RobustCont r olSystem Design Advanced St at e SpaceTechniques Second Edition,Revised and Expanded Chia-Chi Tsui DeVly Znstitute ofTechnology Long Island City, New York, U.S.A. Copyright2004 by Marcel Dekker, Inc. All Rights Reserved.Althoughgreat carehasbeentakentoprovideaccurateandcurrent information,neither the author(s) nor the publisher, nor anyone else associated with thispublication, shall beliableforanyloss, damage, or liabilitydirectlyor indirectlycausedorallegedtobecausedbythisbook. Thematerial containedhereinisnotintendedtoprovidespecicadviceorrecommendationsforanyspecicsituation.Trademarknotice: Product or corporatenames maybetrademarks or registeredtrademarksandareusedonlyforidenticationandexplanationwithout intenttoinfringe.LibraryofCongressCataloging-in-PublicationDataAcatalogrecordforthisbookisavailablefromtheLibraryofCongress.ISBN:0-8247-4869-7Thisbookisprintedonacid-freepaper.HeadquartersMarcelDekker,Inc.,270MadisonAvenue,NewYork,NY10016,U.S.A.tel:212-696-9000;fax:212-685-4540DistributionandCustomerServiceMarcelDekker,Inc.,CimarronRoad,Monticello,NewYork12701,U.S.A.tel:800-228-1160;fax:845-796-1772EasternHemisphereDistributionMarcelDekkerAG,Hutgasse4,Postfach812,CH-4001Basel,Switzerlandtel:41-61-260-6300;fax:41-61-260-6333WorldWideWebhttp://www.dekker.comThepublisheroffersdiscountsonthisbookwhenorderedinbulkquantities. Formore information, write to Special Sales/Professional Marketing at the headquartersaddressabove.Copyright # 2004byMarcelDekker,Inc.AllRightsReserved.Neither this book nor any part may be reproduced or transmitted in any form or byany means, electronic or mechanical, including photocopying, microlming, andrecording, or by any information storage and retrieval system, without permission inwritingfromthe publisher.Currentprinting(lastdigit):10 9 8 7 6 5 4 3 2 1PRINTEDINTHEUNITEDSTATESOFAMERICACopyright2004 by Marcel Dekker, Inc. All Rights Reserved.CONTROL ENGINEERING ASeries of Reference Books andTextbooks 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. Editors NEIL MUNRO, PH.D., D.Sc. Professor Applied Control Engineering University of Manchester Institute of Science and Technology Manchester, United Kingdom FRANK L. LEWIS, PH.D. Moncrief-O'Donnell Endowed Chair and Associate Director of Research Automation & Robotics Research Institute University of Texas, Arlington Nonlinear Control ofElectric Machinery,DarrenM.Dawson, JunHu, and Timothy C. Burg Computational Intelligence in Control Engineering, Robert E. King Quantitative Feedback Theory:Fundamentals and Applications,Con- stantine H. Houpis and Steven J. Rasmussen Self-Learning Control of Finite Markov Chains, A. S. Poznyak, K. Najim, and E. Gomez-Ramirez Robust Control and Filtering for Time-Delay Systems, MagdiS.Mah- moud Classical Feedback Control: With MATLAB, Boris J.Lurie and Paul J. Enright OptimalControlofSingularlyPerturbedLinearSystemsand Applications:High-Accuracy Techniques, ZoranGajic'andMyo-Taeg Lim Engineering SystemDynamics: AUnified Graph-Centered Approach, Forbes T. Brown AdvancedProcessIdentificationandControl,Ensolkonenand Kaddour Najim Modern Control Engineering, P. N. Paraskevopoulos Sliding Mode Control in Engineering, edited byWilfrid Perruquetti and Jean Pierre Barbot ActuatorSaturationControl, edited byVikram KapilaandKarolos M. Grigoriadis Copyright2004 by Marcel Dekker, Inc. All Rights Reserved.13.NonlinearControlSystems,ZoranVukic,LjubomirKuljaCa,Dali DonlagiC, Sejid Tesnjak 14.Linear Control System Analysis and Design with MATLAB: Fifth Edition, Revised andExpanded, JohnJ.DAzzo,Constanfine H.Houpis,and Stuart N. Sheldon 15.Robot Manipulator Control: Theory and Practice, Second Edition, Re- vised and Expanded, Frank L. Lewis, Darren M.Dawson, andChaouki T. Abdallah 16.RobustControlSystemDesign:AdvancedStateSpaceTechniques, Second