IEEE TRANSACTIONS ON NEURAL NETWORKS AND LEARNING SYSTEMS, VOL. 25, NO. 12, DECEMBER 2014 2129

Adaptive Neural Control for a Class of NonlinearTime-Varying Delay Systems With

Unknown HysteresisZhi Liu, Guanyu Lai, Yun Zhang, Xin Chen, and Chun Lung Philip Chen, Fellow, IEEE

Abstract— This paper investigates the fusion of unknowndirection hysteresis model with adaptive neural control tech-niques in face of time-delayed continuous time nonlinear systemswithout strict-feedback form. Compared with previous workson the hysteresis phenomenon, the direction of the modifiedBouc–Wen hysteresis model investigated in the literature isunknown. To reduce the computation burden in adaptationmechanism, an optimized adaptation method is successfullyapplied to the control design. Based on the Lyapunov–Krasovskiimethod, two neural-network-based adaptive control algorithmsare constructed to guarantee that all the system states andadaptive parameters remain bounded, and the tracking errorconverges to an adjustable neighborhood of the origin. In final,some numerical examples are provided to validate the effective-ness of the proposed control methods.

Index Terms— Adaptive neural control, Bouc–Wen hysteresis,nonlinear control, nonstrict-feedback system, unknown directionhysteresis.

I. INTRODUCTION

OVER THE past few decades, approximation-based adap-tive backstepping control for nonlinear systems has

received a great deal of attention, and some significant resultsare presented in [1]–[20]. Among them, [1]–[7] are forstrict-feedback systems in single-input and single-output formand [8]–[11] research the multiple-input and multiple-output(MIMO) nonlinear systems, while the nonlinear systems withimmeasurable states or time delays are investigated in [4],

Manuscript received May 14, 2013; accepted February 5, 2014. Date ofpublication February 25, 2014; date of current version November 17, 2014.This work was supported in part by the National Natural Science Foundationof China under Project 60974047 and Project U1134004, in part by the NaturalScience Foundation of Guangdong Province under Grant S2012010008967,in part by the Science Fund for Distinguished Young Scholars under GrantS20120011437, in part by the 2011 Zhujiang New Star, in part by theMinistry of education of New Century Excellent Talent under Grant NCET-12-0637, in part by the 973 Program of China under Grant 2011CB013104,in part by the Doctoral Fund of Ministry of Education of China under Grant20124420130001, and in part by the University of Macau Multiyear ResearchGrants.

Z. Liu, G. Lai, and Y. Zhang are with the Faculty of Automation,Guangdong University of Technology, Guangzhou 510006, China (e-mail:lz@gdut.edu.cn; lgy124@foxmail.com; yz@gdut.edu.cn).

X. Chen is with the College of Mechanical and Electrical Engineering,Guangdong University of Technology, Guangdong 510006, China (e-mail:chenx@gdut.edu.cn).

C. L. P. Chen is with the Faculty of Science and Technology, University ofMacau, Macau 99999, China (e-mail: philip.chen@ieee.org).

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.2014.2305717

[5], and [13]–[20]. In these control schemes, fuzzy logicsystems (FLSs) or neural networks (NNs) are employed toapproximate the unknown nonlinear functions. Based on thebackstepping or dynamic surface control (DSC) technique,adaptive fuzzy controller or adaptive neural controller are con-structed to guarantee a good tracking performance. However,all the aforementioned control schemes are only suitable tothe nonlinear systems in strict-feedback form, which is in facta special case of the nonstrict-feedback form proposed in [21]recently.

Different from the conventional strict-feedback systems,the semistrict-feedback systems, or the affine pure-feedbacksystems, the unknown nonlinear functions in the nonstrict-feedback system contain whole state variable, i.e., they cannotbe directly compensated by the approximators with currentstates. In addition, the virtual control signals require to bethe functions of current states to guarantee their existence.To deal with these difficulties, Chen et al. [21] have proposeda variable separation technique. Based on the Lyapunov syn-thesis and the FLSs technique, an adaptive control schemewas constructed in [21] to guarantee the stability of nonstrict-feedback control system. Although much progress has beenmade in adaptive control field in [1]–[19] and [21], somechallenging problems still remain. In practical application, theactuator of control system is often subjected to the nonsmoothcharacteristics, such as the hysteresis nonlinearity in [29], [30],[34]–[39], [41], [45], [57], and [74], the saturation nonlinearityin [22]–[28], [31], and [33], the dead-zone nonlinearity in[20], [43], and [44], and the backlash nonlinearity in [40].Especially, the hysteresis phenomenon in the physical systemsand devices severely limits the tracking performance and easilycauses the instability [57]. Therefore, control of nonlinearsystems preceded by the nonsmooth hysteresis characteristichas become one of the active topics.

To address the control problem of nonsmooth hysteresisnonlinearity in the actuator, the authors in [30], [34]–[39],[41], [45], [57], and [74] have made outstanding contributionsin the fusion of hysteresis models with available robust controltechniques. These hysteresis models can be classified intofour classes, which are the backlash-like hysteresis model in[30], [34], [45], and [74], the conventional Prandtl–Ishlinskii(P–I) hysteresis model in [35] and [36], the generalized P–Ihysteresis model in [37] and [42], and the modified Bouc–Wenhysteresis model in [41], respectively. In [34], the backlash-like hysteresis model was first proposed to describe a class of

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2130 IEEE TRANSACTIONS ON NEURAL NETWORKS AND LEARNING SYSTEMS, VOL. 25, NO. 12, DECEMBER 2014

special hysteresis phenomenon in the actuator. By treating theeffect of backlash-like hysteresis as a bounded disturbance,the proposed model can be conveniently fused with availablerobust control techniques. Furthermore, Su et al. [35] havefused a more general conventional P–I hysteresis model withadaptive variable structure control approach. Based on theP–I model, an adaptive control algorithm was proposed in[35] to guarantee a good tracking performance. To enlargethe class of applicable systems, Ren et al. [36] considereda class of MIMO nonlinear systems with the conventionalP–I hysteresis input and time-varying delays. By using theNNs technique to approximate the unknown nonlinear func-tions, the restriction of the linearly parameterized unknownuncertainties can be removed. The effect of time-varyingdelays can be canceled by designing an appropriate Lyapunov–Krasovskii function. Therefore, an approximation-based adap-tive control scheme was developed in [36] to guarantee thesemiglobally uniformly ultimately bounded of control system.To accommodate more general classes of hysteresis shapes, ageneralized P–I hysteresis model was proposed in [42]. Forthe conventional P–I hysteresis model, the different hysteresisshapes are described by adjusting the density function only.For the generalized case, both the density function and theinput function can be utilized to formulate the different hys-teresis shapes, i.e., compared with the conventional case, thegeneralized P–I hysteresis model can capture the hysteresisphenomenon more accurately. Ren et al. [37] considered aclass of nonaffine pure-feedback nonlinear systems with thegeneralized P–I hysteresis input. To deal with the nonaffineproblem in the presence of nonsmooth characteristic of hys-teresis, the mean-value theorem was introduced. By using thebackstepping and NNs techniques, the control approach in [37]can guarantee the uniformly ultimately bounded of control sys-tem, and the tracking error converges to a small neighborhoodof the origin. To relax the restrictions about the density func-tion in P–I hysteresis model, a modified Bouc–Wen hysteresismodel was first proposed in [41]. By constructing a perfectinverse to compensate the effect of hysteresis nonlinearity, anadaptive control approach was provided in [41] to guaranteea good tracking performance. However, the directions of theaforementioned four classes of hysteresis models require tobe known. If the direction of hysteresis phenomenon in theactuator is unknown, the control approaches mentioned abovecannot be directly applied.

Motivated by the aforementioned researches, this paperfocuses on the problem of adaptive neural control for aclass of continuous time (CT) nonstrict-feedback systems withunknown direction modified Bouc–Wen hysteresis input andtime-varying delays. The main contributions in this literaturecan be summarized as follows.

1) The hysteresis models studied in [29], [30], [34]–[39],[41], [45], [57], and [74] are known direction hystere-sis, i.e., though the hysteresis parameter (it refers tothe coefficient or the function governing the hysteresisdirection) may be unknown, its sign must be known asprior knowledge. If the hysteresis direction is unknown,the aforementioned control schemes cannot be directlyapplied. In this paper, we investigate the fusion of

the unknown direction hysteresis with adaptive neuralcontrol approaches.

2) Compared with the unknown nonlinear functions ofcurrent states in conventional strict-feedback systems,the unknown functions in the nonstrict-feedback systemcontain whole state variables, i.e., more NN nodesshould be utilized to obtain a good approximated per-formance. If we still employ the conventional adaptationmechanism, it will lead to unacceptable large learningtime as the number of NN nodes increases. To solvethese difficulties in face of the control system withoutstrict-feedback form, an optimized method is included inthe adaptation mechanism in this literature. By employ-ing the optimized adaptation method, the number ofadaptive parameters does not increase as the NN nodesincrease, and the less knowledge about the centers andwidths of basis functions are needed for the controllerdesign.

