Automatic Control Systems by Benjamin C. Kuo

KUO

BENJAMIN

C.

KUO

utomatic

control

Syste

THIRD

EDITION

7

HI

OX

2D

-'.

PRLNIlUt

HALL

Automatic

Control

Systems

Third

Edition

BENJAMIN

C.

KUO

ProfessorofElectricalEngineering UniversityofIllinoisatUrbana-Champaign

(,^-s

LibraryofCongress CataloginginPublicationData

Kuo,BenjaminC

Automaticcontrol systems. Includes index.

1. Automaticcontrol. 2.Controltheory.

I.Title. TJ213.K83541975 629.8'3 74-26544 ISBN0-13-054973-8

METROPOLITAN

BOROUGH

OF

W1GAN

DEPT.

OF

LEISURE LIBRARIES Ace. No,

10067*2

*?

A-•:

o

5m^?1^;,

:

©

1975 byPrentice-Hall,Inc.

EnglewoodCliffs,

New

Jersey

Allrightsreserved.

No

partofthis

book

may

bereproducedinany formorby any means

without permissioninwritingfromthe publisher.

\

10 9 8 7 6 5 4

PrintedintheUnitedStatesofAmerica

PRENTICE-HALL INTERNATIONAL,

INC., London

PRENTICE-HALL OF AUSTRALIA,

PTY. LTD., Sydney

PRENTICE-HALL OF

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LTD., Toronto

PRENTICE-HALL OF

INDIA PRIVATE LIMITED,

New

Delhi

Contents

Preface

j

x

1.

Introduction

1.1 ControlSystems 1

1.2 WhatIsFeedbackand WhatAreItsEffects ? 1.3 TypesofFeedbackControlSystems 11

2.

Mathematical

Foundation

₁₅

2.7 Introduction 15

2.2 Complex-Variable Concept 15 2.3 Laplace Transform 18

2.4 InverseLaplace TransformbyPartial-FractionExpansion 21 2.5 Application of Laplace Transformtothe Solution of Linear Ordinary

DifferentialEquations 25 2.6 ElementaryMatrixTheory 26 2.7 MatrixAlgebra 32

2.8 z-Transform 39

vi / Contents

3.

Transfer

Function

and

Signal

Flow Graphs

51 3.1 Introduction 51

3.2 TransferFunctionsof LinearSystems 51 3.3 ImpulseResponseof LinearSystems 55 3.4 Block Diagrams 58

3.5 SignalFlowGraphs 64

3.6

Summary

of Basic Properties of SignalFlowGraphs 66 3.7 Definitions forSignalFlow Graphs 67

3.8 Signal-Flow-Graph Algebra 69

3.9 Examplesof the Construction of SignalFlowGraphs 71

3.10 General Gain FormulaforSignalFlowGraphs 75

3.11 Application of theGeneralGain FormulatoBlockDiagrams 80 3.12 TransferFunctions of Discrete-DataSystems 81

4.

State-Variable Characterization of

Dynamic

Systems

95

4.

1

Introductiontothe StateConcept 95

4.2 State EquationsandtheDynamicEquations 97 4.3 Matrix Representation of StateEquations 99 4.4 State TransitionMatrix 101

4.5 StateTransitionEquation 103

4.6 RelationshipBetweenState Equationsand High-OrderDifferentialEquations 107

4.7 TransformationtoPhase-Variable CanonicalForm 109

4.8 RelationshipBetweenStateEquationsandTransferFunctions 115 4.9 CharacteristicEquation, Eigenvalues,andEigenvectors 117 4.10 Diagonalization of the

A

Matrix(SimilarityTransformation) 118 4.11 Jordan CanonicalForm 123

4.12 StateDiagram 126

4.13 Decompositionof TransferFunctions 136

4.14 TransformationintoModal Form 141 4.15 Controllabilityof LinearSystems 144 4.16 Observabilityof LinearSystems 152

4.17 Relationship

Among

Controllability, Observability,

andTransferFunctions 156

4.18 Nonlinear State EquationsandTheir Linearization 158 4.19 State Equations of Linear Discrete-DataSystems 161 4.20 z-Transform Solution of Discrete StateEquations 165 4.21 StateDiagramforDiscrete-DataSystems 167 4.22 StateDiagramsforSamp/ed-Data Systems 171 4.23 State Equations of LinearTime-Varying Systems 173

5.

Mathematical Modeling

of Physical

Systems

187

5. 1

Introduction 187

5.2 Equations ofElectricalNetworks 188

5.3 ModelingofMechanical System Elements 190 5.4 Equations ofMechanical Systems 203 5.5 Error-SensingDevicesin ControlSystems 208 5.6 Tachometers 219

Contents / vii

5.7

DC

MotorsinControlSystems 220 5.8 Two-PhaseInductionMotor 225 5.9 StepMotors 228

5.10 Tension-ControlSystem 235 5.11 Edge-GuideControlSystem 237

5.12 Systemswith TransportationLags 242

5.13 Sun-Seeker System 243

6.

Time-

Domain

Analysis of

Control

Systems

₂₅₉

6. 1

Introduction 259

6.2 Typical Test SignalsforTimeResponseof ControlSystems 260

6.3 Time-DomainPerformanceofControlSystems—Steady-StateResponse 262 6.4 Time-DomainPerformanceofControlSystems—TransientResponse 271 6.5 TransientResponseofaSecond-OrderSystem 273

6.6 TimeResponseofaPositional ControlSystem 284 6.7 Effectsof Derivative ControlontheTimeResponseof

FeedbackControlSystems 295

6.8 EffectsofIntegralControlonthe TimeResponse ofFeedbackControlSystems 300

6.9 RateFeedbackorTachometerFeedbackControl 302

6.10 ControlbyState-VariableFeedback 305

7. Stability

of

Control

Systems

₃₁₆

7.1 Introduction 316

7.2 Stability.CharacteristicEquation,andtheState TransitionMatrix 317

7.3 Stabilityof Linear Time-InvariantSystemswith Inputs 319 7.4 MethodsofDeterminingStabilityofLinear ControlSystems 321 7.5 Routh-HurwitzCriterion 322

7.6 NyquistCriterion 330

7.7 Applicationof theNyquistCriterion

344

7.8 Effectsof Additional PolesandZerosG(s)H(s) ontheShape of theNyquistLocus 352

7.9 StabilityofMultiloopSystems 356

7.10 StabilityofLinear ControlSystemswith Time Delays 360 7.11 Stabilityof NonlinearSystems—Popov'sCriterion 363

8.

Root

Locus Techniques

₃₇₅

8. 1

Introduction 375

8.2 Basic Conditions of theRootLoci 376 8.3 Construction of theCompleteRootLoci 380 8.4 Applicationof theRootLocus Techniquetothe

Solution ofRootsofaPolynomial 412

8.5

Some

ImportantAspectsof theConstruction oftheRootLoci 417 8.6 RootContour—Multiple-ParameterVariation 424

8.7 RootLociofSystemswithPure Time Delay 434 8.8 RelationshipBetween RootLociandthe Polar Plot 444 8.9 RootLociof Discrete-Data ControlSystems 447

viii / Contents

9.

Frequency-Domain

Analysis of

Control

Systems

459

9. 1

Introduction 459

9.2 Frequency-DomainCharacteristics 462

9.3

M

_p.CO_p,andtheBandwidthof aSecond-Order System 464

9.4 EffectsofAddingaZerototheOpen-LoopTransferFunction 467 9.5 EffectsofAddingaPoletotheOpen-LoopTransferFunction 471 9.6 Relative Stability—GainMargin,PhaseMargin,and

M

_p 473 9.7 RelativeStability

As

RelatedtotheSlope of

theMagnitude Curveof theBodePlot 483 9.8 Constant

M

LociintheG(jOi) -Plane 485 9.9 ConstantPhase LociintheG{jCO)-Plane 489

9.10 Constant

M

and

N

LociintheMagnitude-_Versus-PhasePlane—

TheNichols Chart 490

9.11 Closed-Loop FrequencyResponseAnalysis ofNonunityFeedbackSystems 496

9.12 SensitivityStudiesintheFrequencyDomain 497

10.

