NONLINEAR SYSTEM IDENTIFICATION USING DISCRETE-TIME NEURAL NETWORKS WITH STABLE LEARNING ALGORITHM

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1 IEAR SYSEM IDEIFICAI USIG DISCREE-IME EURA ERKS IH SABE EARIG AGRIHM all Korob Mohad Dl Insttut of Probl Solvng XYZ Unvrst Intllgnt Conol dsgn & ptzaton of copl Ssts atonal Engnrng School of Sfa - EIS B.P Sfa unsa orob_tall@ahoo.fr ohad.dl@ns.rnu.tn Mohad Chtourou Intllgnt Conol dsgn & ptzaton of copl Ssts atonal Engnrng School of Sfa - EIS B.P Sfa unsa ohad.chtourou@ns.rnu.tn Kwords: Absact: Stablt nural ntwors dntfcaton bacpropagaton algorth consand larnng rat apunov approach. hs papr prsnts a stabl nural st dntfcaton for nonlnar ssts. An nput output dscrt t rprsntaton s consdrd. o a pror nowldg about th nonlnarts of th sst s assud. h proposd larnng rul s th bacpropagaton algorth undr th condton that th larnng rat blongs to a spcfd rang dfnng th stablt doan. Satsfng such condton unstabl phnonon durng th larnng procss s avodd. A apunov analss s ad n ordr to act th nw updatng forulaton whch contan a st of nqualt consants. In th consand larnng rat algorth th larnng rat s updatd at ach aton b an quaton drvd usng th stablt condtons. As a cas stud dntfcaton of two dscrt t ssts s consdrd to donsat th ffctvnss of th proposd algorth. Sulaton rsults concrnng th consdrd ssts ar prsntd. IRDUCI h ara of sst dntfcaton has rcvd sgnfcant attnton ovr th past dcads and now t s a farl atur fld wth an powrful thods avalabl at th dsposal of conol ngnrs. nln sst dntfcaton thods to dat ar basd on rcursv thods such as last squars for ost ssts that ar prssd as lnar n th paras IP. o ovrco ths IP assupton nural ntwors s ar now plod for sst dntfcaton snc ths ntwors larn copl appngs fro a st of apls. Du to approaton proprts as wll as th nhrnt adaptaton faturs of ths ntwors prsnt a potntall appalng alnatv to odlng of nonlnar ssts. Morovr fro a practcal prspctv th assv parallls and fast adaptablt of plntatons provd addtonal ncntvs for furthr nvstgaton. Svral approachs hav bn prsntd for sst dntfcaton wthout usng and usng arndra and Parthasarath 990 Bosovc and arndra 995. Most of th dvlopnts ar don n contnuous t du to th splct of drvng adaptaton schs. o th conar vr fw rsults ar avalabl for th sst dntfcaton n dscrt t usng s. Howvr ost of th schs for sst dntfcaton usng hav bn donsatd through prcal studs or convrgnc of th output rror s shown undr dal condtons Chng-Hang and al 00. thrs Sadgh 993 hav shown th stablt of th ovrall sst or convrgnc of th output rror usng lnart n th paras assupton. Both rcurrnt and dnac n whch th has ts own dnacs hav bn usd for sst dntfcaton. 35

2 ICIC Innatonal Confrnc on Inforatcs n Conol Autoaton and Robotcs Most dntfcaton schs usng thr ultlar fdforward or rcurrnt nclud dntfr sucturs whch do not guarant th bounddnss of th dntfcaton rror of th sst undr nondal condtons vn n th opn-loop confguraton. Rcnt rsults show that nural ntwor tchnqu ss to b vr ffctv to dntf a broad catgor of copl nonlnar ssts whn coplt odl nforaton cannot b obtand. apunov approach can b usd drctl to obtan stabl anng algorths for contnuous-t nural ntwors G Hang Zhang 00 Kosatopoulos Polcarpou Chrstodoulou Ioannou 995 Yu Pozna 00. h stablt of nural ntwors can b found n Fng Mchl 999and Suns andwall D Moor 997. h stablt of larnng algorths was dscussd n n Gupta 999 and Polcarpou Ioannou 99. It s wll nown that noral dntfcaton algorths ar stabl for dal plants Ioannou Sun 004. In th prsnc of dsturbancs or unodld dnacs ths adaptv procdurs can go to nstablt asl. h lac of robustnss n paras dntfcaton was donsatd n E. Barn 99 and bca a hot ssu n 980s. Svral robust odfcaton tchnqus wr proposd n Ioannou Sun 004. h wght adustng algorths of nural ntwors s a tp of paras dntfcaton th noral gradnt algorth s stabl whn nural ntwor odl can atch th nonlnar plant actl Polcarpou Ioannou 99. Gnrall so odfcatons to th noral gradnt algorth or bacpropagaton should b appld such that th larnng procss s stabl. For apl n. n M.M. Gupta 999 so hard rsctons wr addd n th larnng law n Polcarpou Ioannou 99 th dnac bacpropagaton was odfd wth stablt consants. h papr s organzd as follows. Scton II dscrbs th adoptd dntfcaton sch. In scton III and through a stablt analss a consand larnng rat algorth s proposd to provd stabl adaptv updatng procss. two spl sulaton apls gv th ffctvnss of th suggstd algorth n scton I. PREIMIARIES h an concrn of ths scton s to noduc th fdfarward nural ntwor adoptd archtchtur and so concpts of bacpropagaton anng algorth. Consdr th followng dscrt-t nput-output nonlnar sst [... n u... u ] f h nural odl for th plant can b prssd as ˆ F[ Y ] hr Y... n u u u and and s th wght para vctor for th nural odl. A tpcal ultlar fdfarward nural ntwor s shown n Fgur whr I s th th nuron nput s th th nuron output and ndcat nurons w s th wght btwn nuron and nuron. For th th nuron th nonlnar actv functon s dfnd as f 3 h output of th consdrd s n I w ; f I ;. I ' ; f I ;... Fgur : Fdforward nural odl. 4 ranng th nural odl conssts on th adustnt th wght paras so that th nural odl ulats th nonlnar plant dnacs. Input output apls ar obtand fro th opraton hstor of th plant. 35

