Control Systems: an Introduction

469 Pages·2023·16.474 MB·English

by  Hassan K. Khalil

Books International Lithocase: 8 by 10 cover with 0.922 spine for 468 pages on 620ppi paper
HASSAN K.
CONTROL SYSTEMS: AN INTRODUCTION
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KHALIL
Professor Hassan K. Khalil is Distinguished Universi- C
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ty Professor Emeritus of Electrical and Computer
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Engineering at Michigan State University. He joined
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MSU in 1978, shortly after completing a Ph.D.
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Illinois, which was preceded by B.S. and M.S. R
Page is 19.05 x 12.00 in.
degrees, also in electrical engineering, from Cairo
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University in Egypt. Professor Khalil has consulted
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for General Motors and Delco Products and has

published over 120 journal papers on singular S
perturbation methods and nonlinear control. He is Y
also the author of several highly cited books, includ-
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ing High-Gain Observers in Nonlinear Feedback Con- T
trol (SIAM 2017), Nonlinear Control (Pearson 2015),
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and Nonlinear Systems (Macmillan 1992; Prentice M
Hall 1996 & 2002), and a coauthor of Singular
Perturbation Methods in Control: Analysis and Design S
(Academic Press 1986; SIAM 1999). :

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Professor Khalil was named IEEE Fellow in 1989 and IFAC Fellow in 2007, and has received
numerous international honors and awards, including the 1989 IEEE-CSS George S. Axelby
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+
Outstanding Paper Award, the 2000 AACC Ragazzini Education Award, the 2002 IFAC N
∑
Control Engineering Textbook Prize, the 2004 AACC O. Hugo Schuck Best Paper Award, T
and the 2015 IEEE-CSS Bode Lecture Prize. At MSU he received the 1995 Distinguished R
Faculty Award and the 2020 Withrow Teaching Excellence Award. He has served as Associ- O
ate Editor of the IEEE Transactions on Automatic Control, Automatica, and Neural Networks,
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and as Editor of Automatica for nonlinear systems and control. He was Program Chair of
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the 1988 ACC, and General Chair of the 1994 ACC.
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ISBN 978-1-60785-826-3
9 781607 858263
HASSAN K. KHALIL
CONTROL SYSTEMS:
AN INTRODUCTION
Hassan K. Khalil
Michigan State University
Copyright © 202
This book is published by Michigan Publishing under an agree ment with the author. It is
made available free of charge in electronic form to any student or instructor interested in the
subject matter.
Published intheUnitedStatesofAmericaby Michigan Publishing.
Manufactured intheUnitedStatesofAmerica.
ISBN 978-1-60785-826-3 (print)
ISBN 978-1-60785-827-0 (OA)
The Free ECE Textbook Initiative is sponsored by the ECE
Department at the University of Michigan.

