Applied Mathematical Modelling

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1 Appled Mathematcal Modellng 35 (0) Contents lsts avalable at ScenceDrect Appled Mathematcal Modellng journal homepage: A novel scheme for the desgn of backsteppng control for a class of nonlnear systems q Hongl Sh School of Bomedcal Engneerng, Captal Medcal Unversty, Bejng 00069, Chna artcle nfo abstract Artcle hstory: Receved 6 May 00 Receved n revsed form September 00 Accepted October 00 Avalable onlne 6 October 00 Keywords: Adaptve control Backsteppng Neural network Parameter estmaton Mnmax functonal approxmaton error (MFAE) A novel scheme s proposed for the desgn of backsteppng control for a class of state-feedback nonlnear systems. In the desgn, the unknown nonlnear functons are approxmated by the neural networks (NNs) dentfcaton models. he Lyapunov functon of every subsystem conssts of the trackng error and the estmaton errors of NN weght parameters. he adaptve gans are dynamcally determned n a structural way nstead of keepng them constants, whch can guarantee system stablty and parameter estmaton convergence. When the modelng errors are avalable, the ndrect backsteppng control s proposed, whch can guarantee the functonal approxmaton error wll converge to a rather small neghborhood of the mnmax functonal approxmaton error. When the modelng errors are not avalable, the drect backsteppng control s proposed, where only the trackng error s necessary. he smulaton results show the effectveness of the proposed schemes. Ó 00 Elsever Inc. All rghts reserved.. Background In recent adaptve and robust control lteratures for nonlnear systems, backsteppng consttutes an mportant desgn scheme [ 5]. he backsteppng approach provdes a systematc framework for the desgn of regulator and tracker, sutable for a large class of state-feedback nonlnear systems. he essence of backsteppng control s that some approprate state varables are recursvely treated as the pseudocontrol sgnals for lower dmenson subsystems. he frst pseudocontrol sgnal s desgned wth the am to reduce the error between the desred trajectory and the actual output value, whle the pseudocontrol sgnal of another subsystem s desgned to reduce the error between the pseudocontrol sgnal and the actual state value n the precedng desgn stages. When ths recursve procedure termnates, a feedback desgn for the true control nput results. Generally, the applcaton of adaptve and robust technques s lmted by lack of accurate system dynamcs. Some general dentfcaton models are utlzed to elmnate uncertantes of dynamcs, and NNs become the general choce [6 8]. heoretcally, as long as a suffcent number of neurons are employed, a radal bass functon (RBF) NN can approxmate any contnuous functon to an arbtrary accuracy on any compact set [9,0]. As a result, many nonlnear control approaches had been presented that combne backsteppng wth NNs n the last few years [ 7]. Although sgnfcant progress has been made n backsteppng desgn scheme, there are stll some problems that need to be solved for practcal mplementatons. For example, n order to avod the controller sngularty problem, the gan functons g ðx Þð ¼ ; ;...; nþ (see ()) n Secton are usually assumed to be constants or known functons n some lteratures, whch q hs work have been supported by Natonal Natural Scence Foundaton of Chna (NSFC) under Grant No and Bejng Natural Scence Foundaton under Grant No el.: E-mal address: shl@ccmu.edu.cn X/$ - see front matter Ó 00 Elsever Inc. All rghts reserved. do:0.06/j.apm

