Automatica 45 (2009) 1554–1560

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Automatica

journal homepage: www.elsevier.com/locate/automatica

Brief paper

Novel adaptive neural control design for nonlinear MIMO time-delay systemsI

Bing Chen a,∗, Xiaoping Liu b, Kefu Liu b, Chong Lin aa Institute of Complexity Science, Qingdao University, Qingdao, 266071, PR Chinab Faculty of Engineering, Lakehead University, Thunder Bay, On, P7B 5E1, Canada

a r t i c l e i n f o

Article history:Received 2 April 2008Received in revised form24 October 2008Accepted 22 February 2009Available online 25 March 2009

Keywords:Nonlinear systems with time-delaysBacksteppingAdaptive neural controlOutput tracking

a b s t r a c t

In this paper, we address the problem of adaptive neural control for a class of multi-input multi-output(MIMO) nonlinear time-delay systems in block-triangular form. Based on a neural network (NN) onlineapproximation model, a novel adaptive neural controller is obtained by constructing a novel quadratic-type Lyapunov–Krasovskii functional, which not only efficiently avoids the controller singularity, but alsorelaxes the restriction on unknown virtual control coefficients. The merit of the suggested controllerdesign scheme is that the number of online adapted parameters is independent of the number of nodesof the neural networks, which reduces the number of the online adaptive learning laws considerably. Theproposed controller guarantees that all closed-loop signals remain bounded, while the output trackingerror dynamics converges to a neighborhood of the origin. A simulation example is given to illustrate thedesign procedure and performance of the proposed method.

© 2009 Elsevier Ltd. All rights reserved.

1. Introduction

During the past two decades, neural network method hasattracted considerable attention because of its inherent capabilityfor modelling and controlling highly uncertain, nonlinear andcomplex systems. In Zhang, Xie, Wang, and Zheng (2007), neuralnetwork method was applied to solve chaotic synchronizationproblem. Theworks in Psillakis and Alexandridis (2007) and Zhangand Wang (2008) concerned stochastic control using neuralnetworks. Nonlinear adaptive neural control was consideredin Chen and Narendra (2001) and Yang and Calise (2007) forcontinuous nonlinear systems, and Zhu and Guo (2004) fornonlinear discrete-time systems. In Xu and Tan (2007), waveletnetwork-based adaptive control approach was developed. Lan andHuang (2007) proposed an approximate output regulation schemefor discrete-time nonlinear systems based on neural networks. Theproblems of tracking and stabilization for nonlinear systems vianeural network method with backstepping have been extensivelystudied, and some significant results on these control issues havebeen reported (Farrell (1997), Gao and Selmic (2006), Ge, Hang,and Zhang (2000), Ge, Hong, and Lee (2004), Ge and Wang (2004),

I The material in this paper was not presented at any conference. This paper wasrecommended for publication in revised form by Associate Editor Shuzhi Sam Geunder the direction of Editor Miroslav Krstic.∗ Corresponding author. Tel.: +86 532 85953607; fax: +86 532 85953672.E-mail addresses: chenbing1958@yahoo.com.cn (B. Chen),

xiaoping.liu@lakeheadu.ca (X. Liu), kfiu@lakeheadu.ca (K. Liu),linchong_2004@hotmail.com (C. Lin).

0005-1098/$ – see front matter© 2009 Elsevier Ltd. All rights reserved.doi:10.1016/j.automatica.2009.02.021

Ho, Zhang, and Xu (2001), Narendra and Lewis (2001), Niu, Lam,andWang (2005), Psillakis and Alexandridis (2007), Vapnik (2000),Wang and Hill (2006), Zhang and Huang (2000), Zhang, Ge, andHuang (1999), Zhou, Lam, Feng, and Daniel (2005)).Recently, in view of possible time-delays in practical systems,

approximation-based adaptive neural network control has beenalso addressed in Ge, Hong, and Lee (2003) for nonlinear SISOtime-delay systems with constant virtual control coefficients. Anadaptive neural controller has been developed to guarantee thatall the closed-loop signals remain bounded, meanwhile outputtracking is achieved. Such a result, with the use of Nussbaum-type functions, has been extended to the case of unknown virtualcontrol coefficients in Ge et al. (2004). In Ho, Li, and Niu (2005),based on a wavelet neural network online approximation model,a state feedback adaptive controller is proposed by constructingan integral-type Lyapunov–Krasovskii functional. More recently,in Ge and Tee (2007), neural control has been addressed forMIMO nonlinear time-delay systems in block-triangular form. Thesuggested adaptive neural controller guarantees that the trackingerrors converge to a small neighborhood of the origin, and at thesame time, all other signals in the closed-loop systemare bounded.Though various approximation-based neural control design

