
Step response for 2nd-order system
for various damping ratio
ƒ Undamped
2
y(t)
y(t)
ƒ Underdamped
1
0
0
ƒ Critically damped
ƒ Overdamped
DC gain
1.5
1
0.5
0
0
5
10
15
5
Step response for 2nd-order system
Underdamped case
6
Peak value/time: Underdamped case
ƒ Math expression of y(t) for underdamped case
1 .6
1 .4
1 .2
1
0 .8
0 .6
0 .4
0 .2
0
0
5
10
15
Damped natural frequency
7
8
Properties of 2nd-order system
Some remarks
ƒ Percent overshoot depends on ζ, but NOT ωn.
ƒ From 2nd-order transfer function, analytic
expressions of delay & rise time are hard to
obtain.
ƒ Time constant is 1/(ζωn), indicating
convergence speed.
ƒ For ζ>1, we cannot define peak time, peak
value, percent overshoot.
(5%)
(2%)
9
P.O. vs. damping ratio
10
Pole locations of G
ƒ Poles (0<ζ<1)
ƒ Damping ratio
Next, we clarify the influence of
pole location on step response.
11
12
Influence of real part of poles
Influence of imag. part of poles
ƒ Oscillation frequency ωd increases.
ƒ Settling time ts decreases.
ts
13
Influence of angle of poles
14
An example
ƒ Over/under-shoot decreases.
ƒ Require 5% settling time ts < tsm (given):
Im
Re
15
16
An example (cont’d)
An example (cont’d)
ƒ Require PO < POm (given):
ƒ Combination of two requirements
&
Im
Im
Re
Re
17
18
Exercises
(Use a calculator if necessary.)
Summary
ƒ Transient response of 2nd-order system is
characterized by
ƒ Read the related topics from the textbook.
1. For the system below with ζ=0.6, ωn=5
(rad/sec), obtain
ƒ Damping ratio ζ & undamped natural frequency ωn
ƒ Pole locations
ƒ Delay time and rise time are not so easy to
characterize, and thus not covered in this course.
ƒ For transient responses of high order systems,
we need computer simulations.
ƒ Next, Root locus
19
•
•
Percent overshoot ?
5% settling time ?
20
Exercises
2. For the system below, design K1 and K2 s.t.
ƒ
ƒ
Percent overshoot is at most 20%?
Peak time is at most 1 sec.?
ƒ
With designed K1 and K2, what is 5% settling time?
21