A Nonlinear ILC Schemes for Nonlinear Dynamic Systems To Improve Convergence Speed

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1 IJCSI Inernaonal Journal of Compuer Scence Iue, Vol. 9, Iue 3, No, ay ISSN (Onlne): A Nonlnear ILC Scheme for Nonlnear Dynamc Syem o Improve Convergence Speed Hoen Babaee, Alreza Khorav Faculy of Elecrcal and Compuer Engneerng, Nouhrvan Unvery of echnology,babol , Iran Faculy of Elecrcal and Compuer Engneerng, Nouhrvan Unvery of echnology,babol , Iran Abrac In h paper we udy nonlnear Ierave Learnng Conrol (ILC) cheme for nonlnear dynamc yem. he mehod are propoed n h paper uch a Newon-ype and Secan- ype, for mprove he peed n nonlnear ILC cheme yem. In nonlnear Ierave Learnng Conrol mehod peed faer han he lnear ILC cheme yem. he yem aken no conderaon he one phae conrol of he wched relucance moor (SR) and mulaon reul how ha he peed of he Newon-ype ILC cheme he fae and he Secan-ype faer han he lnear-ype bu lower han he Newon-ype. On he oher hand nonlnear ILC cheme requre more pror knowledge abou he yem. Keyword: ILC, Newon-ype, Secan-ype, Convergence Speed.. Inroducon he peed of he lnear-ype ILC cheme approved by a pove eady raon (le han one) of rackng error beween wo repeaed eraon. h due o he ue of lnear- ype updang law wh a conan learnng gan whch eraondependen. Some echnque n Numercal Analy have been ued n he learnng conrol degn [, ]. Recenly he Newon-ype ILC [3] and Qua-Newon ype ILC cheme [4] have been propoed and explored argeng a acheve a faer peed. In h paper we exended he reul o he general dynamc yem [5]. In order o mprove he peed, he Newon-ype ILC cheme fr preened n he area of erave learnng conrol o provde a correpondng mehod o lnear-ype ILC cheme [5, 6]. A nonlnear gan ake he place of conan gan and proven o be able o peed up he learnng peed [7, 8]. However he faer acheved a he prce of he need for more pror nformaon. he lnear-ype ILC only ued he yem oupu gnal, whle he Newon-ype ILC need he baed dervave of he yem npu-oupu mappng. he Secan-ype ILC cheme, on he oher hand, approxmae he Newon-ype ILC cheme how ha he peed of he Secan-ype ILC cheme beween he lnear-ype ILC cheme and he Newonype ILC cheme. A well known, boh he Newon and Secan mehod, n Numercal Analy, can only guaranee a faer peed around he ably,.e., a rercve condon requred [9-]. In pracce, h condon can hardly be me. In order o exend he range of learnng, he lnear-ype ILC cheme employed when he oupu of he dynamc yem oude a parcular error bound [, 3]. When nde he rackng error bound, he Newon-ype ILC cheme or Secan-ype ILC cheme wll ake over he learnng, expedon he learnng peed conderably whle reanng he learnng propery[4,5]. h paper organzed a follow: he problem formulaed n econ. he Newon-ype ILC cheme preened n econ3. In econ 4, he Secan-ype cheme addreed. Secon 5 provde an llurave example for comparon. hen concluon follow n econ 6.. Problem Saemen Le u look a he followng dynamc yem x ( ) f( X( ), u ( ), ) x()=x () y() g( X(), u(),), Copyrgh (c) Inernaonal Journal of Compuer Scence Iue. All Rgh Reerved.

