Q Shen, P Shi, T Zhang, CC Lim: Novel neural control for a class of uncertain pure‐feedback systems, 

IEEE Transactions on Neural Networks and Learning Systems, vol. 25, no. 4, April 2014, pp 718‐727. 

 

The material posted here is solely for research and scholastic purposes. This material is posted to 

ensure timely dissemination of scholarly and technical work. Copyright and all rights therein are 

retained by authors or by other copyright holders, such as IEEE. All persons accessing this 

information are expected to adhere to the terms and constraints invoked by each author's copyright.  

718 IEEE TRANSACTIONS ON NEURAL NETWORKS AND LEARNING SYSTEMS, VOL. 25, NO. 4, APRIL 2014

Novel Neural Control for a Class of UncertainPure-Feedback Systems

Qikun Shen, Peng Shi, Senior Member, IEEE, Tianping Zhang, and Cheng-Chew Lim, Senior Member, IEEE

Abstract— This paper is concerned with the problem of adap-tive neural tracking control for a class of uncertain pure-feedbacknonlinear systems. Using the implicit function theorem and back-stepping technique, a practical robust adaptive neural controlscheme is proposed to guarantee that the tracking error convergesto an adjusted neighborhood of the origin by choosing appropri-ate design parameters. In contrast to conventional Lyapunov-based design techniques, an alternative Lyapunov function isconstructed for the development of control law and learning algo-rithms. Differing from the existing results in the literature, thecontrol scheme does not need to compute the derivatives of virtualcontrol signals at each step in backstepping design procedures.Furthermore, the scheme requires the desired trajectory and itsfirst derivative rather than its first n derivatives. In addition, theuseful property of the basis function of the radial basis function,which will be used in control design, is explored. Simulationresults illustrate the effectiveness of the proposed techniques.

Index Terms— Adaptive control, neural control, pure feedback.

I. INTRODUCTION

AS A UNIVERSAL function approximator, neural net-works (NNs) have been widely used in robust adaptive

control of uncertain nonlinear systems, owing to their learningand adaptation abilities [1], [2], [5], [6], [14]–[17], [21], [22],[24]–[29], [31]–[33]. For uncertain strict-feedback systems,many approximation-based adaptive backstepping controllershave been developed over the past 10 years. The work in[1]–[6] considered the adaptive fuzzy or neural controlproblem of uncertain nonlinear systems, which includedsingle-input–single-output (SISO) systems and multiple-input–multiple-output (MIMO) systems. In [23], the problem ofadaptive fuzzy output feedback control for a class of uncertainMIMO systems with immeasurable system states was inves-tigated, and the stability of the resulting closed-loop adaptive

Manuscript received January 1, 2013; revised August 13, 2013; acceptedAugust 26, 2013. Date of publication September 16, 2013; date of currentversion March 10, 2014. This work was supported in part by the NationalNatural Science Foundation of China under Grant 61174046, in part by the111 Project under Grant B12018, and in part by the Engineering and PhysicalSciences Research Council, U.K., under Grant EP/F029195.

Q. Shen and T. Zhang are with the College of Information Engineering,Yangzhou University, Yangzhou 225127, China (e-mail: qkshen@yzu.edu.cn;tpzhang@yzu.edu.cn).

P. Shi is with the School of Electrical and Electronic Engineering, The Uni-versity of Adelaide, Adelaide 5005, Australia, and also with the Collegeof Engineering and Science, Victoria University, Melbourne 8001, Australia(e-mail: peng.shi@adelaide.edu.au).

C. C. Lim is with the School of Electrical and Electronic Engi-neering, The University of Adelaide, Adelaide 5005, Australia (e-mail:cclim@eleceng.adelaide.edu.au).

Color versions of one or more of the figures in this paper are availableonline at http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TNNLS.2013.2280728

control system was analyzed. It has been proved that adaptivebackstepping technique is a powerful tool to solve tracking orregulation control problems of unknown nonlinear systems inor transformable to parameter strict-feedback form [7], wherefuzzy logic systems (FLSs) or NNs are used to approximateunknown nonlinear functions [8]–[13], [34], [35].

As shown in [14]–[23], however, because pure-feedbacknonlinear systems have no affine appearance of the statevariables to be used as virtual controls and actual control, suchsystems have a more representative form than strict-feedbacknonlinear systems. In addition, the existing results concern-ing approximation-based adaptive backstepping control foruncertain strict-feedback nonlinear systems cannot be directlyapplied to those uncertain pure-feedback nonlinear systems.Thus, the controller design of nonlinear pure-feedback systemsis more difficult and challenging.

Recently, adaptive backstepping control approaches havebeen developed in [14]–[18] for uncertain SISO pure-feedback nonlinear systems. Adaptive neural-networked con-trol schemes were proposed in [14] and [15] for a classof pure-feedback nonlinear systems, where the controlledsystems were supposed to be an affine nonlinear for avoidingthe algebraic loop problem. Direct adaptive fuzzy or neuralcontrol schemes in [16]–[20] were proposed for uncertainpure-feedback nonlinear systems, where the implicit functiontheorem was exploited to assert the existence of continuousdesired feedback controllers, and NNs or FLSs were usedto approximate these desired feedback controllers. In theseschemes, the bounds and signs of the derivatives of the non-linear functions for all the variables were, however, assumed tobe known. An adaptive fuzzy backstepping control approachin [23] was developed for a class of uncertain MIMO pure-feedback nonlinear systems. The proposed approach not onlyrelaxed the restrictive conditions in [16]–[19], but also avoidedthe algebraic loop problem as well by introducing the filteredsignals into the backstepping control design. More recently,adaptive neural backstepping control approaches have beendeveloped in [21] and [22] for a class of uncertain nonlinearpure-feedback systems by incorporating a dynamics surfacetechnique into the control design; thus, the proposed controlapproaches became much simpler than those in [13], [16], and[18]–[20].

However, the aforementioned approaches required theknowledge of the desired trajectory yd(t) and the first n deriv-atives, i.e., y(i)

d (t), i = 1, 2, . . . , n should be available. Itis important to note that in some important applications(e.g., land vehicle or aircraft), the desired trajectory may be

2162-237X © 2013 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission.See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.

SHEN et al.: NOVEL NEURAL CONTROL FOR A CLASS OF UNCERTAIN PURE-FEEDBACK SYSTEMS 719

generated by a planner, an outer loop, or a user input devicethat does not provide higher derivatives. Relaxing the assump-tion motivates us for this paper. In a standard backsteppingdesign procedure, the analytical computation of the first deriv-atives of virtual control signals αi (i = 1, 2, . . . , n), i.e., αi , isnecessary. As the system dimension increases, the computa-tion of αi becomes increasingly complicated. This limits thetheoretical results’ field of practical applications. How to avoidthe computation of αi is a crucial issue in controller design,which is another motivation of this paper.

In this paper, NNs are first to approximate the unknownfunctions. Then, using the implicit function theorem andthe backstepping technique, a robust adaptive neural controlscheme is proposed to guarantee that the tracking error con-verges to an adjusted neighborhood of the origin by choosingappropriate design parameters. Compared with the existingliterature, the following contributions are worth to be empha-sized.

1) In contrast with the existing results such as [14]–[23]where the desired trajectory yd(t) and the first n deriva-tives, i.e, y(i)

d (t), i = 0, 1, . . . , n should be available, thecontrol scheme presented in this paper only needs theknowledge of the desired trajectory yd(t) and the firstderivative yd(t), which is more reasonable in practicalapplications. The theoretical results of this paper are thusvaluable in a wide field of practical applications.

2) Compared with the existing literatures such as[14]–[18], the control scheme does not need to computethe derivatives of virtual control signals at each stepin backstepping design procedures, which decreases thecomputation complexity.

3) Different from the conventional Lyapunov-based designtechniques, an alternative Lyapunov function is con-structed for the development of control law and learningalgorithms.

The rest of this paper is organized as follows. In Section II,the problem statement and description of NNs are presented.In Section III, a neural controller is designed and the stabilityanalysis of the closed-loop system is developed. Numericalsimulation results are presented to demonstrate the effec-tiveness of the proposed technique in Section IV. Finally,Section V draws the conclusion.

