Adaptive Output Neural Network Control for a Class of ... - IEEE Xplore

This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. IEEE TRANSACTIONS ON NEURAL NETWORKS AND LEARNING SYSTEMS

1

Adaptive Output Neural Network Control for a Class of Stochastic Nonlinear Systems With Dead-Zone Nonlinearities Li-Bing Wu and Guang-Hong Yang, Senior Member, IEEE

Abstract— This paper investigates the problem of adaptive output neural network (NN) control for a class of stochastic nonaffine and nonlinear systems with actuator dead-zone inputs. First, based on the intermediate value theorem, a novel design scheme that converts the nonaffine system into the corresponding affine system is developed. In particular, the priori knowledge of the bound of the derivative of the nonaffine and nonlinear functions is removed; then, by employing NNs to approximate the appropriate nonlinear functions, the corresponding adaptive NN tracking controller with the adjustable parameter updated laws is designed through a backstepping technique. Furthermore, it is shown that all the closed-loop signals are bounded in probability, and the system output tracking error can converge to a small neighborhood in the sense of a mean quartic value. Finally, experimental simulations are provided to demonstrate the efficiency of the proposed adaptive NN tracking control method. Index Terms— Adaptive neural control, backstepping design, dead-zone nonlinearities, stochastic nonaffine and nonlinear systems.

I. I NTRODUCTION

A

S IS well known, a backstepping technique, which provides a promising way to improve the transient and tracking performance of adaptive systems by tuning design parameters, has been widely used to design adaptive controllers for uncertain nonlinear systems [1]. Because of these advantages, studies on an adaptive backstepping tracking control design for uncertain nonlinear systems have attracted considerable attention over the past few decades [2]–[6]. Meanwhile, in view of their universal approximation properties and adaption abilities, neural networks (NNs) and fuzzy logic systems (FLSs) have been extensively applied to

Manuscript received April 13, 2015; revised November 18, 2015; accepted November 19, 2015. This work was supported in part by the Funds of National Science of China under Grant 61273148, Grant 61403070, and Grant 61473067, in part by the Funds for the Author of National Excellent Doctoral Dissertation, China, under Grant 201157, in part by the Fundamental Research Funds for the Central Universities under Grant N110804001, and in part by the IAPI Fundamental Research Funds under Grant 2013ZCX01-01. (Corresponding author: Guang-Hong Yang.) L.-B. Wu is with the College of Information Science and Engineering, Northeastern University, Shenyang 110004, China, and also with the School of Sciences, University of Science and Technology at Liaoning, Anshan 114051, China (e-mail: [email protected]). G.-H. Yang is with the College of Information Science and Engineering, Northeastern University, Shenyang 110004, China, and also with the State Key Laboratory of Synthetical Automation for Process Industries, Northeastern University, Shenyang 110004, China (e-mail: yangguanghong@ ise.neu.edu.cn). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TNNLS.2015.2503004

the modeling and control design for the uncertain nonlinear systems. In the past few decades, a number of adaptive fuzzy or NN control approaches have been studied (see, for instance, [7]–[15] and the references therein). With the help of backstepping techniques, Zhou et al. [14] proposed an adaptive fuzzy output control design method for a class of single-input and single-output nonlinear time-delays systems, and a fuzzy output-feedback dynamic surface control scheme is developed in [15] for a class of multi-input and multi-output nonlinear systems with partial tracking errors constrained. In addition, there are many stochastic nonlinear systems control methods and results [16]–[28] due to the fact that stochastic disturbance, which is usually a source of instability of control systems, often exists in practical systems. Typically, based on a quartic Lyapunov function, the corresponding adaptive backstepping design schemes are successfully applied to stochastic nonlinear systems [16]–[20] and high-order stochastic nonlinear systems [21], [22]. Recently, with the help of the only one adaptive parameter needed to be tuned online, the NN controller proposed in [23] ensures that the tracking error converges to an arbitrarily small neighborhood within the origin in the sense of a mean quartic value, and all signals of the closed-loop system are bounded in probability. In [24]–[26], decentralized adaptive output-feedback controllers for large-scale stochastic nonlinear systems with unmeasured states were developed. By using an input-driven filter to estimate the unavailable states, Zhou et al. [27] proposed an adaptive NN control scheme for a class of strictfeedback nonlinear stochastic systems with unknown time delays. Moreover, in [28], an adaptive fuzzy fault-tolerant control problem was considered for a class of uncertain stochastic nonlinear systems with actuator failures, which include loss of effectiveness and lock-in-place. On the other hand, it is well known that the nonaffine and nonlinear systems, which represent a more general class of triangular systems, can describe a number of practical control problems, such as the flight control systems [29], the mechanical systems [30], and the chemical reactions processes [31]. Meanwhile, it should be pointed out that most of the existing works were focused on the affine nonlinear systems rather than the nonaffine ones. The main reason is that no affine appearance of the state variables can be used as virtual control signals and actual control input, and hence, it makes the control system design more difficult and complicated. Some significant control approaches have been

2162-237X © 2015 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.

This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. 2

presented for uncertain nonaffine and nonlinear systems. In [32], by using the mean value theorem, an adaptive NN control scheme was proposed for a class of uncertain pure-feedback nonlinear systems. Karimi et al. [33], [34] considered decentralized neural-based adaptive control approaches for large-scale nonaffine and nonlinear systems with unknown nonlinear interconnections and unknown nonlinear functions. More recently, based on the implicit function theorem and disturbance observer, direct adaptive NNs control approaches were investigated for uncertain nonaffine and nonlinear systems in [35]. Under the partial persistent excitation condition and with an appropriate state transformation, Dai et al. [36] studied the problem of an adaptive NN learning control scheme of a class of nonaffine and nonlinear systems. In [37], the problem of adaptive fuzzy tracking control for a class of pure-feedback stochastic nonlinear systems with input saturation was studied. Subsequently, Wu and Yang [38] considered adaptive fuzzy tracking control schemes for a class of uncertain nonaffine and nonlinear systems with mismatched external disturbances, parameter uncertainties, and nonsymmetric dead-zone inputs. Nevertheless, it is noted that the assumption 0 < f 1 ≤ (∂ f (x, u)/∂u) ≤ f 2 is used in [32]–[38], so that the implicit function theorem or the mean value theorem holds, and the corresponding affine controller can be effectively designed using NNs or FLSs to approximate the unknown nonlinear functions. It is obvious that this assumption is relatively strict for applying to a system control design, because the differential condition ∂ f (x, u)/∂u is not always positive or negative for practical nonaffine and nonlinear systems. Meanwhile, as an important nonsmooth nonlinearity, dead-zone exists in a wide range of industrial and mechanics devices, and then severely impacts on a system performance. Consequently, the adaptive control approaches in [39]–[42] have been developed to compensate for the dead-zone nonlinearities. In particular, for the nonlinear systems considered in [32]–[38], if the deadzone constraints u d occur, then the partial differential of f (x, u) with respect to u d would not exist. For this reason, it is important to weaken the differential condition 0 < f 1 ≤ (∂ f (x, u)/∂u) ≤ f 2 for nonaffine and nonlinear systems. The above considerations motivate our study work. In particular, inspired by [37] and [41], this paper studies the problem of adaptive output NN control for a class of nonaffine stochastic nonlinear systems with actuator dead-zone inputs. Moreover, compared with the existing results, the main contributions of this paper are as follows. 1) The approximation-based adaptive control scheme is extended to stochastic nonaffine and nonlinear systems with an actuator dead-zone nonlinearity. 2) In the case of only one measurable output variable, a novel NN observer, which includes the nonaffine and nonlinear functions F(x, ˆ u d ) over the proper compact sets, is constructed to gain all the estimation states, and the adaptive output-feedback control scheme is developed based on the backstepping method. 3) In a technique, the main difficulty for an output control scheme is of the design of the virtual control v i = xˆi+1 − z i+1 , i = 1, 2, . . . , n − 1, in which xˆ i+1

IEEE TRANSACTIONS ON NEURAL NETWORKS AND LEARNING SYSTEMS

is the estimation state of the NN observer (27) in the later. In this paper, the proposed control scheme requires only one adaptive parameter (46) to be adjusted online for the stochastic nonaffine and nonlinear systems (1) with an unknown deadzone input (6) in the later, and the computational burden is significantly alleviated in this way. In addition, instead of the differential condition 0 < f 1 ≤ (∂ f (x, u)/∂u) ≤ f2 for nonaffine and nonlinear systems as shown in [32]–[38], and by employing a weaker assumption condition given in (10) and (11) of Assumption 1 in the later, a novel adaptive output NN tracking controller with proper parameter updated laws is developed to guarantee that all the closed-loop signals are bounded in probability, and the system output tracking error can converge to a small neighborhood in the sense of a mean quartic value. The rest of this paper is organized as follows. In Section II, the control objective and the preliminaries results are given. The NN observer design problem is addressed in Section III. In Section IV, a novel adaptive output NN control scheme is presented. Simulation studies are then provided in Section V to verify the effectiveness of our proposed control approach. Finally, the conclusion is drawn in Section VI. II. P RELIMINARIES AND P ROBLEM S TATEMENT A. Stochastic Nonaffine and Nonlinear System Descriptions Consider the following stochastic nonaffine and nonlinear systems described by the form: d x i = (x i+1 + f i (x¯i ))dt + giT (y)dω, 1 ≤ i ≤ n − 1 d x n = f (x, u)dt + gnT (y)dω y = x1