3) Compared with the nonlinear systems in Brunovskyform or nonaffine pure-feedback form studied in[34]–[41], the control system investigated in this paperis the CT nonstrict-feedback form, which is in fact amore general case [21]. Therefore, the investigation inthis paper enlarges the class of applicable systems withhysteresis phenomenon.

II. PROBLEM FORMULATION AND SOME PRELIMINARIES

A. Nonlinear Control Problem

Consider the following nonstrict-feedback system withunknown hysteresis input and time-varying delays:

x = Ax + f (x) + h(x(τ (t))) + Bu + �(t, x)

u = H (v), y = CT x (1)

where

A =[

0 In−10 0

], B = [0, . . . , 0, 1]T ∈ Rn

C = [1, 0, . . . , 0]T ∈ Rn

f (x) = [. . . , fi (x), . . . , fn(x)]T , i = 1, . . . , n − 1

h(x(τ (t))) = [. . . , hi (x(t − τi (t))), . . . , hn(x(t − τn(t)))]T

�(t, x) = [. . . , ωi (t, x), . . . , ωn(t, x)]T (2)

where x = [x1, x2, . . . , xn]T ∈ Rn are the whole statevariables of control system. f (x) ∈ Rn , v ∈ R, and y(t) ∈ Rdenote the unknown smooth function of whole states, systeminput, and system output, respectively. �(t, x) ∈ Rn are theexternal disturbances and h(x(τ (t))) ∈ Rn are the unknownsmooth functions of time-varying time-delayed states. In−1 isan identity matrix. Here, the control signal is subjected to theunknown hysteresis nonlinearity H (v) : R → R given in thelater.

Remark 1: Different from the conventional strict-feedbacknonlinear systems, the unknown functions fi (x), hi (x), andωi (t, x) in the differentiation (1) are evidently the functions ofwhole state variables. Hence, these terms cannot be dealt withas done in most of the previous works, where they are approx-imated by the NNs of current states. To solve the difficulty,

LIU et al.: ADAPTIVE NEURAL CONTROL FOR A CLASS OF NONLINEAR TIME-VARYING DELAY SYSTEMS 2131

Chen et al. [21] have proposed a variable separation technique.However, the fusion of unknown direction hysteresis withthese adaptive control techniques is still a challenging prob-lem, which directly motivates the investigation of this paper.

Remark 2: The hysteresis phenomenon has been inves-tigated in many previous works, such as the backlash-likehysteresis model in [30], [34], [45], and [74], the conven-tional P–I hysteresis model in [35] and [36], the generalizedP–I hysteresis model in [37] and [42], and the modifiedBouc–Wen hysteresis model in [41]. However, the directionsof these hysteresis models require to be known. If the practicalhysteresis direction is unknown, the aforementioned controlschemes cannot be directly applied. To the best of our knowl-edge, it is the first time, in the literature, that the fusionof unknown direction hysteresis with adaptive neural controltechniques is investigated.

The objective of this paper is to design a robust adaptiveneural controller for system (1) to guarantee two points.

1) All the system states and the adaptive parameters arebounded.

2) The tracking error z1(t) = y(t) − yr (t) converges toan adjustable neighborhood of the origin, where y(t) isthe output of control system and yr (t) is the referencesignal. Specifically, for any ε > 0, there exists a timevalue T (ε) > 0 such that z2

1(t) ≤ ε,∀t > T .

Assumption 1: For 1 ≤ i ≤ n, there exists an unknownpositive function �i (x) such that

|ωi (t, x)| ≤ �i (x). (3)

Remark 3: Similar external disturbances �i (t, x) wereconsidered in [69], where they should satisfy the restriction|wi (t, x)| ≤ �i (xi ), xi = [x1, . . . , xi ]T ∈ Ri . From thispoint of view, Assumption 1 is in fact a relaxed version of therestriction in [69]. Therefore, the external disturbances studiedin this paper are more general.

Assumption 2: For 1 ≤ i ≤ n, consider τi (t) is thetime-varying time delay. It requires to satisfy the followinginequalities:

0 < τi (t) ≤ τmax τi ≤ τmax < 1 (4)

where τmax and τmax are known constants.

B. Hysteresis Nonlinearity

A modified Bouc–Wen hysteresis nonlinearity can be for-mulated as follows [41]:

H (v) = μ1v + μ2ζ (5)

where μ1 and μ2 are the constants with the same signs, i.e.,sign(μ1) = sign(μ2). ζ can be specified as

ζ = v − β|v||ζ |n−1ζ − χv|ζ |n = v f (ζ, v), ζ(t0) = 0 (6)

where β > |χ |, n > 1. The shape and amplitude of thehysteresis can be described by the parameters β while thesmoothness from initial slope to asymptote’s slope can bedescribed by the parameter n. The auxiliary variable ζ is

TABLE I

SIMULATION PARAMETERS

Fig. 1. Trajectories of positive direction hysteresis and negative directionhysteresis.

bounded by the constant ζ , i.e., |ζ | ≤ ζ = n√

1/(β + χ). Thefunction f (ζ, v) can be formulated as follows:

f (ζ, v) = 1 − sign(v)β|ζ |n−1ζ − χ |ζ |n. (7)

Remark 4: It should be noted that the sign of μ1 governs thedirection of hysteresis. The issue can be clearly illustrated bythe following numerical example. The simulation parametersare given in Table I. By selecting the input signal v(t) ofhysteresis actuator as v(t) = sin(2t), the graphical result isshown in Fig. 1.

C. Neural Networks

In most of the robust adaptive neural control schemes, theradial basis function neural networks (RBFNNs) are utilizedto deal with unknown nonlinear terms as the approximationproperty. The RBFNNs have the form θT S(Z), where θ =[θ1, . . . , θN ]T ∈ RN is the weight vector with the integer N ,S(Z) = [S1(Z), . . . , SN (Z)]T ∈ RN is the vector of knownbasis function, and Z = [Z1, . . . , Zq ]T ∈ Rq is the inputvector of approximator, see in Fig. 2. Noting that the GaussianRBFNNs have the basis function Si (Z), which uses thestructure of Gaussian function and it can be formulated asfollows:

Si (Z) = exp−(Z − μi )

T (Z − μi )

η2i

, i = 1, 2, . . . , N (8)

where μi = [μi1, . . . , μiq ]T is the center of the receptive fieldand ηi is the width of the Gaussian function. It was proved thatRBFNNs can approximate any nonlinear continuous function

2132 IEEE TRANSACTIONS ON NEURAL NETWORKS AND LEARNING SYSTEMS, VOL. 25, NO. 12, DECEMBER 2014

Fig. 2. Structure of RBFNN.

f j (Z)( j = 1, . . . , n) over a compact set Z ∈ � with anyaccuracy. Therefore, the following lemma holds.

Lemma 1: For any ε > 0, there exists an NN θT S(Z) suchthat

supZ∈�

| f j (Z) − θT S(Z)| ≤ ε (9)

where � is a compact set of Z . The universal approximationproperty can be well utilized to deal with the unknownnonlinear terms.

D. Nussbaum Function

A function N(ϑ) is the Nussbaum function if it satisfies

limk→±∞ sup

1

k

∫ k

0N(s)ds = ∞ (10)

limk→±∞ inf

1

k

∫ k

0N(s)ds = −∞. (11)

Thus, we know that many functions satisfy the aforemen-tioned conditions such as exp(ϑ2) cos ϑ , ϑ2 cos ϑ .

Lemma 2: Let V (·) and ϑ(·) be smooth functions defined on[t0, t f ) with V (t) ≥ 0,∀t ∈ [t0, t f ), N(·) is an even smoothNussbaum-type function. If the following inequality holds for∀t ∈ [t0, t f ):

V (t) ≤ h1 +∫ t

t0(ξ(t)N(ϑ) + 1)ϑe−h2(t−τ )dτ (12)

where h1 and h2 are positive scalars. ξ(t) is a boundedfunction, which takes values at the interval � : [l−, l+]and 0 /∈ �. A conclusion can be made that V (t), ϑ(t),∫ t

t0(ξ N(ϑ)+1)ϑdτ must be bounded on [t0, t f ). Furthermore,

t f can be extended to infinity, i.e., t f → ∞ if the solution ofthe resulting closed-loop system is bounded [20].

III. ADAPTIVE NEURAL CONTROL DESIGN

BASED ON LYAPUNOV SYNTHESIS

We use the backstepping and NN techniques to constructthe adaptive neural control schemes. The coordinate transfor-mations are defined as follows:

z1=y − yr (13)

zi=xi − αi−1 2 ≤ i ≤ n (14)

where yr is the reference signal and assuming its time deriv-atives up to nth order remain bounded and continuous.

