Course roadmap
MECH466: Automatic Control
Modeling
Analysis
Laplace transform
Lecture 10
RouthRouth-Hurwitz criterion: Control examples
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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Stability summary (review)
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Routh-Hurwitz criterion (review)
Let si be poles of
rational G. Then, G is …
ƒ (BIBO, asymptotically) stable if
Re(si)<0 for all i.
ƒ marginally stable if
ƒ Re(si)<=0 for all i, and
ƒ simple root for Re(si)=0
ƒ unstable if
The number of roots
in the right halfhalf-plane
is equal to
the number of sign changes
in the first column of Routh array.
it is neither stable nor
marginally stable.
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1
Why no proof in textbooks?
Why no proof in textbooks? (cont’d)
An Elementary Derivation of the Routh–Hurwitz Criterion
Ming-Tzu Ho, Aniruddha Datta, and S. P. Bhattacharyya
IEEE Transactions on Automatic Control
vol. 43, no. 3, 1998, pp. 405-409.
“most undergraduate students are exposed to the
Routh–
Routh–Hurwitz criterion in their first introductory
controls course. This exposure, however, is at the
purely algorithmic level in the sense that no attempt
is made whatsoever to explain why or how such an
algorithm works.”
works.”
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“The principal reason for this is that the classical
proof of the RouthRouth-Hurwitz criterion relies on the
notion of Cauchy indexes and Sturm’
Sturm’s theorem,
both of which are beyond the scope of
undergraduate students.”
students.”
“RouthRouth-Hurwitz criterion has become one of the few
results in control theory that most control engineers
are compelled to accept on faith.”
faith.”
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Example 1
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Example 1: K(s)=K
ƒ Characteristic equation
ƒ Routh array
ƒ Design K(s)
K(s) that stabilizes the closedclosed-loop
system for the following cases.
ƒ K(s)
K(s) = K (constant)
ƒ K(s)
K(s) = KP+KI/s (PI (Proportional(Proportional-Integral) controller)
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2
Example 1: K(s)=KP+KI/s
Example 1: Range of (KP,KI)
ƒ Characteristic equation
ƒ From Routh array,
3.5
ƒ Routh array
3
2.5
2
1.5
1
0.5
0
-1
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Example 1: K(s)=KP+KI/s (cont’d)
ƒ Select KP=3 (<9)
ƒ Routh array (cont’
(cont’d)
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0
1
2
3
4
5
6
7
8
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Example 1: What happens if KP=KI=3
ƒ Auxiliary equation
2
1.8
1.6
1.4
1.2
ƒ Oscillation frequency
1
0.8
0.6
ƒ Period
ƒ If we select different KP, the range of KI changes.
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0.4
0.2
0
0
2
4
6
8
10
12
14
16
18
20
Unit step response
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Announcements
Example 2
ƒ Midterm exam: October 19 (Friday) 1212-12:50pm
ƒ New lab dates
ƒ Group B, Lab #2 (for Thanksgiving holiday):
ƒ
ƒ
ƒ
• B4B4-B6: Oct.10 (Wed), 4pm4pm-6pm (Report due: Oct.24)
• B1B1-B3: Oct.10 (Wed), 6pm6pm-8pm (Report due: Oct.24)
Group A, Lab #5 (for Remembrance day):
• A1A1-A6: Nov. 21 (Wed), 4pm4pm-6pm (Report due: Dec.4)
Contact Mr. Roland Lang (hxlang@mech.ubc.ca
(hxlang@mech.ubc.ca)) if you
need time rearrangement.
To hand in lab report other than lab time, hand in it to the
instructor (In such cases, due time is 6pm).
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ƒ Determine the range of K and a that stabilize the
closedclosed-loop system.
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Example 2 (cont’d)
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Example 2 (cont’d)
ƒ Characteristic equation
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4
Example 2 (cont’d)
Summary and Exercises
ƒ Routh array
ƒ Control examples for RouthRouth-Hurwitz criterion
ƒ P controller gain range for stability
ƒ PI controller gain range for stability
ƒ Oscillation frequency
ƒ Characteristic equation
ƒ Next
ƒ Time domain specifications
ƒ If K=35, oscillation frequency is obtained by the
auxiliary equation
ƒ Exercises
ƒ Read Section 66-5 again.
ƒ Problems 66-9, 66-10.
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More example 1
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More example 2
Routh array
Routh array
Derivative of auxiliary poly.
Derivative of auxiliary poly.
4
2
4
(Auxiliary poly. is a factor of Q(s).)
Q(s).)
2
No sign changes
in the first column
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No root in OPEN(!) RHP
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No sign changes
in the first column
No root in OPEN(!) RHP
19
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5
More example 3
Routh array
4
0
One sign changes
in the first column
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Derivative of auxiliary poly.
One root in OPEN(!) RHP
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6