coppeufrjbr basiliodeeufrjbr Abstract Normalization is the 64257rst step in the design of multivariable controllers within the Characteristic Locus Method ie if the plant transfer matrix is far from normal at the frequency band of interest then a no ID: 23620
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- ±°²¯ - K(s) - G(s) - 6 ¡+R (s)C Figure1:Closed-loopfeedbackcontrolsystem.InordertobeabletousethegeneralizedNyquiststabilitycriterionasadesigntool,acon-trollersuchthat)=issought.Thisconditionissatis¯edprovidingthatsharethesameeigenvectoranddualeigenvectorframesand,therefore,theeigenvaluesoftheproductareequaltotheproductoftheeigenvaluesof.Thus,toachieveclosed-loopstabilityandtosatisfyperformancerequirements,suchastrack-ing,disturbancerejectionandgoodtransientre-sponse,onecanmanipulatetheeigenfunctionsoftheopen-looptransfermatrix,choosingade-quatelytheeigenfunctionsofthecontrollerprovidingthattheeigenvectorsofarethose.However,insomecases,thecharacter-isticlocicanbeverysensitivetouncertaintiesintheplanttransferfunctionandtherefore,inthesecases,theCLMisnotareliabletoolforthedesignofamultivariablecontrolsystem.Inor-dertoshowthisDoyleandStein(1981)considerastablematrixperturbation,,atanypointinthecon¯gurationofFig.1,andshowthattheclosed-loopsystemremainsstableifandonlyiffMG(j!multipliedbyyI+T¡1(j!islessthan,forallfrequencies,wherenotesthemaximumsingularvalueofamatrixandisthereturnratiomatrixforthepointwheretheuncertaintyisconsidered.SincetheCLMisbasedontheeigenvaluesoftheopen-looptrans-fermatrix[or,equivalently,andnotonitssingularvalues,thenitisshowninDoyleandStein(1981)thattheCLMcanonlybee®ectivefromtherobustnesspointofviewonlyiftheplanttransfermatrixisnormal.Thede¯nitionofnormalmatrixisasfollows.De¯nition1Amatrixisnormalifitcommuteswithitsconjugatetranspose,InordertodealwiththesensitivityproblemofthecharacteristiclociwithintheCLM,itispro-posedinBasilioandSahate(2000),thedesignofacontrollerintwostages.The¯rststageisthedesignofadynamicnormalizingprecom-pensator,,aprecompensatorwhichap-proximatelynormalizestheprecompensatedplantj!j!atthefrequencyrangeofinterest.Thesecondstageisthedesignofacontroller,,thatcommuteswithandalso Ingeneral,itismorecommontoconsidertheuncer-taintyattheplantinputorattheplantoutput,whichcorrespondsto)=)=,re-spectively.internallystabilizesthefeedbacksystem.Withtheviewtode¯ningapproximatelynormalmatri-ces,itisproposedinBasilioandSahate(2000)thefollowingmeasureofthedeviationfromnor-mality:)= wheredenotestheFrobeniusnorm,which,foramatrixisde¯nedas:wheredenotesthetraceofamatrix.OtherwaysofmeasuringtheapproximatenormalityofacomplexmatrixispresentedinHungandMac-Farlane(1982).Oneofthesemeasuresisbasedonthefollowingtheorem.Theorem1Amatrixisnormalifandonlyifithasanorthonormalsetofeigen-vectors.Proof:SeeHornandJohnson(1990).Theorem1leadstothefollowingde¯nitionofapproximatelynormalmatrices.De¯nition2Amatrixissaidtobeanapproximatelynormalmatrixif)=1+wheredenotesconditionnumber,istheeigenvectormatrixof,andde¯nesthedegreeofnormalityofTheconditionnumberoftheeigenvectorma-trixofj!j!j!j!canbeusedtoprovideanupperboundforameasureoftherobustnessoftheclosed-loopsystemwithrespecttoperturbationsattheplantoutputandinput,respectively.Thiswillbedonewiththehelpofthefollowingresult.Lemma2(SmallGainTheorem)De¯ne)=[astheplanttransfermatrixwithmultiplica-tiveperturbationattheplantoutput,whereeMG(j!,andisanon-negativerealfunction.Then,stabilizesifandonlyifstabilizesj!j!)