Edition, Revised and Expanded, Chia-Chi Tsui AdditionalVolumes in Preparation Copyright2004 by Marcel Dekker, Inc. All Rights Reserved.ToSusanandJamesandShaneCopyright2004 by Marcel Dekker, Inc. All Rights Reserved.SeriesIntroductionManytextbookshavebeenwrittenoncontrol engineering, describingnewtechniquesforcontrollingsystems,ornewandbetterwaysofmathematic-ally formulating existing methods to solve the ever-increasing complexproblemsfacedbypracticingengineers.However,fewofthesebooksfullyaddress the applications aspects of control engineering. It is the intention ofthisnewseriestoredressthissituation.Theseries will stress applicationsissues,and not justthemathematicsofcontrolengineering.Itwillprovidetextsthatpresentnotonlybothnewand well-established techniques, but also detailed examples of theapplicationof thesemethodstothesolutionof real-worldproblems. Theauthors will be drawnfromboth the academic worldandthe relevantapplicationssectors.Copyright2004 by Marcel Dekker, Inc. All Rights Reserved.There are already many exciting examples of the application of controltechniques in the established elds of electrical, mechanical (includingaerospace), andchemical engineering. We have only tolookaroundintodays highly automated society to see the use of advanced roboticstechniquesinthemanufacturingindustries; theuseof automatedcontrolandnavigationsystemsinairandsurfacetransportsystems;theincreasinguse of intelligent control systems inthe many artifacts available tothedomestic consumer market; and the reliable supply of water, gas, andelectricalpowertothedomesticconsumerandtoindustry.However,thereare currently many challenging problems that couldbenet fromwiderexposuretotheapplicabilityofcontrol methodologies, andthesystematicsystems-orientedbasisinherentintheapplicationofcontroltechniques.This series presents books that drawon expertise fromboth theacademic world and the applications domains, and will be useful not only asacademically recommended course texts but also as handbooks forpractitioners in many applications domains. Robust Control Systems isanotheroutstandingentryinDekkersControlEngineeringseries.Copyright2004 by Marcel Dekker, Inc. All Rights Reserved.PrefaceThis secondeditionof Robust Control SystemDesign introduces a newdesign approach to modern control systems. This design approachguarantees, fortherst time, thefull realizationof robustnesspropertiesof generalized state feedback control for most open-loop system conditions.Stateandgeneralizedstatefeedbackcontrol canachievefeedbacksystemperformanceandrobustnessfarmoreeffectivelythanotherbasicformsofcontrol. Performance and robustness (versus model uncertainty and controldisturbance) are mutuallycontradictory, yet theyare the keypropertiesrequired by practical control systems. Hence, this design approach not onlyenrichestheexistingmoderncontrol systemdesigntheory, butalsomakespossibleitswideapplication.Copyright2004 by Marcel Dekker, Inc. All Rights Reserved.Modern(orstatespace) control theorywasdevelopedinthe1960s.The theory has evolved such that the state feedback control and itsimplementing observer are designed separately (following the so-calledseparation principle [Wil, 1995]). With this existing design approach,althoughthe direct state feedbacksystemcanbedesignedtohave goodperformanceandrobustness, almostall theactual correspondingobserverfeedback systems have entirely different robustness. In the newdesignapproachpresentedhere, thestatefeedbackcontrol anditsimplementingobserver are designed together. More explicitly, the state feedback control isdesignedbasedontheresultsof itsimplementingobserver. Theresultingstate feedback control is the generalized state feedback control [Tsui, 1999b].This fundamentally new approach guaranteesfor all open-loopsystems with more outputs