Assumption 3: For 1 ≤ i ≤ n, there exists an unknownpositive function qi j such that

|hi (x(t))| ≤n∑

j=1

|z j (t)|qi j (x(t)) (15)

where x(t) = [x1(t), . . . , xn(t)]T are the whole state variablesof control system. z j is the coordinate transformation, i.e.,z1 = y − yr and z j = x j − α j−1(2 ≤ j ≤ n), where α j−1 isthe virtual controller given latter.

Remark 5: A similar assumption was made in [64]. How-ever, the time-delayed term hi (·) studied in [64] is a functionof current states, i.e., hi (·) = hi (xi ), where xi = [x1, . . . , xi ]T .In this paper, we extend the time-delayed term to hi (x),which contains the whole state variables. Therefore, the controlscheme proposed in this paper can be applied more extensively.

At each step of control design, we utilize the

Wi = Ni ||θi ||22 = NiθTi θi , i = 1, . . . , n

where θi = [θi,1, . . . , θi,Ni ]T and Ni are the weight vector andthe number of neurons in i th hidden layer, respectively. Theoperator ||·||2 represents the Euclidean norm of column vector.From the definition, we know that Wi is an unknown constant.The adaptive parameter Wi is utilized to estimate Wi and theestimated error is Wi = Wi − Wi . By directly estimating thenorm of weight vector θi , there are only n adaptive parametersneeded to adjust online. Compared with the control algorithmsproposed in [36] and [37], the computation in adaptationmechanism utilized in the literature can be reduced from∑n

i=1 Ni to n.In the following analysis, two control schemes are proposed

in the Section III-A and III-B, respectively. The control schemein Section III-A is for the nonlinear systems with knowndirection hysteresis, i.e., hysteresis parameter μ1 is unknownbut its sign requires to be known as prior knowledge. InSection III-B, we propose an adaptive neural control algorithmfor the nonlinear systems with unknown direction hysteresis.

A. Known Direction Hysteresis

To facilitate the control design, the following assumption isneeded.

Assumption 4: Consider the direction of unknown hysteresisis known, i.e., μ1 is an unknown parameter but its sign mustbe known. Without loss of generality, it is further assumedsign(μ1) > 0.

Theorem 1: Consider the nonstrict-feedback nonlinear sys-tem in (1), under the Assumptions 1–4, by selecting the virtualcontrollers α j , the auxiliary controller v and the actual controlsignal v as follows:

⎧⎪⎪⎨⎪⎪⎩

α j = −(k j + n2

4 j + 0.5)z j − 1

2a2jz j W j − z j−1

v = −(kn + n2

4n + 1)zn − 1

2a2n

znWn − zn−1

v = ev

(16)

LIU et al.: ADAPTIVE NEURAL CONTROL FOR A CLASS OF NONLINEAR TIME-VARYING DELAY SYSTEMS 2133

where j = 1, . . . , n−1 and z0 = −yr . Define a parameter e ase = 1/μ1, and its estimated value is e = 1/μ1. The estimatederror is defined as e = e− e. The adaptive laws are selected as⎧⎨

⎩˙Wi = ri

2a2i

z2i − k0i Wi (i = 1, . . . , n)

˙e = −ηzn v − η0e.(17)

We can construct a robust adaptive neural control scheme toguarantee that all the system states and adaptive parameters arebounded. By increasing the designed parameters k1, . . . , kn ,k01, . . . , k0n, η, η0, r1, . . . , rn while reducing a1, . . . , an , thetracking error limitation can be adjusted arbitrarily small.

Proof: Backstepping technique is employed to demon-strate this theorem.

Step 1: Differentiating two sides of (13), we have

z1 = f1(x) + h1(x(t − τ1)) + x2 + ω1(t, x) − yr . (18)

The Lyapunov–Krasovskii function can be chosen asfollows:

V1=n∑

i=1

n∑j=1

1

1 − τmaxe−λ(t−τi )

∫ t

t−τi

eλsz2j (s)q

2i j (x(s))ds

+1

2z2

1 + 1

2r1W 2

1 . (19)

Differentiating V1 in (19) yields

V1 = z1( f1(x) + h1(x(t−τ1)) + x2 + ω1(t, x)− yr)

− 1

r1W1

˙W1 + 1

1−τmax

n∑i=1

n∑j=1

eλτi z2j (t)q

2i j (x(t))

− (1−τi)

1−τmax

n∑i=1

n∑j=1

z2j (t−τi )q

2i j (x(t−τi ))

−n∑

i=1

n∑j=1

·λ(1−τi)

1−τmaxe−λ(t−τi )

∫ t

t−τi

eλsz2j (s)q

2i j (x(s))ds.

(20)

Invoking Assumption 2, one has

−λ(1 − τi ) ≤ −λ(1 − τmax). (21)

By rearranging the terms

n∑i=1

n∑j=1

eλτi z2j (t)q

2i j (x(t)) =

n∑j=1

n∑i=1

eλτi z2j (t)q

2i j (x(t))

and combining with (20) and (21), we have

V1 ≤ z1( f1(x) + h1(x(t − τ1)) + z2 + α1 + ω1(t, x) − yr )

− 1

r1W1

˙W1 + 1

1 − τmax

n∑j=1

n∑i=1

eλτi z2j (t)q

2i j (x(t))

−n∑

i=1

n∑j=1

z2j (t − τi )q

2i j (x(t − τi )) − λ(1 − τmax)

×n∑

i=1

n∑j=1

e−λ(t−τi )

1 − τmax

∫ t

t−τi

eλsz2j (s)q

2i j (x(s))ds. (22)

Noting that

z1h1(x(t − τ1)) ≤ |z1|n∑

j=1

|z j (t − τ1)|q1 j (x(t − τ1))

≤ n2

4z2

1 + 1

n

n∑j=1

z2j (t − τ1)q

21 j (x(t − τ1)).

(23)

Similarly

z1ω1(t, x) ≤ 1

2|z1| + 1

2|z1|� 2

1 (x). (24)

We define the nonlinear term σ1 as follows:

σ1 =| f1(x)| + 1

2+ 1

2� 2

1 (x) +

n∑i=1

|z1|eλτmaxq2i1(x(t))

1 − τmax. (25)

With the conclusion of (9), we use the RBFNN θT1 S1 to

approximate the nonlinear term σ1 and the approximated erroris δ1 which satisfies |δ1| ≤ ε1. Combining with the conditionST

1 S1 ≤ N1, we have

|z1|σ1 =|z1|(θT

1 S1 + δ1)

≤ 1

2a21

z21||θ1||2ST

1 S1 + 1

2a2

1 + 1

2z2

1 + 1

2ε2

1

≤ 1

2a21

z21W1 + 1

2a2

1 + 1

2z2

1 + 1

2ε2

1. (26)

Substituting (23)–(26) into (22), we have

V1 ≤− k1z21 + z1z2 + 1

2a2

1 + 1

2ε2

1 + W1

r1

(r1

2a21

z21 − ˙W1

)

+ 1

1 − τmax

n∑j=2

n∑i=1

eλτi z2j (t)q

2i j (x(t)) − n − 1

n

×n∑

j=1

z2j (t − τ1)q

21 j (x(t − τ1)) − λ(1 − τmax)

×n∑

i=1

n∑j=1

e−λ(t−τi )

1 − τmax

∫ t

t−τi

eλsz2j (s)q

2i j (x(s))ds

−n∑

i=2

n∑j=1

z2j (t − τi )q

2i j (x(t − τi )). (27)

Step k(2 ≤ k ≤ n − 1): Differentiating two sides of (14)yields

zk= fk(x) + hk(x(t − τk)) + xk+1 + ωk(t, x) − αk−1. (28)

The Lyapunov–Krasovskii function is chosen as

Vk = Vk−1 + 1

2z2

k + 1

2rkW 2

k . (29)

Differentiating Vk in (29), we have

Vk = Vk−1 + zk( fk(x) + hk(x(t − τk)) + xk+1

+ ωk(t, x) − αk−1) − 1

rkWk

˙Wk

= Vk−1 + zk( fk(x) + hk(x(t − τk)) + zk+1 + αk

+ ωk(t, x) − αk−1) − 1

rkWk

˙Wk . (30)

2134 IEEE TRANSACTIONS ON NEURAL NETWORKS AND LEARNING SYSTEMS, VOL. 25, NO. 12, DECEMBER 2014

Since

zkhk(x(t − τk)) ≤ |zk |n∑

j=1

|z j (t − τk)|qkj (x(t − τk))

≤ n2

4 · kz2

k + k

n

n∑j=1

z2j (t − τk)q

2kj (x(t − τk)).

