Adaptive Output-feedback Regulation for Nonlinear Delayed Systems ...

International Journal of Automation and Computing

05(1), January 2008, 103-108 DOI: 10.1007/s11633-008-0103-2

Adaptive Output-feedback Regulation for Nonlinear Delayed Systems Using Neural Network Wei-Sheng Chen∗

Jun-Min Li

Department of Applied Mathematics, Xidian University, Xi0 an 710071, PRC

Abstract: A novel adaptive neural network (NN) output-feedback regulation algorithm for a class of nonlinear time-varying timedelay systems is proposed. Both the designed observer and controller are independent of time delay. Different from the existing results, where the upper bounding functions of time-delay terms are assumed to be known, we only use an NN to compensate for all unknown upper bounding functions without that assumption. The proposed design method is proved to be able to guarantee semi-global uniform ultimate boundedness of all the signals in the closed system, and the system output is proved to converge to a small neighborhood of the origin. The simulation results verify the effectiveness of the control scheme. Keywords:

1

Adaptive, neural network (NN), output-feedback, nonlinear time-delay systems, backstepping.

Introduction

Time delay is often found in various engineering systems, such as electrical networks, microwave oscillators, nuclear reactors, etc[1] . It is a source of instability and a cause of poor performance. Therefore, the controller design and stability analysis for time-delay nonlinear systems have attracted quite a number of researchers over the past years, and some interesting results have been obtained[2−9] . References [2, 3] tried to solve the problem of robust stabilization in strict-feedback time-delay (stochastic) nonlinear systems by constructively using the appropriate LyapunovKrasovskii function. However, the main results were shown to be wrong[4, 5] . Recently, the aforementioned problem was partially solved by recursively constructing the LyapunovRazumikhin function[6] . In [7-9], the stabilization problem of output-feedback time-delay (stochastic) nonlinear systems was solved by constructing a proper LyapunovKrasovskii function. However, in these results[2, 3, 6−9] , the time-delay nonlinear functions were assumed to be bounded by known functions. On the other hand, the adaptive neural network control (ANNC) method has been successfully applied to some classes of unknown nonlinear systems such as strictfeedback systems[10−15] , output-feedback systems[16−17] , etc. Because of their inherent approximation capabilities, neural networks (NNs) are often used as nonlinear approximators for unknown nonlinearities in ANNC. In contrast to the earlier NN control schemes[18−19] where optimization techniques were used to derive the weight adaptive laws, the main advantage of ANNC lies in the fact that the weight adaptive laws are obtained with Lyapunov synthesis method and therefore the stability of the closed-loop system is guaranteed. Recently, ANNC has been extended to nonlinear time-delay systems[20−22] where the time delay or the upper bound of time delay is assumed to be known. To the best of our knowledge, in the literature, the outputManuscript received September 1, 2006; revised February 10, 2007. This work was supported by National Natural Science Foundation of China (NSFC) (No. 60374015). *Corresponding author. E-mail address: [email protected]

feedback ANNC for nonlinear systems with completely unknown time delay has never been considered when the system states are unmeasurable. When the system states are unavailable, applying the ANNC method to the output-feedback control of nonlinear systems with unknown time delay is a challenging subject. In this paper, we try to solve this open problem. The main contributions are as follows. First, only an NN is employed to approximate all upper bounding functions of the system, so that the requirement that the upper bounding functions are assumed to be known[9] is removed and the controller design is simplified. Second, by successfully constructing a Lyapunov-Krasovskii function, the semi-global uniform ultimate boundedness of all the signals in the closed-loop system is guaranteed, and the system output is proved to converge to a small neighborhood of the origin. Finally, the assumption on time delay is relaxed, and time delays are assumed to be time-varying and unknown in this paper. The rest of this paper is organized as follows. The problem description and preliminaries are given in Section 2. An observer and an adaptive NN controller are designed in Section 3. A simulation example is studied to illustrate the feasibility of the proposed control scheme in Section 4. In Section 5, we conclude the work of this paper.