2 89 H. Sh / Appled Mathematcal Modellng 35 (0) cannot be satsfed n many plants. In some works, for example n[], the gan functons are assumed to be unknown, however, the dscontnuous projectons wth fcttous bounds have to be appled to avod the possble weght dvergence of NNs durng on-lne tunng. In [5], gan functons are also assumed to be unknown. However, due to the ntegral-type Lyapunov functon s ntroduced, the approach s complcated and dffcult to be used n practce. In one of recent reports[6], the gan functons and ther dervatves are assumed to be bounded wth explct bounds. In [7], only a class of second-order nonlnear systems s consdered. Recently, a drect backsteppng control usng fuzzy logc system was proposed n [,5], whch can avod the sngularty problem smartly. However, the parameter estmaton remans as a problem. In ths paper, an alternatve scheme for the desgn of backsteppng control s proposed, n whch the parameter estmaton s regarded as the most mportant task and the desgn focus on t. In the desgn, the adaptve gans are consstently tuned accordng to dentfcaton results of the precedng stages to guarantee convergence of the trackng error and parameter estmaton. wo smlar control schemes, the drect and the ndrect backsteppng, are presented. he latter s sutable for the plants whose dervatves of state varable are avalable, whch can guarantee the functonal approxmaton error wll converge to the small neghborhoods of the mnmax functonal approxmaton error. he former s sutable for the plants that only the state varables are avalable, whch can guarantee the approxmaton error s bounded. Both schemes can guarantee the trackng error wll converge nto certan small range around the desred trajectory... System descrpton he model of many practcal nonlnear systems, for example, the rgd robots and motors, can be expressed n a specal state-feedback form as follows _x ¼ f ðx Þþg ðx Þx þ ; 6 6 n ; _x n ¼ f n ðx n Þþg n ðx n Þu; ðþ y ¼ x ; where x, ½x ; x ;...; x Š R ; ¼ ;...; n; u R; y R, are the state varables, system nput and output, respectvely, whch are all assumed to be avalable for measurement; f () and g (), =,...,n, are smooth nonlnear functons that contan both parametrc and nonparametrc uncertantes. g () s usually referred to as the gan functon. he control objectve s to desgn an adaptve control nput u so that the output y follows a desred trajectory y d wth the constrant that all sgnals n the closed-loop system are sem-globally unformly ultmately bounded. It s assumed that y d and ts dervatves up to the (n + ) th order are all bounded. he controllablty of system () requres that g () 0, =,..., n. Snce they are smooth functons, there s the followng assumpton. Assumpton. g s strctly ether postve or negatve, and ts sgns are known. From the above assumpton, wthout losng generalty, n ths paper we assume g >0, =,..., n... RBF NNs o dentfy unknown nonlnear functon f () and g (), some unversal dentfcaton models can be appled, e.g. NNs, fuzzy logcal systems and wavelet networks. For RBF NNs, the dentfcaton model of a smooth square-ntegrable functon f ðzþ L ðrþ can be expressed as followng f ðzþ ¼h nðzþþeðzþ; where e(z) s the so-called NN functonal approxmaton error; z R m s the nput of NNs; h =[h,h,...,h l ] R l s the weght collecton to be determned, l s the node number of RBF NNs; n(z)=[n (z),n (z),...,n l (z)] s the bass functon vector. n (z) s usually chosen as the Gaussan functon! n ðzþ ¼exp kz l k g ; ¼ ; ;...; l; ð3þ where kk denotes the Eucldan norm; l =[l,l,...,l m ], l j R, j =,..., m, s the center of n (z), and g s the wdth, whch are usually selected accordng to the pror nformaton about f(z). heoretcally, t can been proven that any contnuous functon can be unformly approxmated to any desred accuracy over a compact set by the sngle-hdden-layer RBF NNs as long as a suffcent number of neurons are employed. hs unversal approxmaton capablty of RBF NNs has enabled researchers to model certan complex nonlnear systems effectvely through varous judcous use of NNs [7], n whch the functonal approxmaton error had been neglected n general. However, the approxmaton error cannot always be small enough to be neglected n the practces. In ths paper, we ntroduce a unform bound of t, whch s known as the mnmax functonal approxmaton error (MFAE) and denoted as D, D ¼kf ðxþh nðxþk ; where kk denotes the nfnte norm. In next secton, we wll consder the ndrect backsteppng control usng RBF NNs. ðþ

