Iterative learning control with initial rectifying action for nonlinear continuous systems X.-D. Li 1 T.W.S. Chow 2 J.K.L. Ho 3 J.

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1 Publishd in IET Control Thory and Applications Rcivd on 24th Dcmbr 27 Rvisd on 21st May 28 doi: 1.149/it-cta:27486 ISSN Itrativ larning control with initial rctifying action for nonlinar continuous systms X.-D. Li 1 T.W.S. Chow 2 J.K.L. Ho 3 J. Zhang 1 1 School of Information Scinc and Tchnology, Sun Yat-Sn Univrsity, Guangzhou, Popl s Rpublic of China 2 Dpartmnt of Elctronic Enginring, City Univrsity of Hong Kong, 83 Tat Ch Avnu, Kowloon, Hong Kong, Popl s Rpublic of China 3 Dpartmnt of Manufacturing Enginring and Enginring Managmnt, City Univrsity of Hong Kong, 83 Tat Ch Avnu, Kowloon, Hong Kong, Popl s Rpublic of China lixd@mail.sysu.du.cn Abstract: A nw itrativ larning control (ILC) mthod with initial rctifying action for nonlinar continuous multivariabl systms is prsntd. Unlik gnral ILC tchniqus, th proposd ILC approach allows initial outputs of an ILC systm at diffrnt itrations to fluctuat randomly around th initial valu of th dsird output. Th proposd stratgy includs an initial rctifying action of ILC on a vry small initial tim intrval, and pursus th rfrnc trajctory tracking byond th initial tim intrval. Th output tracking rror byond th initial tim intrval can b drivn to a rsidual st whos siz dpnds on th stimation rror of input matrix. A numrical xampl is usd to illustrat th ffctivnss of th proposd ILC approach. 1 Introduction Itrativ larning control (ILC) is a vrsatil control tchniqu to improv transint rspons of a systm oprating rptitivly ovr a fixd tim intrval. On of th important faturs of ILC is that it rquirs lss a priori knowldg about th controlld systm in th cours of dsign. This maks ILC incrasingly important in control applications, such as robot manipulators and disk driv systms that ar mostly dsignd for rptitiv tasks. Until now, thr hav bn a lot of ILC algorithms rportd. But most of th xisting ILC works assum that th initial outputs of an ILC systm at diffrnt itrations wr kpt invariant [1, 2], which mans that th ILC systm must hav a fixd initial rror at diffrnt itrations. In som cass, a mor stringnt condition of zro initial rror is vn rquird [3 6]. All of ths hypothsiss ar difficult to b implmntd in practic, as locating opration rptitivly in an ILC application can rsult in irrgular drifts of initial outputs at diffrnt itrations to th initial valu of th dsird output. Clarly, th study of ILC problm applid to dynamical systms with non-fixd initial rrors is ssntial. Th difficulty of ILC with non-fixd initial rror mainly lis in th dynamical complxity of ILC in which initial itrativ rror intracts with not only th systm dynamics but also th itrativ larning procss. L and Bin [7] rportd som novl undsirabl phnomnon du to th mismatch in th initial conditions in ILC procss. Thy pointd out that th control systm could bcom unstabl whn th initial output at ach itration was diffrnt from th prvious initial output. Thr ar a fw rsarch paprs [8 1] discussing th robustnss of ILC algorithms to non-fixd initial rrors, but th bound of robustnss is normally too larg and cannot b adjustd for practically tracking a rfrnc trajctory. As a rsult, an initial rctifying