2

System description and preliminaries

Let us consider a class of nonlinear time-delay systems with structure  x˙ 1 = x2 + f1 (¯ x1 ) + h1 (t, y, y(t − τ1 (t)))     ..    . x˙ n−1     x˙ n    y

= = =

xn + fn−1 (¯ xn−1 ) + hn−1 (t, y, y(t − τn−1 (t))) u + fn (¯ xn ) + hn (t, y, y(t − τn (t))) x1 (1)

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International Journal of Automation and Computing 05(1), January 2008

where xi ∈ R, u ∈ R, and y ∈ R represent the state, control input and output of the system, respectively; x ¯i is defined as x ¯i = [x1 , · · · , xi ]T ∈ Ri ; fi (·) : Ri → R is a known smooth nonlinear function with fi (0) = 0, but hi (·) : R3 → R is an unknown smooth nonlinear function with hi (t, 0, 0) = 0; τi (t) is the time-varying time delay of output y, and only the output y is available. In this paper, k · k denotes the 2-norm, and λmax (B) and λmin (B) denote the largest and smallest eigenvalues of a positive definite matrix B, respectively. Remark 1. There are many systems governed by the differential equation with the form (1) such as cold rolling mills[1] , and the two-stage chemical reactor with delayed recycle streams[9] . Some interesting results have been obtained under some assumptions on the nonlinear time-delay function hi (t, y, y(t − τi (t))). For example, hi (t, y, y(t − τi (t))) is bounded by known functions[9] . However, the output-feedback ANNC for system (1) has not been solved in the literature, when the upper bounding functions are completely unknown. The control objective of this paper is to design an outputfeedback ANNC law u(t) for system (1) such that the closed-loop system is semi-globally asymptotically stable while keeping all signals uniformly bounded. In this paper, an unknown smooth nonlinear function G(y) : R→ R will be approximated on a compact set Ω by the following radial basis function (RBF) NN[10] : G(y) = W T S(y) + δ(y)

(2)

where S(y) = [s1 (y), · · · , sl (y)]T : Ω → Rl , is a known smooth vector function with the NN node number l > 1. The basis function si (y) (1 ≤ i ≤ l) is chosen as the commonly used Gaussian function with the form si (y) = exp[−(y − µi )2 /η 2 ], where µi ∈ Ω and η > 0 are the center and the width of the basis function si (y), respectively. The optimal weight vector W = [w1 , · · · , wl ]T is defined as ½ ¯ ¯¾ ¯ ˆ T S(y)¯¯ W = arg min sup ¯G(y) − W (3) ˆ ∈Rl W

y∈Ω

and δ(y) is the NN inherent approximation error, which can be arbitrarily decreased by increasing NN node number l. We make the following assumptions on the approximation error δ(y) and system (1). Assumption 1. On the compact set Ω, the inherent approximation error δ(y) is assumed to be bounded with |δ(y)| ≤ θ, where the unknown parameter θ denotes the smallest upper bound of |δ(y)| with θ ≥ 0. Assumption 2. fi (·) (i = 1, · · · , n) satisfies Lipschitz condition. That is, there exist known positive constants bi,j such that the following inequality holds. ¯ kfi (ϑ¯i ) − fi (ϑˆi )k ≤

i X

bi,j kϑj − ϑˆj k

(4)

j=1

¯ where ϑ¯i = [ϑ1 , · · · , ϑi ]T ∈ Ri and ϑˆi = [ϑˆ1 , · · · , ϑˆi ]T ∈ Ri . Assumption 3. hi (t, y, y(t − τi (t))) satisfies the following inequality: |hi (t, y, y(t − τi (t)))| ≤ |y|hi,1 (y) + |y(t − τi (t))| · hi,2 (y(t − τi (t)))

(5)

where hi,1 (y) and hi,2 (y(t − τi (t))) are completely unknown nonlinear functions. Assumption 4. Time delay τi (t) (i = 1, · · · , n) satisfies the following inequality: τ˙i (t) ≤ ηi < 1.

