Backstepping-based decentralized bounded-H∞ adaptive neural control for a class of large-scale stochastic nonlinear systems

Available online at www.sciencedirect.com

Journal of the Franklin Institute 356 (2019) 8049–8079 www.elsevier.com/locate/jfranklin

Backstepping-based decentralized bounded-H∞ adaptive neural control for a class of large-scale stochastic nonlinear systems Hui Liu a, Xiaohua Li a,∗, Xiaoping Liu b,c, Huanqing Wang c a School

of Electronic and Information Engineering, University of Science and Technology Liaoning, Anshan 114051, China b Faculty of Engineering, Lakehead University, Thunder Bay, ON, Canada c School of Information and Electrical Engineering, Shandong Jianzhu University, Jinan 250101, China Received 19 September 2018; received in revised form 29 May 2019; accepted 26 June 2019 Available online 12 July 2019

Abstract In this paper, a novel decentralized adaptive neural control approach based on the backstepping technique is proposed to design a decentralized H∞ adaptive neural controller for a class of stochastic large-scale nonlinear systems with external disturbances and unknown nonlinear functions. RBF neural networks are utilized to approximate the packaged unknown nonlinearities. A novel concept with regard to bounded-H∞ performance is proposed. It can be applied to solve an H∞ control problem for a class of stochastic nonlinear systems. The constant terms appeared in stability analysis are dealt with by using Gronwall inequality, so that H∞ performance criterion is satisfied. The assumption that the approximation errors of neural networks must be square-integrable in some literature can be eliminated. The design process for decentralized bounded-H∞ controllers is given. The proposed control scheme guarantees that all the signals in the resulting closed-loop large-scale system are uniformly ultimately bounded in probability, and each subsystem possesses disturbance attenuation performance for external disturbances. Finally, the simulation results are provided to illustrate the effectiveness and feasibility of the proposed approach. © 2019 Published by Elsevier Ltd on behalf of The Franklin Institute.

∗

Corresponding author. E-mail addresses: [email protected] (H. Liu), [email protected] (X. Li), [email protected] (X. Liu), [email protected] (H. Wang). https://doi.org/10.1016/j.jfranklin.2019.06.043 0016-0032/© 2019 Published by Elsevier Ltd on behalf of The Franklin Institute.

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1. Introduction As we know, the stochastic disturbances in control systems are usually the instable source of systems. They often exist widely in practical systems. In order to improve the system stability and control accuracy, the research on stability and control of stochastic systems is presented as a meaningful issue and has received an extensive attention [1–4]. A fundamental method for the controller design of stochastic systems has been proposed in [5]. Some research results on the controller design of the general stochastic nonlinear systems have been obtained, see [6–11]. When some nonlinear functions in general stochastic systems are completely or partly unknown, radial basis function neural networks or fuzzy logic systems are used to approximate the unknown functions. And the backstepping technique [12–14] and the Lyapunov theory were utilized to design adaptive controllers for stochastic systems [7–11]. For some unmeasured states in stochastic nonlinear systems, observers were constructed to design adaptive output feedback controllers for stochastic nonlinear systems [11,15–17]. It is worth noting that the above research results are only for non-interconnected stochastic nonlinear systems. For large-scale stochastic nonlinear systems, some research results have been developed as well. It is well known that large-scale systems are composed of interconnected subsystems, which can be found in our society, such as power systems, digital communication networks, aerospace systems, urban traffic networks and multiagent systems. Nowadays, the decentralized control design problem for large-scale stochastic nonlinear systems has received a great deal of attention. Because of the complexity of interconnected systems, the control problem is challenging technically [18]. Considering physical restrictions on information exchange among subsystems, a decentralized control method is usually adopted to obtain local controllers for every subsystem of stochastic large-scale systems [19]. In some literature, this decentralized control problem for large-scale stochastic nonlinear systems was studied based on the backstepping method, see [20–31]. A decentralized adaptive output feedback controller was presented for a class of stochastic nonlinear interconnected systems with both parametric uncertainties and unknown nonlinear interactions [20]. A type of decentralized adaptive controllers were designed for uncertain large-scale stochastic strict-feedback nonlinear systems by using neural or fuzzy logic systems [21–24]. Compared with [21] in which the control direction was assumed to be known, [23] studied the adaptive neural control problem for a class of uncertain multi-input and multi-output stochastic nonlinear systems with unknown control directions. In [25], the decentralized control design problem for a class of large-scale nonaffine stochastic systems was investigated. And in [24,26–28], state observers were constructed to estimate the unmeasurable states of the systems so that decentralized output feedback controllers can be developed. The decentralized control problem of large-scale stochastic nonlinear systems with time-delays was studied in [30] and [31]. The unknown time-varying delays can be compensated by using integral-type Lyapunov–Krasovskii functionals in the design process. Because controlled systems are often influenced by external disturbances, the research on H∞ control problem for stochastic nonlinear systems is necessary. At present, the H∞ controllers are obtained mainly for stochastic nonlinear systems by solving linear matrix inequalities, Hamilton–Jacobi equalities, Hamilton–Jacobi inequalities, or Hamilton–Jacobi– Isaacs inequalities, see [32–35]. So far, only two papers have been found on decentralized H∞ control of large-scale stochastic systems. One studied a decentralized coherent robust H∞ quantum controller design problem for large-scale uncertain linear complex quantum stochastic systems [36], the other investigated the problem of robust decentralized H∞ output feedback

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controller design for a class of stochastic large-scale uncertain systems with time-delays [37]. In [37], stochastic white noises were only viewed as external disturbances. However, both papers investigated the H∞ control problem for linear stochastic systems. As far as we know, there is no literature studying the decentralized H∞ control problem for large-scale stochastic nonlinear systems until now. Motivated by the aforementioned concerns, the decentralized H∞ adaptive neural control problem for a class of strict-feedback stochastic large-scale systems with unknown nonlinear functions and external disturbances is investigated based on backstepping method in this paper. RBF neural networks are utilized to approximate the packaged unknown nonlinearities. The upper bound of the norm of weight vectors of neural networks and the approximation errors are adaptively estimated, respectively. As a result, the presented decentralized H∞ adaptive control scheme requires only two adaptive parameters for each subsystem, and eliminates the existing assumption that the approximation error must be square-integrable in some literature, such as [38] and [39]. The designed decentralized bounded-H∞ adaptive controllers can guarantee that all the signals in the resulting large-scale system are uniformly ultimately bounded in probability. At the same time, each subsystem has an H∞ disturbance attenuation performance for external disturbances. The simulation results are given to demonstrate the feasibility of the proposed approach. The main advantages of this paper can be summarized as follows: (1) In this paper, a novel concept with regard to bounded-H∞ performance is proposed. A decentralized bounded-H∞ adaptive neural control scheme for a class of large-scale stochastic strict-feedback systems is given. The new concept can be applied to solve an H∞ control problem for a class of stochastic nonlinear systems. The constant terms appeared in stability analysis are dealt with by using Gronwall inequality. (2) It is the first time to investigate the decentralized H∞ adaptive neural control problem for large-scale stochastic strict-feedback nonlinear systems. A novel decentralized bounded-H∞ adaptive neural control approach based on backstepping technique is proposed. (3) The method to estimate the upper bounds of norm of all the approximation errors of neural networks is proposed for the first time, which can be used to eliminate the assumption that approximation errors of neural networks must be square-integrable in some literature. (4) The effect of external disturbances can be attenuated by H∞ performance in each subsystem. (5) In our design, only two adaptive parameters need to be estimated online for an ni -order subsystem, so the implementation of the controllers is easier in practical applications. The rest of this paper is organized as follows. Section 2 gives the system descriptions and preliminaries. In Section 3, a decentralized bounded-H∞ adaptive neural controller design scheme is presented. Section 4 provides a simulation example to show the effectiveness of the proposed controllers. Finally, we make some concluding remarks.

