Adaptive NN control of uncertain nonlinear pure-feedback systems

Automatica 38 (2002) 671–682

www.elsevier.com/locate/automatica

Adaptive NN control of uncertain nonlinear pure-feedback systems
S.S. Ge ∗ , C. Wang1 Department of Electrical & Computer Engineering, National University of Singapore, Singapore 117576, Singapore Received 22 September 2000; received in revised form 2 April 2001; accepted 8 October 2001

Abstract This paper is concerned with the control of nonlinear pure-feedback systems with unknown nonlinear functions. This problem is considered di/cult to be dealt with in the control literature, mainly because that the triangular structure of pure-feedback systems has no a/ne appearance of the variables to be used as virtual controls. To overcome this di/culty, implicit function theorem is 0rstly exploited to assert the existence of the continuous desired virtual controls. NN approximators are then used to approximate the continuous desired virtual controls and desired practical control. With mild assumptions on the partial derivatives of the unknown functions, the developed adaptive NN control schemes achieve semi-global uniform ultimate boundedness of all the signals in the closed-loop. The control performance of the closed-loop system is guaranteed by suitably choosing the design parameters. ? 2002 Elsevier Science Ltd. All rights reserved. Keywords: Adaptive neural control; Uncertain pure-feedback system; Backstepping

1. Introduction In the past decade, interest in adaptive control of nonlinear systems has been ever increasing, and many signi0cant developments have been achieved. As a breakthrough in nonlinear control area, adaptive backstepping was introduced to achieve global stability and asymptotic tracking for a large class of nonlinear systems in the parametric strict-feedback form by Kanellakopoulos, Kokotovic, and Morse (1991). Later, the overparametrization problem was successfully eliminated in Krsti;c, Kanellakopoulos, and Kokotovic (1992) through the tuning function method. In an e
for nonlinear systems with a triangular structure. To accommodate uncertainties, robust adaptive backstepping control has been studied for nonlinear strict-feedback systems with time-varying disturbances and static or dynamic uncertainties in Freeman and Kokotovi;c (1996), Yao and Tomizuka (1997), Jiang and Praly (1998) and Pan and Basar (1998) (to name just a few). On the other hand, adaptive neural control schemes have been found to be particularly useful for the control of highly uncertain, nonlinear and complex systems (see Lewis, Jagannathan, & Yeildirek, 1999; Ge, Hang, Lee, & Zhang, 2001 and the references therein). In the earlier NN control schemes, optimization techniques were mainly used to derive parameter adaptation laws with little analytical results for stability and performance. To overcome these problems, some elegant adaptive NN control approaches have been proposed based on Lyapunov’s stability theory (Narendra & Parthasarathy, 1990; Polycarpou & Ioannou, 1992; Sanner & Slotine, 1992; Rovithakis & Christodoulou, 1994; Chen & Khalil, 1995; Yesidirek & Lewis, 1995; Spooner & Passino, 1996). However, one limitation of these schemes is that they can only be applied to nonlinear systems where certain types of matching conditions are required to be satis0ed. Using the idea of adaptive backstepping design (Krsti;c, Kanellakopoulos, & Kokotovi;c, 1995), several neural-based adaptive controllers (Polycarpou &

0005-1098/02/$ - see front matter ? 2002 Elsevier Science Ltd. All rights reserved. PII: S 0 0 0 5 - 1 0 9 8 ( 0 1 ) 0 0 2 5 4 - 0

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S.S. Ge, C. Wang / Automatica 38 (2002) 671–682

Mears, 1998; Ge et al., 2001) have been investigated for some classes of nonlinear systems in the following strict-feedback form without the requirement of matching conditions x˙i = fi (xPi ) + gi (xPi )xi+1 ; x˙n = fn (xPn ) + gn (xPn )u;

1 6 i 6 n − 1; n ¿ 2;

(1)

y = x1 ; where xPi = [x1 ; : : : ; xi ]T ∈ Ri ; i = 1; : : : ; n; u ∈ R; y ∈ R are state variables, system input and output, respectively; fi (·) and gi (·), i=1; : : : ; n are unknown smooth functions. In Polycarpou and Mears (1998), an indirect adaptive NN control scheme was presented for system (1) with the a/ne terms gi (xPi ) = 1; i = 1; : : : ; n − 1, and gn (xPn ) = g being an unknown constant. The unknown functions fi (xPi ); i=1; : : : ; n are 0rstly approximated on-line by neural networks, then a stabilizing controller is constructed based on the approximation. Through the introduction of novel integral Lyapunov functions, direct adaptive neural network control was proposed for system (1) (Ge et al., 2001), in which the possible controller singularity problem usually met in adaptive control is avoided without using projection. While the nonlinear strict-feedback systems have been much investigated via backstepping design, only a few results are available in the literature for the control of nonlinear pure-feedback systems (Nam & Arapostations, 1988; Kanellakopoulos et al., 1991; Seto et al., 1994; Krsti;c et al., 1995). The pure-feedback system represents a more general class of triangular systems which have no a/ne appearance of the variables to be used as virtual controls. In practice, there are many systems falling into this category, such as mechanical systems (Ferrara & Giacomini, 2000), aircraft Sight control system (Hunt & Meyer, 1997), biochemical process (Krsti;c et al., 1995), Du/ng oscillator (Dong, Chen, & Chen, 1997), etc. As indicated in Krsti;c et al. (1995), it was quite restrictive to 0nd the explicit virtual controls to stabilize the pure-feedback systems by using integrator backstepping. In Kanellakopoulos et al. (1991); Krsti;c et al. (1995), while excellent results are given for global stabilization of parametric strict-feedback systems, only local stability is achieved in a well de0ned region around origin for parametric pure-feedback systems. By imposing additional restrictions on the nonlinearities, global stability is obtained for a special case of the parametric pure-feedback systems in Seto et al. (1994). Note that in Kanellakopoulos et al. (1991); Seto et al. (1994); Krsti;c et al. (1995), the nonlinearities are known smooth functions, and the unknown parameters occur linearly. In this paper, adaptive NN control schemes are proposed for the following uncertain nonlinear pure-feedback

systems:  x˙i = fi (xPi ; xi+1 ); 1 6 i 6 n − 2;     x˙n−1 = fn−1 (xPn−1 ) + gn−1 (xPn−1 )x n ; 1 :  x˙n = fn (xPn ) + gn (xPn−1 )u; n ¿ 3;    y = x1 and

 x˙i = fi (xPi ; xi+1 );    

1 6 i 6 n − 2; x˙n−1 = fn−1 (xPn−1 ) + gn−1 (xPn−1 )x n ; 2 :  x˙n = fn (xPn ) + gn (xPn )u; n ¿ 3;    y = x1 ;

(2)