iv
Preface
I wrote this book in response to the initiative established by Professor FawwazUlaby of the
UniversityofMichigantohavetextbooksavailabletostudentsasfreePDFfilesorreasonably
priced printed versions. The book is based on my experience teaching control courses at
MichiganStateUniversityfor42years.Iamgratefultomanypeople.Istartwithmystudents
and colleagues at Michigan State University who played an important role in shaping the
contentofthebook.ProfessorVaibhavSrivastavadidmeagreatfavorbypreservingmyECE
313classnotes,whichIdidn’tkeepatthetimeofmyretirement.
I am grateful to Professor Ulaby, not only for his initiative but also for providing help
throughout the project. I thank Professor Necmiye Ozay of the University of Michigan for
reviewing the book and providing feedback, Andrew Yagle for his language editing, and
RichardCarnesforhisbookcompositing anddesign.
ThebookwastypesetusingLATEX.ComputationsweredoneusingMATLAB.Thefigures
weregenerated usingMATLAB,thegraphics toolofLATEX,orTikZ.
HASSAN KHALIL
v
Contents
Preface iv
1 Introduction 1
2 TransferFunctionModels 9
2-1 LinearSystems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2-2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2-2.1 ElectricCircuits . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2-2.2 Mechanical Systems . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2-2.3 Electromechanical Systems . . . . . . . . . . . . . . . . . . . . . . 37
2-3 BlockDiagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
2-4 Linearization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
3 StabilityandDynamicResponse 64
3-1 BIBOStability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
3-2 Routh-Hurwitz Criterion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
3-3 DynamicResponse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
3-4 StepResponse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
3-4.1 First-OrderTransferFunctions . . . . . . . . . . . . . . . . . . . . . 81
3-4.2 Second-Order TransferFunctions . . . . . . . . . . . . . . . . . . . 83
3-4.3 EffectofZeros . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
3-4.4 Approximation ofHigher-Order TransferFunctions . . . . . . . . . . 93
4 FeedbackControlSystems 107
4-1 Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
4-2 TheRootLocusMethod . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
4-2.1 RulesforConstructing theRootLocus . . . . . . . . . . . . . . . . . 111
4-2.2 RefinementoftheRootLocus . . . . . . . . . . . . . . . . . . . . . 122
4-3 Steady-State Error . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
4-3.1 UnityFeedbackSystem . . . . . . . . . . . . . . . . . . . . . . . . 127
4-3.2 Non-UnityFeedbackSystem . . . . . . . . . . . . . . . . . . . . . . 131
4-4 EffectofDisturbance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
4-5 PlantParameterUncertainty . . . . . . . . . . . . . . . . . . . . . . . . . . 136
4-6 EffectofMeasurement Noise . . . . . . . . . . . . . . . . . . . . . . . . . . 139
vi
4-7 ControlConstraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
4-8 WhyUseClosed-LoopInsteadofOpen-Loop? . . . . . . . . . . . . . . . . 142
5 Time-DomainDesign 154
5-1 DesignSpecifications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
5-2 DesignbyGainAdjustment. . . . . . . . . . . . . . . . . . . . . . . . . . . 156
5-3 DesignbyCascadeCompensation . . . . . . . . . . . . . . . . . . . . . . . 163
5-4 LeadandPDController . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165
5-4.1 LeadCompensator . . . . . . . . . . . . . . . . . . . . . . . . . . . 165
5-4.2 Proportional Derivative(PD)Controller . . . . . . . . . . . . . . . . 173
5-5 LagandPIController . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179
5-5.1 LagCompensator . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179
5-5.2 Proportional-Integral (PI)Controller . . . . . . . . . . . . . . . . . . 181
5-6 Lead-LagandPIDController . . . . . . . . . . . . . . . . . . . . . . . . . . 186
5-6.1 Lead-LagCompensator . . . . . . . . . . . . . . . . . . . . . . . . . 187
5-6.2 Proportional-Integral-Derivative (PID)Controller . . . . . . . . . . . 190
5-7 TuningPIDControllers andAnti-Windup Schemes . . . . . . . . . . . . . . 194
6 FrequencyResponseMethods 210
6-1 PolarPlot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211
6-2 BodePlots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217
6-2.1 AsymptoticBodePlots . . . . . . . . . . . . . . . . . . . . . . . . . 220
6-2.2 Underdamped Second-Order TransferFunctions . . . . . . . . . . . 228
6-2.3 ErrorConstantsfromBodePlots . . . . . . . . . . . . . . . . . . . . 231
6-3 Time-DelaySystems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233
6-4 NyquistCriterion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234
6-4.1 ContourMapping . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234
6-4.2 Cauchy’sPrincipleoftheArgument . . . . . . . . . . . . . . . . . . 235
6-4.3 StabilityofFeedbackSystems . . . . . . . . . . . . . . . . . . . . . 238
6-4.4 Application oftheNyquistCriterionwithGainAdjustment . . . . . . 240
6-4.5 TheCaseWhenGH(s)HasPolesontheImaginary Axis . . . . . . . 243
6-5 RelativeStability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249