(1)

where x = [x 1, x 2 , . . . , x n ]T ∈ Rn is the system state and u ∈ R and y ∈ R are the actual control input and system output, respectively. x¯i = [x 1 , x 2 , . . . , x i ]T , i = 1, 2, . . . , n − 1, and ω denotes an r -dimensional standard Brownian motion defined on the complete probability space (, F , P), where  is a sample space, F is a σ -field, and P is a probability measure. f i (·) : Ri → R and gi (·) : R → Rr stand for unknown smooth system functions with f i (0) = 0 and gi (0) = 0, i = 1, 2, . . . , n − 1, respectively. The nonlinear function f (·, ·) : Rn × R → R is assumed to be known and continuous. Remark 1: For many practical systems, the existences of stochastic disturbances are inevitable. For example, the flight control systems [29], the mechanical systems [30], and the chemical reactions processes [31] can be in the form of the nonaffine and nonlinear systems (1). In addition, Wang et al. [23] and Zhou et al. [27] considered the tracking control problem based on the state feedback and the output feedback for a state time delay of the affine form of system (1), respectively. Before proceeding, the following definitions and lemmas are presented. B. Stochastic Stability Consider the following stochastic system: d x = f (x)dt + h(x)dω

(2)

This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. WU AND YANG: ADAPTIVE OUTPUT NN CONTROL FOR A CLASS OF STOCHASTIC NONLINEAR SYSTEMS

where x and ω are defined in (1), and f (·) : Rn → Rn and h(·) : Rn → Rn×r are locally Lipschitz functions in x and satisfy f (0) = 0 and h(0) = 0, respectively. Definition 1: For any given V (x) ∈ C 2 , associated with the stochastic differential equation (5), define the differential operator L as follows:     1 ∂2V ∂V T f (x) + Tr h T (x) 2 h(x) (3) LV = ∂x 2 ∂x where Tr(A) is the trace of A. Remark 2: As shown in [24] and [37], the term (1/2)T r {h T (x)(∂ 2 V /∂ x 2 )h(x)} is called the I t oˆ correction term, in which the second differential ∂ 2 V /∂ x 2 makes the controller design much more difficult than that of the deterministic system. Definition 2 [44]: The solution process {x(t)|t ≥ 0} of a stochastic system (2) is said to be bounded in probability, if limn→∞ supt ≥0 P{x(t) > n} = 0, where P{A} denotes the probability of event A. Lemma 1 [45]: Consider the stochastic system (2). If there exists a positive definite, radially unbounded, twice continuously differentiable Lyapunov function V (·) : Rn → R, and constants ρ > 0 and γ > 0 such that the following inequality: LV ≤ −ρV + γ

(4)

then system (2) has a unique solution almost surely, and system (2) is bounded in probability. Lemma 2 (Young’s Inequality [17]): For ∀(x, y) ∈ R2 , the following inequality holds: xy ≤

εp p 1 |x| + q |y|q p qε

(5)

where ε > 0, p > 1, q > 1, and (1/ p) + (1/q) = 1. C. Dead-Zone Model For practical engineering system, an actuator dead-zone nonlinearity exists in a wide range of industrial and mechanics processes, which often severely affects system stability and performance. As stated in [5], [38], [41], and [42], the following unknown nonsymmetric dead-zone input nonlinearity model: ⎧ ⎪ ⎨m r (u − br ), u ≥ br (6) u d = N[u] = 0, bl ≤ u ≤ br ⎪ ⎩ m l (u − bl ), u ≤ bl can be written as u d (t) = mu(t) + d0 (u) ⎧

⎪ ⎨−m r br , u ≥ br ml , u ≤ 0 m= d0 (u) = −mu, bl ≤ u ≤ br (7) ⎪ mr , u ≥ 0 ⎩ −m l bl , u ≤ bl where u(t) is the actual input, u d (t) is unavailable for measurement, and N[u] is the actuator nonlinear operator. Besides, the unknown parameters m r and m l are the right and left slopes of the dead-zone, and br and bl are the breakpoints of the input nonlinearity, respectively. It is essentially stressed that d0 (u) is also bounded.

3

Hence, the dynamics of system (1) with a dead-zone nonlinearity (6) is written by the following form: d x i = (x i+1 + f i (x¯i ))dt + giT (y)dω, d x n = F(x, u d )dt +

1≤i ≤n−1

gnT (y)dω

y = x1 where

(8)

⎧ ⎪ ⎨ f (x, m r (u − br )), u ≥ br F(x, u d ) = f (x, 0), bl ≤ u ≤ br ⎪ ⎩ f (x, m l (u − bl )), u ≤ bl .

The control objective of this paper is to design an local adaptive output NN controller u(t) with the corresponding parameter update laws to ensure that the system output y follows a desired reference signal yd in the sense of a mean quartic value, and all the closed-loop signals are bounded in probability. Then, to ensure the achievement of the output tracking objective, in what follows, the necessary assumptions are introduced for the stochastic system (8) with an unknown dead-zone input. Assumption 1: For each 1 ≤ i ≤ n, there exists an unknown positive constant g¯ i such that gi (y) ≤ g¯ i φi (y), where φi (·) : R → R is a bounded nonnegative smooth function with φi (0) = 0. Assumption 2: Let x be a compact set, for the continuous function f (x, u) in stochastic nonaffine and nonlinear systems (1) with a dead-zone input (2), fixed u ∈ R, for any x, y ∈ x ⊂ Rn , there exists a positive constant L satisfying | f (x, u) − f (y, u)| ≤ Lx − y.

(9)

Besides, fixed x ∈ x , for any u, v ∈ R, there exist constants Mi and Ni , i = 1, 2 such that f (x, u) − f (x, v) ≥ M1 (u − v) + N1 if u ≥ v

(10)

f (x, u) − f (x, v) ≤ M2 (u − v) + N2 if u < v

(11)

where Mi , i = 1, 2 are two positive constants. Remark 3: It can be seen that Assumption 1 is quite standard, which means that the nonlinear stochastic disturbance is nonlinear parameterized, which is commonly used in the control design for deterministic or stochastic nonlinear systems. From the perspective of practical system, Assumption 2 is reasonable and implies that the control input gain is bounded within a compact set. It is noted that Assumption 2 relaxes the strict differential condition 0 < f 1 ≤ (∂ f (x, u)/∂u) ≤ f2 used in [32]–[38], respectively. That is, under this differential condition, if u ≥ v, according to the mean value theorem, it is obtained that f (x, u) − f (x, v) = ∂ f (x, u + θ (u − v))/ ∂u(u − v) ≥ f1 (u − v), θ ∈ (0, 1). Consequently, taking M1 = f 1 and N1 = 0, then (10) holds. Similarly, it follows from u < v and M2 = f 1 , N2 = 0 that (11) yields. To facilitate a control system design from a nonaffine form to an affine form, Lemma 3 will be used in the subsequent development. Lemma 3: Let x be a compact set, then, for all (x, u), (x, v) ∈ x × R, there always exist

This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. 4

IEEE TRANSACTIONS ON NEURAL NETWORKS AND LEARNING SYSTEMS

two constant C1 and C2 such that f (x, u) − f (x, v) = M(u − v) + N where

M=

and

D. RBF NNs Approximation (12)

f (Z ) = W ∗T S(Z ) + δ(Z ), δ(Z ) ≤ δ

M1 , u ≥ v M2 , u < v

W ∗ = arg min

W ∈R M

with Mi and Ni defined in (10) and (11), and θi ∈ (0, 1), i = 1, 2. Proof: Let u ⊂ R be a compact set. Since x is also a compact set and f (x, u) is continuous, it implies that f (x, u) − f (x, v) is continuous over the compact x × u . Denote Fmax and Fmin are the maximum and minimum of f (x, u)− f (x, v) on x × u , respectively, it is easy to obtain that (13)

N1 + M1 (u − v) ≤ f (x, u) − f (x, v) ≤ Fmax + M1 (u − v). (14) Similarly, if u < v, we also have Fmin + M2 (u − v) ≤ f (x, u) − f (x, v) ≤ N2 + M2 (u − v). (15) Invoking the intermediate value theorem from (14) and (15) yields f (x, u) − f (x, v) = M1 (u − v) + θ1 N1

+ θ2 Fmin , u < v

sup | f (Z ) − W T S(Z )|

Z ∈ Z

(19)

where W = [w1 , w2 , . . . , w M ]T is the weight vector, M is the number of RBF NN nodes with M > 1, and S(Z ) = [s1 (Z ), s2 (Z ), . . . , s M (Z )]T is the basis function regression vector with 

si (Z ) = exp − (Z − μi )T (Z − μi )/νi2 , i = 1, 2, . . . , M (20) where μi = [μi1 , μi2 , . . . , μim ]T is the center of the receptive field and νi is the width of the Gaussian function. III. NN S TATE O BSERVER D ESIGN

Accordingly, if u ≥ v, it follows from (10) and (13) that:

+ (1 − θ1 )Fmax , u ≥ v f (x, u) − f (x, v) = M2 (u − v) + (1 − θ2 )N2

(18)

where Z ∈  Z ⊂ Rm is the input vector with  Z being a compact set, δ(Z ) is the approximation error, and W ∗ is the optimal constant weight vector and is defined as 

θ1 N1 + (1 − θ1 )C1 , u ≥ v N= θ2 C2 + (1 − θ1 )N2 , u < v

Fmin ≤ f (x, u) − f (x, v) ≤ Fmax .

In this paper, the following radial basis function (RBF) NNs are used to appropriate the nonlinear smooth function f (·) : Rm → R. That is, for any a desired level of accuracy δ > 0, there exists an RBF NN W ∗T S(Z ) such that

(16) (17)

θi ∈ (0, 1), i = 1, 2. Taking Fmax ≤ C1 and Fmin ≥ C2 , it can be concluded from (16) and (17) that (12) holds. This completes the proof. Remark 4: Lemma 3 plays an important role in the design from a nonaffine form to an affine form. On the one hand, without using the implicit function theorem in [32]–[38], which needs the strict differential condition 0 < f 1 ≤ (∂ f (x, u)/∂u) ≤ f 2 to deal with the nonaffine term f (x, u), the affine form (12) is derived using an intermediate value theorem of a continuous function. On the other hand, if the dead-zone constraints u d = mu + d(u) occur, then the partial differential of f (x, u) with respect to u d would not exist. Therefore, it is advantageous to design an adaptive controller for compensating a dead-zone input in the case of removing the differential condition.