3 H. Sh / Appled Mathematcal Modellng 35 (0) Indrect backsteppng control Generally, the dea behnd backsteppng control s lke ths. Frst, x, =,..., n, are treated as the fcttous control sgnals, whch are denoted as x,v, =,..., n, respectvely. In each stage of desgn, the fcttous control sgnal x,v s desgned wth the am to reduce the error jx,v x j formed n the prevous desgn stage. Fnally, an actual control u s desgned to make the error between jx n,v x n j as small as possble. Snce x,v = y d,(x,v x ) becomes the trackng error. In desgn, RBF NNs are utlzed to approxmate the nonlnear functons f ðx Þ and g ðx Þ n each step. he detaled desgn procedure we proposed s descrbed as follows. Step Desgn a vrtual control nput x,v to mnmze the trackng error e = y d x. Recall that _x ¼ f ðx Þþg ðx Þx : By treatng x as a vrtual control nput and usng the feedback lnearzaton method [8], the nomnal control nput x d s desgned as follows x d ¼ g ðx Þ ½f ðx Þþ_y d þ k e Š; where k > 0 s a desgn constant, whch s usually referred to as the adaptve gan. However, g (x ) and f (x ) are unknown, the estmates are utlzed n constructng fcttous control nput, x ;v ¼ ^g ðx Þ ½^f ðx Þþ_x v þ k e Š; ð6þ where ^g ðx Þ and ^f ðx Þ are the estmates of g (x ) and f (x ), respectvely. For the convenence of stablty analyss n followng steps, k s selected as followng ðþ ð5þ k ¼ þ k0 þ k ; ð7þ where k 0 and k are two postve constants. Defne e = x,v x, substtutng the vrtual control (6) nto subsystem () yelds the followng error dynamcs _e þ k e ¼ ^f ðx Þf ðx Þþ^g ðx Þx ;v g ðx Þx ¼ h ;f n ;f ðx Þf ðx Þþh ;g n ;gðx Þx g ðx Þx þ h ;g n ;gðx Þe ; where n,f (x ) and n,g (x ) are the bass functon vectors n approxmatng f (x ) and g (x ) usng dentfcaton model (), respectvely, whch are abbrevated to n,f and n,g n the next; h,g and h,f are the correspondng weght parameters. Denote h ; h, h ¼ h ;f ; h ;g n ¼ a n ;f ; n ;g x then _e þ k e ¼ / n r ;f ðxþr ;g ðx Þx þ h ;g n ;ge ¼ / n r ðx Þþh ;g n ;ge ; ð8þ where r,f (x ) and r,g (x ) are the functonal approxmaton errors usng the optmal parameters, whch are abbrevated h ; to r,f and r ;g ; r ¼ r ;f þ r ;g x ; / ¼ h h ; h ¼ h ;f ; h ;g h ;f and h ;g denote the optmal parameters n approxmatng f (x ) and g (x ), respectvely. Suppose the unversal mnmax functonal approxmaton error n approxmatng g (x ) and f (x ) are D f and D g,.e., D f ¼kf ;f ðxþn ;f h ;f k ; D g ¼kf ;g ðxþn ;g h ;g k ; ¼ ; ;...; n. In order to dscuss stablty of the subsystem, consder followng Lyapunov functon V ;I ¼ e þ b / / ; ð9þ where the ndex I n V,I stands for the ndrect verson of backsteppng control; b s a random postve constants. In ths stage, we propose the followng adaptaton law for adjustng the parameters, _h ¼ n h _e þ k e h ;g b n ;ge þ D ; ð0þ where D ¼ ðd f þjx jd g Þsg _e þ k e h ;g n ;ge, 8 >< ; x > 0; sgðxþ ¼ 0; x ¼ 0; >: ; x < 0: Snce _ / ¼ _ h ; _e þ k e h ;g n ;ge ¼ / n r (see (8)), sgðd Þ¼sg _e þ k e h ;g n ;ge ¼ sg / n r, _/ ¼ n / b n r þ D :

4 896 H. Sh / Appled Mathematcal Modellng 35 (0) he tme dervatve of V,I s as followng usng the adaptve law (0) _V ;I ¼k e þ e / n r þ e h ;g n ;ge / n / n r þ D ¼k 0 e þ e h ;g n ;ge k e e e / n r þ / n r ¼k 0 e þ e h ;g n ;ge e / n r k e Snce jr j < jd j,sg(r + D ) = sg (D ). herefore, f j/ n j P jr j; we have sg / n r ¼ sg / n ; sg / n ðr þ D Þ ¼ sg / n r sgðd Þ¼sgðD ÞsgðD Þ¼; j/ n ðr þ D Þj P r ;.e., f e = 0 and j/ n j P jr j; _ V 6 0. On the other hand, f t results j/ n j < jr j j/ n ðr þ D Þj < 3jr j : þ r / n ðr þ D Þ: þ / n r / n ðx Þ / n r þ D hus, f k e > r and e ¼ 0; V _ ;I < 0. In general, jr j s very small and k can be chosen