Multivariable Feedback Control - Analysis and Design

585 Pages·2003·3.94 MB·English

by  Sigurd Skogestad

MULTIVARIABLE
FEEDBACK CONTROL
Analysis and design
SNoirgwuegriadnUSnikveorsgityeosftSacidenceandTechnology
IUaninversPityoosftLleeicteshtewr aite
SecondEdition
Thisversion:August29,2001
JOHN WILEY & SONS
. . . .
Chichester New York Brisbane Toronto Singapore
ii MULTIVARIABLEFEEDBACK CONTROL
BORGHEIM,anengineer:
Herregud, en kan da ikke gjøre noe bedre enn leke i denne
velsignedeverden.Jegsyneshelelivetersomenlek,jeg!
Goodheavens,onecan’tdoanythingbetterthanplayinthis
blessedworld.Thewholeoflifeseemslikeplayingtome!
Act one, LITTLE EYOLF, Henrik Ibsen.
iv MULTIVARIABLEFEEDBACK CONTROL
CONTENTS
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii
CONTENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v
PREFACE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix
1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Theprocessofcontrolsystemdesign . . . . . . . . . . . . . . . . . 1
1.2 Thecontrolproblem . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.3 Transferfunctions . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.4 Scaling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.5 Derivinglinearmodels . . . . . . . . . . . . . . . . . . . . . . . . 8
1.6 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2 CLASSICALFEEDBACKCONTROL . . . . . . . . . . . . . . . . . 15
2.1 Frequencyresponse . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.2 Feedbackcontrol . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.3 Closed-loopstability . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.4 Evaluatingclosed-loopperformance . . . . . . . . . . . . . . . . . 27
2.5 Controllerdesign . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
2.6 Loopshaping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
2.7 Shapingclosed-looptransferfunctions . . . . . . . . . . . . . . . . 55
2.8 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
3 INTRODUCTIONTOMULTIVARIABLECONTROL . . . . . . . . 63
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
3.2 TransferfunctionsforMIMOsystems . . . . . . . . . . . . . . . . 64
3.3 Multivariablefrequencyresponseanalysis . . . . . . . . . . . . . . 68
3.4 Controlofmultivariableplants . . . . . . . . . . . . . . . . . . . . 79
3.5 IntroductiontomultivariableRHP-zeros . . . . . . . . . . . . . . . 84
3.6 ConditionnumberandRGA . . . . . . . . . . . . . . . . . . . . . 87
vi MULTIVARIABLEFEEDBACK CONTROL
3.7 IntroductiontoMIMOrobustness . . . . . . . . . . . . . . . . . . 91
3.8 Generalcontrolproblemformulation . . . . . . . . . . . . . . . . . 98
3.9 Additionalexercises. . . . . . . . . . . . . . . . . . . . . . . . . . 110
3.10 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
4 ELEMENTSOFLINEARSYSTEMTHEORY . . . . . . . . . . . . . 113
4.1 Systemdescriptions . . . . . . . . . . . . . . . . . . . . . . . . . . 113
4.2 Statecontrollabilityandstateobservability. . . . . . . . . . . . . . 122
4.3 Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
4.4 Poles. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
4.5 Zeros . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
4.6 Someremarksonpolesandzeros. . . . . . . . . . . . . . . . . . . 135
4.7 Internalstabilityoffeedbacksystems. . . . . . . . . . . . . . . . . 139
4.8 Stabilizingcontrollers . . . . . . . . . . . . . . . . . . . . . . . . . 143
4.9 Stabilityanalysisinthefrequencydomain . . . . . . . . . . . . . . 145
4.10 Systemnorms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152
4.11 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158
5 LIMITATIONSONPERFORMANCEINSISOSYSTEMS . . . . . . 159