(6)

Remark 2. Assumption 1 determines that the result obtained in this paper is semi-global. In a way, it is valid so long as the neural input vector remains in Ω, where Ω and θ can be arbitrarily large. In the special case where Assumption 1 holds for all y ∈ R, the stability result obtained in this paper becomes global. Remark 3. Assumption 2 that comes from [9], and (4) make it possible to design the observer for system (1) by LMI method, which will be addressed later. Remark 4. Assumption 3 is different from that of [9], where the upper bounding functions hi,1 (y) and hi,2 (y(t − τi (t))) are assumed to be known. However, this assumption is not required in this paper and NN is used to compensate for these unknown upper bounding functions. Remark 5. Assumption 4 is also from [9]; it will be used to construct a Lyapunov-Krasovskii function. By (6), the following inequality holds: −

1 − τ˙i (t) ≤ −1, 1 − ηi

i = 1, · · · , n.

(7)

This nice property could be used to deal with the derivative of the Lyapunov-Krasovskii function, as will be seen in (23) later.

3 3.1

Observer and ANNC design Observer design

For the observer design, we compactly rewrite system (1) as ( ¯ + F (x) + H(t, y, y¯(t − τn (t))) + en u x˙ = Ax (8) y = x1 where x = [x1 , · · · , xn ]T is the state vector, y¯(t − τn (t)) = [y(t − τ1 (t)), · · · , y(t − τn (t))]T , en = [0, · · · , 0, 1]T ∈ Rn ,     0 f1 (¯ x1 )  .    .. , , A¯ =  F (x) =  .  .. In−1    0 ··· 0 fn (¯ xn )   h1 (t, y, y(t − τ1 (t)))   ..  H(t, y, y¯(t − τn (t))) =  .   hn (t, y, y(t − τn (t))) where I is a unit matrix. The observer for system (8) is designed as x ˆ˙ = Aˆ x + F (ˆ x) + ky + en u (9) where x ˆ = [ˆ x1 , · · · , x ˆn ]T is the estimate of state vecT ¯ tor x, A = A − ke1 with k = [k1 , · · · , kn ]T and e1 = ¯ ¯ [1, 0, · · · , 0]T ∈ Rn , F (ˆ x) = [f1 (x ˆ1 ), · · · , fn (x ˆn )]T with ¯ x ˆi = [ˆ x1 , · · · , x ˆi ]T . To guarantee the stability of the closedloop system, the vector k is chosen to validate the following inequality: AT P + P A + β −1 P 2 + βρ2 I < −Q

(10)

W. S. Chen and J. M. Li/ Adaptive Output-feedback Regulation for Nonlinear Delayed Systems Using Neural Network

in which scalar ρ is defined in (12) below, matrices P and Q are positive definite matrices, and β is a positive scalar. Remark 6. Inequality (10) is equivalent to the following LMI based on Schur compliment lemma[9] : "

#

P A¯ + M B + B T M T + A¯T + βρ2 I + Q P

P −βI

0 and Γ > 0 are the adaptive gains, −σ θˆ and ˆ are leakage terms with the modified coefficient σ > 0. −σ W Remark 8. The leakage terms in the parameter adaptive laws were used in some papers on ANNC[11, 14, 15, 20−22] . The introduction of the leakage terms in (20) is to prevent adaptive laws with high gains. The size of modified coefficient σ will decide the size of the regulating error and the approximation domain of NN. Substituting (17)–(19) into (16), we have the error sys-

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International Journal of Automation and Computing 05(1), January 2008

Rn , π > 0, every underlined term in (23) satisfies

tem   y˙            z˙i   

=

(I) : 2εT P (F (x) − F (ˆ x)) ≤

h1 (t, y, y(t − τ1 (t)))

