IEEE TRANSACTIONS ON NEURAL NETWORKS AND LEARNING SYSTEMS, VOL. 26, NO. 11, NOVEMBER 2015

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Decentralized Output Feedback Adaptive NN Tracking Control for Time-Delay Stochastic Nonlinear Systems With Prescribed Performance Changchun Hua, Liuliu Zhang, and Xinping Guan

Abstract— This paper studies the dynamic output feedback tracking control problem for stochastic interconnected time-delay systems with the prescribed performance. The subsystems are in the form of triangular structure. First, we design a reducedorder observer independent of time delay to estimate the unmeasured state variables online instead of the traditional full-order observer. Then, a new state transformation is proposed in consideration of the prescribed performance requirement. Using neural network to approximate the composite unknown nonlinear function, the corresponding decentralized output tracking controller is designed. It is strictly proved that the resulting closed-loop system is stable in probability in the sense of uniformly ultimately boundedness and that both transient-state and steady-state performances are preserved. Finally, a simulation example is given, and the result shows the effectiveness of the proposed control design method. Index Terms— Neural network (NN) approach, prescribed performance control (PPC), reduced-order observer design, stochastic interconnected time-delay system.

I. I NTRODUCTION

I

N THE past decades, research on stochastic time-delay systems has received considerable attention due to the fact that the existence of stochastic disturbance and time delay may severely degrade the closed-loop system performance, and even make the system unstable. For linear stochastic time-delay systems, many significant achievements have been developed, for example, stability analysis, H∞ analysis, and robust stabilization have been investigated in [1]–[6].

Manuscript received January 19, 2014; revised October 15, 2014 and January 9, 2015; accepted January 10, 2015. Date of publication March 18, 2015; date of current version October 16, 2015. This work was supported in part by the Hundred Excellent Innovation Talents Support Program of Hebei Province, in part by the Applied Basis Research Project under Grant 13961806D, in part by the Top Talents Project of Hebei Province, and in part by the National Natural Science Foundation of China under Grant 61290322, Grant 61273222, Grant 61322303, Grant 61473248, and Grant 61403335. C. Hua and L. Zhang are with the Institute of Electrical Engineering, Yanshan University, Qinhuangdao 066004, China (e-mail: [email protected]; [email protected]). X. Guan is with the Department of Automation, Shanghai Jiao Tong University, Shanghai 200030, China and also with the Institute of Electrical Engineering, Yanshan University, Qinghuangdao City 066004, China (e-mail: [email protected]). 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.2015.2392946

In addition, numerous control approaches for stochastic nonlinear time-delay systems have been developed to guarantee the stability in probability. The problems of robust stochastic stabilization and robust H∞ control for uncertain stochastic time-delay systems were investigated in [7] and [8] via state feedback control. Ever since the first result on dynamic output feedback stabilization for stochastic systems without time delay in [9], the problem of output feedback controller design for stochastic nonlinear time-delay systems has been studied in [10]–[14]. Recently, exponential stability for nonlinear stochastic time-delay systems was proved in [15]–[17], stabilization for stochastic high-order nonlinear time-delay systems was discussed in [18] and [19], and approximation-based adaptive neural network (NN) control approach was proposed for stochastic nonlinear time-delay systems in [20] and [21]. Based on backstepping method, in [4] and [22], the decentralized delay-independent controller was designed for large-scale interconnected stochastic systems. It is well known that the prescribed performance control (PPC) demands the convergence rate no less than a prescribed value, exhibiting a maximum overshoot less than a sufficiently small constant and the output or tracking error approaching to an arbitrarily small residual set. This control idea was first put forward in [23], which studied the robust adaptive control problem for feedback linearizable multiple input and multiple output nonlinear systems. Furthermore, in-depth studies in this area were presented in [24]–[28]. The robust adaptive control for single input and single output strict feedback nonlinear system with PPC was investigated in [24]. Using NN, the output feedback of robust adaptive control was considered in [25] with PPC based on the approximation passivation approach. In [26]–[28], the PPC was used to restrict the output performance in robot control. However, to the best of our knowledge, there is no work done on the output feedback control for stochastic interconnected time-delay system and how to apply this method to investigating the stochastic nonlinear time-delay systems is a challenging subject. The goal of this paper is to design the output feedback tracking controllers for a class of stochastic interconnected nonlinear time-delay systems with the prescribed performance

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IEEE TRANSACTIONS ON NEURAL NETWORKS AND LEARNING SYSTEMS, VOL. 26, NO. 11, NOVEMBER 2015

requirement. First, the reduced-order observer is designed to estimate the unmeasurable state variables online. Since the interconnections are unknown, the radial basis function (RBF) NN is proposed to approximate a composite nonlinear function for each subsystem. A new state transformation is given in consideration of the prescribed performance. Then, the output feedback controller is constructed with the use of backstepping method. Based on Lyapunov stability theory, it is shown that the designed decentralized controller renders the large-scale system stable in probability. Finally, a simulation example is presented to verify the effectiveness and the advantages of the proposed technique. The rest of this paper is organized as follows. In Section II, we give the system description along with the preliminary knowledge. The reduced-order observer is designed in Section III-A. The new state transformation is developed with PPC in Section III-B. In Section III-C, the output feedbacktracking controller is designed with NN approach, and the stability of the closed-loop system is analyzed. Then, a simulation example shows the effectiveness of the proposed method in Section IV. Finally, the conclusion is drawn in Section V. II. S YSTEM F ORMULATION AND P RELIMINARIES

Fig. 1.

Structure of the stochastic large-scale system.

Fig. 2.

Prescribed performance requirement.

Fig. 3.

Block diagram of the stochastic decentralized PPC problem.

A. Problem Formulation In this paper, we consider the following stochastic large-scale system: ⎧ ⎪ ⎪d x i j (t) = [x i( j +1) (t) + h i j (x i j (t))+ f i j (y1 (t), . . . , y N (t), ⎪ ⎪ ⎪ y1 (t − di j 1 (t)), . . . , y N (t − di j N (t)))]dt ⎪ ⎪ ⎪ ⎪ ⎪ + gi j (y1 (t), . . . , y N (t), y1 (t − τi j 1 (t)), . . . , ⎪ ⎪ ⎪ ⎨ y N (t − τi j N (t)))di , j = 1, . . . , n i − 1 ⎪ = [u d x ini i (t) + h ini (x ini (t)) + f ini (y1 (t), . . . , y N (t), ⎪ ⎪ ⎪ ⎪ ⎪ y1 (t − dini 1 (t)), . . . , y N (t − dini N (t)))]dt ⎪ ⎪ ⎪ ⎪ + gini (y1 (t), . . . , y N (t), y1 (t − τini 1 (t)), . . . , ⎪ ⎪ ⎪ ⎩ y N (t − τini N (t)))di , yi (t) = x i1 (t) (1) where 1 ≤ i ≤ N, N is the total number of the subsystems; x i j (t) ∈ , u i (t) ∈ , and yi (t) ∈  are the state variables, control input, and output of the i th subsystem, respectively, and n i is the dimension of the i th subsystem; i is an independent ri -dimensional standard Wiener process, where ri  1 is a positive integer; h i j (·) is a known smooth nonlinear function, and f i j (·) and gi j (·) are the nonlinear interconnections, which are smooth and unknown; the structure of (1) is shown in Fig. 1. x i j (t) = [x i1 (t), x i2 (t), . . . , x i j (t)]T , and only the output yi is available for measurement; τi j k (t) and di j k (t) are the time-varying delays satisfying τ˙i j k (t) ≤ τ¯i j k (t) < 1 and d˙i j k (t) ≤ d¯i j k (t) < 1, where τ¯i j (t) and d¯i j k are positive scalars, and k is a positive integer. The PPC problem for (1) will be investigated and the observer-based output feedback NN controller will be proposed. For each subsystem, the tracking error is defined as y˜i = yi (t)−yid (t), where yid (t) is a given reference signal and we assume that yid (t) is n i times continuously differentiable.