5.1 Input-OutputControllability . . . . . . . . . . . . . . . . . . . . . 159
5.2 Perfectcontrolandplantinversion . . . . . . . . . . . . . . . . . . 163
5.3 Constraintson and . . . . . . . . . . . . . . . . . . . . . . . . 164
5.4 IdealISEoptimSalconTtrol . . . . . . . . . . . . . . . . . . . . . . . 171
5.5 Limitationsimposedbytimedelays . . . . . . . . . . . . . . . . . 172
5.6 LimitationsimposedbyRHP-zeros. . . . . . . . . . . . . . . . . . 173
5.7 RHP-zerosamdnon-causalcontrollers . . . . . . . . . . . . . . . . 180
5.8 Limitationsimposedbyunstable(RHP)poles . . . . . . . . . . . . 182
5.9 Combinedunstable(RHP)polesandzeros . . . . . . . . . . . . . . 185
5.10 Performancerequirementsimposedbydisturbancesandcommands 187
5.11 Limitationsimposedbyinputconstraints. . . . . . . . . . . . . . . 189
5.12 Limitationsimposedbyphaselag . . . . . . . . . . . . . . . . . . 193
5.13 Limitationsimposedbyuncertainty . . . . . . . . . . . . . . . . . 194
5.14 Summary:Controllabilityanalysiswithfeedbackcontrol . . . . . . 196
5.15 Summary:Controllabilityanalysiswithfeedforwardcontrol . . . . 199
5.16 Applicationsofcontrollabilityanalysis . . . . . . . . . . . . . . . . 201
5.17 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210
6 LIMITATIONSONPERFORMANCEINMIMOSYSTEMS . . . . . 213
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213
6.2 Constraintson and . . . . . . . . . . . . . . . . . . . . . . . . 214
6.3 FunctionalcontSrollabiTlity . . . . . . . . . . . . . . . . . . . . . . . 218
6.4 Limitationsimposedbytimedelays . . . . . . . . . . . . . . . . . 219
6.5 LimitationsimposedbyRHP-zeros. . . . . . . . . . . . . . . . . . 220
CONTENTS vii
6.6 Limitationsimposedbyunstable(RHP)poles . . . . . . . . . . . . 223
6.7 RHP-polescombinedwithRHP-zeros . . . . . . . . . . . . . . . . 224
6.8 Performancerequirementsimposedbydisturbances . . . . . . . . . 226
6.9 Limitationsimposedbyinputconstraints. . . . . . . . . . . . . . . 228
6.10 Limitationsimposedbyuncertainty . . . . . . . . . . . . . . . . . 234
6.11 MIMOInput-outputcontrollability . . . . . . . . . . . . . . . . . . 246
6.12 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251
7 UNCERTAINTYANDROBUSTNESSFORSISOSYSTEMS . . . . 253
7.1 Introductiontorobustness. . . . . . . . . . . . . . . . . . . . . . . 253
7.2 Representinguncertainty . . . . . . . . . . . . . . . . . . . . . . . 255
7.3 Parametricuncertainty . . . . . . . . . . . . . . . . . . . . . . . . 258
7.4 Representinguncertaintyinthefrequencydomain . . . . . . . . . . 259
7.5 SISORobuststability . . . . . . . . . . . . . . . . . . . . . . . . . 270
7.6 SISORobustperformance . . . . . . . . . . . . . . . . . . . . . . 277
7.7 Examplesofparametricuncertainty . . . . . . . . . . . . . . . . . 284
7.8 Additionalexercises. . . . . . . . . . . . . . . . . . . . . . . . . . 289
7.9 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291
8 ROBUSTSTABILITYANDPERFORMANCEANALYSIS . . . . . . 293
8.1 Generalcontrolconfigurationwithuncertainty . . . . . . . . . . . . 293
8.2 Representinguncertainty . . . . . . . . . . . . . . . . . . . . . . . 296
8.3 Obtaining , and . . . . . . . . . . . . . . . . . . . . . . . . 303
8.4 DefinitionsPofNrobustMstabilityandrobustperformance . . . . . . . . 305
8.5 Robuststabilityofthe -structure . . . . . . . . . . . . . . . . . 307
8.6 RSforcomplexunstrucMtu(cid:1)reduncertainty . . . . . . . . . . . . . . . 309
8.7 RSwithstructureduncertainty:Motivation. . . . . . . . . . . . . . 312
8.8 Thestructuredsingularvalue . . . . . . . . . . . . . . . . . . . . . 314
8.9 Robuststabilitywithstructureduncertainty . . . . . . . . . . . . . 322
8.10 Robustperformance. . . . . . . . . . . . . . . . . . . . . . . . . . 326