[j!j! forallProof:SeeVidyasagar(1985).Itiswellknownthatsingularvaluesareupperboundsforthemodulioftheeigenvaluesofama-trix,andthus,anecessaryconditionforrobuststabilityoftheclosed-loopsystem,givenintermsoftheeigenvaluesoftheopen-loopsystemateachfrequency,canbegivenas:j!j! 1+j!j! ¹G(!);(5) beasingularvaluedecompositionof,where§=;i=1;:::;m.ThenKGarebothnormalifandonlyifforsomenonsingulardiagonalmatrixProof:SeeHungandMacFarlane(1982).Accordingtolemma5,thecharacteristiclociareattheirleastsensitivitytosmallperturba-tionsattheplantinputandoutputifandonlyifthecontrollerhasthestructuregivenbyEq.(12).However,inthispaper,thecontrollerisde¯nedbyEq.(10),whichimpliesthatmustsatisfysomeconditionssuchthatbebothnormal,asstatedbelow.Theorem6Supposearebothofrankandletthesingularvaluedecompo-sitionofbegivenbyEq.(11).Inadditionlet,wherecommutesexactlywith.Then,KGarebothnormalmatri-cesifandonlyifforsomenonsingulardiagonalmatrixProof:SeeMoreiraandBasilio(2005).Accordingtotheorem6,atagivenfre-quency,toachievetheexactnormalityofj!j!j!j!oneshouldchooseanynonsingulardiagonalmatrixobtainaccordingtoEq.(13).However,inthispaper,itisconsideredthecomputationofanormalizingstaticprecompensator,whichimpliesthatmustbearealmatrix.Itcanbeveri¯edthat,sincej!j!,then,onlyforspecialcasesitispossibletoobtainreal.InHungandMacFarlane(1982),thedesignofastaticprecompensatorwiththestructuregivenbyEq.(13)isconsideredintwocaseswhereitispos-sibletoobtainrealprecompensators,namelythat,atd.c.frequency(=0)andatveryhighfre-quencies(!1).Inthesesituationsj!j!.However,theneedfornormal-izationismorecriticalatfrequencieswherethecharacteristiclociareclosetothecriticalpoint1+,atintermediatefrequencies,asex-plainedinsection2.Althoughexactnormalitiesofj!j!j!j!atintermediatefrequenciesareusuallynotachievable,itismorerealistictoaimatapproximatenormality.Thefollowingresultsrelatestheapproximatenormalityofj!j!j!j!totheapproximatenormalityofj!j!,respectively.Theorem7LetbegivenbyEq.(10)andletbeagivenfrequency.Thenj!j!isapproximatelynormalifj!isapproximatelynormal.Proof:Theproofisstraightforwardandwillbeomitted.Theorem8Underthesameassumptionsofthe-orem7,andassumingthatalleigenvaluesofj!aredistinct,thenj!j!isalsoapproximatelynormalifj!isapproxi-matelynormal.Proof:Letasingularvaluedecompositionofj!begivenbyEq.(11)andde¯nej!)=j!j!foranonsingularmatrix.FromEqs.(11)and(14),itfollowsthat:UMYMYSupposenowthathasthefollowingspectraldecomposition:whereisadiagonalmatrix,isaneigen-vectormatrixof.Then,ac-cordingtoEqs.(15)and(16),canbewrittenYWSince,byassumption,commutesexactlywith,ithasthesameeigenvectormatrixasand,therefore,aspectraldecompositionofbegivenas:YWwhereisadiagonalmatrixformedwiththeeigenvaluesof.Thus,accordingtoEqs.(14),(18)and(11),KGcanbewrittenas:KGUMWNoticethat,accordingtoEq.(16),§=,andthus,Eq.(19),turnsouttobe:KGUMWThus,accordingtoEq.(20),foreverycommuta-tivecontroller,aneigenvectormatrixofKGgivenbyUMW.Inparticular,for,aspectraldecompositionofisgivenby:UMWSupposingthatalleigenvaluesofaredistinct,thentheeigenvectormatrixofisequaltoUMWscaledtohavetheleastconditionnum-beraspossible(BraatzandMorari,1994).There-fore,iftheeigenvectormatrixofhascondi-tionnumberapproximatelyequaltoone,,ifisapproximatelynormal,thenKGisalsoapproximatelynormal.Theorems6,7and8leadtotheformulationofthefollowingoptimizationproblemwiththeview 4ExamplesConsiderthetransfermatrixofanautomotivegasturbine(NobakhtiandMunro,2004))==gij(s)],where)=+0 +1+0)= +12+13+2)=+2+0 +9+9+1)=+25+9 +20+116+111+18andsuppose,asspeci¯edinNobakhtiandMunro(2004),thattheclosed-loopbandwidthbeequalto1rad/s.Theconditionnumberoftheeigenvectorma-trixofj!isshowninFig.2(solidline),whereitcanbeseenthatitislargerthanoneatthede-siredbandwidthfrequency(itisapproximately2.6at0.4rad/s).Asaconsequence,itisnecessarytodesignanormalizingprecompensator.1)=beasingularvaluedecompositionof,wherearegivenas:1)=5612+08857+01)=5356+0Formingthematrix,accordingtoEq.