Chapter PredictionError Identication of LPV Systems Present and Beyond Roland T oth Peter - PDF document

R.Tothetal.models.However,suchaprocedurerequiresadetailedprocessknowledgefromspecialists.Toassembletheexistingknowledgeintoacoherentandcompactmathematicaldescriptionisnotonlyachallengingtaskbutitusuallyresultsinatoocomplexmodelasitishardtodistinguishrelevanteffectsfromnegligibleterms.Theselectionoftheschedulingvariableitselfisalsooftenrestrictedbythewayofmodelconstructionandlikelydifferentchoicesfollowfromlinearization-basedordirectconversionbasedmethods,see[].Therefore,modelingisoftenfoundtobeverylaboriousandexpensive.Ifthespecialist’sknowledgeislacking,likeincaseofpoorlyunderstoodsystems,thederivationofamodelfromrstprinciplesisevenimpossible.Moreover,certainquantities,likecoefcients,rates,etc.,requiredforthemodel,arelikelyunknownandhavetobeestimatedbyperformingdedicatedexperiments.DescriptionsofsystemscanalternativelybederivedbysystemidentiÞcation(ID),wheretheestimationofadynamicalmodelisaccomplisheddirectlyfrommeasuredinput–outputdata.TheexpertÕsknowledgestillhasamajorrole,asitgivesthebasicsourceofinformationindecisionsonparametrization,model-structureselection,experimentdesign,andtheactualwayofderivingtheestimate.Thisknowledgealsohelpsinjudgingthequalityandapplicabilityoftheobtainedmodels.Evenifsystemidenticationrequireshumaninterventionandexpert’sknowledgetoarriveatappropriatemodels,italsogivesageneralframeworkinwhichmostofthestepscanbeautomated,providingalesslaboriousandcostintensivemodelingprocess.Inthecurrentliterature,manyLPVidenticationapproacheshavebeendevel-opedusingmodelstructuresthatareformulatedintermsofstate-space(SS)andlinear-fractionalrepresentations(LFR),e.g.,[inputÐoutputrepresentations,e.g.,[],orseries-expansionforms,e.g.,[].Mostoftheexistingapproachesuseadiscrete-timesettingandcommonlyassumestaticdependenceontheschedulingvariable,with.Here,staticdependencemeansdependenceofthemodelcoefcientsonlyontheinstantaneousvalueof.Forarecentoverviewoftheavailablemethods,see[Recently,LPV–IOmodelstructuresbasedmethodshavegotaconsiderableattentionastheyappeartohavecertainadvantagesw.r.t.otheridenticationapproachesoftheeld.Oneofthemajorbenetsisthatidenticationofthisrepresentation-basedmodelstructurescanbeaddressedviatheextensionoftheLTIprediction-error(PE)framework[].Inoppositionwithotherapproaches,thisenablesthestochasticanalysisoftheestimates,treatmentofgeneralnoisemodels[],experimentaldesign[],modelstructureselection,anddirectidenticationoftheinvolveddependencies[]ofteninacomputationallyattractivemannerandalsoincontinuoustime[].Moreover,inthissettingitisalsorelativelyeasytoidentifymodelswithdynamicdependence(dependenceofthecoefcientsontimeshiftedinstancesof),whichisoftenrequiredforhighaccuracyidenticationofnonlinearsystems(see[]).However,themainstreamLPVcontrol-synthesisapproachesarebasedonmodelsdenedinanSSoranLFRform,hencethedeliveredIOmodelneedstobeconvertedtosuchrepresentationforms.Duetothefactthatmultiplicationwithanytimeoperatorisnotcommutativeover R.Tothetal.whereistheinput,istheoutput,andistheschedulingvariableoftheLPVsystemrepresentedby(isthe(forward)time-shiftoperator,i.e.,0andfunctionsof(instantaneousvalueof)whichiscalledstaticdependence.Thefunctionscanhavearbitrarycomplexityrangingfromsimplelineartorationalorrealmeromorphicdependence.Toguaranteewell-posednessof(),itisoftenassumedthatallareboundedonInidentication,weaimtoestimateadynamicalmodelofthesystembasedonmeasureddata,whichcorrespondstotheestimationofeachin().Toformulateestimationofthesefunctions,itisattractivetointroduceaspolynomialsinwithvaryingcoefcientsand,inspiredbytheLTIsystemtheory,towrite