on the identification and control of dynamical systems using neural networks

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1、452IEEETRANSACTIONSONNEURALNETWORKS,VOL.8,NO.2,MARCH1997OntheªIdentificationandControlofCommentsonªStochasticChoiceofBasisFunctionsDynamicalSystemsUsingNeuralNetworksºinAdaptiveFunctionApproximationandtheFunctional-LinkNetºErnestoRios-PatronandRichardD.BraatzJin-YanLiandTommyW.S.ChowA

2、bstractÐItisnotedthat[1,p.15,Example2]hasathirdequilibriumAbstractÐThispaperincludessomecommentsandamendmentsofthestatecorrespondingtothepoint(0.5,0.5).above-mentionedpaper.Subsequently,Theorem1intheabove-mentionedpaperhasbeenrevised.Thesignificantchangeoftheoriginaltheoremisthespaceo

3、fthethresholdsinthehiddenlayer.TherevisedtheoremsaysI.REMARKSthatthethresholdsofhiddenunits,b0,shouldbe0w01y00u0,wherew0=w^0;w^0=(^w01;111;w^0d),y0=(y01;111;y0d),andu0beIn[1],NarendraandParthasarathyperformanadmirablestudyindependentanduniformlydistributedinVd=[0;]2[0;]d01,oftheappli

4、cationofneuralnetworksforidentificationandcontrol.Id,and[02d;2d],respectively.Weagreewiththestatementoftheauthorsof[1,p.15],thatfornonlinearprocesses,ªSomepriorinformationconcerningtheI.INTRODUCTIONinput±outputbehavioroftheplantisneededbeforeidentificationcan1Theabove-mentionedpaperhas

5、introducedtherandomvectorbeundertaken.Thisincludesthenumberofequilibriumstatesoftheversionofthefunctional-link(RVFL)net.IgelnikandPaoshowunforcedsystemandtheirstabilityproperties....ºTheauthorsthenthefunctionapproximationcapabilityofRVFLbyastochasticstatethattheequilibriumstatesofthe

6、unforcedsystemapproachbasedonanlimit-integralrepresentationofthefunctiontobeapproximatedwithsubsequentevaluationoftheintegralbyyp(k)yp(k01)(yp(k)+2:5)theMonteCarlomethod.Thisstochasticapproachisdemonstratedyp(k+1)=(1)1+yp2(k)+yp2(k01)tobeanefficientapproximationmethodofmultivariatefun

7、ctionsaccordingtoitstheoreticaljustificationandsimulationresults.ThemostdistinctivecharacteristicofRVFListhatpartsofparametersofare(yp(k);yp(k01))=(0;0)and(2,2).RVFL,i.e.,theweightsandthresholdsofhiddenlayerareselectedWewouldliketonotethatthissystemhasathirdequilibriumstate,randomly,i

8、ndependently