Introduction

to

Control

Systems

Design

504

10.1 Introduction

504

10.2 ClassicalDesignof ControlSystems 510 10.3 Phase-Lead Compensation 515

10.4 Phase-Lag Compensation 535 10.5 Lag-LeadCompensation 552

10.6 Bridged-TNetworkCompensation

557

11.

Introduction to

Optimal Control

572

11.1 Introduction 572

11.2 AnalyticalDesign 574

11.3 ParameterOptimization 583

11.4 DesignofSystemwith SpecificEigenvalues—

An

Application ofControllability 585

11.5 Designof State Observers 588 11.6 OptimalLinear RegulatorDesign 599 11.7 Design withPartialStateFeedback 615

APPENDIX

A

Frequency-Domain

Plots

626

A.1 Polar PlotsofTransferFunctions 627

A.2 BodePlot (CornerPlot) ofa TransferFunction 633 A.3 Magnitude-Versus-PhasePlot 643

APPENDIX

B

Laplace

Transform

Table

645

APPENDIX C

Lagrange's

Multiplier

Method

650

Preface

The

firstedition ofthis book, publishedin 1962,

was

characterized

by

having chapters

on

sampled-data

and

nonlinearcontrol systems.

The

treatment ofthe analysis

and

design ofcontrol systems

was

allclassical.

The

two major

changesinthe secondedition, publishedin 1967,werethe inclusionofthestatevariabletechnique

and

the integrationofthe discrete-data systems with the continuous data system.

The

chapter

on

nonlinear systems

was

eliminated in the second edition to the disappointment of

some

users of thattext.

At

thetime ofthe revision theauthorfeltthat acomprehensive

treat-ment

on

the subject_{of nonlinear systems could not be}

_made

_effectively with the available space.

The

third editionis stillwritten as

an

introductorytextforaseniorcourse

on

control systems.

Although

agreat dealhas

happened

inthe area of

modern

control theoryinthe past ten years, preparing suitable material for a

modern

course

on

introductory control systems remains a difficulttask.

The

problem

is

a complicated

one

becauseitisdifficulttoteachthe topics concerned with

new

developmentsin

modern

controltheory atthe undergraduatelevel.

The

unique situation in control_{systems has been} that

many

ofthepractical problemsare

still being solved in the industry

by

the classical methods.

While

some

ofthe techniques in

modern

control theory are

much

more

powerful

and

can solve

more

complex

problems, there are often

more

restrictions

when

it

comes

to practicalapplications ofthe solutions.

However,

itshould be recognized that

a

modern

control engineer _{should have an understanding of the classical}

as well as the

modern

controlmethods.

The

latter willenhance

and broaden

one's perspective in solving a practical problem. It is the author's opinionthat one shouldstrikea balanceintheteaching ofcontrolsystems theoryatthebeginning

x / Preface

and

intermediate levels. Therefore in this current edition, equal emphasis is

placed

on

the classical

methods and

the

modern

control theory.

A

number

of introductory

books

with titles involving

modern

control

theory have been published in recent years.

Some

authors have attempted to unify

and

integrate theclassicalcontrolwiththe

modern

control,but according to the critics

and

reviews,

most

have failed.

Although

such a goal is highly

desirable, ifonly

from

the standpoint ofpresentation, there does not

seem

to be a

good

solution. Itis possible that the objective

may

not be achieved until

new

theories

and

new

techniques are developed for this purpose.

The

fact remains that control systems, in

some

way,

may

be regarded as a science of learning

how

to solve one

problem—

control, in

many

different ways. These different

ways

ofsolution

may

be

compared and weighed

against each other, but it

may

not be possible to unify allthe approaches.

The

approach

used in

this text is topresent the classical

method

and

the

modern

approach

indepen-dently,

and whenever

possible, the

two

approaches are considered as

alterna-tives,

and

the advantages

and

disadvantages of each are weighed.

Many

illustrative examplesare carried out

by

both methods.

Many

existing text

books on

control systems have been criticized fornot including adequate practical problems.

One

reason for this is, perhaps, that

many

text

book

writers are theorists,

who

lack the practical

background and

experience necessary to provide real-life examples.

Another

reason is that the difficulty inthe controlsystemsareais

compounded

by

the factthat

most

real-lifeproblemsare highlycomplex,

and

arerarelysuitableasillustrativeexamples

at the introductory level. Usually,

much

of the realism is lost

by

simplifying the

problem

to fit the nice theorems

and

design techniques developed in the textmaterial. Nevertheless, the majorityofthe students taking a controlsystem course atthe seniorlevel

do

not pursue a graduate career,

and

they

must

put their

knowledge

to

immediate

use in their

new

employment.

It is extremely important for these students, as well as those

who

will continue, to gain

an

actual feel of

what

a real control system is like. Therefore, the author has introduced a

number

ofpractical examples in various fields in this text.

The

homework

problemsalso reflectthe attempt ofthis text to provide

more

real-lifeproblems.

The

following features ofthis

new

edition are

emphasized

by comparison

withthefirst

two

editions

:

1.

Equal

emphasis

on

classical

and

modern

control theory.

2. Inclusionof sampled-data

and

nonlinear systems.

3. Practical system examples

and

homework

problems.

The

material assembled in this

book

is

an outgrowth

of a senior-level

control system course taught

by

the author at the University ofIllinois at

Urbana-Champaign

for

many

years.

Moreover,

this

book

iswritten in astyle

adaptablefor self-study

and

reference.

Chapter 1 presentsthe basicconcept ofcontrol systems.

The

definition of

founda-Preface / xi

tion

and

preliminaries.

The

subjects included are Laplace transform, z-trans-form, matrixalgebra,

and

the applications ofthe_{transform methods. Transfer} function

and

signal flow graphs are discussed in Chapter 3. Chapter 4 intro-duces the state variable

approach

to dynamical systems.

The

concepts

and

definitionsofcontrollability

and

observability areintroducedatthe early stage These subjectsarelaterbeing usedfor the analysis

and

designoflinearcontrol systems. Chapter 5 discusses the mathematical

modeling

ofphysical systems. Here, the emphasis is

on

_{electromechanical systems. Typical transducers}

and

control systems used in practice are illustrated.

The

treatment cannot be exhaustive as there are

numerous

typesofdevices

and

control systems. Chapter 6 gives the time response considerations ofcontrol systems.

Both

the classical

and

the

modern

approach

are used.

Some

simple design considerationsin the

time

domain

arepointed out. Chapters7, 8,

and

9 dealwithtopics

on

stability, root locus,

and

frequency response ofcontrol systems.

InChapter 10,the designofcontrolsystemsisdiscussed,

and

the

approach

is basicallyclassical.

Chapter

11 contains

some

ofthe optimalcontrol subjects which,inthe author's opinion,can be taughtattheundergraduateleveliftime permits.

The

text does contain

more

material than can be covered in one semester.

One

ofthe difficulties in preparing this

book

was

the weighing of

what

subjects to cover.

To

keep the

book

to a reasonable length,

some

subjects,

which

were in the original draft,

had

to be left out of the final manuscript! These includedthe treatment ofsignalflow graphs

and time-domain

analysis, of _{discrete-data systems, the second}

_method

of Liapunov's stability

method!

describing function analysis, stateplane analysis,

and

a

few

selected topics

on

implementing optimal control.