2. Problem formulation and some preliminaries 2.1. System description In this section, a class of large-scale stochastic strict-feedback nonlinear systems with N subsystems is considered. Each subsystem includes external disturbances, and the interconnections are linked only by outputs. The ith subsystem is described by the ITô differential

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equation as follows.      ⎧ ⎪ ⎪ dxi, j (t ) = gi, j x¯i, j xi, j+1 + fi, j x¯i, j T+ hi, j (y¯) ⎪ ⎪ + ⎨  ϕi,j x¯i, j i, j (t ) d t +ψi, j (y¯)d wi , dxi,ni (t ) = gi,ni x¯i,ni ui + fi,ni x¯i,ni + hi,ni (y¯) ⎪ T ⎪ + ϕi,ni x¯i,ni i,ni (t ) d t + ψi,n (y¯)d wi , ⎪ ⎪ i ⎩ yi = xi,1 .

(1)

 T where x¯i, j = xi,1 (t ), xi,2 (t ), . . . , xi, j (t ) ∈ R j , i = 1, 2, . . . , N , j = 1, 2, . . . , ni − 1, y¯ =  T y1 , y2 , . . . , yN . x¯i,ni = [xi,1 (t ), xi,2 (t ), . . . , xi,ni (t )]T ∈ Rni , ui ∈ R and yi (t) ∈ R are the state variables, the scalar control input and the output of the ith subsystem, respectively. wi is an ri -dimensional independent Wiener process defined on a complete probability space (, F, {Ft }t ≥ 0 , P) with  being a sample space, F being a σ -field, {Ft }t ≥ 0 being a filtration and P being a probability measure. gi,j , fi,j , ϕ i,j : Rj → R ( j = 1, 2, . . . , ni ) are the unknown smooth nonlinear functions, hi, j (y¯) : RN → R, and ψi, j (y¯) : RN → Rri , ( j = 1, 2, . . . , ni ) are the unknown smooth interconnected functions between the ith subsystem and other subsystems with fi, j (0) = hi, j (0) = ϕi, j (0) = ψi, j (0) = 0. i, j (t ) ∈ LF2 ([0, T ], R) stand for the nonzero external disturbance signals of the ith subsystem. In this paper, the backstepping technique will be used to design a decentralized boundedH∞ adaptive neural controller for the large-scale system according to the given disturbance attenuation constant γ i of the ith subsystem. The control objective of this paper is as follows: Design a decentralized bounded-H∞ adaptive neural controller for every subsystem by using the backstepping technique, the designed controller can guarantee that the large-scale system has an H∞ disturbance attenuation performance, and all the signals of the closed-loop large-scale system are uniformly ultimately bounded in probability. To facilitate the design for the decentralized bounded-H∞ adaptive neural controllers in Section 3, the following assumptions are necessary for the large-scale system (1). Assumption 1 [39]. For the uncertain smooth interconnection nonlinear functions hi, j (y¯) and ψi, j (y¯) in Eq. (1), there exist unknown smooth functions h¯ i, j,l (yl ) and ψ¯ i, j,l (yl ) satisfying N 



hi, j (y¯) 2 ≤ yl2 h¯ i,2 j,l (yl ), l=1 N  ψi, j (y¯) 2 ≤ yl2 ψ¯ i,2 j,l (yl ), l=1

where h¯ i, j,l (0) = 0 and ψ¯ i, j,l (0) = 0.   Assumption 2 [40]. The signs of the unknown smooth function gi, j x¯i, j are known, and are assumed as strictly positive or negative in the system (1). Without loss of generality, we  further assume that gi, j x¯i, j > 0, and there exist constant bi,m such that   gi, j x¯i, j ≥ bi,m > 0, where j = 1, 2, . . . , ni .

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2.2. Preliminaries Consider a stochastic system described by Itô-type differential equation. d x = f (x)d t + h(x)d w,

(2)

where x ∈ R is the system state, w denotes the r-dimensional standard Brownian motion, f( · ) and h(·) are continuous and satisfy locally Lipshitz condition with f (0) = h(0) = 0. n

Definition 1 [5]. For any given positive definite, radially, unbounded, twice continuously differentiable function V(x) ∈ C2 , we define the infinitesimal generator of stochastic differential equation (2) as follows.

 ∂V 1 ∂ 2V LV = f + T r hT 2 h , ∂x 2 ∂x where Tr{A} is the trace of matrix A. Definition 2 [39,41,42]. For the stochastic system (1), if there exist a control law ui and Lyapunov function Vi , and there exists a finite t so that  t     t   N N N  yi (s )2 ds ≤ i (s )2 ds + E E γi2 E [Vi (x (0 ) )], i=1

0

i=1

0

i=1

LF2 ([0, T ], R)

where yi denotes the system output, i (t ) ∈ is the nonzero external disturbance, γ i is a prescribed attenuation level, then the system is called to have a bounded-H∞ performance. Definition 3 [39]. For the large-scale nonlinear system (1) with Assumption 1, if there exists a decentralized control law vector u = [u1 , u2 , . . . , uN ]T , such that the large-scale nonlinear system is stable whether the interconnections between subsystems exist or not, then the system (1) is called to be connectively stable. Lemma 1 (Young’s inequality) [26]. For any positive number ε > 0, the following inequality holds. 1 εp x y ≤ |x | p + q |y|q , ∀(x , y) ∈ R2 , p qε where p > 1, q > 1 are constants and ( p − 1 )(q − 1 ) = 1. Lemma 2 (Cauchy–Bunyakovsky–Schwarz’s inequality) [43]. If a¯ = (a1 , a2 , . . . , an ) and b¯ = (b1 , b2 , . . . , bn ) are sequences of real numbers, then  n 2 n n    ak bk ≤ ak2 b2k , k=1

k=1

k=1

with equality if and only if the sequences a¯ and b¯ are proportional, i.e., there is a r ∈ R such that ak = rbk for each k ∈ {1, 2, . . . , n}. Lemma 3 [21]. Consider the stochastic system (2). If there exists a positive definite, radially unbounded, twice continuously differentiable Lyapunov function V: Rn → R, and constants a > 0, b > 0 such that LV ≤ −aV + b,