(32):3069009210598400248006410017405237010370andcomputingnowthesmallesteigenvalueof,andtheeigenvectorassociatedwith ,oneobtains:=0 6514052380Therefore,theoptimalnormalizingprecompen-satorisgivenby:Theconditionnumbersoftheeigenvectormatricesj!j!arealsobothequalforallfrequencies,and,inFig.2,itcanbeseenthattheconditionnumberoftheeigenvectormatrixofj!(dash-dottedline),isequaltooneat1rad/sasdesired,andalsoapproximatelyequaltooneatthevicinityof1rad/s.Therefore,theCLMcannowbee®ectivelyappliedtotheprecompen-satedplantAcknowledgementThisworkhasbeenpartiallysupportedbytheBrazilianResearchCouncil(CNPq). Figure2:Conditionnumbersoftheeigenvec-tormatricesofj!(solidline),andj!(dash-dottedline)forexample2.ReferencesBasilio,J.C.andSahate,J.A.(2000).Anormalizingprecompensatorforthedesignofe®ectiveandreli-ablecommutativecontrollers,InternationalJournalofControl:1280{1297.Braatz,R.D.andMorari,M.(1994).Minimizingtheeuclidianconditionnumber,SIAMJ.ControlandOptimization32:1763{1768.Doyle,J.C.andStein,G.(1981).Multivariablefeed-backdesign:Conceptsforaclassic/modernsyn-thesis,IEEE-TransactionsonAutomaticControl:4{16.Horn,R.A.andJohnson,C.R.(1990).MatrixAnalysisCambridgeUniversityPress.Hung,Y.S.andMacFarlane,A.G.J.(1982).MultivariableFeedback:AQuasi-ClassicalApproach,Springer-Verlag.MacFarlane,A.G.J.andBelletruti,J.J.(1973).Thechar-acteristiclocusdesignmethod,Automatica:575{588.MacFarlane,A.G.J.andPostlethwaite,I.(1977).Thegen-eralizednyquiststabilitycriterionandmultivariablerootloci,InternationalJournalofControl:81{127.Moreira,M.V.(2006).Umacontribui»c~aopormeiodepr¶e-compensa»c~aonormalizanteeparametriza»c~aoparaoefetivoprojetodecontroladoresutilizandoom¶etododolugarcaracter¶³stico,D.Sc.Thesis,UFRJ/COPPE-ProgramadeEngenhariaEl¶etrica.Moreira,M.V.andBasilio,J.C.(2004).Controladorescomutativosestabilizantesparaplantasinst¶aveis,CongressoBrasileirodeAutom¶atica,Gramado:RS.Moreira,M.V.andBasilio,J.C.(2005).Designofnormal-izingprecompensatorsviaalignmentofoutput-inputprincipaldirections,44thIEEEConferenceonDeci-sionandControlandEuropeanControlConferenceSevilla:Spain,pp.2170{2175.Moreira,M.V.,Basilio,J.C.andKouvaritakis,B.(2005).Rationalstabilizingcommutativecontrollersforun-stableplant,44thIEEEConferenceonDecisionandControlandEuropeanControlConference,Sevilla:Spain,pp.7864{7869.Moreira,M.V.,Basilio,J.C.andKouvaritakis,B.(2006).Rationalstabilisingcommutativecontrollers:para-meterisationandcharacterisationofdegreesoffree-dom,InternationalJournalofControl(acceptedforpublication)Nobakhti,A.andMunro,N.(2004).Anewmethodforsin-gularvalueloopshapingindesignofmultiple-channelcontrollers,IEEE-TransactionsonAutomaticCon-trol49:249{253.Postlethwaite,I.(1982).Sensitivityofthecharacteristicgainloci,Automatica:709{712.Vidyasagar,M.(1985).ControlSystemSynthesis:AFac-torizationApproach,MITPress.Wilkinson,J.H.