Adaptive Tracking of SISO Nonlinear Systems Using Multilayered Neural Networks

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1992 ACC/WA2 Adative Trackig of SISO Noliear Systems Usig Multilayered Neural Networks L. Ji, P.RN.Nikiforuk, M.M.Guta Itelliget Systems Research Laboratory College of Egieerig, Uiversity of Saskatchewa
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1992 ACC/WA2 Adative Trackig of SISO Noliear Systems Usig Multilayered Neural Networks L. Ji, P.RN.Nikiforuk, M.M.Guta Itelliget Systems Research Laboratory College of Egieerig, Uiversity of Saskatchewa Saskatoo, Saskatchewa, Caada S7N OWO Abstract Multilayered eural etwork. (MNNs) are used i this aer to costruct the oliear learig cotrol systems for a class of ukow oliear systems i a caoical form. A adative outut trackig architecture is roosed usig the oututs of two thme-layered eural etworks which are traied to aroximate a ukow oliear lat to ay desired degree of accuracy y usig the back-roagatio method. The weight udatig algorithm is reseted usig the gradiet descet method with a dead-zoe fuctio. Covergece of the error idex durig the weight traiig is also show. The closed system is roved to be stable, with outut trackig error covergig to a eighborhood of the origi. The effectiveess of the cotrol scheme roosed is illustrated through simulatios. 1. Itroductio Advace i the area of artificial eural etworks has rovided the otetial for ew aroaches to the cotrol of systems with comlex, ukow, ad oliear dyamics. The mai otetial of the eural etworks for cotrol alicatios ca be summarized as follows: they could be used to a roximate ay cotiuous liear or oliear maig; they erform this aroximatio through learig; arallel rocessig ad fault tolerace are easy to be accomlished. Oe of the most oular eural etwork architectures for cotrol urose is the multilayered eural etwork (MNN) with the error back-roagatio (BP) algorithm, ad it is roved that a three-layered eural etwork usig the back-roagatio algorithm ca aroximate a wide rage of oliear fuctios to ay desired degree of accuracy [21-[4]. To avoid the modelig difficulties, a umber of multilayered eural etwork based cotrollers have bee roosed [6]-(9]. Regardig a cotrol system as a maig of cotrol iuts ito observatio oututs, a aroriate maig is realized by a MNN which is traied so that a desired resose is obtaied. For such kid of adative ad learig cotrol systems usig the MNN, the weights of the etwork eed to be udated usig the etwork's outut error idex, ad the learig cotrol law is costructed based o the outut of the MNN. Therefore, the cetral research toics i the field of eural etwork cotrol methods are the roblems of the covergece of the weight traiig schemes ad the stability of the closed loo cotrol systems. I this aer, a adative oliear trackig scheme is roosed; this scheme combies the feedback liearizig cotrol method ad the multilayered eural etwork learig techique. I Sectio 2, the desig aroach of the oliear cotrol law is reseted based o the oliear system models with caoical form. The basic structure of the three-layered eural etworks which are traiable to aroximate the ukow oliear model is discussed, ad the weight udate algorithm with a dead-zoe fuctio is aalyzed i Sectio 3. The covergece of the weight traiig algorithm is roved i Sectio 4. The results o the feedback stabifity of the closed loo system usig the MNN cotrol law are reseted i Sectio 5. The comuter simulatio results are illustrated i Sectio Noliear Cotrol Formulatio Assume that a sigle-iut ad sigle-outut oliear system