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then the system has a unique solution almost surely, and the system is bounded in probability. Corollary 1. Consider the stochastic system (2). For any given a positive definite, radially unbounded, twice continuously differentiable Lyapunov function V: Rn → R, if there are a function: V = V + m, and constants a > 0, b > 0, m > 0, such that LV ≤ −aV + b,

(3)

then the system has a unique solution almost surely, and the system is bounded in probability. Proof. According to Eq. (3), the following inequality holds LV = L (V + m ) ≤ −a (V + m ) + b which results LV ≤ −aV + b,

(4)

It is obvious that Eq. (4) satisfies Lemma 3, then Corollary 1 is proved.  Lemma 4 Gronwall inequality [44]. Let x, ψ and χ be real continuous functions defined in t ∈ [a, b], and satisfying χ (t) ≥ 0. If the following inequality holds,  t x(t ) ≤ ψ (t ) + χ (s)x (s )ds, a

then



t

x(t ) ≤ ψ (t ) +

χ (s)ψ (s )e

t s

χ (u)du

ds.

a

Corollary 2. Let x(t), ψ(t) and χ (t) be stochastic variable functions defined in t ∈ [a, b], and E[χ (t)] ≥ 0. If the following inequality holds,  t E [x(t )] ≤ E [ψ (t )] + E [χ (s )]E [x (s )]ds, a

then



E [x(t )] ≤ E [ψ (t )] +

t

E [χ (s )]E [ψ (s )]e

E



t s

χ (u)du



ds.

(5)

a

Specially, if χ ≥ 0 is a constant, Eq. (5) can be rewritten as  t E [x(t )] ≤ E [ψ (t )] + χ eχ (t−s) E [ψ (s )]ds.

(6)

a

Proof. According to the proof of Gronwall inequality mentioned in [44], let us consider the t function y (t ) := a E [χ (s )] E[x(s)]ds, t ∈ [a, b]. Then, y (a ) = 0, and y (t ) = E [χ (t )]E [x (t )] ≤ E [χ (t )]E [ψ (t )] + E [χ (t )]y (t ), t ∈ (a, b). −E



t

χ (s )ds



a By multiplication with e > 0, it follows that      t t d −E a χ (s )ds −E a χ (s )ds y (t )e ≤ E [χ (t )]E [ψ (t )]e . dt

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By integration on [a, t], one has  t   u t −E a χ (s )ds y (t )e ≤ E [χ (u )]E [ψ (u)]e−E [ a χ (s )ds] du, a

which results in  t   t E χ (s )ds y (t ) ≤ E [χ (u )]E [ψ (u)]e u du, t ∈ [a, b]. a

Since E [x (t )] ≤ E [ψ (t )] + y (t ), Eq. (5) in Corollary 2 is proved.  Obviously, if χ ≥ 0 is a constant, Eq. (6) holds. In this paper, the following RBF NN [8] is used to approximate a continuous function f¯(Z ) : Rq → R f¯nn (Z ) = W T S (Z ),

(7)

where the input vector Z ∈ Z ⊂ R , weight vector W = [w1 , w2 , . . . , wl ] ∈ R , l > 1 is the number of nodes in the neural network, S (Z ) = [s1 (Z ), s2 (Z ), . . . , sl (Z )]T ∈ Rl means the basis function vector and si (Z) is chosen to be a commonly used Gaussian function defined by   −(Z − μi )T (Z − μi ) , i = 1, 2, . . . , l, si (Z ) = exp (8) η2 T

q

l

where μi = [μi1 , μi2 , . . . , μiq ]T is the center of the receptive field, and η is the width of Gaussian function. As the method in [45], the following RBF NN is used to approximate a continuous unknown smooth nonlinear function f¯(Z ), over a compact set Z ⊂ Rq , as f¯(Z ) = W ∗T S (Z ) + δ(Z ), |δ| ≤ δmax , where W∗ is the ideal constant weights vector, δ is the approximation error, and δ max is the minimum upper bound of the approximation error. It has been verified in [46] that a RBF neural network can approximate f¯(Z ) with arbitrary accuracy if the sufficiently large number of nodes l is taken. Lemma 5 [45] . Consider the RBF networks with Gaussian basis function (7) and (8), let ρ = 21 mini = j μi − μ j , then the upper bound of S(Z) can be expressed as S(Z ) ≤

∞ 

3q (k + 2 )q−1 e−2ρ

k /η2

2 2

:= s.

k=0

It has been shown that the constant s in Lemma 5 is a finite value, and is unrelated with the variable Z and the dimension of the neural weight vector. 3. Decentralized bounded-H∞ adaptive neural controller design In this section, the decentralized bounded-H∞ adaptive neural controllers for the system (1) will be designed step by step. The design is composed of 5 procedures: 1) The interconnected terms are separated according to Assumption 1. 2) The unknown nonlinear terms are amplified according to Lemma 1 and Lemma 2. 3) All the terms related to each output are assigned to its corresponding subsystem. 4) The adaptive neural H∞ controllers are designed. 5) Gronwall

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inequality is used to deal with the constant terms appeared in stability analysis in order to prove H∞ performance. Firstly, a coordinate transformation is introduced as zi, j (t ) = xi, j (t ) − ai, j−1 (x¯i, j−1 , βˆi , θˆi ),

(9)

where αi,0 = 0, βˆi denotes the estimate value of unknown parameter β i , and θˆi denotes the estimate value of unknown parameter θ i . β˜i = βi − βˆi and θ˜i = θi − θˆi are estimation errors. β i and θ i are specified as   2 βi = max Wi, j ; j = 1, 2, . . . , ni , (10)   θi = max δi,2 j max ; j = 1, 2, . . . , ni .