(1965).TheAlgebraicEigenvalueProb-lem,ClarendonPress,Oxford. OPTIMALDESIGNOFSTATICNORMALIZINGPRECOMPENSATORSMarcosVicenteMoreira,JoaoCarlosBasilioCOPPE-ProgramadeEngenhariaEl¶etricaUniversidadeFederaldoRiodeJaneiroRiodeJaneiro,RJ,BrasilEmails:moreira@pee.coppe.ufrj.br,basilio@dee.ufrj.brAbstract|Normalizationisthe¯rststepinthedesignofmultivariablecontrollerswithintheCharacteristicLocusMethod,,iftheplanttransfermatrixisfarfromnormalatthefrequencybandofinterest,thenanormalizingprecompensatorshouldbedesignedinordertomaketheprecompensatedplantapproximatelynormalatthisfrequencyband.Thisproblemisaddressedinthispaper,whereanoptimalstaticnormalizingprecompensatorisproposed.Thedesignstrategyisbasedonanoptimizationproblemwithnoconstraintsontheprecompensatorstructure,andwhosesolutioncanbeobtaineddirectlyfromaneigenvalueproblem.Keywords|Linearsystems,Multivariablesystems,Optimization,Frequencydomain.1IntroductionTheCharacteristicLocusMethod(CLM)isanef-fectivetoolforthedesignofmultivariablecon-trolsystems,allowingcon°ictingdesignobjec-tives,suchasstabilityandperformance,bead-dressedbyasinglesystemrepresentation:namelythat,thefrequencyresponseoftheeigenfunctionsoftheopen-loopsystem(characteristicloci).TheessenceoftheCLM(MacFarlaneandBelletruti,1973)istoconstructacommutativecontroller,,acontrollerwiththesameeigen-vectoranddual-eigenvectormatrices(frames)astheplanttransfermatrixandtomanipulatethecontrollereigenfunctionssoastoachieveclosed-loopstability,satisfyingthegeneralizedNyquiststabilitycriterion(MacFarlaneandPostlethwaite,1977),andtosatisfyperformancerequirementssuchastracking,disturbancerejectionandgoodtransientresponse.However,itiswellknown(DoyleandStein,1981;Postlethwaite,1982;Wilkinson,1965)thatforplantswhosefrequencyresponsesarenotnor-malatacertainfrequencyband,thecharacter-isticlociareverysensitivetoperturbationsattheplantinputandoutputatthesefrequencies.Therefore,inordertodealwiththecharacteristiclocussensitivityproblem,thedesignofmultivari-ablecontrollerswithintheCLM,mustbedoneintwostages(BasilioandSahate,2000;MoreiraandBasilio,2005):¯rst,precompensatetheplantinordertoapproximatelynormalizeitinthenec-essaryfrequencyrangeandthen,applytheCLMtotheprecompensatedsystem.Aprecompensationschemewiththeviewtomakingtheprecompensatedplantasnormalaspossiblehasrecentlybeenpresented(BasilioandSahate,2000),wherethedesignofanor-malizingdynamicprecompensatorisproposed.ThemaindisadvantagesoftheprecompensationmethodproposedinBasilioandSahate(2000)are:thecharacteristiclocussensitivitywithre-specttoperturbationsattheplantinputarenotconsideredintheformulationoftheoptimizationproblem,andanoptimizationproblemisfor-mulatedandsolvedforaspeci¯cprecompensatorstructure.Withtheviewtoconsideringthesensitivityofthecharacteristiclociwithrespecttopertur-bationsatboththeplantinputandoutput,itisproposedinMoreiraandBasilio(2005)anopti-mizationproblembasedonameasureofthemis-alignmentbetweentheinputandoutputprincipaldirectionsoftheplant.Althoughthisprecompen-sationschemehasledtoinsensitivecharacteris-ticlocifortheprecompensatedplantwhenanex-actcommutativecontrollerisconsidered,italsoimposesconstraintsontheprecompensatorstruc-ture.Inthispaper,thedesignofanoptimalsta-ticprecompensatorwiththeviewtomakingtheprecompensatedplantapproximatelynormalatthevicinityofthecrossoverfrequencyispro-posed.Thisnewmethodisbasedonanoptimiza-tionproblemwhichhasnoconstraintsonthepre-compensatorstructure.Thesolutiontotheop-timizationproblemisverysimpleandcanbeex-pressedasaneigenvalueproblem.Anotheradvan-tageofthismethodisthatitimprovesthechar-acteristiclocussensitivityindiceswithrespecttoperturbationsatboththeplantinputandoutputwhenanexactcommutativecontroller(MoreiraandBasilio,2004;Moreiraetal.,2005;Moreiraetal.,2006)isconsidered.2Mainconceptsbethetransfermatricesoftheplantandcontroller,respectively.Accord-ingtothegeneralizedNyquiststabilitycriterion(MacFarlaneandPostlethwaite,1977),thefeed-backsystemofFig.1willbestableifandonlyifthenetsumofanti-clockwiseencirclementsofthecriticalpoint1+,bythecharacteristicloci,isequaltothenumberofunstablepolesof - ±°²¯ - K(s) - G(s) - 6 ¡+R (s)C Figure1:Closed-loopfeedbackcontrolsystem.InordertobeabletousethegeneralizedNyquiststabilitycriterionasadesigntool,acon-trollersuchthat)=issought.Thisconditionissatis¯edprovidingthatsharethesameeigenvectoranddualeigenvectorframesand,therefore,theeigenvaluesoftheproductareequaltotheproductoftheeigenvaluesof.Thus,toachieveclosed-loopstabilityandtosatisfyperformancerequirements,suchastrack-ing,disturbancerejectionandgoodtransientre-sponse,onecanmanipulatetheeigenfunctionsoftheopen-looptransfermatrix,choosingade-quatelytheeigenfunctionsofthecontrollerprovidingthattheeigenvectorsofarethose.However,insomecases,thecharacter-isticlocicanbeverysensitivetouncertaintiesintheplanttransferfunctionandtherefore,inthesecases,theCLMisnotareliabletoolforthedesignofamultivariablecontrolsystem.Inor-dertoshowthisDoyleandStein(1981)considerastablematrixperturbation,,atanypointinthecon¯gurationofFig.1,andshowthattheclosed-loopsystemremainsstableifandonlyiffMG(j!multipliedbyyI+T¡1(j!islessthan,forallfrequencies,wherenotesthemaximumsingularvalueofamatrixandisthereturnratiomatrixforthepointwheretheuncertaintyisconsidered.SincetheCLMisbasedontheeigenvaluesoftheopen-looptrans-fermatrix[or,equivalently,andnotonitssingularvalues,thenitisshowninDoyleandStein(1981)thattheCLMcanonlybee®ectivefromtherobustnesspointofviewonlyiftheplanttransfermatrixisnormal.Thede¯nitionofnormalmatrixisasfollows.De¯nition1Amatrixisnormalifitcommuteswithitsconjugatetranspose,InordertodealwiththesensitivityproblemofthecharacteristiclociwithintheCLM,itispro-posedinBasilioandSahate(2000),thedesignofacontrollerintwostages.The¯rststageisthedesignofadynamicnormalizingprecom-pensator,,aprecompensatorwhichap-proximatelynormalizestheprecompensatedplantj!j!atthefrequencyrangeofinterest.Thesecondstageisthedesignofacontroller,,thatcommuteswithandalso Ingeneral,itismorecommontoconsidertheuncer-taintyattheplantinputorattheplantoutput,whichcorrespondsto)=)=,re-spectively.internallystabilizesthefeedbacksystem.Withtheviewtode¯ningapproximatelynormalmatri-ces,itisproposedinBasilioandSahate(2000)thefollowingmeasureofthedeviationfromnor-mality:)= wheredenotestheFrobeniusnorm,which,foramatrixisde¯nedas:wheredenotesthetraceofamatrix.OtherwaysofmeasuringtheapproximatenormalityofacomplexmatrixispresentedinHungandMac-Farlane(1982).Oneofthesemeasuresisbasedonthefollowingtheorem.Theorem1Amatrixisnormalifandonlyifithasanorthonormalsetofeigen-vectors.Proof:SeeHornandJohnson(1990).Theorem1leadstothefollowingde¯nitionofapproximatelynormalmatrices.De¯nition2Amatrixissaidtobeanapproximatelynormalmatrixif)=1+wheredenotesconditionnumber,istheeigenvectormatrixof,andde¯nesthedegreeofnormalityofTheconditionnumberoftheeigenvectorma-trixofj!j!j!j!canbeusedtoprovideanupperboundforameasureoftherobustnessoftheclosed-loopsystemwithrespecttoperturbationsattheplantoutputandinput,respectively.Thiswillbedonewiththehelpofthefollowingresult.Lemma2(SmallGainTheorem)De¯ne)=[astheplanttransfermatrixwithmultiplica-tiveperturbationattheplantoutput,whereeMG(j!,andisanon-negativerealfunction.Then,stabilizesifandonlyifstabilizesj!j!)