II. PROBLEM STATEMENT AND DESCRIPTION OF NNS

In this section, we will first formulate the tracking controlproblem of a class of pure-feedback systems. Then, NNs areintroduced and its mathematical description is given.

A. Problem Statement

Consider the following uncertain nonlinear dynamic systemsin pure-feedback form:

⎧⎪⎨

⎪⎩

xi = fi (xi , xi+1) + di (xn, t), i = 1, 2, . . . , n − 1

xn = fn(xn, u) + dn(xn, t)

y = x1

(1)

where xi = (x1, . . . , xi )T ∈ Ri , i = 1, 2, . . . , n − 1, xn =

(x1, . . . , xn)T ∈ Rn , u ∈ R, and y ∈ R are the system

states, input, and output, respectively, fi (xi , xi+1) ∈ R, i =1, 2, . . . , n are the unknown smooth functions with xn+1 = u,di (xn, t), i = 1, 2, . . . , n, denote the unknown external distur-bances.

Control objective is to design an adaptive controller ufor (1), such that output y follows as accurately the desiredtrajectory yd(t) as possible regardless of unknown externaldisturbances di (xn, t), i = 1, 2, . . . , n.

To design the appropriate controller, the following lemmaand some assumptions are given.

Lemma 1 [30]: For ∀x ∈ R, |x | − tanh(x/δ)x ≤ 0.2785δ,where δ > 0 ∈ R.

Assumption 1: There exist unknown constants Md,i > 0 ∈ R,i = 1, 2, . . . , n such that |di(xn, t)| ≤ Md,i .

Assumption 2: The signs of ∂ fi (xi , xi+1)/∂xi+1, i =1, 2, . . . , n are known, and there exist unknown constants gil ,i = 1, 2, . . . , n, l = 0, 1, 2, such that 0 < gi0 ≤|∂ fi (xi , xi+1)/∂xi+1| ≤ gi1, ∂2 fi (xi , xi+1)/∂[xi+1]2 ≤ gi2.Without the loss of generality, it is assumed that∂ fi (xi , xi+1)/∂xi+1 > 0. Define g

0= min{g10, g20, . . . , gn0},

g1 = max{g11, g21, . . . , gn1}, and g2 = max{g12, g22, . . . ,gn2}.

Assumption 3: The desired trajectory yd(t) is available, andits first derivative yd(t) is bounded, i.e., |yd(t)| ≤ My , ∀t ≥ 0,where My > 0 ∈ R is an unknown constant.

Remark 1: Assumption 2 seems to be restrictive. However,in [17] and [18], the system models considered to satisfythe assumption. For the strict-feedback systems with theassumption that control gain is bounded, i.e., xi = fi (xi ) +gi(xi )xi+1 + di (xn, t), xn = fn(xn) + gn(xn)u + dn(xn, t) andgi0 ≤ gi (xi) ≤ gi1, i = 1, 2, . . . , n, Assumption 2 is naturallysatisfied, namely 0 < gi0 ≤ ∂[ fi (xi ) + gi(xi )xi+1]/∂xi+1 ≤gi1 and 0 = ∂2[ fi (xi ) + gi(xi )xi+1]/∂[xi+1]2 ≤ gi2. Obvi-ously, if the boundedness assumption is extended to pure-feedback systems, one has directly Assumption 2. Hence,Assumption 2 can be seen as the extended version. It is worthnoting that the purpose of gil is for analysis, and the exactvalue of gil is not requested to be known in the controllerdesign.

Remark 2: In the literatures, the existing results concerningtrajectory tracking problem [14]–[21] require the classicalassumption that the desired trajectory yd(t) and the first nderivatives, i.e., y(i)

d (t), i = 0, 1, . . . , n should be available.As stated in Section I, in some important applications (e.g.,land vehicle or aircraft), the desired trajectory may be gen-erated by a planner, an outer loop, or a user input devicethat does not provide higher derivatives. Thus, in such cases,these results do not work. Assumption 3 in this paper is morereasonable in practical applications.

B. Description of NNs

NNs have been widely used in modeling and controllingof nonlinear systems because of their capabilities of nonlinearfunction approximation, learning, and fault tolerance. The fea-sibility of applying NNs to unknown dynamic systems controlhas been demonstrated in many studies. In this paper, radial

720 IEEE TRANSACTIONS ON NEURAL NETWORKS AND LEARNING SYSTEMS, VOL. 25, NO. 4, APRIL 2014

basis function NNs presented in [5]

h(Z , W ) = W T S(Z)

shall be used to approximate a continuous function h(Z):R p → R, where Z = (z1, . . . , z p)T ∈ R p is the input vectorwith p being the NN input dimension, the weight vector Wand the basis function vector S(Z) are defined as

W = (w1, . . . , wN )T

where N is the number of the NNs nodes

S(Z) = (s1(Z), . . . , sN (Z))T

si (Z) = exp

⎛

⎜⎜⎜⎝

−

p∑

j=1(z j − ai, j )

2

(ci )2

⎞

⎟⎟⎟⎠

where ci > 0 is the width of the receptive field and ai, j ∈ R,i = 1, 2, . . . , N , j = 1, 2, . . . , i , are the center of the Gaussianfunction. Let

�W = {W: ||W || ≤ wm}W ∗

i = arg minW∈�W

[

supz∈�Z

∣∣h(Z , W ) − h(Z)

∣∣

]

where wm > 0 is a design parameter and �Z is a sufficientlylarge compact set. For a continuous function h(Z), it can beobtained h(Z) = W ∗T S(Z)+ε(Z), where ε(Z) is the optimalapproximation error. From the universal approximation resultsstated in [5], we know that NNs can approximate any contin-uous function to any accuracy on compact set �Z .

In this paper, there are n NNs to be, respectively, usedto approximate n unknown continuous functions ϕi (Zi ),i = 1, 2, . . . , n. To distinguish these NNs, the correspondingnotations of the weight vector and the basis functionvector are modified slightly, i.e., ϕi (Zi ) = W∗T

i Si (Zi ) +εi (Zi ), where Si (Zi) = [si,1(Zi ), . . . , si,N (Zi )]T ,si, j (Zi ) = exp(−∑pi

j=1 (Zi − ai,h)2/(ci,h, j )2), and

Wi = [wi,1, . . . , wi,N ]T , pi is the dimension of Zi ,i = 1, 2, . . . , n, h = 1, 2, . . . , N .

Assumption 4: There exist unknown constants Mε,i >0 ∈ R,i = 1, 2, . . . , n, such that |εi(Zi )| ≤ Mε,i .

III. DESIGN OF ADAPTIVE NNS CONTROLLER AND

STABILITY ANALYSIS

The recursive design procedure contains n steps. FromSteps 1 to n − 1, virtual control αi is designed at each step.Finally, an overall control law u = αn is constructed at Step n.

In the following, for notational simplicity, we will use • todenote •(·). For example, fi is the abbreviation of fi (xi , xi+1).

Define

zi = xi − αi−1, i = 1, 2, . . . , n

where α0 = yd , αi , i = 1, 2, . . . , n−1 are the virtual controls,which will be designed at each step in backstepping designprocedures.

The time derivative of zi is

zi = xi − αi−1 = fi (xi , xi+1) + di − αi−1.

From Assumption 2, we know ∂ fi (xi , xi+1)/∂xi+1 ≥ g0

> 0

for all (x Ti , xi+1)

T ∈ Ri+1. From the implicit functiontheorem, for every value of xi , there exists a smooth idealcontrol input α∗

i (xi) such that fi (xi , α∗i ) = 0.

By the mean value theorem, one has

fi (xi , xi+1) = fi (xi , α∗i ) +

∫ 1

0

∂ fi (xi , xi+1,λ)

∂xi+1,λdλ(xi+1 − α∗

i )

where λ ∈ [0, 1] and xi+1,λ = λxi+1 +(1−λ)α∗i . Furthermore,

one has

zi = ai (xi , xi+1)(zi+1 + αi − α∗i ) + di − αi−1 (2)

where ai (xi , xi+1) = ∫ 10 ∂ fi (xi , xi+1,λ)/∂xi+1,λdλ

and αi−1 = ∑i−1j=2 ∂αi−1/∂x j + ∂αi−1/∂ bi−1,1 +

∑i−1j=2

∑ jl=1 ∂αi−1/∂ bi−1,2.