than inputs or with at least one stabletransmission zerothe same loop transfer function and therefore thesame robustness of the observer feedbacksystemandthe correspondingdirectstatefeedbacksystem.Most open-loop systems satisfyeither of thesetwoconditions.Forallotheropen-loopsystems, thisapproachguaranteesthat thedifferencebetweenthelooptransfer functions of theabovetwofeedbacksystemsbekeptminimalinasimpleleast-squaresense.Modernandclassicalcontroltheoriesarethetwomajorcomponentsofcontrolsystemstheory.Comparedwithclassicalcontroltheory,moderncontrol theorycandescribeasinglesystemsperformanceandrobustnessmore accurately, but it lacks a clear concept of feedback system robustness,suchas the loop transfer functionof classical controltheory.By fully usingthe concept of loop transfer functions, the approach exploits the advantagesof both classical and modern control theories. This approach guarantees therobustness and loop transfer function of classical control theory, whiledesigning this loop transfer function much more effectively (thoughindirectly) usingmoderncontrol designtechniques. Thus it achieves bothgoodrobustnessandperformanceforfeedbackcontrolsystems.Ifthersteditionofthisbookemphasizedtherstoftheabovetwoadvantages (i.e., thetruerealizationof robustness properties of feedbackcontrol), thenthissecondeditionhighlightsthesecondof theabovetwoadvantagesthe far more effective design of high performance androbustnessfeedbackcontrolitself.A usefulcontroltheoryshould providegeneraland effectiveguidanceon complicated control system design. To achieve this, the designformulationmustfullyaddressbothperformanceandrobustness. Itmustalsoexploitfullytheexistingdesignfreedomandapplyageneral, simple,andexplicit designprocedure. Theapproachpresentedheretrulysatisesthese requirements. Since this book concentrates on this new designapproach and its relevant analysis, other analytical control theory results areCopyright2004 by Marcel Dekker, Inc. All Rights Reserved.presentedwithanemphasis ontheir physical meanings, insteadof theirdetailedmathematicalderivationsandproofs.Thefollowinglistshowsseveralofthebooksmostimportantresults.Withtheexceptionofthethirditem,theseresultsarenotpresentedinanyotherbooks:1. The rst general dynamic output feedback compensator that canimplement state or generalizedstate feedbackcontrol, anditsdesign procedure. The feedback system of this compensator is therst general feedback systemthat has the same robustnessproperties of its corresponding direct state feedback system(Chapters3to6).2. Asystematic, simple, andexpliciteigenvalueassignment proce-dure usingstatic output feedbackcontrol or generalizedstatefeedback control (Section 8.1). This procedure enables thesystematiceigenvectorassignmentproceduresofthisbook, andisgeneral tomost open-loopsystemconditionsifbasedonthegeneralizedstatefeedbackcontrolofthisbook.3. Eigenvector assignment procedures that can fully use thefreedomof this assignment. Both numerical algorithms andanalyticalproceduresarepresented(Section8.2).4. A general failure detection, isolation, and accommodationcompensator that is capable of considering system modeluncertainty andmeasurement noise, andits systematic designprocedure(Chapter10).5. The simplest possibleformulation, andatrulysystematic andgeneral procedure, of minimal order observer design (Chapter 7).6. Solutionof thematrixequationTA FT=LC[matrixpair(A, C)isobservableandeigenvaluesofmatrixFarearbitrarilyassigned]. Thissolutionisgeneral andhasall eigenvaluesof Fandall rowsof Tcompletelydecoupled(FisinJordanform).This solution uniquely enables the full use of the remainingfreedom of this matrix equation, which is fundamentallyimportant in most of the basic design problems of moderncontroltheory(Chapters5to8,10).7. Thebasicdesignconceptofgeneratingastatefeedbackcontrolsignal