(3)

where xPi = [x1 ; : : : ; xi ]T ∈ Ri ; i = 1; : : : ; n; u ∈ R; y ∈ R are state variables, system input and output, respectively; fi (xPi ; xi+1 ) (i = 1; : : : ; n − 2), fj (·) and gj (·) (j = n − 1; n) are unknown smooth functions. Here the di
S.S. Ge, C. Wang / Automatica 38 (2002) 671–682

ultimately bounded, and (ii) the output y follows a desired trajectory yd generated from the following smooth, bounded reference model: x˙di = fdi (xd );

1 6 i 6 m;

(4)

yd = xd1 ;

where xd = [xd1 ; xd2 ; : : : ; xdm ]T ∈ Rm are the states, yd ∈ R is the system output, fdi (·); i = 1; 2; : : : ; m are known smooth nonlinear functions. Assume that xd remain bounded, i.e., xd ∈ d ; ∀t ¿ 0. n

Lemma 1. Assume that f(x; u) : R × R → R is continuously di0erentiable ∀(x; u) ∈ Rn × R; and there exists a positive constant d such that @f(x; u)=@u ¿ d ¿ 0; ∀(x; u) ∈ Rn × R. Then there exists a continuous (smooth) function u∗ = u(x) such that f(x; u∗ ) = 0. Proof. Firstly; it can be proven that for every x ∈ Rn ; there exists a unique u(x) ∈ R such that f(x; u(x))=0. Because the partial derivative @f(x; u)=@u has a lower positive bound d; i.e.; @f(x; u)=@u ¿ d ¿ 0; ∀(x; u) ∈ Rn × R; if f(x; 0) = c = 0; then by mean value theorem (Apostol; 1963);      |c| @f(x; u)  |c| = f(x; 0) + −0 f x; d @u u=u∗1 d 

f x; −

¿ f(x; 0) + |c| ¿ 0;

|c|

d



 @f(x; u)  = f(x; 0) + @u 

¡ f(x; 0) − |c| 6 0;

 2 u=u∗

−

|c|

d

673

large number of NN nodes such that all the signals in the closed-loop remain bounded. In control engineering, radial basis function (RBF) neural networks (NNs) are usually used as a tool for modeling nonlinear functions because of their good capabilities in function approximation. They belong to a class of linearly parameterized networks. For comprehensive treatment of NN approximation, see (Ge et al., 2001). In this paper, the following RBF NN (Haykin, 1999) is used to approximate the continuous function h(Z) : Rq → R, hnn (Z) = W T S(Z);

(6)

where the input vector Z ∈  ⊂ Rq , weight vector W = [w1 ; w2 ; : : : ; wl ]T ∈ Rl , the NN node number l ¿ 1; and S(Z)=[s1 (Z); : : : ; sl (Z)]T , with si (Z) being chosen as the commonly used Gaussian functions, which have the form  −(Z − i )T (Z − i ) si (Z) = exp ; i = 1; 2; : : : ; l; (7) 2 i

where i = [i1 ; i2 ; : : : ; iq ]T is the center of the receptive 0eld and i is the width of the Gaussian function. It has been proven that network (6) can approximate any continuous function over a compact set Z ⊂ Rq to arbitrary any accuracy as h(Z) = W ∗T S(Z) + !;

∀Z ∈ Z ;

(8)

where W ∗ is ideal constant weights, and ! is the approximation error.

 −0

(5)

where u∗1 ∈ (0; |c|=d) and u∗2 ∈ (−|c|=d; 0). Therefore; by intermediate value theorem (Apostol; 1963); there exists u ∈ (−|c|=d; |c|=d) such that f(x; u) = 0. If f(x; 0) = c = 0; then u = 0 is the trivial solution satisfying f(x; u) = 0. Secondly; it can be shown that u(x) is a continuous (smooth) function of x. Fix x0 ; apply implicit function theorem (Apostol; 1963) to f(x; u) around the point (x0 ; u(x0 )). Because the positivity of partial derivative to u; it can be seen that there exists a continuous (smooth) function u1 (x) around x = x0 ; such that f(x; u1 (x)) = 0 around x=x0 . The strict increasing of f(x; u) with respect to u leads to u1 (x) = u(x) for every x. Therefore; u(x) is continuous (smooth) in x around x0 . Since x0 is arbitrary; it is clear that u(x) is a continuous (smooth) function in x. Note that NN approximation is only guaranteed within some compact sets in the derivation of the adaptive neural controller. Accordingly, the stability results obtained in this work are semi-global in the sense that, as long as the input variables of the NNs remain within some compact sets, where the compact sets can be made as large as desired, there exists controller(s) with su/ciently

Assumption 1. There exist ideal constant weights W ∗ such that |!| 6 !∗ with constant !∗ ¿ 0 for all Z ∈ Z . The ideal weight vector W ∗ is an “arti0cial” quantity required for analytical purposes. W ∗ is de0ned as the value of W that minimizes |!| for all Z ∈ Z ⊂ Rq , i.e.,

 ∗ T W , arg minl sup |h(Z) − W S(Z)| : (9) W ∈R

Z∈Z

In the following, let · denote the 2-norm, and "max (B) and "min (B) denote the largest and smallest eigenvalues of a square matrix B, respectively. 3. Direct adaptive NN control for
1 For the control of pure-feedback systems 1 and 2 , de0ne gi (xPi ; xi+1 ) := @fi (xPi ; xi+1 )=@xi+1 ; i = 1; : : : ; n − 2, which are also unknown nonlinear functions. Assumption 2. The signs of gi (xPi ; xi+1 ); i = 1; : : : ; n − 2; gn−1 (xPn−1 ) and gn (xPn−1 ) are known; and there exist constants gi1 ¿ gi0 ¿ 0 such that (i) |gi (·)| ¿ gi0 ¿ 0; ∀xPn ∈ Rn ; and (ii) |gi (·)| 6 gi1 ¡ ∞; ∀xPn ∈ xPn ⊂ Rn where xPn is a compact region; i = 1; : : : ; n.

674

S.S. Ge, C. Wang / Automatica 38 (2002) 671–682

The above assumption implies that partial derivatives gi (·), i = 1; : : : ; n are strictly either positive or negative. Without losing generality, it is assumed that gi (·) ¿ gi0 ¿ 0, ∀xPi ∈ Ri . Accordingly, the derivatives of gi (·); i = 1; : : : ; n are given by g˙i (xPi ; xi+1 ) =

i+1 @gi (xPi ; xi+1 )

@xk

k=1

=

i+1 @gi (xPi ; xi+1 )

@xk

k=1

g˙n−1 (xPn−1 ) =

@xk

n−2 k=1

+ g˙n (xPn−1 ) =

n−1 k=1

=

n−2 k=1

'1 = −x˙d1 + k1 z1 ;

x˙k

where k1 ¿ 0 is a constant. It is clear that '1 is a function of x1 and xd . Considering the fact that @'1 =@x2 = 0, the following inequality holds

According to Lemma 1, by viewing x2 as a virtual control input, for every value of x1 and '1 , there exists a smooth ideal control input x2 = 1∗ (x1 ; '1 ) such that f1 (x1 ; 1∗ ) + '1 = 0:

