Design of normalizing precompensators via alignment of outputinput principal directions - PDF document

Designofnormalizingprecompensatorsviaalignmentofoutput-inputprincipaldirectionsMarcosV.MoreiraandJoaoC.Basilio„NormalizationisacrucialrequirementfortheeffectivenessofmultivariablecontrolsystemdesignwithintheCharacteristicLocusMethod.Previousworkaddressesthisproblembysolvinganoptimizationproblemformulatedinordertoincreasenormality;itsformulation,however,donotconsidertheclosed-loopsystemrobustnesswithrespecttoper-turbationsattheplantinput.Inthispaperadifferentapproachtothedesignofnormalizingprecompensatorswillbeproposed.Itisbasedontheminimizationofacostfunctionrepresenting  K(s)
G(s)
 Fig.1.Closed-loopfeedbackcontrolsystemCLM,beingcomposedoftwostages:“rst,precompensatetheplantinordertoapproximatelynormalizeitinthenecessaryfrequencyrangeandthen,applytheCLMtotheprecompensatedsystem.Aprecompensationschemewiththeviewtomakingtheprecompensatedplantasnormalaspossiblehasrecentlybeenpresented[3],whereitwasproposedanormalizing Besidestheunavoidableproblemofirrationaleigenvectormatrices,theCLMmayalsosufferfromsensitivityproblemwhentheplanttransfermatrixisfarfromnormallThismeansthatwhentheplantisfarfromnormalatacertainfrequencyband,itisnecessarytodesignaprec-ompensatorinordertonormalizetheprecompensatedplant.Onceisapproximatelynormalatthenecessaryfrequencyrange,thenacommutativecontrollercanbedesignedeffectivelyviatheCLM.Therefore,thedesignofanormalizingprecompensatormustbethe“rststageofthedesignofmultivariablecontrolsystemswithintheCLM[3].Theimportanceofnormalityforthesensitivityofthecharacteristiclociispresentedin[5]and[15].Supposingastablematrixperturbation,atanypointinthecon“gurationofFig.1,itisshownin[5]thattheclosed-loopsystemremainsstableifthemaximumsingularvaluemultipliedbythemaximumsingularvalueofofI+T(j
)Š1]Š1islessthan1,forallfrequencies,whereisthereturnratiomatrixforthepointwheretheloopwasbroken.Theconsequenceofthisfactisthat[15]thecharacteristiclociareleastsensitivetoperturbationsattheplantoutputandinput,ifthereturnratiomatricesforthosepoints,namely,respectively,arebothnormalatthenecessaryfrequencyrange.Noticethatifisnormalforallfrequenciesandifisanexactcommutativecontroller,thenarenormal.However,ifisnotnormalatacertainfrequencyband,andanormalizingprecompensatorisdesigned,thenitisnecessarytodesignacontrollerwhichcommuteswithandthusthecontrollertobeimplementeddoesnotnecessarilyleadtorobustnesswithrespecttosmallperturbationsattheplantinput.Toovercomethisproblem,thedesignofaprecompensatorthatmakesbothproximatelynormalisproposedinthispaper.III.ANORMALIZINGPRECOMPENSATORA.ProblemformulationAdirectapproachtotheproblemofdesigninganormal-izingprecompensatorforaplanthasbeenpresentedin[3],whereaprecompensatorhasbeendesignedtomakethetransfermatrixapproximatelynormalinthenecessaryfrequencyrange.Themeasureofthedeviationfromnormalityofacomplexmatrix,usedin[3],hasbeende“nedas: denotestheFrobeniusnorm,which,foramatrixisde“nedas:Amatrixisnormalifitcommuteswithitsconjugatetranspose,denotesthetraceofamatrix.Theidealfrequencyresponseoftheprecompensator(),heredenotedas,hasbeenobtained,bysolving,fora“nitenumberoffrequencies,...,,theoptimizationproblemminnG(j
k)Kp(j