is give i the followig caoical form Ii X.i+I, i, 2,.. - I i Y f(x)+ g(x)u Xi x E R is the state variable u E R is the cotrol iut, f(.) ad g(.) are the oliear cotiuous fuctios o R , g(.) is bouded away from zero, ad y E R is the outut. Let the desired outut yd yd(t) be a cotiuously differetiable fuctio o [, +co), ad its first derivatives vv.) ) be uiformly bouded. The roblem to be addressed cosists of fidig a cotrol u(t) that will force the outut y(t) to track asymtotically the desired outut yd(t), i.e., lim (y(t -yd(t)) (2) t-o Sice g(x) i system (1) is bouded away from zero, its iverse is well defied ad the most coveiet structure for a oliear feedback cotrol is the oe i which the iuit variable u is set equal to U - E Ck-lXk - f(x) + V 1. a(x) 4-3(x)v g(zt) - E Ck.lX1 - f(x) I al$x)kki g(xt) g(zt) (4) ad v is a ew cotrol iut to be desiged for the uirose of outut trackig. Sice oliear feedback cotrol law (3) liearizes the state equatio (1), it ca easily be verified that the system (1) is trasformed ito the liear system as follows riw+1, X-1 2,.., - I Z -ZCik Xk'+ V k1 Y X1 (1) (3) (5) 56 Let c(t) y(t) - yd(t) be the outut trackig error, ad the ew cotrol v be chose as kyd _:-' dtk (6) c, 1, the the outut trackig error e(t) satisfies the followig liear error equatioi e ) + C-le(-l) coe (7) If oe chooses real umbers co,c,c. -I SUCIh that all the roots of the olyomial (s) s + c,1&s-' co have egative real art, the desired outut Yd(t) aid its derivatives yl3()e. y(1)(t ) are asymtotically tracked. Oe ca set AX (AX1,AX21.Ax- - (e e3, '...,e(1)). Thei the error equatio (7 ) ca be rewritte as C A CAx (8) I -CO -Cl --C2... -C-/ is a Hurwitz matrix. Oe maby roceed witlh the followig Lyauov fuctio Vo(Ax) AxTPoAx (1) Po is a symmetric, ositive defiite matrix solutio of the followig Lyauov equatio POC + CT _I (11) Evaluatig its time derivative aloig the state trajectories of the error system (8), oe obtai TPu AXAxA (12) 3. Adative Trackig usig MNNs Recetly, several studies have ideedetly foud that a three-layered eural etwork usig the back-roagatio algorithm ca aroximate a wide rage of oliear fuctios to ay desired degree of accuracy. I this sectio, the multilayered eural etworks are used to costruct a oliear cotroller for the urose of adatively trackig the desired outut yd(t). Suose that the cotiuous fuctios f(x) ad g(x) are ukow, let the oliear system (1) be modeled by the eural etwork x. x*5+1,i 1, 2,... f(z,) -i- + 1)u Y* aw7 I (9) (13) I the case that (13) is a three-layered eural etwork as it is show i Fig.1, the f(x, w) ad 9(x, 1) ca be rereseted as f(x,sw) E2wiH(2 wij,x+sbi) 11 j1 4(x, l) 2 lih(2 I2jx, + it) il jl (14) (15) wo ad I are the weights of the three-layered euiral etworks show i Fig.1. The ad q are the umber of oiliear hidde euros of the eural etworks corresodig to f(za, w) ad 4(x, 1), resectively. The fuctio H i ( 14 ) ad (15) is the hyerbolic taget fuctio Iut layer Fig. 1 tditzl WiZ2 W2i-1Ii H(x) e + e, (16) Hidde layer The three-layered eural etworks ib zi H1(2 tvjjxj + i'i) Wi X/ ~~~j1 Fig.2 The ith euro i the hidde-layer Based o the aalytic results of Hecht-Nielses42] about the caability of layered eural etworks to aroximate oliear fuctios, oe ca guaratee that f(x, UJ) ad 4(x, 2) are comlex eough (i.e. cotai eough euros ) to be able to aroximate f(x) ad g(x) to the desired accuracy. The weights ws ad I are ukow, ad w(t) ad l(t) rereset the estimates of so ad 1. Let the error that wiu be used to trai the etworks be defied as e*(t) y*(f) (y) ftx, w(t)) +4( 1(t))u- (d (17) The, w(t) ad