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外文原文及译文

ARiccatiEquationApproachtotheStabilizationofUncertainLinearSystems

IANR．PETERSENandCHRISTOPHERV．HOLLO

Abstract

Thispaperpresentsamethodfordesigningafeedbackcontrollawtostabilizeaclassofuncertainlinearsystems.Thesystemsunderconsiderationcontainuncertainparameterswhosevaluesareknownonlytowithagivencompactboundingset．Furthermore，theseuncertainparametersmaybetime-varying.ThemethodusedtoestablishasymptoticstabilityoftheclosedloopsystemobtainedwhenthefeedbackcontrolisappliedinvolvestheuseofaquadraticLyapunovfunction.ThemaincontributionofthispaperinvolvesthedevelopmentofacomputationallyfeasiblealgorithmfortheconstructionofasuitablequadraticLyapunovfunction,OncetheLyapunovfunctionhasbeenobtained,itusedtoconstructthestabilizingfeedbackcontrollaw.ThefundamentalideabehindthealgorithmpresentedinvolvesconstructinganupperboundfortheLyapunovderivativecorrespondingtotheclosedloopsystem.Thisupperboundisaquadraticform.Byusingthisupperboundingprocedure,asuitableLyapunovfunctioncanbefoundbysolvingacertainmatrixRiccatiequation.AnothermajorcontributionofthispaperistheidentificationofclassesofsystemsforwhichthesuccessofthealgorithmisbothnecessaryandsufficientfortheexistenceofasuitablequadraticLyapunovfunction．

Keywords:

Feedbackcontrol;Uncertainlinearsystems;Lyapunovmethods;Riccatiequation

1.INTRODUCTION

Thispaperdealswiththeproblemofdesigningacontrollerwhennoaccuratemodelisavailablefortheprocesstobecontrolled.Specifically,theproblemorstabilizinganuncertainlinearsystemusingstatefeedbackcontrolisconsidered.Inthiscasetheuncertainlinearsystemconsistsofalinearsystemcontainingparameterswholevaluesareunknownbutbounded.Thatis,thevaluesoftheseuncertainparametersareknowntobecontainedwithgivencompactboundingsets.Furthermore,theseuncertainparametersareassumedtovarywithtime.

Theproblemofstabilizinguncertainlinearsystemsofthistypehasattractedaconsiderableamountorinterestinrecentyears.InLeitman（1979,1981）andGutmanandPalmoor（1982）,theuncertaintyinthesystemisassumedtosatisfythesocalled“matchingconditions".Thesematchingconditionsconstitutesufficientconditionsforagivenuncertainsystemtobestabilizable.InCorlessandLeitmann（1981）andBarmish,CorlessandLeltmann（1983）,thisapproachisextendedtouncertainnon—linearsystems.However,evenforuncertainlinearsystemsthematchingconditionsareknowntobeundulyrestrictive.Indeed,IthasbeenshowninBarmishandLeitmann（1982）andHollotandBarmish（1980）thatthereexistmanyuncertainlinearsystemswhichfailtosatisfythematchingconditionsandyetareneverthelessstabilizable.Consequently,recentresearcheffortshavebeendirectedtowardsdevelopingcontrolschemeswhichwillstabilizealargerclassofsystemthanthosewhichsatisfythematchingconditions;e.g.BarmishandLeitmann（1982）,HollotandBarmish（1980）,Thorpandbarmish（1981）,Barmlsh（1982,1985）andHoLLot（1984）.Themainaimofthispaperistoenlargetheclassofuncertainlinearsystemsforwhichonecanconstructastabilizingfeedbackcontrollaw.Itshouldbenotedhowever,thatincontrasttoCorlessandLeitmann（1981）Barmish,CorlessandLeitmann（1983）andPetersenandBarmish（1986）,attentionwillberestrictedtouncertainlinearsystemshere.

Lyapunovoflaw,entertothe1990snon-linearcontrolledfieldsucceedinwillitbetheeightiesthe20thcenturywhilebeingexcellentwhilebeingstupidItisamaindesignmethodwithstupidandexcellentandcalmnon-linearsystem.Whileutilizingthiskindofmethodtodesignthestupidexcellentcomposuresystem,supposeatfirsttheuncertaintyexistingisunknownintherealsystem,butbelongtoacertainsetthatdescribes,namelytheuncertainfactorcanshowinordertothereisunknownparameterofthecircle,gainunknownperturbationfunctiontohavecircleandaccuseofmarkoftargetclaimmodelconstructaproperLyapunovfunction,makeitswholesystemofassurancesteadytoanyelementwhileassemblinguncertain.Justbecauseofthiskindofgenerality,nomatterusedforanalyzingthestabilityorusingforbeingcalmandcomprehensive,lackcontractility.Peopleattemptripetheoryisitreachthenon-linearsystemtodelaymorelinearsystem.Introducedthenon-linearsystem.Inrecentyearstothesteps,inthenon-linearsystem,themeaninginthestepsliesinithasdescribedtheessenceofthenon-linearstructureofthesystem.Forimitatingthenon-linearsystempenetrated,canutilizetheconceptofrelativestepstodivideintolinearandtwonon-linearpartssystematically,partitisnon-linearcanview,itislinearforpartcanhaveaccusedofcanwatchaswellas,systemthatformlikethiszerosubsystemnotdynamic,havingproveditundertheone-dimensionalsituation,iftheasymptoticstabilityoftheoverallsituationofzerodynamicsubsystem,sowholesystemcanbeexactlybookednearerandnearerwiththeoverallsituation.Feedbackastostepslinearizationcombinestogether,receiveandwellcontroltheresultsuchasdocument[1].