(11)

For simplicity, the time variable t and the state vector x¯i, j will be omitted from the corresponding functions, for example, gi,1 (xi,1 ) is denoted by gi,1 . Step 1: According to the coordinate transformation (9), the system (1) can be rewritten as   d zi,1 = gi,1 xi,2 + fi,1 + hi,1 (y¯) + ϕi,1 i,1 d t + ψi,T1 (y¯)d wi . Consider a Lyapunov function candidate as   1 1 1 Vi,1 zi,1 , β˜i , θ˜i = hi bi,m zi,4 1 + hi b2i,m β˜i2 + hi b2 θ˜2 , 4 2Ki,1 2Ki,2 i,m i where hi is a design parameter defined by ni hi = 2 + 1, γi

(12)

(13)

γ i represents the given disturbance attenuation constant, ni is order of the ith subsystem, Ki,1 and Ki,2 are positive design parameters. According to Definition 1, Eq. (12) can be rewritten as:    1 ∂Vi,21 3 LVi,1 = hi bi,m zi,1 gi,1 xi,2 + fi,1 + hi,1 (y¯) + ϕi,1 i,1 + T r ψi,T1 (y¯) 2 ψi,1 (y¯) 2 ∂zi,1 hi 2 ˜ ˙ˆ hi 2 ˜ ˙ˆ bi,m βi βi − b θi θi Ki,1 Ki,2 i,m 2   3 = hi bi,m zi,3 1 gi,1 zi,2 + gi,1 αi,1 + fi,1 + hi,1 (y¯) + ϕi,1 i,1 + hi bi,m zi,2 1 ψi,1 (y¯) 2 hi 2 ˜ ˆ˙ hi 2 ˜ ˆ˙ − b βi βi − b θi θi . Ki,1 i,m Ki,2 i,m −

(14)

Using Lemma 1, the following inequalities are true. hi bi,m zi,3 1 gi,1 zi,2 ≤

1 3 hi bi,m gi,1 zi,4 1 + hi bi,m gi,1 zi,4 2 , 4 4

hi bi,m zi,3 1 ϕi,1 i,1 ≤

1 2 2 6 hi b ϕ z + hi i,21 . 4 i,m i,1 i,1

(15)

(16)

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Combining Lemma 1 and Lemma 2, we have 1 3 hi bi,m zi,4 1 + hi bi,m hi,4 1 (y¯) 4 4  N 2  1 3 ≤ hi bi,m zi,4 1 + hi bi,m yl2 h¯ i,1,l (yl ) 4 4 l=1

hi bi,m zi,3 1 hi,1 (y¯) ≤

 1 3 hi bi,m zi,4 1 + hi bi,m N yl4 h¯ i,4 1,l (yl ), 4 4 l=1 N

≤

(17)

2 4 3 3 hi bi,m zi,2 1 ψi,1 (y¯) ≤ 3hi bi,m zi,4 1 + hi bi,m ψi,1 (y¯) 2 16 4 3 2 4 ≤ 3hi bi,m zi,1 + hi bi,m ψi,1 (y¯) 16  1 3 9 4 hi zi,1 + hi3 b2i,m zi,4 1 + hi bi,m N yl4 ψ¯ i,41,l (yl ). 2 2 16 l=1 N

≤

(18)

Now, substituting Eqs. (15)–(18) into Eq. (14) gives   1 1 hi 2 ˜ ˙ˆ ˙ LVi,1 ≤ hi bi,m zi,3 1 gi,1 αi,1 +i,1 − hi b2 β˜i βˆi − b θi θi + hi bi,m gi,1 zi,4 2 + hi i,21 Ki,1 i,m Ki,2 i,m 4   1 1 3 hi bi,m N yl4 h¯ i,4 1,l (yl ) + hi3 b2i,m zi,4 1 + hi bi,m N yl4 ψ¯ i,41,l (yl ), 4 2 16 l=1 l=1 N

+

N

(19)

where i,1 =

3 3 1 9 gi,1 zi,1 + fi,1 + zi,1 + bi,m ϕi,2 1 zi,3 1 + zi,1 . 4 4 4 2bi,m

Step j(2 ≤ j < ni ): Using the coordinate transformation equation (9), one has dzi, j

 T j−1    ∂αi, j−1 = gi, j xi, j+1 + fi, j + hi, j (y¯) +ϕi, j i, j − ηi, j−1 d t + ψi, j (y¯) − ψi,k (y¯) d wi , ∂xi,k k=1

where ηi, j−1 =

j−1 j−1    ∂αi, j−1 ∂αi, j−1  gi,k xi,k+1 + fi,k + hi,k (y¯) ∂xi,k ∂xi,k k=1 k=1

+

j−1  ∂αi, j−1 k=1

∂xi,k

ϕi,k i,k +

j−1 ∂αi, j−1 ˙ˆ ∂αi, j−1 ˙ˆ 1  ∂ 2 αi, j−1 T βi + θi + ψ (y¯)ψi,q (y¯). 2 p,q=1 ∂ xi,p ∂ xi,q i,p ∂ βˆi ∂ θˆi

Select the following Lyapunov function   1 Vi, j zi, j = hi bi,m zi,4 j , 4

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then

  LVi, j = hi bi,m zi,3 j gi, j xi, j+1 + fi, j + hi, j (y¯) + ϕi, j i, j − ηi, j−1 ⎧ T  ⎫ j−1 j−1 ⎬   ∂αi, j−1 ∂ 2Vi, j ∂αi, j−1 1 ⎨ + Tr ψi, j (y¯) − ψi,k (y¯) ψ y ¯ − ψ y ¯ ( ) ( ) i, j i,k ⎭ 2 ⎩ ∂xi,k ∂xi,k ∂zi,2 j k=1 k=1  j−1   ∂αi, j−1  3 gi,k xi,k+1 + fi,k = hi bi,m zi, j gi, j zi, j+1 + gi, j αi, j + fi, j + hi, j (y¯) + ϕi, j i, j − ∂xi,k k=1 j−1 1  ∂ 2 αi, j−1 T ψ (y¯)ψi,q (y¯) ∂xi,k ∂xi,k 2 p,q=1 ∂ xi,p ∂ xi,q i,p k=1 k=1 2  j−1  ∂αi, j−1 ∂αi, j−1 ˙ˆ ∂αi, j−1 ˙ˆ 3 2 − βi − θi + hi bi,m zi, j ψi, j (y¯) − ψi,k (y¯) . 2 ∂xi,k ∂ βˆi ∂ θˆi

−

j−1  ∂αi, j−1

hi,k (y¯) −

j−1  ∂αi, j−1

ϕi,k i,k −

(20)

k=1

The following inequalities hold according to Lemma 1. 1 3 hi bi,m zi,3 j gi, j zi, j+1 ≤ hi bi,m gi, j zi,4 j + hi bi,m gi, j zi,4 j+1 , 4 4 hi bi,m zi,3 j ϕi, j i, j ≤ − hi bi,m zi,3 j

1 2 2 6 hi b ϕ z + hi i,2 j , 4 i,m i, j i, j

j−1  ∂αi, j−1 k=1

∂xi,k

ϕi,k i,k ≤

2 j−1  j−1  1 2  ∂αi, j−1 2 hi bi,m ϕi,k zi,6 j + hi i,k . 4 ∂x i,k k=1 k=1

(21) (22)

(23)