To address the transient and steady-state performance index, we choose a positive decreasing smooth function μi (t) = (μi0 − μi∞ )e−ki t + μi∞ with ki > 0 and μi0 > μi∞ > 0. One knows that μi (t) = μi0 for t = 0 and μi (t) = μi∞ for t (→)∞. The objective of this paper is to construct the decentralized output feedback controller u i (t) in such a way that the following conditions are met. 1) P1: The trajectory of the tracking error y˜i satisfies −δ i μi (t) < y˜i (t) < δ¯i μi (t), where δ i and δ¯i are positive constants chosen based on the prescribed performance requirement, as shown in Fig. 2. 2) P2: All the state variables of the closed-loop system are bounded in probability. The block diagram of the stochastic decentralized PPC problem for two interconnected subsystems is given in Fig. 3. Remark 1: System (1) studied in this paper is in the form of triangular structure. Many practical systems are in this form and many systems can be transformed into this form, such as the chemical reactors, networked robots, and so forth. This

HUA et al.: DECENTRALIZED OUTPUT FEEDBACK ADAPTIVE NN TRACKING CONTROL

class of nonlinear systems has been extensively considered in [9], [20], [21], [29]–[31], and the references therein. For the control design, we impose the following assumptions on (1). Assumption 1: Nonlinear interconnection functions f i j (·), gi j (·) satisfy | f i j (y1 (t), . . . , y N (t), y1 (t − di j 1 (t)), . . . , y N (t − di j N (t)))|4 ≤

N  

f i4j k (yk (t)) + f¯i4j k (yk (t − di j k (t)))



k=1

|gi j (y1 (t), . . . , y N (t), y1 (t − τi j 1 (t)), . . . , y N (t − τi j N (t)))|4 ≤

N  

gi4j k (yk (t)) + g¯ i4j k (yk (t − τi j k (t)))



k=1

where f i j k (·), f¯i j k (·), gi j k (·), and g¯ i j k (·) are unknown smooth positive definite functions with f i j k (0) = f¯i j k (0) = gi j k (0) = g¯ i j k (0) = 0. Assumption 2: The smooth function h i j (·) satisfies x i j (t))| ≤ ρi j x i j (t) − x i j (t) |h i j (x i j (t)) − h i j (

(2)

where j = 2, . . . , n i , ρi j is a known positive parameter and x i j (t) = [xˆi1 (t), xˆi2 (t), . . . , xˆi j (t)]T , x i j (t) is the estimate of the unmeasurable state x i j (t). Remark 2: In Assumption 1, the uncertain functions are required to be bounded by the functions of output variables. This is a general assumption on the output feedback controller design and many engineering systems satisfy this condition [20], [29], [30]. For the smoothness of the functions f i j k (·), f¯i j k (·), gi j k (·), and g¯ i j k (·), we have f i4j k (yk (t)) ≤ y˜k4 κi j k ( y˜k ) + κ¯ i j k (ykd ) f¯i4j k (yk (t − di j k (t))) ≤ y˜k4 (t − di j k (t))φi j k ( y˜k (t − di j k (t))) + φ¯ i j k (ykd (t − di j k (t))) gi4j k (yk (t)) ≤ y˜k4 ϕi j k ( y˜k ) + ϕ¯ i j k (ykd )g¯ i4j k (yk (t − τi j k (t))) ≤ y˜k4 (t − τi j k (t))i j k ( y˜k (t − τi j k (t))) + ¯ i j k (ykd (t − τi j k (t))) where κi j k (·), κ¯ i j k (·), φi j k (·), φ¯ i j k (·), ϕi j k (·), ϕ¯i j k (·), i j k (·), and ¯ i j k (·) are the unknown nonlinear functions. With the above inequalities and Assumption 1, one has | f i j (y1 (t), . . . , y N (t), y1 (t − di j 1 (t)), . . . , y N (t − di j N (t)))|4 ≤

N  

y˜k4 (t − di j k (t))φi j k ( y˜k (t − di j k (t)))

k=1

 + y˜k4 κi j k ( y˜k ) + i j

(3)

|gi j (y1 (t), . . . , y N (t), y1 (t − τi j 1 (t)), . . . , y N (t − τi j N (t)))|4 ≤

N 



y˜k4 (t − τi j k (t))i j k ( y˜k (t − τi j k (t))



k=1

+ y˜k4 ϕi j k ( y˜k )) + ˜i j

(4)

where i j and ˜i j are positive scalars satisfying i j  κ¯ i j k (ykd ) + φ¯ i j k (ykd (t − di j k (t))) and ˜i j  ϕ¯ i j k (ykd ) + ¯ i j k (ykd (t − τi j k (t))). We will use (3) and (4) for the decentralized controller design.

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B. Basic Knowledge on Stochastic System Consider the stochastic system d x(t) = f (x(t))dt + g(x(t))d

(5)

is the system state,  is an r -dimentional where x ∈ standard Wiener process, and f (·) and g(·) are locally Lipschitz functions and satisfy f (0) = g(0) = 0. Definition 1 [10]: Consider (5) and suppose there exists a positive definite, radially unbounded, twice continuously differentiable function V (x), then the differential operator

and derivative operator d are defined as follows:

 2 1 ∂V T∂ V f + Tr g g (6)

V (x) = ∂x 2 ∂x2 ∂V d V = V dt + (7) gd ∂x where Tr{A} represents the trace of a matrix A. Lemma 1 (Stochastic LaSalle Theorem) [32]: For (5), if there exist V (x) ∈ C 2 , α1 (x), α2 (x) ∈ k∞ , positive-definite and radially unbounded function W (x), and constant c > 0 such that α1 (x) ≤ V (x) ≤ α2 (x) and V (x) ≤ −W (x) + c, the solution of (5) is bounded in probability. n