action should b considrd in ILC dsign. It is worth noting that thr hav bn works [11, 12] incorporating an initial rctifying action into ILC dsign to achiv complt tracking of a rfrnc trajctory ovr a spcifid intrval, but thy rquir that th initial rror b a fixd valu. Thy will not work whn a random or nonfixd initial rror is ncountrd. Hithrto, th rsarch on ILC applid to dynamical systms with non-fixd initial rrors still rmains an opn and important issu. Rcntly, a fw studis [13 17] hav bn focussd on th ILC problm of dynamical systms with non-fixd initial rrors. In [13], a mthod calld ILC with multi-modal input was proposd to tackl th ILC problm for linar IET Control Thory Appl., 29, Vol. 3, No. 1, pp doi: 1.149/it-cta:27486 & Th Institution of Enginring and Tchnology 29

2 continuous systms with non-fixd initial rrors. But it is computationally complx bcaus th dtrmind input is, in fact, th synthsis of th multi-modal ILC input with varying initial condition. In [14], undr a spcific condition, an avrag oprator-basd PD-typ ILC controllr for linar continuous systms with non-fixd initial rrors was invstigatd. In [17], an ILC controllr for linar discrt systms with non-fixd initial rrors was invstigatd using 2-D systm thory. Dspit its ability to handl variabl initial conditions, it is only ffctiv whn th tracking tim intrval is sufficintly larg. For ILC on nonlinar systms, Xu and Yan [15] discussd th inhrnt rlationship btwn diffrnt initial conditions and th corrsponding larning convrgnc (or bounddnss) proprty undr a Lyapunov-basd ILC mthod. Howvr, th proposd ILC tchniqu [15] is only suitabl for a class of simpl first-ordr nonlinar continuous systms with unit input gain. Furthrmor, a dirct adaptiv ILC approach basd on a fuzzy nural ntwork was prsntd in [16] for a class of nonlinar systms with non-fixd initial rrors. Using th proposd adaptiv ILC approach, th norm of stat tracking rror will asymptotically convrg to a tunabl rsidual st as itration gos to infinit. Th main objctiv of this papr is to dscrib how an initial rctifying action can b combind into th convntional D- typ ILC tchniqu for nonlinar continuous multivariabl (NCM) systms with non-fixd initial rrors. Byond th dsignatd initial tim intrval, th stablishd ILC tchniqu is abl to driv th output tracking rror to a rsidual st whos siz dpnds on th stimation rror of input matrix. It is also important to not that a prfct rfrnc trajctory tracking byond th initial tim intrval can b achivd whn an accurat knowldg on input matrix at th initial tim intrval is availabl. Th organisation of this papr is as follows. Sction 2 prsnts th ILC problm formulation and th ILC rul with initial rctifying action. Sction 3 invstigats th larning convrgnc of th proposd ILC rul undr nonfixd initial rrors. And th simulation rsults ar illustratd in Sction 4. Finally, Sction 5 concluds this papr. 2 Problm formulation and ILC rul with initial rctifying action Considr th following class of NCM systms prforming rptitiv tasks ovr a fixd tim intrval t [ [, T ] _x k (t) f (x k (t), t) þ B(t) u k (t) y k (t) C(t) x k (t) (1a) (1b) whr k dnots th kth rptitiv opration of th systm; x k (t) [ R n, u k (t) [ R m (m n), and y k (t) [ R p ar th stat, control input and output of th systm, rspctivly; B(t) [ R nm and C(t) [ R pn ar tim-variant