Lemma 1 and Lemma 2 are used in Eq. (20). It can be easily proved that the following inequalities are true. 1 3 hi bi,m zi,3 j hi, j (y¯) ≤ hi bi,m zi,4 j + hi bi,m hi,4 j (y¯) 4 4 N  1 3 ≤ hi bi,m zi,4 j + hi bi,m N yl4 h¯ i,4 j,l (yl ), (24) 4 4 l=1 −hi bi,m zi,3 j

j−1  ∂αi, j−1 k=1

∂xi,k

4 j−1  j−1   ∂αi, j−1 3 4 1 3 4 hi bi,m zi, j + hi bi,m hi,k (y¯) 4 ∂x 4 i,k k=1 k=1 4 j−1  j−1 N    ∂αi, j−1 3 4 1 3 4 ≤ hi bi,m zi, j + hi bi,m N yl4 h¯ i,k,l (yl ), (25) 4 ∂x 4 i,k k=1 k=1 l=1

hi,k (y¯) ≤

j−1  ∂ 2 αi, j−1 T 1 3 − hi bi,m zi, j ψ (y¯)ψi,q (y¯) 2 ∂ xi,p ∂ xi,q i,p p,q=1

≤

2 j−1  2 j−1   ∂ αi, j−1 1 1 ψi,p (y¯) 4 hi bi,m zi,6 j + hi bi,m ( j − 1 ) 4 ∂ xi,p ∂ xi,q 4 p,q=1 p=1

≤

2 j−1  2 j−1 N    ∂ αi, j−1 1 1 4 hi bi,m zi,6 j + hi bi,m ( j − 1 )N yl4 ψ¯ i,k,l (yl ), 4 ∂ x ∂ x 4 i,p i,q p,q=1 k=1 l=1

(26)

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8059

2 j−1  ∂α 3 i, j−1 hi bi,m zi,2 j ψi, j (y¯) − ψi,k (y¯) 2 ∂xi,k k=1  

2 j−1





∂α 3 2 2 i, j−1

≤ hi bi,m zi,2 j j ψi, j (y¯) +

∂x ψi,k (y¯) 2 i,k k=1

j−1

j−1  

∂αi, j−1 4 4 4 3 2 3 2 4



ψi,k (y¯) 4 ≤ 3hi bi,m zi, j + j hi bi,m ψi, j (y¯) +3hi bi,m

∂x zi, j + 16 j hi bi,m 16 i,k k=1 k=1

j−1

j  ∂αi, j−1 4  3 2

4 ψi,k (y¯) 4 ≤ 3hi2 bi,m zi,4 j + 3hi bi,m

∂x zi, j + 16 j hi bi,m i,k k=1 k=1

4 j−1

j N   

∂αi, j−1 4 1 3 2 4 3 2 9 4 4



≤ hi zi, j + hi bi,m zi, j + 3hi bi,m z + j h b N yl4 ψ¯ i,k,l (yl ). i i,m i, j

∂x

2 2 16 i,k k=1 k=1 l=1 (27) Thus, the inequalities (20)–(27) imply that N    1 1 4 gi, j αi, j + i, j + hi bi,m gi, j zi,4 j+1 + hi bi,m N yl4 h¯ i,k,l ( yl ) 4 4 k=1 l=1 j

LVi, j ≤

hi bi,m zi,3 j + hi

j  k=1

 1 1 4 + hi3 b2i,m zi,4 j + hi bi,m ( j − 1 )N yl4 ψ¯ i,k,l ( yl ) 2 4 k=1 l=1 j−1

2 i,k

 3 2 4 + j hi bi,m N yl4 ψ¯ i,k,l (yl ), 16 k=1 l=1 j

N

N

(28)

where i, j

2 j−1   ∂αi, j−1 3 3 1 1 2 3 = gi, j zi, j + fi, j + zi, j + bi,m ϕi, j zi, j + bi,m ϕi,k zi,3 j 4 4 4 4 ∂x i,k k=1 4 j−1 j−1   ∂αi, j−1   3  ∂αi, j−1 3 gi,k xi,k+1 + fi,k + − zi, j ∂xi,k 4 k=1 ∂xi,k k=1

2 j−1  j−1



∂αi, j−1 4 ∂αi, j−1 ˙ˆ ∂αi, j−1 ˙ˆ 1  ∂ 2 αi, j−1 9 3



+ zi, j − βi − θi + zi, j + 3

∂x zi, j . 4 p,q=1 ∂ xi,p ∂ xi,q 2bi,m i,k ∂ βˆi ∂ θˆi k=1

Step ni : With the coordinate transformation, the ni th equation is obtained as shown below.

dzi,ni

 T n i −1   ∂αi,ni −1 = gi,ni ui + fi,ni + hi,ni (y¯) + ϕi,ni i,ni − ηi,ni −1 d t + ψi,ni (y¯) − ψi,k (y¯) d wi , ∂xi,k k=1

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where ηi,ni −1 =

n i −1 k=1

n −1

i   ∂αi,ni −1  ∂αi,ni −1 ∂αi,ni −1 ˙ˆ gi,k xi,k+1 + fi,k + hi,k (y¯) + θi ∂xi,k ∂x i,k ∂ θˆi k=1

ni −1 n i −1 1  ∂ 2 αi,ni −1 T ∂αi,ni −1 ∂αi,ni −1 ˙ˆ + ψ (y¯)ψi,q (y¯) + ϕi,k i,k + βi . 2 p,q=1 ∂ xi,p ∂ xi,q i,p ∂x i,k ∂ βˆi k=1

Choose a Lyapunov function candidate as   1 4 Vi,ni zi,ni = hi bi,m zi,n + mi . i 4 Remark 1. The small positive constant mi is introduced for the proof of the H∞ disturbance attenuation performance only, which is not required to be known for the controller design. Then, by means of Definition 1, one has   3 gi,ni ui + fi,ni + hi,ni (y¯) +ϕi,ni i,ni − ηi,ni −1 LVi,ni = hi bi,m zi,n i ⎧ T  ⎫ n n i −1 i −1 ⎬ 1 ⎨ ∂αi,ni −1 ∂ 2Vi,ni ∂αi,ni −1 ψi,ni (y¯) − ψi,k (y¯) ψ y ¯ − ψ y ¯ + Tr ( ) ( ) i,ni i,k 2 ⎭ 2 ⎩ ∂xi,k ∂xi,k ∂zi,n i k=1 k=1  n i −1  ∂αi,ni −1  3 gi,k xi,k+1 + fi,k = hi bi,m zi,ni gi,ni ui + fi,ni + hi,ni (y¯) + ϕi,ni i,ni − ∂x i,k k=1 n i −1

n ni −1 i −1 ∂αi,ni −1 ∂αi,ni −1 ∂ 2 αi,ni −1 T 1  − hi,k (y¯) − ϕi,k i,k − ψ (y¯)ψi,q (y¯) ∂xi,k ∂xi,k 2 p,q=1 ∂ xi,p ∂ xi,q i,p k=1 k=1 2  n i −1 3 ∂αi,ni −1 ∂αi,ni −1 ˙ˆ ∂αi,ni −1 ˙ˆ 2 − βi − θi + hi bi,m zi,ni ψi,ni (y¯) − ψi,k (y¯) . 2 ∂xi,k ∂ βˆi ∂ θˆi