3 IEAR SYSEM IDEIFICAI USIG DISCREE-IME EURA ERKS IH SABE EARIG AGRIHM Usng th gradnt dcnt th wght connctng nuron to nuron s updatd as 5 hr [ ] s th larnng rat. h partal drvatvs ar calculatd wth rspct to th vctor of wghts. f ' w f ' I I f ' I 6 Bacpropagaton algorth has bco th ost popular on for anng of th ultlar prcpon. Gnrall so odfcatons to th noral gradnt algorth or bacpropagaton should b appld such that th larnng procss s stabl. For apl n B. Egardt 979 so hard rsctons wr addd n th larnng law n.a.k. Suns. andwall B. D Moor 997 th dnac bacpropagaton was odfd wth stablt consants. 3 SABIIY AAYSIS In th latur th apunov snthss Z.P. ang Y. ang 00. Yu X. 00 conssts on th slcton of a postv functon canddat whch lad to th coputaton of an adaptaton law nsurng t s dcrscnc. & 0 for contnuous ssts and Δ 0 for dscrt t ssts. Undr ths assuptons th functon s calld apunov functon and garant th stablt of th sst. ur obctv s th dnaton of a stablzng adaptaton law nsurng th stablt of th dntfcaton sch prsntd blow and th boundnss of th output sgnals. h followng assuptons ar ad for sst Assupton. h unnown nonlnar functon f s contnuous and dffrntabl. Assupton. Sst output can b asurd and ts ntal valus ar assud to ran n a copact st Ω hor h stablt of th dntfcaton sch s guarantd for a larnng rat vrfng th followng nqualt : 0 7 hr ar rspctvl th vctor wght btwn th nputs and th hddn lar and th vctor wght btwn th hddn lar and th outputs lar. dnot th th nput and th th hddn nuron. 3. Proof Consdrng th apunov functon: 8 hr. dnots th a ac opraton. dnots th optal vctor wght btwn th nputs and th hddn lar. dnots th optal vctor wght btwn th hddn lar and th outputs. h coputaton of th Δ prsson lads to : [ ] Δ [ ] h adoptd adaptaton law s th gradnt algorth. hav: 9 hr th partal drvatvs ar prssd as 0 ur fld of nst covrs th blac bo ssts. h partal drvatvs dnotng th sst dnac ar approatd as follow: 353

4 h approatd partal drvatvs ar gvn through: Adoptng th varabls A and B dfnd b: A B h Δ prsson s calculatd as : B A Δ β α hr α β h stablt condton 0 Δ s satsfd onl f : 0 β α 3 Solvng ths scond dgr quaton lad to th stablshnt of th condton 7 : 0 Δ f satsfs th followng condton : s 0 whr Fgur : Evoluton of th sst output and th nural odl output doan stablt. 4 SIMUAI RESUS In ths scton two dscrt t ssts ar consdrd to donsat th ffctvnss of th rsult dscussd blow. 4. Frst rdr Sst h consdrd sst s a faous on n th ltatur of adaptv nural conol and dntfcaton. h dscrt nput-output quaton s dfnd b: Sst ural odl { } n s ICIC Innatonal Confrnc on Inforatcs n Conol Autoaton and Robotcs 354