It is should be pointed out that the state x 1 of the stochastic system (1) is only available and the other states x 2 , x 3 , . . . , x n are not measured. Therefore, to achieve the output-feedback control objective, an NN state observer should be established to obtain the estimates of unmeasured states. Furthermore, the stochastic nonaffine and nonlinear systems (1) can be rewritten in the following form:   n−1  Bi f i (x¯i )+ B F(x, u d ) dt + G(x)dw d x = Ax + K y + y = Cx

i=1

(21)

where K = [l1 , l2 , . . . , ln ]T , B = [0, 0, . . . , 1]T , Bi = [0, . . . , 0,  1 , 0, . . . , 0]T , i = 1, 2, . . . , n − 1, G(x) = [g1(y), ith

g2 (y), . . . , gn (y)]T , C = [1, 0, . . . , 0], and ⎡ ⎤ −l1 1 0 ··· 0 ⎢ −l2 0 1 ··· 0⎥ ⎢ ⎥ ⎢ ⎥ .. .. A=⎢ ⎥. . . ⎢ ⎥ ⎣ −ln−1 0 0 · · · 1 ⎦ −ln 0 0 ··· 0

(22)

In view of the structure of A and by choosing suitable parameters li , i = 1, 2, . . . , n, A can be designed as a Hurwitz matrix. Furthermore, for any given positive definite matrix Q = Q T > 0, there exists a positive definite matrix P = P T > 0 such that the following Lyapunov equation: A T P + P A = −Q.

(23)

Meanwhile, from (10) and according to [42], the nonlinear smooth functions fi (x¯i ), i = 1, 2, . . . , n − 1 in (1) can be approximated by the following NNs: fˆi (x¯i |Wi ) = WiT Si (Z i )

(24)

This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. WU AND YANG: ADAPTIVE OUTPUT NN CONTROL FOR A CLASS OF STOCHASTIC NONLINEAR SYSTEMS

then we have fˆi (xˆ¯i |Wi ) = WiT Si ( Zˆ i ) (25) xˆ¯i = [xˆ1 , xˆ2 , . . . , xˆi ]T is the estimate vector of [x 1 , x 2 , . . . , x i ]T , Z i = x¯i , and Zˆ i = xˆ¯i , i = 1,

where x¯i = 2, . . . , n −1. Moreover, with the help of (18), the optimal NNs parameter vectors Wi∗ and corresponding to NNs approximation errors δi∗ are defined as  ∗ sup | fˆi (x¯ˆi |Wi ) − f i (x¯i )| Wi = arg min 

Wi ∈i

 ∗

( x¯ i , xˆ¯ i )∈ i

δi∗ = fˆi xˆ¯i |Wi − f i (x¯i )

(26)

where i ⊂ R N and i ⊂ R2i are two compact sets for i = 1, 2, . . . , n − 1. Assumption 3 [5], [13]: For each NNs approximation error δi∗ , there exists a unknown positive constant δ¯i∗ satisfying |δi∗ | ≤ δ¯i∗ , i = 1, 2, . . . , n − 1. Consequently, the NN state observer for (8) is designed as follows: n−1  Bi fˆi (xˆ¯i |Wi ) + B F(x, ˆ ud ) x˙ˆ = A xˆ + K y +

Next, according to Assumption 3, combining the facts S T ( Zˆ i )S( Zˆ i ) ≤ s [24] and considering the following inequalities: 2e T P

n−1  i=1

n−1    1 W˜ iT W˜ i Bi W˜ iT S( Zˆ i ) ≤ P2 e2 + τ s 2 τ i=1

(31) 2e T P

n−1 

(27)

then, combining (8) and (27), the observer error stochastic equation can be obtained  n−1  de = Ae + Bi ( f i (x¯i ) − fˆi (xˆ¯i |Wi )) i=1  +B(F(x, u d ) − F(x, ˆ u d )) dt + G(x)dw  =

Ae +

n−1 

  Bi W˜ iT Si ( Zˆ i ) + δi∗

i=1



+B(F(x, u d ) − F(x, ˆ u d )) dt + G(x)dw

(28)

where e = x − xˆ is the observer error vector and W˜ i = Wi∗ − Wi , i = 1, 2, . . . , n − 1 are the parameter error vectors. Next, based on the continuous nonaffine and nonlinear conditions (9)–(11) of Assumption 2, the stability of the state estimation error is given in Lemma 4. Lemma 4: Consider the Lyapunov candidate V0 = e T Pe for the observer error system (28), then the following inequality holds: LV0 ≤ −e T (Q − (α + 1)I )e + β(y) + τ s 2

n−1 

W˜ iT W˜ i

(29)

i=1

where + (1/τ )P2 and β(y) = n α 2 =2 2LPB2  n−1 ¯ ∗2 P i=1 g¯ i φi (y) + P i=1 δi . Proof: Using (23), (28), and Assumption 2, and taking the differential operator of V0 gives LV0 ≤ −e T Qe + 2e T PBLe+2e T P

n−1 

  Bi W˜ iT S( Zˆ i )+δi∗

i=1

+ T r {G T (x)PG(x)}.

(30)

Bi δi∗ ≤ e2 + P2

i=1

n−1 

δ¯i∗2

(32)

i=1

 where τ is a positive design parameter and s = ∞ k=0 3q(k + 2 2 2 2)q−1 e−2ρ k /η . Moreover, by the definition of T r {A}, we have T r {G T (x)PG(x)} ≤ P

n 

g¯ i2 φi2 (y).

(33)

i=1

From (31)–(33), (30) becomes LV0 ≤ −e T (Q − (α + 1)I )e + β(y) + τ s 2

n−1 

W˜ iT W˜ i

(34)

i=1

i=1

yˆ = C xˆ

5

where (1/τ )P2 and β(y) = n α 2 =2 2LPB 2 + n−1 ¯∗2 P i=1 g¯ i φi (y) + P i=1 δi . This completes the proof. IV. A DAPTIVE NN O UTPUT T RACKING C ONTROLLER D ESIGN In this section, the adaptive NN output tracking control scheme will be developed for the stochastic nonaffine and nonlinear systems (1) with a dead-zone nonlinearity (6). With the help of a backstepping design method, the recursive design procedure contains n steps. It should be pointed out that from Step 1 to Step n − 1, the continuous nonlinear functions involved in the virtual control, which are used to be approximated by NNs over the corresponding compacts, are constructed at each step. Finally, the overall control input u is designed at Step n. To achieve the output tracking control objective, the change of coordinates is given as follows: z 1 = y − yd z i = xˆi − v i−1 , i = 2, 3, . . . , n

(35) (36)

where yd is the desired trajectory, v i−1 is the virtual control function in Step i − 1, and the controller u will be designed in Step n. In addition, for convenience, denote (i) (i) (i) y¯d = [yd , y˙d , . . . , yd ]T and Wˆ¯ i = [W1 , W2 , . . . , Wi ]T , i = 1, 2, . . . , n − 1. In the following, the standard backstepping method will be used to design an adaptive NN controller. Step 1: According to (21) and (35), as well as I t oˆ formula, we have dz 1 = (x 2 + f 1 (x 1 ) − y˙d )dt + g1T (y)dw   = xˆ2 + e2 + W1T S1 ( Zˆ 1 ) + W˜ 1T S1 ( Zˆ 1 ) + δ1∗ − y˙d dt + g1T (y)dw.

(37)

This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. 6

IEEE TRANSACTIONS ON NEURAL NETWORKS AND LEARNING SYSTEMS

Consider the Lyapunov function candidate as 1 1 2 1 ˜T ˜ W W1 V1 = V0 + z 14 + θ˜ + 4 2γ0 2γ1 1

(38)

where γ0 and γ1 are positive design parameters. W˜ i = Wi∗ − Wi , i = 1, 2, . . . , n − 1 are the parameter errors defined in (28). Besides, for the convenience of a controller design, denote θ = max1≤i≤n {W¯ i 2 } with the weight vectors W¯ i , i = 1, 2, . . . , n being given in later, θˆ is the estimation of the unknown parameter θ , and θ˜ = θˆ − θ is the parameter error with θˆ being the estimation of θ . Consequently, based on (29), taking the differential operator of (38) gives   LV1 = LV0 +z 13 xˆ2 + e2 + W1T S1 ( Zˆ 1 )+ W˜ 1T S1 ( Zˆ 1 )+δ1∗ − y˙d 1 3 1 + z 12 g1 (y)2 + θ˜ θ˙ˆ − W˜ 1T W˙ 1 2 γ0 γ1 ≤ −e T (Q − (α + 1)I )e + β(y) + z 13 e2   + z 13 xˆ2 + W1T S1 ( Zˆ 1 ) + W˜ 1T S1 ( Zˆ 1 ) + δ1∗ − y˙d 3 1 1 + z 12 g1 (y)2 + θ˜ θˆ˙ − W˜ 1T W˙ 1 2 γ0 γ1 n−1  W˜ iT W˜ i . + τ s2 (39) i=1

Noting that the facts z 13 e2 ≤ (1/2)z 16 + (1/2)e2 and z 12 g1 (y)2 ≤ (1/2)m 21 + (1/2m 21 )z 14 g¯ 14 φ14 (y), one can obtain     3 LV1 ≤ −e T Q − α + I e + β(y) 2   3 + z 13 xˆ2 +W1T S1 ( Zˆ 1 ) + F1 ( Z¯ˆ 1 ) − z 14 +z 13 W˜ 1T S1 ( Zˆ 1 ) 4 n−1  1 ˜T ˙ 3 2 1 ˙ 2 ˜ ˆ + m 1 + θ θ − W1 W1 + τ s W˜ iT W˜ i (40) 4 γ0 γ1 i=1

where m 1 is a design parameter, and F1 ( Zˆ¯ 1 ) = (3/4m 21 )z 1 g¯ i4 φi4 (y)+(1/2)z 13 +(3/4)z 1 +δ1∗ − y˙d with Zˆ¯ 1 being ¯ ˆ , which is a compact set defined Zˆ¯ 1 = [x 1 , yd , y˙d ]T ∈  Z¯ 1

on R3 . Invoking (18), the RBF NN W¯ 1T S1 ( Zˆ¯ 1 ) is used to approximate the unknown nonlinear function F1 ( Zˆ¯ 1 ) over the compact ¯ ˆ . That is to say, for any given positive constant ε1 , we set  Z¯ 1 have F ( Z¯ˆ ) = W¯ T S ( Z¯ˆ ) + δ¯ ( Z¯ˆ ), |δ¯ ( Z¯ˆ )| ≤ ε (41) 1