C. Neural Network (NN) Approximation In the control engineering, RBF NN is usually used as a tool for modeling nonlinear functions because of their good capabilities in function approximation. In this paper, the following RBF NN is used to approximate the continuous function β(Z (t)) : q → : ˆ (t)) = θ T ζ(Z (t)) β(Z where Z (t) ∈  Z ⊂ q is the input vector and q is the NN input dimension. Weight vector θ = [θ1 , θ2 , . . . , θl ]T ∈ l , NN node number l > 1, and ζ(Z (t)) = [ζ1 (Z (t)), . . . , ζl (Z (t))]T , with ζi (Z (t)) chosen as the commonly used Gaussian functions, which is in the following form:

−(Z − μi )T (Z − μi ) ζi (Z (t)) = exp ηi2 where μi = [μi1 , μi2 , . . . , μiq ] is the center of the receptive field and ηi is the width of the Gaussian function. It has been proved that the NN can approximate any continuous function over a compact set  Z ⊂ q to arbitrary accuracy as β(Z (t)) = θ ∗T ζ(Z (t)) + δ ∀Z ∈  Z where θ ∗ is an ideal constant weight and δ is the approximation error. The ideal weight vector θ ∗ is an artificial quantity required for analytical purpose. θ ∗ is defined as the value of θ that minimizes |δ| for all Z ∈  Z in a compact region θ ∗ := arg min {sup |β(Z (t)) − θ T ζ(Z (t))|}. θ∈l

NN approximation idea has been used extensively for control design with unknown system functions.

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In [31] and [33], NN state feedback controllers were constructed for time-delay nonlinear systems. Observer-based output feedback controller was designed in [34]. With the unknown control direction, Du et al. [35] designed the NN controller for nonlinear systems with a triangular structure. We use the NN to approximate the unknown interconnection functions in (1), and NN-based output feedback control methodology is developed for the stochastic large-scale system. III. C ONTROLLER D ESIGN In this section, the reduced-order observer-based decentralized controller will be designed for interconnected system (1). First, the reduced-order observer is designed; then, the controller is constructed based on backstepping approach. A. Reduced-Order Observer Design To estimate the unmeasured state variables, we propose the reduced-order observer for each subsystem as follows: ⎧ λ˙ i j (t) = λi( j +1) (t) + ki( j +1) x i1 (t) ⎪ ⎪ ⎪ ⎪ ⎪ − ki j (h i1 (x i1 (t)) + ki2 x i1 + λi2 ) ⎪ ⎪ ⎪ ⎪ ⎪ + h˜ i j (x i1 (t), λ¯ i j (t)) ⎨ i = 1, . . . , N, j = 2, . . . , n i − 1 (8) ⎪ ⎪ ⎪ ˜ ˙ λini (t) = u i (t) + h ini (x i1 (t), λ¯ ini (t)) ⎪ ⎪ ⎪ ⎪ ⎪ − kini (h i1 (x i1 (t)) + ki2 x i1 + λi2 ) ⎪ ⎪ ⎩ xˆi j (t) = λi j (t) + ki j x i1 (t) where ki j is positive constant defined later and h˜ i j (x 1 (t), λ¯ i j (t)) = h i j (x i1 (t), . . . , λi j (t) + ki j x i1 (t)). The estimation error is defined as ei j (t) = x i j (t) − xˆi j (t).

(9)

From (1) and (8), we can obtain dei = [ Ai ei + Hi + Fi ]dt + G i d

(10)

where ei = [ei2 , . . . , eini ]T ⎡ 1 0 0 ··· −ki2 ⎢ −ki3 0 1 0 ··· ⎢ ⎢ . . . . ⎢ .. .. .. ... .. Ai = ⎢ ⎢ ⎣ −ki(ni −1) 0 0 0 ··· −kini 0 0 0 0      Hi = h i2 x i2 (t) − h i2 xˆ i2 (t) , . . . ,    T h ini x ini (t) − h ini xˆ ini (t)

0 0 .. 0 0

.

⎤ 0 0⎥ ⎥ .. ⎥ ⎥ .⎥ ⎥ 1⎦

Vie ≤ 2ci0 eiT Pi ei eiT Pi (Ai ei + Hi + Fi )  T     + T r G iT 2ci0 eiT Pi eiT Pi + ci0 eiT Pi ei Pi G i ≤ −(ci0 γi0 λmin (Pi ) − 2ci0 ρi Pi 2 − ci1 − ci2 )ei 4 ⎤ ⎡ ni ni   + 8c¯i1 (n i − 1) ⎣ fi4j + ki4j f i14 ⎦ j =2

j =2

j =2

j =2

⎤ ⎡ ni ni   4⎦ + 8c¯i2 (n i − 1) ⎣ gi4j + ki4j gi1

(14)

 −3 , where ρi = ( nj i=2 ρi2j )1/2 , c¯i1 = 4−4 33 (2ci0 Pi 2 )4 ci1 −1 2 4 c¯i2 = 9ci0 Pi  ci2 /4; ci1 and ci2 are positive constants determined in (41). Remark 3: The full-order observer was often designed for output feedback (see [29], [30], and the references therein). While in this paper, we introduce the reduced-order observers instead of full-order ones. Compared with the full-order observer, the structure and computation complexity are simplified. B. Prescribed Performance Transformation To render the tracking error satisfying the prescribed performance requirement, we choose a function Si (εi ) = (δ¯i eεi − δ i e−εi /eεi + e−εi ), then y˜i (t) = μi (t)Si (εi ). Since the function Si (εi ) is strictly monotonic increasing, its inverse function exists as   y˜i 1 si + δ i εi = Si−1 = ln μi (t) 2 δ¯i − si where si = ( y˜i /μi (t)). With Definition 1, the dynamics is that   y˜i μ˙ i T − y˙id +m i gi1 gi1 dt dεi = ri x i2 (t) +h i1 (x i1 (t)) + f i1 − μi (15) + ri gi1 di with ri = (1/2μi )[(1/si + δ i ) − (1/si − δ¯i )] and m i = 1/μi [(1/si + δ i ) + (1/si − δ¯i )]. For the consideration of the nonlinear transformation and zero equilibrium point problem, choosing state transformation z i1 = εi − (1/2) ln(δ i /δ¯i ) gives   y˜i μ˙ i T − y˙id +m i gi1 gi1 dt dz i1 = ri x i2 (t) +h i1 (x i1 (t)) + f i1 − μi + ri gi1 di . (16)

0

Fi = [−ki2 f i1 + f i2 , . . . , −kini fi1 + f ini ]T

(11)

G i = [−ki2 gi1 + gi2 , . . . , −kini gi1 + gini ]

(12)

T

For the error system, we consider the Lyapunov function 2 ci0  T ei Pi ei (13) Vie = 2 where ci0 is a positive constant. Using (10), (13), and Assumption 2, we have

and Ai is designed to satisfy that Pi Ai + AiT Pi ≤ −γi0 I , where Pi is a positive definite matrix and γi0 is a given positive parameter.