The

author feels that the inclusion of these subjects

would add

materially to the spirit of the text, but at the cost of a higherprice.

The

author _{wishes to express his sincere appreciation}

to

Dean

W.

L. Everitt (emeritus), Professors E. C. Jordan, O. L.

Gaddy, and

E.

W.

Ernst, ofthe UniversityofIllinois,fortheir

encouragement and

interest intheproject!

The

author is grateful to Dr.

Andrew

Sage ofthe Universityof Virginia

and

Dr.G. Singh ofthe UniversityofIllinoisfortheir_{valuable suggestions. Special} thanksalsogoesto Mrs. Jane Carlton

who

typed a

good

portion ofthe

manu-script

and

gave herinvaluable assistance in proofreading.

Benjamin

C.

Kuo

Urbana, Illinois

1

Introduction

1.1 Control

Systems

Inrecentyears,automaticcontrolsystems

have assumed an

increasingly impor-tant roleinthe

_{development and advancement}

_of

_modern

_civilization

and

tech-nology. Domestically,automaticcontrolsinheating

and

airconditioning systems regulate the temperature

and

the humidity of

modern homes

for comfortable

living. Industrially, automaticcontrol systems are

found

in

numerous

applica-tions, suchas quality controlof

manufactured

products, automation,

machine

tool control,

modern

space technology

and

weapon

systems,

computer

systems, transportation systems,

and

robotics.

Even

such problemsasinventorycontrol, social

and economic

systemscontrol,

and

environmental

and

hydrological sys-temscontrol

may

be

approached from

thetheory of automaticcontrol.

The

basic control system concept

may

be described

by

the simple block

diagram

shown

in Fig. 1-1.

The

objectiveofthesystemisto control the variable c inaprescribed

manner

by

the actuatingsignale

through

theelements ofthe control system.

In

more

common

terms, the controlled variableisthe outputofthe system,

and

the actuatingsignalistheinput.

As

a simple example,inthe steering control of

an

automobile, the directionofthe

two

frontwheels

may

be regardedas the controlled variablec,theoutput.

The

positionofthe steeringwheelisthe input, the actuatingsignale.

The

controlled processorsystemin thiscaseis

composed

ofthe steering

mechanisms,

including the

dynamics

ofthe entire automobile.

However,

ifthe objectiveis to control the speed ofthe _{automobile, then} the

amount

ofpressure exerted

on

the acceleratoristhe actuating signal, withthe

2 / Introduction Chap.1 Actuating Controlled signal e _Control system variablec (Input) (Output) Fig.1-1. Basic control system.

Thereare

many

situations

where

several variables aretobecontrolled simul-taneously

by

a

number

ofinputs.

Such

systems are referredto as multivariabk systems.

Open-Loop

Control

Systems (Nonfeedback

Systems)

The word

automaticimplies that thereisacertain

amount

ofsophistication

inthe control system.

By

automatic,itgenerally

means

that thesystemisusually

capable of adaptingto avariety of operatingconditions

and

is able torespond to aclassofinputssatisfactorily.

However,

not

any

typeofcontrol system has theautomaticfeature. Usually, theautomaticfeatureis achieved

by

feeding the

outputvariable

back and comparing

itwiththe

command

signal.

When

a system does not havethe feedbackstructure, itis called

an

open-loop system,

which

is

the simplest

and

most

economicaltypeofcontrol system. Unfortunately, open-loopcontrolsystemslackaccuracy

and

versatility

and

can be usedin

none

but the simplest typesofapplications.

Consider, for example, control of the furnace for

home

heating. Let us

assume

thatthe furnaceis equipped only with a timing device,

which

controls

the

on and

off periods of the furnace.

To

regulate the temperature to the properlevel, the

human

operator

must

estimate the

amount

of time required for thefurnace to stay

on and

then setthe timeraccordingly.

When

the preset time is up, the furnaceis turned off.

However,

it is quite likely that thehouse temperatureiseither

above

or

below

the desired value,

owing

toinaccuracyin the estimate.

Without

further deliberation,itisquite apparentthatthistype of control is inaccurate

and

unreliable.

One

reason forthe inaccuracylies inthe

factthat one

may

not

know

the exactcharacteristics ofthe furnace.

The

other factor is that

one

has

no

control over the outdoor temperature,

which

has a definite bearing

on

the indoor temperature. This also points to

an

important disadvantage ofthe performance of

an

open-loop control system, in that the systemis not capable of adaptingto variationsinenvironmentalconditions or

to external disturbances. In the caseofthefurnacecontrol, perhaps

an

experi-enced person can providecontrol foracertain desiredtemperatureinthehouse; but if the doors or

windows

are

opened

or closed intermittently during the operating period, thefinal temperature inside the housewill not be accurately regulated

by

theopen-loopcontrol.

An

electric

washing machine

is anothertypical

example

of an open-loop

system,becausethe

amount

of

wash

timeisentirelydetermined

by

the

judgment

and

estimation of the

human

operator.

A

true automatic electric

washing

machine

should havethe

means

of checkingthecleanliness ofthe clothes con-tinuously

and

turn itselfoff

when

the desired degree ofcleanliness isreached.

Sec. 1.1 _Control

Systems /3 elements ofthe closed-loop control systems.Ingeneral,theelements of

an

open-loopcontrolsystemare represented

by

the block

diagram

ofFig. 1-2.

An

input signal or

command

r is applied to the controller,

whose

output acts as the actuatingsignale;the actuatingsignalthenactuates the controlled process

and

hopefullywill _{drive the controlled variable cto the desired} value. Reference Actuating signale Controlled inputr Controller Controlled process variablec (Output)

Fig.1-2. Block diagramofanopen-loop control system. Closed-Loop Control

Systems

(Feedback Control Systems)

What

is missing in the open-loop control system for

more

accurate

and

more

adaptablecontrolisalinkor feedback

from

theoutputtotheinput ofthe system. Inorderto obtain

more

accurate control, the controlledsignalc(t)

must

be fed

back

and compared

with the reference input,

and an

actuating signal proportionalto the difference oftheoutput

and

theinput

must

besentthrough the systemto correct theerror.

A

system with

one

or

more

feedback pathslike thatjust describedis called a closed-loop system.

Human

beings areprobably the

most complex and

sophisticated feedback control system in existence.

A

human

being

may

be consideredtobe acontrol system with

many

inputs

and

outputs,_{capable of}carryingout highly

complex

operations.

To

illustratethe

human

beingasa feedbackcontrol system, letusconsider that the objectiveistoreachfor

an

object

on

adesk.

As

one

isreachingfor the object, the brain _{sends out a} signal to the

arm

to perform the task.

The

eyes serve asasensing device

which

feeds

back

continuouslythe positionofthehand.

The

distance

between

the

hand

and

the objectisthe error,

which

is eventually brought to zero as the

hand

reaches the object. This is a typical

example

of closed-loopcontrol.

However,

if

one

is told toreach forthe object

and

thenis

blindfolded,

one

can only reach

toward

the object

by

estimatingitsexact posi-tion.It is_{quite possible that the object}

_may

_{be missed}

by

awide margin.

With

the eyes blindfolded, thefeedback pathisbroken,

and

the

human

is operating as an open-loopsystem.

The

example

ofthe reaching of

an

object

by

a

human

beingisdescribed

by

the block

diagram

shown

in Fig. 1-3.

