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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
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
andyetsucceedforanother.Atthemoment,thereisnosystematicmethodforchoosingthebestrank-onedecompositionsandtherefore,thiswouldconstituteanimportantareaforfutureresearch.Afinalobservationconcernstheboundsontheuncertainparameters,ithasbeenassumedthateachparametersatisfiesthesamebound.Forexample,onehas
ratherthanseparatebounds
.Thisassumptioncanbemadewithoutlossorgenerality.Indeed,anyvariationintheuncertainboundscanbeeliminatedbysuitablescalingofthematrices
..TheweightingmatricesQandR.
Associatedwiththesystem(
)arethepositivedefinitesymmetricweightingmatrices
.Thesematricesarechosenbythedesigner.ItwillbeseeninSection4thatthesematricesareanalogoustotheweightingmatricesintheclassicallinearquadraticregulatorproblem.Theformaldefinitionofquadraticstabilizabilitynowpresented.
Definition2.1.Thesystem(
)issaidtobequadraticallystabilizableifthereexistscontinuousfeedbackcontrol
withP(0)=0,ann
n:
positivedefinitesymmetricmatrixPandaconstant>
0suchthatthefoll