Remark 4: The tracking error y˜i (t) is required to satisfy −δ i μi (t) < y˜i (t) < δ¯i μi (t). The prescribed performance index is not symmetric because of δ i = δ¯i . One knows that εi = (1/2) ln(δ i /δ¯i ) when y˜i = 0. We define a new state transformation z i1 = εi − (1/2) ln(δ i /δ¯i ) to deal with this problem, which will result in z i1 = 0 at y˜i = 0.

HUA et al.: DECENTRALIZED OUTPUT FEEDBACK ADAPTIVE NN TRACKING CONTROL

C. Adaptive Controller Design With NN Approximation Based on the designed reduced-order observer, we now consider the decentralized output feedback controller design. With (1) and (8), one has the composite system as follows: ⎧ dei = [ Aiei + Hi + Fi ]dt + G i di ⎪ ⎪ ⎪ ⎪ y˜i μ˙ i ⎪ ⎪ ⎪ dz i1 = ri x i2 (t) + h i1 (x i1 (t)) + f i1 − ⎪ ⎪ μi ⎪  ⎪ ⎪ ⎪ ⎪ T ⎪ − y˙id + m i gi1 gi1 dt + ri gi1 di ⎪ ⎨ (17) λ˙ i j (t) = λi( j +1) (t) + ki( j +1) x i1 (t) ⎪ ⎪ ⎪ ⎪ − ki j (h i1 (x i1 (t)) + ki2 x i1 + λi2 ) ⎪ ⎪ ⎪ ⎪ ⎪ + h˜ i j (x i1 (t), λ¯ i j (t)), j = 2, . . . , n i − 1 ⎪ ⎪ ⎪ ⎪ ˙ ⎪ λini (t) = u i (t) + h˜ ini (x i1 (t), λ¯ ini (t)) ⎪ ⎪ ⎩ − kini (h i1 (x i1 (t)) + ki2 x i1 + λi2 ). For (17), we transformation: 

introduce

the

following

coordinate

z i1 = z i1 z i j = λi j (t) − αi( j −1)

(18)

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Step 1: With (6) and (21), one obtains   y˜i μ˙ i 3 T − y˙id + m i gi1 gi1

Hi1 ≤ z i1 ri x i2 (t) + h i1 + f i1 − μi 3 2  2 T   ∗ ˆ T −1 ˆ. + z i1 Tr ri gi1 gi1 − θi − θi (t) i θ i (t) 2  y˜i μ˙ i 3 ri αi1 + ki2 x i1 + h i1 − − y˙id ≤ z i1 μi  4/3 4/3 4/3 + ci10 ri + c¯i11ri + ci12ri + ci14ri4  2 2 4 4 4 + c¯i13 i1 z i1 z i1 + c¯i10 z i2 + ci11 ei2 + c¯i12 f i14 4 4 3 3 +c¯i14 gi1 + ci13 gi1 + z i1 βi ( y˜i ) − z i1 βi ( y˜i ) · T −1  ∗ − θi − θˆi (t) i θˆi (t)

where ci j 0 , ci j 1 , ci j 2 , ci j 3 , and ci j 4 are positive constants and 3 −4/3 c −1/3 , c¯i j 0 = 4−4 33 ci−3 j 0 , c¯i j 1 = (27/256c0 ), ci j = 3 · 4 ij1 −1 −1 c¯i j 2 = 4−4 33 ci−3 j 2 , c¯i j 3 = 1/4ci j 3 , c¯i j 4 = 9/16ci j 4 , i1 = ri m i , 2 2 2 and i j = (1/2)(∂ αi( j −1) /∂z i1 )ri + (∂αi( j −1) /∂z i1 )ri m i ; βi ( y˜i ) is the unknown nonlinear function defined as nk  N  N   3 z i1 βi ( y˜i ) = qk1 y˜i4 κkj i ( y˜i ) + pk1 y˜i4 ϕkj i ( y˜i ) i=1 k=1 j =2

qk1 y˜i4 (t)φkj i ( y˜i (t)) 1 − d¯kj i  pk1 + y˜i4 (t)kj i (yi (t)) 1 − τ¯kj i

+

where function αi( j −1) is the virtual control input to be designed. Then, for the i th subsystem, we choose the Lyapunov–Krasovskii functional as Ui = Vie + Mi + Wi

(24)

(19)

+

N  N   qk2 y˜i4 κk1i ( y˜i ) + pk2 y˜i4 ϕk1i ( y˜i ) i=1 k=1

qk2 y˜i4 (t)φk1i ( y˜i (t)) 1 − d¯k1i  pk2 y˜i4 (t)k1i (yi (t)) . (25) + 1 − τ¯k1i

where

+ Mi =

ni 

Hi j

(20)

j =1

 T  1 4 1 z (t) + θi∗ − θˆi (t) i−1 θi∗ − θˆi (t) (21) 4 i1 2 1 (22) Hi j = z i4j (t) 4

Hi1 =

and Wi =

nk N  



t t −dkj i (t )

k=1 j =2

 +

+

 N  k=1

t

t −τkj i (t )

t t −dk1i (t )



+

qk1 y˜i4 (ξ )φkj i ( y˜i (ξ )) 1 − d¯kj i

  pk1 4 y˜ (ξ )kj i ( y˜i (ξ )) dξ 1 − τ¯kj i i

qk2 y˜i4 (ξ )φk1i ( y˜i (ξ )) 1 − d¯k1i

t

t −τk1i (t )

  pk2 4 y˜ (ξ )k1i ( y˜i (ξ )) dξ 1 − τ¯k1i i (23)

in which i is a given positive matrix, and qk1 , qk1 , pk1 , and pk2 are positive constants determined later. Now, we use the backstepping method to design the dynamic output feedback controller.

Now, we use RBF NNs to approximate unknown βi ( y˜i ). With βi ( y˜i ) = θi∗T ζi ( y˜i ) + δi , one has  ∗T  3 3 θi ζi ( y˜i ) + δi βi ( y˜i ) = z i1 z i1  ∗ T 3 3 ˆT θi − θˆi ζi ( y˜i ) + z i1 ≤ z i1 θi ζi ( y˜i ) ai 6 1 2 + z i1 + δ (26) 2 2ai i where ai is a given positive constant. We choose the adaptive law as ·

3 ζi ( y˜i ) − li i θˆi (t) θˆi = i z i1

(27)

where li is a positive scalar. The virtual controller is designed as y˜i μ˙ i μi 3 ai z i1 θˆ T ζi ( y˜i ) − − i 2ri ri

αi1 = −ki2 x i1 − h i1 +

ai1 z i1 + y˙id − ri  1/3 1/3 1/3 − ci10 ri + c¯i11 ri + ci12 ri  2 z2 c¯i13 i1 i1 + + ci14ri3 z i1 . ri

(28)

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 4/3  ∂αi( j −1) − ci j 0 + c¯i j 3 i2j z i2j + c¯i j 1 ri ∂z i1  4/3  4 ∂αi( j −1) ∂αi( j −1) + ci j 2 ri + ci j 4 ri ∂z ∂z i1  i1 + c¯i( j −1)0 z i j (31)

Substituting (26)–(28) in (24) gives 4 4 4

Hi1 ≤ c¯i10 z i2 + ci11 ei2 + c¯i12 f i14 + c¯i14 gi1 1 2 4 3 4 + ci13 gi1 − z i1 βi ( y˜i ) − ai1 z i1 + δ 2ai i  T + li θi∗ − θˆi θˆi (t).

