Parametric Identification of Hammerstein Systems With Consistency Results Using Stochastic Inputs

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1、474IEEETRANSACTIONSONAUTOMATICCONTROL,VOL.55,NO.2,FEBRUARY2010ParametricIdentificationofHammersteinSystems[15]–[17],theidentificationproblemofthefollowingHammersteinWithConsistencyResultsUsingStochasticInputssystemisconsidered:Wen-XiaoZhao1110(1)0Abs

2、tract—Theextendedleastsquaresalgorithmisappliedtoidentifywhere,111aretheunknowncoefficientsofthepolyno-theHammersteinsystem,wherethenonlinearstaticfunctionisex-mial.Themethodsusedin[2],[15],[16]areit-pressedasalinearcombinationofbasicfunctionswithunknowncoeffi-erativ

3、e.Letusdenotetheunknownparameters,111,,cients.Strongconsistencyoftheestimatesisestablishedandtheirconver-gentratesareobtainedaswell.111,,111byforsimplicityofnotations.In[15],forthedatasetwithfixedsize,theestimateswithIndexTerms—ARMAX,extendedleasts

4、quares(ELS),Hammersteinsomeinitialvalueareobtainedbyminimizingsomecostfunc-system,strongconsistency.tion00I.INTRODUCTION(2)TheHammersteinsystemconsistingofastaticfunction1fol-lowedbyalineardynamicsystem(seeFig.1)isacommo

5、nlyusedandtheprocess(2)isrepeateduntiltheestimatesconverge.However,modelinengineeringpractice.Therearemanyequipmentsorphe-thisalgorithmmaydivergeifisnotappropriatelychosenasnomenacanbemodeledasHammersteinsystems.Forexample,theshownin[16].Thisdifficultyisovercomein[2]byanormalize

6、dtech-distillationcolumninchemicalengineering[8]andthepowerampli-nique.However,thesealgorithmsarenonrecursive,i.e.,toupdatethefierinelectronicscircuits[12]canbothbemodeledasHammersteinestimateswhilenewdataarriving,onehastocarryoutthewholeiter-systems.Inbiologicalcybernetics,theHammerst

7、einsystemisausefulativeprocedure(2).toolformodelingthedynamicsofthecortexofhumanandanimalsTheinstrumentalvariablemethodisusedin[17]toidentifysystem[11].Soitsidentificationissuenaturallyreceivesmuchattentionfrom(1).Buttheselectionofinstrumentalvariableswithrequiredproper-boththeoristsan

8、dengi