ME451: Control Systems
Relative stability
Dr. Jongeun Choi
Department of Mechanical Engineering
Michigan State University
Gain margin (GM)
ƒ Phase crossover
frequency ωp:
ƒ Gain margin (in dB)
ƒ Indicates how much
OL gain can be
multiplied without
violating CL stability.
Nyquist plot of L(s)
L(s)
Examples of GM
Im
Im
Re
Re
Reason why GM is inadequate
Same gain margin,
but different relative stability
Gain margin is often inadequate
to indicate relative stability
Phase margin!
Phase margin (PM)
ƒ Gain crossover
frequency ωg:
ƒ Phase margin
Nyquist plot of L(s)
L(s)
ƒ Indicates how much
OL phase can be
added without
violating CL stability.
Example 9.5
Relative stability on Bode plot
ωg
GM
ωp
PM
Frequency shaping (Loop shaping)
C(s)
G(s)
Controller Stable plant
Reshape Bode plot of G(jω)
into a “desired” shape of
by a series connection of
appropriate C(s).
Typical shaping goal (review)
SteadySteady-state accuracy
Sensitivity
Disturbance rejection
Transient
Response speed
Noise
reduction
Relative stability
Relative stability
Noise
reduction
Transient
Overshoot
Notes on Bode plot
ƒ Advantages
ƒ Without computer, Bode plot can be sketched easily.
ƒ GM, PM, crossover frequencies are easily determined
on Bode plot.
ƒ Controller design on Bode plot is simple. (Next week)
ƒ Disadvantage
ƒ If OL system is unstable, we cannot use Bode plot for
stability analysis.
Example 9.6
0
GM
-50
-100
0
10
ωg
1
10
ωp
2
10
3
10
PM
-180
-270
0
1
10
10
2
10
3
10
Relative stability with time delay
20
GM
0
-20 -1
10
0
10
-90
Time delay reduces
relative stability!
PM
-180
-1
10
0
10
Delay time
Bode plot of a time delay (review)
1
0.5
0
-0 . 5
-1
-2
10
10
-1
10
0
10
1
10
2
Huge phase lag!
0
-2 0 0 0
-4 0 0 0
-6 0 0 0
-2
10
10
-1
10
0
10
1
10
2
The phase lag causes instability of the closed-loop
system, and thus, the difficulty in control.