λ + 2ζ ³ ∂αi−1 ´2 zi − −zi−1 − ci zi + zi+1 − 2 ∂y ³ ´ ∂αi−1 ε2 + h1 (t, y, y(t − τ1 (t))) ∂y λ + 2ζ ³ ∂αn−1 ´2 −zn−1 − cn zn − zn − 2 ∂y ´ ∂αn−1 ³ ε2 + h1 (t, y, y(t − τ1 (t))) . ∂y (21)

=

       z˙n        

3.3

³ ´ ˆ T S(y) + θˆ + ε2 + −c1 y + z2 − y W

=

Stability analysis and main results

Consider the following Lyapunov-Krasovskii function: n 1 2 1 X 2 1 ˜ T −1 ˜ 1 y + zi + W Γ W + γ −1 θ˜2 + 2 2 i=2 2 2 Z t n y 2 (Θ)h21,2 (y(Θ))dΘ+ 2ζ(1 − η1 ) t−τ1 (t) Z t n X 2λ y 2 (Θ)h2j,2 (y(Θ))dΘ. (22) 1 − η j t−τj (t) j=1

V = εT P ε +

With (11) and (21), the derivative of V is given by V˙ ≤ εT (P A + AT P )ε + 2εT P (F (x) − F (ˆ x)) + | {z } (I) T

2

2ε P H(t, y, y¯(t − τn (t))) −c1 y + yz2 − | {z } (II)

ˆ + y(ε2 + h1 (t, y, y(t − τ1 (t)))) + ˆ T S(y) + θ] y 2 [W | {z } λ + 2ζ ³ ∂αi−1 ´2 2 − zi−1 zi − + zi zi+1 − zi − 2 ∂y i=2 ´i ∂αi−1 ³ ε2 + h1 (t, y, y(t − τ1 (t))) − zi ∂y | {z } ci zi2

λ + 2ζ ³ ∂αn−1 ´2 2 + zn − 2 ∂y ³ ´ ∂αn−1 zn ε2 + h1 (t, y, y(t − τ1 (t))) − ∂y | {z }

zn−1 zn − cn zn2 −

T

(VI) n X n 2λ 2 y 2 (t)h21,2 (y(t)) + y (t)h2j,2 (y(t))− 2ζ(1 − η1 ) 1 − ηj j=1

n 2 y (t − τ1 (t))h21,2 (y(t − τ1 (t)))− 2ζ n X 2λy 2 (t − τj (t))h2j,2 (y(t − τj (t))).

2λ

n X

y 2 (t − τj (t))h2j,2 (y(t − τj (t)))

(25)

j=1

(III) : y(ε2 + h1 (t, y, y(t − τ1 (t)))) ≤ λ + 2ζ 2 1 2 2 1 T ε ε+ y + y h1,1 (y)+ 2λ 2 2ζ 1 2 y (t − τ1 (t))h21,2 (y(t − τ1 (t))) (26) 2ζ ³ ´ ∂αi−1 (IV) : − zi ε2 + h1 (t, y, y(t − τ1 (t))) ≤ ∂y 1 2 2 1 T λ + 2ζ ³ ∂αi−1 ´2 2 ε ε+ zi + y h1,1 (y)+ 2λ 2 ∂y 2ζ 1 2 y (t − τ1 (t))h21,2 (y(t − τ1 (t))) (27) 2ζ ³ ´ ∂αn−1 ε2 + h1 (t, y, y(t − τ1 (t))) ≤ (V) : − zn ∂y 1 T λ + 2ζ ³ ∂αn−1 ´2 2 1 2 2 ε ε+ zn + y h1,1 (y)+ 2λ 2 ∂y 2ζ 1 2 y (t − τ1 (t))h21,2 (y(t − τ1 (t))) (28) 2ζ ˆ ≤ − σ (W ˜ TW ˆ + θ˜θ) ˜ TW ˜ + θ˜2 )+ (VI) : σ(W 2 σ (W T W + θ2 ). (29) 2 ³ 1 n ´ 1 I ε− V˙ ≤ εT P A + AT P + P P + βρ2 I + P P + β λ 2λ n ³ ´ X ci zi2 − y 2 W T S(y) + θ + i=1

