ME451: Control Systems

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 Nyquist plot of L(s)
OL gain can be
multiplied without
violating CL stability.

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)

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)

Steady-state accuracy Transient
Sensitivity Response speed
Disturbance rejection

Noise Relative stability
reduction
Noise
reduction

Relative stability 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 ωg 1 ωp 2 3
-100
10 10 10
0 103

10

-180 PM

-270 101 102
100

Relative stability with time delay

20

GM

0

-20 PM 0

-1 10

10 Time delay reduces
-90 relative stability!

-180 0

-1 10

10 Delay time

Bode plot of a time delay (review)

1

0.5

0

-0 . 5

-1 1 0 -1
Huge phase lag!10-2 100 101 102

0

-2 0 0 0

-4 0 0 0

-6 0 0 0 1 0 -1 100 101 102
1 0 -2

The phase lag causes instability of the closed-loop
system, and thus, the difficulty in control.