Course roadmap
ME451: Control Systems
Modeling
Analysis
Laplace transform
Lecture 11
Routh-Hurwitz criterion: Control examples
Transfer function
Models for systems
• electrical
• mechanical
• electromechanical
Block diagrams
Linearization
Dr. Jongeun Choi
Department of Mechanical Engineering
Michigan State University
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
1
Stability summary (review)
2
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.
3
4
Example 1
Example 1: K(s)=K
ƒ Characteristic equation
ƒ Routh array
ƒ Design K(s) that stabilizes the closed-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)
5
Example 1: K(s)=KP+KI/s
6
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
7
0
1
2
3
4
5
6
7
8
9
8
Example 1: K(s)=KP+KI/s (cont’d)
ƒ Select KP=3 (<9)
ƒ Routh array (cont’d)
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.
0.4
0.2
0
0
2
4
6
8
10
9
Example 2
12
14
16
18
20
Unit step response
10
Example 2 (cont’d)
ƒ Determine the range of K and a that stabilize the
closed-loop system.
11
12
Example 2 (cont’d)
Example 2 (cont’d)
ƒ Characteristic equation
ƒ Routh array
ƒ If K=35, oscillation frequency is obtained by the
auxiliary equation
13
Summary and Exercises
14
More example 1
ƒ Control examples for Routh-Hurwitz criterion
ƒ P controller gain range for stability
Routh array
ƒ PI controller gain range for stability
Derivative of auxiliary poly.
ƒ Oscillation frequency
ƒ Characteristic equation
ƒ Next
2
ƒ Time domain specifications
(Auxiliary poly. is a factor of Q(s).)
Q(s).)
ƒ Exercises
No sign changes
in the first column
15
No root in OPEN(!) RHP
16
More example 2
More example 3
Routh array
Routh array
Derivative of auxiliary poly.
4
4
2
Derivative of auxiliary poly.
0
4
No sign changes
in the first column
No root in OPEN(!) RHP
17
One sign changes
in the first column
One root in OPEN(!) RHP
18