matrics; th nonlinar function f (, ) : R n [, T ] 7! R n is locally Lipschitz in x k, that is, for all t [ [, T ] and k, thr xists a constant L f such that kf (x kþ1 (t), t) f (x k (t), t)k L f kx kþ1 (t) x k (t)k (2) whr th norm k.k will b dfind latr. An xampl of systm (1) can b rfrrd to th PM synchronous motor studid in [18]. Givn a rfrnc output trajctory y r (t) and an initial control input u (t) att [ [, T ], an ILC dsign for systm (1) is closly rlatd to its boundary conditions x k () for k, 1, 2... Unlik gnral ILC litraturs, w assum that x k () is boundd and diffrnt for k, 1, 2..., and y k () fluctuats randomly around y r () within a bound in this study. Basd on th givn boundary conditions, our ILC objctiv is to itrativly dtrmin a control input squnc fu k (t)g, t [ [, T ], such that as k gos to infinit, th output tracking rror y r (t) y k (t) ovr a spcifid tim intrval t [ [h, T ] can b drivn to a rsidual st whos siz dpnds on th stimation rror of input matrix B(t), whr h is a spcifid small ral numbr btwn and T. In many ILC applications, th modl information on th controlld systm, spcially th input matrix B(t), can b wll known. In this cas, our ILC dsign with th control objctiv can achiv an ffctiv rfrnc trajctory tracking. Th output tracking rror in ILC procss is dnotd as k (t) y r (t) y k (t) (3) Whn th initial ILC rrors k () for k, 1, 2...ar nonfixd, but boundd, it has bn shown in [9] that a wllknown D-typ ILC rul u kþ1 (t) u k (t) þ K (t)_ k (t) (4) can nsur th bounddnss of th output tracking rror k (t), t [ [, T ]ask gos to infinit. In practic, this bound is usually too larg to b applicabl to track a rfrnc trajctory. In this papr, a rctifying action is thn combind into a D-typ ILC rul (4), and th following ILC rul for control calculation is applid u kþ1 (t) u k (t) þ K (t)_ k (t) þ u h (t) ^B(t) 1 R X k () (5) whr ^B(t) dnots th stimation of B(t); ^B(t) 1 R invrs of ^B(t); and u h ðþ t 8 2 h 1 t <, t [ ½, hþ h :, t [ ½h, T is th right (6) X k () ^B()K () k () þ x k () x kþ1 () (7) For th convninc of convrgnc analysis of th proposd ILC approach, th following dfinitions and lmma on 5 IET Control Thory Appl., 29, Vol. 3, No. 1, pp & Th Institution of Enginring and Tchnology 29 doi: 1.149/it-cta:27486

3 norms ar givn ksk max js ðþ i j 1in kgk max 1im kqðþk t l X n t[ ½,T j1 jg i, j j! lt kqt ðþk, l. whr s [ s (1) s (2) s (n) ] T is a vctor, G [g i, j ] [ R mn is a matrix and q(t), t [ [, T ] is a ral function vctor/matrix. Spcially, in this papr, w us kq(t)k l j t[[h,t ] t[[h,t ] lt kq(t)k (8) to dfin l-norm of q(t) on th tim intrval t [ [h, T ]. On th rlationship btwn th norm k.k and th l-norm k.k l of a vctor x(t), t [ [, T ], w hav th following lmma 1. Lmma 1: t[ ½,T lt kxðþkdt t 1 l kxt ðþk l (9) Th proof of Lmma 1 is an immdiat consqunc of th norm k.k l, and thrfor is omittd. 