methods have been proposed for delay-free systems or delayedsystems, there are still some issues which need to be furtheraddressed. When a neural network is used as an approximator, toget the sufficient approximation accuracy, a large number of NNnodes should be adopted. As a result, a great number of adaptationparameters are required to be adapted online simultaneously.This makes the learning time become unacceptably large. Inaddition, in the above-mentioned adaptive neural control design,

B. Chen et al. / Automatica 45 (2009) 1554–1560 1555

even though an integral-type Lyapunov function has been usedto successfully avoid the controller singularity problem, it hasthe following drawbacks: The upper bounds for unknown virtualcontrol coefficients must be some known functions and realcontrol coefficients should be independent of some specified statevariables and control inputs. This may limit the applicability of theapproach to certain practical systems.The above observation motivates the research in this paper. A

novel adaptive neural control design procedure is proposed forMIMO nonlinear time-delay systems in block-triangular form. Itis shown that the proposed neural adaptive control scheme notonly avoids the controller singularity problem, but also reducesthe number of adaptation parameters considerably. As a matter offact, to control anm input system, there are onlym adaptive laws.In addition, by adopting a new Lyapunov functional, the existingrestrictions on the virtual or real control coefficients are removed.The proposed control scheme guarantees the boundedness of allthe signals in the closed-loop system, at the same time outputtracking is achieved.

2. Problem formulation and preliminaries

Consider the following MIMO nonlinear time-delay systemwith block-triangular structure.

xj,ij = fj,ij(xj,ij)+ gj,ij(xj,ij)xj,ij+1 + hj,ij(xτj,ij ),

xj,mj = fj,mj(X, uj−1)+ gj,mj(X, uj−1)uj + hj,mj(Xτ ),

yj = xj,1

(1)

for j = 1, 2, . . . , n, ij = 1, 2, . . . ,mj − 1, where xj,ij =[xj,1, . . . , xj,ij ]

T∈ Rij is the state vector for the first ij differential

equations of the jth subsystem, y = [y1, y2, . . . , yn]T ∈ Rn is theoutput, uj = [u1, . . . , uj]T are the inputs for the first j subsystems,fj,ij(.), gj,ij(.) and hj,ij(.) are unknown smooth nonlinear functions,X = [xT1, . . . , x

Tn]T with xj = [xj,1, . . . , xj,mj ]

T , xτj,ij = xj,ij(t − τj,ij)denotes the delayed state, and xτj,ij and Xτ are defined as

xτj,ij = [xτj,1 , . . . , xτj,ij ]T ,

Xτ = [xτ1,1 , . . . , xτ1,n1 , . . . , xτn,1 , . . . , xτn,mn ]T

and τj,ij is the unknown constant time-delay. For t ∈ [−τj,ij,0], letxj,ij(t) = φj,ij(t), which are assumed to be smooth and bounded.

Remark 1. In plant (1), each control gain function gj,mj(.) containsall state variables and the inputs of the previous subsystems. Thisis apparently different from the case in Ge and Tee (2007), wheregj,mj do not contain the control input ul (1 ≤ l < j) and the statevariables xk,mk (k = j+1, . . . , n). Obviously, the system consideredhere is more general.

As usually done, the following assumptions are made forsystem (1).

Assumption 1. The desired trajectories ydj, j = 1, 2, . . . , n, andtheir time derivatives up to the nth order, are continuous andbounded.

Assumption 2. The signs of gj,ij(.) are known and there exist someunknown constant gj0 and unknown smooth functions gj,ij(.) suchthat 0 < gj0 ≤

∣∣gj,ij(.)∣∣ ≤ gj,ij(.) <∞.Remark 2. Apparently, Assumption 2 implies that gj,ij(.) is strictlypositive or negative. Without loss of generality, we further assumegj,ij(.) > gj0 > 0. In addition, notice that in Assumption 2 the

upper bounds gj,ij(.) are unknown. Therefore, Assumption 2 in thispaper is less restrictive than that in Ho et al. (2005) and Ge andTee (2007), where gj,ij(.) are required to be known for constructingcontrol laws.