2 IJCSI Inernaonal Journal of Compuer Scence Iue, Vol. 9, Iue 3, No, ay ISSN (Onlne): Where X, uu and [, ] a compac n ube of R and U a compac eparaon of R, he dynamc yem () repeaable over [, ], and afe he followng aumpon:[] Aumpon : he vecor-valued f(x, u, ) and calar g(x, u, ) are a lea wce connuouly dfferenable n he compac e U[, ] wh repec o x, u and correpondngly. Remark. From he connuy of f and g n Aumpon, he followng reul can be obaned drecly:. Nonlnear funcon f(x, u, ) and fr dervave wh repec o x and u are bounded n he compac e Ω, herefore, here ex a conan f uch ha: f ( x, u,) f( x, u,) f x x u u (). Nonlnear funcon g(x, u, ) and fr dervave wh repec o x and u are bounded n he compac e Ω, herefore, here ex a conan g uch ha: g( x, u,) g( x, u,) g x x u u (3) 3. Denoe g((),(),) x u gx ((),(), u gxx( ), g ( ), xu x x u g((),(),) x u g() u Where gx, g, gxu and g xx, are bounded by ome fne conan x,, xu and xx repecvely n he compac e Ω. Remark.Conder Aumpon, more rercve a requre no only Lpchz connuy condon for boh f () and g () bu alo he boundedne of he econd dervave for boh f () and g (). I can be een laer ha, uch a rercve requremen needed n order o acheve a faer peed. Aumpon. I aumed ha: g, ( xu,, ). u Remark3. gu Repreen he yem gan, whch could be nonlnear n Ω. Aumpon3.he dencal nalzaon condon hold for all eraon,.e. x () x, Z 3. Convergence Analy for Lnear-ype ILC Scheme he analy for lnear-ype ILC cheme gven wh aumpon.e. he unquene of ud (), n he followng we how ha, whou he requremen of uch an aumpon, he lnear-ype ILC cheme ll work [8]. heorem. For he dynamc yem (), aocaed wh he dered ou-pu yd () and he lnear-ype ILC cheme, he monoonc of wh he meweghed norm rcly guaraneed n he eraon doman f () afed Proof: Denoe x x x, u u u, ( x v x, u vu, ) where v. Applyng aylor heorem, [, ], we have gx (, u, ) gx ( xu, u, ) =g(x, u, ) qygu( ) x gx( ). =y ( ) qy g ( ) x g ( ) u x he rackng error a he (+) h eraon : y y y y = y qygu( ) x gx( ) (5) =[-q gu( )] y [ x x ] gx( ) Accordng o Aumpon and Aumpon 3, follow ha: x x [ f( x, u, ) f( x, u, )] d akng norm o boh de of he above equaon, nocng he Lpchz connuy condon of f(x (), u (), ), follow ha: x x f ( x x u u ) d f x x u u d f x x q y d Applyng Gronwall Lemma: f x x f e q y d Hence f e = fe q y (6) f fe q e d y (7) (4) Copyrgh (c) Inernaonal Journal of Compuer Scence Iue. All Rgh Reerved.

3 IJCSI Inernaonal Journal of Compuer Scence Iue, Vol. 9, Iue 3, No, ay ISSN (Onlne): x x max e x x [, ] f e max fe q y [, ] fe = fe q y akng norm of (5) yeld: y q g ( ) y x x. u x akng he me-weghed norm of he above nequaly, accordng o Aumpon and (8), we have: fe y qgu( ) x q fe y (9) (8) We defne qg u ( ), he correpondng fe equal o x q fe, (9) can be wren a: y ( ) y () Wh a uffcenly large, he lnear-ype ILC cheme and he peed : y QL,lm up () y h conclude he proof. Remark4. I obvou ha he npu equence and oupu equence have he ame peed. herefore, he fae peed can be acheved n he preence of uncerany g u, by olvng he mn-max problem. he opmal gan q wh he fae peed J c( ). Remark5. he fae peed depend on wo parameer and. Even he opmal echnque employed, f, J.e. he learnng wll be exremely low. On he oher hand, f gu avalable, accordng o (9) we can chooe q, whch can brng J g o he lowe level. h u mple obervaon movae u o develop he Newonype ILC cheme whch can conderably mprove he peed.[] 4. he Newon-ype ILC Scheme A fxed learnng gan q ulzed n he lnear-ype ILC approach, whch lm he learnng rae. A fa alway preferred n real applcaon. A maller Q-facor mple a faer rae. he Newon-ype approach well known n Numercal Analy a guaranee a fa peed for memory-le erave proce. he Newon-ype ILC approach, orgnaed from he ame dea, nroduced n he degn of erave learnng conrol law for nonlnear non-affne dynamc yem, amng a mprovng he peed of he learnng proce n he neghbourhood of he dered rajecory. In he ac erave proce, he Newonype cheme guaranee a fa peed [3]. For he dynamc erave proce, nce a nonlnear learnng facor q ued, Newon-ype ILC can alo acheve a much faer peed han ha of he lnear-ype ILC cheme n he ene of Q-facor [4]. Noe ha Q-facor of a local concep. In order o wden he range of learnng, he lnear-ype ILC cheme employed when he oupu of he dynamc yem oude a pecfed error bound. When nde he rackng error bound, he Newon-ype ILC cheme wll ake over he learnng job, mprove he learnng peed gnfcanly whle reanng he learnng propery [5]. Suppoe gu() known. From Aumpon u g ( ) (x,u,).he Newonype ILC cheme conruced a u() qy() f y LN : u ( ) y () u () ele gu, () ( ) Where L N and for he Newon-ype erave proce, q he robu opmal learnng gan for he lnear-ype ILC, and y () y () y () d heorem. For he dynamc yem (), he conrol updang law () wll make he rackng error a. Newon-ype ILC Q-faer y han lnear-ype ILC cheme n he ene ha QL (,) QL (,) N Remark6. Le u explan he Newon-ype ILC cheme furher by a lnear me-nvaran yem x Ax bu y cxdu Copyrgh (c) Inernaonal Journal of Compuer Scence Iue. All Rgh Reerved.