k)],subjecttoconstraintsonthepre-compensatorstructureandonthemodulusofitsentries.Inthesequel,rationalandstabletransferfunctionshavebeenobtainedfortheelementsofwiththeviewtoapproximatingthefrequencyresponseofitselementstothoseobtainedfor.Thestructureforusedin[3]isofapermuteddiagonalmatrixwithconstraintsonthemodulusofitsentriestoguaranteethatthemaximumsingularvalueoftheprecompensatorbelessthanorequalto1.Oneofthemaindrawbacksofthisprecompensationmethodisthatthesensitivityofthecharacteristiclocitoperturbationsattheplantinput,whenhasnotbeenconsidered.Indeed,ifisnormalatacertainfrequency,thenisnor-mal,butisnotnecessarilynormalandthecharacteristiclocicanbeverysensitivetoperturbationsattheplantinput.Withtheviewtoconsideringthenormalizationofbothitwasintroducedin[8]thesocalledreversed-frame-normalizing-controllers(RFNC),whosetheoreticaljusti“cationisgivenbelow.Lemma1:Supposearebothofrankandletbeasingularvaluedecompositionof,wherediag,...,.ThenarebothnormalifandonlyifforsomenonsingulardiagonalmatrixProof:See[8].Accordingtolemma1,thecharacteristiclociareattheirleastsensitivetosmallperturbationsattheplantinputandoutputifandonlyifthesingularvectorframesofarethoseoftakeninreversedorder.However,inthispaper,thecontrollerisde“nedbyEq.(1),whichimpliesthatmusthaveaspeci“cstructuresothatbebothnormalatthefrequenciesofinterest.Theorem1:SupposearebothofrankandletthesingularvaluedecompositionofbegivenbyEq.(4).Inadditionlet,wherecommutesexactlywith.Then,arebothnormalmatricesifandonlyif,forsomenonsingulardiagonalmatrixProof:)Ifarenormal,then,accordingtolemma1,Therefore,fromEqs.(4)and(5),,whichisaspectraldecompositionfor.Suppos-ingthatisanexactcommutativecontroller,thensharethesameeigenvectormatrices,whichmeansthata spectraldecompositionofcanbewrittenasisadiagonalmatrix,andthus,Eq.(6)becomesTherefore,de“ning,yieldstheresult.)Theproofisstraightforwardandwillbeomitted.Theorem1showsthattheleastsensitivityofthechar-acteristiclociwithrespecttoperturbationsattheplantinputandoutput,atagivenfrequency,isachievedwhenthenormalizingprecompensatorissuchthatthematrixisdiagonal.However,sincenormalmatricesarearelativelysmallsetcomparedtoapproximatelynormalmatrices,theninsteadoftryingtoachievetheexactnormalityoftheprecompensatedplant,itismorerealisticto“ndwaystoapproximatelynormalizebothreturnratiomatrices,,withaprecompensator,atthefrequenciesofinterest.Todoso,thefollowingresultisneeded.Lemma2:.Thenisnormalifandonlyhasacompleteorthonormalsetofeigenvectors.Proof:See[7].Lemma2suggestsawayofmeasuringhowclosetonormalagivenmatrixis[8],[3],namelyisapproximatelynormaliftheconditionnumberofitseigenvectormatrixisapproximatelyequaltoone.Inthispaper,itwillbeshownthatwiththeprecompensatorstructuregivenbyisaunitarymatrix,itispossi-bletoachievethesamedegreeofnormalityfor,inthesensethattheconditionnumberoftheireigenvectormatricesareequal,asshowninthesequel.Lemma3:havethesingularvaluedecom-positiongivenbyEq.(4),andde“nethecomplexmatrixisnormalifandonlyifisnormal.Proof:Theproofisstraightforwardandwillbeomitted.Theorem2:begivenbyEqs.(1)and(8)foragivenfrequency,respectively,andassumethatcommutesexactlywith.Ifisapproximatelynormal,inthesensethatitseigenvectormatrixhasconditionnumberapproximatelyequaltoone,thenalsoapproximatelynormal.Proof:Supposethathasthefollowingspectralde-composition:isadiagonalmatrix,istheeigenvectormatrixof.Then,canbewrittenas:andsinceisaunitarymatrix,theconditionnumberofitseigenvectormatrixisequaltotheconditionnumberoftheeigenvectormatrixoffYWM]=C[WM],wheree.]denotesconditionnumber.Therefore,itsuf“cestoprovethattheconditionnumbersoftheeigenvectormatricesofarebothequaltooWM].Sincecommutesexactlywith,ithasthesameeigenvectormatrixasandthus:Thusaspectraldecompositionforcanbegivenas:Toshowthatisalsoapproximatelynormalwhenisapproximatelynormal,notice,initiallythatsince,whereisaunitarymatrix,thenisaunitarymatrix.Therefore,usingEqs.(9),(12)and(4),oneobtainsUMWAccordingto(10),andthus:SubstitutingEq.(16)inEq.(15)andmakingyields:FromEqs.(13)and(17)itcanbeeasilyseenthattheconditionnumbersoftheeigenvectormatricesofarebothequaltooWM].