3 Larning convrgnc undr non-fixd initial rrors In this sction, th convrgnc rsults of ILC rul (5) ar providd in Thorm 1. Thorm 1: For an NCM systm (1), pos that th rfrnc output y r (t) is diffrntiabl, and th initial ILC rror k () varis randomly within a bound. If thr xists a matrix K(t) to mak t[ ½,T ki Ct ðþbt ðþkt ðþkr, 1 (1) and th matrix B(t) att [ [, h] is right invrtibl; thn, th ILC rul (5) can nsur lim k k (t)k l j t[[h,t ] lm C(M E MX þ M MK ) k!1 (1 r)( ) (11) whr B(t) ^B(t) 1 R I þ E(t), t[[,h] ke(t)k M E, t[[,t ] kc(t)k M C, k ^B() B()k M, k kx k () kmx, k kk () k ()k M K,, r, 1, and l is a boundd valu. Espcially, whn th input matrix B(t) at t [ [, h] is accuratly known, w hav lim k!1 k k (t) k l j t[[h,t ]. Proof: From (1) and (5), w hav x kþ1 (t) x k (t) þ _x kþ1 (t) dt _x k (t) dt þ [x kþ1 () x k ()] þ B(t)[u kþ1 (t) u k (t)] dt þ [x kþ1 () x k ()] B(t)K (t)_ k (t) dt þ u h (t)b(t) ^B(t) 1 R dt X k () þ [x kþ1 () x k ()] [ f (x kþ1 (t), t) f (x k (t), t)] dt þ B(t)K (t) k (t) B()K () k () ð t dt k (t) dt þ u h (t)[i þ E(t)]dt X k () þ [x kþ1 () x k ()] (12) On th othr hand, from (1) and (3) kþ1 (t) y r (t) y kþ1 (t) k (t) [y kþ1 (t) y k (t)] k (t) C(t)[x kþ1 (t) x k (t)] (13) Substituting (12) into (13), w hav kþ1 (t) [I C(t)B(t)K (t)] k (t) C(t) þ C(t) C(t) dt k (t)dt u h (t)[i þ E(t)] dt X k () þ C(t)B()K () k () C(t)[x kþ1 () x k ()] (14) As t [ [h, T ], considring that Ð t u h (t)dt Ð h u h (t)dt 1 IET Control Thory Appl., 29, Vol. 3, No. 1, pp doi: 1.149/it-cta:27486 & Th Institution of Enginring and Tchnology 29

4 from (6), (12) and (14) can b, rspctivly, writtn as x kþ1 (t) x k (t) þ B(t)K (t) k (t) þ ð h dt k (t)dt u h (t)e(t)dt X k () þ [ ^B() B()]K () k () kþ1 (t) [I C(t)B(t)K (t)] k (t) C(t) þ C(t) ð h [f (x kþ1 (t), t) f (x k (t), t)] dt dt k (t)dt C(t) u h (t)e(t)dt X k () C(t)[ ^B() (15) B()]K () k () (16) Now, lt us indpndntly invstigat th proprtis of (15) and (16) on th whol tim intrval t [ [, T ]. Taking th norms on two sids of (15) and (16), rspctivly, w hav kx kþ1 (t) x k (t)k L f kx kþ1 (t) x k (t)k dt þ M 1 k k (t)kþm 2 k k (t)k dt þ M E M X þ M M K (17) k kþ1 (t)k r k k (t)kþm C L f kx kþ1 (t) x k (t)k dt þ M C M 2 k k (t)k dt þ M C M E M X þ M CM M K (18) whr t[[,t ] kb(t)k (t)k M 1, t[[,t ] k[d(b(t)k (t))=dt] km 2, and M E, M, M C, MX, M K, r ar dfind as in Thorm 1. Multiplying (17) and (18) by 2lt, rspctivly, whr l. L f,whav kx kþ1 ðþ x t k ðþk t l L f þ M 1 k k ðþk t l þ M 2 t[ ½,T t[ ½,T lt kx kþ1 ðþ x t k ðþkdt t lt k k ðþkdt t þ M E M X þ M M K (19) k kþ1 ðþk t l r k k ðþk t l þ M C L f lt kx kþ1 ðþ x t k ðþkdt t t[ ½,T t[ ½,T lt k k ðþkdt t þ M C M E MX þ M C M 2 þ M C M M K (2) Applying (9) of Lmma 1 to (19) and (2), w can obtain th following inquality kx kþ1 ðþ x t k t ðþk l lm 1 þ M 2 þ lm EM X k k ðþk t l k kþ1 (t)k l r þ M CM 2 k l k (t)k l þ M CL f l Substituting (21) into (22), w obtain k kþ1 (t)k l rk k (t)k l þ lm CM E M X kx kþ1 (t) x k (t)k l þ lm M K (21) þ M C M E M X þ M CM M K (22) þ lm CM M K (23) whr rr þ(m C M 2 =l)þ(m C L f (M 1 þ(m 2 =l))=(l L f )). Whn r, 1, w can slct l to b sufficintly larg such that r, 1. Equation (23) is a contraction, thrfor lim k k (t)k l lm C(M E MX þ M MK ) k!1 (1 r)( ) (24) On th othr hand, from th dfinition of th l-norm, lim k!1 k k (t)k l j t[[h,t ] lim k!1 k k (t)k l. Considring (24), w hav lim k k (t)k l j t[[h,t ] lm C(M E MX þ M MK ) k!1 (1 r)( ) (25) Furthrmor, if th input matrix B(t) at t [ [, h] is accuratly known, w hav M E M. It can b dirctly drivd from (25) that lim k!1 k k (t)k l j t[[h,t ]. From th abov dduction, it is important to not that (25) is obtaind from th invstigation of systm (15) and (16) on th tim intrval t [ [, T ]. Systm (15) and (16) is not quivalnt to th original systm (12) and (14) at t [ [, T ], but thy ar quivalnt on th tim intrval t [ [h, T ]. Thy should xhibit th sam proprtis at t [ [h, T ]. Thrfor (25) is also suitabl to systm (12) and (14) at t [ [h, T ]. Thorm 1 is provd. A 52 IET Control Thory Appl., 29, Vol. 3, No. 1, pp & Th Institution of Enginring and Tchnology 29 doi: 1.149/it-cta:27486