[j!j! forallProof:SeeVidyasagar(1985).Itiswellknownthatsingularvaluesareupperboundsforthemodulioftheeigenvaluesofama-trix,andthus,anecessaryconditionforrobuststabilityoftheclosed-loopsystem,givenintermsoftheeigenvaluesoftheopen-loopsystemateachfrequency,canbegivenas:j!j! 1+j!j! ¹G(!);(5) Inordertoobtainnowasu±cientcondition,thefollowingresultisneeded.Lemma3Letbeasquarematrixoforderandsupposethat,where,isaspectraldecompositionof.Then, Proof:Noticethat=¹g· ¾ Lemmas2and3provideasu±cientconditionfortherobustnessoftheclosed-loopsystem,asfollows.Theorem4stabilizesj!j! 1+j!j! )[1+foreach,where)==W(j!withj!denotingtheeigenvectormatrixofj!j!,thenstabilizes,givenbyEq.(3).Proof:SeeMoreira(2006)Remark1Similarresultstothosegiveninthe-orem4andinequality(5)canbeobtainedcon-sideringamultiplicativeperturbationattheplantinput,)==I+MG(s)].Inthiscasetheupperboundsarebasedontheeigenvaluesofj!j!andontheconditionnumberoftheeigenvectormatrixofj!j!Lemma2,theorem4andinequality(5)pro-videallthenecessaryinformationforthedesignofreliablecommutativecontrollers,asfarasnormal-ityoftheopen-looptransfermatrixisconcerned.Noticethat:Athighfrequenciesj!j!,andthussG(j!j!=1;:::;mTherefore,conditions(5)and(8),become,re-spectively:² ² (1+;²whicharesatis¯edforarbitrarilylargevalues.Thisimpliesthatthereisnoneedfornormalizationathighfrequencies.Atlowfrequencies,duetointegralaction,j!j!becomesin¯niteandthus,ac-cordingtoinequality(4),becomesanecessaryandsu±cientconditionfortherobustnessoftheclosed-loopsystem.Thisimpliesthatthesu±cientconditiongivenbyinequality(8)becomes º(!)]1,andthus,normalizationisnotacrucialre-quirementforlowsensitivityofthecharac-teristiclociatlowfrequencies.Atthefrequenciesnearthecrossoverfre-quency,condition(8)isalsonecessarywhen,ifj!j!becomesapproximatelynormal.Thediscussionaboveallowsustoconcludethatitsu±cestouseastaticprecompensator,in-steadofadynamicnormalizingprecompensatorasdesignedinBasilioandSahate(2000)andMoreiraandBasilio(2005).Aspointedoutintheintroduction,normaliza-tionshouldbeunderstoodasthe¯rststepinthedesignofcommutativecontrollers,,ifj!isnotnormalatthefrequencybandnearthecrossoverfrequency,thenastaticnormalizingpre-compensatorwillbedesignedinordertomakej!approximatelynormalinthisfrequencyband.Afterthat,acontrollerthatcom-muteswithmustbedesigned.Intheend,thecontrollertobeimplementedwillbegivenas:)=Thispaperdealswiththedesignofanopti-malstaticprecompensatorthatmakesbothj!j!j!j!approxi-matelynormalatthefrequencyrangeofinterest.3Anormalizingprecompensator3.1ProblemformulationInthispaper,itisassumedmultiplicativepertur-bationstomodeluncertaintiesintheplanttrans-fermatrix.Thischoiceofrepresentationisprefer-ableoveranadditiverepresentation(DoyleandStein,1981),sincethesameuncertaintyrepresen-tationcanbeappliedtoboththeplantthecompensatedsystem.Asaconse-quence,robustnessoftheclosed-loopsystemmaybeconsideredonlyatthe¯nalstageofthecon-trollerdesign,namely,thedesignofthecommu-tativecontrollerWiththeviewtoconsideringthenormaliza-tionofbothitisin-troducedinHungandMacFarlane(1982)thesocalledreversed-frame-normalizing-controllers(RFNC),whereitcanbeseenthatbotharenormalifandonlyifthesingular-vectorframesofarethoseoftakeninreversedorder,asstatedinthefollowinglemma.Lemma5Supposearebothofrankandlet beasingularvaluedecompositionof,where§=;i=1;:::;m.ThenKGarebothnormalifandonlyifforsomenonsingulardiagonalmatrixProof:SeeHungandMacFarlane(1982).Accordingtolemma5,thecharacteristiclociareattheirleastsensitivitytosmallperturba-tionsattheplantinputandoutputifandonlyifthecontrollerhasthestructuregivenbyEq.