(29)

k=1

Similar to Eqs. (22) and (23), (30) and (31) can be obtained. 1 3 2 6 2 hi bi,m zi,n ϕ i,ni ≤ hi b2i,m ϕi,n z + hi i,n , i i,ni i i,ni i 4 3 − hi bi,m zi,n i

n i −1 k=1

2 n n i −1  i −1 ∂αi,ni −1 ∂αi,ni −1 1 6 2 ϕi,k i,k ≤ hi b2i,m ϕi,k zi,n + h i,k . i i ∂xi,k 4 ∂x i,k k=1 k=1

(30)

(31)

By making use of the same method as Eqs. (24)–(27), it can be verified that the following are true. N  3 1 3 4 4 hi bi,m zi,n h y ¯ ≤ h b N yl4 h¯ i,n (yl ) + hi bi,m zi,n , (32) ( ) i,n i i,m i i i ,l i 4 4 l=1 3 −hi bi,m zi,n i

n i −1 k=1

4 n i −1  ∂αi,ni −1 3 4 ∂αi,ni −1 3 hi,k (y¯) ≤ hi bi,m zi,ni ∂xi,k 4 ∂xi,k k=1 n −1 N

i   1 4 + hi bi,m N yl4 h¯ i,k,l (yl ), 4 k=1 l=1

(33)

H. Liu, X. Li and X. Liu et al. / Journal of the Franklin Institute 356 (2019) 8049–8079

8061

2 n n i −1 i −1  ∂ 2 αi,ni −1 ∂ 2 αi,ni −1 T 1 1 3 6 − hi bi,m zi,n ψ y ¯ ψ y ¯ ≤ h b zi,n ( ) ( ) i,q i i,m i,p i i 2 ∂ x ∂ x 4 ∂ x ∂ x i,p i,q i,p i,q p,q=1 p,q=1 n −1 N

i   1 4 + hi bi,m (ni − 1 )N yl4 ψ¯ i,k,l (yl ), 4 k=1 l=1

(34)

2



n n i −1 i −1

∂αi,ni −1 4 4 3 ∂αi,ni −1 1 3 2 4 9 4 2

z hi bi,m zi,ni ψi,ni (y¯) − ψi,k (y¯) ≤ hi zi,n + h b z +3 h b i i,m

∂x

i,ni i 2 ∂xi,k 2 2 i i,m i,ni i,k k=1 k=1 i   3 2 4 ni hi bi,m N yl4 ψ¯ i,k,l (yl ). 16 k=1 l=1

n

+

N

(35)

Furthermore, combining Eq. (29) with the inequalities (30)–(35), we have ni  ni N     1 3 2 4 1 3 4¯4 2 g + LVi,ni ≤ hi bi,m zi,n u +  h b z + h b N y y + h i,k h ( ) i,n i i,n i i,m l i i i i i,m i,n l i,k,l i i 2 4 k=1 l=1 k=1 n −1 N

i i     1 3 4 4 + hi bi,m (ni − 1 )N yl4 ψ¯ i,k,l yl4 ψ¯ i,k,l (yl ) + ni2 hi bi,m N (yl ), 4 16 k=1 l=1 k=1 l=1

n

N

(36)

where i,ni

2 n i −1  ∂αi,ni −1 3 1 1 2 3 3 = fi,ni + zi,ni + bi,m ϕi,ni zi,ni + bi,m ϕi,k zi,n i 4 4 4 ∂x i,k k=1 4 n i −1 i −1   3 n ∂αi,ni −1 3 ∂αi,ni −1  gi,k xi,k+1 + fi,k + − zi,ni ∂xi,k 4 k=1 ∂xi,k k=1 2 ni −1  2 ∂ αi,ni −1 1  9 ∂αi,ni −1 ˙ˆ ∂αi,ni −1 ˙ˆ 3 + zi,n − βi − θi + zi,ni i ˆ ˆ 4 p,q=1 ∂ xi,p ∂ xi,q 2b i,m ∂ βi ∂ θi



n i −1

∂αi,ni −1 4

zi,n . +3 i

∂x

i,k k=1

Now, the total Lyapunov function of the ith subsystem is chosen as Vi (Zi ) =

ni 

Vi, j =

j=1

ni  1 j=1

4

hi bi,m zi,4 j +

1 1 hi b2i,m β˜i2 + hi b2 θ˜2 + mi , 2Ki,1 2Ki,2 i,m i

(37)

 T where Zi = zi,1 , zi,2 , . . . , zi,ni , β˜i , θ˜i . Because mi > 0, one can verify that Eq. (38) holds. Vi (Zi ) > 0, So, LVi =

ni  j=1

LVi, j .

(38)

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Considering the interconnected terms of the subsystem in Eqs. (19), (28) and (36), the following can be proved. n −1

i i      1 1 1 4 4 hi bi,m N yl4 h¯ i,4 1,l (yl ) + hi bi,m N yl4 h¯ i,k,l (yl ) + hi bi,m N yl4 h¯ i,k,l (yl ) 4 4 4 j=2 k=1 l=1 l=1 k=1 l=1

N

n

N

N

i   1 4 hi bi,m N yl4 h¯ i,k,l (yl ), 4 d=1 k=1 l=1

n

=

j

d

N

(39)

j−1 N n n N i −1 i −1    1 1 4 4 hi bi,m N yl4 ψ¯ i,k,l yl4 ψ¯ i,k,l ( j − 1) (yl ) + hi bi,m (ni − 1 )N ( yl ) 4 4 j=2 k=1 l=1 k=1 l=1

=

ni d−1  N   1 4 hi bi,m N yl4 ψ¯ i,k,l (d − 1 ) (yl ), 4 d=2 k=1 l=1 n −1

(40)

i    3 3 4 hi bi,m N yl4 ψ¯ i,41,l (yl ) + hi bi,m N j2 yl4 ψ¯ i,k,l ( yl ) 16 16 j=2 l=1 k=1 l=1

N

N

i  i    3 2 3 4 4 ni hi bi,m N yl4 ψ¯ i,k,l hi bi,m N d2 yl4 ψ¯ i,k,l ( yl ) = (yl ). 16 16 k=1 l=1 d=1 k=1 l=1

n

+

j

n

N

d

N

(41)