1

1

1

1

1

1

1

1

3 1 1 + z 13 W˜ 1T S1 ( Zˆ 1 ) + m 21 + n 21 + ε14 4 2 4  1 1 ˙ θ˜ θˆ − W˜ 1T W˙ 1 + τ s 2 W˜ iT W˜ i . γ0 γ1 n−1

+

(43)

i=1

Consequently, using Lemma 2 with p = 3/4, q = 4 again, the following inequality holds: 3 4 1 4 z + z . (44) 4 1 4 2 Choose the virtual control function v 1 with respect to the adaptive laws θ and W1 as follows: z 13 z 2 ≤

v 1 = −k1 z 1 −

1 3 z 1 θˆ S1 ( Zˆ¯ 1 )2 − W1T S1 ( Zˆ 1 ) 2n 21

 γ0 z 6 Si ( Zˆ¯ i )2 θ˙ˆ = −k0 θˆ + 2 i 2n i i=1

(45)

n

W˙ 1 = −γ1 σ1 W1 + γ1 z 13 S1 ( Zˆ 1 )

(46) (47)

where k0 , k1 , and σ1 are the positive design parameters. Next, combining (44), substituting (45)–(47) into (43) yields     3 I e+β LV1 ≤ −e T Q − α + 2   1 4 1 3 γ0 6 ˙ ˆ 4 2 ¯ ˜ ˆ + z 2 − c1 z 1 + θ θ − 2 z 1 S1 ( Z 1 ) + m 21 4 γ0 4 2n 1  1 1 W˜ iT W˜ i + n 21 + ε14 + σ1 W˜ 1T W1 + τ s 2 2 4 n−1

(48)

i=1

1

with δ¯1 ( Z¯ 1 ) being the approximation error. Moreover, it follows from Lemma 2 and the definition of θ that: W¯ T z 13 F1 ( Zˆ¯ 1 ) = z 13 1 W¯ 1 S1 ( Zˆ¯ 1 ) + z 13 δ¯1 ( Zˆ¯ 1 ) W¯ 1  1 6 ¯ 2 ≤ z 1 W1  S1 ( Z¯ˆ 1 )2 2n 21 1 1  3 4/3 1 ¯4 ˆ¯ z + n 21 + + δ1 ( Z 1 ) 2 4/3 1 4 1 6 1 3 1 2 ˆ ≤ z 1 θ S1 ( Z¯ 1 ) + n 21 + z 14 + ε14 (42) 2 4 4 2n 21 with n 1 being a design parameter.

Meanwhile, it follows from Assumption 1 that   P ni=1 g¯ i2 φi2 (y) ≤ P ni=1 g¯ i2 φi∗2 with φi∗ being an ¯ ˆ . upper bound of φi (y) over the corresponding compact  Z¯ 1  n ∗2 ¯ Consequently, setting β = P i=1 g¯ i2 φi∗2 +P2 n−1 δ i=1 i , and combining (41) and (42), (40) becomes     3 I e + β + z 13 z 2 LV1 ≤ −e T Q − α + 2   1 3 ˆ 3 T 2 ˆ ¯ + z 1 v 1 + W1 S1 ( Z 1 ) + 2 z 1 θ S1 ( Z 1 ) 2n 1

where c1 = k1 − (3/4) > 0 is a design parameter, and the function (1/4)z 24 will be handled in the next step. Step 2: Together (21) with (36) and I t oˆ formula, one can obtain that   ∂v 1 ˙ˆ dz 2 = xˆ3 + l2 e1 + W2T S2 ( Zˆ 2 ) dt − θ dt ∂ θˆ 1  ∂v 1 (i+1) ∂v 1 ˙ W1 dt y dt − − (i) d ∂ W1 i=0 ∂yd −

∂v 1 1 ∂ 2v1 g1 (y)2 dt − d x1 2 ∂ x 12 ∂ x1

This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. WU AND YANG: ADAPTIVE OUTPUT NN CONTROL FOR A CLASS OF STOCHASTIC NONLINEAR SYSTEMS



∂v 1 ˙  ∂v 1 (i+1) = xˆ3 + l2 e1 + W2T S2 ( Zˆ 2 ) − y θˆ − (i) d ∂ θˆ i=0 ∂yd  ∂v 1 ˙ ∂v 1 1 ∂ 2 v1 2 − (x 2 + f 1 (x 1 ))− g1 (y) dt W1 − ∂ W1 ∂ x1 2 ∂ x 12 ∂v 1 T − g (y)dw. (49) ∂ x1 1 1

¯ is the approximation error. Similar to the calcuwhere δ¯2 ( Z) lation of (41)–(43) in Step 1, (52) becomes LV2 ≤ −e T (Q − (α + 2)I )e + β − c1 z 14   1 γ0 6 ˙ 2 ˆ ¯ ˜ ˆ + θ θ − 2 z 1 S1 ( Z 1 ) + z 23 z 3 γ0 2n 1   1 + z 23 v 2 + W2T S2 ( Zˆ 2 ) + 2 z 23 θ S2 ( Zˆ¯ 2 )2 2n 2

Consider the Lyapunov function as follows: 1 1 ˜T ˜ V2 = V1 + z 24 + W W2 4 2γ2 2

7

(50)

+

with γ2 being the positive design parameter. According to (49), the differential operator of V2 can be shown as follows:

3 2 1 2 1 4 mi + ni + εi 4 2 4 2

2

2

i=1

i=1

i=1

 1 ˜T ˙ W2 W2 + σ1 W˜ 1T W1 + τ s 2 W˜ iT W˜ i . γ2 n−1

−

(54)

i=1

LV2 ≤ LV1



+ z 23 xˆ3 + l2 e1 + W2T S2 ( Zˆ 2 ) −

Consequently, by using the fact that z 23 z 3 ≤ (3/4)z 24 +(1/4)z 34, the virtual control function v 2 with respect to the adaptive law W2 is chosen as follows:

∂v 1 ˙ˆ θ ∂ θˆ

1  ∂v 1

∂v 1 ˙ ∂v 1 − W1 − − ∂ W1 ∂ x1  ∂v 1 3 1 ∂ 2 v1 ×(xˆ2 + f 1 (x 1 )) − g1 (y)2 − z e2 2 ∂ x 12 ∂ x1 2   3 2  ∂v 1 2 1 + z2  g1 (y)2 − W˜ 2T W˙ 2 . (51) 2 ∂ x1  γ2 y (i+1) (i) d ∂y i=0 d

v 2 = −k2 z 2 −

W˙ 2 = −γ2 σ2 W2

where m 2 is a design parameter, and F2 ( Zˆ¯ 2 ) = (3/4m 22 )z 2 |∂v 1 /∂ x 1 |4 g¯ 14 φ14 (y) + (1/2)|∂v 1 /∂ x 1 |2 z 23 + l2 e1 − ˆ θ˙ˆ − (∂v 1 /∂ W1 )W˙ 1 − (∂v 1 /∂ x 1 )(xˆ2 + f 1 (x 1 )) − (∂v 1 /∂ θ) 1 (i) (i+1) + (3/4)z 2 with Zˆ¯ 2 being Zˆ¯ 2 = i=0 (∂v 1 /∂yd )yd ¯ ˆ , which is a compact set [x 1 , xˆ1 , θˆ , W1 , yd , y˙d , y¨d ]T ∈  ¯ Z2

defined on R N+6 . From (18), for any given constant ε2 > 0, the RBF NN W¯ 2T S2 ( Zˆ¯ 2 ) is employed to approximate the unknown nonlinear ¯ ˆ as follows: function F2 ( Zˆ¯ 2 ) over the compact set  ¯

LV2 ≤ −e (Q − (α + 2)I )e + β −

(53)

2 

c j z 4j

j =1

⎞ 2  γ0 6 1 1 + θ˜ ⎝θ˙ˆ − z S j ( Zˆ¯ j )2 ⎠ + z 34 2 j γ0 4 2n j j =1 ⎛

+ +

3 2 1 2 1 4 mi + ni + εi 4 2 4 2

2

i=1

i=1

2 

σi W˜ iT Wi + τ s 2

i=1

2

i=1

n−1 

W˜ iT W˜ i

(57)

i=1

where c2 = k2 − (3/4) > 0 is a design parameter, and the function (1/4)z 34 will be handled in the next step. Step i (3 ≤ i ≤ n − 1): Recursively, invoking (55) yields   ˆ yd , . . . , y (i−1) . v i−1 = v i−1 x 1 , xˆ2 , . . . , xˆi−1 ,W1 , . . . ,Wi−1 θ, d Consequently, it can be inferred from (21), (27), and (36), and the form of v i−1 that   dz i = xˆ i+1 + li e1 + WiT Si ( Zˆ i ) dt ⎛ ⎞ i−1 i−1   ∂v ∂v ∂v i−1 ˆ˙ i−1 ( j +1) i−1 ˙ ⎠ y − −⎝ W j dt θ− ( j) d ∂Wj ∂ θˆ ∂y j =0

−

i−1  j =2

Z2

F2 ( Zˆ¯ 2 ) = W¯ 2T S2 ( Zˆ¯ 2 ) + δ¯2 ( Zˆ¯ 2 ), |δ¯2 ( Zˆ¯ 2 )| ≤ ε2

(56)

T

(1/2m 22 )z 24 |∂v 1 /∂ x 1 |4 g¯ 14 φ14 (y) and using (48), (51) becomes

i=1

(55)

with k2 and σ2 are the positive design parameters. Substituting (55) and (56) into (54) gives

Invoking the inequalities −(∂v 1 /∂ x 1 )z 23 e2 ≤ (1/2)e2 + (1/2)|∂v 1 /∂ x 1 |2 z 26 and z 22 |∂v 1 /∂ x 1 |2 g1 (y)2 ≤ (1/2)m 22 +