However,in()relatestoatransferfunctionifandonlyifaconstantsignal,i.e.,forall,where.Thisisjustiedbythefactthatifissubstitutedwiththecomplexvariable,thenisamixedfrequency–timerelationship.IfdenotetheZ-transformofthesignalsonanappropriateregionofconvergence,thenhasameaningifandonlyif,associatedwith,isaconstant(not-varyingwithtime).Furthermore,isill-denedalsoasanoperatorbecausemultiplicationwithisnotcommutativeovertime-dependentcoefcientssuch,i.e.,))=.Therefore,multiplicationfromtheleftorrighthasdifferentmeaning.In(),itisambiguouswhetherleftorrightmultiplicationisintendedtoderivethisrationaloperatorform.Currentlynotheoreticalframeworkisavailable(totheauthor’sknowledge)tohandlerationaltime-operatorformswithtime-dependentcoefcients(suchaframeworkdoesexistincaseofconstantcoefcients,i.e.,intheLTIcase,see[Toovercomethis“representation”problem,ithasbeenshownin[]thatthedynamicmappingbetweencanbecharacterizedasaconvolutioninvolving.Thisso-calledimpulseresponserepresentation(IRR)isgivenas isarealmeromorphicfunctionifanalyticand R.Tothetal.2.4ClassicalModelStructuresNextwecaninvestigatehowtheclassicalmodelstructurescanbeformulatedintheintroducedLPV–PEsetting.Tofollowtheclassicalformulations,wewillconstructboththeprocessandthenoisecomponentsusinganLPV–IOrepresentationform.Forthesakeofsimplicity,wetreatthesemodelstructuressuchthattheircoefcientshaveonlystaticdependenceon.Extensionofthesedenitionsusingcoefcientswithdynamicdependencefollowsnaturally.2.4.1ProcessModelConsidertheparametrizedmodel,wheretheprocesspart,whoseIRRisgivenby,isdenedbyHereisthenoise-freeoutputoftheprocesspart,0istheinputdelayandthe-dependentpolynomials0,areparametrizedas:1.Inthisparticularparametrization,whichiscalledlinearparametrizationarepriorigivenfunctions(chosenbytheuser)whichareboundedon)+(,representstheunknownparameterstobeestimatedfortheprocesspart.Notethatparametrizationsotherthan()arepos-sible;however,theadvantageof()isthatalargenumberoffunctionaldepen- 2Prediction-ErrorIdenticationofLPVSystems:PresentandBeyond35i.e.,thedifferencebetweenandthepredictedmodeloutput,whichistheprimarygoalinthePEsetting.Considerthe-truecasewith,and(incaseofcausaldependence).Since(impliesthatischaracterizedfor1w.r.t.theinformationsetBasedon(),itisnotcomplicatedtoshowthatunderthegiveninformationset,theone-step-aheadpredictionofw.r.t.thelossisThiscorrespondstotheLPVformoftheclassicalresultoftheLTIcase(see[Notethatinasimilarmanner,-step-aheadpredictorscanalsobeformulatedinthissetting.Foradetailedproof,see[2.3.5ParametrizedModelsandEstimationNow,introduceanLPVparametrizedmodelintheformofwherearetheIRRsoftheprocesspart,denotedasandthenoisepart,denotedas,ofthemodelstructure,respectively,andaretheparameterstobeestimated.Notethattheseparametersarenotnecessarilyassociatedwiththeparametrizationofimpulseresponsecoefcientsdirectly,butcancorrespondtotheparametrizationofthecoefcientsoftheprocessandnoisemodelsgiveninanSSorIOform.ThentheseparametrizedstructuresarerepresentedbytheIRRs:.Alsointroduce,thevectorofindependentparametersinDenotethecollectionofallprocessandnoisemodelswiththeconsideredparametrizationandsimilarlydenotetheoverallparametrizedmodel()as.Then,basedonthemodelstructure,themodelset,denotedas,takestheformThissetcorrespondstothesetofcandidatemodelsinwhichweseekthemodelthatexplainsdatagatheredfromthebest,underagivencriterion.Wedenote,whenthedata-generatingsystemisinthemodelset,i.e.,suchthat 2Prediction-ErrorIdenticationofLPVSystems:PresentandBeyond33sense,lim