To obtain the above control objective, αi , i = 1, 2, n areemployed as follows:

αi = −hi (Zi ) − W Ti Si (Zi ) (3)

where hi (Zi ) = 1/2 + ki + bi,1zi w2i + bi,2zi wi +

tanh(zi/δε)Mε,i , Zi = (x Ti , zi , 1)

T, W T

i S(Zi ) is used toapproximate a function ϕi (Zi ) defined later, which is approx-imated by NNs as ϕi (Zi ) = W∗T

i Si (Zi ) + εi (Zi ), Wi isan estimate of W∗

i at time t , bi,1, bi,2, and Mε,i are theestimates of unknown constants bi,1 = g2/(8g2

0), bi,2 =

∑il=1 g2Md,l/(4g2

0), and Mε,i , wi is defined in Appendix A,

and ki > 0 ∈ R and δε > 0 ∈ R are the design parameters.It is necessary to point again out that αn is the actual

controller.Define the following adaptive laws:

˙W i =

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

−ηW Si (Zi )zi , if ||Wi || < wm or

||Wi || = wm and − W Ti Si (Zi )zi ≤ 0

−ηW Si (Zi )zi + ηW Wi W Ti Si (Zi )zi/||Wi ||2,

if ||Wi || = wm and − W Ti Si (Zi )zi > 0

(4)

˙Mε,i = ηε tanh

(zi

δε

)

zi (5)

˙bi,1 = ηbi z2i w

2i (6)

˙bi,2 = ηbz2i wi (7)

where ηW > 0 ∈ R, ηε > 0 ∈ R and ηb > 0 ∈ R are theadaptive rates and wm > 0 ∈ R is a design parameter.

Now, to prove the convergence of the closed-loop system,the following theorem is given.

Theorem 1: Consider the closed-loop system consistingof (1) under Assumptions 1–4. If the virtual and actual controllaws (3) and the adaptation laws (4)–(7) are employed, then thetracking error can be made arbitrarily small by choosing appro-priate parameters δl , l = d, W, ε, y and ki , i = 1, 2, . . . , n.

Proof: The proof procedure contains n steps.Step 1: Define the following Lyapunov function:

V ′1(t) =

∫ z1

0

[∫ 1

0β1dλ

]−1

σdσ

SHEN et al.: NOVEL NEURAL CONTROL FOR A CLASS OF UNCERTAIN PURE-FEEDBACK SYSTEMS 721

where β1 = ∂ f1(x1, x2λσ )/∂x2λσ , x2λσ = λx2σ + (1 − λ)α∗1 ,

x2σ = −h1(σ ) − W T1 S(Z1σ ), and Z1σ = (x1, σ, 1)T . From

Appendix C, one has

V ′1 ≤ z1 z1

a1(x1,x2)+ g2

2g20

∫ z10 |ST

1 (Z1σ )˙W 1|σdσ + g2z2

14g2

0

× w1|( f1 + d1)| + g2z21

4g20

∫ 10

∣∣∣∂α∗

1∂xi

( f1 + d1)∣∣∣ λdλ. (8)

Substituting (2) into (8) and considering Assumption 3 yields

V ′1 ≤ z1(z2 + α1 − α∗

1 ) + |z1|Md,1

g0

+ |z1|My + g2z21

4g20

w1| f1|

+ g2

2g20

∫ z1

0|ST

1 (Z1σ )˙W 1|σdσ + g2z2

1

4g20

∫ 1

0

∣∣∣∣∂α∗

1

∂xif1

∣∣∣∣ λdλ

+(

g2z21

4g20

w1 + g2z21

4g20

∫ 1

0

∣∣∣∣∂α∗

1

∂xi

∣∣∣∣λdλ

)

Md,1. (9)

By the adaptive law (4), it is easily seen that ˙W1 dependson Z1. From Appendix B, one has

|ST1 (Z1)

˙W 1| ≤ ηW z1|(

N + ||W1||21||W1||2

)

≤ 2ηW N |z1|.

Furthermore, one has

g2

2g20

∫ z1

0

∣∣∣ST

1 (Z1σ )˙W 1

∣∣∣ σdσ ≤ g2

2g20

∫ z1

02ηW N |z1|σdσ

= ηWg2 N

g20

|z31|.

Thus, (9) is further derived as

V ′1 ≤ z1z2 + z1α1 − z1α

∗1 + |z1|Md,1

g0

+ ηWg2 N

g20

|z31|

+ |z1|My + g2z21

4g20

w1| f1| + g2z21

4g20

∫ 1

0

∣∣∣∣∂α∗

1

∂xif1

∣∣∣∣ λdλ

+(

g2z21

4g20

w1 + g2z21

4g20

∫ 1

0

∣∣∣∣∂α∗

1

∂xi

∣∣∣∣ λdλ

)

Md,1. (10)

From Lemma 1, one has |z1|Md,1/g0

≤tanh(z1/δd)z1 Md,1/g

0+ 0.2785δd , ηW g2 N/g2

0|z3

1| ≤tanh(z1/δW )ηW g2 Nz3

1/g20

+ 0.2785δW , and |z1|My ≤tanh(z1/δy)z1 My +0.2785δy, where δd > 0 ∈ R, δW > 0 ∈ R,and δy > 0 ∈ R are the design parameters. Thus

V ′1 ≤ 1

2z2

1+1

2z2

2 + z1α1 + b1,1z21w

21 + b1,2z2

1w1

+ g2z21

8g20

f 21 + g2z2

1

4g20

∫ 1

0

∣∣∣∣∂α∗

1

∂xif1

∣∣∣∣ λdλ

+0.2785(δd + δW + δy) + z1ϕ1(Z1) (11)

where b1,1 = g2/(8g20), b1,2 = Md,1 g2/(4g2

0), and

ϕ1(Z1) = tanh

(z1

δd

)Md,1

g0

+ tanh

(z1

δW

)

ηWg2 Nz2

1

g20

− α∗1

+ tanh

(z1

δy

)

z1 My + g2z1

4g20

∫ 1

0

∣∣∣∣∂α∗

1

∂xi

∣∣∣∣ λdλMd,1.

(12)

Since ϕ1(Z1) is a continuous function with respect to Z1, it canbe approximated by NNs as follows: ϕ1(Z1) = W ∗T

1 S1(Z1)+ε1(Z1). From Assumption 4, one further has

V ′1 ≤ 1

2z2

1+1

2z2

2 + z1α1 + b1,1z21w

21

+ b1,2z21w1 + z1W∗T

1 S1(Z1) + z1Mε,1 + g2z21

8g20

f 21

+ g2z21

2g20

∫ 1

0

∣∣∣∣∂α∗

1

∂xif1

∣∣∣∣ λdλ + 0.2785(δd + δW + δy). (13)

Because |z1|Mε,1 ≤ tanh(z1/δε)z1 Mε,1 + 0.2785δε, whereδε > 0 ∈ R is a design parameter, (18) can be further derivedas

V1 ≤ 1

2z2

1+1

2z2

2 + z1α1 + b1,1z21w

21 + b1,2z2

1w1 + z1

× W∗T1 S1(Z1) + tanh

(z1

δε

)

z1 Mε,1 + g2z21

8g20

(

f 21 + 4

×∫ 1

0

∣∣∣∣∂α∗

1

∂xif1

∣∣∣∣ λdλ

)

+ 0.2785(δd + δW + δy + δε). (14)

Substituting virtual control α1 (3) into (14) yields

V ′1 ≤ −k1z2

1 + 1

2z2

2 + z1W T1 S(Z1) + tanh

(z1

δε

)

× z1 Mε,1 + b1,1z21w

21 + b1,2z2

1w1 + ρ1 + 0.2785δ (15)

where W1 = W∗1 − W , Mε,1 = Mε,1 − Mε,1, b1,1 = b1,1 − b1,1,

b1,2 = b1,2 − b1,2, δ = δd + δW + δy + δε , and

ρ1 = f 21 g2z2

1/(8g20) + (g2z2

1

∫ 1

0|(∂α∗

1)/(∂xi ) f1|λdλ)/(2g20).