as a large postve constant, t s reasonable to regard the system wll converge to a very small neghborhood of the reference sgnal. Consder a Lyapunov functon formed by the estmaton error of NN weght parameter V / ¼ / : ðþ By the adaptve law (0), the tme dervatve of V / becomes _V / ¼ / b n / n r þ D : ðþ It can be proven n a smlar way that when j/ n j P jr j; V _ / < 0,.e., the adaptve law (0) can also guarantee the functonal approxmaton error consstently converges to a very small neghborhood of MFAE. In the next step, a vrtual control nput x 3,v s desgn to drve je j as small as possble. hrough out ths paper, we defne e = x,v x, and denote ^g ðx Þ and ^f ðx Þ as the estmates of g ðx Þ and f ðx Þ, respectvely. We denote n ;f ðx Þ and n ;g ðx Þ (abbrevated to n,f and n,g ) as bass functon vectors n approxmatng f ðx Þ and g ðx Þ, respectvely; h,g and h,f are the correspondng h ; h ; h, weghts; n ¼ n ;f ; n ;gx þ h ¼ h ;f ; h ;g / ¼ h h ; h ¼ h ;f ; h ;g where h ;g and h ;f are the nomnal optmal weghts. We also denote D ¼ ðd f þjx þ jd g Þsgð_e þ k e h ;g n ;ge þ Þ; r ¼ r ;f þ r ;g x þ. In order to ensure that ^g ðx Þ always keeps the same sgn wth g ðx Þ ( ts sgn s known accordng to Assumpton ),.e., to avod control sngularty problem, some addtonal adjustment s necessary n updatng h,g. For example, f g ðx Þ > 0, we propose the followng teratve adjustment, h ;g ¼ h ;g þ c l ; f h ;g n ;g 6 0; ð3þ * where c > 0 s a qute small constant; l = [,,...,] R l, l s the dmenson of h,g.in[9], the sngularty problem s solved by remanng all elements of h,g n a compact set X R +. In fact, g ðx Þ > 0 does not necessarly mean that all the elements of optmal weght h ;g reman postve, therefore, the scheme prevents the parameters from approachng ther optmal values n some way. Obvously, the addtonal adjustment (3) do no harm to system stablty. In the adaptve law (0), the dervatve of trackng error, _e, s employed, whch means the dervatve of the measurement value, _x, must be avalable. In some stuatons, the dervatves of state varable can be obtaned by certan specal sensors, for example, the rotary acceleratons are obtaned by the rotary accelerometers n the aerocrafts. In many stuatons, however, the dervatves of measurement value cannot be obtaned readly due to measurement nose. Another approach of parameter adjustment wll be presented n the next secton, n whch only the measurement values are necessary. Step In ths step, a vrtual control nput x 3,v s desgn to drve je j as small as possble. In a smlar way, x 3,v s desgned as follows x 3;v ¼ h ^g ðx Þ ^f ðx Þþ_x ;v þ k ðtþe ; ðþ

5 H. Sh / Appled Mathematcal Modellng 35 (0) where k (t)=k s selected as h k ðtþ ¼k 0 þ k þ þ h ;g n ;g k 0 ; ð5þ where k 0 and k are two postve constants. Smlarly, substtutng () nto the correspondng subsystem yelds followng error dynamcs _e þ k e ¼ ^f ðx Þf ðx Þþ^g ðx Þx 3;v g ðx Þx 3 ¼ / n þ h ;g n ;ge 3 r : Consder the followng Lyapunov canddate V ;I ¼ V ;I þ e þ b / / : Smlarly, the adaptaton law s proposed as followng _h ¼ n _e þ k e h ;g b n ;ge 3 þ D : ð6þ he tme dervatve of V,I becomes e 3 _V ;I ¼ V _ ;I þ e _e þ b / _ h / ¼ e h / n þ r / n ðr þ D Þ k 0 e e h ;g n ;g n 6 ;g 7 ;ge þ 5 e e / n r þ / n / n r þ D þ r k e k e k0 e þ e h ;g n ;ge 3 3 h ¼ e h / n e þ r / n þr ff 6 k 0 e h ;g n ;ge 7 ff 5 / n ðr þ D Þ/ n ðr þ D Þ k e k e þ r þ r k0 e þ e h ;g n ;ge 3 : Smlarly, t results V _ 6 0fe 3 = 0 and j/ n j > jr j; ¼ ;, or f e 3 ¼ 0; j/ n j < jr j and k e > r ; ¼ ;. he next step s to make je 3 j as small as possble. Step (3 6 6 n ) In a smlar fashon, the vrtual control sgnal x +,v s desgned to mnmze je j, whch s x þ;v ¼ h ^g ðx Þ ^f ðx Þþ_x ;v þ k ðtþe : ð7þ Consder the Lyapunov functon canddate V ;I ¼ V ;I þ e þ b / / : k (t) s selected as k ðtþ ¼k þ k 0 þ þ ½h ;g n ;gš ; ð8þ b k 0 where k 0 and k are two postve constants. Smlarly, the