@gn−1 (xPn−1 ) fk (xPk+1 ) @xk

(13)

Using mean value theorem (Apostol, 1963), there exists "1 (0 ¡ "1 ¡ 1) such that

@gn−1 (xPn−1 ) [fn−1 (xPn−1 )+gn−1 (xPn−1 )x n ]; @x n−1

@gn (xPn−1 ) x˙k @xk @gn (xPn−1 ) fk (xPk+1 ) @xk

@gn (xPn−1 ) [fn−1 (xPn−1 ) + gn−1 (xPn−1 )x n ]: + @x n−1 (10)

f1 (x1 ; x2 ) = f1 (x1 ; 1∗ ) + g1"1 (x2 − 1∗ );

Clearly, they only depend on states xPn . Because fi (·); i= 1; : : : ; n − 1; gn−1 (·) and gn (·) are assumed to be smooth functions, g˙i (·); i = 1; : : : ; n are therefore bounded within a compact set. Accordingly, the following assumption is made for g˙i (·); i = 1; : : : ; n. Assumption 3. There exist constants gid ¿ 0 such that |g˙i (·)| 6 gid ; ∀xPn ∈ xPn ⊂ Rn where xPn is a compact region; i = 1; : : : ; n. In this section, direct adaptive NN design is presented by using RBF NN. At each recursive step i, an intermediate desired feedback control i∗ is 0rst shown to exist which possesses some desired stabilizing properties, and then the ith-order subsystem is stabilized with respect to a Lyapunov function Vi by the design of a stabilizing function i , where an RBF neural network is employed to approximate the unknown part in intermediate desired feedback control i∗ . The control law u is designed in the last step. Step 1: De0ne z1 = x1 − xd1 . Its derivative is (11)

(14)

where g1"1 := g1 (x1 ; x2"1 ) with x2"1 = "1 x2 + (1 − "1 ) 1∗ . Note that Assumption 2 on g1 (x1 ; x2 ) is still valid for g1"1 . Since g1"1 is a function of x1 ; x2 and 1∗ , from (12) and (13), it can be seen that 1∗ is a function of x1 and xd , the derivation of g˙1"1 is g˙1"1 =

2 @g1" k=1

=

@xk

2 @g1" k=1

z˙1 = f1 (x1 ; x2 ) − x˙d1 :

(12)

@[f1 (x1 ; x2 ) + '1 ] ¿ g10 ¿ 0: @x2

fk (xPk+1 ); i = 1; : : : ; n − 2;

n−1 @gn−1 (xPn−1 ) k=1

=

x˙k

From Assumption 2, we know that @f1 (x1 ; x2 )=@x2 ¿ g10 ¿ 0 for all (x1 ; x2 ) ∈ R2 . De0ne '1 as

@xk

1

x˙k +

m @g1" k=1

1

1

@xdk

fk (xPk+1 ) +

x˙dk

m @g1" k=1

1

@xdk

fdk (xd ):

Since xd is assumed to be bounded, similar to Assumption 3, it is reasonable to assume that |g˙1"1 | is bounded by some constant within some compact region. For uniformity of presentation, we assume that |g˙1"1 (·)| 6 g1d , ∀xPn ∈ xPn and xd ∈ d , where g1d is the same as in Assumption 3 for |g˙1 |. In other words, Assumption 3 on g˙1 (x1 ; x2 ) is valid for g˙1"1 . Combining (11) – (14) yields z˙1 = −k1 z1 + g1"1 (x2 − 1∗ ):

(15)

By employing an RBF neural network W1T S1 (Z1 ) to approximate 1∗ (x1 ; '1 ), where Z1 = [x1 ; x˙d1 ; z1 ]T ∈ 1 ⊂ R3 ,

1∗ can be expressed as

1∗ = W1∗T S1 (Z1 ) + !1 ;

(16)

where W1∗ denotes the ideal constant weights, and |!1 | 6 !1∗ is the approximation error with constant !1∗ ¿ 0. Let Wˆ 1 be the estimate of W1∗ . De0ne z2 = x2 − 1 and let T

1 = −c1 z1 + Wˆ 1 S1 (Z1 );

(17)

S.S. Ge, C. Wang / Automatica 38 (2002) 671–682

675

where c1 is a positive constant to be speci0ed later. Then, the dynamics of z1 is governed by

From Assumption 2, we know that @f2 (xP2 ; x3 )=@x3 ¿ g20 ¿ 0 for all xP3 ∈ R3 . De0ne

z˙1 = −k1 z1 + g1"1 (z2 + 1 − 1∗ )

'2 = − ˙1 + k2 z2 ;

= −k1 z1 + g1"1 [z2 − c1 z1 +

T W˜ 1 S1 (Z1 )

− !1 ];

(18)

where W˜ 1 = Wˆ 1 − W1∗ . Through out this paper, de0ne (˜· ) = (ˆ· ) − (·)∗ . Consider the Lyapunov function candidate V1 =

1 2 1 ˜ T −1 ˜ z + W ( W 1: 2g1"1 1 2 1 1

(19)

The derivative of V1 is T z1 z˙1 g˙1"1 z12 V˙ 1 = − + W˜ 1 (1−1 Wˆ˙ 1 2 g1"1 2g1" 1

(20)

(21)

where )1 ¿ 0 is a small constant. Let c1 = c10 + c11 , with c10 and c11 ¿ 0. Then, Eq. (20) becomes 

g ˙ k 1 1" z 2 + z1 z2 − c10 + 2 1 z12 V˙ 1 = − g1"1 1 2g1"1 − c11 z12 − z1 !1 −

T )1 W˜ 1 Wˆ 1 :

@[f2 (xP2 ; x3 ) + '2 ] ¿ g20 ¿ 0: @x3 According to Lemma 1, by viewing x3 as a virtual control input, for every value of xP2 and '2 , there exists a smooth ideal control input x3 = 2∗ (xP2 ; '2 ) such that

(22)

f2 (xP2 ; x3 ) = f2 (xP2 ; 2∗ ) + g2"2 (x3 − 2∗ );

g˙2"2 = =

+

)1 W˜ 1 2 )1 W1∗ 2 + ; 6− 2 2 !12 !∗2 6 1 : 4c11 4c11

(23)

2 2 ))z12 6 − (c10 − (g1d =2g10 ))z12 , Because −(c10 + (g˙1"1 =2g1" 1 ∗ by choosing c10 large enough such that c10 := c10 − 2 (g1d =2g10 ) ¿ 0, the derivative of V1 satis0es

k1 2 )1 W˜ 1 2 ∗ 2 V˙ 1 ¡ − z1 + z1 z2 − c10 z1 − g1"1 2 )1 W1∗ 2 !∗2 + + 1 : 2 4c11

@xk

2

x˙k +

m @g2" k=1

1

2

@xdk

fk (xPk+1 ) +

(24)