LV2 ≤ −e T (Q − (α + 2)I )e + β − c1 z 14   1 ˜ ˙ˆ γ0 6 ˆ 2 ¯ + θ θ − 2 z 1 S1 ( Z 1 ) γ0 2n 1   + z 23 xˆ3 + W2T S2 ( Zˆ 2 ) + F2 ( Z¯ˆ 2 ) 3 3 1 3 1 1 − z 24 + m 22 − W˜ 2T W˙ 2 + m 21 + n 21 + ε14 4 4 γ2 4 2 4 n−1  + σ1 W˜ 1T W1 + τ s 2 (52) W˜ iT W˜ i

1 3 z 2 θˆ S2 ( Zˆ¯ 2 )2 − W2T S2 ( Zˆ 2 ) 2n 22

−

d

j =1

 ∂v i−1  xˆ j +1 + W jT S j ( Zˆ j ) + l j e1 dt ∂ xˆ j

1 ∂ 2 v i−1 ∂v i−1 dy g1 (y)2 dt − 2 ∂ x 12 ∂y

This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. 8

IEEE TRANSACTIONS ON NEURAL NETWORKS AND LEARNING SYSTEMS

⎛

∂v i−1 ˙ˆ  ∂v i−1 ( j +1) = ⎝ xˆi+1 +li e1 +WiT Si ( Zˆ i )− y θ− ( j) d ∂ θˆ ∂y i−1

j =0

−

i−1  j =1

d

∂v i−1 ∂v i−1 ˙ Wj − (x 2 + f 1 (x 1 )) ∂Wj ∂ x1

 ∂v i−1 1 ∂ 2 v i−1 2 − g (y) − 1 2 ∂ x 12 ∂ xˆ j j =2 ⎞   ∂v i−1 T × xˆ j +1 +W jT S j ( Zˆ j ) + l j e1 ⎠ dt − g (y)dw. ∂ x1 1 i−1

(58) In the following, choose the Lyapunov function as: 1 1 ˜T ˜ Vi = Vi−1 + z i4 + W Wi 4 2γi i

(59)

with γi being the positive design parameter. From (57) and (58), taking the differential operator of Vi yields LVi ≤ LVi−1 + z i3 ⎛ × ⎝xˆi+1 + li e1 + WiT Si ( Zˆ i ) − −

i−1  ∂v i

( j +1)

y ( j) d

j =0 ∂yd

−

∂v i−1 ˙ˆ θ ∂ θˆ

i−1  ∂v i−1 j =1

∂Wj

∂v i−1 W˙ j − ∂ x1

(1/2)m 2i + (1/2m 2i )z i4 |∂v i−1 /∂ x 1 |4 g¯ 14 φ14 (y), (60) becomes    i−1  i Q− α+ +1 I e+β − c j z 4j 2 j =1 ⎞ ⎛ i−1  γ0 6 1 + θ˜ ⎝θ˙ˆ − z S j ( Zˆ¯ j )2 ⎠ 2 j γ0 2n j j =1 

j =1

+

i−1  j =1

j =1

σ j W˜ jT W j + τ s 2

1 + 4

n−1  i=1

ε4j

i−1

i−1

Zˆ¯ i

d

Zi

Fi ( Zˆ¯ i ) = W¯ iT Si ( Z¯ˆ i ) + δ¯i ( Z¯ˆ i ), |δ¯i ( Zˆ¯ i )| ≤ εi

(62)

where δ¯i ( Z¯ ) is the approximation error. Similar to the calculation of (54) in Step 2, (61) becomes    i−1  i LVi ≤ −e Q− α+ +1 I e+β − c j z 4j 2 j =1 ⎛ ⎞ i−1  γ0 1 + θ˜ ⎝θ˙ˆ − z 6 S j ( Zˆ¯ j )2 ⎠ + z i3 z i+1 2 j γ0 2n j j =1   1 ˆ 3 T 3 2 + z i v i + Wi Si ( Zˆ i ) + 2 z i θ Si ( Z¯ i ) 2n i 

T

3 2 1 2 1 4 1 ˜T ˙ mj + nj + εj Wi Wi + γi 4 2 4 j =1

+

i 

σ j W˜ jT W j + τ s 2

j =1

n−1 

i

i

j =1

j =1

W˜ iT W˜ i .

(63)

i=1

4 , By using Young’s inequality z i3 z i+1 ≤ (3/4)z i4 + (1/4)z i+1 the virtual control function v i with respect to the adaptive law Wi is chosen as follows:

v i = −ki z i − W˙ i = −γi σi Wi

1 3 z i θˆ Si ( Zˆ¯ i )2 − WiT Si ( Zˆ i ) 2n 2i

(64) (65)

with ki and σi are the positive design parameters. Substituting (64) and (65) into (63) yields    i  i c j z 4j Q− α+ +1 I e+β − LVi ≤ −e 2 j =1 ⎞ ⎛ i  γ0 1 1 4 + θ˜ ⎝θ˙ˆ − z 6 S j ( Zˆ¯ j )2 ⎠ + z i+1 2 j γ0 4 2n j j =1 + +

j =1

W˜ iT W˜ i

1

T

i   3 3 2 + z i3 xˆi+1 + WiT Si ( Zˆ i ) + Fi ( Zˆ¯ i ) − z i4 + mj 4 4

n 2j

i

a compact set defined on R2i+(i−1)N+2 . Next, for any given constant εi > 0, the RBF NN W¯ iT Si ( Zˆ¯ i ) is employed to approximate the unknown nonlinear function ¯ ˆ as follows: Fi ( Zˆ¯ i ) over the compact set  ¯



LVi ≤ −e T

1 1 − W˜ iT W˙ i + γi 2

i

=

i

By applying mathematical induction procedures and noting the fact that −(∂v i−1 /∂ x 1 )z i3 e2 ≤ (1/2)e2 + (1/2)|∂v i−1 /∂ x 1 |2 z i6 and z i2 |∂v i−1 /∂ x 1 |2 g1 (y)2 ≤

i−1 

(3/4m 2i )z i |∂v i−1 /∂ x 1 |4 g¯ 14 φ14 (y) + ˆ θ˙ˆ (1/2)|∂v i−1 /∂ x 1 |2 z i3 + l i e1 − (∂v i−1 /∂ θ) − i−1 ˙ (∂v /∂ W ) W − (∂v /∂ x )( x ˆ + f (x )) − i−1 j j i−1 1 2 1 1 j =1 i−1 i−1 ( j ) ( j +1) (∂v /∂y )y − (∂v /∂ x ˆ )( i−1 i−1 j xˆ j +1 + j =0 j =2 d d T W j S j ( Zˆ j )+l j e1 )+(3/4)z i with m i being a design parameter (i) ¯ , which is and Zˆ¯ being Zˆ¯ = [x , xˆ¯ , θˆ , Wˆ¯ , y¯ ]T ∈ 

−

i−1  ∂v i−1 1 ∂ 2 v i−1 2 ×(xˆ2 + f1 (x 1 )) − g (y) − 1 2 2 ∂ x1 ∂ xˆ j j =2 ⎞   ∂v i−1 3 × xˆ j +1 + W jT S j ( Zˆ j ) + l j e1 ⎠ − z e2 ∂ x1 2   3  ∂v i−1 2 1 + z 22  g1 (y)2 − W˜ iT W˙ i . (60)  2 ∂ x1 γi

i−1 

Fi ( Zˆ¯ i )

where

3 2 1 2 1 4 mj + nj + εj 4 2 4 i

i

j =1

j =1

i  j =1

(61)

σ j W˜ jT Wi + τ s 2

i

j =1

n−1 

W˜ jT W˜ j

(66)

j =1

where ci = ki − (3/4) > 0 is a design parameter, and the 4 function (1/4)z i+1 will be handled in the next step.

This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. WU AND YANG: ADAPTIVE OUTPUT NN CONTROL FOR A CLASS OF STOCHASTIC NONLINEAR SYSTEMS

Step n: The actual control input will be constructed in the final step. Using (21) and (36), and invoking I t oˆ formula gives ⎛ n−1 ∂v n−1 ˙  ∂v n−1 ( j +1) dz n = ⎝F(x, ˆ u d ) + l n e1 − y θˆ − ( j) d ∂ θˆ ∂y j =0

−

n−1  ∂v n−1 j =0

∂Wj

W˙ j −

i−1  ∂v i−1  j =2

∂ xˆ j

d

 xˆ j +1 +W jT S j ( Zˆ j )+l j e1

d

∂v n−1 ∂v n−1 e2 − (xˆ2 + f 1 (x 1 )) ∂ x1 ∂ x1  1 ∂ 2 v n−1 ∂v n−1 T − g1 (y)2 dt − g (y)dw. 2 ∂ x 12 ∂ x1 1

(67)

1 (68) Vn = Vn−1 + z n4 4 with γn being the positive design parameter. From (68), taking the differential operator of Vn gives LVn ≤ LVn−1 ⎛ ∂v n−1 ˙ˆ ˆ u d ) + l n e1 − + z n3 ⎝ F(x, θ− ∂ θˆ

j =1

n−1 

( j +1) y ( j) d

j =0 ∂yd

∂v n−1 ∂v n−1 ˙ Wj − (xˆ2 + f1 (x 1 )) ∂Wj ∂ x1 n−1

⎞   ∂v n−1 3 × xˆ j +1 + W jT S j ( Zˆ j ) + l j e1 ⎠ − z e2 ∂ x1 n (69)