Assumption 3. There exist positive functions Q j,ijj,l (xτj,l) for l =

1, 2, . . . , ij such that∣∣∣hj,ij(xτj,ij )∣∣∣ ≤∑ij

l=1 Qj,ijj,l (xτj,l).

In this paper, the following radial basis function (RBF) NNswill be used as an approximator to approximate an unknowncontinuous function. As pointed out in Sanner and Slotine (1992),for a given ε > 0 and any continuous function f (Z) defined onΩZ ⊂ Rn, there exists an NNW T S(Z) such that

f (Z) = W T S(Z)+ δ(Z), |δ(Z)| ≤ ε, (2)

where Z ∈ ΩZ ⊂ Rn is the input vector, W = [w1, w2, . . . , wl]Tis the weight vector, l > 1 is the number of the NN nodes andS(Z) = [s1(Z), . . . , sl(Z)]T is defined by

si = exp[−(Z − µi)T (Z − µi)

φ2i

], i = 1, 2, . . . , l

withµi = [µi1, µi2, . . . , µin]T the center of the receptive field andφi the width of the Gaussian function.

3. Adaptive NN control design

This section is devoted to developing a novel adaptive NNcontrol design procedure. The design procedure for the jthsubsystem is composed of mj design steps. In each step, the radialbasis function NN W Tj,ijS(Zj,ij) will be used to approximate the

unknown nonlinear function fj,ij(Zj,ij). Thus, define an unknownconstant as

θj =1gj0max

∥∥Wj,ij∥∥2 : 1 ≤ ij ≤ mj,where the constant gj0 is defined as in Assumption 2, functionfj,ij and vector Zj,ij will be specified in each step. Furthermore, forj = 1, . . . , n and ij = 1, . . . ,mj− 1, the virtual control laws αj,ij(.)are chosen as follows:

αj,ij = −

(kj,ij +

12

)zj,ij −

12a2j,ij

θjzj,ijST (Zj,ij)S(Zj,ij), (3)

where kj,ij > 0 and aj,ij > 0 are design parameters, θj is theestimation of the unknown constant θj, S(.) is the basis functionvector, and the variables zj,ij are defined by

zj,ij = xj,ij − αj,ij−1, zj,1 = xj,1 − ydj (4)

for j = 1, . . . , n and ij = 2, . . . ,mj. The adaptive laws˙θ j for

j = 1, . . . , n are given by

˙θj(t) =

mj∑ij=1

rj2a2j,ij

z2j,ijST (Zj,ij)S(Zj,ij)− bjθj, (5)

where rj > 0 and bj > 0 are design parameters.

Step (j.1 (1 ≤ j ≤ n)). For the first differential equation of the jthsubsystem, consider the Lyapunov function as follows:

Vzj,1 =12z2j,1 +

gj02rjθ2j ,

where zj,1 = xj,1 − ydj, θj = θj − θj, θj is the estimate of θj. Then,the time derivative of Vzj,1 is given by

1556 B. Chen et al. / Automatica 45 (2009) 1554–1560

Vzj,1 = zj,1(fj,1 + gj,1αj,1 − ydj + hj,1(xτj,1)

)+ zj,1gj,1zj,2 −

gj0rjθj˙θ j. (6)

With Assumption 3, completing the square gives

zj,1hj,1(xτj,1) ≤∣∣zj,1∣∣Q j,1j,1 (xτj,1) ≤ 12 z2j,1 + 12 [Q j,1j,1 (xτj,1)]2 .

Substituting this inequality into (6) yields

Vzj,1 ≤ zj,1

(fj,1 + gj,1αj,1 − ydj +

12zj,1

)+ zj,1gj,1zj,2 +

12

[Q j,1j,1 (xτj,1)

]2−gj0rjθj˙θ j. (7)

To deal with the delay term in (7), consider the Lyapunov–Krasovskii functional as follows: Vj,1 = Vzj1 + VUj,1 with

VUj,1 =∫ t

t−τj,1

12

[Q j,1j,1 (x(s))

]2ds.