@g2"2 ˆ˙ W1 @Wˆ 1

x˙dk +

m @g1" k=1

1

@xdk

fdk (xd )

@g2"2 [(1 (−S1 (Z1 )z1 − )1 Wˆ 1 )]: @Wˆ 1

Similar to Step 1, it is reasonable to assume that |g˙2"2 | is bounded by some constant within some compact region. For uniformity of presentation, we assume that |g˙2"2 (·)| 6 g2d ; ∀xPn ∈ xPn , xd ∈ d and Wˆ 1 ∈ W1 , where g2d is the same as in Assumption 3 for |g˙2 |. In other words, Assumption 3 on g˙2 (xP2 ; x3 ) is valid for g˙2"2 . Subsequently, such an assumption is also made for g˙i"i in the following steps. Combining (25) – (28) yields z˙2 = −k2 z2 + g2"2 (x3 − 2∗ ):

(29)

Since 1 is a function of x1 , xd and Wˆ 1 , ˙1 is given by @ 1 @ 1 @ 1 ˆ˙ x˙1 + x˙d + W1 @x1 @xd @Wˆ 1 @ 1 = f1 (xP2 ) + *1 ; @x1

˙1 =

Step 2: The derivative of z2 = x2 − 1 is z˙2 = f2 (xP2 ; x3 ) − ˙1 :

@xk

2 @g1" k=1

T

− c11 z12 − z1 !1 6

3 @g2" k=1

−)1 W˜ 1 Wˆ 1 = −)1 W˜ 1 (W˜ 1 + W1∗ )

6 −)1 W˜ 1 2 + )1 W˜ 1

W1∗

(28)

where g2"2 := g2 (xP2 ; x3"2 ) with x3"2 = "2 x3 + (1 − "2 ) 2∗ . Note that Assumption 2 on g2 (xP2 ; x3 ) is still valid for g2"2 . Since g2"2 is a function of x1 ; x2 ; x3 and 2∗ , and from (26) and (27), it can be seen that 2∗ is a function of xP2 , z2 and

˙1 , i.e., a function of x1 ; x2 ; xd and Wˆ 1 , the derivation of g˙2"2 is

By completion of squares, the following inequalities hold T

(27)

Using mean value theorem (Apostol, 1963), there exists "2 (0 ¡ "2 ¡ 1) such that

Consider the following adaptation law Wˆ˙ 1 = W˜˙ 1 = (1 [ − S1 (Z1 )z1 − )1 Wˆ 1 ];

where k2 ¿ 0 is a constant. Considering the fact that @'2 =@x3 = 0, the following inequality holds

f2 (xP2 ; 2∗ ) + '2 = 0:

g˙ k1 2 =− z + z1 z2 − c1 z12 − 1"2 1 z12 − z1 !1 g1"1 1 2g1"1 T T + W˜ 1 S1 (Z1 )z1 + W˜ 1 (1−1 Wˆ˙ 1 :

(26)

(25)

(30)

676

S.S. Ge, C. Wang / Automatica 38 (2002) 671–682

where *1 =

@ 1 @ 1 x˙d + [(1 (−S1 (Z1 )z1 − )1 Wˆ 1 )] @xd @Wˆ 1

(31)

is computable. By employing an RBF neural network W2T S2 (Z2 ) to approximate 2∗ (xP2 ; '2 ), where Z2 =[xPT2 ; @ 1 =@x1 ; *1 ; z2 ]T ∈ 2 ⊂ R5 ; 2∗ can be expressed as

2∗ = W2∗T S2 (Z2 ) + !2 ;

V˙ 2 ¡ z2 z3 −

Remark 1. From the de0nitions of '2 in (26); and ˙1 in (30); it can be seen that 2∗ (xP2 ; '2 ) in (28) is a function of xP2 ; xd ; z2 and Wˆ 1 . However; neural network weights Wˆ 1 are not recommended to be used as inputs to the NN controller under construction. This is because of the curse of dimensionality of RBF NN which may result in T the number of NN inputs to be too large for Wˆ 2 S2 (Z2 ). By de0ning the intermediate variables @ 1 =@x1 and *1 T which are computable; the NN approximation Wˆ 2 S2 (Z2 ) for 2∗ (xP2 ; '2 ) can be computed by using the minimal number of NN inputs Z2 = [xPT2 ; @ 1 =@x1 ; *1 ; z2 ]T . Note also that the variable vector xd does not appear in Z2 ; because it has been combined into the intermediate variable *1 . Since W2∗ is unknown, 2∗ cannot be realized in practice. Let Wˆ 2 be the estimate of W2∗ . De0ne z3 = x3 − 2 and let T

2 = −z1 − c2 z2 + Wˆ 2 S2 (Z2 );

(33)

where c2 is a positive constant to be speci0ed later. Then, the dynamics of z2 is governed by z˙2 = −k2 z2 + g2"2 (z3 + 2 − 2∗ ) T

= −k2 z2 + g2"2 [z3 − z1 − c2 z2 + W˜ 2 S2 (Z2 ) − !2 ]: (34)

(35)

The derivative of V2 is

2 )j Wj∗ 2

2

j=1

2

∗ 2 cj0 zj −

j=1

+

j=1

2 !j∗2 j=1

2 )j W˜ j 2

4cj1

2

;

(38)

∗ 2 where c20 := c20 − (g2d =2g20 ) ¿ 0. Step i (3 6 i 6 n − 2): The derivative of zi =xi − i−1 is

z˙i = fi (xPi ; xi+1 ) − ˙i−1 :

(39)

From Assumption 2, we know that @fi (xPi ; xi+1 )=@xi+1 ¿ gi0 ¿ 0 for all xPi+1 ∈ Ri+1 . De0ne 'i = − ˙i−1 + ki zi ;

(40)

where ki ¿ 0 is a constant. Considering the fact that @'i =@xi+1 = 0, the following inequality holds @[fi (xPi ; xi+1 ) + 'i ] ¿ gi0 ¿ 0: @xi+1 According to Lemma 1, by viewing xi+1 as a virtual control input, for every value of xPi and 'i , there exists a smooth ideal control input xi+1 = i∗ (xPi ; 'i ) such that fi (xPi ; i∗ ) + 'i = 0:

(41)

Using mean value theorem (Apostol, 1963), there exists "i (0 ¡ "i ¡ 1) such that fi (xPi ; xi+1 ) = fi (xPi ; i∗ ) + gi"i (xi+1 − i∗ );

(42)

where gi"i := gi (xPi ; x(i+1)"i ) with x(i+1)"i ="i xi+1 +(1−"i ) i∗ . Note that Assumptions 2 and 3 on gi (xPi ; xi+1 ) is still valid for gi"i . Combining (39) – (42) yields (43)

Since i−1 is a function of xPi−1 , xd and Wˆ 1 ; : : : ; Wˆ i−1 , ˙i−1 is given by i−1 @ i−1 k=1

