MECH466 L17 NyquistCriterion

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2008/09 MECH466 : Automatic Control 1

MECH466: Automatic Control

Dr. Ryozo Nagamune

Department of Mechanical Engineering

University of British Columbia

University of British Columbia Lecture 17

Lecture 17 Nyquist

Nyquiststability criterionstability criterion

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

Frequency response (review)

Steady state output Steady state output

FrequencyFrequencyis same as the input frequencyis same as the input frequency

AmplitudeAmplitudeis that of input (A) multiplied byis that of input (A) multiplied by

PhasePhaseshiftsshifts

Frequency response functionFrequency response function(FRF): (FRF): G(jG(jωω))

OpenOpen--loop FRF contains closedloop FRF contains closed--loop stability info.loop stability info.

Gain

G(s

G(s))

y(t

y(t))

Stable Stable

Stability of feedback system

Consider the feedback systemConsider the feedback system

Fundamental questionsFundamental questions

If G and C and H are stable, is closedIf G and C and H are stable, is closed--loop system loop system

always

alwaysstablestable??

If G and C and H are unstable, is closedIf G and C and H are unstable, is closed--loop system loop system always

alwaysunstableunstable??

G(s

G(s))

C(s

C(s))

H(s

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Closed

-

loop stability criterion

ClosedClosed--loop stability can be determined by the loop stability can be determined by the roots of the

roots of the characteristic equationcharacteristic equation

CL system is stable if the Ch. CL system is stable if the Ch. EqEq. has all roots in . has all roots in the open left half plane.

the open left half plane.

How to check the stability?How to check the stability?

Compute all the roots.Compute all the roots.

RouthRouth--Hurwitz stability criterionHurwitz stability criterion

NyquistNyquiststability criterionstability criterion

Nyquist

plot

NyquistNyquistpathpath

(very large)

NyquistNyquistplotplot

s s

L(s

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

Re

Im

Re

Im

Example of Bode &

Nyquist

plots

First order systemFirst order system

10-2 10-1 100 101 102 -40

-20 0

-80 -60 -40 -20 0

Bode plot

Bode plot NyquistNyquistplotplot

-0.5 0 0.5 1

Example of Bode &

Nyquist

plots

Second order systemSecond order system

Bode plot

Bode plot

10-2 10-1 100 101 102 -100

-50 0

-150 -100 -50 0

Nyquist

Nyquistplotplot

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Example of Bode &

Nyquist

plots

Third order systemThird order system

Bode plot

Bode plot

10-2 10-1 100 101 102 -150

-100 -50 0

10-2 10-1 100 101 102 -200

-100 0

Nyquist

Nyquistplotplot

-1 -0.5 0 0.5 1 -1

-0.5 0 0.5 1

Nyquist

stability criterion

Z: # of CL poles in open RHPZ: # of CL poles in open RHP

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

N: # of clockwise encirclement of N: # of clockwise encirclement of --1 1 by

by NyquistNyquistplot of OL transfer function plot of OL transfer function L(sL(s)) (counted by using

(counted by using NyquistNyquistplot of plot of L(sL(s))))

Remark:

Remark:N=N=--1: a counter1: a counter--clockwise encirclementclockwise encirclement

Encirclements in

Nyquist

plot

ClockwiseClockwise

Re

Im

CounterCounter--clockwiseclockwise

Re

Im

Remark

If If NyquistNyquistplot passes the point plot passes the point --1, it means that 1, it means that the closed

the closed--loop system has a pole on the loop system has a pole on the imaginary axis (and thus, not stable). imaginary axis (and thus, not stable).

Re

Im

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2008/09 MECH466 : Automatic Control 13 -1.5 -1 -0.5 0 0.5 1

-1 -0.5 0 0.5 1

-1.5 -1 -0.5 0 0.5 1 -1

-0.5 0 0.5 1

Example for 2nd order

L(s

)

For

For L(sL(s) of 2) of 2ndndorder and constant numerator, order and constant numerator,

gain increase never lead to unstable CL system!

Gain increase

-5 0 5 10 15

-15 -10 -5 0 5 10 15

-2 -1.5 -1 -0.5 0 0.5 1 -1

-0.5 0 0.5 1

Example for 3rd order

L(s

)

Gain increase

Gain increase

For

For L(sL(s) with relative degree 3, ) with relative degree 3,

gain increase eventually lead to unstable CL system!