³ λ + 2ζ 2

+

n X n h2j,1 (y)+ h21,2 (y) + 2λ 2ζ(1 − η1 ) j=1

σ ˜T˜ σ (W W + θ˜2 ) + (W T W + θ2 ). 2 2

˜ + σ(W ˆ + ˜ S(y)y − θy ˜ W ˆ + θ˜θ) W | {z } 2

n X 1 T ε P P ε + 2λy 2 h2j,1 (y)+ λ j=1

n ´ X n 2 2λ h1,1 (y) + h2j,2 (y) − 2ζ 1 − ηj j=1

(V) 2

(II) : 2εT P H(t, y, y¯(t − τn (t))) ≤

y2

(IV)

T

(24)

Substituting (24)-(29) into (23), we have

(III) n−1 Xh

1 T ε P P ε + βρ2 εT ε β

(30)

The underlined function in (30) is denoted by G(y), which can be approximated on a compact set Ω G(y) = W T S(y) + δ(y)

(31)

with the approximation error |δ(y)| ≤ θ. Substituting (10) and (31) into (30), we have (23)

j=1

We employ (7) in the last two terms. By using (12) and 1 T b b, a, b ∈ (13) and Young0 s inequality aT b ≤ π2 aT a + 2π

¶ n X n 1 PP + I ε− ci zi2 − λ 2λ i=1 σ ˜T˜ σ (W W + θ˜2 ) + (W T W + θ2 ). 2 2 µ

V˙ ≤ εT

−Q+

(32)

W. S. Chen and J. M. Li/ Adaptive Output-feedback Regulation for Nonlinear Delayed Systems Using Neural Network

For given 0 < ν < 1, the parameter λ is selected to be sufficiently large to make the following inequality hold: −Q +

1 1 PP + nI ≤ −νI. λ 2λ

(33)

Then, we have V˙ ≤ − νεT ε −

n X

ci zi2 −

i=1

σ (W T W + θ2 ). 2

(34)

The main results are stated as follows. Theorem 1. Under Assumptions 1–4, consider the closed-loop system consisting of system (1), control law (20) and adaptive laws (9). Assume that there exist a sufficiently large compact set Ω ⊂ R, such that y ∈ Ω for all t ≥ 0. Then, for bounded initial conditions, there are the following properties. ˆ x ˆ , θ, 1) All signals ε, y, z2 , · · · , zn , W ˆ, x and u in the closed-loop system are semi-globally uniformly ultimately bounded and eventually converge to the compact set D, which is specified as n ¯ n X ¯ ˆx ˆ , θ, D = ε, y, z2 , · · · , zn , W ˆ, x, u ¯ νεT ε + ci zi2 +

o σ ˜T˜ σ (W W + θ˜2 ) ≤ (W T W + θ2 ) . 2 2

Remark 9. The results obtained in Theorem 1 show that the size of the compact set D, the output y and observer error ε depend on the design parameter σ. Especially, if the design parameter σ is chosen as σ = 0, we have limt→∞ y(t) = 0, and limt→∞ ε(t) = 0.