5 Rmark 1: Equation (25) shows that th ILC rul (5) is abl to driv th ILC rror at t [ [h, T ] into a bound. But, it is worth noting that aftr th paramtr l is slctd, th bound is mainly dcidd by paramtrs M E and M, which ar rlatd to th stimation rror with rspct to th input matrix B(t). Thrfor compard with th robust ffct of th D-typ ILC rul (4), th bound can b controlld to a much smallr lvl whn B(t) is wll stimatd. This point will b bttr illustratd by th xampl prsntd in th nxt sction. Rmark 2: From th ILC rul (5) and th dfinition (6) of u h (t), it is shown that th ILC rul (5) puts an initial rctifying action on a small initial tim intrval [, h] and pursus th rfrnc trajctory tracking at t [ [h, T ]. Th function u h (t) spcifis th initial rctifying action on [, h]. A lss valu h in u h (t) may rsult in a largr control input. Thus, th slction of h should b don basd on th trad-off btwn th rsulting control input and th tracking rror. In addition, Thorm 1 rquirs that th input matrix B(t) is right invrtibl and stimabl. But th corrsponding tim intrval is only a small initial tim intrval [, h], not th ntir tim intrval [, T]. Also, in ordr to achiv prfct tracking at t [ [h, T ] namly lim k!1 k k (t)k l j t[[h,t ], w nd th input matrix, B(t), to b accuratly known on th initial tim intrval [, h] only. Rmark 3: Rgarding th slction of th larning gain matrix K(t) in th ILC rul (5), Thorm 1 givs us a thortical guidlin that K(t) should satisfy (1). In particular, lt K (t) a( ^C(t) ^B(t)) T [ ^C(t) ^B(t)( ^C(t) ^B(t)) T ] 1 if ^C(t) ^B(t) is of full-row rank, whr ^C(t) and ^B(t) ar stimations to C(t) and B(t), rspctivly. W can find a [ (1, 2 1) and 1 [ (, 1) so that t[[,t ] ki C(t)B(t)K (t)k r, 1. 4 Illustrativ xampl Considr an ILC problm of th following NCM systm d x (1) :5 sin x (1) þ :8 cos x (2) >< dt x (2) 4 5 þ Bt ðþu sin x (1) x (2) >: y Ct ðþ x (1) x (2) (26) 2:5 sin(1t) whr Bt ðþ, C(t) :5t þ : :3t Th dsird output y r (t) is dscribd by th quation y r (t) 15(t t 2 ), t [ [, 1] (27) In ordr to vrify our ILC approach for NCM systms with non-fixd initial rrors, w randomis th initial stats of systm (26) x (1) () and x (2) (), which vary btwn 21 and 1 at diffrnt itrations of ILC procss randomly, as shown in Tabl 1. Th inducd non-fixd initial ILC rrors, k (), ar also listd in Tabl 1. Suppos that th accurat information on paramtrs B(t) and C(t) in systm (26) is unavailabl, and only thir 2:3 :7 sin(1t) stimation ^B(t) and ^C(t) ar givn as :15 1:2 :45t and :6t þ :15 1:2, rspctivly. In th ILC procss of systm (26), w st th initial input of ILC as u (t), t [ [, 1], and h.5. W pursu th tracking of th dsird output y r (t) on th intrval t [ [:5, 1]. Th accuracy of tracking is valuatd by th following maximum absolut rror of tracking EE jy r (t) y(t)j t[[h,1] To bttr illustrat th initial rctifying action of our Tabl 1 Non-fixd initial stats and initial rrors of ILC tracking prformanc at diffrnt itrations k x k () :36 :59 :42 :17 :31 :93 :73 :13 :41 :77 :81 :4 :84 :54 :75 :42 k () k x k () :38 :93 :6 :83 :71 :8 :27 :23 :84 :67 :33 :21 :15 :52 :78 :82 k () IET Control Thory Appl., 29, Vol. 3, No. 1, pp doi: 1.149/it-cta:27486 & Th Institution of Enginring and Tchnology 29