(12).However,inthispaper,thecontrollerisde¯nedbyEq.(10),whichimpliesthatmustsatisfysomeconditionssuchthatbebothnormal,asstatedbelow.Theorem6Supposearebothofrankandletthesingularvaluedecompo-sitionofbegivenbyEq.(11).Inadditionlet,wherecommutesexactlywith.Then,KGarebothnormalmatri-cesifandonlyifforsomenonsingulardiagonalmatrixProof:SeeMoreiraandBasilio(2005).Accordingtotheorem6,atagivenfre-quency,toachievetheexactnormalityofj!j!j!j!oneshouldchooseanynonsingulardiagonalmatrixobtainaccordingtoEq.(13).However,inthispaper,itisconsideredthecomputationofanormalizingstaticprecompensator,whichimpliesthatmustbearealmatrix.Itcanbeveri¯edthat,sincej!j!,then,onlyforspecialcasesitispossibletoobtainreal.InHungandMacFarlane(1982),thedesignofastaticprecompensatorwiththestructuregivenbyEq.(13)isconsideredintwocaseswhereitispos-sibletoobtainrealprecompensators,namelythat,atd.c.frequency(=0)andatveryhighfre-quencies(!1).Inthesesituationsj!j!.However,theneedfornormal-izationismorecriticalatfrequencieswherethecharacteristiclociareclosetothecriticalpoint1+,atintermediatefrequencies,asex-plainedinsection2.Althoughexactnormalitiesofj!j!j!j!atintermediatefrequenciesareusuallynotachievable,itismorerealistictoaimatapproximatenormality.Thefollowingresultsrelatestheapproximatenormalityofj!j!j!j!totheapproximatenormalityofj!j!,respectively.Theorem7LetbegivenbyEq.(10)andletbeagivenfrequency.Thenj!j!isapproximatelynormalifj!isapproximatelynormal.Proof:Theproofisstraightforwardandwillbeomitted.Theorem8Underthesameassumptionsofthe-orem7,andassumingthatalleigenvaluesofj!aredistinct,thenj!j!isalsoapproximatelynormalifj!isapproxi-matelynormal.Proof:Letasingularvaluedecompositionofj!begivenbyEq.(11)andde¯nej!)=j!j!foranonsingularmatrix.FromEqs.(11)and(14),itfollowsthat:UMYMYSupposenowthathasthefollowingspectraldecomposition:whereisadiagonalmatrix,isaneigen-vectormatrixof.Then,ac-cordingtoEqs.(15)and(16),canbewrittenYWSince,byassumption,commutesexactlywith,ithasthesameeigenvectormatrixasand,therefore,aspectraldecompositionofbegivenas:YWwhereisadiagonalmatrixformedwiththeeigenvaluesof.Thus,accordingtoEqs.(14),(18)and(11),KGcanbewrittenas:KGUMWNoticethat,accordingtoEq.(16),§=,andthus,Eq.(19),turnsouttobe:KGUMWThus,accordingtoEq.(20),foreverycommuta-tivecontroller,aneigenvectormatrixofKGgivenbyUMW.Inparticular,for,aspectraldecompositionofisgivenby:UMWSupposingthatalleigenvaluesofaredistinct,thentheeigenvectormatrixofisequaltoUMWscaledtohavetheleastconditionnum-beraspossible(BraatzandMorari,1994).There-fore,iftheeigenvectormatrixofhascondi-tionnumberapproximatelyequaltoone,,ifisapproximatelynormal,thenKGisalsoapproximatelynormal.Theorems6,7and8leadtotheformulationofthefollowingoptimizationproblemwiththeview to¯ndingarealprecompensatorthatapprox-imatelynormalizesj!andj!atagivenfrequencysubjecttobeanonsingulardiago-nalmatrix,and,wherej!j!areobtained,accordingtoEq.(11),fromasingularvaluedecompositionofj!3.2SolutiontotheoptimizationproblemDe¯nethecostfunctionofproblem(22)as:©)=Then,applyingthede¯nitionofFrobeniusnorm,givenbyEq.(2),inEq.(23),andaftersomesimplealgebraicmanipulations,oneobtains:©)=Re(24)whereRedenotestherealpartofacomplexmatrix.Let),and)bethereal(imaginary)partsof,and,respec-tively.Thus:j!)