(VQTAM), which uses Kohonen’s Self-Organizing Map

PAPERSUBMITTEDTOTHESPECIALISSUEOFIEEETNNONTEMPORALCODING1IdenticationandControlofDynamicalSystemsUsingtheSelf-OrganizingMapGuilhermeA.Barreto,member,IEEE,andAluizioF.R.AraujoAbstract|Inthispaper,weintroduceageneralmodelingtechnique,calledVector-QuantizedTemporalAssociativeMem-ory(VQTAM),whichusesKohonen'sSelf-OrganizingMap(SOM)asanalternativetoMLPandRBFneuralmodelsfordynamicalsystemidenticationandcontrol.Wedemon-stratethattheestimationerrorsdecreaseastheSOMtrain-ingproceeds,allowingtheVQTAMschemetobeunderstoodasaself-supervisedgradient-basederrorreductionmethod.Theperformanceoftheproposedapproachisevaluatedonavarietyofcomplextasks,namely:(i)timeseriesprediction,(ii)identicationofSISO/MIMOsystems,and(iii)nonlin-earpredictivecontrol.Foralltasks,thesimulationresultsproducedbytheSOMareasaccurateasthoseproducedbytheMLPnetwork,andbetterthanthoseproducedbytheRBFnetwork.TheSOMhasalsoshowntobelesssensitivetoweightinitializationthanMLPnetworks.WeconcludethepaperbydiscussingthemainpropertiesoftheVQTAMandtheirrelationshipstootherwell-establishedmethodsfordynamicalsystemidentication.Wealsosuggestdirectionsforfurtherwork.Keywords|Self-organizingmaps,timedelays,temporalassociativememory,functionapproximation,timeseriesprediction,predictivecontrol.I.IntroductionDYNAMICALsystemidenticationisthedisciplinein-terestedinbuildingmathematicalmodelsofnonlinearsystems,startingfromexperimentaltimeseriesdata,mea-surements,orobservations[1].Typically,acertainlinearornonlinearmodelstructurewhichcontainsunknownpa-rametersischosenbytheuser.Ingeneral,theparametersshouldbecomputedsothattheerrorsbetweenestimated(orpredicted)andactualoutputsofthesystemshouldbeminimizedinordertocapturethedynamicsofthesystemascloseaspossible.Theresultingmodelcanbeusedasatoolforanalysis,simulation,prediction,monitoring,diag-nosis,andcontrolsystemdesign.Articialneuralnetwork(ANN)modelshavebeensuc-cessfullyappliedtotheidenticationandcontrolofavari-etyofnonlineardynamicalsystems,suchaschemical,eco-nomical,biologicalortechnologicalprocesses[2],[3],[4],[5],[6],[7],[8].Suchachievementsaremainlyduetoanumberoftheoreticandempiricalstudiesshowingthatsupervisedfeedforwardarchitectures,suchasthemulti-layerperceptron(MLP)ortheradialbasisfunction(RBF)networks,canapproximatearbitrarilywellanycontinuousinput-outputmapping(see[9],[10]forsurveys).Inthispaper,weproposeasystemidenticationtech-niquewhichusestheSelf-OrganizingMap(SOM)[11]G.A.BarretoiswiththeDepartmentofTeleinformaticsEngineer-ing,FederalUniversityofCeara(UFC),Fortaleza-CE,Brazil.E-mail:[e-mail]ujoiswiththeCenterofInformatics,FederalUniversityofPernambuco(UFPE),Recife-PE,Brazil.E-mail:[e-mail],insteadoftheusualsuper-visedones(MLPandRBF).Thistechnique,calledVector-QuantizedTemporalAssociativeMemory(VQTAM),showsthattheSOMcanbesuccessfullyusedtoapproximatedynamicalinput-outputmappings,withminormodica-tionsintheoriginalalgorithm.TheSOMisanunsuper-visedneuralalgorithmdesignedtobuildarepresentationofneighborhood(spatial)relationshipsamongvectorsofanunlabelleddataset.TheneuronsintheSOMareputtogetherinanoutputlayer,A,inone-,two-oreventhree-dimensionalarrays.Eachneuroni2Ahasaweightvec-torwi2nwiththesamedimensionoftheinputvectorx2n.Thenetworkweightsaretrainedaccordingtoacompetitive-cooperativeschemeinwhichtheweightvec-torsofawinningneuronanditsneighborsintheoutputarrayareupdatedafterthepresentationofaninputvec-tor.Usually,atrainedSOMisusedforclusteringanddatavisualizationpurposes.TheSOMwaspreviouslyappliedtolearnstaticinput-outputmappings[12],[13],[14],whichareusuallyrepre-sentedas:y(t)=g(u(t));(1)inwhichthecurrentoutputy(t)dependssolelyonthecur-rentinputu(t).Inthispaper,however,weareinterestedinsystemswhichcanbemodeledbythefollowingnonlineardiscrete-timedierenceequation:y(t+1)=f[y(t);:::;y(tny+1);u(t);:::;u(tnu+1)](2)wherenyandnuarethe(memory)ordersofthedynamicalmodel.Asstatedin(2),thesystemoutputy2qattimet+1depends,inthesensedenedbythenonlinearmapf(
),onthepastnyoutputvaluesandonthepastnuvaluesoftheinputu2p.Inmanysituations,itisalsodesirabletoapproximatetheinversemappingofanonlinearplant1:u(t)=f1[y(t+1);y(t);:::;y(tny+1);u(t1);:::;u(tnu+1)](3)Insystemidentication,thegoalistoobtainestimatesoff(
)andf1(
)fromavailabletimeseriesdatafu(t);y(t)g,t=1;:::;N.FortheSOMandotherunsupervisednetworkstobeabletolearndynamicalmappings,theymusthavesometypeofshort-termmemory(STM)mechanism[15],[16].Thatis,theSOMshouldbecapableoftemporarilystoring1Whenusedfordirect-inversecontrol,thetermy(t+1)in(3)isreplacedbyareferencesignal,r(t+1),whichisusuallyassumedtobeavailableattimet.PAPERSUBMITTEDTOTHESPECIALISSUEOFIEEETNNONTEMPORALCODING2pastinformationaboutthesysteminputandoutputvec-tors.ThereareseveralSTMmodels,suchasdelaylines,leakyintegrators,reaction-diusionmechanismsandfeed-backloops,whichcanbeincorporatedintotheSOMtoallowittoapproximateadynamicalmappingf(
)oritsinversef1(
)[17],[18].Inordertodrawaparallelwithstandardsystemidenticationapproaches,welimitour-se