(29)

Step j : According to Definition 1, one has ⎛

results in 4

Hi j ≤ −ai j z i4j − c¯i( j −1)0 z i4j + c¯i j 0 z i( j +1)

Hi j ≤ z i3j ⎝αi j + z i( j +1) + ki( j +1) x i1 + h˜ i j

− ki j (h i1 + ki2 x i1 + λi2 ) −

4 4 4 + ci j 1 ei2 + c¯i j 2 f i14 + ci j 3 gi1 + c¯i j 4 gi1 .

j −1  ∂αi( j −1) k=2

∂λik ⎞

λ˙ ik

Step n i : For j = n i , we design the controller as u i (t) = −h˜ ini + kini (h i1 + ki2 x i1 + λi2 ) n i −1 ∂αi(ni −1) ∂αi(ni −1) + + λ˙ ik ∂t ∂λik

∂αi( j −1) ∂αi( j −1) · T − i j gi1 gi1 − θˆi ⎠ ∂t ∂ θˆi  3 ∂αi( j −1) − zi j ri λi2 + ki2 x i1 + ei2 ∂z i1  yi μ˙ i + h i1 + f i1 − − y˙id μi  2  T  3 2 ∂αi( j −1) ri T r gi1 gi1 + zi j 2 ∂z i1  ∂αi( j −1) · ≤ z i3j αi j + ki( j +1) x i1 + h˜ i j − θˆi ∂ θˆi −

− ki j (h i1 + ki2 x i1 + λi2 ) −

j −1  ∂αi( j −1) k=2



∂λik

k=2

∂αi( j −1) ∂t   ∂α yi μ˙ i i( j −1) ri λi2 + ki2 x i1 + h i1 − − y˙id −z i3j ∂z i1 μi   4/3 ∂αi( j −1) + c¯i j 1 ri ∂z i1  4/3 ∂αi( j −1) + ci j 0 + ci j 2 ri + c¯i j 3 i2j z i2j ∂z i1  4  ∂αi( j −1) 4 + ci j 4 ri z i4j + c¯i j 0 z i( j +1) ∂z i1 +

c¯i j 2 fi14

4 + ci j 3 gi1

4 + c¯i j 4 gi1 .

Thus, the choice  ∂αi( j −1) αi j = − ki( j +1) x i1 + h˜ i j − ∂t

+

k=2

∂αi( j −1) λ˙ ik − ai j z i j ∂λik

4 4 4

Hini ≤ −aini z in − c¯i(ni −1)0 z in + cini 1 ei2 i i 4 4 + c¯ini 2 f i14 + cini 3 gi1 + c¯ini 4 gi1 .

i=1

≤

N 

⎛

j =1

⎛

⎝ − ⎝ci0 γi0 λmin (Pi ) − 2ci0 ρi Pi 2 − ci1

i=1

− ci2 − + qi1



(34)

With (14), (29), (32), and (34), one has ⎛ ⎛ ⎞⎞ ni N   ⎝Vie +

⎝ Hi j ⎠ ⎠

ni 

⎞ ci j 1⎠ ei 4 −

j =1

·

(33)

With (33), we have

(30)

∂αi( j −1) θˆi − ki j (h i1 + ki2 x i1 + λi2 ) − ∂ θˆi   ∂αi( j −1) yi μ˙ i ri λi2 + ki2 x i1 + h i1 − − y˙id + ∂z i1 μi j −1 

∂αi( j −1) · ∂αi(ni −1) + ri θˆi + ˆ ∂z i1 ∂θ  i  y˜i μ˙ i × λi2 + ki2 x i1 + h i1 − − y˙id μi   4/3 ∂αi(ni −1) ri − aini z ini − c¯ini 1 ∂z i1  4/3 ∂αi(ni −1) 2 2 + cini 2 ri + c¯ini 3 in z i ini ∂z i1   4 ∂αi(ni −1) + cini 4 ri + c¯i(ni −1)0 z ini . ∂z i1

λ˙ ik

−

4 + ci j 1 ei2

(32)

ni  j =2

f i4j + qi2 f i14 +

ni 

j =1 n i  pi1 gi4j j =2

⎞   T 3 − z i1 βi (yi ) + li θi∗ − θˆi θˆi (t)⎠

ai j z i4j +

1 2 δ 2ai i

4 + pi2 gi1

(35)

 where qi1 = 8c¯i1 (n i − 1), qi2 = 8c¯i1 (n i − 1) nj i=2 ki4j + n i c¯ , p = 8c¯i2 (n i − 1), and pi2 = 8c¯i2 (n i − 1) nj i=1 i4j 2 i1ni j =2 k i j + j =1 (ci j 3 + c¯i j 4 ).

HUA et al.: DECENTRALIZED OUTPUT FEEDBACK ADAPTIVE NN TRACKING CONTROL

 N  N n k  N n i  N χkj i ( y˜i ) with i=1 k=1 χi j k ( y˜k ) = i=1 j =2  N N  Nk=1 Nj =2 and = i=1 k=1 χi1k ( y˜k ) i=1 k=1 χk1i ( y˜i ), where χi j k (·) and χi1k (·) are nonlinear functions, we have ⎛ ⎛ ni N   ⎝− ⎝ci0 γi0 λmin (Pi ) − ci j 1