l(t) ca be adjusted usig a gradiet search such that e*(t) ca be reduced. Let ir(t) [w T(t),lT(t)IT be the estimated weight coefficiet of the eural etwork model (13), the the udatig rule of the weights is give as r(t +6t) 7(t).1 a9 (18) q is ste size arameter, which affects the rate of coiivergece of the weights durig traiig. O the othier had, it is well-kow that the major disadvatage of the gradiet method is that the rate of covergece of the iterative roceedig ear the miimum will be very slow, i order to imrove the rate of covergece ear the miimum usig a simler method, oe ca emloy a dead-zoe algorithm for udatig the weights, which is roosed by Che, Khalil [8]. Let the error e*(t) be alied as iut to a dead-zoe fuctio D(e*) D(e*) e* - e +do if le*l do if e* do if ec -do 57 The, the outut of the dead-zoe fuctio is used iii the followig udatig rule r(t + Et) w(t) -rd()- ad the estimated oliear feedback law is desiged as Fig.3 u a(z, w(t), I(t)) +4(, I(t))v I -E elckizk - f(z, w(t)) _ 4 ) 9( XzI (t)) x(z,l(t)) - 1 4(Z,l(t)) &(z, w(t), I(t)) Adative oliear trackig usig MNN 4. Covergece of the Weight Traiig (19) (2) (21) (22) For coveiece, let ri r(t + kat), the followig theorem shows that the error e* is descet toward the directio of the origi usig the weight iterative equatio (19). Theorem 1: If there is a weight Trk such that Ie*(xrk-)l do, the, there exists a umber 17 such that [e*(rk + Ar) e*(7r)isig(e*(ra)) (23) - AXr -iyd(e*)o9e*/atr. The roof of the Theorm 1 aears i the roof rocedure of the fouowig Theorem 2 which rovides sufficiet coiditio for the covergece i the case a leil do ca ot be reached i a fiite umber of traiig iteratios. Theorem 2: If the limit of the sequece {vrk} of (22) is r*, the le*(r*)l. do. Proof: The roof is by cotradictio. Assume that e*(r') do. The, there exist umbers ad 7 such that D(e*)[ Or IT[ Or I - (24) for all w1 ad T2 i some eighbourhood of r; If ik is a oit i this eighbourhood the so is rk+1, x,+l Tk - td(e*) oe (25) O the other had, by the first mea value theorem, oe ca show that ec(rkil ) - e*(irk) -D(e')[ (era) )T[8DC(k - OrAk)1 6 1 (26) I equatio (26), the coefficiet of -sq is at least, from (24) ad (25). Thus, i movig from Trk to wr+a, the value of ec(r ) is decreased by at least q. But there are a ifiite umber of terms of the sequece {r;k} i ay eighbourhood of xr-, sice rim rk k--i oo (27) Hece, by reeatig the above argumet for successive values of k, oe fids that e*(r*) - -xo, which cotradicts the fact that Oe*(r*)/lr exists. The origial assumtio that e*(r*) do is, therefore, false. By same roof roceedig, oe ca show that e*(rx) c -do is false too, ad the theorem is roved Results o Feedback Stability I order to aalyze the local stability of the closed loo system, the followig assumtios about the oliear lat (1) ad the desired outut yd(t) are required. Assumtio 1: For ay z E R k1 : 1g(x)j (28) Assumtio 2: For ay t E [,+oo), the desired outut yd(t) ad its first derivatives ()d.. y() are uiformly bouded; that is, Y4(i)(t) mi, i, 1., (29) The followig assumtio is give based o the aalytic results of Becht-Nielse [2J about the caability of the MNNs to aroximate oliear fuctios usig the back-roagatio techique. Assumtio 3: There exist weight cofficiets w ad I such that f(x, w) ad g(x, 1) aroximate the cotiuouis fuctios f(z) ad g(x) with accuracy e o 5, a comact subset of RT, that is 3 w, I s.t. maxif(x,w) - f(x)i f maxtl(t,l)-g(z) l VxEVE (3) (31) Lemmas 1-3 will show that the MNNs with the hyerbolic taget fuctio i the hidde layers satisfy some algebraic