@xk

(fk (xPk+1 )) + *i−1 ;

(44)

where

k2 2 g˙ z2 − z1 z2 + z2 z3 − c2 z22 − 2" z22 − z2 !2 = V˙ 1 − 2 g2"2 2g2" 2 (36)

*i−1 =

i−1 @ i−1 k=1

+

Consider the following adaptation law Wˆ˙ 2 = W˜˙ 2 = (2 [ − S2 (Z2 )z2 − )2 Wˆ 2 ];

+

˙i−1 =

T z2 z˙2 g˙2" z22 V˙ 2 = V˙ 1 + − 2 + W˜ 2 (2−1 Wˆ˙ 2 ; g2"2 2g2"2

T T + W˜ 2 S2 (Z2 )z2 + W˜ 2 (2−1 Wˆ˙ 2 :

gj"

zj2 −

z˙i = −ki zi + gi"i (xi+1 − i∗ ):

Consider the Lyapunov function candidate 1 2 1 ˜ T −1 ˜ z + W ( W 2: 2g2"2 2 2 2 2

2 kj j=1

(32)

where W2∗ denotes the ideal constant weights, and |!2 | 6 !2∗ is the approximation error with constant !2∗ ¿ 0.

V 2 = V1 +

where )2 ¿ 0 is a small constant. Let c2 =c20 +c21 , where c20 and c21 ¿ 0. By using (24), (36) and (37), and with some completion of squares and straightforward derivation similar to those employed in Step 1, the derivative of V2 becomes

@xd

i−1 @ i−1 k=1

(37)

x˙d

@Wˆ k

is computable.

[(k (−Sk (Zk )zk − )k Wˆ k )]

(45)

S.S. Ge, C. Wang / Automatica 38 (2002) 671–682

By employing an RBF neural network WiT Si (Zi ) to approximate i∗ (xPi ; 'i ), where Zi =[xPTi ; @ i−1 =@x1 ; : : : ; @ i−1 =@xi−1 ; *i−1 ; zi ]T ∈ i ⊂ R2i+1 ; i∗ can be expressed as

i∗

=

Wi∗T Si (Zi )

+ !i ;

(46)

where Wi∗ denotes the ideal constant weights, and |!i | 6 !i∗ is the approximation error with constant !i∗ ¿ 0. Since Wi∗ is unknown, i∗ cannot be realized in practice. Let Wˆ i be the estimate of Wi∗ . De0ne zi+1 = xi+1 − i and let T

i = −zi−1 − ci zi + Wˆ i Si (Zi );

(47)

where ci is a positive constant to be speci0ed later. Then, Eq. (43) becomes

there exists a desired feedback control ∗

n−1 = −zn−2 − cn−1 zn−1 1 (fn−1 (xPn−1 ) − ˙n−2 ); − gn−1 (xPn−1 )

˙n−2 =

n−2 @ n−2 k=1

@xk

(gk (xPk )xk+1 + fk (xPk )) + *n−2 ;

n−2 @ n−2 k=1

T

1 2 1 ˜ T −1 ˜ z + W i (i W i : 2gi"i i 2

Vi = Vi−1 +

(49)

The derivative of Vi is

@xd

x˙d +

n−2 @ n−2 k=1

@Wˆ k

[(k (Sk (Zk )zk − )k Wˆ k )]

= V˙ i−1 −

T

T

(50)

Wˆ˙ i = W˜˙ i = (i [ − Si (Zi )zi − )i Wˆ i ];

(51)

where )i ¿ 0 is a small constant. Let ci = ci0 + ci1 , where ci0 and ci1 ¿ 0. By using the equation for V˙ i−1 corresponding to (38), the Eqs. (50) and (51), and with some completion of squares and straightforward derivation similar to those employed in Step 1, the derivative of Vi becomes i kj j=1

+

i j=1

gj"

)j Wj∗ 2 2

zj2 −

i

∗ 2 cj0 zj −

j=1

+

i j=1

!j∗2

4cj1

i )j W˜ j 2 j=1

;

(56)

∗ ∗ T = −zn−2 − cn−1 zn−1 + Wn−1 Sn−1 (Zn−1 ) + !n−1 ;

n−1 (57) ∗ where Wn−1 denotes the ideal constant weights, and

Consider the following adaptation law

V˙ i ¡ zi zi+1 −

1 (fn−1 (xPn−1 ) − ˙n−2 ); gn−1 (xPn−1 )

where Zn−1 = [xPTn−1 ; @ n−2 =@x1 ; : : : ; @ n−2 =@x n−2 ; *n−2 ]T ∗ ∈ n−1 ⊂ R2(n−1) ; n−1 can be expressed as

g˙ ki 2 zi − zi−1 zi + zi zi+1 − ci zi2 − i"2i zi2 gi"i 2gi"i

− zi !i + W˜ i Si (Zi )zi + W˜ i (i−1 Wˆ˙ i :

(55)

is computable. T Sn−1 (Zn−1 ) By employing an RBF neural network Wn−1 ∗ in to approximate the following unknown part of n−1 (53) −

T zi z˙i g˙i"i zi2 − + W˜ i (i−1 Wˆ˙ i V˙ i = V˙ i−1 + 2 gi"i 2gi"i

(54)

where

= −ki zi + gi"i [zi+1 − zi−1 − ci zi + W˜ i Si (Zi ) − !i ]: (48) Consider the Lyapunov function candidate

(53)

where cn−1 is a positive constant to be speci0ed later, gn−1 (xPn−1 ) and fn−1 (xPn−1 ) are unknown smooth functions of xPn−1 , and ˙n−2 is a function of xPn−2 , xd and Wˆ 1 ; : : : ; Wˆ n−2 . Therefore, ˙n−2 can be expressed as

*n−2 =

z˙i = −ki zi + gi"i (zi+1 + i − i∗ )

677

∗ |!n−1 | 6 !n−1 is the approximation error with constant

∗ ¿ 0. !n−1 ∗ ∗ is unknown, n−1 cannot be realized in Since Wn−1 ∗ ˆ . De0ne zn = practice. Let W n−1 be the estimate of Wn−1 x n − n−1 and let T

n−1 = −zn−2 − cn−1 zn−1 + Wˆ n−1 Sn−1 (Zn−1 ):

(58)

Then, the dynamics of zn−1 is governed by z˙n−1 = fn−1 (xPn−1 ) + gn−1 (xPn−1 )(zn + n−1 ) − ˙n−2

2

= gn−1 [zn − zn−2 − cn−1 zn−1 (52)

∗ 2 := ci0 − (gid =2gi0 ) ¿ 0. where ci0 Step n − 1: The derivative of zn−1 = x n−1 − n−2 is z˙n−1 =fn−1 (xPn−1 )+gn−1 (xPn−1 )x n − ˙n−2 . By viewing x n as a virtual control to stabilize the (z1 ; : : : ; zn−1 )-subsystem,