So,   1 hi 2 ˜ ˙ˆ hi 2 ˜ ˙ˆ LVi ≤ hi bi,m zi,3 1 gi,1 αi,1 + i,1 − bi,m βi βi − bi,m θi θi + hi3 b2i,m zi,4 1 Ki,1 Ki,2 2    n i −1 1 1 3 2 4 3 zi, j gi, j αi, j + gi, j−1 zi, j + i, j + hi bi,m zi, j + hi bi,m 4 2 j=2   ni  1 1 3 2 4 3 2 +hi bi,m zi,n g + u + g z +  h b z + h (ni + 1 − k )i,k i,n i i,n −1 i,n i,n i i i i i i i,m i,n i i 4 2 k=1 ni  ni d  N d−1  N    1 1 4¯4 4 + hi bi,m N yl hi,k,l (yl ) + hi bi,m N yl4 ψ¯ i,k,l (d − 1 ) ( yl ) 4 4 d=1 k=1 l=1 d=2 k=1 l=1 i   3 4 hi bi,m N d2 yl4 ψ¯ i,k,l (yl ). 16 d=1 k=1 l=1

n

+

d

N

Now, the following auxiliary functions are defined for the analysis of disturbance attenuation.   2 Hi = LVi + hi2 zi,1 − γi2 i 2 ⎛ ⎞ ni  = LVi + hi2 ⎝zi,2 1 − γi2 i,2 j ⎠, j=1

 T where i = i,1 , i,2 , . . . , i,ni .

H. Liu, X. Li and X. Liu et al. / Journal of the Franklin Institute 356 (2019) 8049–8079

8063

Furthermore, the Lyapunov function of the whole large-scale nonlinear system is defined as V (Z ) =

N 

Vi ,

(42)

i=1

where Z = [Z1 , Z2 , . . . , ZN ]T . Then, LV =

N 

LVi .

(43)

i=1

In the procedure of computing Eq. (43), the some interconnected items will appear for the whole large-scale system. According to the equalities (39)–(41), the interconnected items can be handled as follows [21]. i  l  1  1  4 4 N hi bi,m yl4 h¯ i,k,l hl bl,m zi,4 1 h¯ l,k,i ( yl ) = N (yi ), 4 i=1 d=1 k=1 l=1 4 i=1 l=1 d=1 k=1

n

N

d

N

N

N

n

d

ni nl N d−1  N N N d−1   1  1  4 4 N hi bi,m yl4 ψ¯ i,k,l hl bl,m zi,4 1 ψ¯ l,k,i (d − 1 ) ( yl ) = N (d − 1 ) (yi ), 4 i=1 d=2 4 i=1 l=1 d=2 k=1 l=1 k=1 i l   3  3  4 4 N d2 hi bi,m yl4 ψ¯ i,k,l d2 hl bl,m zi,4 1 ψ¯ l,k,i ( yl ) = N (yi ). 16 i=1 d=1 k=1 l=1 16 i=1 l=1 d=1 k=1

N

n

d

N

N

N

n

d

Therefore,  N    1 3 2 4 hi 2 ˜ ˙ˆ hi 2 ˜ ˙ˆ 3 hi bi,m zi,1 gi,1 αi,1 + i,1 − LV ≤ b βi βi − b θi θi + hi bi,m zi,1 Ki,1 i,m Ki,2 i,m 2 i=1   N n N i −1    1   3 hi bi,m zi,3 j gi, j αi, j + i, j + hi3 b2i,m zi,4 j + hi bi,m zi,n + gi,ni ui + i,ni i 2 i=1 j=2 i=1   ni N  1 3 2 4 2 + hi bi,m zi,ni + hi (ni + 1 − k )i,k , 2 i=1 k=1

where l   1 4 N hl bl,m zi,1 h¯ l,k,i ( yi ) 4hi bi,m l=1 d=1 k=1

N

i,1 =i,1 +

n

d

nl N  d−1   1 4 + N hl bl,m zi,1 ψ¯ l,k,i (d − 1 ) ( yi ) 4hi bi,m l=1 d=2 k=1 l   3 4 + N d2 hl bl,m zi,1 ψ¯ l,k,i (yi ), 16hi bi,m l=1 d=1 k=1

N

n

d

(44)

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i, j =

1 gi, j−1 zi, j + i, j , 4

1 gi,n −1 zi,ni + i,ni . 4 i Define the following auxiliary function

i,ni =

H=

N 

Hi = LV +

i=1

N 

⎛ ⎞ ni N     2 hi2 zi,1 − γi2 i 2 = LV + hi2 ⎝zi,2 1 − γi2 i,2 j ⎠.

i=1

i=1

(45)

j=1

According to Lemma 1, one has hi2 zi,2 1 ≤

1 4 4 h z + Mi . 4Mi i i,1

(46)

Remark 2. Here, Mi > 0 is an arbitrarily small constant. The purpose of choosing the constant Mi is to make the constant terms appearing in the stability analysis be small enough. Because Mi is not used to construct the controller, so its value is not required to be known. Substituting Eqs. (44) and (46) into Eq. (45), produces H≤

   3   1 1 1 ˙ ˙ zi,1 gi,1 αi,1 + f¯i,1 Zi,1 − hi b2i,m β˜i βˆi − hi b2i,m θ˜i θˆi + hi3 b2i,m zi,4 1 Ki,1 Ki,2 2 i=1   N n i −1       ∂αi, j−1 ˙ hi bi,m zi,3 j gi, j αi, j + f¯i, j Zi, j + hi bi,m zi,3 j ξi, j Zi, j − + βˆi ∂ βˆi

N 

hi bi,m

i=1 j=2

   N     ∂αi, j−1 ˙ˆ 1 3 hi bi,m zi,n gi,ni ui + f¯i,ni Zi,ni θi + hi3 b2i,m zi,4 j + i 2 ∂ θˆi i=1     N   ∂αi,ni −1 ˙ 1 3 2 4 ∂αi,ni −1 ˙ˆ 3 ˆ +hi bi,m zi,ni ξi,ni Zi,ni − βi − θi + hi bi,m zi,ni + Mi 2 ∂ βˆi ∂ θˆi −

i=1

ni N     2 + hi ni + 1 − k − hi γi2 i,k ,

(47)

i=1 k=1

where   f¯i,1 Zi,1 = i,1 +

hi3 zi,1 , 4Mi bi,m

    f¯i, j Zi, j = i, j − ξi, j Zi, j ,     f¯i,ni Zi,ni = i,ni − ξi,ni Zi,ni . where Zi,1 =zi,1 , Zi, j = [zi,1 , zi,2 , . . . , zi, j , βˆi , θˆi ]T , Zi,ni = [zi,1 , zi,2 , . . . , zi,ni , βˆi , θˆi ]T . ξ i,j (Zi,j ) and ξi,ni Zi,ni are the auxiliary functions, which are introduced here to prove the inequality (57) and will be defined later. Since the smooth functions gi,j , fi,j , hi, j (y¯),ϕ i,j , ψi, j (y¯) are unknown, the RBF neural networks are employed to approximate the unknown packaged