error dynamcs becomes _e þ k ðtþe ¼ ^f ðx Þf ðx Þþ^g ðx Þx þ;v g ðx Þx þ ¼ / n r þ h ;g n ;ge þ : he proposed adaptaton law s smlar to those n above steps _h ¼ n _e þ k e h ;g b n ;ge þ þ D : ð9þ By same completon of squares smlar to those employed n the prevous steps, the tme dervatve of V becomes _V ;I ¼ X e 3 s / s n s þ r s þ k e s r s þ / s n sðr s þ D s Þ X 6 k 0 se s h s;g n s;ge sþ 7 5 k 0 e þ e h ;g n ;ge þ : s¼ s¼ he next step s to make je n j as small as possble. Step n In the fnal step, the true control u s desgned to mnmze je n j n a way that s qute smlar to those employed n vrtual control. uðtþ ¼ ^g n ðx n Þ ð^f n ðx n Þþ_x n;v þ k n e n Þ; ð0þ k 0 k 0 s k 0

6 898 H. Sh / Appled Mathematcal Modellng 35 (0) where k n (t) s selected as k n ðtþ ¼ h þ k n þ n;g n n;g : ðþ k 0 n he error dynamcs becomes _e n þ k n e n ¼ ^f n ðx n Þf n ðx n Þþ^g n ðx Þu g n ðx n Þu ¼ / n n n r n ; h. where n n ¼ n n;f ; n n;g u he overall Lyapunov functon s defned as V I ¼ V n;i þ e n þ b n/ n / n: A smlar adaptaton law s proposed _h n ¼ n n b n ð_e n þ k n e n þ D n Þ: ðþ he tme dervatve of V I s as follows _V I ¼ Xn e 3 / n þ r þ k e r þ / n ðr þ D Þ Xn 6 k 0 e h ;g n ;ge þ 7 5 k 0 n e : ¼ ¼ herefore, f j/ n j P jr j,orfj/ n j < jr j and P n ¼ k e k 0 P P n ¼r, t results _ V 6 0. Snce jr j s very small generally, the system wll converge to a very small range around the reference sgnal. Smlarly, t can be proven the approxmaton error of every functon wll converge to a very small neghborhood of ts mnmax functonal approxmaton error. When the approxmaton errors, r,f and r,g, =,..., n, are small enough to be neglected by choosng proper RBF NNs, the adaptve laws and stablty analyss becomes rather smple. For example, just let r = 0 and D = 0, the control and adaptve laws can guarantee system stablty and convergence of parameter estmaton. Obvously, the constant term n the varable gans (7), (5), (8) and () can be selected as any other random postve number, and the system can be regulated n a qute smlar way, only the adaptvelaw becomes somewhat complcated. Usually, / n s referred to as the modelng error n [9]. Snce the modelng error / n r ¼ _e þ k e h ;g n ;ge þ s employed n parameter adjustment, we refer to the scheme presented n ths secton as the ndrect backsteppng control. 3. Drect backsteppng control In the above approach, the dervatves of the measurement values are utlzed n the adaptve laws for parameter adjustment, for example, n (9). However, the measurement s usually dsturbed by varous noses n practcal plant. he dfferental calculaton always enlarges the measurement dsturbance dramatcally, whch perhaps cause serous dstorton n parameter adjustment. herefore, an alternatve adaptaton law s proposed, where only the measurement values are used. Frst of all, consder the followng Lyapunov functon of frst-degree subsystem V ;D ¼ e þ b ;f / ;f / ;f þ b ;g/ ;g / ;g; where the ndex D n V,D stands for the drect verson of backsteppng control; b,f and b,g are random postve constants. In ths case, the adaptve gan k n the Eq. (6) s select as k ¼ k 0 þ k ; where k 0 and k are two postve constants. A new adaptaton law s presented as followngs _h ;f ¼ n b e ¼ / _ ;f ; ;f _h ;g ¼ n b x e ¼ / _ ;g : ;g ð3þ ðþ By the error dynamcs (8), usng the adaptve law (), the tme dervatve of V,D becomes _V ;D ¼k e r k 0 e þ e h ;g n ;ge : herefore, f e = 0 and k e > jr j, then _ V ;D 6 0. Smlarly, the Lyapunov functon n th ( 6 < n) step s selected as V ;D ¼ V ;D þ e þ b ;f / ;f / ;f þ b ;g/ ;g / ;g;

7 H. Sh / Appled Mathematcal Modellng 35 (0) where b,f and b,g are two postve constants. he adaptve gan k (t) s also dynamcally selected, k ðtþ ¼k 0 þ k þ ðh ;g n ;gþ ; k 0 where k 0 and k are two postve constants. he adaptaton law s smlar to () Fg.. System output and the desred trajectory. Fg.. g ðxþ and g ðxþ and the estmaton results.