roerties o the comact set S. Lemma 1: The hyerbolic taget fuctio H(x) satisfies the followig roerties: 1. H(x) is strictly icreasig; for each XI, X2 E R such that XI x2 it is true that H(xl) H(z2); 2. H(x) is uiformly liear growth; there exists costat 31 such that IH(x)l dtlxl for all x E R. Proof Oe ca easily show that the first art is true based o the defiitio of the fuctio H(x). Note that H1(z) is uiformly Lischitz [13], there exists a costat fi1 such that for all XI, X2 E R hece 111(x) - H(x2)1. t3ixi - X21 (32) IH(xI)l - 11H(X2)1. 1(l)- H(Z2)1. flixi - XZ21 (33) Note that H(), if oe sets x2, the the seod( art is imlied. Lemma 2: There exist costats j31 52w3, ad 1311, i21 such that the eural etwork models (14) ad (15) o E, a comact subset of R, satisfy the followig co(litios 58 Ihl(X, W)l /1u11AII + 2w 14i(x,L1)j /ulfrit + 32 VX EE Proof: By Lemma 1, we have P Ifx(,w)I ZI II?-IIH(ZE wi,ux + tbi)i i, (w,i,w...iwijj)t, ad Ei Iti(ol(iluillxII + li?, ) il - 13iwIlill + /2w Vr fer 11w E 11ItIIk1tI7, /2w ZIlWillti l t~~~~~~1 (3i) (35) (36) (37) hece, the roof is comlete. Lemma 3: There exist costats ki, k2 such that the eural etwork 4(x, 1) o the comact set S of R satisfies Proof: ki 1d1(x,l)j k2 Note that 4l(,11) Vx E S (38) Yx E (39) Hece, it imlies that 41(r,l) , or 4I(xr,) for all x E E, meawhile, 4I(x,l) is the cotiuous fuctio o the comact set E, hece, there exist the ozero maximtim aid miimum of 41(z,r) o E, which meas that (38) is true. The folowig theorem will give the local stability of the adative learig cotrol system o the comact set S. Theorem 3: Uder the assumtios 1-3, there exists a costat 6(s) such that the outut trackiig error of the oliear system (1) o the comact set usig the eural etwork cotrol law (2) is cofied to a eighborhood of the origi defied by IFAz*II 6. Proof: The error dyamics of the system uder the eural etwork cotrol law fi ca be obtaied as CA: + bl(x, w, I) + b2(, l)v Ax (e, e(l),., e(-l))t, ad bl(, w, ( 2(x, I) g(x)(q(x,1) - (x)) Comutig the time derivative of Lyauov fuctio V state trajectories of the error dyamics (4), oe obtais (4) (41) (42) alog Vo dvo,cax+ bi(,w,1)+ b2(x,)v -ATAx + 211PoIIAxIlg(x)(&(x, w- 1) - c(x))i +211 Pol11 zx11lg(x)(x3(x, 1) - O(x))I vl (43) O the other had, by Assumtio 3 ad Lemmas 2,3 oe has the - g(x)(d(r,w,l) a(r))j ad ad - E Ck-IXk + f(x, t) E CkI Xk-f(Z) g(x)( k'_ + - g(z) ) 2 ICkI1XkI + If(X, wu)i + 1(X,l)I kil(,q IICIIIiIl + /31wllIll + /32w X11 + t321 jiatlj6ll + 62e Ig(x)(x(1)-T (r)i I(r.T, l)og(r)l ce 61 a lieul + 1W' + olu k, C (Ca, e,.., )T 1d ( yd, y4) therefore, oe may easily see that itj 2 illydli+2+i2i ki Vo -IIAl-TI(IINAII6-64) b3 1-21jPojei, Poll(62 + 1$) Thus Vo ca be assured oicreasig wheever IlAxil 6 64/63, so the outut trackig error is cofied to a eighborhood of Ax defied by IlAztI 6, which ca be arbitrarily small as -O. 6. Simulatio Results I this sectio, the above oliear learig cotrol scheme is illustrated o a secod-order SISO oliear lat. Cosider a sigle-lik maiulator which is described by T(t) mlpg(t) + vo(t) + mglcos6(t) (44) (45) (46) (47) (48) the legth 1, mass m, frictio coefficiet u ad gravity costat are I 1 m, m 2. kg, v 1. kg m2/s, ad g 9.8 kg mi/s2. Let the coefficiets of the error equatio (7) be chose as cl 8 ad co 3. The ad eural etworks are traied to aroximate the oliear fuctio -9.8cosx1.5X2 ad costat.5. Assume that the - f etwork has 2 hidde euros, the etwork has 5 lhidde euros, the arameters of the weight udatig law ( 19) are set as.5 ad do.1, ad the iitial values of all weiglits of the f ad etworks