U ≤

From Assumption 1, we have ⎛ ⎞ ni ni N    4⎠ ⎝qi1 f i4j + qi2 f i14 + pi1 gi4j + pi2 gi1 j =2

i=1

≤

j =2

ni  N N  

 qi1 y˜k4 κi j k ( y˜k )

i=1 j =2 k=1

+ y˜k4 (t − di j k )φi j k ( y˜k (t − di j k )) +

N  N 

qi2 y˜k4 κi1k ( y˜k )

ni  N  N 



−

ni  j =1



pi1 y˜k4 ϕi j k ( y˜k )

i=1 j =2 k=1

+ y˜k4 (t − τi j k (t))i j k ( y˜k (t − τi j k (t))) N N  

+



⎞ # # 1 2 3 ai j z i4j − li #θi∗ − θˆi # − z i1 βi ( y˜i )⎠ 2

nk  N  N   i=1 k=1 j =2

 pi2 y˜k4 i1k ( y˜k )

i=1 k=1

 + y˜k4 (t − τi1k (t))i1k ( y˜k (t − τi1k (t))) ⎞ ni ni N    ⎝ qi1 i j + qi2 i1 + pi1 ˜i j + pi2 ˜i1⎠. (36) + ⎛

j =2

i=1

+

N  N   qk2 y˜i4 κk1i ( y˜i ) +

Derivating Wi along (23), one has W˙ i ≤

k=1 j =2

≤

qk1 y˜i4 (t)φkj i ( y˜i (t)) 1 − d¯kj i

N   k=1

qk2 y˜i4 (t)φk1i ( y˜i (t)) 1 − d¯k1i

(37) Forthe whole system, we choose Lyapunov function as N Ui . Considering that U = i=1 #2 1 # #2  T 1 # li θi∗ − θˆi θˆi ≤ − li #θi∗ − θˆi # + li #θi∗ # 2 2

(38)

and setting ni  #2 1 # 1 2 li #θi∗ # + δi + qi1 i j 2 2ai j =2

+ qi2 i1 + pi1

ni  j =2

⎛

⎛

 pk2 4 y˜ (t)k1i (yi (t)) 1 − τ¯k1i i

⎝ − ⎝ci0 γi0 λmin (Pi ) − 2ci0 ρi Pi 2 − ci1 − ci2

−

ni  j =1

⎞ ci j 1 ⎠ ei 4 − ⎞

ni 

ai j z i4j

j =1

#2 1 # − li #θi∗ − θˆi # + σ⎠ i . 2

(40)

We choose the proper parameters ci0 , ci1 , ci2 , and ci j 1 such that the positive definite matrix Pi satisfies

− qk2 y˜i4 (t − dk1i (t))φk1i ( y˜i (t − dk1i (t))) pk2 + y˜ 4 (t)k1i (yi (t)) 1 − τ¯k1i i  4 − pk2 y˜i (t − τk1i (t))k1i ( y˜i (t − τk1i (t))) .

σi =

N 

qk2 y˜i4 (t)φk1i ( y˜i (t)) 1 − d¯k1i

+ pk2 y˜i4 ϕk1i ( y˜i ) +

i=1

− qk1 y˜i4 (t − dkj i (t))φkj i ( y˜i (t − dkj i (t))) pk1 y˜ 4 (t)kj i (yi (t)) + 1 − τ¯kj i i  4 − pk1 y˜i (t − τkj i (t))kj i ( y˜i (t − τkj i (t))) +

qk1 y˜i4 (t)φkj i ( y˜i (t))+qk1 y˜i4 κkj i ( y˜i ) 1 − d¯kj i pk1 + y˜ 4 (t)kj i (yi (t)) 1 − τ¯kj i i  + pk1 y˜i4 ϕkj i ( y˜i )

i=1 k=1

j =2

nk  N  

⎞

−2ci0 ρi Pi 2 − ci1 − ci2 ⎠ ei 4 + σi



+ y˜k4 (t − di1k (t))φi1k ( y˜k (t − di1k (t)))

+

j =1

i=1



i=1 k=1

+

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˜i j + pi2 ˜i1

(39)

ci0 γi0 λmin (Pi ) − 2ci0 ρi Pi 2 − ci1 −ci2 −

ni 

ci j 1 > 0. (41)

j =1

Furthermore, one has ⎛ ⎞ ni N   # # ⎝− 1 li #θi∗ − θˆi #2 −

U ≤ ai j z i4j + σi⎠. 2 i=1

(42)

j =1

Now, we present the main result of this paper. Theorem 1: For (1), under Assumptions 1 and 2, the reduced-order observers (8) and the dynamic controller (28), (31), and (33) with adaptive law (27) can get the objectives P1 and P2 of this paper. Proof: With Lemma 1 and (42), we can obtain Theorem 1 directly. Remark 5: In this section, we consider the case that the bounds of nonlinear uncertain interconnections f i j (·) and gi j (·) are unknown. The NN is used to approximate the composite unknown nonlinear function for each subsystem.

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For the case that the functions are known, the proposed method is also feasible. Similar to the design procedure, we design the first virtual control input as y˜i μ˙ i ai1 z i1 + y˙id − αi1 = −ki2 x i1 − h i1 + μi ri  βi ( y˜i ) 1/3 1/3 1/3 − − ci10 ri + c¯i11ri + ci12 ri ri  2 z2 c¯i13 i1 i1 3 + + ci14ri z i1 ri which is a little different from the one in former case. Then, the controller is constructed as follows: u i (t) = −h˜ ini + kini (h i1 + ki2 x i1 + λi2 ) n −1

i  ∂αi(ni −1) ∂αi(ni −1) + λ˙ ik − aini z ini ∂t ∂λik k=2   ∂αi(ni −1) y˜i μ˙ i + ri λi2 + ki2 x i1 + h i1 − − y˙id ∂z i1 μi   4/3 ∂αi(ni −1) 4/3 − c¯ini 1ri ri ∂z i1   ∂αi(ni −1) 4/3 2 2 + cini 2 ri + c¯ini 3 in z i ini ∂z i1  4  ∂αi(ni −1) + cini 4 ri + c¯i(ni −1)0 z ini . ∂z i1

+

It is easy to prove that the closed-loop system achieves the prescribed performance requirement with the designed controller. Remark 6: It is well known that control for stochastic nonlinear interconnected systems is more difficult compared with the control for single deterministic systems. By constructing the new Lyapunov functional (19)–(23), we successfully design the decentralized memoryless output feedback controller with backstepping method. Remark 7: In this paper, we employ the RBF NN for the control design of nonlinear systems with unknown interconnections. In the control design of each subsystem, the unknown function βi ( y˜i ) in (25) is approximated by RBF NN. With this NN approximation, the decentralized adaptive controller is proposed. The NN plays an important role for our control design. The RBF NN contains two parts: 1) a RBF vector and 2) a weight vector. The element of RBF vector is RBF basis function, which is chosen as the commonly used Gaussian function based on the unknown smooth functions to be approximated. The second part is weight vector, which has an idea value. The adaptive law is designed such that the weight vector is tuned online to approximate the idea vector. Remark 8: In this paper, the nonlinear functions are not known in each subsystem. Instead of using NN to approximate single nonlinear function, we employ one NN for each subsystem to approximate a composite nonlinear function, which will reduce the complexity of control design. Only N NNs are used for the large-scale time-delay systems with N subsystems. Remark 9: We propose a reduced-order observer-based output feedback control method for a class of large-scale