Now, consider the following Lyapunov function:

V1 = V ′1 + W T

1 W1/(2ηW )+ M2ε,1/(2ηε)+ (b2

1,1 + b21,2)/(2ηb)

where ηW > 0 ∈ R and ηε > 0 ∈ R. Its time derivative is

V1 ≤ −k1z21 + z2

2/2 + ρ1 + 0.2785δ + b1,1z21w

21 + b1,2

× z21w1 + z1W T

1 S(Z1) + tanh

(z1

δε

)

z1 Mε,1 − W T1

× ˙W 1/ηW − Mε,1˙Mε,1/ηε + (b1,1

˙b1,1 + b1,2˙b1,2)/ηb.

(16)

Substituting adaptive laws (4)–(7) into (16) yields

V1 ≤ −k1z21 + (1/2)z2

2 + ρ1 + 0.2785δ. (17)

Note that ρ1 will be considered in the final step.Step i(2 ≤ i ≤ n − 1): Define the following Lyapunov

function:

V ′i (t) =

∫ zi

0

[∫ 1

0βi dλ

]−1

σdσ

where βi = ∂ fi (xi , xi+1,λσ )/∂xi+1,λσ , xi+1,λσ = λxi+1,σ +(1 − λ)α∗

i , xi+1,σ = −hi (σ ) − W Ti S(Ziσ ), and Ziσ =

722 IEEE TRANSACTIONS ON NEURAL NETWORKS AND LEARNING SYSTEMS, VOL. 25, NO. 4, APRIL 2014

(x Ti , σ, 1)

T, i = 1, 2, . . . , n, xn+1 = u, from Appendix C,

one has

V ′i ≤ zi zi

ai (xi , xi+1)+ g2

2g20

∫ zi

0|ST

i (Ziσ )˙W i |σdσ + g2z2

i

4g20

wi

×i∑

l=1

|( fl + dl)| + g2z2i

4g20

∫ 1

0

i∑

l=1

∣∣∣∣∂α∗

i

∂xl( fl + dl)

∣∣∣∣λdλ.

(18)

Substituting (2) into (18) yields

V ′i ≤ 1

2(z2

i + z2i+1) + ziαi − ziα

∗i + |zi |Md,i

g0

+ g2

2g20

×∫ zi

0|ST

i (Ziσ )˙W i |σdσ − zi αi−1 +

i∑

l=1

g2z2i

4g20

wi | fl |

+i∑

l=1

g2z2i

4g20

wi Md,l +i∑

l=1

g2z2i

4g20

∫ 1

0

∣∣∣∣∂α∗

i

∂xlfl

∣∣∣∣λdλ

+i∑

l=1

g2z2i

4g20

∫ 1

0

∣∣∣∣∂α∗

i

∂xl

∣∣∣∣ λdλ · Md,l . (19)

From Appendix B, one has |STi (Zi)

˙W i | ≤ 2ηW N |zi |.Furthermore, one has

g2/2g20

∫ zi

0|ST

i (Ziσ )˙W i |σdσ ≤ g2/2g2

0

∫ zi

02ηW N |zi |σdσ

= ηW (g2 N/g20)|z3

i |.Thus, (19) is further derived as

V ′i ≤ 1

2(z2

i + z2i+1) + ziαi − ziα

∗i + |zi |Md,i

g0

+ηWg2 N

g20

|z3i | + g2

8g20

i z2i w

2i +

i∑

l=1

g2z2i

8g20

f 2l

+i∑

l=1

g2Md,l

4g20

z2i wi +

i∑

l=1

g2z2i

4g20

∫ 1

0

∣∣∣∣∂α∗

i

∂xlfl

∣∣∣∣λdλ

+i∑

l=1

g2z2i

4g20

∫ 1

0

∣∣∣∣∂α∗

i

∂xl

∣∣∣∣ λdλ · Md,l − zi αi−1.

Since |zi |Md,i/g0

≤ tanh(zi/δd)zi Md,i/g0

+ 0.2785δd andηW g2 N/g2

0|z3

i | ≤ tanh(zi/δW )ηW g2 Nz3i /g2

0+ 0.2785δW , one

has

V ′i ≤ 1

2(z2

i + z2i+1) + ziαi + 0.2785(δd + δW ) + ziϕi

×(Zi ) − zi αi−1 + g2

8g20

i z2i w

2i +

i∑

l=1

g2Md,l

4g20

z2i wi

+i∑

l=1

g2z2i

8g20

f 2l +

i∑

l=1

g2z2i

4g20

∫ 1

0

∣∣∣∣∂α∗

i

∂xlfl

∣∣∣∣λdλ (20)

where

ϕi (Zi ) = −α∗i + tanh

(zi

δd

)zi Md,i

g0

+ tanh

(zi

δW

)

ηW

· g2 Nz3i

g20

+i∑

l=1

Md,lg2z2

i

4g20

∫ 1

0

∣∣∣∣∂α∗

i

∂xl

∣∣∣∣λdλ. (21)

Since ϕi (Zi ) is a continuous function with respect to Zi , it canbe approximated by NNs as ϕi (Zi) = W∗T

i Si (Zi )) + εi (Zi ).Therefore, we derive (20) as

V ′i ≤ 1

2(z2

i + z2i+1) + ziαi + 0.2785(δd + δW + δε) + zi

× W ∗Ti Si (Zi )) + tanh

(zi

δε

)

zi Mε,i

+ bi,1i z2i w

2i + bi,2z2

i wi

+i∑

l=1

g2z2i

8g20

f 2l +

i∑

l=1

g2z2i

4g20

∫ 1

0

∣∣∣∣∂α∗

i

∂xlfl

∣∣∣∣ λdλ − zi αi−1.

(22)

Substituting virtual control αi (3) into (22) yields

V ′i ≤ −ki z

2i + 1

2z2

i+1 + 0.2785δ − zi αi−1 + zi WTi

× Si (Z1σ ) + tanh(zi )zi Mε,i + bi,1i z2i w

2i + bi,2z2

i wi

+i∑

l=1

g2z2i

8g20

f 2l +

i∑

l=1

g2z2i

4g20

∫ 1

0

∣∣∣∣∂α∗

i

∂xlfl

∣∣∣∣ λdλ

where Wi = W∗i − Wi and Mε,i = Mε,i − Mε,i .

Now, consider Lyapunov function

Vi = V ′i + W T

i Wi

2ηW+ M2

ε,i

2ηε+ b2

i,1 + b2i,2

2ηb.

Its time derivative is

Vi = V ′i − W T

i˙W i

ηW− Mε,i

˙Mε,i

ηε− bi,1

˙bi,1 + bi,2˙bi,2

ηb. (23)

Substituting adaptive laws (4)–(7) into (23) yields

Vi ≤ −ki z2i + 1

2z2

i+1 + ρi − zi αi−1 + 0.2785δ (24)

where

ρi =i∑

l=1

g2z2i

8g20

f 2l +

i∑

l=1

g2z2i

4g20

∫ 1

0

∣∣∣∣∂α∗

i

∂xlfl

∣∣∣∣ λdλ.

Remark 3: From the aforementioned analysis, we can findthat the virtual control law αi , i = 1, 2, . . . , n − 1 are thecontinuous functions of system state xi and intermediate vari-able zi . Since xi and zi are available, the first derivative of αi ,i.e., αi can be obtained by analytical computation. However,as stated in Section I, as the system dimension increases, thecomputation of αi becomes increasingly complicated. Becausevirtual control αi contains αi−1 in the existing literatures, thereexists such computation complexity. In this paper, however, itis easily seen that αi−1 does not occur in αi , and thus suchcomputation complexity is reduced.

Step n: Define the following Lyapunov function:

V ′n(t) =

∫ zn

0

[∫ 1

0βndλ

]−1

σdσ

where βn = ∂ fn(xn, uλσ )/∂uλσ , uλσ = λuσ + (1 − λ)α∗n ,

uσ = −hn(σ )σ + W Tn Sn(Znσ ), and Znσ = (x T

n , Z Tn−1, σ, 1)

T.