Inallthereferencescitedabovedealingwithuncertainlinearsystems,thestabilityoftheclose-loopuncertainsystemisestablishedusingaquadraticLyapunovfunction.Thismotivatestheconceptofquadraticstabilizabilitywhichisformalizedinsection2;seealsoBarmish（1985）.Furthermore,inBarmlsh（1985）andPetersen（1983）,itisshowinsection2;seealsoBarmish（1985）.Furthermore,inBarmish（1985）andPetersen（1983）,itisshownthattheproblemofstabilizinganuncertainlinearsystemcanbereducedtotheproblemorconstructingasuitablequadraticLyapunovfunctionforthesystemconsequentlyamajorportionofthispaperisdevotedtothisproblem.VariousaspectsoftheproblemorconstructingsuitablequadraticLyapunovfunctionshavebeeninvestigatedinHollotandBarmish（1980）,ThorpandBarmish（1981）andHollot（1984）,ChangandPeng（1972）,Noldus（1982）andPetcrsen（1983）.OneapproachtofindingasuitbalequadraticLyapunovfunctioninvolvessolvingan“augmented”matrixRiccatiequationwhichhasbeenspeciallyconstructedtoaccountfortheuncertaintyinthesystem;e.g.ChangandPeng（1972）andNoldus（1982）.TheresultspresentedinthispapergobeyondNoldus（l982）inthatuncertaintyisallowedinboththe“A”matrixand“B”matrix.Furthermore,anumberofclassesofuncertainsystemsareidentified,forwhichthesuccessofthismethodbecomesnecessaryandsufficientfortheexistenceofasuitablequadraticLyapunovfunction.ThefundamentalideabehindtheapproachinvolvesconstructingaquadraticformwhichservesasanupperboundfortheLyapunovderivativecorrespondingtotheclosedloopuncertainsystem.Thisproceduremotivatestheintroductionorthetermquadraticboundmethodtodescribetheprocedureusedinthispaper.

ThebenefitofquadraticboundindstemsfromthefactthatacandidatequadraticLyapunovfunctioncaneasilybeobtainedbysolvingamatrixRiccatiequation.Forthespecialcaseorsystemswithoutuncertainty,this“augmented”Riccatiequationreducestothe“ordinary”Riccctiequationwhicharisesinthelinearquadraticregulatorproblem,e.g.AndersonandMoore（1971）.Hence,theprocedurepresentedinthepapercanberegardedasbeinganextensionofthelinearquadraticregulatordesignprocedure.

2.SYSTEMANDDEFINITIONS

Aclassofuncertainlinearsystemsdescribedbythestateequations

where

isthestate,

isthecontroland

and

arevectorsofuncertainparameters,isconsidered.Thefunctionsr（·）ands（·）arerestrictedtobeLebessuemeasurablevectorfunctions.Furthermore,thematrices

and

areassumedtoberankonematricesoftheform

and

intheabovedescription

and

denotethecomponentofthevectorsr（t）ands（t）respectively.

Remarks:

Notethatanarbitraryn

nmatrix

canalwaysbedecomposedasthesumofrankonematrices;i.e.forthesystem（∑）,onecanwrite

withrankone

.Consequently,if

isreplacedby

andtheconstraint

isincludedforalliandjthenthis"overbounding”oftheuncertaintieswillresultinasystemwhichsatisfiestherank-oneassumption.Moreover,stabilizabilityofthis"larger"systemwillimplystabiliabilltyfor（Z）.Atthispoint,observethattherankonedecompositionsforthe

and

arenotunique.Forexample,

canbemultipliedbyanyscalarif

isdividedbythesamescalar.Thisfactrepresentsoneofthemainweaknessesortheapproach.Thatis,thequadraticboundmethoddescribedinthesequelmayfailforonedecompositionof

and

andyetsucceedforanother.Atthemoment,thereisnosystematicmethodforchoosingthebestrank-onedecompositionsandtherefore,thiswouldconstituteanimportantareaforfutureresearch.Afinalobservationconcernstheboundsontheuncertainparameters,ithasbeenassumedthateachparametersatisfiesthesamebound.Forexample,onehas

ratherthanseparatebounds

.Thisassumptioncanbemadewithoutlossorgenerality.Indeed,anyvariationintheuncertainboundscanbeeliminatedbysuitablescalingofthematrices

and

..TheweightingmatricesQandR.

Associatedwiththesystem（

）arethepositivedefinitesymmetricweightingmatrices

and

.Thesematricesarechosenbythedesigner.ItwillbeseeninSection4thatthesematricesareanalogoustotheweightingmatricesintheclassicallinearquadraticregulatorproblem.Theformaldefinitionofquadraticstabilizabilitynowpresented.

Definition2.1.Thesystem（

）issaidtobequadraticallystabilizableifthereexistscontinuousfeedbackcontrol

withP（0）=0,ann

n:

positivedefinitesymmetricmatrixPandaconstant>0suchthatthefoll