6-5.1 StabilityMargins . . . . . . . . . . . . . . . . . . . . . . . . . . . . 250
6-5.2 StabilityMarginsfromBodePlots . . . . . . . . . . . . . . . . . . . 251
6-6 Relations betweenFrequency-Domain andTime-DomainResponses . . . . . 256
7 Frequency-DomainDesign 277
7-1 DesignSpecifications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 278
7-2 DesignbyGainAdjustment. . . . . . . . . . . . . . . . . . . . . . . . . . . 280
7-2.1 GainAdjustmenttoAchieveaDesiredCrossoverFrequency . . . . . 280
7-2.2 GainAdjustmenttoAchieveaDesiredPhaseMargin . . . . . . . . . 281
7-2.3 GainAdjustmenttoAchieveaDesiredK ,K ,orK . . . . . . . . . 282
p υ a
7-2.4 GainAdjustmenttoShapetheLowFrequencyGain . . . . . . . . . 284
7-3 LeadCompensation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287
vii
7-3.1 Design of a Lead Compensator to Increase the Phase Margin at a
DesiredCrossoverFrequency . . . . . . . . . . . . . . . . . . . . . 290
7-3.2 DesignofaLeadCompensator toIncreasethePhaseMarginwithout
Changing theLow-FrequencyGain . . . . . . . . . . . . . . . . . . 292
7-4 LagCompensation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295
7-4.1 Design of a Lag Compensator to Increase the Low-Frequency Gain
withMinimalEffectonthePhaseMarginoftheUncompensatedSystem297
7-5 PICompensation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 299
7-5.1 DesignofaPIControllertoIncreasetheTypeoftheFeedbackSystem
andAchieveaDesiredPhaseMargin . . . . . . . . . . . . . . . . . 299
7-6 Lead-LagCompensation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302
7-7 Advantages oftheFrequency-Domain Approach . . . . . . . . . . . . . . . 306
8 State-SpaceMethods 317
8-1 State-SpaceModels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 318
8-1.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 321
8-1.2 Linearization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 324
8-2 SolutionoftheStateEquation . . . . . . . . . . . . . . . . . . . . . . . . . 327
8-3 Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 330
8-4 Controllability andObservability . . . . . . . . . . . . . . . . . . . . . . . . 333
8-4.1 Controllability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333
8-4.2 Observability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 338
8-4.3 Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 340
8-5 TransferFunctionofaState-SpaceModel . . . . . . . . . . . . . . . . . . . 340
8-6 State-SpaceModelofaTransferFunction . . . . . . . . . . . . . . . . . . . 344
8-6.1 Controllable CanonicalForm . . . . . . . . . . . . . . . . . . . . . . 344
8-6.2 Observable CanonicalForm . . . . . . . . . . . . . . . . . . . . . . 347
8-6.3 ParallelRealization . . . . . . . . . . . . . . . . . . . . . . . . . . . 349
8-7 State-SpaceDesign . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 352
8-7.1 StateFeedbackControl . . . . . . . . . . . . . . . . . . . . . . . . . 353
8-7.2 ObserversandOutputFeedbackControl . . . . . . . . . . . . . . . . 358
8-7.3 Set-PointRegulation . . . . . . . . . . . . . . . . . . . . . . . . . . 361
9 DigitalControl 377
9-1 DigitalControlSystems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 378
9-2 Discrete-TimeSystems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 381
9-3 Discrete-TimeEquivalentofDigitalControlSystem . . . . . . . . . . . . . . 387
9-4 StabilityofDigitalControlSystems . . . . . . . . . . . . . . . . . . . . . . 393
9-4.1 Calculation oftheClosed-LoopPoles . . . . . . . . . . . . . . . . . 394
9-4.2 RootLocusMethod . . . . . . . . . . . . . . . . . . . . . . . . . . 394
9-4.3 Routh-Hurwitz Criterion . . . . . . . . . . . . . . . . . . . . . . . . 397
9-4.4 NyquistCriterion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 398
9-5 Steady-State Error . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 403
9-6 Sample-Frequency Selection . . . . . . . . . . . . . . . . . . . . . . . . . . 405
9-6.1 SamplingTheory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 405
viii
9-6.2 Antialiasing Filter(Prefilter) . . . . . . . . . . . . . . . . . . . . . . 407
9-6.3 MethodsforChoosingω . . . . . . . . . . . . . . . . . . . . . . . 408
s
9-7 DesignbyEmulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 411
9-8 Discrete-TimeDesign . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 416
9-8.1 RootLocusDesign . . . . . . . . . . . . . . . . . . . . . . . . . . . 416
9-8.2 Frequency-Domain Design . . . . . . . . . . . . . . . . . . . . . . . 419
9-9 Quantization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 422
A ReviewofLaplaceTransform 438
B ElementsofMatrixAnalysis 442
C Laplaceandz-Transform Tables 444
D MATLABTutorial 446
D-1 TransferFunctionDefinitionandManipulation . . . . . . . . . . . . . . . . 446
D-2 RootLocusAnalysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 447
D-3 ErrorConstants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 447
D-4 Frequency Response . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 448
D-5 State-SpaceCalculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 448
D-6 DigitalControl . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 449
E Symbols 451
Bibliography 453
Index 456
1
Chapter
Introduction
1