MECH466 L18 RelativeStability

(1)

2008/09 MECH466 : Automatic Control 1

MECH466: Automatic Control

Dr. Ryozo Nagamune

Department of Mechanical Engineering

University of British Columbia

Lecture 18

Nyquist

stability criterion: Examples

Relative stability

Course roadmap

Laplace transform Laplace transform Transfer function Transfer function

Models for systems Models for systems •

•electricalelectrical •

•mechanicalmechanical •

•electromechanicalelectromechanical

Linearization Linearization

Modeling

Modeling AnalysisAnalysis DesignDesign

Time response Time response •

•TransientTransient •

•Steady stateSteady state

Frequency response Frequency response •

•Bode plotBode plot

Stability Stability •

•Routh-Routh-HurwitzHurwitz •

•NyquistNyquist

Design specs Design specs Root locus Root locus Frequency domain Frequency domain

PID & Lead PID & Lead--laglag

Design examples Design examples

Matlab

Matlabsimulations & laboratoriessimulations & laboratories

Nyquist

plot (review)

Nyquist

path

(very large)

Nyquist

plot

s s

L(s

L(s) when s ) when s moves on moves on Nyquist Nyquistpath path

Re Re Im Im Re Re Im Im

Nyquist

stability criterion (review)

Z: # of CL poles in open RHP

P: # of OL poles in open RHP (given)

N: # of clockwise encirclement of

-

1 by

Nyquist

plot of

L(s

)

G(s

G(s))

C(s

C(s))

H(s

(2)

Nyquist

criterion: A special case

(review)

IF P=0 (i.e., if

L(s

) has no pole in open RHP)

We assume P=0 from now on!

Examples when P=0

Re

Im

Re

Im

CL stable

CL unstable

Example of

L(s

) with an integrator

When open loop poles

lie on the imaginary

axis, we

modify

Nyquist

path

path, by

, by

detouring to the right of

the poles.

Ex:

(very large)

s s

Re

Im

(very small)

L(s

) for modified

Nyquist

path

For small |s|,

When s moves as

L(s

) moves as

Re

Im

(very small)

s s

L(s L(s))

Note: If

Note: If L(sL(s) has no open RHP pole,) has no open RHP pole, we are interested in (and draw)

we are interested in (and draw) NyquistNyquistplot plot ONLY around the critical point

(3)

Bode plot of

L(s

)

dB

deg

--2020

--4040

Example of

L(s

) with an integrator

-1 -0.8 -0.6 -0.4 -0.2 0 -15

-10 -5 0 5 10 15

Example of

L(s

) with double integrator

-5 -4 -3 -2 -1 0 -5

0 5

Bode plot of

L(s

)

dB

deg

--4040

(4)

L(s

) with a time

-

delay

-1 0 1 2 3

-3 -2.5 -2 -1.5 -1 -0.5 0 0.5

Routh

Routh--Hurwitz is NOT applicable!Hurwitz is NOT applicable!

1 0- 2 1 0- 1 1 00 1 01 1 02 -1

-0 . 5 0 0 . 5 1

1 0- 2 1 0- 1 1 00 1 01 1 02 -6 0 0 0

-4 0 0 0 -2 0 0 0 0

Bode plot of a time delay (review)

Huge phase lag! Huge phase lag!

The phase lag causes instability

The phase lag causes instabilityof the closedof the closed--loop loop

system, and thus, the difficulty in control.

Announcements

Lab 4 starts next week. See the schedule in

MECH466_LabSchedule_0809 (Mar 5).pdf MECH466_LabSchedule_0809 (Mar 5).pdf

Final exam

Date & time: April 21 (Tue), 3:30pmDate & time: April 21 (Tue), 3:30pm--6pm6pm

Place: CHBE101Place: CHBE101

Policy changePolicy change

•

• One letterOne letter--size cheat sheetsize cheat sheet(you can use both (you can use both sides of the sheet) is allowed.

sides of the sheet) is allowed.

•

• The cheat sheet The cheat sheet must be written by hand by must be written by hand by

yourself!

Some remarks

Nyquist

stability criterion gives not only

absolute

but also

relative stability

.

Absolute stabilityAbsolute stability: Is the closed: Is the closed--loop system stable or loop system stable or not? (Answer is yes or no.)

not? (Answer is yes or no.)

Relative stabilityRelative stability: How : How ““muchmuch””is the closedis the closed--loop loop system stable? (Margin of safety)

system stable? (Margin of safety)

Relative stability is important because a

mathematical model is never accurate.

How to measure relative stability?

Use a Use a ““distancedistance””from the critical point from the critical point --1.1.

(5)

Gain margin (GM)

Phase crossover

frequency

ωp

ω

p

:

Gain margin

(in dB)

Indicates how much Indicates how much OL gain can be

OL gain can be

multiplied without

violating CL stability.

violating CL stability. NyquistNyquistplot of L(splot of L(s))

Examples of GM

Re

Im

Re

Im

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)

Gain crossover

frequency

ωg

ω

g

:

Phase margin

Indicates how much Indicates how much

OL phase lag can be

added without

violating CL stability.

Nyquist

(6)

An example

Notes on

Nyquist

plot

Advantages

NyquistNyquistplot can be used for study of closedplot can be used for study of closed--loop loop stability, for open loop systems which is unstable and

stability, for open loop systems which is unstable and

includes time

includes time--delay.delay.

Disadvantage

Controller design on Controller design on NyquistNyquistplot is difficult. plot is difficult. (Controller design on Bode plot is much simpler.)

(Controller design on Bode plot is much simpler.)

We translate GM and PM defined in

We translate GM and PM defined in NyquistNyquistplot plot into those in Bode plot!

into those in Bode plot!

Relative stability on Bode plot

ω

gg

ω

pp

GM

PM

Notes on Bode plot

Advantages

Without computer, Bode plot can be sketched easily.Without computer, Bode plot can be sketched easily.

GM, PM, crossover frequencies are easily determined GM, PM, crossover frequencies are easily determined

on Bode plot.

Controller design on Bode plot is simple. (Next week)Controller design on Bode plot is simple. (Next week)

Disadvantage

If OL system is unstable, we cannot use Bode plot for If OL system is unstable, we cannot use Bode plot for

closed

(7)

100 101 102 103

-100 -50 0

100 101 102 103

-180

-270

An example

ω

gg

ω

pp

GM

PM

10-1 100

-20 0 20

10-1 100

-180 -90

Relative stability with time delay

PM

GM

Time delay reduces

relative stability!

Delay time Delay time

Unstable closed

-

loop case

ω

gg

ω

pp

GM

PM

Summary and exercises

Relative stability: Closeness of

Nyquist

plot to

the critical point

-

1

Gain margin, phase crossover frequencyGain margin, phase crossover frequency

Phase margin, gain crossover frequencyPhase margin, gain crossover frequency

Relative stability on Bode plot

We normally emphasize PM in controller design.

Exercises

Read Sections 10.6 & 10.7.Read Sections 10.6 & 10.7.

Obtain GM, PM for openObtain GM, PM for open--loop Bode plots in Figures loop Bode plots in Figures P10.7, P10.8, P10.10 (pages 593