existsforagivenandforallduetotheconvergencepropertiesofandtheindependenceof.Hencequaliesasaquasi-stationarysignal(see[]forthedetailedproof).2.3.2One-Step-AheadPredictionofInordertoformulatetheestimationofparametrizedLPVmodelsof()inaprediction-errorsetting,itisnecessarytoderiveapredictorof.Thesimplestcaseistocharacterizeaone-step-aheadpredictor,forwhichitisessentialtoclarifyhowwecanpredictatagiventime-stepifwehaveobserved1.Intermsof(meaningthatandagiventrajectoryofdenes.Notethatifeachdependsonlyonthecurrentandthebackward-time-shiftedvaluesof,e.g.,,etc.,whichiscalledcausaldynamicdependencethenonlytheknowledgeofissufcienttocharacterize.IncasethenoiseprocesshasanLPV–IOrepresentationintermsof()withonlystatic-dependence,thentheequivalentIRRform()isguaranteedtohaveonlycausaldependence[].Inmostpracticalapplications,causaldynamicdependenceonisquiterealistic.Tofollowtheclassicalconceptofdeningthepredictionof,assumethatobservationsofaregiven.Underthisinformationset,ourobjectiveistocomputetheone-step-aheadpredictionofw.r.t.the-loss:argminIn[]itisshownthatifisfullyknown,then()isequaltoIteasilyfollowsthat()alsominimizesthemean-squarederroroftheprediction.Ofcourseitisnotpracticaltoassumethatisknown.Toformulatepredictionintermsofobservationof,itisrequiredthatastableinverse,i.e.,thereexistsamonicconvergentLPV–IRRdenotedassuchthat1.Notethatifsuchaexists,thenitisabi-lateralinverse R.Tothetal.2.5.1Prediction-ErrorBasedIdentiÞcationintheARXCaseIntermsoftheconsideredglobalsetting,weaimatthedirectminimizationof(intermsoftheparametrizedmodelstructure(–c)usingadatasetwhereisvarying.Thisdatasetisassumedtobeinformativew.r.t.(–c)tohaveawell-posedproblemforidentication.Tofulllourestimationobjective,severalapproachescanbeintroducedforthevariousmodelstructuresgiveninSect..Forthesakeofclarity,wewillstudytheseestimationapproachesstep-by-step,startingfromthemostsimplestcaseofARXmodelswheretheestimationcanbeaddressedviasimplelinearregression.2.5.1.1LinearRegressionConsidertheLPV–ARXmodelstructure().Aparticularpropertyofthisstructurewiththelinearparametrization()ofisthatthepredictor(islinearintheparameters,see(),andhencecanbewrittenaswhere···ŠAs()isalinearregressionequation,thusbydeningg(1)···(N)]andY=[y(0)···y(N)],theminimumof()isuniqueandequaltoifrank,where N  Š11 istheregularizedMooreÐPenrosepseudoinverse.ThisapproachissummarizedintermsofAlgorithm.Equation)hasbeenusedinmanyworks,e.g.,[],toestimateLPV–IOmodels,however,intheintroducedPEframeworkitisjustiedthat()istheminimizerof()incaseofanLPV–ARXmodelstructure.Itisalsoimportanttomentionthat()canbealsoconsideredasanLTImultiple-outputmultiple-input(MIMO)ARXmodelwith“virtual”inputandoutputsignalsTogetaninsightofthestochasticbehavioroftheLSestimator(),assumethatandconsiderthe“optimal”residualerror,whichbasedon()is 2Prediction-ErrorIdenticationofLPVSystems:PresentandBeyond37Themodelstructureissaidgloballyidentiableatifthesameholdsforarbitrarylarge.ItiscalledgloballyidentiableifitisgloballyidentiableatallAnotherconditionforuniqueness()istheinformativityofthedatasetDeÞnition2.2(Informativedata,basedon[Foramodelstructuredenedby()withaparameterdomain,aquasi-stationarydatasetiscalledinformative,ifforanybeingthegeneralizedexpectationoperator,impliesthatIntermsofthesedenitions,ifthemodelsetisgloballyidentiable(notwodifferentparametersgiverisetothesamepredictor)andthedatasetisinformative,thenhasaglobaloptimuminthestatisticalsense.2.3.7ConsistencyandConvergenceWhenapplyingthequadraticIDcriterion(),theasymptoticpropertiesoftheresultingparameterestimatecanbeanalyzedinthesituationwhensimilarlyasintheLTIcase.Considerthefollowingdenitionsofconsistencyandconvergence.DeÞnition2.3(Convergence).Foraninformativedatasetandmodelstructure,theparameterestimateiscalledconvergentifimpliesthatwithprobabilityone,i.e.,1,whereargminNotethatconvergenceimpliesthattheasymptoticparameterestimateisinde-pendentoftheparticularnoiserealizationinthedatasequenceandislocallyidentiableatDeÞnition2.4(Consistency).Formodelstructurewithmodelsetandadatasetwhichisinformativew.r.t.,aconvergentparameterestimateiscalledconsistentifeitherofthefollowingconditionsholds:If,thenIfbut,thenWewillinvestigatethesepropertiesw.r.t.theparticularidenticationapproachesweconsiderinSect. Thenotation isadoptedfromtheprediction-errorframeworkof R.Tothetal.2.5.1.2InstrumentalVariableApproachTheoriginalaimofinstrumentalvariablemethodsistocopewiththefactthatinmostcases,isacoloredprocess.Theideaistointroduceaninstrumentwhichisusedtoproduceaconsistentestimateindependentlyonthenoisemodeltaken.TheIVestimateisgivenaswhichimpliesthatTherefore,andsimilarlytotheLSsolution,isaconsistentestimateof0andThereisaconsiderableamountoffreedominthechoiceofaninstrumentrespectingtheseconditions.IntheLTIcontext,thechoiceoftheinstrumenthasbeenwidelystudiedandmostoftheadvancedIVmethodsoffersimilarperformanceasextendedLSmethodsorotherPEminimizationmethods(see[]).Aparticularlyinterestingfactisthat,undertheARXmodelassumption,thevarianceoftheIVestimateisminimaliftheinstrumentischosenasthenoise-freeversionoftheregressor[].Inotherwords,whendirectlyapplyingtheIVtheorytotheLPV–ARXmodel()(theLPV–ARXmodelcanbeseenasanLTImodel),theoptimalIVestimateisgivenbywheretheoptimalinstrumentisdenedas:···ŠHeredenotesthenoise-freeoutputofthedata-generatingsystemwhichisaprioriunknowninpractice.Consequently,oftenanestimateofisappliedasaninstrument,likethesimulatedoutputofapreviouslyobtainedmodelestimatewhichinturncanberenediteratively.Notethatif,thenboththeIVsolution 2Prediction-ErrorIdenticationofLPVSystems:PresentandBeyond53understoodingeneral.Intermsofapplicationoftheseapproaches,ithasaprimeimportancethatconvergenceisquitesensitiveonthemodelingassumptionandthelargenessofparametrizationwhicharetypicallyill-choseninmostapplications.Nevertheless,theintroducedschemes,iftheyconverge,providecomputationallyefcientestimationapproachesintheconsideredcontext.2.5.2.2NonlinearOptimizationAlternatively,minimizationof()isavailablebygeneralnonlinearoptimizationmethods,likegradient-basedminimizationwhichcanbeapplieddirectlycom-putingthepartialderivativesofthepredictor()w.r.t..EventheadvancedLSQNONLINapproachofMATLABcanbedirectlyusedtoobtainanestimate.Astheapplicationofthesenonlinearoptimizationschemesonlyextendstothesolutionoftheunderlayingoptimizationproblem,theseapproachesarenotpresentedindetail.However,therearetwoparticulardifcultiesthatcanhindertheapplicationofnonlinearoptimizationschemes:1.Incaseofover-parametrizationoftheschedulingdependencies,thenumberofpossiblesaddlepointsof()canseriouslyincreasewhichcanslowdownorevenpreventtheconvergencetotheglobaloptimum.2.Incaseoflarge-scalesystems,thecomputationaltimecanbesubstantialcomparedtootherapproaches.2.5.2.3InstrumentalVariableApproachAsthealternativeofthepreviousestimationmethod,wecanalsointroduceaninstrumentalvariableapproachthatmakespossiblethedirectidenticationofLPV–BJmodelswith-independentnoisepart.HenceitimprovesconsiderablytheachievablevarianceoftheIV4methodincaseofmorecomplicatednoiseprocesses.ToderivesuchanimprovedIVscheme,werststartwithrewritingtheprocessequation()aswhereisanLTIlter,˜.NotethatinthiswaytheprocesspartisrewrittenasaMISOLTImodelwithinputsignals.However,thisisnotarepresentationoftheoriginalLPVbehaviorof)asitcontainslumpedoutputterms.Asasecondstep,assumethatthenoisepartisnotdependenton,henceitismodeledasastableLTIlter,whichisatechnicalassumptionweneedtotaketoderivetheintendedapproach.Giventhefactthatthepolynomialoperatorcommutesin 