without estimating all state variables, andthe general-ization of this design concept from function observers only to allfeedbackcompensators(Chapters3to10).8. The complete unication of two existing basic feedbackstructuresof moderncontrol theorythezeroinput gainstateCopyright2004 by Marcel Dekker, Inc. All Rights Reserved.observer feedback structure and the static output feedbackstructure(Section6.3).9. A more generally accurate robust stability measure that isexpressedintermsofthesensitivitiesofeachsystempole. Thisanalytical measure can be used to guide systematic feedbacksystemdesign(Sections2.2.2and8.2).10. Comparisonof computational complexityandthereforetrack-ability (ability to adjust the original design formulation based onthe nal andnumerical designresults) of all feedbackcontroldesigntechniques(Section9.3).11. Emphasis on the distinct advantages of high performance/robustness control design using eigenstructure assignmenttechniques over the techniques for the direct design of looptransferfunctions(Chapters2,3,8,9).12. The concept of adaptive control andits applicationinfailureaccommodationandcontrol(Section10.2).The rst ve of the above results are actual design results. The last seven arenew theoreticalresultsandconcepts thathaveenabledtheestablishmentoftherstveresults. Inotherwords, themainnewresult(result1, thefullrealization of robustness properties of state/generalized state feedbackcontrol) is enabled by some signicant and fundamental developments (suchas results 6to8), andis validatedbythe distinct effectiveness of state/generalizedstatefeedbackcontrol(results2to3and9to11).Thisbookalsoaddressesthecomputational reliabilityofitsanalysisanddesignalgorithms. Thisisbecausepractical control problemsusuallyrequirealarge amount of computation, andunreliablecomputationcanyieldtotallyunreliableresults. Everyeffort hasbeenmadetousereliablecomputational methodsindesignalgorithms, suchasthecomputationofHessenbergform(insteadofthecanonicalform)andoforthogonalmatrixoperation(insteadofelementarymatrixoperation).As aresult, thecomputationrequiredinthis bookis slightlymorecomplicated, but the morereliable results thus obtainedmake the effortworthwhile. It should be noted that the computation of polynomialsrequired by the classical control theory is usually unreliable. Thedevelopment of computational software has alsoeasedconsiderably thecomplexity of computation. Each design procedure is presented in algorithmform, and eachstep ofthesealgorithmscanbeimplementeddirectly by theexistingcomputationalsoftware.Thisbookwill beusefultocontrolsystemdesignersandresearchers.Althoughasolidbackgroundinbasiclinearalgebraisrequired,itrequiresremarkablyless mathematical sophisticationthanother books similar inCopyright2004 by Marcel Dekker, Inc. All Rights Reserved.scope. This book can also be used as a textbook for students who have had arst course (preferably including state space theory) incontrol systems.Multi-inputandmulti-outputsystemsarediscussedthroughout. However,readers will ndthat the results havebeensubstantiallysimpliedtobequite easily understandable, and that the results have been well unied withthesingle-inputandsingle-outputsystemresults. Inaddition, thisbookiscomprehensiveandself-contained,witheverytopicintroducedatthemostbasic level. Thus it couldalsobe usedbyhonor programstudents withbackgroundinsignalsandsystemsonly.An overviewof each chapter follows. Chapter 1 introduces basicsystemmodels andproperties. Chapter 2 analyzes the performance andsensitivityofa singleoverallsystem.Chapter3describesthecriticalroleoflooptransferfunctions on the sensitivityof feedback systems, including theobserver feedbacksystems. Chapter 4proposes thenewdesignapproachandanalyzes its advantages. Chapter 5 presents the solutionof a