As

anotherillustrative

example

of a _{closed-loop control system, Fig.} 1-4

Error Input detector command

_f

_x

_Error Reach for object Controller (brain) Controlled process

(armandhand)

1 Controlled

variable Position

ofhand

4 / Introduction Chap.1

Rudder

Fig. 1-4. Ruddercontrol system.

shows

theblock

diagram

oftheruddercontrolsystem of aship. Inthiscase the objective ofcontrol is the position ofthe rudder,

and

the reference input is

appliedthroughthe steeringwheel.

The

error

between

the relativepositions of the steeringwheel

and

the rudder is the signal,

which

actuates the controller

and

the motor.

When

the rudder is finally aligned with the desired reference direction,the output ofthe error sensoriszero. Letus

assume

thatthe steering wheelpositionisgiven a

sudden

rotationof

R

units, as

shown

bythetimesignal in Fig. l-5(a).

The

positionoftherudderasafunctionoftime,depending

upon

the characteristicsofthe system,

may

typicallybe one ofthe responses

shown

in Fig. l-5(b).Becauseallphysicalsystems haveelectrical

and

mechanicalinertia, the position ofthe rudder cannot respondinstantaneously to a step input, but

will,rather,

move

gradually

toward

thefinaldesiredposition.Often, theresponse will oscillate

about

thefinalposition before settling. Itis apparentthat for the

ruddercontrolitisdesirable tohave anonoscillatory response. 0,(0

R

6e

W

-*-t *~t

(a) (b)

Fig.1-5. (a)Step displacement input of rudder controlsystem,(b)Typical outputresponses.

Sec. 1.1 ControlSystems / 5 Error sensor Input

~^

Error Controller Controlled process Output

J

Feedback elements

Fig. 1-6. _{Basic elements of a feedbackcontrolsystem.}

The

basicelements

and

the block

diagram

ofa closed-loop controlsystem

are

shown

in Fig. 1-6. In general, the configurationof a feedbackcontrolsystem

may

not be constrainedto thatofFig. 1-6. In

complex

systems there

may

be a multitude of feedback loops

and

elementblocks.

Figurel-7(a) illustratestheelements ofa tension controlsystem of a

windup

process.

The

unwind

reel

may

containarollofmaterialsuchaspaperor cable

which

istobesent intoaprocessingunit,suchasa cutteror aprinter,

and

then collectsit

by

windingitonto another roll.

_The

controlsystem in thiscaseis to maintain the tension ofthe material or

web

at acertain prescribed tension to avoid such problems as tearing, stretching, orcreasing.

To

regulate the tension, the

web

is

formed

into ahalf-loop

by

passing it

down

and around

aweightedroller.

The

rollerisattachedtoapivotarm,

which

allowsfree

up-and-down motion

oftheroller.

The

combination oftheroller

and

the pivot

arm

iscalled the dancer.

When

the system is in operation, the

web

normally travels at a constant speed.

The

ideal position ofthe danceris horizontal, producing a

web

tension equalto one-halfofthetotalweight

W

ofthedancerroll.

The

electric brake

on

the

unwind

reel is to generate a restrainingtorque to keep the dancer in the horizontal positionatalltimes.

During

actual operation, because ofexternal disturbances, uncertainties

and

irregularitiesofthe

web

material,

and

the decreaseoftheeffectivediameter ofthe

unwind

reel, the dancer

arm

will not remain horizontal unless

some

scheme

is

employed

toproperly sensethe dancer-armposition

and

control the restraining brakingtorque.

To

obtain the correction of the dancing-arm-position error,

an

angular sensoris used to

measure

the angulardeviation,

and

asignal in proportionto the error is used to control the braking torque through a controller. Figure l-7(b)

shows

a block

diagram

thatillustrates the interconnections between the elements ofthe system.

6 / Introduction

Chap.1

Unwindreel

(decreasingdia.)

Web

processing Windupreel

(increasingdia.) Drivesystem (constantweb speed) (Current) Reference input ~"\ Error Controller Electric brake Unwind process Tension Dancer arm (b)

Fig.1-7. (a) Tension control system, (b) Block diagram depicting the basicelementsand interconnections of a tensioncontrol system.

1.2

What

Is Feedback and

What

AreItsEffects?

The

concept of feedbackplays

an

importantrole incontrol systems.

We

demon-strated in Section 1.1 that feedback is a

major

requirement of a closed-loop control system.

Without

feedback,acontrolsystem

would

not beable to achieve the accuracy

and

reliability that are required in

most

practical applications.

However, from

a

more

rigorous standpoint, the definition

and

thesignificance of feedback are

much

deeper

and

more

difficult to demonstrate than the few examples givenin Section 1.1. In reality, the reasons for usingfeedbackcarry far

more

meaning

thanthe simple

one

of

comparing

the input withthe output inordertoreducetheerror.

The

reductionof systemerrorismerely

one

ofthe

Sec. 1.2 _What

Is FeedbackandWhatAreItsEffects? / 7

feedbackalsohaseffects

on

_{such system performancecharacteristicsasstability,}

bandwidth, overall gain, impedance,

and

sensitivity.

To

understandtheeffectsof feedback

on

acontrol system,it isessentialthat

we

examine

this

phenomenon

with a

broad

mind.

When

feedbackisdeliberately introducedfor thepurpose ofcontrol,itsexistenceiseasily identified.

However,

there are

numerous

situationswherein aphysicalsystem that

we

normally rec-ognize as

an

inherently

nonfeedback

system

may

turn out to have feedback

when

itis observedinacertain

manner.

Ingeneral

we

canstatethat

whenever

a closed sequence of cause-and-effect relation exists

among

the variables of a system,feedback issaid to exist. This viewpointwillinevitablyadmit feedback inalarge

number

of systemsthat ordinarily

would

beidentifiedas

nonfeedback

systems.

However,

with the availability of the feedback

and

control system theory, this general definition of feedbackenables

numerous

systems, with or withoutphysical feedback, tobe studiedin asystematic

way

oncethe existence of feedbackinthe

above-mentioned

senseis established.

We

shall

now

investigate the effects of feedback

on

the various aspects of system performance.

Without

the necessary

background and

mathematical foundation oflinear system theory, at this point

we

can only rely

on

simple static system notation forourdiscussion. Let us consider the simple feedback systemconfiguration

shown

in Fig. 1-8,

where

ris theinputsignal,ctheoutput signal, ethe error,

and

b thefeedbacksignal.

The

parameters

G

and

ZTmay

be consideredas constantgains.

By

simple algebraic manipulationsit is simpleto

show

that theinput-outputrelationofthesystemis

G

M

₌

_t

₌

_FTW

(l-i)

Using

thisbasic relationship ofthefeedback system structure,

we

can uncover

some

ofthe significant effects offeedback.

G

_. r

-

b + + e i -o c ^

H

_ -o

Fig. 1-8. Feedbacksystem. Effectof

Feedback on

OverallGain

As

seen

from

Eq. (1-1),feedbackaffectsthe gain

G

of a

nonfeedback

system

by

afactorof1

+

GH. The

referenceofthefeedbackinthe system ofFig. 1-8

is negative, sincea

minus

signis assigned to thefeedback signal.

The

quantity

GH

may

itselfinclude a

minus

sign, sothe generaleffect of feedbackis thatit

8 / Introduction Chap.1 functionsoffrequency, so the

magnitude

of1

+

GH

may

be greaterthan 1 in

one

frequency range but less than 1 in another. Therefore, feedback could increase the gainofthesysteminone frequency range butdecreaseitinanother. Effectof

Feedback on

Stability

Stabilityisa notionthat describeswhetherthesystemwillbeable to follow the input

command.

In a nonrigorous

manner,

a system is said to be unstable ifits outputis out ofcontrol orincreases without bound.