Accordingly, noting the fact that −(∂v n−1 /∂ x 1 )z n3 e2 ≤ (1/2)e2 + (1/2)|∂v n−1 /∂ x 1 |2 z n6 and z n2 |∂v n−1 / ∂ x 1 |2 g1 (y)2 ≤ (1/2)m 2n + (1/2m 2n )z n4 |∂v n−1 /∂ x 1 |4 g¯ 14 φ14 (y), we have n−1 $ $ % %  n LVn ≤ −e T Q − α + + 1 I e + β − c j z 4j 2 j =1 ⎞ ⎛ n−1  γ0 6 1 + θ˜ ⎝θ˙ˆ − z S j ( Zˆ¯ j )2 ⎠ 2 j γ0 2n j j =1 + z n3 (F(x, ˆ u d ) − F(x, ˆ 0)) + z n3 F˜n ( Zˆ¯ n )

j =1

+

n−1  j =1

σ j W˜ jT W j + τ s 2

j =1

j =1

n−1  i=1

W˜ iT W˜ i

ε4j +

j =1

n−1 

σ j W˜ jT W j + τ s 2

j =1

n−1

j =1

j =1

n−1 

W˜ iT W˜ i

(71)

i=1

where Fn ( Zˆ¯ n ) = F˜n ( Zˆ¯ n ) + Md0 (u) + N. Moreover, for any given constant εn > 0, the RBF NN W¯ nT Sn ( Zˆ¯ n ) is employed to approximate the unknown nonlinear ¯ ˆ as follows: function Fi ( Zˆ¯ n ) over the compact set  ¯

(70)

(72)

where δ¯n ( Z¯ ) is the approximation error. Substituting (72) into (71) and similar to the calculation of (63) in Step i , we have    n−1  n+3 3 I e+β − Q− α+ c j z 4j − z n4 LVn ≤ −e 2 4 j =1 ⎞ ⎛ n−1  γ0 6 1 + θ˜ ⎝θ˙ˆ − z S j ( Zˆ¯ j )2 ⎠ + z n3 2 j γ0 2n j j =1   n n 3 2 1 2 1 × Mmu + 2 z n3 θ Sn ( Zˆ¯ n )2 + mj+ nj 2n n 4 2 

T

j =1

+

1 4

n  j =1

ε4j +

n−1  j =1

σ j W˜ jT W j + τ s 2

n−1 

W˜ iT W˜ i .

j =1

(73)

i=1

It follows from (73) that the actual controller is designed as:   1 u = (min )−1 −kn z n − 2 z n3 θˆ Sn ( Zˆ¯ n )2 2n n

3 2 1 2 1 4 3 mj + nj + εj − z n4 + 4 4 2 4 n−1

1 4

n−1 

n

Fn ( Zˆ¯ n ) = W¯ nT Sn ( Z¯ˆ n ) + δ¯n ( Z¯ˆ n ), |δ¯n ( Zˆ¯ n )| ≤ εn

j =2

n−1

3 3 2 1 2 + z n3 (Mmu + Fn ( Zˆ¯ n )) − z n4 + mj + nj 4 4 2

Zn

 ∂v n−1 1 ∂ 2 v n−1 g1 (y)2 − − 2 2 ∂ x1 ∂ xˆ j

n

   n−1  n+1 +1 I e+β − Q− α+ c j z 4j 2 j =1 ⎛ ⎞ n−1  γ0 6 1 + θ˜ ⎝θˆ˙ − z S j ( Zˆ¯ j )2 ⎠ 2 j γ0 2n j j =1

LVn ≤ −e T

+

∂v n

  3 2  ∂v n−1 2 + zn  g1 (y)2 . 2 ∂ x1 

Z¯ n



Next, the Lyapunov function is as follows:

−

ˆ 0) + (3/4m 2n )z n |∂v n−1 /∂ x 1 |4 g¯ 14 φ14 (y) + where F˜n ( Zˆ¯ n ) = F(x, ˆ θ˙ˆ + l n e1 − (∂v n−1 /∂ θ) − (1/2)|∂v n−1 /∂ x 1 |2 z n3 n−1 n−1 ( j ) ( j +1) ˙ − − j =0 (∂v n /∂yd )yd j =1 (∂v n−1 /∂ W j ) W j + f 1 (x 1 )) − (1/2)(∂ 2 v n−1 /∂ x 12 ) (∂v n−1 /∂ x 1 )(xˆ2  T ˆ g1 (y)2 − n−1 j =2 (∂v n−1 /∂ xˆ j )( xˆ j +1 + W j S j ( Z j ) + l j e1 ) + (3/4)z n with m n being a design parameter and Zˆ¯ n being (n) ¯ ˆ , which is a compact Zˆ¯ n = [x 1 , x¯ˆn−1 , θˆ , Wˆ¯ n−1 , y¯ ]T ∈  set defined on R2n+(n−1)N+2 . Now, invoking (6) and (12) of Lemma 3, (70) becomes

−

n−1 

9

(74)

where min = min{Mm} = min{M1 m l , M1 m r , M2 m l , M2 m r }, and kn is a positive design parameter. Then, invoking (46) and (74), as well as the following

This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. 10

IEEE TRANSACTIONS ON NEURAL NETWORKS AND LEARNING SYSTEMS

inequality: z n3 Mmu = −z n3 Mm(min )−1 kn z n −

1 3 z Mm(min )−1 z n3 θˆ Sn ( Zˆ¯ n )2 2n 2n n

≤ z n3 min kn z n (min )−1 −

1 3 z min z n3 θˆ Sn ( Zˆ¯ n )2 (min )−1 2n 2n n

≤ −kn z n4 −

1 6ˆ z θ Sn ( Zˆ¯ n )2 2n 2n n

(75)

(73) becomes     n  n+1 T I e+β − Q− α+ LVn ≤ −e c j z 4j −k0 θ˜ T θˆ 2 j =1

3 + 4 +

n  j =1

n−1 

1 m 2j + 2

n  j =1

1 n 2j + 4

σ j W˜ jT W j + τ s 2

j =1

n−1 

n 

ε4j

j =1

W˜ iT W˜ i

(76)

1 4 1 2 1  −1 ˜ T ˜ zi + γi Wi Wi . θ˜ + 4 2γ0 2 n

n−1

i=1

i=1

(77)

By using the following inequalities: 2θ˜ T θˆ = θ˜ 2 + θˆ 2 − θ ∗ 2 ≥ θ˜ 2 − θ ∗ 2 & &2 & &2 2 W˜ i Wi = W˜ i 2 + Wi 2 − & Wi∗ & ≥ W˜ i 2 − &Wi∗ & i = 1, 2, . . . , n − 1

(78)

and invoking (75), we have     n+1 LVn ≤ −λmin Q − α + I e2 + β 2 n 

c j z 4j −

j =1

+

−

k0 k0 θ˜ 2 + θ ∗ 2 2 2

3 2 1 2 1 4 mj + nj + εj 4 2 4 n

n

n

j =1

j =1

j =1

n−1 n−1  & σi & 1 &W ∗ &2 (σi − τ s 2 )W˜ i 2 Wi + i 2 2 i=1

≤ −ρVn + γ

Z =

 z i |E

n 

|z i |

4

( ≤ 4 V¯ , 1 ≤ i ≤ n

i=1

i=1

where cn = kn − (3/4) > 0 is a design parameter. On the other hand, it is easy to see that

−



˜ θ˜ | ≤ 2γ0 V¯ } θ˜ = {θ|| (

 n  2 W˜ = W˜ i |E Wi  ≤ 2γ¯ V¯ , 1 ≤ i ≤ n − 1 (80)

i=1

Vn = e T Pe +

Theorem 1: Consider the stochastic nonaffine and nonlinear systems described by (1) with an unknown dead-zone input (6), an adaptive output NN controller (74), and the parameter updated laws (46). Under Assumptions 1–3 and given any initial values z j (0), j = 1, 2, . . . , n, θˆ (0) > 0 belongs to 0 , which is an appropriately chosen compact set. Then, all the closed-loop signals are uniformly bounded in probability. Furthermore, the error signals θ , z j , j = 1, 2, . . . , n remain the compact sets e ,  Z , θ˜ , and W˜ in the sense that   ' e = e|E[e2 ] ≤ V¯ /λmin (P)

i=1

(79)

where ρ = min{(λmin (Q − (α + (n + 1/2))I )/λmax (P)), 2c j , k0 γ0 , γi (σi − τ s 2 )|i = 1, 2, . . . , n − 1, j = 1, 2,  . . . , n} and γ = β + (k0 /2)θ ∗ 2 + (3/4) nj =1 m 2j + (1/2) nj =1 n 2j +   ∗ 2 (1/4) nj =1 ε4j + n−1 i=1 (σi /2)Wi  . So far, by means of the backstepping method, the adaptive output NN control design has been completed, and the main result is presented in Theorem 1.

where V¯ = V |t =0 + (γ /ρ) and γ¯ = min1≤i≤n−1 {γi }. Proof: Based on the above stochastic stability analysis of the closed-loop system, choose the Lyapunov function V = Vn . Then, according to (79), it is easy to see that LVn ≤ −ρVn + γ . Moreover, with the help of [19, Th. 4.1] and [37, eq. (61)–(63)], we have γ (81) E[V (t)] ≤ V |t =0 + ρ  where V |t =0 = e T (0)Pe(0) + (1/2) ni=1 z i4 (0) +  −1 ˜ T ˜ (1/2γ0 )θ˜ 2 (0) + (1/4) n−1 i=1 γi Wi (0) Wi (0). It follows from (77) that (80) holds, which means that all the closed-loop signals θ˜ , e, W˜ i , and z j , i = 1, 2, . . . , n − 1, j = 1, 2, . . . , n are uniformly bounded in probability. This completes the proof. Remark 5: From Step 1 to Step n, it can be concluded that the parameter γ in (79) gradually increases with each step of the backstepping procedure. However, as the similar statements in [4], [5], [10], [23], [37], and [41], it follows from (81) that, by decreasing the values of the design parameters k0 , m j , n i , and σi , the value of the parameter γ can be reduced. Meanwhile, increasing the values of the design parameters γ0 , c j , and γi , or decreasing the value of τ can help to increase the value of the parameter ρ. In this way, the radius of the compact sets e ,  Z , θ˜ , and W˜ can be effectively reduced. It should be pointed out that, if the value of the parameter ρ is bigger, then the control energy will become larger. Therefore, the controller design parameters should be chosen suitably to achieve the better transient performance and the desired control objective. Remark 6: It should be noted that it is difficult to the stochastic nonaffine and nonlinear systems (1) with an unknown dead-zone input (6) by using the differential condition 0 < f 1 ≤ (∂ f (x, u)/∂u) ≤ f 2 , as shown in [32]–[38]. The reason is that the nonaffine function f (x, u) will be not partial differential with respect to an unknown dead-zone input u d = mu + d(u). To overcome this difficulty, the affine

This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. WU AND YANG: ADAPTIVE OUTPUT NN CONTROL FOR A CLASS OF STOCHASTIC NONLINEAR SYSTEMS

11

form (12) is derived using an intermediate value theorem of a continuous function instead of a traditional differential condition. Remark 7: Meanwhile, it is worth mentioning that the proposed method in [37] is based on an assumption that all the states are measurable, and the result obtained in [41] is for deterministic and affine nonlinear system. In this paper, the adaptive output NN control scheme is developed for the stochastic nonaffine and nonlinear systems through the backstepping technique, which denotes more general case. Furthermore, a novel adaptive output NN tracking controller with proper parameter updated laws is developed to guarantee that all the closed-loop signals are bounded in probability, and the system output tracking error can converge to a small neighborhood in the sense of a mean quartic value. V. S IMULATION S TUDIES

Fig. 1.

