MECH466 L14 RLLeadLag

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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 14

Lecture 14

Root locus: Lead

Root locus: Lead--lag compensator designlag compensator design

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

Closed

Closed-

-loop design by root locus

loop design by root locus

Place closedPlace closed--loop poles at desired locationloop poles at desired location

by tuning the gain by tuning the gain C(sC(s)=K.)=K.

If root locus does not pass the desired location, If root locus does not pass the desired location, then reshape the root locus

then reshape the root locus

by adding poles/zeros to by adding poles/zeros to C(sC(s). (How?)). (How?)

G(s

G(s)) C(s

C(s))

Plant Plant Controller

Controller

Compensation Compensation

Fixed! Fixed! Designable!

Designable!

(for time domain specs)

Lead and lag compensators

LeadLeadcompensatorcompensator

G(s

G(s)) C(s

C(s))

Controller

Re

Im

LagLagcompensatorcompensator

Re

Im

The reason why these are called

The reason why these are called ““leadlead””and and ““laglag””will be explained will be explained

in frequency response approach (later in this course).

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Roles of lead & lag compensators

Lead compensatorLead compensator

Improve transient responseImprove transient response

Improve stabilityImprove stability

Lag compensatorLag compensator

Reduce steady state errorReduce steady state error

LeadLead--lag compensatorlag compensator

Take into account all the above issues.Take into account all the above issues.

Lead compensator design

Consider a systemConsider a system

Analysis of CL system for C(sAnalysis of CL system for C(s)=1)=1

Damping ratio Damping ratio ζζ=0.5=0.5

UndampedUndampednatural freq. natural freq. ωωnn=2 =2 rad/srad/s

Performance specificationPerformance specification

G(s

G(s)) C(s

C(s))

Controller

Re

Im

Desired pole

Desired pole

CL pole

with

with C(sC(s)=1)=1

Angle & magnitude conditions (review)

A point s to be on root locus A point s to be on root locus ÅÆÅÆit satisfiesit satisfies

Angle conditionAngle condition

For a point on root locus, gain K is obtained byFor a point on root locus, gain K is obtained by

Magnitude conditionMagnitude condition

Odd number Odd number

Lead compensator design (cont

’

d)

Evaluate

Evaluate G(s) at the desired pole.G(s) at the desired pole.

o

o If If angle conditionangle conditionis satisfied, is satisfied,

compute the corresponding K.

o

o In this example, In this example,

Angle condition is not satisfied.

Re

Im

Angle deficiency

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Lead compensator design (cont

Lead compensator design (cont’

’d)

d)

To compensate angle deficiency, To compensate angle deficiency,

design a lead compensator design a lead compensator C(sC(s))

satisfying satisfying

Re

Im

Desired pole

Desired pole

There are many ways to design such

There are many ways to design such C(sC(s)!)!

How to select pole and zero?

Draw horizontal line PADraw horizontal line PA

Draw line PODraw line PO

Draw bisector PBDraw bisector PB

Draw PC and PDDraw PC and PD

Pole and zero of C(sPole and zero of C(s) are shown in the figure.) are shown in the figure.

Re

Im

Desired pole

Desired pole

P

A

O

B

C

D

--z(=z(=--2.9)2.9)

--p(=p(=--5.4)5.4)

Comparison of root locus

G(sG(s))

Re

Im

G(s)C(sG(s)C(s))

Re

Im

Improved stability! Improved stability!

How to design the gain K?

Open loop transfer functionOpen loop transfer function

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0 1 2 3 4 5

0 0.2 0.4 0.6 0.8 1 1.2 1.4

Comparison of step responses

Uncompensated system ( Uncompensated system (C(sC(s)=1))=1) Compensated system

Compensated system

Lead compensator Lead compensatorgives gives •

•faster transient responsefaster transient response (shorter rise and settling time)

(shorter rise and settling time) •

•improved stabilityimproved stability

0 1 2 3 4 5

Error constants

Step-Step-error constanterror constant

Ramp-Ramp-error constanterror constant

Lag compensator can be used to reduce Lag compensator can be used to reduce

steady

steady--state error.state error.

Unit ramp input Unit ramp input

Ramp response

Roles of lead & lag compensators

Improve transient responseImprove transient response

Improve stabilityImprove stability

Reduce steady state errorReduce steady state error

LeadLead--lag compensatorlag compensator

Take into account all the above issues.Take into account all the above issues.