Differentiating Vj,1 and using (7), the inequality below can beobtained easily.

Vj,1 ≤ zj,1(fj,1(Zj,1)+ gj,1αj,1

)+ zj,1gj,1zj,2 −

gj0rjθj˙θ j

+

[1− 2 tanh2

(zj,1ηj,1

)]Uj,1, (8)

where Zj,1 = [xj,1, ydj, ydj, θj]T ,Uj,1 = 12

[Q j,1j,1 (x)

]2and

fj,1(Zj,1) = fj,1 − ydj +12zj,1 +

2zj,1tanh2

(zj,1ηj,1

)with ηj,1 being a positive constant. As pointed out by Remark 5

in Ge and Tee (2007), the function 1z tanh2(zη

)is well defined at

z = 0 and can be approximated by a neural network. So, theNN W Tj,1S(Zj,1) is utilized to approximate fj,1 such that for givenεj,1 > 0,

fj,1(Zj,1) = W Tj,1S(Zj,1)+ δj,1(Zj,1),∣∣δj,1∣∣ ≤ εj,1,

where δj,1(Zj,1) denotes the approximation error. Furthermore, astraightforward calculation shows that

zj,1 fj,1(Zj,1) ≤12a2j,1

gj0z2j,1θjST (Zj,1)S(Zj,1)

+12a2j,1 +

12gj0z2j,1 +

12ε2j,1g

−1j0 (9)

where aj,1 > 0 is a design parameter. In addition, from (5), it canbe verified that for any initial conditions θj(t0) ≥ 0, θj(t) ≥ 0 forall t ≥ t0. Consequently, it follows that

zj,1gj,1αj,1 ≤ −gj02a2j,1

θjz2j,1ST (Zj,1)S(Zj,1)−

(kj,1 +

12

)gj0z2j1. (10)

Thus, substituting (9) and (10) into (8) results in

Vj,1 ≤ −kj,1gj0z2j,1 +12

(a2j,1 + ε

2j,1g−1j0

)+gj0rjθj

(rj2a2j,1

z2j,1ST (Zj,1)S(Zj,1)−

˙θ j

)

+ zj,1gj,1zj,2 +[1− 2 tanh2

(zj,1ηj,1

)]Uj,1. (11)

Step (j.ij (for ij = 2, . . . ,mj − 1)). Consider the followingLyapunov–Krasovskii functional

Vzj,ij =12z2j,ij .

Differentiating Vzj,ij to get that

Vzj,ij = zj,ij(fj,ij + gj,ijxj,ij+1 − αj,ij−1 + hj,ij(xτj,ij )

). (12)

By Assumption 3 and completion of the square, we have

zj,ijhj,ij(xτj,ij ) ≤ij∑k=1

∣∣zj,ij ∣∣Q j,ijj,k (xτj,k)≤

ij∑k=1

12z2j,ij +

ij∑k=1

12

[Qj,ijj,k (xτj,k)

]2. (13)

Notice that αj,ij−1(Zj,ij−1) can be expressed as

αj,ij−1 =

ij−1∑k=1

∂αj,ij−1

∂xj,k

(fj,k + gj,kxj,k+1

)+

ij−1∑k=1

∂αj,ij−1

∂xj,khj,k(xτj,k)

+

ij−1∑k=0

∂αj,ij−1

∂y(k)djy(k+1)dj +

∂αj,ij−1

∂θj

˙θ j. (14)

Similar to (13), we have

− zj,ij

ij−1∑k=1

∂αj,ij−1

∂xj,khj,k(xτj,k)

≤

ij−1∑k=1

k∑l=1

12zj,ij

[∂αj,ij−1

∂xj,k

]2+

ij−1∑k=1

k∑l=1

12

[Q j,kj,l (xτj,l)

]2. (15)

By utilizing (13)–(15), (12) can be rewritten in the following form.