To

investigatetheeffect of feedback

on

stability,

we

can againreferto the expression inEq. (1-1).If

GH

= -

1,theoutput ofthesystemisinfinitefor

any

finiteinput. Therefore,

we

may

state that feedback can cause a system thatis

originally stableto

become

unstable. Certainly,feedbackis atwo-edged sword;

when

itisimproperlyused,it

can

beharmful.Itshouldbe pointedout,however, that

we

are onlydealingwiththe staticcase here, and,ingeneral

GH

= —

1 is

nottheonly conditionfor instability.

It can be demonstratedthat

one

ofthe advantages of incorporating

feed-back

isthatitcanstabilize

an

unstable system. Let us

assume

thatthefeedback systemin Fig. 1-8 isunstablebecause

GH

=

—1.

If

we

introduceanother

feed-back

loop

through

anegative feedback ofF, as

shown

in Fig. 1-9, the input-outputrelationoftheoverallsystemis

c

G

r

~

I

+GH+GF

( "

It is apparent that although the properties of

G

and

H

are such that the inner-loop feedback system is unstable, because

GH =

—

1, the overallsystem

can bestable

by

properly selectingthe outer-loop feedbackgain F.

-o

G

— i o+ + r b + e + c

—

o +

-

+ -o -o

H

o--o

F

o-Fig.1-9. Feedbacksystem withtwofeedbackloops. Effectof

Feedback on

Sensitivity

Sensitivity considerations often play

an

important role in the design of control systems. Since all physical elements have properties that

change

with environment

and

age,

we

cannot always consider the parameters of a control

Sec-1-2 WhatIsFeedback andWhatAreItsEffects? / 9 system tobe completely stationaryoverthe entireoperatinglifeofthe system.

For

instance, the winding resistance of

an

electric

motor

changes as the tem-perature ofthe

motor

risesduringoperation. In general, a

good

controlsystem should beveryinsensitiveto theseparametervariationswhilestillable to follow the

command

responsively.

We

shall investigate

what

effectfeedback has

on

the sensitivity to parameter variations.

Referring to thesysteminFig. 1-8,

we

consider

G

asa parameterthat

may

vary.

The

sensitivityofthe gain ofthe overallsystem

M

tothe variation in

G

is

defined as

™

_

dM/M

io

_~

~dGjG

^-

3>

where

dM

denotesthe incremental

change

in

M

due

to theincremental

change

inG;

dM/M

and

dG/G

denotethepercentage changein

M

and

G,respectively.

The

expressionofthesensitivityfunction

Sg

can bederived

by

usingEq.(1-1).

We

have

SM

_

dM

G

_

1

io

_~lGM~l+GH

(

M

>

This relation

shows

that thesensitivity function can be

made

arbitrarilysmall

by

increasing

GH,

providedthat the system remains stable. It is apparentthat in

an

open-loop system the gain ofthe system will respond in a one-to-one fashion to the variationin G.

Ingeneral,thesensitivityofthesystemgainofafeedback systemto

param-eter variations depends

on where

the parameter is located.

The

reader

may

derivethesensitivityofthesystemin Fig. 1-8

due

to the variationof

H.

Effectof

Feedback on

ExternalDisturbance or Noise

All physical controlsystemsare subject to

some

typesof extraneoussignals ornoise duringoperation.

Examples

ofthesesignals arethermalnoise voltage inelectronic amplifiers

and

brush or

commutator

noisein electricmotors.

The

effectof feedback

on

noisedependsgreatly

on where

the noiseis

intro-duced

intothe system;

no

general conclusionscan be

made. However,

in

many

situations,feedback can reducethe effectofnoise

on

system performance.

Let usreferto thesystem

shown

inFig. 1-10, in

which

rdenotes the

com-mand

signal

and

n is the noise signal. In the absence offeedback,

H

=

0, the outputcis

c

=

G

x

G

2e

+

G

2n (1-5)

where

e

=

r.

The

signal-to-noise ratio ofthe outputis defined as output

due

tosignal

_

G

x

G

2e

_

_c

e

output

due

tonoise

—

G

2n

~~ l

~n ' '

To

increase the signal-to-noise ratio, evidently

we

should either increase the

magnitude

of G, orerelativeton.Varyingthe

magnitude

of

G

2

would

have

no

effectwhatsoever

on

theratio.

10 / Introduction Chap. 1 + n \h Gi

G

2 r b + e + + e2 c + __

H

_. simultaneouslyis

Fig. 1-10. Feedbacksystem with anoisesignal.

_

Gl

G

2 r_| b£3 n (1-7) *

_

_T

₊

_G,G

2

H

+

1

+

G,G

₂

H

K '

Simply

comparing

Eq. (1-7) with Eq. (1-5)

shows

that the noise

component

in theoutput of Eq. (1-7)isreduced

by

the factor1

+

Gfi,H,

butthesignal

com-ponent

isalsoreduced

by

the

same amount.

The

signal-to-noiseratiois

output

due

to signal

_

G

i

G

2rj(\

+

G^G^H

) ___

g

r_ _(1-%}

output

due

to noise ~~

G

2n/(l

+

G

1

G

2

H)

1

n

and

isthe

same

as thatwithoutfeedback.Inthiscasefeedbackis

shown

tohave

no

direct effect

on

the output signal-to-noise ratio ofthe system in Fig. 1-10.

However,

the application of feedback suggests a possibility of improving the signal-to-noiseratio undercertainconditions. Let us

assume

thatinthe system ofFig. 1-10, ifthe

magnitude

of

G

t is increased to G\

and

that oftheinput r

to r', with all other parameters unchanged,the output

due

to the input signal actingaloneisatthe

same

levelasthat

when

feedbackisabsent.Inother words,

we

let

'1-™

=

^

(1'9)

With

the increased G,, G\, the output

due

to noise acting alone

becomes

which

issmallerthanthe output

due

to n

when

G

t is notincreased.

The

signal-to-noiseratiois

now

G

2nl{\

+

G\G

2

H)

-

n

^

+

°^^>

(1-11)

which

is greaterthan thatofthe system without feedback by a factorof(1

+

G\G

2H).

Seo-1-3

TypesofFeedbackControl Systems / 11 as bandwidth, impedance, transient response,

and

frequency response. These effectswill

become

known

as

one

progresses into theensuingmaterialofthistext.

1.3 TypesofFeedbackControl

Systems

Feedback

control systems

may

be classified in a

number

of ways, depending

upon

the purpose ofthe classification.

For

instance, according to the

method

ofanalysis

and

design,feedback controlsystemsareclassifiedaslinear

and

non-linear,time varyingortimeinvariant.

According

tothe typesofsignal

found

in the system, referenceisoften

made

tocontinuous-data

and

discrete-data systems, or

modulated and unmodulated

systems. Also, with reference to the type of system components,

we

often

come

across descriptionssuchaselectromechanical control systems, hydraulic control systems,

pneumatic

systems,

and

biological control systems. Control systems are oftenclassifiedaccordingtothe

main

pur-pose ofthe system.

A

positional controlsystem

and

a velocity control system control the outputvariablesaccording to the

way

the

names

imply. Ingeneral, there are

many

other

ways

ofidentifying control systems according to

some

specialfeaturesofthe system. Itisimportantthat

some

ofthese

more

common

ways

ofclassifying control systems are

known

so that proper perspective is

gainedbefore

embarking

on

the analysis

and

designofthese systems.

LinearVersus Nonlinear Control

Systems

Thisclassificationis

made

accordingtothe

methods

ofanalysis

and

design. Strictlyspeaking,linear systems

do

notexist in practice, since allphysical sys-temsare nonlinear to

some

extent.Linear feedbackcontrolsystemsare idealized

models

that are fabricated

by

the analyst purely for the simplicity ofanalysis

and

design.