8 900 H. Sh / Appled Mathematcal Modellng 35 (0) _h ;f ¼ n b e ; ;f _h ;g ¼ n b x þ e : ;g ð5þ he tme dervatve of V,D s as followng usng (5) _V ;D ¼ X s¼ k e s þ r 0 s k e þ e h ;g n ;ge þ X s¼ 0 B k 0 se s h s;g n s;ge C A : k 0 s In the last step, the overall Lyapunov functon V n,d, the adaptve gan k 0 n ðtþ, the adaptaton law and the tme dervatve of V n,d become V n;d ¼ V n;d þ b n;f / n;f / n;f þ b n;g/ n;g / n;g; k n ¼ k n þ k0 n þ ½h n;g n n;gš ; k n > 0; k0 n > 0; _h n;f ¼ n b n e n ; n;f _h n;g ¼ n b n ue n ; n;g _V n;d ¼ Xn ¼ k 0 n k e 0 þ r k n e n Xn s¼ 3 6 k 0 se s h s;g n s;ge sþ 7 5 : k 0 s Fg. 3. System output and desred trajectory usng dfferent control schemes. (a) Indrect backsteppng. (b) Drect backsteppng.

9 H. Sh / Appled Mathematcal Modellng 35 (0) he true control u s as (0). Obvously, f P n ¼ ðk e þ r Þ >; _ V n 6 0. Because jr j s very small generally, t s reasonable to thnk that the system wll converge to a very small range around the reference sgnal. Here, only the trackng error of subsystem, e, s employed n parameter adjustment, we refer to the scheme n ths secton as the drect backsteppng control. For ths scheme, t s dffcult to fnd a smple way to guarantee functonal approxmaton error consstently converge to a very small values, for example, there does not exst an equaton smlar to (). It s reasonable to regard that the ndrect backsteppng control wll possess some superorty over the drect one.. Smulaton analyss In ths secton, the proposed backsteppng are appled to regulate two nonlnear affne systems. In the smulatons, we employed Smulnk of Matlab and the Solver optons s ode5. Example. Consder a second-degree state-feedback nonlnear system _x ¼ x = þðþx =0Þx ; _x ¼ x x þ :5 þ cosðx Þ snðx Þu; y ¼ x ; where x and x are the state varables, y s the system output. he ntal states s [x (0),x (0)] = [0,0]. he desred reference sgnal of ths system s y d = sn (0.t). In the smulaton, the ndrect backsteppng control are employed and results are shown n Fgs. and. In approxmatng f, g, NNs contan 6 nodes wth centers of receptve feld l evenly spaces n [5,5]; n approxmatng f, g, they have 3 nodes wth centers evenly spacng n [0,0]. he wdths are all selected as g = 0. he elements of ntal parameters h,f and h,f are all chosen as 0.05, the elements of h,g and h,g are all chosen as 0.5, D f and D g are all selected as 0.; k 0 þ k ¼ k0 þ k ¼ 0:5; b ¼ b ¼ 0:5. Fg.. g 3 ðxþ and ts estmate usng dfferent control schemes. (a) Indrect backsteppng. (b) Drect backsteppng.