Vzj,ij ≤ zj,ij

(fj,ij + gj,ijxj,ij+1 +

ij∑k=1

12zj,ij

−

ij−1∑k=1

∂αj,ij−1

∂xj,k

(fj,k + gj,kxj,k+1

)−

ij−1∑k=0

∂αj,ij−1

∂y(k)djy(k+1)dj

+

ij−1∑k=1

k∑l=1

12z2j,ij

[∂αj,ij−1

∂xj,k

]2)−∂αj,ij−1

∂θj

˙θ j

+

ij∑k=1

12

[Qj,ijj,k (xτj,k)

]2+

ij−1∑k=1

k∑l=1

12

[Q j,kj,l (xτj,l)

]2. (16)

To compensate the delay terms, define an integral function asfollows:

VUj,ij =ij∑k=1

∫ t

t−τj,k

12

[Qj,ijj,k (x(s))

]2ds

+

ij−1∑k=1

k∑l=1

∫ t

t−τj,l

12

[Q j,kj,l (x(s))

]2ds.

Differentiating VUj,ij yields

VUj,ij = Uj,ij −ij∑k=1

12

[Qj,ijj,k (xτj,k)

]2−

ij−1∑k=1

k∑l=1

12

[Q j,kj,l (xτj,l)

]2= zj,ij

2zj,ijtanh2

(zj,ijηj,ij

)Uj,ij +

[1− 2 tanh2

(zj,ijηj,ij

)]Uj,ij

B. Chen et al. / Automatica 45 (2009) 1554–1560 1557

−

ij∑k=1

12

[Qj,ijj,k (xτj,k)

]2−

ij−1∑k=1

k∑l=1

12

[Q j,kj,l (xτj,l)

]2(17)

where Uj,ij =∑ijk=1

12

[Qj,ijj,k (xj,k)

]2+∑ij−1k=1

∑kl=1

12

[Q j,kj,l (xj,l)

]2.

It is clearly seen that by adding (17) to (16) the delay terms arecancelled out. Hence, by utilizing (16) and (17), it can be provedthat the derivative of the Lyapunov–Krasovskii functional Vj,ij =Vzj,ij + VUj,ij satisfies

Vj,ij ≤ zj,ij

(ϕj,ij −

∂αj,ij−1

∂θj

˙θ j

)+

[1− 2 tanh2

(zj,ijηj,ij

)]Uj,ij(x)

+ zj,ij(fj,ij + gj,ijαj,ij

)+ gj,ijzj,ijzj,ij+1, (18)

where the equality Uj,ij = zj,ij2zj,ijtanh2

(zj,ijηj,ij

)Uj,ij(x) +[

1− 2 tanh2(zj,ijηj,ij

)]Uj,ij(x) is used, and

fj,ij(Zj,ij) = fj,ij + zj,ij−1gj,ij−1 +ij∑k=1

12zj,ij

−

ij−1∑k=1

∂αj,ij−1

∂xj,k

(fj,k + gj,kxj,k+1

)−

ij−1∑k=0

∂αj,ij−1

∂y(k)djy(k+1)dj

+

ij−1∑k=1

k∑l=1

12z2j,ij

[∂αj,ij−1

∂xj,k

]2+2zj,ijtanh2

(zj,ijηj,ij

)Uj,ij − ϕj,ij ,(19)

Zj,ij = [xTj,ij , ydj, . . . , y

(ij)dj , θj]

T ,

Uj,ij =ij∑l=1

12

[Qj,ijj,l (xj,l)

]2+

ij−1∑l=1

l∑p=1

12

[Q j,lj,p(xj,p)

]2.

Remark 3. From (5), ˙θ j is evidently a function of the whole statevariable xj. Hence, the term

∂αj,k−1

∂θj

˙θ j cannot be dealt with as done

in Ge and Tee (2007), where it is treated as a part of fj,ij . So, thismakes the control law design more difficult. To overcome thisdifficulty, we introduce a function ϕj,ij to compensate the term∂αj,k−1

∂θj

˙θ j, which will be specified later.

In what follows, the NNW Tj,ijS(Zj,ij) is used to approximate the

unknown fj,ij such that for given εj,ij > 0, the following holds.

fj,ij = WTj,ijS(Zj,ij)+ δj,ij(Zj,ij),

∣∣δj,ij ∣∣ εj,ijwhere δj,ij denotes the approximation error. Then, by following asimilar line used in the procedure from (9) to (10), we have

Vj,ij ≤ −kj,ijgj0z2j,ij +

12

(a2j,ij + ε

2j,ijg−1j0

)+gj0rjθjrj2a2j,ij

z2j,ijST (Zj,ij)S(Zj,ij)+

[1− 2 tanh2

(zj,ijηj,ij

)]Uj,ij(x)

+ zj,ij

(ϕj,ij −

∂αj,ij−1

∂θj

˙θ j

)+ gj,ijzj,ijzj,ij+1. (20)

Step (j.mj). This is the last step for the jth subsystem toconstruct the real control law uj. Consider the following

Lyapunov–Krasovskii functional.