4

σ ˜T˜ (W W + θ˜2 )+ 2

i=1

(35)

2) The system output y(t) and the observer error vector ε(t) satisfy Z 1 t 2 σ lim (W T W + θ2 ) (36) y (Θ)dΘ ≤ t→∞ t 0 2c1 Z t 1 σ lim (W T W + θ2 ). (37) εT (Θ)ε(Θ)dΘ ≤ t→∞ t 0 2ν ˜ u) is ˜ , θ, Proof. From (34), when (ε, y, z2 , · · · , zn ,P W 2 outside the compact set Ω, that is, νεT ε + n i=1 ci zi + 2 T 2 σ ˜ T ˜ σ ˜ ˙ (W W + θ ) > 2 (W W + θ ), we know that V is 2 negative-semidefinite. In the light of the Lyapunov stability theory to the retarded functional differential equation[23] , ˜ and θ˜ are (34) implies that the signals ε, y, z2 , · · · , zn , W semi-globally uniformly ultimately bounded and eventually converge to the compact set D, implying that the signal ˆ and θˆ are bounded. The boundedness of other signals W is proved as follows. Since y and ε1 are bounded, the signal x ˆ1 = y − ε1 is bounded. By (17), the boundedness of ˆ and θ implies that α1 is bounded. From (15), we have y, W ¯ ˆW ˆ ). The boundedness of z2 and α1 x ˆi = zi +αi−1 (y, x ˆi−1 , θ, ¯ ˆW ˆ ) is also implies the boundedness of x ˆ2 . Then α2 (y, x ˆ2 , θ, bounded. Continuing in the same fashion, we obtain the boundedness of x ˆ and u. In the end, from the boundedness of ε and x ˆ, we obtain the boundedness of x. Integrating (34) from 0 to t, we have Z t Z 1 t ˙ 1 σ y 2 (Θ)dΘ ≤ − c1 V (Θ)dΘ + (W T W + θ2 ) t t 2 0 0 1 σ ≤ − V (0) + (W T W + θ2 ). (38) t 2 Let t → ∞. Then (36) can be obtained. In the same fashion, we can obtain (37). ¤

107

Application and simulation

In this section, we consider a practical example of a twostage chemical reactor with delayed recycle streams[9]  1 1 − R2   x˙ 1 (t) = − x1 (t) − K1 x1 (t) + x2 (t)+   Θ1 V1     δ1 (t, x1 (t − τ1 (t)))   1 R1 x˙ 2 (t) = − x2 (t) − K2 x2 (t) + x1 (t − τ2 (t))+  Θ V2 2   F  2  u(t) + δ2 (x1 (t, t − τ2 (t)))    V2   y(t) = x1 (t) (39) where x1 (t) and x2 (t) are the compositions, R1 = 0.5 and R2 = 0.5 are the recycle flow rates, Θ1 = 2 and Θ2 = 2 are the reactor residence times, K1 = 0.3 and K2 = 0.3 are the reaction constants, F2 = 0.5 is the feed rate, V1 = 0.5 and V2 = 0.5 are reactor volumes, δ1 (x1 (t − τ1 (t))) = 6.5(sin t)x21 (t−τ1 (t)) and δ2 (x1 (t−τ2 (t))) = 8.5(sin t)x31 (t− τ2 (t)) are uncertain nonlinear functions with time delay τ1 (t) = 0.8(1 + sin t) and τ2 (t) = 0.8(1 + cos t). Clearly, the aforementioned system satisfies Assumptions 1–4. By the ANNC method proposed in this paper, we design the following observer: ( x ˆ˙ 1 (t) = x ˆ2 (t) − 0.8x1 (t) + k1 (x1 (t) − x ˆ1 (t)) (40) x ˆ˙ 2 (t) = u(t) − 0.8ˆ x2 (t) + k2 (x1 (t) − x ˆ1 (t)). By the LMI in Remark 6 with ρ = 0.8 and Q = I, and the LMI tool box in Matlab, we can obtain # " # " # " 7.1206 351.0482 −149.1343 k1 = . P = , k2 13.1213 −149.1343 88.4575 We design the following control law and the stabilizing function λ + 2ζ ³ ∂α1 ´2 2 u = − y − c2 z2 − z2 + 0.8ˆ x2 − 2 ∂y ∂α1 ∂α1 ˆ˙ ∂α1 ˆ˙ k2 (y − x ˆ1 ) + (ˆ x2 − 0.8y) + θ+ W (41) ˆ ∂y ∂W ∂ θˆ ˆ + 0.8y1 ˆ T S(y) + θ) α1 = − c1 y − y(W (42) where S(y) = [s1 (y), · · · , sl (y)]T , the design parameters are c1 = c2 = 0.2 and λ = ζ = 10, the coordinate transformation is z2 = x ˆ2 − α1 , the basis function is si (y) = exp[−(y − µi )2 /η 2 ] with the node number l = 11, width η = 1 and center µi evenly spaced in [−2, 2], the parameter adaptive laws are designed as ˙ ˆ θˆ = γ(y 2 − σ θ),