(31)

Similarly

zkωk(t, x) ≤ 1

2|zk | + 1

2|zk |� 2

k (x). (32)

The time differentiation of the virtual control signalαk−1 is

αk−1 =k−1∑m=1

∂αk−1

∂xm( fm(x) + hm(x(t − τm)) + xm+1

+ ωm(t, x)) +k−1∑m=1

∂αk−1

Wm

˙Wm +k−1∑m=0

∂αk−1

∂y(m)r

y(m+1)r .

(33)

By analyzing some complex terms in the αk−1, we have

− zk

k−1∑m=1

∂αk−1

∂xmhm(x(t−τm))

≤ |zk |k−1∑m=1

∣∣∣∣∂αk−1

∂xm

∣∣∣∣n∑

j=1

|z j (t−τm)|qmj (x(t−τm))

≤k−1∑m=1

⎡⎣n2

4z2

k

(∂αk−1

∂xm

)2

+ 1

n

n∑j=1

z2j (t−τm)q2

mj (x(t−τm))

⎤⎦

≤k−1∑m=1

n2

4z2

k

(∂αk−1

∂xm

)2

+ 1

n

k−1∑l=1

n∑j=1

z2j (t−τl)q

2l j (x(t−τl)).

(34)

Noting that

−zk

k−1∑m=1

∂αk−1

∂xmωm(t, x)≤

k−1∑m=1

[|zk|2

(∂αk−1

∂xm

)2

+ |zk |2

� 2m(x)

].

(35)

We define σk as follows:

σk = | fk(x)| + 1

2+

k∑m=1

1

2� 2

m(x) +k−1∑m=1

∣∣∣∣∂αk−1

∂xm

∣∣∣∣ | fm(x)|

+k−1∑m=1

(n2

4|zk | + 1

2

) (∂αk−1

∂xm

)2

+k−1∑m=1

∣∣∣∣∂αk−1

∂xm

∣∣∣∣ |xm+1|

+k−1∑m=1

∂αk−1

Wm

˙Wm +k−1∑m=0

∂αk−1

∂y(m)r

y(m+1)r

+ 1

1 − τmax

n∑i=1

eλτmax |zk |q2ik(x(t)). (36)

We employ the RBFNN to approximate the nonlinear termσk , the approximated error is δk which satisfies |δk | ≤ εk .Combining with the condition ST

k Sk ≤ Nk , we have

|zk |σk = |zk |(θT

k Sk + δk) ≤ 1

2a2k

z2k Wk + 1

2a2

k + 1

2z2

k + 1

2ε2

k .

(37)

Substituting (31)–(37) into (30), we have

Vk ≤ −k∑

j=1

k j z2j + zkzk+1 + 1

2

k∑j=1

(a2

j + ε2j

)

+k∑

j=1

W j

r j

(r j

2a2j

z2j − ˙W j

)+

n∑j=k+1

n∑i=1

eλτi z2j (t)q

2i j (x(t))

1 − τmax

−n∑

i=k+1

n∑j=1

z2j (t − τi )q

2i j (x(t − τi )) − n − k

n

·k∑

l=1

n∑j=1

z2j (t − τl)q

2l j (x(t − τl)) − λ(1 − τmax)

·n∑

i=1

n∑j=1

e−λ(t−τi )

1 − τmax

∫ t

t−τi

eλsz2j (s)q

2i j (x(s))ds. (38)

Step n: The Lyapunov–Krasovskii function can be chosenas follows:

Vn = Vn−1 + 1

2z2

n + 1

2rnW 2

n + μ1

2ηe2. (39)

Differentiating two sides of (39), we have

Vn = Vn−1 + zn( fn(x) + hn(x(t − τn)) + H (v)

+ ωn(t, x) − αn−1) − 1

rnWn

˙Wn − μ1

ηe ˙e.

(40)

Since

znhn(x(t − τn)) ≤ n2

4 · nz2

n + n

n

n∑j=1

z2j (t − τk)q

2kj (x(t − τk)).

(41)

Noting that

znωn(t, x) ≤ 1

2|zn| + 1

2|zn |� 2

n (x). (42)

The time differentiation of the virtual controller αn−1 is

αn−1 =n−1∑m=1

∂αn−1

∂xm( fm(x) + hm(x(t − τm)) + xm+1

+ ωm(t, x)) +n−1∑m=1

∂αn−1

Wm

˙Wm +n−1∑m=0

∂αn−1

∂y(m)r

y(m+1)r .

(43)

LIU et al.: ADAPTIVE NEURAL CONTROL FOR A CLASS OF NONLINEAR TIME-VARYING DELAY SYSTEMS 2135

Noting that

−zn

n−1∑m=1

∂αk−1

∂xmhm(x(t−τm))

≤n−1∑m=1

n2

4z2

k

(∂αk−1

∂xm

)2

+ 1

n

n−1∑l=1

n∑j=1

z2j (t−τl)q

2l j (x(t−τl)).

(44)

Subsequently, a compounded nonlinear term σn similar toσk in the equality (36) is also defined. By using the RBFNNto approximate the unknown nonlinear term σn , and theapproximated error is δn satisfying |δn| ≤ εn . Thus, we have

|zn |σn = |zn|(θT

n Sn + δn)

≤ 1

2a2n

z2n Wn + 1

2a2

n + 1

2z2

n + 1

2ε2

n. (45)

The hysteresis input H (v) in (40) can be dealt with asfollows:

zn H (v) = zn(μ1v + μ2ζ ). (46)

Consider μ1 is the unknown constant, we should utilize μ1to estimate μ1. By defining e = 1/μ1 and e = 1/μ1, thus, theestimated error is e = e − e. Combining with Assumption 4yields

zn H (v)= zn(μ1v + μ2ζ ) = znμ1ev + znμ2ζ

= znμ1

(1

μ1− 1

μ1

)v + znμ2ζ

= zn v − znμ1ev + znμ2ζ. (47)

Since

znμ2ζ ≤ 1

2z2

n + 1

2(μ2ζ )2 (48)

where ζ = n√

1/(β + χ).From (16), the auxiliary control signal v is designed as

v = −(

kn + n2

4 · n+ 0.5 + 0.5

)zn − 1

2a2n

zn Wn − zn−1.

(49)

Based on the analysis of (40)–(49), we have

Vn ≤ −n∑

j=1

k j z2j + 1

2

n∑j=1

(a2

j + ε2j

)+n∑

j=1

W j

r j

(r j

2a2j

z2j − ˙W j

)

− λ

n∑i=1

n∑j=1

e−λ(t−τi )

∫ t

t−τi

eλsz2j (s)q

2i j (x(s))ds

− μ1e

(zn v +

˙eη

)+ 1

2(μ2ζ )2. (50)

Noting that

n∑j=1

k0 j

r jW j W j ≤−

n∑j=1

k0 j

2r jW 2

j +n∑

j=1

k0 j

2r jW 2

j (51)

μ1η0

ηee ≤−μ1η0

2ηe2 + μ1η0

2ηe2. (52)

Substituting (51) and (52) into (50), we have

Vn ≤ −n∑

j=1

k j z2j −

n∑j=1

k0 j

2r jW 2

j +n∑

j=1

k0 j

2r jW 2

j + 1

2

n∑j=1

(a2

j +ε2j

)

− λ

n∑i=1

n∑j=1

e−λ(t−τi )

∫ t

t−τi

eλsz2j (s)q

2i j (x(s))ds

+ 1

2(μ2ζ )2 − μ1η0

2ηe2+ μ1η0

2ηe2. (53)

Let P = min{2k1, . . . , 2kn, k01, . . . , k0n, λ(1 − τmax), η0}and

Y =n∑

j=1

k0 j/2r j W 2j + 1/2

n∑j=1

(a2

j + ε2j

)

+ μ1η0/2ηe2 + 1/2(μ2ζ )2.

Consequently, the inequality (53) can be reexpressed as

Vn ≤ −PVn + Y. (54)

Thus, we have

Vn ≤(

Vn(0) − YP

)e−P t + Y

P . (55)

Noting that if t → ∞z2

1 = (y − yr )2 ≤ 2Y

P . (56)

By increasing the parameters k j , k0 j , η, and r j , η0 mean-while reducing the parameters a j , the tracking error limitation2Y/P can be adjusted arbitrarily small. Therefore, the controlscheme proposed in Theorem 1 can guarantee that all thesignals in the closed-loop system remain bounded, and theoutput of the system converges to an adjustable neighborhoodof the reference signal.

Remark 6: With the prerequisite of Assumption 4, a robustadaptive neural controller can be constructed for the nonstrict-feedback system in (1), and the conclusion is presented inTheorem 1. However, from Assumption 4, we know that thesign of μ1 must be known in advance as prior knowledge.To relax the restriction, a Nussbaum-type function is intro-duced to solve the problem of unknown control direction anda novel adaptive neural control scheme is proposed in thefollowing theorem.

B. Unknown Direction Hysteresis

In the section, we investigate the fusion of unknown direc-tion hysteresis with adaptive neural control approach. Thestable control algorithm is presented in the following theorem.