Vj,mj =12z2j,mj + VUj,mj

with VUj,mj =∑nj=1∑mjk=1

∫ tt−τj,k

12

[Qj,mjj,k (x(s))

]2ds+

∑mj−1k=1

∑kl=1∫ t

t−τj,l12

[Q j,kj,l (x(s))

]2ds and zj,mj = xj,mj−αj,mj−1. Then, taking (18)

with ij = mj into account results in that

Vj,mj ≤

[1− 2 tanh2

(zj,mjηj,mj

)]Uj,mj(x)

+ zj,mj

(ϕj,mj −

∂αj,mj−1

∂θj

˙θ j

)+ zj,mj(fj,mj + gj,mjuj), (21)

where fj,mj(Zj,mj) can be defined by (19) with ij = mj, and Zj,mj =

[X, ydj , . . . , y(mj)dj, θ1, . . . , θj]

T . The NNW Tj,mjS(Zj,mj) is employed to

approximate fj,mj such that for given εj,mj > 0, the followingexpression holds.

fj,mj = WTj,mjS(Zj,mj)+ δj,mj(Zj,mj),

∣∣δj,mj ∣∣ εj,mj .At the present step, choose the control law uj as

uj =−12a2j,mj

zj,mj θjST (Zj,mj)S(Zj,mj)−

(kj,mj +

12

)zj,mj . (22)

Again repeating the procedure from (9) to (10) gives

Vj,mj ≤ −kj,mjgj0z2j,mj +

12

(a2j,mj + ε

2j,mjg

−1j0

)+gj0rjθjrj2a2j,mj

z2j,mjST (Zj,mj)S(Zj,mj)

+

[1− 2 tanh2

(zj,mjηj,mj

)]Uj,mj(x)

+ zj,mj

(ϕj,mj −

∂αj,mj−1

∂θj

˙θ j

). (23)

Let V =∑nj=1∑mjk=1 Vj,k. Then, (11), (18) and (23) imply that

V ≤ −n∑j=1

mj∑k=1

kj,kgj0z2j,k +n∑j=1

mj∑k=1

12

(a2j,k + ε

2j,kg−1j0

)+

n∑j=1

gj0rjθj

( mj∑k=1

rj2a2j,k

z2j,kST (Zj,k)S(Zj,k)−

˙θ j

)

+

n∑j=1

mj∑k=1

[1− 2 tanh2

(zj,kηj, k

)]Uj,k(x)

+

n∑j=1

mj∑k=2

zj,k

(ϕj,k −

∂αj,k−1

∂θj

˙θ j

). (24)

So far, we have completed the control law design.

Remark 4. In the existing neural adaptive control design ap-proaches, each weight vector is just the estimated vector.Therefore, the number of the adaptation parameters depends onthe number of the NN nodes. Consequently, if a system contains alarge number of unknown nonlinear functions, or more NN nodesare used to improve the approximation precision, there will be alarge number of adaptation parameters that need to be updated

1558 B. Chen et al. / Automatica 45 (2009) 1554–1560

online simultaneously. This makes the learning time become un-acceptably large. In this paper, inspired by the idea in Yang, Feng,and Ren (2004), instead of estimating each element in the weightvectorW , we estimate the norm of all the weight vectors, so onlyone adaptation learning law is required to control each subsystem.

4. Stability Analysis

In this section, the boundedness of all the signals in the closed-loop system will be proved. The main result is summarized in thefollowing theorem.

Theorem. Consider system (1) satisfying Assumptions 1–3. Supposethat for 1 ≤ j ≤ m, 1 ≤ ij ≤ mj, the packaged unknown functionsfj,ij can be approximated by neural network in the sense that theapproximating error δj,ij are bounded. Then under the action of controllaw (22) and theNNadaptation law (5), all the closed-loop trajectoriesremain bounded.