=j!)+jUj!j!)=j!)+jYj!j!)=©j!)+j!UsingEqs.(25),thenitispossibletowriteEq.(24)asfollows:©)=whereRee(UTR¡jUjYAftersomesimplealgebraicmanipulationsthenEq.(27)canbewrittenas:)+)+(28)Noticethateachtermoftheright-handsideofEq.(28)hasthesamestructuregivenby,whereareequaltotherealorimaginarypartsof,respec-tively.De¯ne,now,thevector asthecolumnvectorformedwiththediagonalentriesofandlet =1;:::;m,denotethecolumnsofandformthevector =hk T1k :::k .Thus,denot-ingthecolumnsofby ,for=1;:::;mtheelementsofby,thenitcanbeveri¯edthat:©)= TA^Á where 1^y11^u ::: m^y1m^u 1y21^u ::: m^y2m.........^u 1^ym1^u ::: Therefore,de¯ningthevectors RandÁ asthecolumnvectorsformedwiththediagonalentriesand,respectively,andformingthevector =hÁ TRÁ ,thenEq.(24)canberewrittenas: ;Á )= Tk ¡2k T~AÁ +Á TÁ whereRRIIRIIRwiththematricesRRIIIRRIformedinaccordancewithEq.(30),wherethe¯rstandsecondindexesrepresent,respectively,thereal(R)orimaginary(I)partofFromEq.(31),itcanbeseenthatifone¯xesthevaluesoftheelementsofvector ,thenthecostfunction ;Á describesaparaboloidwithminimumgivenby: )= TÁ ¡k Tk where =~AÁ ,and,therefore,theproblemofmin-imizing ;Á ,givenbyEq.(31),isequivalenttothefollowingoptimizationproblem: Á T(I¡~AT~A)Á subjectto besuchthatisanonsingularma-trix.Remark2Noticethatthecostfunctionofprob-lem(34)isequivalenttothenormde¯nedinEq.(23).Therefore,thematrixis,ingen-eral,positivede¯nite(itcanalsobepositivesemi-de¯nite).Sincemustbeanonsingularmatrix,thentheEuclidean-normofthevector mustbedif-ferentfromzero.Thus,supposethat =1wheredenotestheEuclidean-norm.Then,usingtheRayleigh'sprinciple(HornandJohn-son,1990)andremark2,itcanbeseenthattheminimumvalueforproblem(34)isachievedfor =v ,where denotestheeigenvectorassoci-atedwiththesmallesteigenvalue,,ofthema-trix.Moreover,theminimumvalueofthecostfunctionofproblem(34)isgivenbyRemark3Ingeneral,theeigenvector associ-atedwiththesmallesteigenvalueleadstoamatrixnonsingular.However,ifthematrixassociatedwith issingular,thenitisalwayspossibletomakealinearcombinationoftheeigenvectorsof,inordertoobtainavector ,suchthat T(I¡~AT~A)¹Á obtainedfrom ,benonsingular.Thus, mustbenormalizedleadingtoacostvalue )= ,where s=¹Á =k¹Á k. 4ExamplesConsiderthetransfermatrixofanautomotivegasturbine(NobakhtiandMunro,2004))==gij(s)],where)=+0 +1+0)= +12+13+2)=+2+0 +9+9+1)=+25+9 +20+116+111+18andsuppose,asspeci¯edinNobakhtiandMunro(2004),thattheclosed-loopbandwidthbeequalto1rad/s.Theconditionnumberoftheeigenvectorma-trixofj!isshowninFig.2(solidline),whereitcanbeseenthatitislargerthanoneatthede-siredbandwidthfrequency(itisapproximately2.6at0.4rad/s).Asaconsequence,itisnecessarytodesignanormalizingprecompensator.1)=beasingularvaluedecompositionof,wherearegivenas:1)=5612+08857+01)=5356+0Formingthematrix,accordingtoEq.(32):3069009210598400248006410017405237010370andcomputingnowthesmallesteigenvalueof,andtheeigenvectorassociatedwith ,oneobtains:=0 6514052380Therefore,theoptimalnormalizingprecompen-satorisgivenby:Theconditionnumbersoftheeigenvectormatricesj!j!arealsobothequalforallfrequencies,and,inFig.2,itcanbeseenthattheconditionnumberoftheeigenvectormatrixofj!(dash-dottedline),isequaltooneat1rad/sasdesired,andalsoapproximatelyequaltooneatthevicinityof1rad/s.Therefore,theCLMcannowbee®ectivelyappliedtotheprecompen-satedplantAcknowledgementThisworkhasbeenpartiallysupportedbytheBrazilianResearchCouncil(CNPq). 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