are chose as.5. The simulatios are erformed usig a fourth-order, fixed stesize Ruge-Kutta algorithm with At.1 sec.. The outut trackiig trajectories obtaied by alyig four differet desired otltllt sigals; ste, sie, square,ad sawtooth, are show i Figs.4-7. It is see that the satisfactory outut trackig erformace has bee achieved through the roosed cotrol scheme, ad, the simulatio results show that the system outuit resose ad covergece of the learig cotrol law are sesitive to the umber of the hidde euros ad the iitial values of tihe weights. 59 9 U II, Si 15 1 S It 1 Fig.4 Fig.5 5SO time (.1 x sec.) Trackig a desired ste outut 5 1 time (.1 x see.) Trackig a desired sie outut 1,...,... S...e.e.t..... i..... S....,, Fig I,.., WOO 15 2 time (.1 x sec.) Trackig a desired square outut t Fig.7 time (.1 x sec.) Trackig a desired sawtooth outult 7. Cocludig Remarks We have made use of the multilayered euiral ietworks to costruct the oliear learig cotrol law for a -class of ukow SISO oliear cotrol systems, ad it is straiglhtforward to exted the MNN cotroller develoed to MIMO ioiliear cotrol systems. The two mai results rovided i this aer are (1). The three-layered eural etworks are itroduced to aroximate a ukow oliear lat usig the error backroagatio, ad a weight traiig algorithm with a dead-zoe fuctio is discussed. (2). The covergece of the weight udate law is guarateed through the theorems roved i Sectio 4, ad the stability of closed loo is show, that is, if there are eough euros i oliear hidde layers of the three-layered eural etworks to be able to aroximate f(x) ad g(x) to the desired accuracy, the the outut trackig error will coverge to a eighborhood of the origi, which ca be arbitrarily smahl withi some desired accuracy of the etwork learig. Refereces [1] R.E. Rumelhart, ad J.L. McCelad, Learig Iteral Reresetatios by Error Proagatio , Parallel Distributed Processig Exloratios i the Microstructure of Cogitio, Vol.1: Foudatios, MIT Pre, [2] R. Hecht-Nielse, Theory of the Back-roagatio Neural Network', roceedig It'l. Joit Cof. o Neural Networks, , Jue [3] E.K.Bhum, ad L-K.Li, Aroxiiatio Theory ad Feedforward Networks, Neural Networks, vol.4, , [4] [5] P.K. Simso, Artifical Neural Systems, Pergamo Press, 19I9. K. Horik, M. Stickcombe, ad H. White, ' Multilayer Feedforward Networks are Uiversal Aroximators , Neural Networks, Vol.2, ,1989. [6] D. Psaltis, A. Sideris, ad A.A. Yaamura, A Multilayered Neural Network Cotroler , IEEE Cotrol System Magazie, Aril 1988, [7] F.-C Che, Back-roagatio Neural Networks for Noliear Self- Tuig Adative Cotrol , IEEE Cotrol System Magazie, Aril 199, [8] F.-C. Che, H.K. Khalil, Adative Cotrol of Noliear Systems Usig Neural Networks-A Dead-Zoe Aroach , Proceedig 1991 America Cotrol Coferece, [9] S.S. Kumar, ad A.Guez, ART Based Adative Pole Placemea for- Neurocotroliers, Neural Networks, vol.4, , [1] J.-J.E. Slotie, Slidig Cotroller Des for Noliear Systems , It'l. Joural of Cotrol, Vol.4, No.2, , [ll] J.-J.E. Slotie, ad S.S.Sastry, Trackig Cottrol of Noliear Systems Usig Slidig Suirfaces with Alicatio to Robot Maiulators', It'l. Joural of Cotrol, Vol.38, No.2, , [12] R.M. Saer, ad J.-J.E. Slotie, Gaussia Networks for Direct Adative Cotrol , Proceedig 1991 America Cotrol Coferece, [13] S.Hui, ad S.IkZak, AalysLs of Sigle Percetrois Learig Caabilities , Proceedig 1991 America Cotrol (o frcrece, [14] A. Isidori, Nolhear Cottrol System, Sriiger Verlag, New York [15] G.R. Walsh, Methods of Otimizatio, Joh Wiley k Sos Ltdl
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