When

themagnitudes ofthesignals ina controlsystem are limited to a range in

which

system

components

exhibitlinear characteristics (i.e., the principleofsuperpositionapplies),thesystemisessentially linear.

But

when

the magnitudes ofthesignalsareextendedoutside therange ofthelinearoperation, depending

upon

theseverityofthe nonlinearity, thesystemshould

no

longerbe consideredlinear.

For

instance,amplifiersusedincontrolsystemsoften exhibit saturation effect

when

their input signals

become

large; the magnetic field of a

motor

usually has saturation properties.

Other

common

nonlinear effects

found

in control systems are the backlash or

dead

play

between

coupledgear

members,

nonlinearcharacteristics in springs, nonlinearfrictionalforceor

tor-que between

moving members, and

so on. Quiteoften,nonlinearcharacteristics are intentionally introducedina control systemto

improve

itsperformance or provide

more

effectivecontrol.

For

instance,to achieve

minimum-time

control, an on-off (bang-bangorrelay)type ofcontrollerisused.Thistypeofcontrolis

found

in

many

missileorspacecraft control systems.

For

instance,inthe attitude controlofmissiles

and

spacecraft, jets are

mounted

on

thesides ofthe vehicle to provide reaction torquefor attitude control. Thesejets are often controlled inafull-onorfull-offfashion, so afixed

amount

ofairis applied

from

a given jetfor a certaintime durationtocontrol the attitudeofthespacevehicle.

12 / Introduction Chap.1

For

linear systems there exists a wealth ofanalytical

and

graphical tech-niques for design

and

analysis purposes.

However,

nonlinearsystems are very difficult totreatmathematically,

and

there are

no

general

methods

that

may

be usedto solvea

wide

class ofnonlinear systems.

Time-InvariantVersus Time-Varying

Systems

When

the parameters of a control system are stationary with respect to time duringtheoperation ofthe system,

we

have atime-invariant system.

Most

physical systems contain elements thatdrift or vary with time to

some

extent. Ifthe variation ofparameteris significantduring theperiod ofoperation, the systemis

termed

atime-varying system.

For

instance, the radius ofthe

unwind

reelofthe tension controlsystemin Fig. 1-7decreaseswith timeas the material

isbeingtransferred to the

windup

reel.

Although

atime-varyingsystem without

nonlinearityis stilla linear system,its analysisis usually

much

more

complex

than that ofthe linear time-invariant systems.

Continuous-Data Control

Systems

A

continuous-data systemis one in

which

the signals at various parts of the system are allfunctionsofthe continuous time variable t.

Among

all con-tinuous-datacontrol systems, thesignals

may

be further classified as ac or dc. Unlikethe generaldefinitionsofac

and

dcsignals usedin electricalengineering, ac

and

dc control systems carry special significances.

When

one refers to

an

accontrolsystemitusually

means

thatthesignalsin the system are

modulated

by

some

kind of

modulation

scheme.

On

the other hand,

when

a dc control system isreferred to, itdoes not

mean

that allthe signals inthe system areof the direct-current type;thenthere

would

be

no

control

movement.

A

dccontrol system simplyimplies that thesignalsareunmodulated, butthey are still ac

by

common

definition.

The

schematic

diagram

of aclosed-loop dc control system

is

shown

in Fig. 1-11. Typical

waveforms

ofthe system inresponse to a step

^^^

Error

6*^)

"r detector

Reference Controlled

input variable

6,

Sec.1.3

TypesofFeedbackControl Systems / 13

function input are

shown

inthefigure.Typical

components

ofa dccontrol

sys-tem

are potentiometers, dc amplifiers, dc motors,

and

dctachometers.

The

schematic

diagram

of atypicalaccontrolsystem is

shown

_{in Fig. 1-12.}

Inthis case thesignals inthesystemaremodulated; thatis, theinformationis transmitted

by an

accarrier signal. Notice that the output controlled variable

still behavessimilar to that ofthedc system ifthe

two

systems have the

same

control objective. In this case the

modulated

signals are

demodulated by

the low-pass characteristics of the control motor. Typical

components

of an ac control systemare synchros, acamplifiers, ac motors, gyroscopes,

and

acceler-ometers.

Inpractice,notallcontrolsystemsarestrictlyoftheac or the dc type.

A

system

may

incorporate a mixture of ac

and

dc components, usingmodulators

and

demodulatorsto

match

thesignals atvarious pointsofthe system.

Synchro transmitter Reference input 0. a-cservomotor

Fig. 1-12. Schematicdiagramofatypicalac closed-loop controlsystem.

Sampled-Data

and

DigitalControl

Systems

Sampled-data

and

digital control systems differ

from

the continuous-data systemsinthat the signals at

one

or

more

points ofthe systemareinthe

form

ofeithera pulsetrain or a digitalcode. Usually,sampled-data systemsreferto a

more

general class of systems

whose

signals are in the

form

ofpulse data,

where

adigitalcontrolsystemreferstothe useofadigital

computer

or controller in the system. In this text the term "discrete-data control system" is used to describebothtypesofsystems.

In general _{a sampled-data systemreceives}

data or information only inter-mittentlyat specificinstantsoftime.

For

instance, the errorsignal in acontrol system

may

besuppliedonlyintermittentlyinthe

form

ofpulses, in

which

case the control system receives

no

information about the error signalduringthe periods between

two

consecutive pulses. Figure 1-13 illustrates

how

a typical sampled-data systemoperates.

A

_{continuous input signalr(t)is}

14 / Introduction Chap.1 Input r(t) eg)

y

e*c> Sampler Data hold (filter) hit) Controlled process c(f)

Fig. 1-13. Block diagramofasampled-datacontrol system.

system.

The

errorsignale(t) is

sampled

by a sampling device, the sampler,

and

the output ofthe samplerisa sequence ofpulses.

The

samplingrateofthe

sam-pler

may

or

may

not be uniform.

There

are

many

advantages ofincorporating samplinginacontrol system,

one

ofthe

most

easilyunderstood ofthese being thatsampling provides time sharing of

an

expensive

equipment

among

several control channels.

Becausedigital

computers

provide

many

advantages in size

and

flexibility,

computer

control has

become

increasingly popular in recentyears.

Many

air-borne systems containdigitalcontrollers thatcan

pack

several

thousand

discrete elementsina space

no

largerthanthesize ofthisbook. Figure 1-14

shows

the basic elements of a digitalautopilot for a guidedmissile.

Digital Attitude of missile coded input Digital computer Digital-to-analog converter Airframe ,, Analog-to-con\erter

2

Mathematical

_Foundation

2.1 Introduction

The

study ofcontrolsystemsreliestoa great extent

on

theuseofapplied

mathe-matics.

For

the study ofclassical_{control theory, the prerequisites include such} subjects as

complex

variable theory, differentialequations, Laplace transform,

and

z-transform.

Modern

control theory,

on

the otherhand,requires consider-ably

more

intensive _{mathematical background. In addition to the}

above-men-tioned subjects,

modern

control theory is based

on

the _{foundation of matrix} theory, settheory,linearalgebra, variationalcalculus, various typesof

mathe-matical

programming, and

so on.

2.2 Complex-Variable

Concept

Complex-variable theoryplays

an

importantrole inthe analysis

and

design of control systems.

When

studying linearcontinuous-data systems, it is essential that

one

understandstheconcept of

complex

variable

and

functionsof a

complex

variable

when

the transfer function

method

is used.