Lead

Lead-

-lag compensator design

lag compensator design

Consider a systemConsider a system

Analysis of CL system for C(sAnalysis of CL system for C(s)=1)=1

RampRamp--error constant error constant KvKv=2=2

Performance specificationPerformance specification

RampRamp--error constant error constant KvKv=50=50

G(s

G(s)) C(s

C(s))

Controller

Re

Im

Desired pole

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0 1 2 3 4 5

0 0.2 0.4 0.6 0.8 1 1.2 1.4

Comparison of step responses

(after lead compensation)

Uncompensated system ( Uncompensated system (C(sC(s)=1))=1) Compensated system

Compensated system

Lead compensator Lead compensatorgives gives •

•faster transient responsefaster transient response (shorter rise and settling time)

(shorter rise and settling time) •

•improved stabilityimproved stability

0 1 2 3 4 5

Error constants

(after lead compensation)

Step-Step-error constanterror constant

Ramp-Ramp-error constanterror constant

Lag compensator can reduce steady

Lag compensator can reduce steady--state error. state error. Unit ramp input Unit ramp input

Ramp response

Ramp response

NOT SATISFACTORY! NOT SATISFACTORY!

How to design lag compensator

We want to increase rampWe want to increase ramp--error constanterror constant

Take, for example, z=10p. Take, for example, z=10p.

We do not want to change CL pole location sWe do not want to change CL pole location s11so so

much (already satisfactory transient). much (already satisfactory transient).

Root locus with lag compensator

Without compensatorWithout compensator

s

s11

With compensatorWith compensator

s

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Guidelines to choose z and p

The zero and the pole of a lag compensator The zero and the pole of a lag compensator should be

should be close to each otherclose to each other, for , for

The pole of a lag compensator should be The pole of a lag compensator should be close close

to the origin

to the origin, to have a large ratio , to have a large ratio z/pz/p, leading to , leading to a large ramp

a large ramp--error constant error constant KvKv..

However, the pole of a lag compensator too However, the pole of a lag compensator too close to the origin may be problematic: close to the origin may be problematic:

Slow settling (due to closedSlow settling (due to closed--loop pole near the origin)loop pole near the origin)

How to design lag compensator

For the desired CL poleFor the desired CL pole

Take a small p (by trial-Take a small p (by trial-andand--error!)error!)

Lead-Lead-lag controllerlag controller

Root locus

- 6 - 4 - 2 0 - 1 5

- 1 0 - 5 0 5 1 0 1 5

- 6 - 4 - 2 0 - 1 5

- 1 0 - 5 0 5 1 0

1 5 R o o t L o c u s

R e a l A x is

Im

agi

nar

y

A

x

is

R o o t L o c u s

R e a l A x is

Im

agi

nar

y

A

x

is

With lead compensator

With lead compensator With leadWith lead--lag compensatorlag compensator

Desired pole

Comparison of step responses

0 1 2 3 4 5

0 0.2 0.4 0.6 0.8 1 1.2 1.4

Uncompensated

With lead

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Comparison of ramp responses

0 1 2 3 4 5

Uncompensated

With lead

With lead--lag compensatorlag compensator Unit ramp input

Unit ramp input

Summary and exercises

Controller design based on root locusController design based on root locus

Lead compensator design Lead compensator design to improve stability and to improve stability and transient response.

transient response.

Lag compensator design Lag compensator design to improve steady state to improve steady state error.

error.

LeadLead--lag compensator design lag compensator design to improve stability, to improve stability, transient, SS responses.

transient, SS responses.

Exercises Exercises

Read Sections 9.1Read Sections 9.1--9.4. 9.4.

Solve Problem 9.3 with K=10, and Problem 9.4 with Solve Problem 9.3 with K=10, and Problem 9.4 with

K=17.5.

Compensator realization

One example, using operational amplifiersOne example, using operational amplifiers

R

R22

R

R11

C

C22

-+

+

C

C11

-+

+

R

R33

R

R44

v

vii(t(t))

v

voo(t(t))

Compensator realization (cont

Compensator realization (cont’

’d)

d)

Transfer functionTransfer function

LeadLeadcompensatorcompensator

Re

Im

LagLagcompensatorcompensator

Re

Im

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Suggestion on Lab 4 schedule

Current scheduleCurrent schedule

Suggested scheduleSuggested schedule

Please email me during this week if this suggestion is Please email me during this week if this suggestion is

inconvenient for you.

If your group wants to perform Lab 4 on Tuesday March If your group wants to perform Lab 4 on Tuesday March

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