Course roadmap
MECH466: Automatic Control
Modeling
Analysis
Laplace transform
Lecture 18
Nyquist stability criterion: Examples
Relative stability
Transfer function
Models for systems
• electrical
• mechanical
• electromechanical
Dr. Ryozo Nagamune
Department of Mechanical Engineering
University of British Columbia
Linearization
Time response
• Transient
• Steady state
Frequency response
• Bode plot
Stability
• RouthRouth-Hurwitz
• Nyquist
Design
Design specs
Root locus
Frequency domain
PID & LeadLead-lag
Design examples
Matlab simulations & laboratories
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Nyquist plot (review)
ƒ Nyquist path
Im
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Im
2
Nyquist stability criterion (review)
ƒ Nyquist plot L(s)
L(s) when s
s
MECH466 : Automatic Control
C(s)
C(s)
moves on
Nyquist path
G(s)
G(s)
H(s)
H(s)
Re
Re
ƒ Z: # of CL poles in open RHP
ƒ P: # of OL poles in open RHP (given)
ƒ N: # of clockwise encirclement of -1 by Nyquist
plot of L(s)
L(s)
(very large)
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1
Nyquist criterion: A special case
(review)
Examples when P=0
Im
Im
Re
ƒ IF P=0 (i.e., if L(s)
L(s) has no pole in open RHP)
CL stable
We assume P=0 from now on!
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Example of L(s) with an integrator
ƒ When open loop poles
lie on the imaginary
Im
axis, we modify
Nyquist path,
path, by
detouring to the right of
the poles.
(very small)
Ex:
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Re
CL unstable
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L(s) for modified Nyquist path
ƒ For small |s|,
Im
s
s
Re
(very small)
ƒ When s moves as
L(s)
L(s) moves as
Re
L(s)
L(s)
(very large)
Note: If L(s)
L(s) has no open RHP pole,
we are interested in (and draw) Nyquist plot
ONLY around the critical point -1.
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2
Bode plot of L(s)
Example of L(s) with an integrator
dB
-20
15
10
5
-40
0
deg
-5
-10
-15
-1
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-0.8
-0.6
-0.4
-0.2
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0
10
Bode plot of L(s)
Example of L(s) with double integrator
dB
-40
5
-60
deg
0
-5
-5
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-4
-3
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-2
-1
0
11
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3
L(s) with a time-delay
Bode plot of a time delay (review)
0.5
1
0.5
0
0
-0 . 5
-0.5
-1
-2
10
-1
10
-1
10
0
10
1
10
2
Huge phase lag!
0
-1.5
-2 0 0 0
-2
-4 0 0 0
-2.5
-3
-6 0 0 0
-2
10
-1
0
1
2
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-1
10
0
10
1
10
2
The phase lag causes instability of the closedclosed-loop
system, and thus, the difficulty in control.
3
RouthRouth-Hurwitz is NOT applicable!
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Announcements
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Some remarks
ƒ Lab 4 starts next week. See the schedule in
ƒ Nyquist stability criterion gives not only absolute
but also relative stability.
stability.
MECH466_LabSchedule_0809 (Mar 5).pdf
ƒ Final exam
ƒ Absolute stability:
stability: Is the closedclosed-loop system stable or
not? (Answer is yes or no.)
ƒ Date & time: April 21 (Tue), 3:30pm3:30pm-6pm
ƒ Relative stability:
stability: How “much”
much” is the closedclosed-loop
ƒ Place: CHBE101
system stable? (Margin of safety)
ƒ Policy change
ƒ Relative stability is important because a
mathematical model is never accurate.
ƒ How to measure relative stability?
• One letterletter-size cheat sheet (you can use both
sides of the sheet) is allowed.
• The cheat sheet must be written by hand by
yourself!
ƒ Use a “distance”
distance” from the critical point -1.
ƒ Gain margin (GM) & Phase margin (PM)
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Gain margin (GM)
Examples of GM
Im
ƒ Phase crossover
frequency ωp:
ƒ Gain margin (in dB)
Im
Re
Re
ƒ Indicates how much
OL gain can be
multiplied without
violating CL stability.
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Nyquist plot of L(s)
L(s)
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Reason why GM is inadequate
Same gain margin,
but different relative stability
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Phase margin (PM)
ƒ Gain crossover
frequency ωg:
ƒ Phase margin
Gain margin is often inadequate
to indicate relative stability
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Nyquist plot of L(s)
L(s)
ƒ Indicates how much
OL phase lag can be
added without
violating CL stability.
Phase margin!
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19
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An example
Notes on Nyquist plot
ƒ Advantages
ƒ Nyquist plot can be used for study of closedclosed-loop
stability, for open loop systems which is unstable and
includes timetime-delay.
ƒ Disadvantage
ƒ Controller design on Nyquist plot is difficult.
(Controller design on Bode plot is much simpler.)
We translate GM and PM defined in Nyquist plot
into those in Bode plot!
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Relative stability on Bode plot
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Notes on Bode plot
ƒ Advantages
ƒ Without computer, Bode plot can be sketched easily.
ωg
ƒ GM, PM, crossover frequencies are easily determined
on Bode plot.
GM
ƒ Controller design on Bode plot is simple. (Next week)
ƒ Disadvantage
ωp
ƒ If OL system is unstable, we cannot use Bode plot for
closedclosed-loop stability analysis.
PM
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An example
Relative stability with time delay
20
0
GM
GM
0
-50
-20 -1
10
-100
10
0
ωg
1
10
ωp
2
0
10
3
10
10
-90
Time delay reduces
relative stability!
PM
-180
PM
-180
-270
10
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-1
0
10
0
1
10
2
10
3
10
Delay time
10
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Unstable closed-loop case
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Summary and exercises
ƒ Relative stability: Closeness of Nyquist plot to
the critical point -1
GM
ƒ Gain margin, phase crossover frequency
ƒ Phase margin, gain crossover frequency
ωp
ƒ
ƒ
ƒ
ωg
Relative stability on Bode plot
We normally emphasize PM in controller design.
Exercises
ƒ Read Sections 10.6 & 10.7.
ƒ Obtain GM, PM for openopen-loop Bode plots in Figures
P10.7, P10.8, P10.10 (pages 593593-595).
PM
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