System output y and the reference signal yd of Example 1.

Fig. 2.

Output tracking errors z 1 (t) and z 2 (t) of Example 1.

Fig. 3.

Response curves of W1 of Example 1.

In this section, compared with the existing methods, two numerical examples are considered to illustrate the advantages of the presented adaptive neural tracking control technique. A. Example 1 In this section, considering the following two-order stochastic nonaffine and nonlinear systems with a dead-zone nonlinearity:   0.3x 1 dt − x 13 sin(x 1 )dw d x1 = x2 − 1 + x 12  √  2 (x 1 + x 2 ) d x 2 = u + 0.5 cos(u) + sin dt 4   + x 22 cos(x 1 ) + 0.5 dw y = x1

(82)

where the reference signal is assumed to be yd = sin(t), and the dead-zone parameters are chosen as m l = 2.6, m r = 1.6, br = 0.9, and bl = −1.5. Correspondingly, from (27), (45)–(47), and (74), the RBNN basis of the Gaussian function is 

si (Z j ) = exp − (Z j − μi, j )T (Z j − μi, j )/νi,2 j i = 1, 2, . . . , M, j = 1, 2, . . . , N

(83)

where the nodes M = 10, the input vector numbers N = 3, and the width νi, j = 2, i = 1, 2, . . . , 10, j = 1, 2, 3. In particular, taking Z 1 = xˆ1 , Z 2 = [x 1 , yd , y˙d ]T and Z 3 = [x 1 , xˆ1 , W1 , θˆ , yd , y˙d , y¨d ]T with the center of the receptive fields μi,1 = −1 + 2i /11, μi,2 = [−2 + 4i /11, −4 + 8i /11, −6 + 12i /11]T and μi,3 = [−1 + 2i /11, −2 + 4i /11, . . . , −16 + 32i /11]T , i = 1, 2, . . . , 10, respectively. Furthermore, the simulation parameters are selected as l1 = 10, l2 = 8, k1 = k2 = 10, n 1 = n 2 = 8, γ0 = 10, γ1 = 15, σ1 = 10, min = 0.8, and the initial values are chosen as x(0) = [−2, 3]T , x(0) ˆ = [5, −6]T , θˆ (0) = 2, W1 (0) = [0, 0, 1, 0, 1, 0, 1, 0, 1, −5]T . The simulation results are shown in Figs. 1–5. From Fig. 1, it can be seen that the

system output y and the reference signal yd . Fig. 2 shows that the tracking performance is satisfactory and the tracking error sufficiently small for the uncertain system (1) with a nonsymmetric dead-zone nonlinearity. The boundedness of a parameter estimate θˆ and Wˆ 1 , as well as the designed control u is shown in Figs. 3–5, respectively.

This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. 12

IEEE TRANSACTIONS ON NEURAL NETWORKS AND LEARNING SYSTEMS

Fig. 4.

Response curves of θˆ of Example 1.

Fig. 7.

Output tracking errors z 1 (t) and z 2 (t) of Example 2.

Fig. 5.

Response curves of the control input u(t) of Example 1.

Fig. 8.

Response curves of W1 of Example 2.

Fig. 6.

System output y and the reference signal yd of Example 2.

B. Example 2 In this section, a second-order inverted pendulum model from [43] is used, and the corresponding stochastic nonlinear model is d x 1 = x 2 dt − x 13 cos(x 1 )dw d x 2 = (a1 sin(x 1 ) + a2 x 2 + b(tanh(u) + 0.5u))dt + sin(x 1 x 2 )dw y = x1 (84)

where a1 = 24.527, a2 = −0.107, and b = 12.5. The reference signal is assumed to be yd = sin(t) − cos(0.5t), and the dead-zone parameters are chosen as m l = 1.5, m r = 0.6, br = 0.8, and bl = −1. In addition, take the RBFs and the corresponding radial basis function neural networks parameters the same as the example 1. The initial values as ˆ ˆ = [−1, 2]T , θ(0) = 5, x(0) = [−5, 5]T , x(0) T W1 (0) = [1, 0, 1, 0, 1, 0, 1, 0, 1, 0] , and the other simulation parameters are chosen as l1 = 25, l2 = 18, k1 = k2 = 3, n 1 = n 2 = 1, γ0 = 3, γ1 = 15, σ1 = 10, and min = 5. The simulation results are shown in Figs. 6–10. Fig. 6 shows the system output y and the reference signal yd . From Fig. 7, it can be seen that the tracking performance is satisfactory and the tracking error sufficiently small for the stochastic nonlinear system (1) with a dead-zone input. The boundedness of a parameter estimate θˆ and Wˆ 1 , as well as the designed control u is shown in Figs. 8–10, respectively. Remark 8: In views of the above simulation results, the adaptive output NN tracking methods can effectively solve the tracking control problem of the uncertain stochastic nonlinear system (1) with an unknown dead-zone nonlinearity (7). In particular, the control methods of nonaffine and nonlinear systems proposed in [32]–[38] cannot be applied to solve

This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. WU AND YANG: ADAPTIVE OUTPUT NN CONTROL FOR A CLASS OF STOCHASTIC NONLINEAR SYSTEMS

Fig. 9.

Fig. 10.

Response curves of θˆ of Example 2.

Response curves of the control input u(t) of Example 2.

the control problem of the stochastic nonaffine and nonlinear systems (82) and (84). Besides, the proposed control design method in this paper is more suitable for practical applications. VI. C ONCLUSION This paper studies the adaptive output NN control problem for a class of stochastic nonaffine and nonlinear systems in the presence of unknown dead-zone inputs. Based on NNs to approximate the appropriate nonlinear functions, and by employing the intermediate value theorem instead of a differential condition for a nonaffine function, a novel adaptive NN tracking controller with the adjustable parameter updated laws is designed through a backstepping technique. It is also proved that all the closed-loop signals are uniformly bounded in probability, and the system output tracking error can converge to a small neighborhood in the sense of a mean quartic value through a stochastic Lyapunov-based analysis. Finally, two numerical examples are provided to show the efficiency of the presented adaptive neural tracking control design approach. R EFERENCES [1] M. Krstic, I. Kanellakopoulos, and P. V. Kokotovic, Nonlinear and Adaptive Control Design. New York, NY, USA: Wiley, 1995.