8.11 Application:RPwithinputuncertainty . . . . . . . . . . . . . . . . 330
8.12 -synthesisand -iteration . . . . . . . . . . . . . . . . . . . . 339
8.13 F(cid:22)urtherremarksDonK . . . . . . . . . . . . . . . . . . . . . . . . . 348
8.14 Conclusion . . . .(cid:22). . . . . . . . . . . . . . . . . . . . . . . . . . 351
9 CONTROLLERDESIGN . . . . . . . . . . . . . . . . . . . . . . . . . 355
9.1 Trade-offsinMIMOfeedbackdesign . . . . . . . . . . . . . . . . 355
9.2 LQGcontrol. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 359
9.3 and control . . . . . . . . . . . . . . . . . . . . . . . . . . 368
9.4 2 loop-shapingdesign . . . . . . . . . . . . . . . . . . . . . . . 382
H H1
9.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 403
H1
10 CONTROLSTRUCTUREDESIGN . . . . . . . . . . . . . . . . . . . 405
viii MULTIVARIABLEFEEDBACK CONTROL
10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 405
10.2 Optimizationandcontrol . . . . . . . . . . . . . . . . . . . . . . . 407
10.3 Selectionofcontrolledoutputs . . . . . . . . . . . . . . . . . . . . 410
10.4 Selectionofmanipulationsandmeasurements . . . . . . . . . . . . 416
10.5 RGAfornon-squareplant . . . . . . . . . . . . . . . . . . . . . . 418
10.6 Controlconfigurationelements . . . . . . . . . . . . . . . . . . . . 420
10.7 Hierarchicalandpartialcontrol . . . . . . . . . . . . . . . . . . . . 429
10.8 Decentralizedfeedbackcontrol . . . . . . . . . . . . . . . . . . . . 441
10.9 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 458
11 MODELREDUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . 459
11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 459
11.2 Truncationandresidualization . . . . . . . . . . . . . . . . . . . . 460
11.3 Balancedrealizations . . . . . . . . . . . . . . . . . . . . . . . . . 462
11.4 Balancedtruncationandbalancedresidualization . . . . . . . . . . 463
11.5 OptimalHankelnormapproximation. . . . . . . . . . . . . . . . . 464
11.6 Twopracticalexamples . . . . . . . . . . . . . . . . . . . . . . . . 467
11.7 Reductionofunstablemodels. . . . . . . . . . . . . . . . . . . . . 476
11.8 ModelreductionusingMATLAB. . . . . . . . . . . . . . . . . . . 477
11.9 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 478
12 CASESTUDIES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 479
12.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 479
12.2 Helicoptercontrol . . . . . . . . . . . . . . . . . . . . . . . . . . . 480
12.3 Aero-enginecontrol . . . . . . . . . . . . . . . . . . . . . . . . . . 490
12.4 Distillationprocess . . . . . . . . . . . . . . . . . . . . . . . . . . 500
12.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 506
A MATRIXTHEORYANDNORMS . . . . . . . . . . . . . . . . . . . . 509
A.1 Basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 509
A.2 Eigenvaluesandeigenvectors . . . . . . . . . . . . . . . . . . . . . 512
A.3 SingularValueDecomposition . . . . . . . . . . . . . . . . . . . . 515
A.4 RelativeGainArray . . . . . . . . . . . . . . . . . . . . . . . . . . 522
A.5 Norms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 526
A.6 Factorizationofthesensitivityfunction . . . . . . . . . . . . . . . 539
A.7 Linearfractionaltransformations . . . . . . . . . . . . . . . . . . . 541
B PROJECTWORKandSAMPLEEXAM . . . . . . . . . . . . . . . . 545
B.1 Projectwork . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 545
B.2 Sampleexam . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 546
BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 551
INDEX . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 561
PREFACE
This is a book on practical feedback control and not on system theory generally.
Feedbackisusedincontrolsystemstochangethedynamicsofthesystem(usually
tomaketheresponsestableandsufficientlyfast),andtoreducethesensitivityofthe
systemtosignaluncertainty(disturbances)andmodeluncertainty.Importanttopics
coveredinthebook,include
classicalfrequency-domainmethods
analysis of directions in multivariable systems using the singular value
(cid:15)
decomposition
(cid:15)
input-outputcontrollability(inherentcontrollimitationsintheplant)
modeluncertaintyandrobustness
(cid:15)
performancerequirements
(cid:15)
methodsforcontrollerdesignandmodelreduction
(cid:15)
controlstructureselectionanddecentralizedcontrol
(cid:15)
(cid:15)
Thetreatmentisforlinearsystems.Thetheoryisthenmuchsimplerandmorewell
developed, and a large amount of practical experience tells us that in many cases
linear controllers designed using linear methods provide satisfactory performance
whenappliedtorealnonlinearplants.
Wehaveattemptedtokeepthemathematicsatareasonablysimplelevel,andwe
emphasizeresultsthatenhanceinsightandintuition.Thedesignmethodscurrently
available for linear systems are well developed, and with associated software it
is relatively straightforward to design controllers for most multivariable plants.
However,withoutinsightandintuitionitisdifficulttojudgeasolution,andtoknow
howtoproceed(e.g.howtochangeweights)inordertoimproveadesign.
The book is appropriate for use as a text for an introductory graduate course
in multivariable control or for an advanced undergraduate course. We also think
it will be useful for engineers who want to understand multivariable control, its
limitations, and how it can be applied in practice. There are numerous worked
examples,exercisesandcasestudieswhichmakefrequentuseofMATLABTM1.
MATLABisaregisteredtrademarkofTheMathWorks,Inc.
1
x MULTIVARIABLEFEEDBACK CONTROL
The prerequisites for reading this book are an introductory course in classical
single-input single-output (SISO) control and some elementary knowledge of
matricesandlinearalgebra.Partsofthebookcanbestudiedalone,andprovidean
appropriatebackgroundforanumberoflinearcontrolcoursesatbothundergraduate
andgraduatelevels:classicalloop-shapingcontrol,anintroductiontomultivariable
control, advanced multivariable control, robust control, controller design, control
structuredesignandcontrollabilityanalysis.
Thebookispartlybasedonagraduatemultivariablecontrolcoursegivenbythe
first authorin the CyberneticsDepartmentat the NorwegianUniversityofScience
and Technology in Trondheim. About 10 students from Electrical, Chemical and
Mechanical Engineering have taken the course each year since 1989. The course
has usually consisted of 3 lectures a week for 12 weeks. In addition to regular
assignments,thestudents havebeenrequiredto completea 50hourdesignproject
using MATLAB. In AppendixB, a project outlineis giventogetherwith a sample
exam.
Examplesand internet
Most of the numerical examples have been solved using MATLAB. Some sample
files areincludedin thetexttoillustrate thesteps involved.Mostofthese files use
the -toolbox,andsometheRobustControltoolbox,butinmostcasestheproblems
coul(cid:22)dhavebeensolvedeasilyusingothersoftwarepackages.
Thefollowingareavailableovertheinternet:
MATLABfilesforexamplesandfigures
Solutionstoselectedexercises
(cid:15)
Linearstate-spacemodelsforplantsusedinthecasestudies
(cid:15)
Corrections,commentstochapters,extraexercisesandexamsets
(cid:15)
(cid:15)
Thisinformationcanbeaccessedfromtheauthors’homepages:
http://www.chembio.ntnu.no/users/skoge
http://www.le.ac.uk/engineering/staff/Postlethwaite
(cid:15)
(cid:15)
Commentsand questions
Pleasesendquestions,errorsandanycommentsyoumayhavetotheauthors.Their
emailaddressesare:
[email protected]
[email protected]
(cid:15)
(cid:15)