SHEN et al.: NOVEL NEURAL CONTROL FOR A CLASS OF UNCERTAIN PURE-FEEDBACK SYSTEMS 723

From Appendix C, one has

V ′n ≤ zn zn

an(xn, uλ)+ g2

2g20

[ ∫ zn

0|ST

n (Znσ )˙W n |σdσ

+ 4(2n − 1)z2nw

2n + 2z2

nwn

n∑

l=1

Md,l + My

+ 2z2nwn

n−1∑

l=1

Md,l + ρn

]

(25)

where

ρn = g2

8g20

[

z2n

n∑

l=1

f 2l + 4

n∑

l=1

(∣∣∣∣∂α∗

n

∂xlfl

∣∣∣∣ + Md,l

)

+ z2n f 2

1 +n−1∑

l=2

( fl − αl−1)2

+ 4n−1∑

l=2

(∣∣∣∣∂α∗

n

∂zlfl − αl−1

∣∣∣∣ + Md,l

)

+∣∣∣∣∂α∗

n

∂z1fl

∣∣∣∣ +

∣∣∣∣∂α∗

n

∂z1

∣∣∣∣ My

]

.

Substituting (2) into (25) yields

V ′n ≤ znu − znα∗

n + |zn|Md,n

g0

+ |zn||αn−1|g

0

+ g2

2g20

×∫ zn

0|ST

n (Znσ )˙W n|σdσ + g2

8g20

(2n − 1)z2nw

2n

+ g2z2n

4g20

wn

(n∑

l=1

Md,l + My +n−1∑

l=1

Md,l

)

+ ρn .

(26)

From Appendix B, one has |STn (Zn)

˙W n| ≤ 2ηW N |zn |, furtherhas

g2

2g20

∫ zn

0|ST

n (Zn)˙W n |σdσ ≤ ηW

g2 N

g20

|z3n|. (27)

Substituting (27) into (26) yields

V ′n ≤ znu − znα∗

n + |zn|Md,n

g0

+ ηWg2 N

g20

|z3n|

+ |zn||αn−1|g0

+ g2

8g20

(2n − 1)z2nw

2n + ρn

+ g2z2n

4g20

wn

(n∑

l=1

Md,l + My +n−1∑

l=1

Md,l

)

. (28)

Since (|zn|Md,n/g0) ≤ tanh(zn/δd)(zn Md,n/g

0) + 0.2785δd

and ηW × (g2 N/g20)|z3

n | ≤ tanh((zn/δW ))ηW (g2 Nz3n/g2

0) +

0.2785δW , (28) can be further derived as

V ′n ≤ znu − znα∗

n + 0.2785(δd + δW ) + g2

8g20

[

(2n − 1)z2nw

2n

+ 2z2nwn

(n∑

l=1

Md,l +My +n−1∑

l=1

Md,l

)]

ϕ′n(Zn)+ρn

where ϕ′n(Zn) = −znα

∗n + tanh(zn/δd )zn Md,n/g

0+

ηW tanh(zn/δW )· g2 N/g20z3

n + |zn ||αn−1|/g0.

Define the following Lyapunov function:

V ′ =n−1∑

i=1

Vi + V ′n.

Differentiating V ′ with respect to time t , one has

V ′ ≤ −n−1∑

i=1

ki z2i + (n − 1)0.2785δ + 0.2785(δd + δW )

+ϕ′′n(Zn) + 1

2z2

n + g2

8g20

(2n − 1)z2nw

2n

+ g2z2n

4g20

wn

(n∑

l=1

Md,l + My +n−1∑

l=1

Md,l

)

+ znu

where ϕ′′n(Zn) = ϕ ′

n(Zn) + ∑ni=1 ρi − ∑n−1

i=1 zi αi .Remark 4: Note that the control problem considered in this

paper is to design an adaptive controller u for (1), such thatthe output y follows as accurately the desired trajectory yd aspossible. Since zi = xi − αi−1, i = 1, 2, . . . , n with α0 = yd ,the control problem is to design u such that z1 is as small aspossible. On the other hand, from a practical point of view,once the tracking error reaches the origin, no control actionshould be in order to consume less power. As zn = 0 is hardto detect because of the existence of measurement noise, it ismore practical to relax our control objective of convergenceto a ball rather than to the origin [5]. Similar to [5], let usdefine �z as �z := { zn | |zn | ≤ cz}, where cz > 0 ∈ R isa constant that can be chosen arbitrarily small. Hence, thefollowing analysis is developed in two cases.

Case 1 (zn /∈ �z): Obviously, |zn| > cz > 0 in this case.Function ϕ′′

n thus can be described as znϕ′′

n(Zn)/zn . Let

ϕn(Zn) = ϕ ′′n(Zn)

zn(29)

which is approximated by NNs as ϕn(Zn) = W ∗Tn Sn(Zn) +

εn(Zn), one has

V ′ ≤ −n−1∑

i=1

ki z2i + (n − 1)0.2785δ

+ 0.2785(δd + δW ) + zn W∗Tn Sn(Zn) + zn Mε,n

+ 1

2z2

n + bn,1(2n − 1)z2nw2

n + bn,2z2nwn + znu.

Substituting control law u = αn (3) into the above inequalityyields

V ′ ≤ −n∑

i=1

ki z2i + 0.2785nδ + zn W T

n Sn(Zn)

+zn Mε,n + bn,1nz2nw2

n + bn,2z2nwn

where Wn = W∗n − Wn , Mε,n = Mε,n − Mε,n .

Now, consider Lyapunov function

V = V ′ + 1

2ηWW T

n Wn + 1

2ηεM2

ε,n + 1

2ηb(b2

n,1 + b2n,2).

724 IEEE TRANSACTIONS ON NEURAL NETWORKS AND LEARNING SYSTEMS, VOL. 25, NO. 4, APRIL 2014

Fig. 1. System output y = x1 and desired reference signal yd .

Its time derivative is

V = V ′ − W Tn

˙W n

ηW− Mε,n

˙Mε,n

ηε− bn,1

˙bn,1 + bn,2˙bn,2

ηb.

(30)

Substituting adaptive laws (4)–(7) into (30) yields V ≤−∑n

i=1 ki z2i + 0.2785nδ. It is shown that Vi < 0 as long

as zi remains outside the compact set �z = {zi |∑ni=1 ki z2

i ≤0.2785nδ}. According to an extension of Lyapunov theorem,the signals zi , i = 1, 2, . . . , n are bounded for all the timeand converge to a small enough neighborhood of the originby choosing appropriate parameters δ and ki , i = 1, 2, . . . , n,which means that the system output y(t) can track well thegiven reference signal yd(t) with the tracking error convergingto a small enough neighborhood of the origin.

Case 2 (zn ∈ �z): In this case, just as stated in Remark 4,the tracking control objective has been obtained, which meansthat no control action should be taken, namely u = 0.

From Cases 1 and 2, the control law u should be redesignedas follows:

u ={

− hn(Zn) − W Tn Sn(Zn), zn /∈ �z

0, zn ∈ �z.(31)

From the above analysis, it is easy to obtain that underactual control law (31) and adaptive laws (4)–(7), the trackingerror can be made arbitrarily small by choosing appropriateparameters, such that the output y of (1) can track the givenreference signal yd . The proof is completed.

IV. SIMULATION RESULTS

In this example, a system is described as follows:⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

x1 = x1 + x2 + 1.2 cos(x2) + d1

x2 = x3 + d3

x3 = (x22 + 2x1 + x3) sin(x2)

+ (1 − 0.5 sin(x2))(u + sin(u)) + d2

y = x1

(32)

where the external disturbances are given as d1 = d2 =0.2 sin(π t) and d3 = 0.2 cos(π t). It is easy to find thatg

0= 0.2, g1 = 1.5, and g2 = 1.5.

Fig. 2. Tracking error z1 = y − yd .

Fig. 3. Control signal u.

The control objective is for the system output y to track thedesired reference signal yd . Here, the desired reference signalis taken as yd(t) = 0.4 sin(π t).