basicmatrixequation. Thissolutionisusedthroughout theremainingchapters(except Chapter 9). Chapter 6 presents the design of the dynamic part of theobserversuchthatforanystatefeedbackcontrol signal generatedbythisobserver, the looptransfer functionof this control is alsofullyrealized.Chapter7presentsthedesignofthefunctionobserver,whichgeneratesanarbitrarily given state feedback control signal, with minimizedobserverorder. Chapter8presentstheeigenvalue/vectorassignmentcontrol designmethods.Chapter9introducesthelinearquadraticoptimalcontroldesignmethods. Both designs of Chapters 8 and 9 will determine the output part oftheobserverofChapter6, aswell asthetarget closed-loopsystemlooptransfer function. Comparison of various designs reveals two distinctadvantagesofeigenstructureassignmentdesign. Chapter10dealswiththedesignofageneralfailuredetection,isolation,and(adaptive)accommoda-tioncompensatorthatiscapableofconsideringsystemmodel uncertaintyandmeasurement noise. This compensator has the compatible structureofandcanbe implementedincoordinationwiththe normal (free ofmajorfailure) robust control compensatorof thisbook. Thereisaset ofsimpleexercisesattheendofeachchapter.To make the book self-contained, Appendix Aprovides a simpleintroductiontotherelevantmathematical backgroundmaterial. AppendixBliststhemathematicalmodelsofeightreal-worldsystemsforsynthesizeddesignpractice.Iwouldliketothankeveryonewhohelpedme, especiallyduringmystudentyears.IalsothankmyformerstudentRezaShahriar,whoassistedwithsomeofthecomputergraphics.Chia-ChiTsuiCopyright2004 by Marcel Dekker, Inc. All Rights Reserved.ContentsSeriesIntroductionPreface1. SystemMathematicalModelsandBasicProperties1.1 TwoKindsofMathematicalModels1.2 EigenstructureDecompositionofaStateSpaceModel1.3 SystemOrder,Controllability,andObservability1.4 SystemPolesandZerosExercisesCopyright2004 by Marcel Dekker, Inc. All Rights Reserved.2 Single-SystemPerformanceandSensitivity2.1 SystemPerformance2.2 SystemSensitivityandRobustnessConclusionExercises3 FeedbackSystemSensitivity3.1 SensitivityandLoopTransferFunctionofFeedbackSystems3.2 SensitivityofFeedbackSystemsofModernControlTheorySummary4 ANewFeedbackControlDesignApproach4.1 BasicDesignConceptofObserversDirectGenerationofStateFeedbackControlSignalWithoutExplicitSystemStates4.2 PerformanceofObserverFeedbackSystemsSeparationProperty4.3 TheCurrentStateofLTRObserverDesign4.4 ANewDesignApproachandNewFeedbackStructureADynamicOutputFeedbackCompensatorthatGeneratesState/GeneralizedStateFeedbackControlSignalExercises5 SolutionofMatrixEquationTAFT=LC5.1 ComputationofaSystemsObservableHessenbergForm5.2 SolvingMatrixEquationTAFT=LCExercises6 Observer(DynamicPart)DesignforRobustnessRealization6.1 SolutionofMatrixEquationTB=06.2 AnalysisandExamplesofThisDesignSolution6.3 CompleteUnicationofTwoExistingBasicModernControlSystemStructures6.4 ObserverOrderAdjustmenttoTradeoffBetweenPerformanceandRobustnessExercisesCopyright2004 by Marcel Dekker, Inc. All Rights Reserved.7 ObserverDesignforMinimizedOrder7.1 DesignFormulation7.2 DesignAlgorithmandItsAnalysis7.3 ExamplesandSignicanceofThisDesignExercises8 DesignofFeedbackControlEigenstructureAssignment8.1 SelectionandPlacementofFeedbackSystemPoles8.2 EigenvectorAssignmentSummaryExercises9 DesignofFeedbackControlQuadraticOptimalControl9.1 DesignofDirectStateFeedbackControl9.2 DesignofGeneralizedStateFeedbackControl9.3 ComparisonandConclusionofFeedbackControlDesignsExercises10 DesignofFailureDetection,Isolation,andAccommodationCompensators10.1 FailureDetectionandIsolation10.2 AdaptiveStateFeedbackControlforFailureAccommodation10.3 TheTreatmentofModelUncertaintyandMeasurementNoiseExercisesAppendixA:RelevantLinearAlgebraandNumericalLinearAlgebraAppendixB:DesignProjectsandProblemsReferencesCopyright2004 by Marcel Dekker, Inc. All Rights Reserved.