stochastic systems with time delays. For delay-free stochastic system [di j k (t) = τi j k (t) = 0 in (1)], there is still no result reported to deal with the PPC control problem. The proposed method is also feasible for the delay-free case. With the chosen Lyapunov function Ui = Vie + Mi , we can design the decentralized NN controller via the same procedure. From the above synthesis, we can obtain the following controller design procedure. Step 1: For (1), check Assumptions 1 and 2. If the assumptions are satisfied, obtain constant ρi j and functions φi j k (·), κi j k (·), i j k , and ϕi j k based on (2)–(4). Step 2: Choose a positive definite matrix Pi and solve linear matrix inequality to obtain observer gain ki j and constants ci0 , γi0 , ci1 , ci2 , and ci j 1 ; then, according to (8), design the reduced-order observer. Step 3: Choose function μi (t) and obtain the prescribed performance transformation according to (16). Step 4: Choose the RBF function ζi j ( y˜i ) and further obtain ζi ( y˜i ). Based on the proposed backstepping method, design the virtual control input αi1 (·) (28) with the corresponding adaptive law; here, parameter li should be selected small enough, whereas parameter ai1 should be big enough. Via the step-by-step method, the output feedback controller u i (t) (33) is finally constructed. IV. S IMULATION E XAMPLE In this section, to illustrate the validity of the proposed controller, we consider the bounded stabilization in probability problem for the stochastic large-scale system consisting of two chemical reactors [36]–[38], which is described by   ⎧ 1 1 − Ri,2 ⎪ ⎪ d x (t) = − x − K x + x + f ⎪ i1 i1 i,1 i1 i2 i,1 ⎪ Ci,1 Vi,1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎨  × dt + gi,1 di 1 Fi,2 d x i2 (t) = − x i2 − K i,2 x i2 + u i + fi,2 ⎪ ⎪ Ci,2 Vi,2 ⎪ ⎪ ⎪ ⎪ ⎪ × dt + gi,2 di ⎪ ⎪ ⎩ yi = x i1 where i = 1, 2, x i1 and x i2 are the compositions, Ri,2 is the recycle flow rate, Ci,1 and Ci,2 are the reactor residence times, K i,1 and K i,2 are the reactor constants, Fi,2 is the feed rate, Vi,1 and Vi,2 are reactor volumes, f i,1 and f i,2 are uncertain nonlinear functions with inter connected time delays, gi,1 and gi,2 are the nonlinear stochastic disturbance, and yi is the output of the i th subsystem, respectively. For the simulation, we choose the system parameters as Ci,1 = Ci,2 = 2, K i,1 = K i,2 = 0.5, and Ri,2 = Vi,1 = Vi,2 = Fi,2 = 0.5. The nonlinear functions are set as f 11 = y1 sin(y1 + y2 ), g11 = y1 (t − τ111(t)) sin(y2 (t − 2 2 2 τ112 (t))), f 12 = y1 (t − d121)e−y2 , g12 = y1 e−(y1 +y2 ) , f 21 = y2 sin(y1 +y2 ), g21 = y2 (t −τ212 (t)) sin(y1 (t −τ211 (t))), 2 2 2 f 22 = y2 (t − d222)e−y1 , and g22 = 0.1y2 e−(y1 +y2 ) . The time-delay functions are set as di j k (t) = τi j k (t) = 0.2(1 + sin t). We consider the tracking signal as yid (t) = 0.5 + 0.5 sin(10t).

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The reduced-order observers are designed as λ˙ 12 (t) = u 1 (t) − 2.25x 11 − 2.5λ12 λ˙ 22 (t) = u 2 (t) − 2.25x 21 − 2.5λ22 with k12 = k22 = 1.5, γi0 = 3, Pi = 1, ci1 = 0.5, ci2 = 0.2, ci11 = ci21 = 0.1, and ci0 = 1. The performance function is selected as μ(t) = μ1 (t) = μ2 (t) = 0.9e−10t + 0.1 and δ 1 = δ¯1 = δ 2 = δ¯2 = 1. Because the nonlinear functions are unknown, the adaptive NN will be used to cope with this problem. The virtual control is designed as

Fig. 4.

Response of output of subsystem1.

Fig. 5.

Response of output of subsystem2.

Fig. 6.

Response of the tracking error performances of subsystem1.

Fig. 7.

Response of the tracking error performances of subsystem2.

z3 y˜i μ˙ i + y˙id − 5z i1 − i1 μi 2ri   θˆ T ζi ( y˜i ) 1/3 2 − 2.1ri + 0.25m 2i ri z i1 + 0.5ri3 z i1 − i ri

αi1 = −0.5x i1 +

where ri = −μi /( y˜i2 − μ2i ), m i = 2 y˜i /( y˜i2 − μ2i ). The adaptive law is chosen as ·

3 θˆi = z i1 ζi ( y˜i ) − 10θˆi (t)

where

⎡

⎤

exp(−0.5 ∗ (yi − 8)ˆ2) ⎢ exp(−0.5 ∗ (yi − 4)ˆ2) ⎥ ⎥ ⎢ ⎥ ⎢ ξiT (yi ) = ⎢ exp(−0.5 ∗ (yi )ˆ2) ⎥ ⎥ ⎢ ⎣ exp(−0.5 ∗ (yi + 4)ˆ2) ⎦ exp(−0.5 ∗ (yi + 8)ˆ2) and the controller is u i (t) = 1.5x i1 + λi2 + 1.5(0.5x i1 + λi2 ) ∂αi1 ˆ· ∂αi1 + + θi − 5z ini ∂t ∂ θˆ  i  ∂αi1 yi μ˙ i + ri λi2 (t) + 0.5x i1 (t) − − y˙id ∂z i1 μi    4/3 4  ∂αi1 ∂αi1 2 2 +0.25ini z ini +0.5 ri − 1.6 z ini . ∂z i1 ∂z i1 The initial values are chosen as x 12 (0) = 0.01, x 22 (0) = 0.8, λ12 = 0.1, λ22 = 0.1, x 11 (t) = 0.5, and x 21 (t) = 0.6 for t ∈ [−0.2, 0], θˆi = [0, 0.8, 0.7, 0.6, 0.5]. The simulation results are shown in Figs. 4–7 with the horizontal axis as the time. The response of output curves and the tracking signal curves for each subsystem are shown in Figs. 4 and 5, respectively. Figs. 6 and 7 show the response curves of tracking errors and the bounds of the prescribed performance from which we can see that the required performances are achieved. To validate the improved performance with the proposed schemes, the tracking error profile under the control of conventional backstepping method without PPC is also plotted in Figs. 6 and 7. It can be observed that the convergence speed, the overshoot, and the steady-state tracking error can be further reduced to the preset range with the proposed schemes, and then, the transient-state and steady-state performances are enhanced.