ˆ˙ = Γ(y 2 S(y) − σ W ˆ) W

(43)

with the adaptive gains γ = 2.5, Γ = 2.5I and σ = 0.05. In our simulation, the initial values are chosen as x1 (0) = 0.5, x2 (0) = −0.5. The simulation results are shown in Fig. 1, from which we can see that the controller renders the resulting closed-loop system asymptotically stable and all signals of the closed-loop system are bounded.

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International Journal of Automation and Computing 05(1), January 2008

Fig. 1

5

Simulation results of the closed-loop system

Conclusions

In this paper, the ANNC method is extended to a class of unknown nonlinear time-delay systems. Only an NN is used to compensate for all unknown upper bounding functions, and thus the assumption on the nonlinear time delay terms is relaxed. The semi-global asymptotical stability of the closed-loop system is guaranteed. One of the future tasks is to extend this result to the setting of tracking control.

References [1] S. L. Niculescu. Delay Effects on Stability: A Robust Control Approach, Springer-Verlay, London, UK, 2001. [2] S. K. Nguang. Robust Stabilization of a Class of Time-delay Nonlinear Systems. IEEE Transactions on Automatic Control, vol. 45, no. 4, pp. 756–762, 2000. [3] Y. S. Fu, Z. H. Tian, S. J. Shi. State Feedback Stabilization for a Class of Stochastic Time-delay Nonlinear Systems. IEEE Transactions on Automatic Control, vol. 48, no. 2, pp. 282–286, 2003. [4] S. S. Zhou, G. Feng, S. K. Nguang. Comments on “Robust Stabilization of a Class of Time-delay Nonlinear Systems”. IEEE Transactions on Automatic Control, vol. 47, no. 9, pp. 1586–1586, 2002. [5] C. C. Hua, X. P. Guan. Comments on “State Feedback Stabilization for a Class of Stochastic Time-delay Nonlinear Systems”. IEEE Transactions on Automatic Control, vol. 49, no. 7, pp. 1216–1216, 2004. [6] X. H. Jiao, T. L. Shen. Adaptive Feedback Control of Nonlinear Time-delay Dystems: The LaSalle-Razumikhinbased Approach. IEEE Transactions on Automatic Control, vol. 50, no. 11, pp. 1909–1913, 2005. [7] Y. S. Fu, Z. H. Tian, S. J. Shi. Output Feedback Stabilization for Time-delay Nonlinear Systems. Acta Automatica Sinica, vol. 28, no. 5, pp. 802–805, 2002. (in Chinese) [8] Y. S. Fu, Z. H. Tian, S. J. Shi. Output Feedback Stabilization for a Class of Stochastic Time-delay Nonlinear Systems. IEEE Transactions on Automatica Control, vol. 50, no. 6, pp. 847–851, 2005. [9] C. C. Hua, X. P Guan, P. Shi. Robust Backstepping Control for a Class of Time Delay Systems. IEEE Transactionsons Automatica Control, vol. 50, no. 6, pp. 894–899, 2005. [10] M. M. Polycarpou, M. J. Mears. Stable Adaptive Tracking of Uncertain Systems Using Nonlinearly Parameterized Online Approximators. International Journal of Control, vol. 70, no. 3, pp. 363–384, 1998. [11] T. Zhang, S. S. Ge, C. C. Huang. Adaptive Neural Network Control for Strict-feedback Nonlinear Systems Using Backstepping Design. Automatica, vol. 36, no. 12, pp. 1835– 1846, 2000.