13

[2] C. Wen and Y. C. Soh, “Decentralized adaptive control using integrator backstepping,” Automatica, vol. 33, pp. 1719–1724, Sep. 1997. [3] S. S. Ge and J. Wang, “Robust adaptive tracking for time-varying uncertain nonlinear systems with unknown control coefficients,” IEEE Trans. Autom. Control, vol. 48, no. 8, pp. 1463–1469, Aug. 2003. [4] B. Chen, X. P. Liu, S. S. Ge, and C. Lin, “Adaptive fuzzy control of a class of nonlinear systems by fuzzy approximation approach,” IEEE Trans. Fuzzy Syst., vol. 20, no. 6, pp. 1012–1021, Dec. 2012. [5] S. Tong and Y. Li, “Adaptive fuzzy output feedback tracking backstepping control of strict-feedback nonlinear systems with unknown dead zones,” IEEE Trans. Neural Netw., vol. 20, no. 1, pp. 168–180, Feb. 2012. [6] J. Zhou, C. Wen, and G. Yang, “Adaptive backstepping stabilization of nonlinear uncertain systems with quantized input signal,” IEEE Trans. Autom. Control, vol. 59, no. 2, pp. 460–464, Feb. 2014. [7] D. Wang and J. Huang, “Neural network-based adaptive dynamic surface control for a class of uncertain nonlinear systems in strict-feedback form,” IEEE Trans. Neural Netw., vol. 16, no. 1, pp. 195–202, Jan. 2005. [8] M. M. Arefi, M. R. Jahed-Motlagh, and H. R. Karimi, “Adaptive neural stabilizing controller for a class of mismatched uncertain nonlinear systems by state and output feedback,” IEEE Trans. Cybern., vol. 45, no. 8, pp. 1587–1596, Aug. 2015. [9] S. C. Tong, Y. M. Li, and H.-G. Zhuang, “Adaptive neural network decentralized backstepping output-feedback control for nonlinear largescale systems with time delays,” IEEE Trans. Neural Netw., vol. 22, no. 7, pp. 1073–1086, Jul. 2011. [10] S. S. Ge and J. Zhang, “Neural-network control of nonaffine nonlinear system with zero dynamics by state and output feedback,” IEEE Trans. Neural Netw., vol. 14, no. 4, pp. 900–918, Jul. 2003. [11] Q. Shen, B. Jiang, P. Shi, and J. Zhao, “Cooperative adaptive fuzzy tracking control for networked unknown nonlinear multiagent systems with time-varying actuator faults,” IEEE Trans. Fuzzy Syst., vol. 22, no. 3, pp. 494–504, Jun. 2014. [12] C. L. P. Chen, Y.-J. Liu, and G.-X. Wen, “Fuzzy neural network-based adaptive control for a class of uncertain nonlinear stochastic systems,” IEEE Trans. Cybern., vol. 44, no. 5, pp. 583–593, May 2014. [13] B. Chen, X. Liu, K. Liu, and C. Lin, “Direct adaptive fuzzy control of nonlinear strict-feedback systems,” Automatica, vol. 45, no. 6, pp. 1530–1535, Feb. 2009. [14] Q. Zhou, P. Shi, S. Xu, and H. Li, “Adaptive output feedback control for nonlinear time-delay systems by fuzzy approximation approach,” IEEE Trans. Fuzzy Syst., vol. 21, no. 2, pp. 301–313, Apr. 2013. [15] S. Tong, S. Sui, and Y. Li, “Fuzzy adaptive output feedback control of MIMO nonlinear systems with partial tracking errors constrained,” IEEE Trans. Fuzzy Syst., vol. 23, no. 4, pp. 729–742, Aug. 2015. [16] Z. Pan and T. Basar, “Backstepping controller design for nonlinear stochastic systems under a risk-sensitive cost criterion,” SIAM J. Control Optim., vol. 37, no. 3, pp. 957–995, Feb. 1999. [17] H.-B. Ji and H.-S. Xi, “Adaptive output-feedback tracking of stochastic nonlinear systems,” IEEE Trans. Autom. Control, vol. 51, no. 2, pp. 355–360, Feb. 2006. [18] T. Zhang and X. Xia, “Adaptive output feedback tracking control of stochastic nonlinear systems with dynamic uncertainties,” Int. J. Robust Nonlinear Control, vol. 25, no. 9, pp. 1282–1300, Jun. 2015. [19] H. Deng, M. Krsti´c, and R. J. Williams, “Stabilization of stochastic nonlinear systems driven by noise of unknown covariance,” IEEE Trans. Autom. Control, vol. 46, no. 8, pp. 1237–1253, Aug. 2001. [20] Z.-J. Wu, X.-J. Xie, P. Shi, and Y.-Q. Xia, “Backstepping controller design for a class of stochastic nonlinear systems with Markovian switching,” Automatica, vol. 45, no. 4, pp. 997–1004, Apr. 2009. [21] X.-J. Xie, N. Duan, and X. Yu, “State-feedback control of high-order stochastic nonlinear systems with SiISS inverse dynamics,” IEEE Trans. Autom. Control, vol. 56, no. 8, pp. 1921–1926, Aug. 2011. [22] X.-J. Xie and N. Duan, “Output tracking of high-order stochastic nonlinear systems with application to benchmark mechanical system,” IEEE Trans. Autom. Control, vol. 55, no. 5, pp. 1197–1202, May 2010. [23] H.-Q. Wang, B. Chen, and C. Lin, “Adaptive neural tracking control for a class of stochastic nonlinear systems,” Int. J. Robust Nonlinear Control, vol. 24, no. 7, pp. 1262–1280, May 2014. [24] S.-J. Liu, J.-F. Zhang, and Z.-P. Jiang, “Decentralized adaptive outputfeedback stabilization for large-scale stochastic nonlinear systems,” Automatica, vol. 43, no. 2, pp. 238–251, Feb. 2007. [25] C. Hua, L. Zhang, and X. Guan, “Decentralized output feedback adaptive NN tracking control for time-delay stochastic nonlinear systems with prescribed performance,” IEEE Trans. Neural Netw. Learn. Syst., vol. 26, no. 11, pp. 2749–2759, Nov. 2015.

This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. 14

[26] Q. Zhou, P. Shi, H. Liu, and S. Xu, “Neural-network-based decentralized adaptive output-feedback control for large-scale stochastic nonlinear systems,” IEEE Trans. Syst., Man, Cybern. B, Cybern., vol. 42, no. 6, pp. 1608–1619, Dec. 2012. [27] Q. Zhou, P. Shi, S. Xu, and H. Li, “Observer-based adaptive neural network control for nonlinear stochastic systems with time delay,” IEEE Trans. Neural Netw. Learn. Syst., vol. 24, no. 1, pp. 71–80, Jan. 2013. [28] S. Tong, T. Wang, and Y. Li, “Fuzzy adaptive actuator failure compensation control of uncertain stochastic nonlinear systems with unmodeled dynamics,” IEEE Trans. Fuzzy Syst., vol. 22, no. 3, pp. 563–574, Jun. 2014. [29] S. A. Al-Hiddabi and N. H. McClamroch, “Tracking and maneuver regulation control for nonlinear nonminimum phase systems: Application to flight control,” IEEE Trans. Control Syst. Technol., vol. 10, no. 6, pp. 780–792, Nov. 2002. [30] A. S. Shiriaev, H. Ludvigsen, O. Egeland, and A. L. Fradkov, “Swinging up of non-affine in control pendulum,” in Proc. Amer. Control Conf., San Diego, CA, USA, 1999, pp. 4039–4044. [31] S. S. Ge, C. C. Hang, and T. Zhang, “Nonlinear adaptive control using neural networks and its application to CSTR systems,” J. Process Control, vol. 9, no. 4, pp. 313–323, Aug. 1999. [32] H. Du, H. Shao, and P. Yao, “Adaptive neural network control for a class of low-triangular-structured nonlinear systems,” IEEE Trans. Neural Netw., vol. 17, no. 2, pp. 509–514, Mar. 2006. [33] B. Karimi, M. B. Menhaj, M. Karimi-Ghartemani, and I. Saboori, “Decentralized adaptive control of large-scale affine and nonaffine nonlinear systems,” IEEE Trans. Instrum. Meas., vol. 58, no. 8, pp. 2459–2467, Aug. 2009. [34] B. Karimi and M. B. Menhaj, “Non-affine nonlinear adaptive control of decentralized large-scale systems using neural networks,” Inf. Sci., vol. 180, no. 17, pp. 3335–3347, Sep. 2010. [35] M. Chen and S. S. Ge, “Direct adaptive neural control for a class of uncertain nonaffine nonlinear systems based on disturbance observer,” IEEE Trans. Cybern., vol. 43, no. 4, pp. 1213–1225, Aug. 2013. [36] S.-L. Dai, C. Wang, and M. Wang, “Dynamic learning from adaptive neural network control of a class of nonaffine nonlinear systems,” IEEE Trans. Neural Netw. Learn. Syst., vol. 25, no. 1, pp. 111–123, Jan. 2014. [37] H. Wang, B. Chen, X. Liu, K. Liu, and C. Lin, “Robust adaptive fuzzy tracking control for pure-feedback stochastic nonlinear systems with input constraints,” IEEE Trans. Cybern., vol. 43, no. 6, pp. 2093–2104, Dec. 2013. [38] L.-B. Wu and G.-H. Yang, “Adaptive fuzzy tracking control for a class of uncertain nonaffine nonlinear systems with dead-zone inputs,” Fuzzy Sets Syst., doi: 10.1016/j.fss.2015.05.006. [39] X.-S. Wang, C.-Y. Su, and H. Hong, “Robust adaptive control of a class of nonlinear systems with unknown dead-zone,” Automatica, vol. 40, no. 3, pp. 407–413, 2004. [40] H.-J. Ma and G.-H. Yang, “Adaptive output control of uncertain nonlinear systems with non-symmetric dead-zone input,” Automatica, vol. 46, no. 2, pp. 413–420, 2010. [41] S. Tong and Y. Li, “Adaptive fuzzy output feedback control of MIMO nonlinear systems with unknown dead-zone inputs,” IEEE Trans. Fuzzy Syst., vol. 21, no. 1, pp. 134–146, Feb. 2013.

IEEE TRANSACTIONS ON NEURAL NETWORKS AND LEARNING SYSTEMS

[42] Z. Zhang, S. Xu, and B. Zhang, “Asymptotic tracking control of uncertain nonlinear systems with unknown actuator nonlinearity,” IEEE Trans. Autom. Control, vol. 59, no. 5, pp. 1336–1341, May 2014. [43] T. Bian, Y. Jiang, and Z.-P. Jiang, “Adaptive dynamic programming and optimal control of nonlinear nonaffine systems,” Automatica, vol. 50, no. 10, pp. 2624–2632, Oct. 2014. [44] R. Z. Khasminskii, Stochastic Stability of Differential Equations. Norwell, MA, USA: Kluwer, 1980. [45] H. E. Psillakis and A. T. Alexandridis, “Adaptive neural tracking for a class of SISO uncertain and stochastic nonlinear systems,” in Proc. 44th IEEE Conf. Decision Control, Eur. Control Conf., Seville, Spain, Dec. 2005, pp. 7822–7827.

Li-Bing Wu received the B.S. degree from the Department of Mathematics, Jinzhou Normal College, Jinzhou, China, in 2004, and the M.S. degree in basic mathematics from Northeastern University, Shenyang, China, in 2007, where he is currently pursuing the Ph.D. degree in control theory and control engineering. He is currently a Lecturer with the Department of Mathematics, School of Sciences, University of Science and Technology at Liaoning, Anshan, China. His current research interests include adaptive control, fault-tolerant control, nonlinear control, and fault diagnosis.

Guang-Hong Yang (SM’04) received the B.S. and M.S. degrees from Northeast University, Shenyang, China, in 1983 and 1986, respectively, and the Ph.D. degree in control engineering from Northeastern University, in 1994. He was a Lecturer/Associate Professor with Northeastern University from 1986 to 1995. He joined Nanyang Technological University, Singapore, in 1996, as a Post-Doctoral Fellow. From 2001 to 2005, he was a Research Scientist/Senior Research Scientist with the National University of Singapore, Singapore. He is currently a Professor with the College of Information Science and Engineering, Northeastern University. His current research interests include fault-tolerant control, fault detection and isolation, nonfragile control systems design, and robust control. Dr. Yang is an Associate Editor of the International Journal of Control, Automation, and Systems, the International Journal of Systems Science, the IET Control Theory & Applications, and the IEEE T RANSACTIONS ON F UZZY S YSTEMS .