Theorem 2: Consider the nonstrict-feedback system in (1),under the Assumptions 1–3, if we choose the virtual con-trollers α j , the auxiliary controller v , the control law v, andthe adaptive laws as follows:⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎩

α j = −(k j + n2

4 j + 0.5)z j − 12a2

jz j W j − z j−1

v = −(kn + n2

4n + 1)zn − 12a2

nznWn − zn−1

v = −N(ϑ)v

ϑ = −γ vzn˙Wi = ri

2a2i

z2i − k0i Wi (i = 1, . . . , n; j = 1, . . . , n − 1)

(57)

2136 IEEE TRANSACTIONS ON NEURAL NETWORKS AND LEARNING SYSTEMS, VOL. 25, NO. 12, DECEMBER 2014

where z0 = −yr . We can construct a robust adaptive neuralcontrol scheme to guarantee that all system states and adaptiveparameters are bounded. By increasing the designed para-meters k1, . . . , kn , k01, . . . , k0n , γ, r1, . . . , rn while reducinga1, . . . , an , the tracking error limitation can be adjusted arbi-trarily small.

Remark 7: It should be highlighted that Assumption 4 is notnecessary to the control scheme in Theorem 2. Therefore, thecontrol algorithm in Theorem 2 can be applied to nonstrict-feedback system with the unknown direction hysteresis.

Proof: We employ the backstepping technique to demon-strate this theorem. The first n − 1 steps are similar tothe aforementioned procedures. At nth step, the Lyapunov–Krasovskii function is chosen as follows:

Vn = Vn−1 + 1

2z2

n + 1

2rnW 2

n . (58)

Differentiating two sides of (58), we have

Vn = Vn−1 + zn( fn(x) + hn(x(t − τn)) + H (v)

+ ωn(t, x) − αn−1) − 1

rnWn

˙Wn . (59)

Noting that

zn H (v) = zn(μ1v + μ2ζ ). (60)

Consider the sign of μ1 is unknown, we introduce aNussbaum-type function N(ϑ) as N(ϑ) = ϑ2 cos ϑ . Thecontrol signal v is constructed as v = −N(ϑ)v . Subsequently,we have

zn H (v) = −(μ1 N(ϑ) + 1)zn v + zn v + znμ2ζ (61)

where v is the auxiliary control signal needed to be furtherdesigned. The variable ϑ is obtained by the differentiationequation ϑ = −γ vzn . With these analysis, equality (61) canbe expressed as

zn H (v) = 1

γ(μ1 N(ϑ) + 1)ϑ + zn v + znμ2ζ. (62)

The auxiliary controller v and the adaptive parameters Wi

are designed in (57), respectively. By repeating the proceduresof (18)–(38), and combining with the analysis of (59)–(62), wehave

Vn ≤−n∑

j=1

k j z2j −

n∑j=1

k0 j

2r jW 2

j +n∑

j=1

k0 j

2r jW 2

j + 1

2

n∑j=1

(a2

j + ε2j

)

− λ

n∑i=1

n∑j=1

e−λ(t−τi )

∫ t

t−τi

eλsz2j (s)q

2i j (x(s))ds

+ 1

γ(μ1 N(ϑ) + 1)ϑ + 1

2(μ2ζ ).2 (63)

Two constants are defined as P = min{2k1, . . . , 2kn,k01, . . . , k0n, λ(1 − τmax)} and

Y =n∑

j=1

k0 j/2r j W 2j + 1/2

n∑j=1

(a2

j + ε2j

) + 1/2(μ2ζ )2.

Subsequently, inequality (63) can be written as the followingform:

Vn ≤ −PVn + Y + 1

γ(μ1 N(ϑ) + 1)ϑ. (64)

By direct integration of the differentiation inequality (64) atthe interval [0, t) yields

Vn ≤ Vn(0)e−P t + YP (1 − e−P t )

+e−P t

γ

∫ t

0(μ1 N(ϑ) + 1)ϑ · ePτ dτ. (65)

From Lemma 2, we can know that the terms Vn , ϑ , and∫ t0 (μ1 N(ϑ) + 1)ϑdτ are bounded at the interval [0, t). Fur-

thermore, the result can be extended to t → ∞. Consequently,let C0 be upper bound of � = ∫ t

0 (μ1 N(ϑ) + 1)ϑe−P(t−τ )dτ .Then, the following inequality holds:

Vn ≤(

Vn(0) − YP

)e−P t + Y

P + C0

γ. (66)

When t → ∞z2

1 = (y − yr )2 ≤ 2Y

P + 2C0

γ. (67)

By increasing the designed parameters k j , λ, γ , and k0 j

meanwhile reducing the parameters a j , the tracking errorlimitation can be adjusted arbitrarily small. Therefore, therestriction in Assumption 4 can be removed by the controlscheme presented in the Theorem 2. Furthermore, the controlalgorithm proposed in the Theorem 2 can guarantee that all thestate variables and adaptive parameters of the control systemare bounded.

IV. SIMULATION STUDY

In the section, three numerical examples are provided tovalidate the effectiveness of the proposed control schemes.Example 1 is a general nonlinear system model with thetime-varying delays and the unknown direction hysteresis.Examples 2 and 3 are practical physical models.

Example 1: Consider the following general third-ordernonstrict-feedback nonlinear system:⎡

⎣x1x2x3

⎤⎦=

⎡⎣0 1 0

0 0 10 0 0

⎤⎦ ·

⎡⎣x1

x2x3

⎤⎦ +

⎡⎣ f1(x)

f2(x)f3(x)

⎤⎦ +

⎡⎣h1(τ1(t))

h2(τ2(t))h3(τ3(t))

⎤⎦

+⎡⎣0.1 sin(t)x1x2

2 x3

0.3 cos(π t)x21 x2

20.1 sin(π t)x1x2

2

⎤⎦ +

⎡⎣0

0u

⎤⎦, u = H (v) (68)

where y = x1 is the output of control system. The unknownnonlinear functions are selected as f1(x) = x2

1 x2 + x3,f2(x) = x2

1 x2 + 2x3, and f3(x) = −7x1 − 10x22 − 5x2,

respectively. It should be noted that the external disturbancesw1 = 0.1 sin(t)x1x2

2 x3, w2 = 0.3 cos(π t)x21 x2

2 , and w3 =0.1 sin(π t)x1x2

2 contain whole state variables and time vari-able simultaneously. The time-delayed terms are selected ash1(τ1(t)) = x2(t − τ1(t))x3(t − τ1(t)), h2(τ2(t)) = 0.5x2

2(t −τ2(t))x3(t − τ2(t)), and h3(τ3(t)) = 0.01x2

1(t − τ3(t))x23

LIU et al.: ADAPTIVE NEURAL CONTROL FOR A CLASS OF NONLINEAR TIME-VARYING DELAY SYSTEMS 2137

TABLE II

INITIAL VALUES AND DESIGNED PARAMETERS

Fig. 3. Trajectories of system output y and the corresponding tracking errorz1 = y − yr .

(t − τ3(t)), respectively. The time delays are selected asτ1(t) = 0.5 + 0.1 sin(t), τ2(t) = 0.5 + 0.3 sin(0.5t), andτ3(t) = 0.5 + 0.2 sin(π t), respectively.

The actual control signal is v, and the reference signal isyr = sin(t) + sin(2t) + cos(3t). The control objective of thissimulation is to construct a robust adaptive neural controlscheme to guarantee that all the system states and adaptiveparameters are bounded, and the tracking error can convergeto an adjustable neighborhood of the origin.

In the simulation, the hysteresis parameters are presentedin Table I, and the initial values of system states and adaptiveparameters are provided in Table II. In addition, the designedparameters are also given in Table II. Based on the theoremderived in this paper and the aforementioned designed para-meters, a robust adaptive neural control scheme is constructed.The simulation results are shown in Figs. 3 and 4, respectively.Fig. 3 shows the tracking performance of control system. Fig. 4shows that the trajectories of system states x2, x3 and thecorresponding actual control signal v.

Example 2: Consider the following inverted pendulum withhysteresis actuator shown in Fig. 5. By writing x1 = θand x2 = θ , the inverted pendulum can be modeled as thefollowing second-order differentiation equation [45]:

x1 = x2

x2 = f (x2) + g(x2)H (v) + d(t) (69)

Fig. 4. Trajectories of system states x2, x3 and actual control signal v .

Fig. 5. Inverted pendulum system with hysteresis actuator.

where

f (x2)= 9.8(mc + m) sin x1 − mlx22 cos x1 sin x1

l(

43 − m cos2 x1

mc+m

)(mc + m)

(70)

g(x2)= cos x1

l(

43 − m cos2 x1

mc+m

)(mc + m)

(71)

d(t) = 3 + 2 cos(2t). (72)

The physical parameters are selected as mc = 1 kg,m = 0.1 kg, and l = 0.5 m, respectively.