Proof. We first determine the functions ϕj,k such that

n∑j=1

mj∑k=2

zj,k

(ϕj,k −

∂αj,k−1

∂θj

˙θ j

)≤ 0. (25)

In light of the definition of ˙θ j and0 < ST (.)S(.) ≤ L (L is thenumberof neural network weights), a straightforward calculation showsthat

−

mj∑k=2

zj,k∂αj,k−1

∂θj

˙θ j ≤

mj∑k=2

zj,k

(bjθj

∂αj,k−1

∂θj

−

k−1∑l=1

∂αj,k−1

∂θj

rj2a2j,lz2j,lS

T (Zj,l)S(Zj,l)

)

+

mj∑k=2

zj,k

(rjL2a2j,k

zj,kk∑l=2

∣∣∣∣∣zj,l ∂αj,l−1∂θj

∣∣∣∣∣).

Thus, by choosing ϕj,k as

ϕj,k = −bjθj∂αj,k−1

∂θj−rjL2a2j,k

zj,kk∑l=2

∣∣∣∣∣zj,l ∂αj,l−1∂θj

∣∣∣∣∣+

k−1∑l=1

∂αj,k−1

∂θj

rj2a2j,lz2j,lS

T (Zj,l)S(Zj,l),

(25) holds. Similarly, the following can be verified easily.n∑j=1

gj0rjθj

( mj∑k=1

rj2a2j,k

z2j,kST (Zj,k)S(Zj,k)−

˙θ j

)

≤

n∑j=1

gj0bj2rj

(−θ2j + θ

2j

). (26)

At the present stage, choose Lyapunov functional as V = Vn,mn .Then, combining (24)–(26) results in

V ≤ −n∑j=1

mj∑k=1

kj,kgj0z2j,k −n∑j=1

gj0bj2rj

θ2j + C

+

n∑j=1

mj∑k=1

[1− 2 tanh2

(zj,kηj,k

)]Uj,k(x), (27)

where C =∑nj=1∑mjk=1

12

(a2j,k + ε

2j,kg−1j0

)+∑nj=1

gj02rjθ2j is a

constant. Thus, by (27) the boundedness follows immediately fromfollowing the same line used in the proof of Theorem 1 in Ge andTee (2007) and Zhou et al. (2005). The proof is thus completed.

Fig. 1. System output y1(t) (‘‘–’’) and the reference yd1(t) (‘‘-.-’’).

5. Simulation example

In this section, a simulation example is used to illustrate theeffectiveness of the proposed adaptive neural control method.Consider the following nonlinear time-delay system.

x1,1 = −x1,1 +(1+ sin2(x1,1)

)x1,2 + x2τ1,1 ,

x1,2 = x1,1x1,2 + x2,1 + x2,2+(1+ sin2(x1,1)+ 0.5 cos2(x2,2)

)u1 + xτ1,2 ,

x2,1 = −x2,1 + x2.2 + xτ2,1 ,

x2,2 = (x1,2 + x2,1)x2,2 − x1,1u1 + (2+ sin2(u1)

− sin(x2,1x2,2 − x1,1))u2 + xτ1,1xτ2,2 , (28)

where xτj,ij = xj,ij(t − τj,ij), for j = 1, 2, ij = 1, 2, and the time-delays are chosen as τ1,1 = 2, τ1,2 = 1.5, τ2,1 = 0.5, and τ2,2 = 1.Clearly, if remove the terms 0.5 cos2(x2,2) in g1,2 and sin2(u1) in

g2,2, respectively, this system is the same as the one studied in GeandTee (2007). Now, because g1,2 contains x2,2 and g2,2 containsu1,the neural control scheme proposed in Ge and Tee (2007) cannotbe used to control this system. Given the reference output signalsas yd1 = 0.5 (sin(t)+ sin(0.5t)) , yd2 = 0.5 sin(t) + sin(0.5t),for the control law (22) and the NN adaptation law (5) we choosethe design parameters as: k1,1 = k1,2 = k2,1 = k2,2 = 30,a1,1 = a1,2 = 3, a2,1 = a2,2 = 1, r1 = r2 = 600, b1 = b2 = 0.075.The simulation is run under the initial conditions xj,ij(ϑ) = 0