10 90 H. Sh / Appled Mathematcal Modellng 35 (0) Fg. shows the ndrect backsteppng control can guarantee the system output converge to a small range around the desred trajectory. Consder the sze of NNs are rather small, the MFAEs cannot be very small. As a result, the trackng error and parameter estmaton error cannot be very well. In the next example, a thrd-degree nonlnear system s smulated, and NNs wth more large szes are employed to mprove trackng accuracy and parameter estmaton accuracy. Example. Consder a thrd-degree state-feedback nonlnear system _x ¼ cosðx Þþexp x x ; _x ¼ snðx Þ exp x =0 þ þ exp x cosðx Þ x 3 ; _x 3 ¼ x x snðx 3 Þþ þ exp x =0 snðx Þ cosðx 3 Þ u; y ¼ x ; where x, x, x 3, and y are the state varables and system output, respectvely. he ntal states s [x (0),x (0), x 3 (0)] = [0,0,0]. he desred reference sgnal also s y d = sn(0.t). he ndrect and drect backsteppng controls s employed and results are shown n Fgs. 3 and. In approxmatng f, g, NNs contan 0 nodes wth centers of receptve feld l evenly spaces n [5,5]; n approxmatng f, g, they have nodes wth centers evenly spacng n [30,30]; n approxmatng f 3, g 3, they have 6 nodes wth centers evenly spacng n [0,0]. he wdths are all selected as g = 0. he elements of ntal parameters h,f, h,f and h 3,f are all chosen as 0., the elements of h,g, h,g and h 3,g are all chosen as, and k 0 þ k ¼ k 0 þ k ¼ k 0 3 þ k ¼. D f and D g are all selected as 0.. In the ndrect backsteppng, we choose b = b =5, b 3 = 0. In the drect backsteppng, we choose k 0 þ k ¼ k 0 þ k ¼ k 0 3 þ k ¼ :; b ;f ¼ ; b ;f ¼ :5; b 3;f ¼ 3; b ;g ¼ 0; b ;g ¼ 5; b 3;g ¼ 60. Fgs. 3 and show both the ndrect and drect backsteppng control can guarantee the plant output wll converge to a rather small range around the desred trajectory. he ndrect backsteppng control possesses some superorty over the drect one n the terms of trackng speed, trackng accuracy and parameter estmaton accuracy. 5. Conclusons Compared wth backsteppng control n current lterature, the nnovaton n ths paper s that the adaptve gans are dynamcally updated accordng to dentfcaton results. Accordng to stablty analyss usng Lyapunov functon, the varable adaptve gans can ensure convergence of the trackng error and functonal approxmaton error. he ndrect drect backsteppng can ensure the functonal approxmaton error converge to a very small neghborhood of ts mnmax functonal approxmaton error consstently. he drect backsteppng s effectve when only the measurement values are avalable. No matter how large a RBF NN or other unversal approxmaton model s employed n dentfyng a unknown nonlnear functon, there always exsts resdual functonal approxmaton error. he proposed schemes can get rd of the potental rsk of system dvergence caused by the approxmaton errors. he smulaton results llustrate both backsteppng control approaches can make a plant to track a desred reference sgnal stably and effcently. When the modelng errors are avalable, the ndrect backsteppng control possesses some superorty over the drect one n the term of trackng speed, trackng accuracy and parameter estmaton accuracy at a neglgble ncreasng n the mplementaton cost and the computatonal complexty. Acknowledgments Authors wll greatly apprecate all the anonymous revewers for ther comments, whch help to mprove the qualty of ths paper. References [] D. Seto, M. Annaswamy, J. Balleul, Adaptve control of nonlnear systems wth a trangular structure, IEEE rans. Autom. Control 39 (99) 7. [] M. Krstc, I. Kanellakopoulos, P.V. Kokotovc, Nonlnear and Adaptve Control Desgn, Wley, New York, 995. [3] P.-Y. Peng, Z.-P. Jang, Stable neural controller desgn for unknown nonlnear systems usng backsteppng, IEEE rans. Neural Netw. (000) [] Bng Chen, Xaopng Lu, Kefu Lu, Chong Ln, Drect adaptve fuzzy control of nonlnear strct-feedback systems, Automatca 5 (009) [5] Mn Wang, Bng Chen, Peng Sh, Adaptve neural control for a class of perturbed strct-feedback nonlnear tme-delay systems, IEEE rans. Syst. Man Cybern. Part B: Cybern. 38 (3) (008) [6] S. Fabr, V. Kadrkamanathan, Dynamc structure neural networks for stable adaptve control of nonlnear systems, IEEE rans. Neural Netw. 7 (996) [7] K.-I. Funahash, On the approxmate realzaton of contnuous mappngs by neural networks, Neural Netw. (989) [8] G.A. Rovthaks, M.A. Chrstodoulou, Adaptve control of unknown plants usng dynamcal neural networks, IEEE rans. Syst. Man Cybern. (99) 00. [9]. Chen, H. Chen, Approxmaton capablty to functons of several varables, nonlnear functonals, and operators by radal bass functon neural networks, IEEE rans. Neural Netw. 6 (995)

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