It should be noted that the simulation model has beenperformed in [45]. However, the works in [45] consideredthe known direction backlash-like hysteresis model. Comparedwith the simulation model in [45], we consider a more generalmodified Bouc–Wen hysteresis model in this paper. Further-more, the hysteresis direction does not require to be known.By constructing the stable control scheme as the Theorem 2,the simulation results are shown in Figs. 6 and 7, respectively.The comparative simulation results can further validate theeffectiveness of the proposed control scheme in this paper.

Example 3: Consider the following piezo-positioning mech-anism subjected to unknown direction hysteresis input:

M y(t) + Dy(t) + Fy(t) = u, u = H (v) (73)

where y, y, and y represent the position, the velocity, and theacceleration, respectively. v is the voltage signal applied tothe piezo-positioning mechanic system. M , D, and F denote

2138 IEEE TRANSACTIONS ON NEURAL NETWORKS AND LEARNING SYSTEMS, VOL. 25, NO. 12, DECEMBER 2014

Fig. 6. Trajectories of the system output y and control signal v .

Fig. 7. Trajectories of adaptive parameters W1 and W2.

Fig. 8. Trajectories of the system output y and control signal v .

the mass, damping, and stiffness coefficients, respectively.These parameters are all unknown. In the simulation, the actualparameter values are chosen as M = 1 kg, D = 0.15 Ns/m,F = 1 M/m. The modified Bouc–Wen hysteresis parametersare μ1 = μ2 = 1, β = 1, χ = 0.5, and n = 2. The initialvalue of state is chosen as y(0) = 1.5. The reference signal isyr = 2 sin(0.5π t).

Remark 8: The piezo-position mechanic model in (73)has been investigated in [41] and [75]. However, the controlschemes in [41] and [75] require both the parameters andthe direction of modified Bouc–Wen hysteresis to be knownas prior knowledge. To relax these restrictions, an adaptivecontrol algorithm is proposed in Theorem 2 for the nonlinearsystems with the unknown direction hysteresis input.

In the simulation, the initial values of adaptive parame-ters are chosen as W1(0) = 0.1, W2(0) = 0.01, andϑ(0) = 0.1, respectively. The designed parameters are selectedas k1 = 15, k2 = 16, a1 = a2 = 10, r1 = 250, r2 =250, γ = 1, k01 = 0.003, and k02 = 0.002, respectively.Therefore, we can construct a neural-network-based adaptive

Fig. 9. Trajectories of adaptive parameters W1 and W2.

controller as Theorem 2. The simulation results are shownin Figs. 8 and 9, respectively. Fig. 8 shows that the track-ing trajectory and the corresponding actual control signal v.Fig. 9 shows the trajectories of adaptive parameters W1 andW2. Clearly, the simulation results can further validate theeffectiveness of the proposed control scheme in this paper.

V. CONCLUSION

In this paper, two NN-based adaptive control algorithmsare proposed for a class of CT nonstrict-feedback systemswith unknown modified Bouc–Wen hysteresis input and timevarying delays. Compared with the previous works on hys-teresis phenomenon, the direction of the modified Bouc–Wenhysteresis model studied in the literature can be unknown.Furthermore, an optimized method is proposed to reduce thecomputation burden in adaptation mechanism. Based on theLyapunov–Krasovskii method, two adaptive neural controlschemes are constructed to guarantee that all the system statesand adaptive parameters remain bounded, and the trackingerror is driven to an adjustable neighborhood of the origin.In addition, three simulation examples are provided to verifythe effectiveness of the proposed control schemes.

ACKNOWLEDGMENT

The authors would like to thank the Associate Editor and theanonymous reviewers for a number of constructive commentsthat have improved the presentation of this paper.

REFERENCES

[1] B. Chen, X. P. Liu, K. F. Liu, and C. Lin, “Direct adaptive fuzzycontrol of nonlinear strict-feedback systems,” Automatica, vol. 45, no. 6,pp. 1530–1535, Jun. 2009.

[2] D. Wang and J. Huang, “Neural network-based adaptive dynamic surfacecontrol for a class of uncertain nonlinear systems in strict-feedbackform,” IEEE Trans. Neural Netw., vol. 16, no. 1, pp. 195–202, Jan. 2005.

[3] D. Swaroop, J. Hedrick, P. Yip, and J. Gerdes, “Dynamic surface controlfor a class of nonlinear systems,” IEEE Trans. Autom. Control, vol. 45,no. 10, pp. 1893–1899, Oct. 2000.

[4] Q. Zhou, P. Shi, J. J. Lu, and S. Y. Xu, “Adaptive output-feedback fuzzytracking control for a class of nonlinear systems,” IEEE Trans. FuzzySyst., vol. 19, no. 5, pp. 972–982, Oct. 2011.

[5] S. C. Tong, X. L. He, and H. G. Zhang, “A combined backstepping andsmall gain approach to robust adaptive fuzzy output feedback control,”IEEE Trans. Fuzzy Syst., vol. 17, no. 5, pp. 1059–1069, Oct. 2009.

LIU et al.: ADAPTIVE NEURAL CONTROL FOR A CLASS OF NONLINEAR TIME-VARYING DELAY SYSTEMS 2139

[6] Y. S. Yang and C. J. Zhou, “Adaptive fuzzy H∞ stabilization forstrict-feedback canonical nonlinear systems via backstepping and small-gain approach,” IEEE Trans. Fuzzy Syst., vol. 13, no. 1, pp. 104–114,Feb. 2005.

[7] Z. P. Jiang and D. J. Hill, “A robust adaptive backstepping schemefor nonlinear systems with unmodeled dynamics,” IEEE Trans. Autom.Control, vol. 44, no. 9, pp. 1705–1711, Sep. 1999.

[8] W. Lin and C. J. Qian, “Semi-global robust stabilization of MIMOnonlinear systems by partial state and dynamic output feedback,” Auto-matica, vol. 37, no. 7, pp. 1093–1101, Jul. 2001.

[9] S. S. Ge and C. Wang, “Adaptive neural control of uncertainMIMO nonlinear systems,” IEEE Trans. Neural Netw., vol. 15, no. 3,pp. 674–692, May 2004.

[10] S. C. Tong and Y. M. Li, “Observer-based fuzzy adaptive control forstrict-feedback nonlinear systems,” Fuzzy Sets Syst., vol. 160, no. 12,pp. 1749–1764, Jun. 2009.

[11] S. C. Tong, C. Y. Li, and Y. M. Li, “Fuzzy adaptive observer backstep-ping control for MIMO nonlinear systems,” Fuzzy Sets Syst., vol. 160,no. 19, pp. 2755–2775, Oct. 2009.

[12] H. Y. Li, X. J. Jing, and H. R. Karimi, “Output-feedback based H-infinitycontrol for active suspension systems with control delay,” IEEE Trans.Ind. Electron., vol. 61, no. 1, pp. 436–446, Jan. 2014.

[13] B. Chen, X. P. Liu, K. Liu, and C. Lin, “Fuzzy-approximation-basedadaptive control of strict-feedback nonlinear systems with time delays,”Automatica, vol. 18, no. 5, pp. 883–892, Oct. 2010.

[14] Y. He, G. P. Liu, and D. Rees, “New delay-dependent stability criteriafor neural networks with time-varying delay,” IEEE Trans. Neural Netw.,vol. 18, no. 1, pp. 310–314, Jan. 2007.

[15] W. Daniel, J. M. Li, and Y. G. Niu, “Adaptive neural control for a class ofnonlinearly parametric time-delay systems,” IEEE Trans. Neural Netw.,vol. 16, no. 3, pp. 625–635, May 2005.

[16] X. H. Jiao and T. L. Shen, “Adaptive feedback control of nonlinear time-delay systems: The LaSalle–Razumikhin–based approach,” IEEE Trans.Autom. Control, vol. 50, no. 11, pp. 1909–1913, Nov. 2005.

[17] S. S. Ge, F. Hong, and T. H. Lee, “Adaptive neural network controlof nonlinear systems with unknown time delays,” IEEE Trans. Autom.Control, vol. 48, no. 11, pp. 2004–2010, Nov. 2003.

[18] B. Chen, X. P. Liu, K. F. Liu, and C. Lin, “Novel adaptive neural controldesign for nonlinear MIMO time-delay systems,” Automatica, vol. 45,no. 6, pp. 1554–1560, Jun. 2009.

[19] M. Jankovic, “Control Lyapunov–Razumikhin functions and robust sta-bilization of time delay systems,” IEEE Trans. Autom. Control, vol. 46,no. 7, pp. 1048–1060, Jul. 2001.

[20] T. P. Zhang and S. S. Ge, “Adaptive neural control of MIMO nonlinearstate time-varying delay systems with unknown dead-zones and gainsigns,” Automatica, vol. 43, no. 6, pp. 1021–1033, Jun. 2007.