[12] Y. H. Li. Robust and Adaptive Backstepping Control for Nonlinear Systems Using RBF Neural Networks. IEEE Transactions on Neural Networks, vol. 15, pp. 3, pp. 693– 701, 2004. [13] Y. S. Yang, C. J. Zhou. Adaptive Fuzzy H∞ Stabilization for Strict-feedback Canonical Nonlinear Systems via Backstepping and Small-gain Approach. IEEE Transactions on Fuzzy Systems, vol. 13, no. 1, pp. 104–114, 2005. [14] D. Wang, J. Huang. Neural Network-based Adaptive Dynamic Surface Control for a Class of Uncertain Nonlinear Sytems in Strict-feedback Form. IEEE Transactions on Neural Networks, vol. 16, no. 1, pp. 195–202, 2005. [15] H. B. Du, H. H. Shao, P. J. Yao. Adaptive Neural Network Control for a Class of Low-triangular-structured Nonlinear Systems. IEEE Transactions on Neural Networks, vol. 17, no. 2, pp. 509–514, 2006. [16] J. Y. Choi, J. A. Farrell. Adaptive Observer Backstepping Control Using Neural Networks. IEEE Transactions on Neural Networks, vol. 12, no. 5, pp. 1103–1113, 2001. [17] J. Stoev, J. Y. Choi, J. Farrell. Adaptive Control for Output Feedback Nonlinear Systems in the Presence of Modeling Errors. Automatica, vol. 38, no. 10, pp. 1761–1767, 2002. [18] K. Funahashi. On the Approximate Realization of Continuous Mappings by Neural Networks. Neural Networks, vol. 2, no. 1, pp. 183–192, 1989. [19] K. S. Narendra, K. Parthasarathy. Identification and Control of Dynamic Systems Using Neural Networks. IEEE Transactions on Neural Networks, vol. 1, no. 1, pp. 4–27, 1990. [20] H. Hong, S. S. Ge, T. H. Lee. Practical Adaptive Neural Control of Nonlinear Systems with Unknown Time Delay. IEEE Transactions on Systems, Man, and Cybernetics – Part B, vol. 35, no. 4, pp. 849–854, 2005. [21] W. S. Chen, J. M. Li. Adaptive Output Feedback Control for Nonlinear Time-delay Systems Using Neural Networks. Journal of Control Theory and Application, vol. 4, no. 4, pp. 313–320, 2006. [22] W. S. Chen, J. M. Li. Adaptive Neural Tracking Control for Unknown Output Feedback Nonlinear Time Delay Systems. Acta Automatica Sinica, vol. 31, no. 5, pp. 799–803, 2005. (in Chinese) [23] J. K. Hale, S. M. Lunel. Introduction to Functional Differential Equations, Springer-Verlay, New York, USA, 1993.

Wei-Sheng Chen received the B. Sc. degree in the Department of Mathematics at Qufu Normal University, Qufu, China, in 2000, and the M. Sc. degree in the Department of Applied Mathematics at Xidian University, Xi0 an, China, in 2004. He is currently a Ph. D. candidate in the Department of Applied Mathematics at Xidian University, China. His research interests include robust and adaptive control, neural network control, nonlinear control, and time-delay control systems. Jun-Min Li received the B. Sc. and M. Sc. degrees from the Department of Applied Mathematics at Xidian University, China, in 1987 and 1989, respectively, and Ph. D. degree in systems engineering from Xi0 an Jiaotong University, Xi0 an, China, in 1997. He is currently a professor in the Department of Applied Mathematics at Xidian University, China. His research interests include robust and adaptive control, optimal control, iterative learning control, and networked control systems.