Complex

Variable

A

complex

variable jis consideredtohave

two components:

areal

compo-nent a,

and an

imaginary

component

co. Graphically, the real

component

is

represented

by an

axisinthe horizontaldirection,

and

theimaginary

component

is

measured

along a vertical axis, in the

complex

j-plane. In other words, a

complex

variable is always defined

by

a point in a

complex

plane that has'

a

a

axis

and

aycoaxis. Figure 2-1 illustratesthe

complex

j-plane, in

which any

16 / Mathematical Foundation Chap.2 /co s-plane OJ] i i i i i i °\

Fig.2-1. Complexj-plane.

arbitrary point, s

=

s

_u

is

denned by

the coordinates

a

== a,

and

co

=

a>„

or simplySi

=

ffj +y'coi.

Functions ofa

Complex

Variable

The

function G(s)is said tobe afunction ofthe

complex

variables

iffor every value of s there is a corresponding value (or there are corresponding

values)ofG(s).Sincesisdefined tohavereal

and

imaginaryparts,the function

G(s)is also represented

by

its real

and

imaginary parts;

that is,

G(j)

=

ReG+yImC

(2-1)

where

Re

G

denotes the real part of G(s)

and

Im

G

represents the imaginary part of

G

Thus, the function G(s) can also berepresented

by

the

complex

G-plane

whose

horizontal axisrepresents

Re

G

and whose

vertical axismeasures theimaginary

component

of G{s). Iffor every value ofs (every pointinthe s-plane) thereis only

one

correspondingvalue for G(s) [one corresponding

point in the G^-plane], G(s)is said to be asingle-valued function,

and

the

mapping

(correspondence)

from

points in the j-plane onto points in the G(s)-plane is described as single valued (Fig. 2-2).

However,

there are

many

functions for

which

the

mapping

from

the function plane to the complex-variable plane

is

x-plane /co . /

ImG

S, =0, +/C0,

a, a

G

0)-plane

ReG

Gfri)

Sec

-2 -2

Complex-VariableConcept / 17

notsinglevalued.

For

instance, given the function

G(J)=

,-(7TT)

<2 "₂

) itis apparentthat for each value ofsthere is only

one

unique corresponding value for G(s).

However,

the reverse is not true; for instance, the point G(s)

=

oo is

mapped

onto

two

points,s

=

and

j

=

—

1,inthej-plane.

AnalyticFunction

A

function G(s) ofthe

complex

variables is calledan analytic function in

a region ofthe s-plane ifthefunction

and

allits derivatives exist in the region.

For

instance, the functiongivenin Eq. (2-2) isanalytic ateverypointinthe

s-plane except at the points s

=

and

s

=

-1.

At

these

two

points the value ofthe function isinfinite.

The

function G(s)

=

s

+

2is analyticatevery point inthefinite .s-plane.

Singularities

and

Poles ofaFunction

The

singularitiesofafunctionare the pointsinthe j-planeat

which

the func-tionoritsderivativesdoes notexist.

A

poleisthe

most

common

type of singu-larity

and

plays a very important role in the studies ofthe classical control theory.

The

definition of apole can be statedas: If afunction G(s)isanalytic

and

single valuedin the _{neighborhood of}s

t, except at s

t, it is saidto have apole _of

orderrat s

=

s,ifthe limit

lim

_[0

-

s,)rG(s)]

hasafinite,nonzerovalue. Inother words,the

denominator

ofG(s)

must

include the factor (s

—

s,)r, so

when

s

=

s„ the function

becomes

_infinite.

If r

=

1,

the poleat j

=

s, iscalleda simplepole.

As

an

example,the function

G(s)

=

l0(s

+

2)

n

xi

°

W

s(s

+

_IX*

+

3)* (2

"₃ >

has apoleof order2 at s

=

-3

and

simple poles at s

=

and

s

= -

1. Itcan also be said that the function is analytic in the j-plane except at these poles. Zeros ofaFunction

The

definition of azeroofafunctioncan bestated as:Ifthefunction G(s)

isanalytic at s

=

st ,itissaidtohave azerooforderrat s

=

slif the limit

!S

[(_*

~

J'>",<7

W]

(2-4)

hasafinite,nonzerovalue.

Or

simply, G(s)hasazero_oforderrat s

=

s,ifl/G(s) has anrth-orderpoleats

=

s,.

For

example,_{the functioninEq.(2-3)hasasimple}

zero ats

=

—2.

If the function under consideration is a rational function ofs, that is,

aquotient of

two

polynomials ofs, thetotal

number

ofpoles equals the total

number

ofzeros, counting_{the multiple-order poles}

_and

_{zeros, ifthe poles}

. , r. ^ • Chap.2

18 / Mathematical Foundation

zerosat infinity

and

atzero aretakeninto account.

The

functioninEq. (2-3)has fourfinite polesat s

=

0,

-1,

-3,

-3;

thereis

one

finite zeroats

=

-2,

but there are three zeros atinfinity, since

limGO)

=

lim^=0

(2"5) s-«. s-<*>S

Therefore, the function has atotal of fourpoles

and

four zerosinthe entire s-plane.

2.3 LaplaceTransform3-5

The

Laplace transformis

one

ofthe mathematical tools used for the solution

of ordinarylinear differentialequations.In

comparison

withtheclassical

method

ofsolvinglinear differential equations, theLaplace transform

method

hasthe following

two

attractive features

:

1.

The

homogeneous

equation

and

the particular integral are solved

in oneoperation.

2.

The

Laplace transform converts the differential equation into an algebraicequationins.Itisthenpossible tomanipulatethe algebraic

equation

by

simple algebraic rules to obtain the solution in the s domain.

The

finalsolutionis obtained

by

taking the inverseLaplace transform.

DefinitionoftheLaplace Transform

Given

the function /(f)

which

satisfiesthe condition

r\f(t)e-°'\dt<oo

(2-6)

J

for

some

finite real a,the Laplace transform of/(f)is defined as

F(s)=

\~_f(t)e-"dt (2-7)

or

m

=

£[/(')] (2-g)

The

variable sis referred to as the Laplace operator,

which

is a

complex

variable; thatis,s

=

a

+jco.

The

definingequation ofEq.(2-7) isalso

known

as the one-sidedLaplace transform,asthe integrationisevaluated

from

to oo

.

This simply

means

thatallinformation containedin/(f) prior tot

=

isignored

or consideredtobe zero.This assumption does notplaceanyserious limitation

on

the applications ofthe Laplace transform to linear system problems, since inthe usual

time-domain

studies, time reference is often chosen atthe instant

t

=

0. Furthermore, for a physical system

when

an

input is applied at t

=

0, the response ofthe system does not startsooner than t

=

0; that is, response

does not precede excitation.

The

followingexamplesserve asillustrations

on

how

Eq.(2-7)

may

be used for the evaluationoftheLaplace transform of a function /(f).

Sec. 2.3 LaplaceTransform / 19

Example

2-1

Let/0)

beaunit step function thatisdefined tohavea constant value

ofunity fort

>

and a zero valuefor/

<

0.Or,

/(/)

=

u.(t) (2-9)

Then

theLaplace transform off(t)is

F(s)

=

£[us(t)]

=

j~

us«)e-"dt

e~"

s (2-10)

Of

course,theLaplace transformgivenbyEq. (2-10)isvalidif

j |u£t)e-"|dt

=

r

|e-«|dt

<

co

whichmeansthattherealpartofs,a,mustbegreaterthanzero.However,in practice,

we

simplyreferto theLaplace transform ofthe unit stepfunctionaslis,andrarelydo

we

have tobe concerned aboutthe regioninwhichthe transformintegral converges absolutely.

Example

2-2 Considertheexponential function /(,)

=

e-", t2; wherea isaconstant.

The

Laplace transform of/(?)is written

=/:

F(s)

=

e--'e-" dt

s

+

a s

+

a (2-11) InverseLaplace Transformation

The

operationofobtaining /(?)

from

theLaplace transformF(s)istermed

the inverse Laplace transformation.

