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1、478IEEETRANSACTIONSONINDUSTRIALELECTRONICS,VOL.47,NO.2,APRIL2000AReal-TimeLearningControlApproachforNonlinearContinuous-TimeSystemUsingRecurrentNeuralNetworksTommyW.S.Chow,Member,IEEE,Xiao-DongLi,andYongFangAbstract—Inthispaper,areal-timeiterativelearningcontrolrequiremanyle
2、arningiterationstoachievetherequiredaccu-(ILC)approachforanonlinearcontinuous-timesystemusingre-racy.Recently,two-dimensional(2-D)systemtheorywasintro-currentneuralnetworks(RNN’s)withtime-varyingweightsispre-ducedtotheILCapproach[4]–[8].Intheapplicationof2-Dsented.TwoRNN’sar
3、eutilizedintheILCsystem.OneisusedtosystemtheorytotheILCtechnique,verypromisingresultsonapproximatethenonlinearsystemandanotherisusedtomimicthedesiredsystemresponse.TheILCruleisobtainedbycombininglinearsystemcontrolhavebeenobtained[4]–[6].In[7],ChowthetwoRNN’stoformaneuralnet
4、workcontrolsystem.Also,aandFangextendedthediscrete-time2-DILCcontroltechniquekindofiterativeRNN’strainingalgorithmisdevelopedbasedontocontinuous-time2-DILCsystems.Morerecently,in[8],Fangthetwo-dimensional(2-D)systemtheory.AnRNNusingthepro-andChowproposedamodifiedlineardiscre
5、te-timeILCruleas-posed2-Dtrainingalgorithmisabletoapproximateanytrajectorysuringthedesiredtrajectorytobeaccuratelytrackedinonlyonetoaveryhighdegreeofaccuracy.SimulationresultsshowthattheproposedILCapproachisveryefficient.Thenewlydeveloped2-Dlearningiteration.In[9],ChowandFan
6、gworkedontheILCRNN’strainingalgorithmsprovidesanewdimensiontotheappli-ofnonlineardiscrete-timesystemusingrecurrentneuralnet-cationofRNN’sinanonlinearcontinuous-timesystem.works(RNN’s)withtime-varyingweights.Intheirwork,theyIndexTerms—Approximation,continuous-timeiterativeder
7、ivedanovel2-DRNN’strainingalgorithmonwhichthelearningcontrol,real-timetrainingalgorithm,recurrentneuraldevelopmentofanefficientnonlinearILCsystemwasbased.networks,two-dimensionalsystem.ThedevelopedRNN’sbasedILCapproachfornonlineardis-crete-timesystemcanachievetherequiredaccu
8、racywithfewerlearningiterationsthancommonnonlinearILCtechniques.De-I.INTROD