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ROBUSTSTABILITYANALYSISOFSYSTEMSUNDERPARAMETRIC UNCERTAINTY By JOSEALVAROLETRA ADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOL OFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENT OFTHEREQUIREMENTSFORTHEDEGREEOF DOCTOROFPHILOSOPHY UNIVERSITYOFFLORIDA 1991 To CarmenLucia and Ariadne ACKNOWLEDGMENTS Iamprofoundlyindebtedtomyadvisorandsupervisorycommitteechairman,Dr.Ha- niphA.Latchman,forhisguidance,permanentsupportandencouragementduringmythree yearsattheUniversityofFlorida. Despitehisseveralotherresponsibilities,Dr.Latchman alwaysfoundtimetodiscussmyworkandgivemehisinsightfulorientation. Iwishtothanktheprofessorswhoservedonmycommittee,Dr. ThomasE.Bullock, Dr.J.Hammer,Dr. A.AntonioArroyoandDr. SpyrosA.Svoronos,fortheirwillingness todiscussandadvicemywork,andforthehighlevelofconsiderationIwasalwaystreated with. IwishtothankthehelpandadviceofDr.G.Basile,myfirstcommitteechairman. IamindebtedtotheEEGraduateCoordinator,Dr. LeonWCouch,andhisstaff,for alltheirassistance.Particularly,IhavetothankMrs. GretaSbrocco,whoalwaysprovided helpfulorientationonadministrativesubjects. Itwasaprivilegetoworkclosetomyex-fellowstudent,Dr. RobertJ.Norris,whose valuableincentiveandhelpInowacknowledge.IalsowishtothankDr.JulioS.Dolceda Silva,oftheBrazilianArmy,forhishelponmyenrollmentandadaptationtotheUniversity. IamgratefultotheExecitoBrasileiro(BrazilianArmy)forconcedingmetheopportu- nityofcomingtotheUniversityofFloridatofurtherpursuemystudies,andtotheCNPq- ConselhoNacionaldeDesenvolvimentoCientificoeTecnologico(ScientificandTechnologi- calNationalDevelopmentAgency-Brazil)forthescholarshipIwasgranted. 111 TABLEOFCONTENTS page ACKNOWLEDGMENTS iii ABSTRACT vi CHAPTERS 1 INTRODUCTION 1 1.1DissertationObjective 1 1.2BriefHistoricalofUncertaintyTreatment 2 1.3StructureoftheDissertation 9 1.4Notation 11 2 NOMINALMODELSANDUNCERTAINTYREPRESENTATION 16 2.1NominalModelsandDefinitions 16 2.2UncertaintyRepresentation 20 2.3Conclusions 38 3 STABILITYANALYSISOFLINEARSYSTEMS 39 3.1Introduction 39 3.2StabilityofStateSpaceSystems 39 3.3StabilityofTransferMatrixModels 45 3.4Frequency-DomainScalingTechniques 63 3.5Conclusions 72 4 LYAPUNOVDIRECTMETHODINTHEPRESENCEOFSTRUCTURED UNCERTAINTY 73 4.1Introduction 73 4.2DependenceofConservatismonPerturbationStructure 76 4.3StabilityUnderStructuredUncertainty 82 4.4MaximizationofStabilityDomains 92 4.5ApplicationofOptimizationOverO 109 4.6Conclusions 113 IV 5 STABILITYUNDERDIAGONALPARAMETRICUNCERTAINTY 115 5.1Introduction 115 5.2DiagonalRepresentationofStateSpacePerturbations 116 5.3ProblemFormulation 122 5.4NecessaryandSufficientConditionsforRobustStability 127 5.5SufficientConditionsforRobustStability 132 5.6NumericalApplication 136 5.7SomeExtensionsofPreviousResults 139 5.8Conclusions 143 6 COMPARISONOFSUFFICIENTPARAMETERNORMBOUNDS 145 6.1Introduction 145 6.2ResultsforProblemswith2and3Parameters 146 6.3ResultsforRandomlyGeneratedMatrices 154 6.4Conclusions 161 7 ITERATIVECONTROLLERROBUSTIFICATION 163 7.1Introduction 163 7.2RobustificationAssociatedtoLyapunovAnalysis 169 7.3RobustificationAssociatedtoFrequency-DomainAnalysis 169 7.4Application 187 7.5Conclusion 195 8 NECESSARYSTABILITYDOMAININTHEPARAMETERSPACE 197 8.1Introduction 197 8.2CharacterizationofaNecessaryStabilityDomain 199 8.3ComputationoftheNecessaryStabilityDomain 202 8.4Applications 209 8.5Conclusions 214 9 CONCLUSION 216 9.1Summary 216 9.2DirectionsforFutureWork 223 REFERENCES 230 BIOGRAPHICALSKETCH 