[21] B. Chen, X. P. Liu, S. S. Ge, and C. Lin, “Adaptive fuzzy control ofa class of nonlinear systems by fuzzy approximation approach,” IEEETrans. Fuzzy Syst., vol. 20, no. 6, pp. 1012–1021, Dec. 2012.

[22] W. Z. Gao and R. R. Selmic, “Neural network control of a class ofnonlinear systems with actuator saturation,” IEEE Trans. Neural Netw.,vol. 17, no. 1, pp. 147–156, Jan. 2006.

[23] M. Chen, S. S. Ge, and B. B. Ren, “Adaptive tracking control ofuncertain MIMO nonlinear systems with input constraints,” Automatica,vol. 47, no. 3, pp. 452–465, Mar. 2011.

[24] C. Y. Wen and J. Zhou, “Robust adaptive control of uncertain nonlinearsystems in the presence of input saturation and external disturbance,”IEEE Trans. Autom. Control, vol. 56, no. 7, pp. 1672–1678, Jul. 2011.

[25] T. S. Li, R. G. Li, and J. F. Li, “Decentralized adaptive neural control ofnonlinear interconnected large-scale systems with unknown time delaysand input saturation,” Neurocomputing, vol. 74, no. 14, pp. 2277–2283,Jul. 2011.

[26] Y. S. Zhong, “Globally stable adaptive system design for minimumphase SISO plants with input saturation,” Automatica, vol. 41, no. 9,pp. 1539–1547, Sep. 2005.

[27] M. Chen, S. S. Ge, and V. Bernard, “Robust adaptive neural networkcontrol for a class of uncertain MIMO nonlinear systems with inputnonlinearities,” IEEE Trans. Neural Netw., vol. 21, no. 5, pp. 796–812,May 2010.

[28] Y. Y. Cao and Z. L. Lin, “Robust stability analysis and fuzzy-schedulingcontrol for nonlinear systems subject to actuator saturation,” IEEE Trans.Fuzzy Syst., vol. 11, no. 1, pp. 57–67, Feb. 2003.

[29] X. Y. Zhang and Y. Lin, “Adaptive tracking control for a class of pure-feedback nonlinear systems including actuator hysteresis and dynamicuncertainties,” IET Control Theory Appl., vol. 5, no. 16, pp. 1868–1880,Apr. 2011.

[30] Y. M. Li, S. C. Tong, and T. S. Li, “Adaptive fuzzy output feedback con-trol of MIMO nonlinear uncertain systems with time-varying delays andunknown backlash-like hysteresis,” Neurocomputing, vol. 93, pp. 56–66,Sep. 2012.

[31] Q. Chen and X. Ren, “Identier-based adaptive neural dynamic sur-face control for uncertain DC–DC buck converter system with inputconstraint,” Commun. Nonlinear Sci. Numer. Simul., vol. 17, no. 4,pp. 1871–1883, Sep. 2012.

[32] S. J. Yoo and J. B. Park, “Decentralized adaptive output-feedbackcontrol for a class of nonlinear large-scale systems with unknown time-varying delayed interactions,” Inf. Sci., vol. 186, no. 1, pp. 222–238,Mar. 2012.

[33] Y. M. Li, T. S. Li, and S. C. Tong, “Robust adaptive fuzzy controlof nonlinear systems with input saturation based on DSC and K-filtertechniques,” in Proc. IEEE World Congr. Comput. Intell., Jun. 2012,pp. 1–7.

[34] C. Y. Su, Y. Stepanenko, J. Svoboda, and T. P. Leung, “Robustadaptive control of a class of nonlinear systems with unknownbacklash-like hysteresis,” IEEE Trans. Autom. Control, vol. 45, no. 12,pp. 2427–2432, Dec. 2000.

[35] C. Y. Su, Q. Q. Wang, X. K. Chen, and S. Rakheja, “Adaptive variablestructure control of a class of nonlinear systems with unknown Prandtl-Ishlinskii hysteresis,” IEEE Trans. Autom. Control, vol. 50, no. 12,pp. 2069–2073, Dec. 2005.

[36] B. B. Ren, S. S. Ge, T. H. Lee, and C. Y. Su, “Adaptive neural controlfor a class of nonlinear systems with uncertain hysteresis inputs andtime-varying state delays,” IEEE Trans. Neural Netw., vol. 20, no. 7,pp. 1148–1164, Jul. 2009.

[37] B. B. Ren, S. S. Ge, C. Y. Su, and T. H. Lee, “Adaptive neural controlfor a class of uncertain nonlinear systems in pure-feedback form withhysteresis input,” IEEE Trans. Syst., Man, Cybern. B, Cybern., vol. 39,no. 2, pp. 431–443, Apr. 2009.

[38] P. P. San, B. B. Ren, S. S. Ge, T. H. Lee, and J. K. Liu, “Adaptiveneural network control of hard disk drives with hysteresis frictionnonlinearity,” IEEE Trans. Control Syst. Technol., vol. 19, no. 2,pp. 351–358, Mar. 2011.

[39] B. B. Ren, S. S. Ge, K. P. Tee, and T. H. Lee, “Adaptive neural control foroutput feedback nonlinear systems using a barrier Lyapunov function,”IEEE Trans. Neural Netw., vol. 21, no. 8, pp. 1339–1345, Aug. 2010.

[40] J. Zhou and C. J. Zhang, “Robust adaptive output control of uncertainnonlinear plants with unknown backlash nonlinearity,” IEEE Trans.Autom. Control, vol. 52, no. 3, pp. 503–509, Mar. 2007.

[41] J. Zhou, C. Y. Wen, and T. S. Li, “Adaptive output feedback control ofuncertain nonlinear systems with hysteresis nonlinearity,” IEEE Trans.Autom. Control, vol. 57, no. 10, pp. 2627–2633, Oct. 2012.

[42] O. Klein and P. Krejci, “Outwards pointing hysteresis operators andasymptotic behaviour of evolution equations,” Nonlinear Anal., RealWorld Appl., vol. 4, no. 5, pp. 755–785, Dec. 2003.

[43] S. C. Tong and Y. M. Li, “Adaptive fuzzy output feedback trackingbackstepping control of strict-feedback nonlinear systems with unknowndead zones,” IEEE Trans. Fuzzy Syst., vol. 20, no. 1, pp. 168–180,Feb. 2012.

[44] S. C. Tong and Y. M. Li, “Adaptive fuzzy output feedback controlof MIMO nonlinear systems with unknown dead-zones,” IEEE Trans.Fuzzy Syst., vol. 21, no. 1, pp. 134–146, Feb. 2013.

[45] Y. M. Li, S. C. Tong, and T. S. Li, “Adaptive fuzzy output feedbackcontrol of uncertain nonlinear systems with unknown backlash-likehysteresis,” Inf. Sci., vol. 198, no. 1, pp. 130–146, Sep. 2012.

[46] Y. J. Liu, S. C. Tong, and W. Wang, “Adaptive fuzzy output trackingcontrol for a class of uncertain nonlinear systems,” Fuzzy Sets Syst.,vol. 160, no. 19, pp. 2727–2754, Oct. 2009.

[47] Y. J. Liu, S. C. Tong, and T. S. Li, “Observer-based adaptive fuzzytracking control for a class of uncertain nonlinear MIMO systems,”Fuzzy Sets Syst., vol. 164, no. 1, pp. 25–44, Jan. 2011.

[48] W. S. Chen and L. C. Jiao, “Adaptive tracking for periodically time-varying and nonlinearly parameterized system using multilayer neuralnetworks,” IEEE Trans. Neural Netw., vol. 21, no. 2, pp. 345–351,Feb. 2010.

[49] S. S. Ge and J. Wang, “Robust adaptive neural control for a class ofperturbed strict feedback nonlinear systems,” IEEE Trans. Neural Netw.,vol. 13, no. 6, pp. 1409–1419, Nov. 2002.

[50] C. C. Hua and Q. Guo, “Adaptive fuzzy output-feedback controllerdesign for nonlinear time-delay systems with unknown control direc-tion,” IEEE Trans. Syst., Man, Cybern. B, Cybern., vol. 39, no. 2,pp. 363–374, Apr. 2009.

2140 IEEE TRANSACTIONS ON NEURAL NETWORKS AND LEARNING SYSTEMS, VOL. 25, NO. 12, DECEMBER 2014

[51] M. Narimani and H. K. Lam, “LMI-based stability analysis of fuzzy-model-based control system using approximated polynomial member-ship functions,” IEEE Trans. Syst., Man, Cybern. B, Cybern., vol. 41,no. 3, pp. 713–724, Jun. 2011.

[52] C. Kwan and F. Lewis, “Robust backstepping control of nonlinearsystems using neural networks,” IEEE Trans. Syst., Man, Cybern. A,Syst., Humans, vol. 30, no. 6, pp. 753–766, Nov. 2000.