T + W˜ n−1 Sn−1 (Zn−1 ) − !n−1 ]:

(59)

Consider the Lyapunov function candidate Vn−1 = Vn−2 +

1 1 T z 2 + W˜ (−1 W˜ n−1 : 2gn−1 (xPn−1 ) n−1 2 n−1 n−1 (60)

678

S.S. Ge, C. Wang / Automatica 38 (2002) 671–682

The derivative of Vn−1 is 2 T zn−1 z˙n−1 g˙n−1 zn−1 −1 ˆ˙ V˙ n−1 = V˙ n−2 + W n−1 − + W˜ n−1 (n−1 2 gn−1 2gn−1

= V˙ n−2 − zn−2 zn−1 + zn−1 zn −

2 cn−1 zn−1

T T −1 ˆ˙ + W˜ n−1 Sn−1 (Zn−1 )zn−1 + W˜ n−1 (n−1 W n−1 :

(61)

Wˆ˙ n−1 = W˜˙ n−1 = (n−1 [ − Sn−1 (Zn−1 )zn−1 − )n−1 Wˆ n−1 ]; (62) where )n−1 ¿ 0 is a small constant. Let cn−1 = c(n−1)0 + c(n−1)1 , where c(n−1)0 and c(n−1)1 ¿ 0. With some completion of squares and straightforward derivation similar to those employed in Step 1, the derivative of Vi becomes gj"

j=1

−

n−1 )k W˜ k 2 k=1

2

+

zj2

−

n−1

∗ 2 ck0 zk

k=1

n−1 )k W ∗ 2 k=1

2

k

+

n−1 ∗ 2 !k k=1

;

4ck1

g(n−1)d ¿ 0: 2 2g(n−1)0

(69)

(63)

(64)

T

u = −zn−1 − cn zn + Wˆ n Sn (Zn );

(71)

where cn is a positive constant to be speci0ed later. Then, Eq. (65) becomes T z˙n = −kn zn + gn [ − zn−1 − cn zn + W˜ n Sn (Zn ) − !n ]: (72)

Consider the Lyapunov function candidate 1 2 1 ˜ T −1 ˜ z + W ( W n: 2gn n 2 n n

T zn z˙n g˙n zn2 − + W˜ n (n−1 Wˆ˙ n V˙ n = V˙ n−1 + 2 gn 2gn g˙ = V˙ n−1 − zn−1 zn − cn zn2 − n2 zn2 − zn !n 2gn T T + W˜ S (Z )z + W˜ (−1 Wˆ˙ : n n

n

n

n

Consider the following adaptation law

z˙n = fn (xPn ) + gn (xPn−1 )u − ˙n−1 :

Wˆ˙ n = W˜˙ n = (n [ − Sn (Zn )zn − )n Wˆ n ];

To stabilize the whole system (z1 ; : : : ; zn ), there exists a desired feedback control u∗ = −zn−1 − cn zn −

1 (fn − ˙n−1 ); gn

(66)

where cn is a positive constant to be speci0ed later. Since n−1 is a function of xPn−1 , xd and Wˆ 1 ; : : : ; Wˆ n−1 ,

˙n−1 is given by

˙n−1 =

n−1 @ n−1 k=1

@xk

fk (xPk+1 ) + *n−1 ;

(67)

where *n−1 =

k=1

@xd

is computable.

x˙d +

n−1 @ n−1 k=1

@Wˆ k

[(k (−Sk (Zk )zk − )k Wˆ k )] (68)

n

n

(74)

(75)

where )n ¿ 0 is a small constant. Let cn = cn0 + cn1 , where cn0 and cn1 ¿ 0. With the completion of squares and straightforward derivation similar to those employed in Step 1, the derivative of Vn satis0es the following inequality V˙ n ¡ −

n−2 kj j=1

+

gj"

zj2 −

n

2

∗ 2 cj0 zj −

+

2

n !j∗ 2

4cj1

n )j W˜ j 2 j=1

2

n )j W˜ j 2 j=1

j=1

n )j Wj∗ 2 j=1

∗ 2 cj0 zj −

j=1

j=1

+

n

n )j Wj∗ 2 j=1

¡− n−1 @ n−1

(73)

The derivative of Vn is

Step n: This is the 0nal step. The derivative of zn = x n − n−1 is (65)

(70)

where Wn∗ denotes the ideal constant weights, and |!n | 6 !n∗ is the approximation error with constant !n∗ ¿ 0. Since Wn∗ is unknown, u∗ cannot be realized in practice. Let Wˆ n be the estimate of Wn∗ , and

Vn = Vn−1 +

where ∗ c(n−1)0 := c(n−1)0 −

1 (fn (xPn ) − ˙n−1 ); gn (xPn−1 )

u∗ = −zn−1 − cn zn + Wn∗ T Sn (Zn ) + !n ;

Consider the following adaptation law

n−2 kj

−

where Zn = [xPTn ; @ n−1 =@x1 ; : : : ; @ n−1 =@x n−1 ; *n−1 ]T ∈ n ⊂ R2n , u∗ can be expressed

g˙ 2 − n−1 zn−1 − zn−1 !n−1 2 2gn−1

V˙ n−1 ¡ zn−1 zn −

By employing an RBF neural network WnT Sn (Zn ) to approximate the following unknown part of u∗ in (66)

+

2

n !j∗ 2 j=1

4cj1

∗ 2 := cn0 − gnd =2gn0 ¿ 0. where cn0

;

(76)

S.S. Ge, C. Wang / Automatica 38 (2002) 671–682

Theorem 1. Consider the closed-loop system consisting of the plant (2); the reference model (4); the controller (71) and the NN weight updating laws (21); (37); (51) and (75). Assume there exists su4ciently large compact sets i ∈ R2i+1 ; i = 1; : : : ; n − 2 and i ∈ R2i ; i = n − 1; n such that Zi ∈ i for all t ¿ 0. Then; all signals in the closed-loop system remain bounded; and the output tracking error y(t) − yd (t) converges to a small neighborhood around zero by appropriately choosing design parameters. Proof. Let + :=

n )k W ∗ 2

2

k=1

k

+

n !k∗ 2 k=1

4ck1

:

∗ ∗ is chosen such that ck0 ¿ ,=2gk0 ; i.e.; ck0 ¿ ,=2gk0 + If ck0 2 gkd =2gk0 ; k = 1; : : : ; n; where , is a positive constant; and )k and (k are chosen such that )k ¿ ,"max {(k−1 }; k = 1; : : : ; n; then from (76) the following inequality holds

V˙ n ¡ −

n

∗ 2 ck0 zk −

k=1

n k=1

)k W˜ k 2 ++ 2

T n , 2 ,W˜ k (k−1 W˜ k ++ 6− z − 2gk0 k k=1 2 k=1  n  T n 1 W˜ k (k−1 W˜ k 2 6 −, z + ++ 2gk k k=1 2 k=1 n