The

inverse Laplace transform of F(s) is

denoted

by

f(t)

=

Z-Vis)]

(2-12)

and

is given

by

the inverse Laplace transformintegral

/(0

=

2^7

/

'

me"ds

_(2-13)

where

cisarealconstantthatisgreaterthantherealpartsofallthesingularities of F(s). Equation (2-13) represents a line integral that is to be evaluated in thej-plane.

However,

for

most

engineeringpurposesthe inverseLaplace

trans-form

operation can be accomplished simply

by

referringto the Laplace

trans-form

table, suchas the one givenin

Appendix

B.

Important

Theorems

oftheLaplace Transform

The

applicationsoftheLaplace transformin

many

instances aresimplified

by

the utilization of the properties of the transform. These properties are presented in the following in the

form

of theorems,

and no

proofs are given.

20 / Mathematical Foundation Chap.2

1

.

MultiplicationbyaConstant

The Laplacetransformoftheproduct of aconstant

k

and

atime func-tionf{t)istheconstant

k

multipliedbytheLaplacetransformoff{t); thatis,

£[kf(t)]

=

kF(s) (2-14)

where

F{s)isthe Laplace transform off{t).

2.

Sum

andDifference

The

Laplacetransformofthe

sum

{or difference) of twotime functions isthe

sum

(or difference) oftheLaplacetransforms ofthe time func-tions; thatis,

£[fi(0

±

UO]

=

F,(s)

±

F

2(s) (2-1 5)

where

F

t{s)

and

F

2{s) are the Laplace transforms of fit)

and/

2(r),

respectively.

3. Differentiation

The

Laplace transform ofthefirstderivative ofa time function f(t)

iss times theLaplace transform off(f) minus the limit off(t) as t

approaches0-\-;thatis,

df(ty dt

=

sF(s)

-

lim /(/)

=

sF(s)

-

/(0+)

(2-16)

Ingeneral,for higher-orderderivatives,

[d'fW] _

L

df

J =_s"F(s) -- lim (-0 + s»-if(t)

+

s»~

ld

l^-\-

. snF(s) --

_^r

1

_/(o+)

-

5""2

/

<u

(o+)

- ..-/

(

_"-"(0+)

(2-17) 4. Integration

The

Laplace transform ofthefirst integral of afunction fit) with respect to timeis theLaplace transform off(t) dividedbys; thatis,

£

S\

f(j)dr

F{s) s

Ingeneral, for «th-orderintegration,

rr...r.

J o J o

Jo

£

... fir)dx

A,

...dtn (2-18)

^

(2-19) 5. Shift inTime

The

Laplacetransformoff{t)delayedbytime

T

isequalto theLaplace transformoff{t) multipliedbye~Tl;thatis,

£[fit

-

T)u

s{t

-

T)]

=

e-T

*F{s) (2-20)

where

us{t

—

T) denotes the unit step function,

which

isshiftedin

Sec-2-4 _{Inverse Laplace}

Transform byPartial-FractionExpansion / 21

6. Initial-ValueTheorem

IftheLaplacetransform offit) isF(s), then

lim f(t)

=

lim sF(s) (2-21) ifthetimelimit exists.

7. Final-ValueTheorem

IftheLaplace transform offit)is

F(s)and

ifsF(s)isanalytic onthe imaginaryaxis

and

inthe righthalfofthe s-plane, then

lim f(t)

=

lim sFis) (2-22)

The

final-value

theorem

is a veryuseful relationinthe analysis

and

design of feedbackcontrol systems, sinceitgives the finalvalue of a time function

by

determiningthebehavior ofits Laplace transform asstends tozero.

However,

thefinal-value

theorem

is notvalidifsFis)contains

any

poles

whose

realpart is zero or positive,

which

is equivalent to the analytic requirement of sFis) stated in thetheorem.

The

followingexamplesillustratethe care that

one

must

takein applyingthefinal-value theorem.

Example

2-3 Considerthefunction F(s)

s(s2

+s+2)

SincesFis) isanalytic ontheimaginary axis andin the right halfofthe.s-plane, the final-valuetheorem

may

beapplied. Therefore,using Eq. (2-22),

limfit)

=

limsFis)

=

lim , ,

5

, „

=

-1 f2-231

Example

2-4 Considerthefunction

F

_&

=

_J2"^p

(2-24) whichis

known

tobetheLaplace transform of/(f)

=

sincot. SincethefunctionsFis) hastwopoles

on

theimaginaryaxis,the final-valuetheoremcannotbeapplied inthis case.In other words, althoughthe final-valuetheorem would yielda value of zeroas thefinalvalue offit),theresultiserroneous.

2.4 Inverse LaplaceTransform byPartial-FractionExpansion 71'

In a great majority ofthe

problems

in control systems, the evaluation ofthe inverse_{Laplace transform does not necessitatethe useof}

the inversionintegral of Eq.(2-13).

The

inverseLaplace transform operationinvolving rational func-tions can be carried out using a Laplace transform table

and

partial-fraction expansion.

When

theLaplace transformsolutionof adifferentialequationisarational functionins, itcan bewritten

X

_^

₌

22 / Mathematical Foundation Chap-2

where

P(s)

and

Q(s) arepolynomials ofs.It is

assumed

thatthe orderof Q(s)

in sisgreaterthanthatofP(s).

The

polynomial Q(s)

may

bewritten

Q(s)

=

sn

+

a^"'

1

+

. ..

+

an-ts

+

a„ (2-26)

where

a,, . . . , a

n arereal coefficients.

The

zeros of Q(s) are either real or in

complex-conjugate pairs, insimple or multiple order.

The

methods

of partial-fraction expansion will

now

be given for the cases ofsimple poles, multiple-order poles,

and complex

poles,ofX(s).

Partial-FractionExpansion

When

AllthePoles ofX(s)

Are

Simple

and

Real Ifallthe polesofX(s) are real

and

simple, Eq. (2-25)can bewritten

X(

S)

=

M

P

^

,

, (2-27)

where

the poles

—

st ,

—

s2, . . . ,

—

s„ are consideredto be real

numbers

in the

present case.

Applying

the partial-fraction expansion technique, Eq. (2-27) is

written

X(

S)

=

-^-

+

-4*-

+

• •

+

T^V

<2 -₂₈ ) v ' _s

+

_{i, s}

+

_s 2 s -1-- s„

The

coefficient,

K„

(i

=

1,2, . . .,n), is determined

by

multiplying bothsides

of Eq. (2-28) or (2-27)

by

the factor(s

+

st)

and

then setting s equal to —s,.

To

find thecoefficient

K

sl,for instance,

we

multiplyboth sidesofEq. (2-27)

by

(j

+

Ji)

and

lets

= —

j,; thatis,

K

s\ (S

+

s

)?&

Pis,

(s2

—

st)(s3

—

Si)... (s„

—

Si)

As

an

illustrative example, considerthe function

(2-29)

(s

+

l)(,s

+

2)(s

+

3)

which

iswritteninthe partial-fractioned

form

The

coefficients

K_

_u

K_

2,

and

K_

3 aredetermined as follows:

*-.

=

[('

+

W*)],-i

=

(

2-~lX3

+

-

3l)

*-2

=

[(.s

+

2)X(s)}s

^

2

=

5(

~

2)

+

3 A"-3

=

[(*

+

3)*(j)],-,

Therefore,Eq.(2-31)

becomes

W)

=

_tti

+

^

-

t^

(2 -₃₅ >

(1-

2X3

-

2) ._ 5(--3)

+

3

(1-

3X2

-

3) 7 6 s

+

2 j

+

3 :

_i

_(2-32)

=

7 (2-33) ,-=

-6

(2-34)