[53] T. Zhang, S. S. Ge, and C. C. Hang, “Adaptive neural network con-trol for strict feedback nonlinear systems using backstepping design,”Automatica, vol. 36, no. 12, pp. 1835–1846, Dec. 2000.

[54] Y. Zhang, P. Y. Peng, and Z. P. Jiang, “Stable neural controller designfor unknown nonlinear systems using backstepping,” IEEE Trans. NeuralNetw., vol. 11, no. 6, pp. 1347–1360, Nov. 2000.

[55] J. Fei and J. Zhou, “Robust adaptive control of MEMS triaxial gyroscopeusing fuzzy compensator,” IEEE Trans. Syst., Man, Cybern. B, Cybern.,vol. 42, no. 6, pp. 1599–1607, Dec. 2012.

[56] W. S. Chen, L. C. Jiao, R. H. Li, and J. Li, “Adaptive backsteppingfuzzy control for nonlinearly parameterized systems with periodic distur-bances,” IEEE Trans. Fuzzy Syst., vol. 18, no. 4, pp. 674–685, Aug. 2010.

[57] G. Tao and P. V. Kokotovic, “Adaptive control of plants with unknownhysteresis,” IEEE Trans. Autom. Control, vol. 40, no. 2, pp. 200–212,Feb. 1995.

[58] W. S. Chen, L. C. Jiao, J. Li, and R. H. Li, “Adaptive NN backsteppingoutput-feedback control for stochastic nonlinear strict-feedback systemswith time-varying delays,” IEEE Trans. Syst., Man, Cybern. B, Cybern.,vol. 40, no. 3, pp. 939–950, Jun. 2010.

[59] C. Hua, X. Guan, and P. Shi, “Robust output feedback tracking controlfor time-delay nonlinear systems using neural network,” IEEE Trans.Neural Netw., vol. 18, no. 2, pp. 495–505, Mar. 2007.

[60] H. Y. Li, X. J. Jing, H. K. Lam, and P. Shi, “Fuzzy sampled-datacontrol for uncertain vehicle suspension systems,” IEEE Trans. Syst.,Man Cybern., 2013, to be published.

[61] B. Chen and X. Liu, “Fuzzy approximate disturbance decoupling ofMIMO nonlinear systems by backstepping and application to chemicalprocesses,” IEEE Trans. Fuzzy Syst., vol. 13, no. 6, pp. 832–847,Dec. 2005.

[62] T. S. Li, D. Wang, and G. F. Feng, “A DSC approach to robust adaptiveNN tracking control for strict-feedback nonlinear systems,” IEEE Trans.Syst., Man, Cybern. B, Cybern., vol. 40, no. 3, pp. 915–927, Jun. 2010.

[63] W. Y. Wang and M. L. Chan, “Adaptive fuzzy control for strict-feedbackcanonical nonlinear systems with H∞ tracking performance,” IEEETrans. Syst., Man, Cybern. B, Cybern., vol. 30, no. 6, pp. 878–885,Dec. 2000.

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

[65] S. Zhou, G. Feng, and C. Feng, “Robust control for a class of uncertainnonlinear systems: Adaptive fuzzy approach based on backstepping,”Fuzzy Sets Syst., vol. 151, no. 1, pp. 1–20, 2005.

[66] S. C. Tong, B. Chen, and Y. F. Wang, “Fuzzy adaptive output feedbackcontrol for MIMO nonlinear systems,” Fuzzy Sets Syst., vol. 156, no. 2,pp. 285–299, 2005.

[67] M. Chadli and H. R. Karimi, “Robust observer design for unknowninputs Takagi-Sugeno models,” IEEE Trans. Fuzzy Syst., vol. 21, no. 1,pp. 158–164, Feb. 2013.

[68] Y. H. Li, S. Qiang, X. Y. Zhuang, and O. Kaynak, “Robust and adaptivebackstepping control for nonlinear system using RBF neural networks,”IEEE Trans. Neural Netw., vol. 15, no. 3, pp. 693–701, May 2004.

[69] M. Wang, X. P. Liu, and P. Shi, “Adaptive neural control of pure-feedback nonlinear time-delay systems via dynamic surface tech-nique,” IEEE Trans. Syst., Man, Cybern. B, Cybern., vol. 41, no. 6,pp. 1681–1692, Dec. 2011.

[70] H. Y. Li, J. Y. Yu, C. Hilton, and H. H. Liu, “Adaptive sliding modecontrol for nonlinear active suspension vehicle systems using T-S fuzzyapproach,” IEEE Trans. Ind. Electron., vol. 60, no. 8, pp. 3328–3338,Aug. 2013.

[71] Q. Zhou, P. Shi, H. H. Liu, and S. Y. Xu, “Neural-network-baseddecentralized adaptive output-feedback control for large-scale stochasticnonlinear systems,” IEEE Trans. Syst., Man, Cybern. B, Cybern., vol. 42,no. 6, pp. 1608–1619, Dec. 2012.

[72] C. Wang, M. Wang, T. F. Liu, and D. J. Hill, “Learning fromISS-modular adaptive NN control of nonlinear strict-feedback systems,”IEEE Trans. Neural Netw., vol. 23, no. 10, pp. 1539–1550, Oct. 2012.

[73] C. Novara, F. Ruiz, and M. Milanese, “Direct filtering: A new approachto optimal filter design for nonlinear systems,” IEEE Trans. Autom.Control, vol. 58, no. 1, pp. 86–99, Jan. 2013.

[74] C. Y. Wen and J. Zhou, “Decentralized adaptive stabilization in thepresence of unknown backlash-like hysteresis,” Automatica, vol. 43,no. 3, pp. 426–440, Mar. 2007.

[75] F. J. Lin, H. J. Shieh, and P. K. Huang, “Adaptive wavelet neural networkcontrol with hysteresis estimation for piezo-positioning mechanism,”IEEE Trans. Neural Netw., vol. 17, no. 2, pp. 432–444, Mar. 2006.

[76] Y. J. Liu and S. C. Tong, “Adaptive fuzzy control for a class of nonlineardiscrete-time systems with backlash,” IEEE Trans. Fuzzy Syst., 2013, tobe published.

[77] C. L. P. Chen, Y. J. Liu, and G. X. Wen, “Fuzzy neural network-basedadaptive control for a class of uncertain nonlinear stochastic systems,”IEEE Trans. Syst., Man, Cybern. B, Cybern., 2013, to be published.

[78] S. C. Tong and H. X. Li, “Fuzzy adaptive sliding-mode control forMIMO nonlinear systems,” IEEE Trans. Fuzzy Syst., vol. 11, no. 3,pp. 354–360, Jun. 2003.

Zhi Liu received the B.S. degree from the HuazhongUniversity of Science and Technology, Wuhan,China, the M.S. degree from Hunan University,Changsha, China, and the Ph.D. degree fromTsinghua University, Beijing, China, all in electricalengineering, in 1997, 2000, and 2004, respectively.

He is currently a Professor with the Department ofAutomation, Guangdong University of Technology,Guangzhou, China. His current research interestsinclude fuzzy logic systems, neural networks, robot-ics, and robust control.

Guanyu Lai received the B.S. degree in electricalengineering and automation, and the M.S. degree incontrol science and engineering from the GuangdongUniversity of Technology, Guangdong, China, in2012 and 2014, respectively, where he is currentlypursuing the Ph.D. degree from the Department ofAutomation.

His current research interests include neural net-works, adaptive fuzzy control, cooperative control,and unmanned aerial vehicle.

Yun Zhang received the B.S. and M.S. degreesin automatic engineering from Hunan University,Changsha, China, and the Ph.D. degree in automaticengineering from the South China University of Sci-ence and Technology, Guangzhou, China, in 1982,1986, and 1998, respectively.

He is currently a Professor with the Department ofAutomation, Guangdong University of Technology,Guangzhou. His current research interests includeintelligent control systems, network systems, andsignal processing.

Xin Chen received the B.S. degree from ChangshaRailway Institute, Changsha, China, the M.S. degreefrom the Harbin Institute of Technology, Harbin,China, and the Ph.D. degree from the Huazhong Uni-versity of Science and Technology, Wuhan, China,all in mechanical engineering, in 1982, 1988, and1995, respectively.

He is currently a Professor with the Department ofMechatronics Engineering, Guangdong University ofTechnology, Guangzhou, China. His current researchinterests include mechatronics and robotics.

Chun Lung Philip Chen (S’88–M’88–SM’94–F’07) received the M.S. degree from the Universityof Michigan, Ann Arbor, MI, USA, and the Ph.D.degree from Purdue University, West Lafayette, IN,USA, all in electrical engineering, in 1985 and 1988,respectively.

He is currently a Chair Professor with the Depart-ment of Computer and Information Science and theDean of the Faculty of Science and Technology,University of Macau, Macau, China. His currentresearch interests include computational intelligence,

systems, and cybernetics.