234 v AbstractofDissertationPresentedtotheGraduateSchool oftheUniversityofFloridainPartialFulfillmentofthe RequirementsfortheDegreeofDoctorofPhilosophy ROBUSTSTABILITYANALYSISOFSYSTEMSUNDERPARAMETRIC UNCERTAINTY By JOSEALVAROLETRA May1991 Chairman: Dr.HaniphA.Latchman MajorDepartment:ElectricalEngineering Intheanalysisofstabilitypropertiesofcontrolsystems,theuncertaintyinmathemati- calmodelsmustbetakenintoaccount.Mainsourcesofuncertaintyarehighorderdynamic phenomenaofthephysicalsystemneglectedinthemodel,andvariationsinsystemparam- eters.Thesubjectofthisworkistheassessmentofstabilityoflinearcontrolsystemsinthe presenceofparametricuncertainty. Statespaceandfrequency-domainmodelsanduncertaintyrepresentationarereviewed, aswellasgeneral conditions fornominal and robust stability. Alsoreviewed arescal- ingtechniquesusedforreducingthedegreeofconservatismoffrequency-domainstability conditions,includingoptimalsimilarityscaling,optimalnon-similarityscalingandPerron scaling. Particularly, theperturbed statespacemodel x(t) = (A+E)x(t)is studied. The nominal matrix Ais assumed asymptotically stable, and the perturbation Eis ofthe form E = YHt=\PkEk, wherepis am—dimensional vectorofsystem parameters, and Ek,k= areconstantmatrices. TheapplicationoftheLyapunovDirectMethod vi forobtainingconditionsonthenormofpwhicharesufficientforrobuststabilityisdiscussed indetail.Anewstabilityconditionon||p||2isgiven,whichispotentiallylessconservative thanavailableresults. TheproblemofthechoiceoftheLyapunovmatrixwhichyields lessconservativestabilityconditionsisformalizedasaconstrainednumericaloptimization problem. Forthecaseoftime-invariant uncertainty, an equivalentfrequency-domainstability problemisformulated,wheretheperturbationisareal,diagonalmatrixobtaineddirectly from thestate spaceperturbation. Sufficient stability conditions arederived from the equivalentformulation,andscalingtechniquesareused,inordertoreduceconservatism. Comparisonofnumericalresultsobtainedforseveralproblemsindicatesthat,fortime- invariantuncertainty,thefrequency-domainapproach,associatedtoPerronscaling,consti- tutesanalternativewhichhasbetterperformancethantheLyapunovDirectMethod.The frequency-domainapproachandcorrespondingstabilityconditionsarealsoshowntobeof advantageiniterativeoptimizationofstaticfeedbackcontrollersoffixedorder. Additionally,aprocedureissuggestedforobtaininganecessarystabilitydomaininthe spaceofplantparameters,startingfromaknownsufficientdomain. Finally,theintegrationofthestabilityanalysistechniquesintorobustcontrollerdesign isdiscussed. vn CHAPTER1 INTRODUCTION 1.1 DissertationObjective Atleasttwocommonaspectsaresharedbythemajorityofthecurrentliteratureon controlsystemsanalysisanddesign,althoughmanydifferentmethodsandtechniquesare nowadaysemployed.Theseaspectsareasfollows: • Focusisplacedonmultivariablesystems; • Uncertaintyinsystemmodelsisexplicitlytakenintoaccount. Theseaspectsconstituteaframeforthepresentdissertation.Thespecificsubjectisthe assessmentofrobuststabilitypropertiesofsystemsunderparametricuncertainty,which findsmotivationinthefollowingconsiderations. Controlsystemsaredesignedtomeetsomeperformancespecifications. Althoughthe formulationofperformancespecificationsdependsontheapproachused,italwaysrequires thatsomequantitativeindicesbesatisfiedbythesystemresponse,whatofcourseimplies inconstraintstothedynamicbehaviorofthesystem. However,itonlymakessensetodiscussthequantitativebehaviorofacontrolsystem ifitsstabilitycanbeassured. Otherwise,thedynamicbehaviorcanbeexpectedtoblow