Theorem2showsthatwiththeprecompensatorstruc-turegivenbyEq.(8)itispossibletoapproximatelynor-,simplybyseekingaprecompensatorthatapproximatelynormalizesatagivenfrequency.Thus,inthispaper,willhavethestructuregivenbyEq.(8).Remark1:Noticethatifisapproximatelynormal,for,thenisalsoapproximatelynormalandvice-versa.Therefore,althoughthecomplexnumbermaynotbeequaltoone,forsimplicity,intherestofthissectionitwillbeassumedthat.Asitwillbeseeninthenextsection,thechoiceofplaysakeyroleintheapproximationofthedesiredfrequencyresponseofbyarationalandstabletransfermatrix.Supposingthatthen,accordingtothe-orem1,thenormalityofachievedifandonlyifisdiagonal.Moreover,itcanbeproventhattheapproximatenormalityisrelatedtoanapproximatediagonalform.Inordertoshowthis,itis“rstnecessarytointroducetheconceptofanalignedmatrix,asfollows.De“nition1:beacomplexmatrix.Ifallpossiblesingularvaluedecompositionsof,aresuchthatarealigned,diag,...,,then,issaidtobealigned. De“nition1leadstothefollowingresult.Lemma4:beacomplexmatrixwithdistinctsingularvalues.Then,isalignedifandonlyifisnormal.Proof:See[8].Fromlemma4,itispossibletostatethefollowingtheorem.Theorem3:beaunitarymatrixandletbeasingularvaluedecompositionof,whereallsingularvaluesaredistinct.Then,theproductisnormal,ifandonlyifthereexistsdiag,...,,suchthatProof:Lettheproductbeanormalmatrix.Thenand,sincebyassumptionisaunitarymatrix,isalsoaunitarymatrix.Therefore,where,maybeseenasasingularvaluedecompositionof.Accordingtolemma4,sincethematrixisnormalandsincehasdistinctsingularvalues,itisalsoaligned.Thus,,namely,Theconversecanbeeasilyprovedbynoticingthatifequalityissatis“edthenisnormal.Theorem3showsthat,whenallsingularvaluesofaredistinctatagivenfrequency,thediagonalformofisalsoanecessaryandsuf“cientcon-ditionfortobenormal,normalifandonlyifitisaligned.Therefore,ifforanydiagonalmatrix,thestaticprecompensator,normalizestheplant.However,thisprecompensatorcannotbe,ingeneral,approximatedatthenecessaryfrequencyrangebyadynamicprecompensator,withallentriesbeingchosenasstableandproperrationaltransferfunctionsasisdonein[3].Thissuggeststhatthesearchofanexactalignmentbetweenshouldbereplacedbythesearchofaunitarymatrixthatmakesapproximatelyaligned.Therefore,itisnecessarytode“neameasureofthedeviationoffromalignment.FromEq.(18)anaturalde“nitionofameasureofdeviationfromalignmentisasfollowsmin0whenisaligned.Thefollowingresultshowsthatifisapproximatelyaligned,0,thenisapproximatelynormal,inthesensethatTheorem4:Accordingtothemeasuresde“nedinEqs.(2)and(20)then,if0thenProof:beasingularvaluedecomposition.Therefore,DividingbothsidesoftheinequalityabovebyandusingthefactthatyieldsThismeasureofalignmentisthesameasthatusedin[8]withadifferentnorm.,whichcompletestheproof.Theorem4showsthatifthereexistsaunitarymatrixthatmakesapproximatelyaligned,thenisalsoapproximatelynormal.Therefore,fromtheorems1,2,3and4,theproblemofdesigningaprecompensatorthatapproximatelynormalizesaplant,atagivenfrequency,canbeformulatedasfollows:Prob.1:minminsubjecttobeaunitarymatrixandeachentryofthemaindiagonalof,for,...,B.SolutionoftheoptimizationproblemUsingthede“nitionofFrobeniusnormandaftersomestraightforwardmanipulation,Eq.