5 IEAR SYSEM IDEIFICAI USIG DISCREE-IME EURA ERKS IH SABE EARIG AGRIHM 3 u 5 For th nural odl a thr-lar was slctd wth two nputs thr hddn and on output nods. Sgodal actvaton functons wr plod n all th nods. h wghts ar ntalzd to sall rando valus. h larnng rat s valuatd at ach aton through 4. It s also rcognzd that th anng prfors vr wll whn th larnng rat s sall. As nput sgnal a snusodal on s chosn whch th prson s dfnd b: 0.05 π u 0.5cos π 6 5 h sulatons ar ralzd n th two cass durng 0 atons. wo larnnng rats valus ar fd n and out of th larnng rat rang prsntd n 7.Sulaton rsults ar gvn through th followng fgurs : [ ϕ ] 05u 50tanh whr 4 u ϕ u 7 h procss dnac s nstng. In fact t has th bhavour of a frst ordr law pass fl for nputs sgnal apltud about 0. th bhavour of a lnar scond ordr sst n th cas of sall apltuds 0 < u < 05 and th bhavour of a non lnar scond ordr sst n th cas of grat nputs apltuds 05 < u < 5 Chng-Hang and al 00. For th nural odl a thr-lar was slctd wth thr nputs thr hddn and on output nods. Sgodal actvaton functons wr plod n all th nods. h wghts ar ntalzd to sall rando valus. h larnng rat para s coputd nstantanousl. As nput sgnal a snusodal on s chosn whch th prson s dfnd b: π u 0.5 cos π 8 3 h sulatons ar ralzd n th two cass. wo larnnng rats valus ar fd n and out of th larnng rat rang prsntd n 7. Sulaton rsults ar gvn through th followng fgurs: Sst ural odl Fgur 3 : Evoluton of th sst output and th nural odl output stablt doan. 4.. Conts Fg and Fg 3 show that f th larnng rat blongs to th rang dfnd n 7 th stablt of th dntfcaton sch s garantd. It s shown through ths sulaton that th dntfcaton obctvs ar satsfd. ut ths varaton doan of th larnng rat th dntfcaton s nstabl and th dntfcaton obctvs ar unrachabl. Sst ural odl Fgur 4: Evoluton of th sst output and th nural odl output stablt doan. 4. Scond rdr Sst h scond apl concrns a dscrt t sst gvn b: 355

6 ICIC Innatonal Confrnc on Inforatcs n Conol Autoaton and Robotcs Fgur 5: Evoluton of th sst output and th nural odl output stablt doan. 4.. Conts Hr w ad a coparatv stud btwn an arbar choc of a larnng rat out sd of th stablt doan and a consand choc vrfng th stablt condton and guarantng acng capablt. h sulaton rsults schow that a larnng rat n th stablt doan nsur th stablt of th dntfcaton sch. 5 CCUSIS Sst ural odl o avod unstabl phnonon durng th larnng procss consand larnng rat algorth s proposd. A stabl adaptv updatng procsss s guarantd. A apunov analss s ad n ordr to act th nw updatng forulatons whch undr nqualt consant. In th consand larnng rat algorth th larnng rat s updatd at ach atv nstant b an quaton drvd usng th stablt condtons. h applcablt of th approach prsntd s llusatd through two sulaton apls. P. A. Ioannou. Sun 004. Robust Adaptv Conol Inforaton Scncs Prntc-Hall Uppr Saddl Rvr.. n M.M. Gupta 999. Stabl dnac bacpropagaton larnng n rcurrnt nural ntwors IEEE rans. ural twors Z. P. ang Y. ang 00. Input-to-stat stablt for dscrt-t nonlnar ssts Autoatca E. B. Kosatopoulos M.M. Polcarpou M.A. Chrstodoulou P.A. Ioannou 995. Hgh-ordr nural ntwor sucturs for dntfcaton of dnacal ssts IEEE rans. ural twors M. M. Polcarpou P.A. Ioannou 99. arnng and convrgnc analss of nural-tp sucturd ntwors IEEE rans. ural twors A. K. Suns. andwall B. D Moor 997. q thor: chcng and posng stablt of rcurrnt nural ntwors for nonlnar odllng IEEE rans. Sgnal Procss spcal ssu on nural ntwors for sgnal procssng Yu X. 00. So stablt proprts of dnac nural ntwors IEEE rans. Crcuts Sst. Part I Yu A.S. Pozna X. 00. Multlar dnac nural ntwors for nonlnar sst on-ln dntfcaton Int.. Conol E. Barn 99. ptsaton for anng nural nts IEEE rans. ural twors Chng-Hang and al 00. Conol of onlnar Dnac Ssts a adaptv PID Conol Sch wth Daturaton Bound Innatonal ournal of Fuzz Ssts ol. 4 o K. S. arndra and K. Parthasarath 990. Idntfcaton and conol of dnacal ssts usng nural ntwors IEEE ransacton ural twors vol. pp.4 7 Mar.. D. Bosovc and K.S.arndra 995. Coparson of lnar nonlnar and nural-ntwor basd adaptv conollrs for a class of fd-batch frntaton procss Autoatca vol. 3 no. 6 pp REFERECES B. Egardt 979. Stablt of adaptv conollrs. n: ctur ots n Conol and Inforaton Scncs 0 Sprngr-rlag Brln. Z. Fng A.. Mchl 999. Robustnss analss of a class of dscrt-t ssts wth applcatons to nural ntwors n: Procdngs of Arcan Conol Confrnc San Dgo pp S.S. G C.C. Hang.H.. Zhang 00: Stabl Adaptv ural twor Conol Kluwr Acadc Boston. 356