6 −,Vn + +:

(77)

Let - := +=, ¿ 0; then (73) satis0es 0 6 Vn (t) ¡ - + (Vn (0) − -) exp(−,t):

(78)

Therefore zi and Wˆ i are uniformly ultimately bounded. Since z1 = x1 − xd1 ; zi = xi − i−1 and xd is bounded; it can be seen that xi is bounded. Using (71); it is concluded that control u is also bounded. Thus; all the signals in the closed-loop system remain bounded. Combining Eq. (73) and inequality (78) yields n 1 k=1

2gk

zk2 ¡ - + (Vn (0) − -) exp(−,t) ¡ - + Vn (0) exp(−,t):

(79)

Let g∗ = max16i6n {gi1 }. Then, the following inequality holds n 1

2g∗ k=1

zk2 6

n 1 k=1

2gk

zk2 ¡ - + Vn (0) exp(−,t)

(80)

n k=1

zk2 ¡ 2g∗ - + 2g∗ Vn (0) exp(−,t);

 which implies that given  ¿ 2g∗ -, there exists T such that for all t ¿ T , the tracking error satis0es |z1 (t)| = |x1 (t) − xd1 (t)| = |y(t) − yd (t)| ¡ ;

Remark 2. In general; implicit functions of the desired virtual controls are very di/cult to be solved; even in the case when the system nonlinearities fi (xPi ; xi+1 ) (i = 1; : : : ; n − 2) are known functions. With the help of NN; there is no need to solve the implicit functions for the explicit virtual controls and the practical controller to cancel the unknown functions in each backstepping design procedure. Remark 3. The adaptive NN controller (71) with adaptation laws (21); (37); (51) and (75) are highly structural; and independent of the complexities of the system nonlinearities. Such a structural property is particularly suitable for parallel processing and hardware implementation in practical applications. Simulation studies have been conducted to verify the e
(81)

(82)

where  is the size of a small residual set which depends on the NN approximation error !i and controller parameters ci , )i and (i . It is easily seen that the increase in the control gain ci , adaptive gain (i and NN node number li will result in a better tracking performance.

z1 = x1 − xdj ;

that is,

679

1 = −c1 z1 + Wˆ 1 S1 (Z1 ); T

i = −zi−1 − ci zi + Wˆ i Si (Zi );

(83)

680

S.S. Ge, C. Wang / Automatica 38 (2002) 671–682

Z1 = [x1 ; x˙d1 ; z1 ]T ∈ 1 ⊂ R3 ;  T @ i−1 @ i−1 Zi = xPTi ; ;:::; ; *i−1 ; zi ∈ i ⊂ R2i+1 ; @x1 @xi−1 2 6 i 6 n − 2;  T @ n−2 @ n−2 T Zn−1 = xPn−1 ; ;:::; ; *n−2 ∈ n−1 ⊂R2(n−1) ; @x1 @x n−2  T @ n−1 @ n−1 T Zn = xPn ; n−1 ; ;:::; ; *n−1 ∈ n ⊂ R2n+1 ; @x1 @x n−1 *i−1 =

i−1 @ i−1

@xk

k=1

i−1

fk (xPk ) +

@ i−1 ˙ @ i−1 x˙d + Wˆ k ; @xd @Wˆ k k=1

2 6 i 6 n;

(84)

where ci ¿ 0 are design constants, RBF NNs WiT Si (Zi ) are used to approximate the unknown functions in the controller design, with Wˆ i being the estimates to Wi∗ , and the adaptation laws being given by Wˆ˙ i = (i [ − Si (Zi )zi − )i Wˆ i ];

(85)

where (i = (iT ¿ 0, and )i ¿ 0, i = 1; : : : ; n are positive constant design parameters. Theorem 2. Consider the closed-loop system consisting of the plant (3); the reference model (4); the controller (83) and the NN weight updating laws (85). Assume there exists su4ciently large compact sets i such that Zi ∈ i for all t ¿ 0. Then; all signals in the closed-loop system remain bounded; and the output tracking error y(t) − yd (t) converges to a small neighborhood around zero by appropriately choosing design parameters. Proof. From the 0rst to the (n − 1)th step; the proof is carried out along the same lines as that of Theorem 1; thus it is omitted here. In the last step; integral-type Lyapunov function is employed in controller design to avoid the possible singularity problem caused by gn (xPn ) in case the quadratic Lyapunov function candidate is chosen. Step n: The derivative of zn = x n − n−1 is z˙n = fn (xPn ) + gn (xPn )u − ˙n−1 :

(86)

De0ne /n (xPn ) = gPn (xPn )=gn (xPn ), and a smooth scalar function 

Vzn = =

0

zn2

zn

)/n (xPn−1 ; ) + n−1 ) d)

 0

1

0/n (xPn−1 ; 0zn + n−1 ) d0:

(87)

Noting that 1 6 /n (xPn−1 ; 0zn + n−1 ) 6 gPn (xPn−1 ; 0zn + n−1 )=gn0 (Assumption 4), the following inequality holds  z2 1 zn2 6 Vzn 6 n 0gPn (xPn−1 ; 0zn + n−1 ) d0: (88) 2 gn0 0 Therefore, Vzn is positive de0nite and radically unbounded with respective to zn . Choose  zn 1 T )/n (xPn−1 ; ) + n−1 ) d) + W˜ n (n−1 W˜ n Vn = Vn−1 + 2 0 (89) as a Lyapunov function candidate. Its time derivative becomes T V˙ n = V˙ n−1 + zn /n (xPn )z˙n + W˜ n (n−1 Wˆ˙ n  zn  @/n (xPn−1 ; ) + n−1 ) ˙ + ) xPn−1 @xPn−1 0 @/n (xPn−1 ; ) + n−1 ) +

˙n−1 d): (90) @ n−1 Since n−1 is a function of xPn−1 ; xd and Wˆ 1 ; : : : ; Wˆ n−1 ;

˙n−1 is given by

˙n−1 =

n−1 @ n−1 k=1

@xk

fk (xPk+1 ) + *n−1 ;

(91)

where *n−1 = n−1 @ n−1 k=1

@xd

x˙d +

n−1 @ n−1 k=1

@Wˆ k

[(k (−Sk (Zk )zk − )k Wˆ k )] (92)

is computable. Using the fact that  zn @/n (xPn−1 ; ) + n−1 ) )

˙n−1 d) @ n−1 0  zn @/n (xPn−1 ; ) + n−1 ) d) ) = ˙n−1 @) 0   zn /n (xPn−1 ; ) + n−1 ) d) ; = ˙n−1 zn /n (xPn ) − 0

we obtain T V˙ n = V˙ n−1 + zn [gPn (xPn )u + hn (Zn )] + W˜ n (n−1 Wˆ˙ n ;

(93)

where hn (Zn ) = /n (xPn )fn (xPn )  1 @/n (xPn−1 ; 0zn + n−1 ) ˙ + zn 0 xPn−1 d0 @xPn−1 0  1 − ˙n−1 /n (xPn−1 ; 0zn + n−1 ) d0 0

