MECH466 L11 RootLocus

(1)

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 11

Lecture 11

Root locus

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

What is Root Locus?

Pole locationPole locationof the feedback system of the feedback system characterizes

characterizes stabilitystabilityand and transient propertiestransient properties..

Consider a feedback system that has one Consider a feedback system that has one

parameter (gain) K>0 to be designed.

Root locusRoot locusgraphically shows how poles of CL graphically shows how poles of CL

L(s

L(s)) K

K

L(s

L(s): open): open--loop TFloop TF

A simple example

Characteristic Characteristic eqeq..

K=0: s=0,K=0: s=0,--22

L(s

L(s))

K

Closed

Closed--loop polesloop poles

Im

(2)

A more complicated example

Characteristic Characteristic eqeq..

It is hard to solve this analytically for each K.It is hard to solve this analytically for each K.

Is there some way to Is there some way to sketch roughlysketch roughlyroot locus root locus by hand? (In

by hand? (In MatlabMatlab, use command , use command ““rlocus.mrlocus.m””..))

L(s

L(s)) K

K

Root locus sketching algorithm

Step 0: Mark openStep 0: Mark open--loop poles and zerosloop poles and zeros

Step 1: On the real axisStep 1: On the real axis

Step 2: AsymptotesStep 2: Asymptotes

Step 3: Breakaway pointsStep 3: Breakaway points

Step 4: Angles of departures and arrivalsStep 4: Angles of departures and arrivals

Root locus: Step 0

Root locus is symmetric Root locus is symmetric w.r.tw.r.t. the real axis.. the real axis.

The number of branches = order of The number of branches = order of L(sL(s))

Mark poles of L with Mark poles of L with ““xx””and zeros of L with and zeros of L with ““oo””..

Re

Im

(3)

Root locus: Step 1 (On the real axis)

RL includes all points on real axis to the left of an RL includes all points on real axis to the left of an odd number of real poles/zeros.

odd number of real poles/zeros.

RL originates from the poles of L and terminates RL originates from the poles of L and terminates at the zeros of L, including infinity zeros.

at the zeros of L, including infinity zeros.

Re

Im

Indicate the direction Indicate the direction with an arrowhead. with an arrowhead.

Root locus sketching algorithm

Root locus: Step 2 (Asymptotes)

Number of asymptotes = relative degree (r) of L:Number of asymptotes = relative degree (r) of L:

Angles of asymptotes areAngles of asymptotes are

Intersections of asymptotesIntersections of asymptotes

Asymptotes

(Not root locus)

Re

Im

(4)

Root locus: Step 3

Breakaway points are among roots ofBreakaway points are among roots of

For each candidate s, check the

For each candidate s, check the positivitypositivityofof

Points where two or more branches meet and break away.

Points where two or more branches meet and break away.

Root locus: Step 3

Re

Im

Breakaway point

Matlab

command “

command

“rlocus.m

rlocus.m”

”

Root Locus

Real Axis

Ima

g

in

a

ry

A

x

is

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

(5)

A simple example: revisited

AsymptotesAsymptotes

Relative degree 2Relative degree 2

IntersectionIntersection

Breakaway pointBreakaway point

L(s

L(s)) K

K

Re

Im

Root locus: Step 0

Root locus is symmetric Root locus is symmetric w.r.tw.r.t. the real axis.. the real axis.

The number of branches = order of The number of branches = order of L(sL(s))

Mark poles of L with Mark poles of L with ““xx””and zeros of L with and zeros of L with ““oo””..

Re

Im

Root locus: Step 1 (Real axis)

RL includes all points on real axis to the left of an RL includes all points on real axis to the left of an odd number of real poles/zeros.

odd number of real poles/zeros.

RL originates from the poles of L and terminates RL originates from the poles of L and terminates at the zeros of L, including infinity zeros.

at the zeros of L, including infinity zeros.

Re

Im

Root locus: Step 2 (Asymptotes)

Number of asymptotes = relative degree (r) of L:Number of asymptotes = relative degree (r) of L:

(6)

Intersections of asymptotesIntersections of asymptotes

Asymptote

(Not root locus)

Re

Im

Root locus: Step 3 (Breakaway)

Breakaway points are among roots ofBreakaway points are among roots of

For each candidate s, check the

For each candidate s, check the positivitypositivityofof

What is this value?

For what K?

Re

Im

Root locus: Step 3 (Breakaway)

Breakaway point

Routh

Routh--Hurwitz!Hurwitz!

Finding K for critical stability

Characteristic equationCharacteristic equation

RouthRoutharrayarray

When K=30When K=30

Stability condition

(7)

Re

Im

Root locus

Breakaway point

Example with complex poles

Re

Im

Breakaway point

After Steps 0,1,2,3, we obtain

How to compute How to compute angle of departure? angle of departure?

Root locus: Step 4

Angle of departure

Angle condition: Angle condition: For a point For a point ““ss””to be on RL,to be on RL,

Re

Im

If s is close to p

(8)

Root locus

Breakaway point

Re

Im

Summary and exercises

Root locusRoot locus

What is root locusWhat is root locus

How to roughly sketch root locusHow to roughly sketch root locus

Sketching root locus relies heavily on experience.Sketching root locus relies heavily on experience.

PRACTICE!

PRACTICE!

To accurately draw root locus, use To accurately draw root locus, use MatlabMatlab..

Next, sketch of proofs for root locus algorithmNext, sketch of proofs for root locus algorithm

ExercisesExercises

Read Sections 8.1Read Sections 8.1--8.6.8.6.

Exercises 0

(9)

Exercises 2

Exercises 3