V. C ONCLUSION The output feedback tracking control problem was investigated for interconnected stochastic time-delay systems with the prescribed performance requirement. The novel state transformation was proposed to deal with the asymmetric

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IEEE TRANSACTIONS ON NEURAL NETWORKS AND LEARNING SYSTEMS, VOL. 26, NO. 11, NOVEMBER 2015

prescribed performance function. The NN was used to approximate the unknown composite interconnection functions; thus, only N NNs were used for the large-scale time-delay systems with N subsystems. The reduced-order observer was designed at first. Then, the observer-based output feedback tracking controller was designed based on backstepping method. The observer and the controller designed were both memoryless, which would render our main result more suitable for practical applications. Finally, simulation was performed and the results showed the effectiveness of the proposed method. R EFERENCES [1] C.-Y. Lu, J. S.-H. Tsai, G.-J. Jong, and T.-J. Su, “An LMI-based approach for robust stabilization of uncertain stochastic systems with time-varying delays,” IEEE Trans. Autom. Control, vol. 48, no. 2, pp. 286–289, Feb. 2003. [2] H. Jia, H. R. Karimi, and Z. Xiang, “Dynamic output feedback passive control of uncertain switched stochastic systems with time-varying delay,” Math. Problems Eng., vol. 2013, pp. 1–10, Mar. 2013. [Online]. Available: http://dx.doi.org/10.1155/2013/281747 [3] Z. Wang, F. Yang, D. W. C. Ho, and X. Liu, “Robust H∞ filtering for stochastic time-delay systems with missing measurements,” IEEE Trans. Signal Process., vol. 54, no. 7, pp. 2579–2587, Jul. 2006. [4] S. Xie and L. Xie, “Stabilization of a class of uncertain large-scale stochastic systems with time delays,” Automatica, vol. 36, no. 1, pp. 161–167, 2000. [5] S. Xu and T. Chen, “Robust H∞ control for uncertain stochastic systems with state delay,” IEEE Trans. Autom. Control, vol. 47, no. 12, pp. 2089–2094, Dec. 2002. [6] X.-J. Li and G.-H. Yang, “Dynamic output feedback control synthesis for stochastic time-delay systems,” Int. J. Syst. Sci., vol. 43, no. 3, pp. 586–595, 2012. [7] S. Xu, P. Shi, Y. Chu, and Y. Zou, “Robust stochastic stabilization and H∞ control of uncertain neutral stochastic time-delay systems,” J. Math. Anal. Appl., vol. 314, no. 1, pp. 1–16, 2006. [8] W. Chen, J. Wu, and L. C. Jiao, “State-feedback stabilization for a class of stochastic time-delay nonlinear systems,” Int. J. Robust Nonlinear Control, vol. 22, no. 17, pp. 1921–1937, 2012. [9] H. Deng and M. Krstic, “Output-feedback stochastic nonlinear stabilization,” IEEE Trans. Autom. Control, vol. 44, no. 2, pp. 328–333, Feb. 1999. [10] Z. Wang and K. J. Burnham, “Robust filtering for a class of stochastic uncertain nonlinear time-delay systems via exponential state estimation,” IEEE Trans. Signal Process., vol. 49, no. 4, pp. 794–804, Apr. 2001. [11] Y. Fu, Z. Tian, and S. Shi, “Output feedback stabilization for a class of stochastic time-delay nonlinear systems,” IEEE Trans. Autom. Control, vol. 50, no. 6, pp. 847–851, Jun. 2005. [12] L. Li, Y. Jia, J. Du, and S. Yuan, “Dynamic output feedback control for a class of stochastic time-delay systems,” in Proc. Amer. Control Conf., St. Louis, MO, USA, Jun. 2009, pp. 5121–5125. [13] S.-J. Liu, S. S. Ge, and J.-F. Zhang, “Adaptive output-feedback control for a class of uncertain stochastic non-linear systems with time delays,” Int. J. Control, vol. 81, no. 8, pp. 1210–1220, 2008. [14] J. Liang, Z. Wang, and X. Liu, “Robust state estimation for two-dimensional stochastic time-delay systems with missing measurements and sensor saturation,” Multidimensional Syst. Signal Process., vol. 25, no. 1, pp. 157–177, 2014. [15] C.-D. Zheng, Q.-H. Shan, H. Zhang, and Z. Wang, “On stabilization of stochastic Cohen–Grossberg neural networks with mode-dependent mixed time-delays and Markovian switching,” IEEE Trans. Neural Netw. Learn. Syst., vol. 24, no. 5, pp. 800–811, May 2013. [16] D. Yue and Q.-L. Han, “Delay-dependent exponential stability of stochastic systems with time-varying delay, nonlinearity, and Markovian switching,” IEEE Trans. Autom. Control, vol. 50, no. 2, pp. 217–222, Feb. 2005. [17] B. Zhang, S. Xu, G. Zong, and Y. Zou, “Delay-dependent exponential stability for uncertain stochastic Hopfield neural networks with timevarying delays,” IEEE Trans. Circuits Syst. I, Reg. Papers, vol. 56, no. 6, pp. 1241–1247, Jun. 2009.

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HUA et al.: DECENTRALIZED OUTPUT FEEDBACK ADAPTIVE NN TRACKING CONTROL

Changchun Hua received the Ph.D. degree in electrical engineering from Yanshan University, Qinhuangdao, China, in 2005. He was a Research Fellow with the National University of Singapore, Singapore, from 2006 to 2007. From 2007 to 2009, he was with Carleton University, Ottawa, ON, Canada, funded by the Province of Ontario Ministry of Research and Innovation Program. From 2009 to 2011, he was with the University of Duisburg-Essen, Essen, Germany, funded by the Alexander von Humboldt Foundation. He has been involved in more than 10 projects supported by the National Natural Science Foundation of China, the National Education Committee Foundation of China, and other important foundations. He is currently a Full Professor with Yanshan University. He has authored or co-authored over 110 papers in mathematical, technical journals, and conferences. His current research interests include nonlinear control systems, control systems design over network, teleoperation systems, and intelligent control.

Liuliu Zhang received the B.Sc. degree in electrical engineering from Yanshan University, Qinhuangdao, China, in 2012, where she is currently pursuing the Ph.D. degree in electrical engineering. Her current research interests include nonlinear system control.

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Xinping Guan received the B.S. degree in mathematics from Harbin Normal University, Harbin, China, in 1986, and the M.S. degree in applied mathematics and the Ph.D. degree in electrical engineering from the Harbin Institute of Technology, Harbin, in 1991 and 1999, respectively. He has finished more than 20 projects supported by the National Natural Science Foundation of China, the National Education Committee Foundation of China, and other important foundations, as a Co-Investigator. He is currently with the Department of Automation, Shanghai Jiao Tong University, Shanghai, China. He is the Cheung Kong Scholars Program Special Appointment Professor. He has co-authored over 200 papers in mathematical, technical journals, and conferences. His current research interests include networked control systems, robust control, and intelligent control for complex systems and their applications. Dr. Guan serves as a reviewer of the Mathematic Review of America, a member of the Council of Chinese Artificial Intelligence Committee, and the Chairman of the Automation Society of Hebei Province in China.