In this simulation, the initial conditions are chosen as:system state vector x(0) = (−0.5, 0.5, 0.1)T , weight vectorsW1(0) ∈ R10, W2(0) ∈ R10, and W3(0) ∈ R10 are takenrandomly in the interval (0,1]; the adaptive parameters areas follows: Mε,1(0) = Mε,2(0) = Mε,3(0) = 0, b1,1(0) =b2,1(0) = b3,1(0) = 0, and b1,2(0) = b2,2(0) = b3,2(0) = 0;the designed parameters are given as follows: ηε = ηW =ηb = 1, δd = δW = δε = δy = 0.01, k1 = k2 = k3 = 0.5, andcz = 0.01; the sample time is 0.08 s.

Figs. 1–3 show partial simulation results, which areobtained by applying the virtual control laws (3), the actualcontrol law (31) and adaptive laws (4)–(7) to (32) for trackingthe desired reference signal yd . Fig. 1 shows the trajectoriesof the system output y and the desired reference signal yd .It can be seen that (32) has good tracking performance in thepresence of the external disturbances and the tracking errorz1 = y −yd converges to a small neighborhood around theorigin, as shown in Fig. 2. Boundedness of the control signalu is shown in Fig. 3.

V. CONCLUSION

In this paper, an adaptive neural tracking control approachhas been developed for a class of uncertain pure-feedbacknonlinear systems. NNs are used to approximate the unknownnonlinear functions. By applying the adaptive backstepping

SHEN et al.: NOVEL NEURAL CONTROL FOR A CLASS OF UNCERTAIN PURE-FEEDBACK SYSTEMS 725

technique and Lyapunov function method, an adaptive neuralcontrol scheme is proposed. It is shown that the tracking errorconverges to an adjustable neighborhood of the origin.

APPENDIX A

From the basis function’s definition, one has

∂si, j (zσ )

∂xl= −2(xl − ai,h, j )

c2i,h, j

exp

(

−∑N

j=1 (Zi − ai,h, j )2

(ci,h, j )2

)

where l = 1, 2, . . . , n. Since

exp

(

−∑N

j=1 (Zi − ai,h, j )2

(ci,h, j )2

)

≤ exp

(−(xl − ai,h, j )

2

(ci,h, j )2

)

one has∣∣∣∣∂si, j (zσ )

∂xl

∣∣∣∣ ≤ 2|xl − ai,h, j |

c2i,h, j

exp

(−(xl − ai,h, j )

2

(ci,h, j )2

)

.

It is easy to find, if 2|xl − ai,h, j |/ci,h, j = (1/2)1/2, then|(∂si, j (ziσ )/∂xl)| has the maximum value

∣∣∣∣∂si, j (Zi)

∂xl

∣∣∣∣ ≤

√2

ci,h, je(− 1

2 ).

Note that ||Wi || ≤ wm , which is guaranteed by the projec-tion adaptive law to be designed in Section III. Hence, everyelement Wi,h of Wi has the property: |wi,h | ≤ wm , wherewm > 0 ∈ R is a design parameter.

Define wi = ∑Nj=1 (

√2/ci,h, j )exp(−1/2)wm ; therefore,

one has∣∣∣∣∂Si (zσ )

∂xl( fl + dl)

∣∣∣∣ ≤ wi | fl + dl |.

APPENDIX B

In Section III, the adaptive law concerning Wi is designedas follows:

˙W i =

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

− ηW Si (Zi )zi , if ||Wi || < wm or

||Wi || = wm and −WT

Si (Zi )zi ≤ 0

− ηW Si (Zi )zi + ηW Wi W Ti Si (Zi )zi ||Wi ||2,

if ||Wi || = wm and − W Ti Si (Zi )zi > 0.

Obviously, ˙W i depends on Zi and zi .

1) If ||Wi || < wm or ||Wi || = wm and W Ti Si (Ziσ )zi ≤ 0,

then |STi (Ziσ )

˙W i | ≤ STi (Ziσ )S(Ziσ )|zi |.

Since |si,l (Ziσ )| < 1, |ST (Ziσ )S(Ziσ )| ≤ N . Therefore

|STi (Ziσ )

˙W i | ≤ 2ηW N |zi |where N is the number of the NNs nodes.

2) If ||Wi || = wm and W Ti Si (Ziσ )zi > 0, then

|STi (Ziσ )

˙W i | ≤ ηW

∣∣∣∣

(

STi (Ziσ )Si (Ziσ )

+ STi (Ziσ )Wi W T

i Si (Ziσ )

||Wi ||2)

zi

∣∣∣∣.

Since |si,h (Ziσ )| < 1, |STi (Ziσ )Si (Ziσ )| ≤ N . There-

fore, one has

STi (Ziσ )Wi W T

i Si (Ziσ ) ≤ ||Wi ||1.Noting that ||Wi ||21 ≤ N ||Wi ||2, one further has

|STi (Ziσ )

˙W i | ≤ ηW |zi |[ N + (||Wi ||21/||Wi ||2)] ≤2ηW N |zi |.

Hence, by 1) and 2), it results in |STi (ziσ )

˙W i | ≤ 2ηW N |zi |.Further, one has

g2

2g20

∫ zi

0|ST

i (Ziσ )˙W i |σdσ ≤ g2

2g20

∫ zi

02ηW N |zi |σdσ

= ηWg2 N

g20

|z3i |.

APPENDIX CSince

V ′i (t) =

∫ zi

0

[∫ 1

0βi dλ

]−1

σdσ

where βi = ∂ fi (xi , xi+1,λσ )/∂xi+1,λσ , xi+1,λσ = λxi+1,σ +(1 − λ)α∗

i , xi+1,σ = −hi (σ ) − W Ti S(Ziσ ), and Ziσ =

(x Ti , σ, 1)

T, i = 1, 2, . . . , n, xn+1 = u, its time derivative is

V ′i = zi

[∫ 1

0βi dλ

]−1

zi

−∫ zi

0

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

i∑

l=1

∫ 10

∂βi∂xi+1,λσ

∂xi+1,λσ

∂xlxldλ

[∫ 10 βi dλ]2

⎫⎪⎪⎪⎬

⎪⎪⎪⎭

σdσ

−∫ zi

0

⎧⎪⎨

⎪⎩

∫ 10

∂βi∂xi+1,λσ

(∂xi+1,λσ

∂Wi

)T ˙W i dλ

[∫ 10 βi dλ]2

⎫⎪⎬

⎪⎭σdσ . (33)

From Assumption 2 and xi = fi +di , i = 1, 2, . . . , n−1, (33)is further derived as

V ′i ≤ zi zi

ai (xi , xi+1)+ 1

g20

∫ z1

0

∣∣∣∣∣

i∑

l=1

∫ 1

0βix dλ

∣∣∣∣∣σdσ

+ 1

g20

∫ z1

0

∣∣∣∣

∫ 1

0βiW dλ

∣∣∣∣σdσ (34)

where βix = (∂βi/∂xi+1,λσ )(∂xi+1,λσ /∂xl)( fl + di )

and βiW = (∂βi/∂xi+1,λσ )· (∂xi+1,λσ /∂Wi )T ˙W i . Since

(∂xi+1,λσ /∂xl) = λ(xi+1,σ /∂xl) + (1 − λ)∂α∗i /∂xl =

λ(∂[W Ti S(Ziσ )]/∂xl) + (1 − λ)(∂α∗

i /∂xl), one has∣∣∣∣∣

i∑

l=1

∫ 1

0βix dλ

∣∣∣∣∣

≤i∑

l=1

∣∣∣∣

∫ 1

0

∂βi

∂xi+1,λσλdλ

∣∣∣∣

×∣∣∣∣∣∣

∫ 1

0

∂[W T

i Si (Ziσ )]

∂xldλ

∣∣∣∣∣∣| fl + dl |

+i∑

l=1

∣∣∣∣

∫ 1

0

∂βi

∂xi+1,λσ(1 − λ)dλ

∣∣∣∣

×∣∣∣∣∂α∗

i

∂xl( fl + dl)

∣∣∣∣ . (35)

726 IEEE TRANSACTIONS ON NEURAL NETWORKS AND LEARNING SYSTEMS, VOL. 25, NO. 4, APRIL 2014

From Assumption 2, (35) can be further derived as∣∣∣∣∣

i∑

l=1

∫ 1

0βi X dλ

∣∣∣∣∣≤ g2

2

i∑

l=1

∣∣∣∣∣

∫ 1

0

∂[W Ti S(zσ )]∂xl

dλ

∣∣∣∣∣| fl + dl |

+ g2

2

i∑

l=1

∣∣∣∣∂α∗

i

∂xi( fi + di )

∣∣∣∣. (36)

From Appendix A, we know that there exists a constant wi >0such that |W T

i (∂[Si (Zi )]/∂xl)| ≤ wi . Furthermore, one has

g2

2

i∑

l=1

∣∣∣∣∣

∫ 1

0

∂[W Ti Si (Ziσ )]∂xl

dλ

∣∣∣∣∣| fl + dl |

≤ g2

2

i∑

l=1

(wi | fl + dl |).