4 IJCSI Inernaonal Journal of Compuer Scence Iue, Vol. 9, Iue 3, No, ay ISSN (Onlne): Where gu lead o fae. d conan. If q g d, whch u, qq, he peed he 5. he Secan-ype ILC Scheme Secan algorhm n Numercal Analy provde a good approxmaon o Newon algorhm, whch movae he nroducon of he Secan-ype ILC cheme. I hould be noed ha Secan mehod n Numercal Analy alo a local concep, he rackng error converge only when he oupu of he yem a fr eraon near he dered rajecory. In h econ, he Secan-ype ILC approach conruced for he non-lnear non-affne dynamc yem n he ene of global : when he oupu of he dynamc yem far away from he dered rajecory, he lnear-ype ILC approach employed o guaranee he of he dered rajecory, he Secan-ype ILC approach employed o enhance he rae.[] he Secan-ype ILC cheme conruced a: L : u() qy() ele ( k ) u () ()( () () f y y u u u () gxu (,,) gxu (,,) & u u, k Where and k (3) L and for he Secan-ype erave proce, a fne conan. Remark7. I can be een ha Secan-ype ILC a econd order cheme, bu nonlnear n naure. heorem3. For he dynamc yem (), he conrol updang law wll make he rackng error a. he Secan-ype ILC Q- y faer han lnear-ype ILC cheme, however lower han he Newon-ype ILC cheme n he ene ha QL (,) QL (,) QL (,) N h how ha he peed of he Secanype ILC cheme n beween he Newon-ype ILC and he lnear-ype ILC cheme. Remark8. In above comparon of he learnng peed, neceary o chooe he maxmum afyng all requremen from he lnear-ype, Newon-ype and Secan-ype cheme. Under uch me-weghed norm, he peed are compared. 6. Illurave Example Conder one phae conrol of he Swched Relucance oor model a follow: dx Nr x d dx J j x u j d j (, ) N dh ( x ) ( x, u ) u h ( x ) e r j j j j j hj ( x) dx h ( x ) L L n x ( j ) / 4 j a u u jhj ( x ) (4) ha, only one phae orque generaed o follow he dered y d.44 Nm. 6.. Lnear-ype ILC Scheme Fr we chooe he lnear-ype ILC cheme o rack he dered oupu y d. Chooeu (). o guaranee he of he cheme, we chooe q 3.9, and =.475.oreover g.495, ( x, u) he mulaon reul hown n Fg (old lne). From he fgure can be een ha he lnear-ype ILC cheme generae a monoonou. he relave error max [,] y y d 6 drop from o whn 5 eraon. Furher more, he curve of he error very cloe o a ragh lne whch ndcae ha he lnear-ype ILC cheme ha he lnear order (geomerc n he log cale). 6.. Newon-ype ILC Scheme For he dynamc yem (4), he mulaon reul of he Newon-ype ILC cheme alo hown n Fg (doed lne).when he rackng error oude he error bound, he of Newon-ype he ame a he lnear-ype ILC. When he rackng error ener he error bound, he rackng error converge n a much faer. Afer 6 eraon, he relave rackng error drop from 4 o. From Fg, can be een ha he rackng error of he lnear-ype ILC cheme ake 7 eraon o reach he ame precon level. Due o he compuaon precon uch a quanzaon or fne word lengh, peed ceae afer he rackng error reache Secan-ype ILC Scheme For he dynamc yem (4), he mulaon reul of Secan-ype ILC cheme alo hown n Fg (dahdoed lne). Le k.5, where he rackng error Copyrgh (c) Inernaonal Journal of Compuer Scence Iue. All Rgh Reerved.