(22)canbewrittenas:De“ningtheunitarymatricesanddenotingeachelementof,thenEq.(23)canbere-writtenas:as:Re(h11)+Re(h22)+...+Re(hmm)].(24)FromEq.(23),itcanbeeasilyseenthateachelementofthemaindiagonalofmultipliesthei-th.Denotingeachelementof,thenthatminimizesthecostfunctiongivenbyEq.(24)issuchthateachcomplexnumbermustberealandpositive.Thus,de“ning,thentheoptimumwillbegivenby,where.Therefore,theminimumvalueforonlyonandisgivenby:by:|t11|+|t22|+...+|tmm|].(25)Consequently,accordingtoEq.(25),theoptimizationprob-lem1,isequivalentto:Prob.2:maxmaxNotice,accordingtoEq.(27)andthede“nitionofthattheprecompensatorthatsolvesproblem2makestheunitarymatrixasdiagonallydominantaspossible.Inthispaper,thestructureadoptedforwillbeofapermuteddiagonalmatrixasdonein[3]andwillbegivenas:!)isamatrixformedwithallpossiblepermutationofthecolumnsoftheidentitymatrix(permu-tationmatrix)andisadiagonalmatrixwhereitsmaindiagonalentriesareequalto1or1toguaranteethatisunitary;noticethat,sincemultiplicationby1doesnotchangetheunitarynatureofamatrix,itispossibletoform .ThischoiceofismotivatedbythefactthatprecompensationusingpermutationmatricesisusualinthedesignofmultivariablecontrollersviaNyquistArrayMethods[13],[6]intheattainmentofdiagonaldominance.Theuseofotherstructuresforwillbethesubjectoffutureresearch.C.Precompensatorimplementationhasbeencomputedforeachfrequencythenecessaryfrequencyrange,thenthenextstepisthede-signofadynamicnormalizingprecompensator.Noticethatisrealandunitary,whilemusthaveasentriesonlyrationalandstabletransferfunctions.Thisshowstheneedforaddingphaseandmodulustoeachnonzeroentry.Notice,however,thatthestaticprecompensatorhasbeenmadeequaltoonlyforsimplicity(remark1),beingactuallyequalto.Therefore,choosingappropriatelythevaluesof,itispossibletoaddthesamephaseandmodulustoeachnonzeroentrywiththeviewtoapproximatingthemtothefrequencyresponseofarationalandstabletransferfunction.Inpractice,however,itisnotnecessarytochooseinordertocomputetheentriesof.Noticethatthiscanbedonebychoosing,foracertainfrequencyband,thesametransferfunctionsfortheentriesofwiththenonzeroentriesofatthisfrequencyband,suchthattheirmoduliapproximatelymatchthemoduliofthenonzeroentriesof.Thisprocedureleadstoadynamicprecompensatorthatapproximatelyalignstheprecompensatedplantprovidingthatalsoaligns.Anexceptionismadeatthevicinityofthefrequencieswherethefrequencyresponsemoduliofjumpfrom1to0orfrom0to1,sinceatthesefrequenciesthemodulioftheentriesofaredifferentfromthoseof.Thisproblemcanbeovercomebyincreasingtheorderofthepoleorzeroassociatedwiththefrequencywherethejumpoccurs;althoughattheexpensesofanincreaseintheorderoftheprecompensator.Inthispaper,withtheviewtomakingtheprecompensatedsystemwiththelowestpossibleorder,onlyapproximationsbylead/lagtransferfunctionswillbeusedtoobtaintherationalprecompensatorTheproceduretoobtainthenormalizingprecompensatorcanbesummarizedinthefollowingalgorithm.Algorithm1:1)Formthe2diagonalmatricesofdimensionwitheither1or1initsmaindiagonal.2)Selecta“nitenumberoffrequenciesandset1and3)If1,chooseapermutationmatrix.Ifformadifferentpermutationmatrixfromtheotherpermutationmatricesalreadyformed.4)Usingde“nedinstep3,compute,foreachoneofthe2,de“nedinstep1,andthecostfunctionmaxFindmaxmaxandthematrixmaxwhichleadstomax5)Make1andrepeatsteps3and4until6)Amongallvaluesofmax,computedinstep4,choosemaxmaxandselectthematricesoptoptwhichleadstomax.Formoptopt7)Set1and1andgobacktostep3.Repeatsteps3to7until8)Findrationalandstabletransferfunctionsforeachen-tryof,suchthatthemagnitudeofthefrequencyresponseofitsentriesapproximatelymatchthoseof,forIV.EXAMPLELetthetransferfunctionmatrixofthelinearizedmodeloftheverticalplanedynamicsofanaircraftbegivenby[13]: (29)29)nij(s)],i,j=1,2,3,andaregivenas:InordertousetheCLMtodesignacommutativecontroller,itis“rstnecessarytoverifyifisclosetonormalinthenecessaryfrequencyrange.Thiscanbedonebycomputingthemeasureofnormality,de“nedinEq.(2),andtheconditionnumberoftheeigenvectormatrixof.ItcanbeseenfromFigs.2(a)and(b)(dashedlines)thatfarfromnormalatlowandhighfrequencies.Itisimportantalsotonotethatatveryhighfrequenciesbecomesnormal,whichoccursbecause.Moreover,inFig.2(c)(dashed-line)itcanalsobeseenthatfarfromalignedatallfrequencies.Therefore,itisnecessarytodesignanormalizingprecompensatorforTheprecompensatordesigniscarriedoutinaccordancewithalgorithm1.Since3(thedimensionof),the“rststepistoform24diagonalmatriceswith11initsmaindiagonal.Inthesequel,itisnecessarytoform6permutationmatrices;thussteps3and4ofalgorithm1willberepeated6timesforeachmatrixobtainedinstep1.Thenextstepistoobtainoptoptforeachfrequencypointinthefrequencyrange,leadingtothedesiredunitarymatrix.Fig.3(x-markedlines)showsthemagnitudeofeachentryofNoticethat,forsuchatheprecompensatedplantisapproximatelyalignedforalmostallfrequenciesascanbeseenfromFig.2(c)(dash-dottedline).Thisimplies,accordingtotheorem4,thatisalsoapproximatelynormalatthesamefrequenciesascanbeseenfromFigs.2(a)and(b)(dash-dottedline).Itisalsoimportanttoremarkthattheconditionnumberoftheeigenvectormatrixof 10 0 (a) 0 2 4 6 8 0 (c)ConditionnumbersFig.2.(a)Measuresofnormality:(dashedline),(dash-dottedline)and(solidline);(b)Conditionnumbersoftheeigenvectormatricesof(dashedline),(dash-dottedline)and(solidline);and(c)Measuresofalignment:(dashed(dash-dottedline)and(solidline).isformostofthefrequencyrangesmallerthan1.3,whichrepresentsasigni“cantimprovementonthenormalityofSimilarconclusionscouldbedrawnfromtheanalysisofaccordingtoFig.2(a)(dash-dottedline).The“nalstepinthedesign(step8)isto“ndstabletransferfunctionsforeachentryof,suchthatthefrequencyresponsemagnitudeofitsentriesapproximatelymatchthenonzeroentriesofshowninFig.3(x-markedlines)foreachfrequency.Noticethat,inthisexample,theentriesofcanbechosentobe“rstordertransferfunctions,suchthattheentriesofapproximatelymatchthenonzeroentriesofatlowfrequencies,andvanishathighfrequencies,andanothertransferfunctionthatapproximatelymatchthenonzeroentriesofathighfrequencies,andvanishesatlowfrequencies.Adynamicprecompensatorthatsatis“estheserequirementsisgivenby: s+0.1s s+50.1 s+0.1s s+50s s+500.1 (30)Noticethatthereisacloseagreementbetweenthemagni-tudesofeachentryof,whichleadstoaapproximatelyunitaryatlowandhighfrequencies.Fig.2(c)showsthatisalsoapproximatelyalignedatalmostallfrequencies(solidline)andthereforeasexpected,approximatelynormalatthesefrequencies;thesameconclusioncanbedrawnfromFigs.2(a)and(b).Magnitudeof Š0.5 Š0.5 Š0.5 Š0.5 Š0.5 Š0.5 Š0.5 Š0.5 Š0.5 Fig.3.Magnitudefortheoptimum(x-markedline)andfortherationalapproximation(solidline).Itisalsoimportanttoremarkthat,exceptatintermediatefrequencies(atthevicinityofthefrequencywheretherearejumpsintheelementsof),themeasuresofnormalityandmisalignmentofareveryclosetothedesiredEFERENCES[1]J.C.BasilioandB.Kouvaritakis,Theuseofrationaleigenvectorapproximationsincommutativecontrollers,ŽInternationalJournalofControl,vol.61,pp.333…356,1995.[2]„„,Designofcausalreversed-frame-normalizingcontrollersusingbicausalexpansions,ŽInternationalJournalofControl,vol.66,pp.1…14,1997.[3]J.C.BasilioandJ.A.Sahate,Anormalizingprecompensatorforthedesignofeffectiveandreliablecommutativecontrollers,ŽInternationalJournalofControl,vol.73,pp.1280…1297,2000.[4]D.J.CloudandB.Kouvaritakis,Commutativecontrollersrevisited:Parallelcomputation,anewleaseoflife,ŽInternationalJournalofControl,vol.45,pp.1335…1370,1987.[5]J.C.DoyleandG.Stein,Multivariablefeedbackdesign:Conceptsforaclassic/modernsynthesis,ŽIEEE-TransactionsonAutomaticControl,vol.26,pp.4…16,1981.[6]J.M.Edmunds,Inputandoutputscalingandreorderingfordiagonaldominanceandblockdiagonaldominance,ŽIEEProc.-ControlTheoryAppl.,vol.145,pp.523…530,1998.[7]F.R.Gantmacher,Thetheoryofmatrices.NewYork:ChelseaPublishingCompany,1959,vol.IandII.[8]Y.S.HungandA.G.J.MacFarlane,MultivariableFeedback:AQuasi-ClassicalApproach.Springer-Verlag,1982.[9]B.KouvaritakisandJ.C.Basilio,Bi-causaleigenvectorsequencesandthedesignofcausalcommutativecontrollers,ŽInternationalJournalofControl,vol.59,pp.1173…1189,1994.[10]A.G.J.MacFarlaneandJ.J.Belletruti,Thecharacteristiclocusdesignmethod,ŽAutomatica,vol.9,pp.575…588,1973.[11]A.G.J.MacFarlaneandB.Kouvaritakis,Adesigntechniqueforlin-earmultivariablefeedbacksystems,ŽInternationalJournalofControlvol.25,pp.837…874,1977.[12]A.G.J.MacFarlaneandI.Postlethwaite,Thegeneralizednyquiststabilitycriterionandmultivariablerootloci,ŽInternationalJournalofControl,vol.25,pp.81…127,1977.[13]J.M.Maciejowski,MultivariableFeedbackDesign.Addison-WesleyPublishingCompany,1989.[14]M.V.Moreira,J.C.Basilio,andB.Kouvaritakis,Rationalstabilizingcommutativecontrollersforunstableplant,Žin44thIEEEConferenceonDecisionandControlandEuropeanControlConference(acceptedforpresentation),2005.[15]I.Postlethwaite,Sensitivityofthecharacteristicgainloci,ŽAutomat-,vol.18,pp.709…712,1982.[16]J.H.Wilkinson,TheAlgebraicEigenvalueProblem.Oxford:ClarendonPress,1965.