Thus, (36) can be derived as∣∣∣∣∣

i∑

l=1

∫ 1

0βix dλ

∣∣∣∣∣≤ g2

2wi

i∑

l=1

|( fl + dl)|

+ g2

2

i∑

l=1

∣∣∣∣∂α∗

i

∂xi( fl + dl)

∣∣∣∣.

Furthermore, one has

1

g20

∫ zi

0

∣∣∣∣∣

i∑

l=1

∫ 1

0βix dλ

∣∣∣∣∣≤ g2z2

i

4g20

wi

i∑

l=1

|( fl + dl)|

+ g2

2g20

∫ zi

0

i∑

l=1

∣∣∣∣∂α∗

i

∂xl( fl + dl)

∣∣∣∣σdσ .

(37)

In addition, since (∂xi+1,λσ /∂Wi ) = λ(∂xi+1,λσ /∂Wi ) =λSi (Ziσ ), one has

∣∣∣∣

∫ 1

0βiW dλ

∣∣∣∣ ≤ g2

2|ST

i (Ziσ )˙W i |.

Furthermore, one has

1

g20

∫ zi

0

∣∣∣∣∣

∫ 1

0

∂βi

∂xi+1,λσ

(∂xi+1,λσ

∂Wi

)T ˙W i dλ

∣∣∣∣∣σdσ

≤ g2

2g20

∫ zi

0|ST

i (Ziσ )˙W i |σdσ.

(38)

It follows from (34), (37), and (38) that

V ′i ≤ zi zi

ai(xi , xi+1)+ g2

2g20

∫ zi

0|ST

i (Ziσ )˙W i |σdσ + g2z2

i

4g20

wi

×i∑

l=1

| fl + dl |+ g2

2g20

∫ z1

0

i∑

l=1

∣∣∣∣∂α∗

i

∂xl( fl + dl)

∣∣∣∣σdσ .

(39)

Since α∗i (xi ) = α∗

i (x1, . . . , xi ), one has

g2

2g20

∫ z1

0

i∑

l=1

∣∣∣∣∂α∗

i

∂xl( fl + dl)

∣∣∣∣σdσ

= g2z2i

4g20

∫ 1

0

i∑

l=1

∣∣∣∣∂α∗

i

∂xl( fl + dl)

∣∣∣∣λdλ.

Hence, (39) can be further derived as follows:

V ′i ≤ zi zi

ai (xi , xi+1)+ g2

2g20

∫ zi

0|ST

i (Ziσ )˙W i |σdσ + g2z2

i

4g20

wi

·i∑

l=1

|( fl + dl)|+ g2z2i

4g20

∫ 1

0

i∑

l=1

∣∣∣∣∂α∗

i

∂xl( fl + dl)

∣∣∣∣λdλ.

(40)

REFERENCES

[1] Y. P. Zhang, P. Y. Peng, and Z. P. Jiang, “Stable neural controllerdesign for unknown nonlinear systems using backstepping,” IEEETrans. Neural Netw., vol. 11, no. 6, pp. 1347–1360, Nov. 2000.

[2] M. Wang, B. Chen, and P. Shi, “Adaptive neural control for a class ofperturbed strict-feedback nonlinear time-delay systems,” IEEE Trans.Syst., Man, Cybern. B, Cybern., vol. 38, no. 3, pp. 721–730, Jun. 2008.

[3] J. Liu, W. Wang, S. C. Tong, and Y.-S. Liu, “Robust adaptive trackingcontrol for nonlinear systems based on bounds of fuzzy approximationparameters,” IEEE Trans. Syst., Man, Cybern. A, Syst. Humans, vol. 40,no. 1, pp. 170–184, Jan. 2010.

[4] H. Lee, “Robust adaptive fuzzy control by backstepping for a class ofMIMO nonlinear systems,” IEEE Trans. Fuzzy Syst., vol. 19, no. 2,pp. 265–275, Apr. 2011.

[5] S. S. Ge and K. P. Tee, “Approximation-based control of nonlinearMIMO time-delay systems,” Automatica, vol. 43, no. 1, pp. 31–43,2007.

[6] T. P. Zhang and S. S. Ge, “Adaptive neural network tracking controlof MIMO nonlinear systems with unknown dead zones and controldirections,” IEEE Trans. Neural Netw., vol. 20, no. 3, pp. 483–497,Mar. 2009.

[7] M. Krstic, I. Kanellakopoulos, and P. Kokotovic, Nonlinear and Adap-tive Control Design. Hoboken, NJ, USA: Wiley, 1995.

[8] T. C. Lin and T. Y. Lee, “Chaos synchronization of uncertain fractional-order chaotic systems with time delay based on adaptive fuzzy slidingmode control,” IEEE Trans. Fuzzy Syst., vol. 19, no. 4, pp. 623–635,Aug. 2011.

[9] Z. J. Li, X. Q. Cao, and N. Ding, “Adaptive fuzzy control forsynchronization of nonlinear teleoperators with stochastic time-varyingcommunication delays,” IEEE Trans. Fuzzy Syst., vol. 19, no. 4,pp. 745–757, Aug. 2011.

[10] Y. P. Pan, M. J. Er, D. P. Huang, and Q. R. Wang, “Adaptive fuzzycontrol with guaranteed convergence of optimal approximation error,”IEEE Trans. Fuzzy Syst., vol. 19, no. 5, pp. 807–818, Oct. 2011.

[11] A. B. Cara, H. Pomares, and I. Rojas, “A new methodology for theonline adaptation of fuzzy self-structuring controllers,” IEEE Trans.Fuzzy Syst., vol. 19, no. 3, pp. 449–464, Jun. 2011.

[12] A. Lemos, W. Caminhas, and F. Gomide, “Multivariable Gaussianevolving fuzzy modeling system,” IEEE Trans. Fuzzy Syst., vol. 19,no. 1, pp. 91–104, Feb. 2011.

[13] Y. C. Hsueh, S. F. Su, C. W. Tao, and C. C. Hsiao, “Robust L2-gaincompensative control for direct-adaptive fuzzy-control-system design,”IEEE Trans. Fuzzy Syst., vol. 18, no. 4, pp. 661–673, Aug. 2010.

[14] D. Wang and J. Huang, “Adaptive neural network control of a classof uncertain nonlinear systems in pure-feedback form,” Automatica,vol. 42, no. 8, pp. 1365–1372, 2002.

[15] S. S. Ge and C. Wang, “Adaptive NN control of uncertain nonlinearpure-feedback systems,” Automatica, vol. 38, no. 4, pp. 671–684, 2002.

[16] C. Wang, D. J. Hill, S. S. Ge, and G. Chen, “An ISS-modular approachfor adaptive neural control of pure-feedback systems,” Automatica,vol. 42, no. 5, pp. 671–684, 2006.

[17] T. P. Zhang and S. S. Ge, “Adaptive dynamic surface control ofnonlinear systems with unknown dead zone in pure feedback form,”Automatica, vol. 44, no. 7, pp. 1895–1903, 2008.

SHEN et al.: NOVEL NEURAL CONTROL FOR A CLASS OF UNCERTAIN PURE-FEEDBACK SYSTEMS 727

[18] T. P. Zhang, H. Wen, and Q. Zhu, “Adaptive fuzzy control of nonlinearsystems in pure-feedback form based on input-to state stability,” IEEETrans. Fuzzy Syst., vol. 18, no. 1, pp. 80–93, Feb. 2010.