2Prediction-ErrorIdenticationofLPVSystems:PresentandBeyond51Thenbyconsideringaregressordenedasbefore(see()),butextendedwith,...,,thepredictor(canberewrittenasThisequationcorrespondstoapseudolinearregression,henceminimizationof)followsbyaniterativeLSapproachwhereanestimateofisgeneratedbyamodelobtainedinapreviousiteration,seeAlgorithm Algorithm3:LPV–ARMAXidentication,iterativeLSglobalmethod Require:adatarecord,theLSidenticationcriterionandtheLPV–ARMAXmodelstructure(2.29a–c)withandlinearparametrization(2.27)withparameterssa1,0···bnb,nd1,0···dnd,n]Rn.Assumethatisinformativew.r.t.(2.29a–c)and(2.29a–c)isgloballyidentiableon1:estimateanARXmodelbyAlgorithmresultingin.Setrepeat3:generateanestimateˆbasedontheresultingmodelofthepreviousstep,i.e.,4:calculatethesignalsandletetx0,0···xna+nb+nd+1,n].5:estimateintermsof.Increaseby1.untilhasconvergedorthemaximumnumberofiterationsisreached.returnestimatedmodel(2.29a NowconsidertheLPV–OEcase.Inthiscase,0and()readasDeneasthenoise-freeoutputoftheLPV–OEmodel.Then,)canberewrittenasThisgivestheideaagaintointroducetheregressor···Štowrite()intheformof().AgainaniterativeLSalgorithm,similartoAlgorithm(),canbeintroducedtoobtainanestimate. 2Prediction-ErrorIdenticationofLPVSystems:PresentandBeyond55besignicantlydecreasedwithrespecttotheIV4method:evenifthenoiseprocessisnotinthenoisemodelsetdened,itismorelikelytobebetterdescribedbyanBJmodelthanbyanARXmodel.Intermsoftheestimation,itisimportanttonoticethatcontainsthenoise-freeoutputterms.Therefore,bymomentaryassumingthatknownaprioriandthatthedata-generatingsystemisinthemodelset,thenthepreviouslydiscussedconditionsforoptimalestimatesleadtothechoiceofoptimalinstrument[···Š···Šwhiletheoptimallterisgivenas Inapracticalsituation,theoptimalinstrument()andlter()areunknownapriori.Therefore,theRIVestimationnormallyinvolvesaniterative(orrelaxation)algorithminwhich,ateachiteration,an“auxiliarymodel”isusedtogenerateanestimateof()and().Thisauxiliarymodelisbasedontheparameterestimatesobtainedatthepreviousiteration.Consequently,ifconvergenceoccurs,theoptimalinstrumentandlterareobtained.Basedonthepreviousconsiderations,theRIValgorithmdedicatedtotheLPVcaseissummarizedinAlgorithmUsingasimilarconcept,theso-calledsimpliÞedRIV(SRIV)methodcanalsobedevelopedfortheestimationofLPV–OEmodels.Asinthiscase1,Step7ofAlgorithmcanbeskipped.Inpracticalcases,itisafairassumptiontoconsiderthatthenoisemodelassumedisincorrectforbothLPV–OEandLPV–BJmodels.Inthiscase,theLPV–SRIValgorithmmightperformaswellastheLPV–RIValgorithm:theBJassumptionmightbemorerealistic,butthisiscompensatedbythereducednumberofparameterstobeestimatedundertheOEassumption.Additionally,boththeRIVandSRIValgorithmscanbealsoextendedtobeapplicableinaclosed-loopsetting[2.6ConclusionByusinganimpulseresponserepresentationofLPVsystems,ithasbeenshowninthischapterthatauniedprediction-errorframeworkfortheidenticationofLPVpolynomialmodelscanbeestablished.WehaveseenthatthisframeworkallowstounderstandtheroleofgeneralnoisemodelsintheLPVsetting,makingpossibletoformulatetheLPVextensionsofclassicalmodelstructuresoftheLTIcase,likeARX,ARMAX,Box–Jenkins,OE,FIR,andseriesexpansionmodels.Further-moreestimationofthesemodelsiscomputationallyratherattractiveandallows