6 Figur 1 Maximum absolut rror of tracking on th intrval t [[.5, 1] at diffrnt itration numbrs using th D-typ ILC rul (4) proposd ILC rul (5) by comparison, first, a D-typ ILC rul (4) without a rctifying action is usd. W st K (t) :6( ^C(t) ^B(t)) T [ ^C(t) ^B(t)( ^C(t) ^B(t)) T ] 1, which maks max t[[,1] ki C(t)B(t)K (t)k, 1. Fig. 1 prsnts th situation of th tracking rror indx EE on th intrval t [ [:5, 1] whn th D-typ ILC rul (4) is xcutd at diffrnt itration numbrs. In Fig. 1, it is shown that th ILC rul (4) is abl to driv th ILC rror into a bound. Th robustnss of th ILC rul (4) to non-fixd initial ILC rrors is thus illustratd. But it is also noticd that th bound is surly too larg for practical application. Nxt, th abov ILC rul (5), which includs an initial rctifying action, is applid. Fig. 2 shows how th tracking rror indx EE on th intrval t [ [:5, 1] varis at diffrnt numbrs of itration. Compard with th rsults shown in Fig. 1, th ILC tracking rror is boundd to a much lowr lvl. Figur 3 Tracking prformanc of th ILC systm outputs on th intrval t [[, 1] at diffrnt itration numbrs using th ILC rul (5) with th known B(t) (Th dottd, dashd dottd and dashd lins rprsnt th systm outputs as th ILC rul (5) is itrativly xcutd two, thr and four tims, rspctivly, and th solid lin rprsnts th dsird output) Finally, givn that th input matrix B(t) is accuratly known on th initial tim intrval [,.5], and th ILC rul (5) is usd, Fig. 3 shows th tracking prformanc of th ILC systm output on th intrval t [ [, 1] whn th ILC rul (5) is itrativly xcutd from th scond to th forth tim. Also, Fig. 4 prsnts th situation of th tracking rror indx EE on th intrval t [ [:5, 1] whn th ILC rul (5) is xcutd at diffrnt itration numbrs. In Fig. 4, it is noticd that th ILC tracking rror on th intrval t [ [:5, 1] can b surly drivn to zro by th ILC rul (5) with th known B(t). A prfct tracking of th dsird output y r (t) on th intrval t [ [:5, 1] is achivd. Th xampl usd in this papr illustrats that th proposd ILC approach for NCM systms with non-fixd Figur 2 Maximum absolut rror of tracking on th intrval t [[.5, 1] at diffrnt itration numbr using th ILC rul (5) Figur 4 Maximum absolut rror of tracking on th intrval t [[.5, 1] at diffrnt itration numbrs using th ILC rul (5) with th known B(t) 54 IET Control Thory Appl., 29, Vol. 3, No. 1, pp & Th Institution of Enginring and Tchnology 29 doi: 1.149/it-cta:27486

7 initial rrors is vry ffctiv. It can ovrcom random initial rrors of ILC systms and can succssfully track th dsird output trajctory byond a small initial tim intrval aftr a small numbr of itrations. 5 Conclusion This papr introducs an initial rctifying action into th convntional D-typ ILC mthod for NCM systms. Th stablishd ILC tchniqu allows th initial output of th ILC systm at diffrnt itrations fluctuating randomly around th initial valu of th dsird output. Th usd ILC stratgy is to st a vry small initial tim intrval and to pursu th rfrnc trajctory tracking byond th initial tim intrval. Th output tracking rror byond th initial tim intrval can b drivn to a rsidual st whos siz dpnds on th stimation rror of input matrix. It is worth noting that whn an accurat knowldg on th input matrix on th initial tim intrval is availabl, a prfct rfrnc trajctory tracking byond th initial tim intrval can vn b achivd. Compard with th convntional ILC mthods, th robustnss of th proposd ILC systm is gratly improvd. 