5 IJCSI Inernaonal Journal of Compuer Scence Iue, Vol. 9, Iue 3, No, ay ISSN (Onlne): ehod,inellgen Compuaon echnology and Auomaon, 9, pp: 8-85 [6] Xu,J.X and Y.an, On he P-ype and newon-ype ILC cheme for dynamc yem wh non-affne-n npu facor, Auomaca,, 38, pp:37-4,. [7]J. Kang, A new erave learnng conrol algorhm wh global for nonlnear yem, Inernaonal Conference on Compuer Applcaon and Syem odellng, ICCAS,, pp: [8]J. Kang, A new erave learnng conrol algorhm wh global for nonlnear yem, Inernaonal Conference on Compuer Applcaon and Syem odellng, ICCAS,,, pp: Fg. Convergence peed of he hree ILC cheme for SR 7. Concluon In h paper we uded nonlnear-ype ILC cheme for nonlnear dynamc yem. he propoed nonlnear mehod uch a Newon-ype and Secan-ype ILC cheme mprove he learnng peed. he order of nonlnear ILC cheme evaluaed n an yemac mehod. I concluded ha he peed of Newon-ype ILC cheme he fae. he Secan-ypee faer han ha of he lnear-ype bu lower han he Newon-ype. Reference [] Jan-Xn.Xu, Yng an, Lnear and Nonlnear Ierave Learnng Conrol, New Haven: Hamlon, arch 3. [] R.wLongman,H.S..Beg and e.all, Learnng Conrol by numercal opmzaon mehod, odelng and mulaon, Inrumen Socey of Amercan, (5),pp: , 998. [3] Xu,J.X and e.all, New ILC algorjm wh mproved for cla of non-affne funcon, proceedng of IEEE conferencee on Decon and conrol, Florda, 998, pp: [4]K.E. Avarachenkov, Ierave learnng conrol baed on qua-newon mehod, proceedng conference on Decon and conrol,florda,998, pp:7-74. [5] J. Kang; W. ang, Ierave Learnng Conrol for Nonlnear Syem Baed on New Updaed Newon [9]Y. Du,J. Sheng-Hong,.Guo,e-Jen Su and Ch. Chen, Oberver-Baed Ierave Learnng Conrol Wh Evoluonary Programmng Algorhm for IO Nonlnear Syem,Inernaonal Journal of Innovave Compung,7,pp: ,. [].Heerje,.o, Nonlnear Ierave Learnng Conrol wh applcaon o lhographc machnery, Conrol Engneerng Pracce, 5, pp: , 7. oude he error bound ( k), he of he Secan-ype he ame a he lnear-ype ILC. When he rackng error ener he error bound, he peed of he rackng error n beween he lnear-ypee ILC and Newon-ype ILC cheme. From Fg, can be een ha he rackng error of he Secan-ype ILC cheme ake eraon o reach he precon levell of. []Ch.Rong-Hu,H.Zhong-Shen Dual-age Opmal Ieravee Learnng Conrol for Nonlnear Non-affne 33(),pp: 6- Dcree-me Syem,Auomaca, 65, 7. [].Volckaer,.Dehl,J.Swever, Ierave Learnng Conrol for Nonlnear Syem wh Inpu Conran and Dconnuly Changng Dynamc,Amercan Conrol Conference (ACC),,pp: [3]Sh. Dong, Ch.Hanfu, Ierave learnng conrol for a cla of nonlnear yem,ieee Conrol Conference, 8,pp:6-64. [4]K.Jn,.Sun, A robu erave learnng conroller degn for uncerann nonlnear yem,inellgen Conrol and Auomaon,,pp:4-9. [5]J. Ku, S.ngxuan; Y.Yongqang, Robu repeve learnng conrol for a cla of me-varyng nonlnear yem,conrol Conference (CCC),, pp: Hoen Babaee receved h B.S. degree n Elecronc Engneerng from Deparmen of Elecrcal and Compuer Engneerng, ShahdBeheh Unvery, ehran, Iran n 9. He currenly maer cence uden of Elecrcal Engneerng Deparmen, Babol (Nouhrvan) Unvery of echnology, Babol, Iran. H reearch nere nclude Inellgen Conrol, Opmal Conrol, and Fuzzy conrol. Alreza Khorav receved he Ph.D. degreee n Conrol Engneerng from Iran Unvery of Scence and echnology (IUS), Iran, n 8. He currenly aan profeor a Elecrcal Engneerng Deparmen, Babol (Nouhrvan) Unvery of echnology, Babol, Iran. H reearch nere nclude robu and opmal conrol, modelng and yem denfcaon and nellgen yem. Copyrgh (c) Inernaonal Journal of Compuer Scence Iue. All Rgh Reerved.