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数学IEEC

数学IEEC

手机3603671840 2014-02-16 评分 0 浏览量 0 0 0 0 暂无简介 简介 举报

简介:本文档为《数学IEECpdf》,可适用于自然科学领域,主题内容包含IEEETRANSACTIONSONAUTOMATICCONIXOL,VOLAC,NO,FEBRUARYANewApproachtothePerfe符等。

IEEETRANSACTIONSONAUTOMATICCONIXOL,VOLAC,NO,FEBRUARYANewApproachtothePerfectRegulationandtheBoundedPeakinginLinearMultivariableControlSystemsHIDENORIKIMURA,MEMBER,NEEAbstraetThispaperpresentsanewapproachtotheperfectregnlation(ps)andtheboundedpeaking(bop)inlinearmultivariablemntrdsystems,basedontbepoleandtheeigenvectorassignmenttechniqueRyspealdng,thepsrepresentsanidealmntroddonwhichredoeesthesettling~etoO,whilethebp~~meanstheboundedovershootinthisidealdtuatiomSimplefrequencydomainchraAionsoftbeprandthebparederivedByrevealsomeinvariancepropertiesdthepaandthebpthatprovideapowtoolforachievingtheprandthebpTheegisteneeconditionfortbeprisderived,wbichtunsouttobeisextendedtoamoregeneralcontrd~~veofattainingttsepsforoneowwhilekeepingthebpforanotheroutputAconditiononMchtheprisrealizedbyanoutputfeedbackisalsoderivedAsimpledesignalgorithmisproposedforarhieringtheprwbbisessentiallytheeigenvectorassignmentpromlureFinally,anapplicationtoarealsystemkldisarsedidenticaltotberesuHobtainedintheoplima!regolatoa~ThisresultIINTRODUCTTONITiswellknownthattheclosedlooppolesofacontrollableplantcanbeassignedarbitrarilybyastatefeedbackIUsingthisfact,wecanmaketheclosedloopresponseasfastasdesired(ofcourse,atthecostofincreasingfeedbackgain)byassigningpoleswithnegativelylargerealpartsHowever,astheresponseismadefasterinthisrespect,itusuallyexhibitsanunboundedlygrowingovershootorundershootthatisunacceptableforpracticalsynthesisToillustratethesituation,consideraplantdescribedbyThestatefeedbackassigningtheclosedlooppoles{p,p)isgivenby#=x,=p,~()Figshowsthephaseplanetrajectoriesoftheclosedloopsystemwithinitialstatex(O)=OlTOneobservesthatthepeakvalueofxgrowsunboundedlyaspincreasesThus,theunboundedovershootisinevitableforalloutputsexceptfory=xlTheaboveproblemofunboundedoutputresponsewasManuscriptreceivedFebruary,revisedSeptember,TheauthoriswiththeDepartmentofControlEngineering,FacultyofEngineeringScience,OsakaUniversity,Toyonaka,Osaka,JapanFigPhasespacetrajectoriesofasecondorderClosedloOpsystemforvariousvaluesofpfirsttreatedtheoreticallybyMitaforsingleinputsingleoutputcasesHeshowedthatthepresenceofthezerosofthetransferfunctionwasthesourceoftheunboundedovershootAccordingto,iftheopenloopsystemiszerofree,wecanincreasetheresponsespeedwithoutcausingexcessiveovershootAnimmediateconsequenceofthisfactisthatthepolezerocancellationpreventsexcessiveoutputresponseinspeedinguptheclosedloopdynamicsTheextensionofthisobservationtomultivariablecasesismoreorlesstheinitialmotivationofthispaperAoneparameterfamilyofthestatefeedbackcontrollawlike()isassumedtorealizeaprescribedsetofrootlociwithrespecttopIfthesquaredintegraloftheoutputconvergestoaspincreases,wesaythatthecontrollawattainstheperfectregulationr)Itisalmostasynonymtothe“infinitelyfastresponse”Similarly,ifthepeakvalueofthenormedoutputresponsecurveremainsboundedasptendstoinfinity,wecallthatthecontrollawsatisfiestheboundedpeaking(bp)Theproblemoftheprhasbeendiscussedsofarmainlyinconnectionwiththesocalledcheapcontrolof$IEEEIEEETRANSACTIONSONAUTOMATICCONTROL,VOLAC,NO,FEBRUARYtheoptimalregulatorllThere,theparameterpin()appearsasrepresentingtherelativeweightontheoutputdeviationtotheweightontheinputpowerinthedefinitionofthecostfunctionalTheprinthiscontextmeansthattheoptimalcosttendstoaspincreasesTheconditionfortheprhasbeenderivedinitsfinalformbyFrancisllTheproblemofthebpwastreatedinthesamecontextinAsdartypeofproblemhasalsobeenstudiedinthefieldofhighgaincontrolThere,theparameterpin(l)appearsasacommonmultiplicativefactorofthegainmatrixSomeinterestingresultsontheasymptoticstructureoftheclosedlooprootlocihavebeenobtained,,buttheasymptoticbehavioroftheclosedloopresponseinthetimedomain,whichisthemainconcernofthedesignobjective,hasbeenstudiedonlyforaspecialcaseInthispaper,weapproachtheproblemoftheprandthebpfromtheviewpointofpoleandeigenvectorassignment,ThetwocharacteristicfeaturesoftheasymptoticclosedloopeigenstructureoftheprwillbedemonstratedFirst,allthefinitetransmissionzerosshouldbecancelledoutexactlyorasymptoticallySecond,acertainnumberoftheclosedlooppolesshouldbeinfinite,theeigensubspacesofwhicharedeterminedaccordingtotheprimestructureoftheplantInthesynthesisofthepr,wecanassignalltheasymptotesoftheinfiniterootlociarbitrarily,aswellastheirorderofgrowthtoThisoffersaconsiderableflexibilityinthesynthesiscomparedwiththecheapoptimalregulator,wherethestructureoftheasymptotesisdeterminedentirelybytheplantitselfSIThisflexibilitydemonstrates,insomesense,asuperiorityofthepoleandtheeigenvectorassignmenttechniquetotheoptimalregulatorAsasynthesismethod,thepoleassignmentismuchmoreintuitiveandcomputationallyeasierthantheoptimalregulatorHowever,sofarithasnotbeentreatedasadesigntoolTheprimaryreasonforthisseemstobethatthereexistmanyfeedbackgainsattainingthesamepoleconfigurationinmultivariablecasesWehavebeenunsuccessfulinfindingtheproperwayofselectingtheoneamongthem,justaswiththeoptimalregulator,wherewehavenodnectwayofdeterminingthecostfunctionalThispaperproposesaprocedurefortheselectionbyspecifyingtheasymptoticpropertiesoftheclosedloopsystem,justaswasdonefortheoptimalregulatorinInSectionIwegivedeprecisedefinitionsoftheprandthebpInSectionIwederivesimplecharacterizationsoftheprandthebpinthefrequencydomainTheyrevealsomeinterestinginvariancepropertiesoftheprandthebpthatprovideakeytoolindevelopingthecomputationalprocedureBasedontheseresults,wederivetheexistenceconditionoftheprbygeometricapproachinSectionIVThisconditionturnsouttobeessentiallyidenticaltotheresultforthecheapoptimalregulatorderivedinllTheresultisgeneralizedinSectionV,whereweconsidertheprandthebpforapairofdifferentoutputsTheresultobtainedtherecontains,inasense,theresultofasaspecialcaseAconditionisalsoderivedunderwhichtheprandthebparerealizedbyoutputfeedbackSectionVIdiscussesthecomputationalaspectsoftheprandthebp,wherethenotionofinteractorintroducedinplaysanessentialroleAnumericalexampleisalsoshowntoillustratetheprocedureInSectionVII,anapplicationtoarealsystemisdiscussed,showingtheeffectivenessofthedesignpMcipledevelopedinthispaperNotationTherealfieldandthecomplexfieldaredenotedbyRand,respectivelyR"denotesthendimensionalvectorspaceonR,Canddenotetheopenlefthalfandtheclosedrighthalfcomplexplain,respectivelyCapitalitalicAdenotesamaporamatrixandthecorrespondingsansserifletterAdenotesitsimageIfAVCVforsomesubspaceV,therestrictionofAonVisdenotedbyAIVimAandkerAdenotetheimageandthekernalofA,respectivelyo(A)denotesthesetofeigevaluesofAThesubspacespannedbyvectorsx,,*,xkisdenotedbyspan{xI,x*,,x}Finally,idenotestheset(,,i)PROBLEMFORMULATIONConsideralinearplantdescribedbyi=AxBu(la)y=cx(lb)wherexEX(R")isthestatevector,UEU(R')istheinputvectorand~EY(R")istheoutputvectorA:XX,B:UXandC:XYareconstantmaps(matrices)Itisassumedthat(A,B)iscontrollableToruleoutthetrivialities,weassumethatBismonicandCisepicLetA,(p),iEn,becomplexvaluedfunctionsdefinedforp>Duetothewellknownresultin,wecanfindcontrollawsparameterizedbyascalarp>u=F,x,:xu,()satisfyingABF,)={MP),Ah),**hm}Aspincreases,Ai(p),iEn,describeasetofnrootloci,whosechoiceiscompletelyatourdisposalWerestrictourattentionentirelytothoserootlocisatisfyingeitherAi(p)yi(pO),YiE()orPIhi(P)Yi(PW),YEQ=()Therelation()impliesthatA,(p)isafiniterootlocuswiththeterminatingpointinCThesetofalltherootlocisatisfying()isdenotedbyA,,Therelation()impliesthatA,(p)isafirstordermfiniterootlocuswiththedirectionoftheasymptoteyiThesetofalltherootlocisatisfying()isdenotedbyAmASfarasthesynthesisisconcerned,thesetwotypesofrootlociaresufficientfortheachievementoftheprHence,wedis~~~KIMURA:REGULATIONANDBOUNDEDPEAKINGINMULTIVARIABLESYSTEMScardalltheinfiniterootlocioflargerorsmaller(thanone)orderthatappearintheanalysisofthehighgaincontrol,Henceforth,weconcentrateon<suchthatU(ABF~)CA,UA,()Wedenotebyuo(AB<)thesubsetofo(AB<)inA,andbyu,(ABFp)thatinA,WesometimesuseaslightlyabusednotationP'U(AB)={p'A,(p),,p'A,(p))NowwemakeanotherimportantassumptiononFp:eachentryofFpisarationalfunctionofpDueto,F,ischosenas<=gg,,g,lfl(~)f(P),"'f(P)'J(~>=(A,(P)zA)'B~,,wherevectorsgiarechosensuchthatJ(p)arelinearlyindependentHence,itisobviousthattheassumptionholdsifA,(p)arerationalfunctionsTherefore,thisassumptiononF,isbynomeansastrongrestrictiononassigningasetofrootlociwithaprescribedasymptoticstructureWedenotethesetofallfeedbackmatrixsatisfyingtheabovetwoassumptionsbyFp(AB),ie,!Fp(A,B)={Fp:a(ABF,)~Ao~A,,~pisrationalinp}Ifthereisnopossibilityofconfusion,wesimplywriteaskP'From()and(),theoutputoftheclosedloopsystemwithinitialstatexiswrittenasYP(~xO)=X(tx(tF,)ACpBFp)'()Nowwedefinetheprandthebpperfectregulationofthesystem(C,A,B)ifDefinitionI:Astatefeedback()inFpattainsthe()wheredenotestheEuclidiannormofamatrixIf()holds,wesimplywrite<(C,A,B)Definition:Astatefeedback()inFpsatisfiestheboundedpeakingforthesystem(C,A,B)if,forsomePo,supSUPIIX(t<()P>Pof>QIf()holds,wesimplywriteFp(C,A,B)Definitioncorrespondstotheprinthecheapoptimalregulator,inwhichtheoptimalcostconvergestoasthepenaltyoninputpowertendstoTherelation()requiresthattheclosedloopresponsebecomesinbpfinitelyfastaspmDefinitionispreciselythesameasthebpwithrespectto,normin,exceptthatinCisconsideredtobetheidentitymatrixItrequiresthatthepeakvalueoftheoutputresponseisuniformlyboundedwithrespecttopOfcourse,astableclosedloopsystemwithboundedpolesonlyalwayssatisfiesthebpTherefore,theproblemofthebparisesonlywhensomeoftheclosedlooppolestendtoinfinityFREQUENCYDOCHARACTERIZATIONOFPRANDBPThemostsuccessfulandestablishedapproachforattackingagroupofproblemsincludingoursisnodoubtthesingularperturbationtheorydevelopedmainlyinthetimedomain,Basedontheassumptionsstatedintheprecedingsection,wecanderiveasimplecharacterizationoftheprandthebpinthefrequencydomainalmostalgebraicallywithouthavingrecoursetosingularperturbationtechniqueDefinetheclosedlooptransferfunctionmatrixfromtheinitialstatetotheoutputbyT(sFp)C(sIABFp)'Notethat,if<EFp,T(sFp)isarationalmatrixinpThisfactplaysanimportantrolethroughoutthesectionSinceT(sFp)istheLaplacetransformofX(tQ,Parseval'sidentityyieldsSincetheLaplacetransformofp*X(p't)isT(pse),againappealingtoParseval'sidentityshowsIntroducinganewvariable=p'tyieldsNowwestatetwoasymptoticpropertiesofaparameterizedrationalmatrix,thefirstofwhichconcernsa"regular"perturbationLemma:LetU(sp)beastrictlypropermatrixins,eachpoleofwhichconvergestoafinitepointinCIfU(sp)convergestoamatrixU(s)forpm,thenL'U(sp)convergesuniformlytoL'U(s),whereL'denotestheinverseLaplacetransformTheproofissimplycarriedoutbyastandardanalyticaltechniqueandisomittedhereLemma:IfFpEFpandT(psF,)O,thenthereexiststrictlypropermatricesUl(sp)andU(sp)inssuchthat()whereUl(sp)convergestoastrictlypropermatrixU,(s)IEEETRANSACTIONSONAUTOMATICCONTROL,VOLAC,NO,FEBRUARYwhosepolesareallinandeachpoleofq(sp),i=,,convergestoafinitepointinProof:SeeAppendixIn(),thefirsttermrepresentsthefast(boundarylayer)modeoftheclosedloopsystem,whilethesecondtermrepresentstheslowmodeNowwearereadytostatethemainresultofthissectionHenceforth,weomitthestatement(poo)inthedescriptionsofthelimitprocessifitisclearfromthecontextTheoremI:F,EffpsatisfiesF,"J'(CyA,B)ifandonlyif,foreachs,T(sF,)o,qpsF,)o()Proof:AssumethatEffpand(C,A,B)Letdet(SIABF,)=,(s~)$~(sp),where$,(sp)and(sp)correspondtoo,(ABF,)andoo(ABF,),respectivelyTheassumption()impliesthat,intherepresentation#,(sp)=ska,(p)sk'*eak(p)withkthenumberofinfiniterootloci,p'a,(p)convergestoapositivenumberforeachjEkThisimpliesthatp)convergestoapositivenumberforeachsObviously,I,L~(~p)convergestoafixedpolynomialNowwewriteqsF,)=asF,),(sP)rco(sP)Dueto()and(),U(jo<),(jwp)OforaeinwFromtheabovereasoning,weconcludethat~~U(joF,)Oforeachw,andtherefore,p(sF,)OforeachsThus,wehaveT(sF,)OforeachsSimilarargumentappliesto()andyieldsT(psF,)OThus,thenecessityhasbeenprovedIf()holdswecandecomposeT(sF,)like()DuetoLemma,~,(p'sp)convergestoafinitematrixU,(O)Therefore,fromthefirstrelationof(),weconcludethatUz(sp)OSinceU,(sp)isarationalmatrixinp,thematrixU(sp)=pU(sp)convergestoafinitestrictlypropermatrixU(s)Hence,wecanrepresentT(sF,)=U,pV(sp)Plca)WsP)Ul(S)YY(sP)U,(S)wherebothU,(s)andU,(s)arestrictlypropermatriceswhosepolesareinLetX,(t,p)=L'q(sp)andX,(t)=L'U,(s),i=,SinceXi(t)representstheoutputofanasymptoticallystablesystem,wehavelmllX,(t)ldt<M,i=l,forsomeconstantMDuetoLemma,X,(tp)convergestoXi(t)uniformlyTherefore,forsufficientlylargep,LmlXi(tp)ldt<M,i=,From(),wehaveX(tF,)=X,(ptp)p'X,(tp)ItfollowsthatI()MLettingpcaestablishesF,(C,A,B)Inananalogousway,wecanderiveaparallelresultforthebpTheorem:F,Effpsatisfies(CyA,B)ifandonlybpif,foreachs,whereT,(s)isastrictlypropermatrixProof:AssumethatF,(C,A,B)Fromthedefinibptionofbp,forallp>poHence,forallResa>,Mlmea'dt=MaLettingaverifiesthatT(s)mustconvergetoastrictlypropermatrixThus,thefirstrelationin()hasbeenshownSinceX(tF,)representstheoutputofanasymptoticallystablelinearsystem,theboundedness()impliesactuallyX(t<MleO'forsufficientlylargep,whereQisanumbersatisfying<CJ<min{Reh,(p)}Therefore,weconcludethat,forsufficientlylargep,forsomenumberM>ODueto(),asimilarargumentusedintheproofof'TheoremshowsT(pjwF,)Oforeachw,andthus,T(psF,)Oforeachs,whichestablishesthesecondrelationof()Assumethat()holdsSincetheconditionofLemmaholds,weconsiderthedecomposition()againThefirsttermvanishesinthelimit,sothat,fromthefirstconditionof(,weconcludethatU,(sp)T,(s)DuetoLemma,X(tp)=L'U(sp)convergesuniformlytox,(t)=L'T=(s)SinceallthepolesofT,(s)arein,sup,,,IIX,(t)ll<MforsomeM>OHence,SUPSUPIIX(CP)II<M()papor>OforsufficientlylargepoAnalogously,X,(p)=LW,(sp)convergesuniformlytox,(t>=L'u,(s),whichisalsouniformlyboundedTherefore,weconcludeKIMURA:REGULATIONANDBOUNDEDPEAKINGINMULTIVARIABLESYSTEMSforsufficientlylargepoTakingtheinverseLaplacetransformof()yieldsx(t<)=X,(pfp)X,(tPITheinequalities()and(),implythatSUPSUPIIX(t)llP>POt>OQsupSUPIIX,(ptP)IISUPSUPIIX,(tP)IIP>Pot,oP>Pot>OQMTheproofisnowcompleteTheoremsandgivecompletecharacterizationsoftheprandthebpinthefrequencydomainSomeanalogousresultsarefoundininrelationtothebpofallthestatetrajectoriesinthecheapoptimalregulatorNowwediscusstheimplicationsoftheabovetheoremsFirst,wenotethatthematrixcanberegardedasastrictlypropermatrixTherefore,()implies()ThisobservationimmediatelyleadstothefollowingCoroZZary:<(CyA,B)alwaysimpliesFpbp(CyA,B)Mathematically,thereexistsasequenceoftimefunctionswhoseLnormconvergeto,whiletheirpeakvaluesgrowunboundedlyTheabovecorollaryshowsthatsuchapathologicalcaseneverappearsintheresponsecurvesoflinearfeedbacksystemsThepralwaysassuresthe“smooth”convergenceoftheresponsecurveRecallthatthesufficiencyproofsofthetheoremsheavily

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