(94)

S.S. Ge, C. Wang / Automatica 38 (2002) 671–682

with



Zn = xPTn ; n−1 ;

@ n−1 @ n−1 @ n−1 ; ;:::; ; *n−1 @x1 @x2 @x n−1

Let

T

+ :=

To stabilize the whole system (z1 ; : : : ; zn ), there exists a desired feedback control 1 [ − zn−1 − cn zn − hn (Zn )]; (95) u∗ = gPn (xPn ) where cn is a positive constant to be speci0ed later. By employing an RBF neural network WnT Sn (Zn ) to approximate −hn (Zn ), u∗ can be expressed 1 u∗ = [ − zn−1 − cn zn + Wn∗ T Sn (Zn ) + !n ]; (96) gPn (xPn ) where Wn∗ denotes the ideal constant weights, and |!n | 6 !n∗ is the approximation error with constant !n∗ ¿ 0. Since Wn∗ is unknown, u∗ cannot be realized in practice. Let Wˆ n be the estimate of Wn∗ , and the controller be chosen as T 1 [ − zn−1 − cn zn + Wˆ n Sn (Zn )]: (97) u= gPn (xPn ) The derivative of Vn becomes V˙ n = V˙ n−1 − zn−1 zn − cn zn2 − zn !n T T + W˜ S (Z )z + W˜ (−1 Wˆ˙ :

(98)

Consider the following adaptation law Wˆ˙ n = W˜˙ n = (n [ − Sn (Zn )zn − )n Wˆ n ];

(99)

n

n

n

n

n

where )n ¿ 0 is a small constant. Let cn =cn0 +cn1 , where cn1 ¿ 0 and  1 , cn0 = 0gPn (xPn−1 ; 0zn + n−1 ) d0 ¿ 0 (100) gn0 0 with , being a positive constant. Then, Eq. (98) becomes T V˙ n = V˙ n−1 − zn−1 zn − cn0 zn2 − cn1 zn2 − zn !n − )n W˜ n Wˆ n : (101)

By using the completion of squares and straightforward derivation similar to those employed in Section 3, the derivative of Vn satis0es V˙ n ¡ −

n−2 kj j=1

+

n−1

n−1

2

+

j=1

4cj1

n )j W˜ j 2

2

j=1

2

+

2

n !j∗ 2

∗ 2 cj0 zj − cn0 zn2 −

n )j Wj∗ 2

n )j W˜ j 2 j=1

j=1

j=1

+

∗ 2 cj0 zj − cn0 zn2 −

j=1

n )j Wj∗ 2 j=1

¡−

gj"

zj2 −

n !j∗ 2 j=1

4cj1

:

2

k=1

∈ n ⊂ R2n+1 :

n n

n )k W ∗ 2 k

+

681

n !k∗ 2 k=1

4ck1

:

∗ ∗ is chosen such that ck0 ¿ ,=2gk0 , i.e., ck0 ¿ ,=2gk0 + If ck0 2 gkd =2gk0 ; k = 1; : : : ; n − 1, where , is a positive constant, and )k and (k are chosen such that )k ¿ ,"max {(k−1 }, k = 1; : : : ; n, then from (102) the following inequality holds n−1 n )k W˜ k 2 ∗ 2 ++ ck0 zk − cn0 zn2 − V˙ n ¡ − 2 k=1 k=1

 n−1 , 2 ,zn2 6− zk − k=1

−

2gk0

gn0

T n ,W˜ k (−1 W˜ k k

k=1

2

1

0

0gPn (xPn−1 ; 0zn + n−1 ) d0

++

6 −,Vn + +:

(103)

Let - := +=, ¿ 0, then (89) satis0es 0 6 Vn (t) ¡ - + (Vn (0) − -) exp(−,t):

(104)

Therefore, following the same procedure in the proof of Theorem 1, it can be concluded that all the signals in the closed-loop system, including xi , Wˆ i (i = 1; : : : ; n) and u remain bounded. Moreover, the output tracking error y(t) − yd (t) converges to a small neighborhood around zero by appropriately choosing design parameters. 5. Conclusion In this paper, direct adaptive NN control schemes are presented for nonlinear pure-feedback systems with unknown nonlinear functions. Implicit function theorem is 0rstly exploited to assert the existence of the continuous desired virtual controls. NN approximators are then used to approximate the continuous desired virtual controls and desired practical control. With mild assumptions on the partial derivatives of the unknown functions, the developed adaptive NN control scheme achieves semi-global uniform ultimate boundedness of all the signals in the closed-loop. Moreover, the output of the system is proven to converge to a small neighborhood of the desired trajectory. The control performance of the closed-loop system is guaranteed by suitably choosing the design parameters. Further research on this topic lies in the exploration of control schemes for pure-feedback system 2 where the (n − 1)th equation is non-a/ne in x n , and=or the nth equation is non-a/ne in u. Acknowledgements

(102)

The authors wish to thank Baozhong Yang for fruitful discussions on implicit function theorem.

682

S.S. Ge, C. Wang / Automatica 38 (2002) 671–682

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S. S. Ge received the B.Sc. degree from Beijing University of Aeronautics and Astronautics (BUAA), Beijing, China, in 1986, and the Ph.D. degree and the Diploma of Imperial College (DIC) from Imperial College of Science, Technology and Medicine, University of London, in 1993. From May 1992 to June 1993, he did his postdoctoral research at Leicester University, England. He has been with the Department of Electrical & Computer Engineering, the National University of Singapore since 1993, and is currently an Associate Professor. He was a visiting sta< in Laboratoire de’Automatique de Grenoble, France in 1996, the University of Melbourne, Australia in 1998–99, and University of Petroleum, China in 2001. He has authored and co-authored over 100 international journal and conference papers, one monograph and co-invented one patent. He served as an Associate Editor on the Conference Editorial Board of the IEEE Control Systems Society in 1998 and 1999, has been serving as an Associate Editor, IEEE Transactions on Control Systems Technology since June 1999, and a Member of the Technical Committee on Intelligent Control of the IEEE Control System Society since 2000. He was the winner of the 1999 National Technology Award, Singapore. He serves as a technical consultant local industry. He is a Senior Member of IEEE. His current research interests are Control of nonlinear systems, Neural Networks and Fuzzy Logic, Real-Time Implementation, Robotics and Arti0cial Intelligence. Cong Wang received the B.E. and M.E. degrees from Department of Automatic Control, Beijing University of Aeronautic & Astronautics, China, in 1989 and 1997, respectively. He has recently 0nished his Ph.D. studies from the Department of Electrical & Computer Engineering, National University of Singapore. He is currently a postdoctoral fellow at the Centre for Chaos Control and Synchronization, City University of Hong Kong. His research interest includes adaptive neural control, neural networks, chaos control and synchronization, and control applications.