[19] M. Wang, S. S. Ge, and K. S. Hong, “Approximation-based adaptivetracking control of pure-feedback nonlinear systems with multipleunknown time-delays,” IEEE Trans. Neural Netw., vol. 21, no. 11,pp. 1804–1816, Nov. 2010.

[20] A. M. Zou and Z. G. Hou, “Adaptive control of a class of nonlinearpure-feedback systems using fuzzy backstepping approach,” IEEETrans. Fuzzy Syst., vol. 16, no. 4, pp. 886–897, Aug. 2008.

[21] M. Wang, X. Liu, and P. Shi, “Adaptive neural control of pure-feedbacknonlinear time-delay systems via dynamic surface technique,” IEEETrans. Syst., Man, Cybern. B, Cybern., vol. 41, no. 6, pp. 1681–1692,Dec. 2011.

[22] D. Wang, “Neural network-based adaptive dynamic surface control ofuncertain nonlinear pure-feedback systems,” Int. J. Robust NonlinearControl, vol. 21, no. 5, pp. 527–541, 2011.

[23] S. C. Tong, Y. M. Li, and P. Shi, “Observer-based adaptive fuzzybackstepping output feedback control of uncertain MIMO pure-feedback nonlinear systems,” IEEE Trans. Fuzzy Syst., vol. 20, no. 4,pp. 771–784, Aug. 2012.

[24] S. S. Ge, F. Hong, and T. H. Lee, “Adaptive neural control of nonlineartime-delay systems with unknown virtual control coefficients,” IEEETrans. Syst., Man, Cybern. B, Cybern., vol. 34, no. 1, pp. 499–516,Feb. 2004.

[25] Z. Wu, J. Yang, and P. Shi, “Adaptive tracking for stochastic nonlinearsystems with Markovian switching,” IEEE Trans. Autom. Control,vol. 55, no. 9, pp. 2135–2141, Sep. 2010.

[26] C. Hua, X. Guan, and P. Shi, “Robust backstepping control for a classof time delayed systems,” IEEE Trans. Autom. Control, vol. 50, no. 6,pp. 894–899, Jun. 2005.

[27] Y. Xia, M. Fu, P. Shi, Z. Wu, and J. Zhang, “Adaptive backsteppingcontroller design for stochastic jump systems,” IEEE Trans. Autom.Control, vol. 54, no. 12, pp. 2853–2859, Dec. 2009.

[28] H. Zhang, Y. Luo, and D. Liu, “Neural-network-based near-optimalcontrol for a class of discrete-time affine nonlinear systems with controlconstraints,” IEEE Trans. Neural Netw., vol. 20, no. 9, pp. 1490–1503,Sep. 2009.

[29] H. Zhang, Z. Wang, and D. Liu, “Global asymptotic stability ofrecurrent neural networks with multiple time varying delays,” IEEETrans. Neural Netw., vol. 19, no. 5, pp. 855–873, May 2008.

[30] M. M. Polycarpou and P. A. Ioannou, “A robust adaptive nonlinearcontrol design,” Automatica, vol. 31, no. 3, pp. 423–427, 1995.

[31] C. Hsu, C. Chiu, and J. Tsai, “Master-slave chaos synchronizationusing an adaptive dynamic sliding-mode neural control system,” Int.J. Innovative Comput., Inf. Control, vol. 8, no. 1A, pp. 121–138, 2012.

[32] R. Wang and J. Li, “Adaptive neural control design for a class ofperturbed nonlinear time-varying delay systems,” Int. J. InnovativeComput., Inf. Control, vol. 8, no. 5B, pp. 3727–3740, 2012.

[33] M. Wang, C. Wang, and S. S. Ge, “A novel ISS-modular adaptiveneural control of pure-feedback nonlinear systems,” Int. J. InnovativeComput., Inf. Control, vol. 7, no. 11, pp. 6559–6570, 2011.

[34] Q. Shen, B. Jiang, and P. Shi, “Fault diagnosis for T-S fuzzy systemswith sensor faults and system performance analysis,” IEEE Trans FuzzySyst., 2013, doi: 10.1109/TFUZZ.2013.2252355.

[35] Q. Shen, B. Jiang, P. Shi, and J. Zhao, “Cooperative adaptive fuzzytracking control for networked unknown nonlinear multi-agent systemswith time-varying actuator faults,” IEEE Trans. Fuzzy Syst., 2013, doi:10.1109/TFUZZ.2013.2260757.

Qikun Shen received the B.Sc. degree in computerscience and applications from the Chinese Univer-sity of Mining and Technology, Xuzhou, China, in1996, and the M.Sc. degree in computer science andapplications from Yangzhou University, Yangzhou,China, in 2007. He is currently pursuing the Ph.D.degree with the College of Automation Engineering,Nanjing University of Aeronautics and Astronautics,Nanjing, China.

He is currently an Associate Professor with theCollege of Information Engineering, Yangzhou Uni-

versity. His current research interests include fault-tolerant control, adaptivecontrol, fuzzy control, and intelligent control.

Peng Shi (M’95–SM’98) received the B.Sc. degreein mathematics from the Harbin Institute of Technol-ogy, Harbin, China, the M.E. degree in systems engi-neering from Harbin Engineering University, Harbin,the Ph.D. degree in electrical engineering from theUniversity of Newcastle, Callaghan, Australia, thePh.D. degree in mathematics from the Universityof South Australia, Adelaide, Australia, and theD.Sc. degree from the University of Glamorgan,Glamorgan, U.K.

He was a Lecturer with Heilongjiang University,Harbin, as a Post-Doctoral and Lecturer with the University of South Australia,a Senior Scientist with the Defence Science and Technology Organisation,Australia, and a Professor with the University of Glamorgan (now TheUniversity of South Wales). He is currently a Professor with The Universityof Adelaide, Adelaide, and Victoria University, Melbourne, Australia. Hiscurrent research interests include system and control theory, computationalintelligence, and operational research.

Dr. Shi is a fellow of the Institution of Engineering and Technology, U.K.,and the Institute of Mathematics and its Applications, U.K. He has been onthe editorial board of a number of journals, including Automatica, the IEEETRANSACTIONS ON AUTOMATIC CONTROL, the IEEE TRANSACTIONS ON

FUZZY SYSTEMS, the IEEE TRANSACTIONS ON CYBERNETICS, the IEEETRANSACTIONS ON CIRCUITS AND SYSTEMS-I, and IEEE ACCESS.

Tianping Zhang received the B.Sc. degree inmathematics from Yangzhou Teachers College,Yangzhou, China, in 1986, the M.Sc. degree inoperations research and control theory from EastChina Normal University, Shanghai, China, in 1992,and the Ph.D. degree in automatic control theoryand applications from Southeast University, Nanjing,China, in 1996.

He is currently a Professor with the Collegeof Information Engineering, Yangzhou University,Yangzhou. From October 2005 to October 2006, he

was a Visiting Scientist with the Department of Electrical and ComputerEngineering, National University of Singapore, Singapore. He has publishedmore than 180 papers in journals and conference proceedings. His currentresearch interests include fuzzy control, adaptive control, intelligent control,and nonlinear control.

Cheng-Chew Lim (M’82–SM’02) received theB.Sc. (Hons.) degree in electronic and electricalengineering and the Ph.D. degree from Loughbor-ough University, Leicestershire, U.K., in 1977 and1981, respectively.

He is an Associate Professor and a Reader ofelectrical and electronic engineering and the Headof School of Electrical and Electronic Engineering,The University of Adelaide, Adelaide, Australia.His current research interests include control sys-tems, machine learning, wireless communications,

and optimization techniques and applications.Dr. Lim is currently serving as the Chairman of the IEEE Chapter on Control

and Aerospace Electronic Systems at the IEEE South Australia Section. Heis serving as an Editorial Board Member for the Journal of Industrial andManagement Optimization and has served as a Guest Editor of a number ofjournals, including Discrete and Continuous Dynamical Systems-Series B.