B

How to count # of encirclement

A ray is drawn from A ray is drawn from --1 point in any convenient 1 point in any convenient direction. Then,

direction. Then,

Re

Im

A

Count N for cases: Count N for cases:

A = A = --11

B = B = --11

Notes on

Nyquist

stability criterion

NyquistNyquiststability criterion allows us to determine stability criterion allows us to determine the

the stability of CL systemstability of CL systemfrom a knowledge of from a knowledge of the

the G(jωG(jω) of OL system.) of OL system.

If an OL system is stable, it requires If an OL system is stable, it requires only only frequency response data

frequency response dataof OL system (TF of OL system (TF model

model L(sL(s) is not necessary).) is not necessary).

It can deal with It can deal with time delay, which time delay, which RouthRouth--Hurwitz Hurwitz criterion cannot.

criterion cannot.

We often draw only half of We often draw only half of NyquistNyquistplot. (The plot. (The other half is mirror image

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Some examples

Unstable Unstable L(sL(s))

Stable Stable L(sL(s))

L(sL(s) with an integrator) with an integrator

L(sL(s) with a double integrator) with a double integrator

L(sL(s) with a time) with a time--delaydelay

Example for unstable

L(s

)

-1.2 -1 -0.8 -0.6 -0.4 -0.2 0 -0.1

-0.05 0 0.05 0.1

Example of unstable

L(s

) (cont

’

d)

-2 -1.5 -1 -0.5 0 -0.1

-0.05 0 0.05 0.1 0.15

-1 -0.8 -0.6 -0.4 -0.2 0 -0.06

-0.04 -0.02 0 0.02 0.04 0.06

Gain increase

Gain increase Gain decreaseGain decrease

Interpretation by root locus

1 1

--22 --33

Re

Im

Open loop gain increase will first stabilize, Open loop gain increase will first stabilize, and then, destabilize the closed

and then, destabilize the closed--loop system.loop system. Stabilizing!Stabilizing!

Destabilizing!

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Nyquist

criterion: A special case

IF P=0 (i.e., if IF P=0 (i.e., if L(sL(s) has no pole in open RHP)) has no pole in open RHP)

This fact is very important since open

This fact is very important since open--loop systems loop systems in many practical problems have no pole in open RHP! in many practical problems have no pole in open RHP!

-1 0 1 2 3

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

Example for stable

L(s

)

-5 0 5 10 15 20 -15

-10 -5 0

CL stable

CL stable CL unstableCL unstable

Interpretation by root locus

--11 --22 --33

Re

Im

Open loop gain increase will Open loop gain increase will destabilize the closed

destabilize the closed--loop system.loop system.

Example of

L(s

) with an integrator

We modify We modify NyquistNyquistpath:path:

(very large)

s s

Re

Im

(very small)

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Example of

L(s

) with an integrator

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

-10 -5 0 5 10 15

L(s

) for modified

Nyquist

path

For small |s|,For small |s|,

When s moves asWhen s moves as L(s

L(s) moves as) 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

ONLY around the critical point --1.1.

Example of

L(s

) with double integrator

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

0 5

L(s

) with a time

-

delay

-1 0 1 2 3 -3

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

Routh

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2008/09 MECH466 : Automatic Control 29 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.

system, and thus, the difficulty in control.

Summary and exercises

NyquistNyquistplot (plot (MatlabMatlabcommand: command: “nyquist.m“nyquist.m””))

NyquistNyquiststability criterion for feedback stabilitystability criterion for feedback stability

Examples for Examples for NyquistNyquiststability criterionstability criterion

Exercises: For (half of) Exercises: For (half of) NyquistNyquistplot below, plot below, count N for each case.

count N for each case. _•_•_A=_A=_-_-₁₁_{(Ans. N=2)}_{(Ans. N=2)}

•

•B=B=--1 1 (Ans. N=0)(Ans. N=0)

•

•C=C=--1 1 (Ans. N=2)(Ans. N=2)

•

•D=D=--1 1 (Ans. N=0)(Ans. N=0)

•

•E=E=--1 1 (Ans. N=2)(Ans. N=2)

•

•F=F=--1 1 (Ans. N=0)(Ans. N=0) A

A

B

C

D

E

F

Exercise

Read Sections 10.3Read Sections 10.3--10.5.10.5.

Exercise: Suppose Exercise: Suppose L(sL(s) is stable and has ) is stable and has Nyquist

Nyquistplot below. Find the range of OL gain plot below. Find the range of OL gain K>0 for which CL system is stable.

K>0 for which CL system is stable.

--0.50.5

--22

--3.33.3

(Ans. 0<K<1/3.3, 1/2<K<2)