upundersomeadmissibleoperatingcondition,thusrenderingthesystemuseless.Stability, therefore,emergesasafundamentalrequirement. Controldesignreliesonmathematicalmodelingofthecontrolledsystem.Unfortunately, therealwaysexistsadegreeofuncertaintybetweenthemodelandthemodeledsystem, 1 2 whichmustbetakenintoaccount.Theexistenceofuncertaintygivesrisetotherequirement ofrobustness,namelytheaptitudeofacontrolsystemforretainingthedesiredbehaviorin spiteoftheuncertainty. Designmethodsdefinitelydependonanalysistechniquesinordertoassesssystemprop- erties,includingrobuststability. Techniquesforrobuststabilityanalysiscountonuncer- taintyrepresentation,whichisdictatedbyseveralfactors,mainlybythecausesofuncer- taintyandavailableinformationonuncertaintystructure. Variationsinsystemparameters aresourcesofanimportantcategoryofperturbations, whichisparticularlysuitableto representationinstatespacemodels. Motivatedbythesefacts, thisdissertationaddressestheproblemofrobust stability analysisinthepresenceofparametricperturbations.Theperturbationwillbeassumedto dependlinearlyonavectorofparameters,thusadmittingthepracticallyimportantcase inwhichoneparameteraffectsseveralentriesofthesystemmatricesinthestatespace representation.Thismodelhasbeenusedinseveralrecentworksinstabilityanalysis. ThedevelopmentofthesubjectisoutlinedinSection1.3.Beforethis,abriefhistorical summaryofthetreatmentofuncertaintyincontroltheoryisgiven. 1.2 BriefHistoricalofUncertaintyTreatment Theneedforcontrolsystemshasbeenlongfeltintheprocessoftechnologicaldevel- opment. Examplesoftheuseofcontrolsystemsdatebacktofourthousandyears[50]. Noteworthyisthefactthatfeedbackprinciplesarefoundeveninthoseearlyexamples. Amongtheseveraladvantagesthatthefeedbackprinciplebringstocontrolsystems,ap- pearsthepropertyofeffectivelycopingwithdisturbancesandsystemuncertainty[31]. 3 ImportanteventsinfeedbackhistoryareregisteredbySage[50]. Amongthemarethe inventionofthemechanicalfly-ballgovernorbyJamesWattin1788,whichwasdeveloped fromearlywindmillregulators,andtheanalysisoffeedbackcontrolsystemspublishedin 1868byMaxwell. In1927,theconceptoffeedbackwasintroducedbyBlackinthedesignofamplifiers forlongdistancetelephonelines;hispioneeringworkiscontainedinthepaper‘Stabilized FeedbackAmplifiers’,publishedin1934. Althoughrobusttouncertaintiescausedbynon- linearityandotherfactors,thefeedbackamplifierpresentedunwantedoscillations. The theoreticalstudyofthisphenomenonledtothedevelopmentoftheregenerationtheoryby Nyquist,whoseworkwaspublishedin1932. TheNyquistcriterion,whichderivesclosed- loopstabilitycharacteristicsfromopen-loopinformation,wouldconstituteafundamental techniqueforfrequency-domainstabilityanalysis. Ensuingdevelopmentsoffrequency-domainconceptsoriginatedfromtheworkofBode, innetworkanalysisandamplifierdesign(1945),whichdemonstratetheexistenceofcon- straintsinthemanipulationofthefrequencyresponseoflineartime-invariantsystems;from theNicholstransformationoftheNyquistdiagram,andfromtherootlocustechniqueof Evans. Thesetofthosetechniquesconstitutewhat becameknownastheclassicalapproach toanalysis and designofSingle-Input, Single-Output (SISO) systems. In theclassical approach,theissueofcopingwith uncertaintyisindirectlyaddressed, byprovidingthe systemwithenoughgainandphasemargins. Thesemarginsensurethatunwantedeffects ofuncertaintywillnotdisruptstability. Inthelate’50s,problemsofmorecomplexnature,mainlyoriginatedbythecontroland guidanceofmissilesandspacevehicles,cameintotheconsiderationofcontrolengineers