6 Acknowldgmnt This work is portd in part by th National Natural Scinc Foundation of China undr Grant , and in part by th Rsarch Grants Council of th Hong Kong Spcial Administrativ Rgion, China undr Projct No. 8734/CityU. 7 Rfrncs [1] KUREK J.E., ZAREMBA M.B.: Itrativ larning control synthsis basd on 2-D systm thory, IEEE Trans. Autom. Control, 1993, 38, pp [2] LI X.-D., HO J.K.L., CHOW T.W.S.: Itrativ larning control for linar tim-variant discrt systms basd on 2-D systm thory, IEE Proc. Control Thory Appl., 25, 152, (1), pp [3] CHIEN C.J.: A discrt itrativ larning control for a class of nonlinar tim-varying systms, IEEE Trans. Autom. Control, 1998, 43, pp [4] HUANG S.N., TAN K.K., LEE T.H.: Ncssary and sufficint condition for convrgnc of itrativ larning algorithm, Automatica, 22, 38, pp [5] LI X.-D., CHOW T.W.S., HO J.K.L.: 2-D systm thory basd itrativ larning control for linar continuous systms with tim-dlays, IEEE Trans. Circuit Syst., I, 25, 52, (7), pp [6] JIANG P., CHEN H., BAMFORTH L.C.A.: A univrsal itrativ larning stabilizr for a class of MIMO systms, Automatica, 26, 42, pp [7] LEE H.S., BIEN Z.: Initial condition problm of larning control, IEE Proc.-D, 1991, 138, pp [8] ARIMOTO S.: Robustnss of larning control for robot manipulators, Proc. IEEE Int. Conf. Robot. Autom., 199, 3, pp [9] HEINZINGER G., FENWICK D., PADEN B., MIYAZAKI F.: Stability of larning control with disturbancs and uncrtain initial conditions, IEEE Trans. Autom. Control, 1992, 37, pp [1] WANG D.: Convrgnc and robustnss of discrt tim nonlinar systms with itrativ larning control, Automatica, 1998, 34, (11), pp [11] SUN M., WANG D.: Itrativ larning control with initial rctifying action, Automatica, 22, 38, (7), pp [12] SUN M., WANG D.: Initial condition issus on itrativ larning control for non-linar systms with tim dlay, Int. J. Syst. Sci., 21, 32, (11), pp [13] LEE H.S., BIEN Z.: Study on robustnss of itrativ larning control with non-zro initial rror, Int. J. Control, 1996, 64, (3), pp [14] PARK K.-H.: An avrag oprator-basd PD-typ itrativ larning control for variabl initial stat rror, IEEE Trans. Autom. Control, 25, 5, (6), pp [15] XU J.-X., YAN R.: On initial conditions in itrativ larning control, IEEE Trans. Autom. Control, 25, 5, (9), pp [16] WANG Y.-C., CHIEN C.-J., TENG C.-C.: Dirct adaptiv itrativ larning control of nonlinar systms using an output-rcurrnt fuzzy nural ntwork, IEEE Trans. Syst., Man, Cybrn. B, Cybrn., 24, 34, (3), pp [17] FANG Y., CHOW T.W.S.: 2-D analysis for itrativ larning controllr for discrt-tim systms with variabl initial conditions, IEEE Trans. Circuit Syst., I: Fundamntal Thory Appl., 23, 5, (5), pp [18] QIAN W., PANDA S.K., XU J.-X.: Spd rippl minimization in PM synchronous motor using itrativ larning control, IEEE Trans. Enrgy Convrs., 25, 2, (1), pp IET Control Thory Appl., 29, Vol. 3, No. 1, pp doi: 1.149/it-cta:27486 & Th Institution of Enginring and Tchnology 29