for −τj,ij ≤ ϑ ≤ 0, j = 1, 2, ij = 1, 2, and [θ1(0), θ2(0)]T =[0, 0]T . The simulation results are shown in Figs. 1–4. Figs. 1and 2 show the corresponding system outputs and the referencesignals. Fig. 3 shows the responses of state variables x1,2 andx2,2, Fig. 3 displays the control input signals u1 and u2, andFig. 4 shows the boundedness of adaptive parameters θ1 andθ2. From the simulation results, it can clearly be seen that theproposed controller guarantees the boundedness of all the signalsin the closed-loop system, and also achieves the good trackingperformance. In addition, to control this system, our methodrequires only two adaptation laws.

6. Conclusion

A novel adaptive neural network tracking control designscheme has been proposed for a class of MIMO nonlinear time-delay systems with block-triangular structure. The suggestedcontrol law guarantees that the tracking errors converge to aneighborhood of the origin and all the other signals in the resultingclosed-loop system remain bounded. Compared with the existing

B. Chen et al. / Automatica 45 (2009) 1554–1560 1559

Fig. 2. System output y2(t) (‘‘–’’) and the reference yd2(t) (‘‘-.-’’).

Fig. 3. The control inputs u1(t) (‘‘–’’) and u2(t) (‘‘-.-’’).

Fig. 4. The adaptive parameters θ1(t) (‘‘–’’) and θ2(t) (‘‘-.-’’).

results, the main advantage of our result is that the restriction onthe control gain functions has been removed and the number ofadaptation parameters has been considerably reduced.

Acknowledgements

This work is partially supported by the Natural Sciences andEngineering Research Council of Canada, and the Natural Science

Foundation of China (60674055, 60774047), and the‘‘Taishan’’scholar program and the Natural Science Foundation of ShandongProvince of Shandong province, China (No. Y2006G04).

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1560 B. Chen et al. / Automatica 45 (2009) 1554–1560

Bing Chen received the B.A. degree in mathematicsfrom Liaoning University, P. R. China, the M.A. degreein mathematics from Harbin Institute of Technology,P. R. China, and the Ph.D. degree in electrical engineeringfrom Northeastern University, P. R. China, in 1982, 1991and 1998, respectively. Currently, he is a Professor withthe Institute of Complexity Science, Qingdao University,Qingdao, P. R. China. His research interests includenonlinear control systems, robust control, and fuzzycontrol theory.

Xiaoping Liu obtained his B. Sci, M.Sci, and Ph.D. degreein electrical engineering from Northeastern University,P. R. China, in 1984, 1987, and 1989, respectively. He spentmore than 10 years in the School of Information Scienceand Engineering at Northeastern University, P. R. China. In2001, he joined the Department of Electrical Engineeringat Lakehead University, Canada. His research interests arenonlinear control systems, singular systems, and robustcontrol. He is a member of the Professional Engineers ofOntario

Kefu Liu is a Full Professor in the Department ofMechanical Engineering at Lakehead University, Canada.He receivedhis B.Eng. andM.Sc. inMechanical Engineeringfrom Central South University of Technology, China in1981 and 1984, respectively, and his Ph.D. degree inMechanical Engineering fromTechnical University of NovaScotia, Canada in 1992. He was assistant professor atSt. Mary’s University, Canada, from 1993 to 1995, andat Dalhousie University, Canada, from 1995 to 1998. Hejoined Lakehead University in 1998. Dr. Liu is a memberof Professional Engineers Ontario. His research interests

include vibration control, control of nonlinear systems, and mechatronics.

Chong Lin (SM’2006) received the B.Sci and the M.Sci inApplied Mathematics from the Northeastern University,P.R.China, in 1989 and 1992, respectively, and the Ph.D.in Electrical and Electronic Engineering from the NanyangTechnological University, Singapore, in 1999.He was a Research Associate with the Department

of Mechanical Engineering, University of Hong Kong, in1999. From 2000 to 2006, he was a Research Fellow withthe Department of Electrical and Computer Engineering,National University of Singapore. Since 2006, he has beena professor with the Institute of Complexity Science,

Qingdao University, China. He has published more than 60 research papers and co-authored two monographs. His current research interests are mainly in systemsanalysis and control, robust control and fuzzy control.