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8065

functions f¯i, j (Zi, j ) (see [7] and [9]) when the decentralized controllers are designed, j = 1, 2, . . . , ni . According to Eq. (7), one has f¯i, j (Zi, j ) = W T Si, j (Zi, j ) + δi, j (Zi, j ), i, j

Combining Lemma 1 and Eq. (10) with Eq. (11) gives   hi bi,m zi,3 j f¯i, j = hi bi,m zi,3 j Wi,Tj Si, j + δi, j ≤

2 hi b2i,m 6 2 hi b2i,m 6 hi ai, j,1 hi ai, j,2 zi, j Wi, j Si,T j Si, j + + zi, j δi, j max + 2ai, j,1 2 2ai, j,2 2

≤

2 1 hi  1 hi b2i,m zi,6 j βi Si,T j Si, j + hi b2i,m zi,6 j θi + ai, j,k . 2ai, j,1 2ai, j,2 2 k=1

(48)

It follows from Eqs. (47) and (48) that N   1 1 hi 2 ˜ ˙ˆ hi bi,m zi,3 1 gi,1 αi,1 + H≤ hi b2i,m zi,6 1 βi Si,T1 Si,1 + hi b2i,m zi,6 1 θi − b βi βi 2ai,1,1 2ai,1,2 Ki,1 i,m i=1   N n i −1  1 hi 2 ˜ ˙ˆ 1 3 2 4 hi bi,m zi,3 j gi, j αi, j + − b θi θi + hi bi,m zi,1 + hi b2i,m zi,6 j βi Si,T j Si, j Ki,2 i,m 2 2a i, j, 1 i=1 j=2      ∂αi, j−1 ˙ ∂αi, j−1 ˙ˆ 1 1 3 2 4 2 6 3 ˆ + hi b z θi + hi bi,m zi, j ξi, j Zi, j − βi − θi + hi bi,m zi, j 2ai, j,2 i,m i, j 2 ∂ βˆi ∂ θˆi  N  1 1 3 6 6 hi bi,m zi,n g u + hi b2i,m zi,n β ST S + hi b2i,m zi,n θ + i i,ni i i i i,ni i,ni i i 2a 2a i,n , 1 i,n , 2 i i i=1      ∂αi,ni −1 ˙ 1 3 2 4 ∂αi,ni −1 ˙ˆ 3 ˆ +hi bi,m zi,ni ξi,ni Zi,ni − βi − θi + hi bi,m zi,ni 2 ∂ βˆi ∂ θˆi ⎛ ⎞ n n N 2 N  i  i      2 ⎝M i + h i + ai, j,k ⎠ + hi ni + 1 − k − hi γi2 i,k . (49) 2 i=1 j=1 k=1 i=1 k=1 Because β i and θ i are not available, the estimation values βˆi and θˆi are used to replace β i and θ i . βi = βˆi + β˜i , θi = θˆi + θ˜i . Therefore, according to Eq. (49), the virtual control laws, true control inputs and adaptive laws can be obtained as follows.   1 3 ˆ T 1 3 ˆ 1 2 αi,1 = − z βi S Si,1 − z θi − h + ci,1 zi,1 , (50) 2ai,1,1 i,1 i,1 2ai,1,2 i,1 2 i αi, j

1 3 ˆ T 1 3 ˆ =− zi, j βi Si, j Si, j − z θi − 2ai, j,1 2ai, j,2 i, j

ui = − ˙ βˆi =

1 2ai,ni ,1

3 ˆ T zi,n βS S i i i,ni i,ni

−

1 2ai,ni ,2

ni  Ki,1 6 T zi,k Si,k Si,k − bi βˆi , 2a i,k, 1 k=1

3 ˆ zi,n θ i i



 1 2 h + ci, j zi, j , 2 i 

−

 1 2 h + ci,ni zi,ni , 2 i

(51)

(52)

(53)

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H. Liu, X. Li and X. Liu et al. / Journal of the Franklin Institute 356 (2019) 8049–8079

10 -3

20

Without disturbances With disturbances

System Output

1

15

0.02 10 0.01 0

5

-0.01 0

0.05

0.1

0

-5 0

0.5

1

1.5

2

2.5

3

3.5

4

t/s Fig. 1. The comparison on output signal y1.

i  Ki,2 6 ˙ θˆi = z − di θˆi , 2ai,k,2 i,k k=1

n

(54)

where j = 2, 3, . . . , ni − 1, ci,1 , ci, j , ci,ni , bi , di are positive design parameters. Remark 3. According to the initial condition βˆi (0 ) ≥ 0, θˆi (0 ) ≥ 0, Eqs. (53) and (54), it is easy to prove that βˆi (t ) ≥ 0 and θˆi (t ) ≥ 0 hold. The two results have been used in this section. Substituting Eqs. (50)–(54) into Eq. (49) yields  1 1 bi β˜i βˆi + di θ˜i θˆi Ki,1 Ki,2 i=1 j=1 i=1 ⎛ ⎞ ni ni  N  N 2      2 ⎝M i + h i + hi ni + 1 − k − hi γi2 i,k + ai, j,k ⎠

H ≤−

ni N  

i=1 k=1

+

N 



hi b2i,m

i=1

j=1 k=1

    ∂αi, j−1 ˙ ∂αi, j−1 ˙ˆ hi bi,m zi,3 j ξi, j Zi, j − βˆi − θi . ∂ βˆi ∂ θˆi j=2

ni N   i=1

hi b2i,m ci, j zi,4 j +

(55)

H. Liu, X. Li and X. Liu et al. / Journal of the Franklin Institute 356 (2019) 8049–8079

8067

0.06 Without disturbances With disturbances

0.05 0.04

System Output

2

0.06 0.03

0.04 0.02

0.02

0 0.01 -0.02 0

0.05

0.1

0 -0.01 -0.02 0

0.5

1

1.5

2

2.5

3

3.5

t/s Fig. 2. The comparison on output signal y2.

From Eqs. (53) and (54), the fifth item in Eq. (55) can be processed as follows.   ∂αi, j−1 ˙ˆ ∂αi, j−1 ˙ˆ hi bi,m zi,3 j − βi − θi ∂ βˆi ∂ θˆi i=1 j=2   j−1 ni N    Ki,1 3 ∂αi, j−1 ˆ 3 ∂αi, j−1 ˆ 3 ∂αi, j−1 6 T = hi bi,m bi zi, j βi + di zi, j θi − zi, j zi,k Si,k Si,k ˆi ˆi ˆi 2a i,k, 1 ∂ β ∂ θ ∂ β i=1 j=2 k=1 ⎞ ⎛ ⎞⎞ j−1 ni ni    ∂α Ki,1 6 T K K i, j−1 i,2 i,2 6 6 ⎠⎠ ⎝ + zi,k Si,k Si,k ⎠ −zi,3 j zi,k + zi,k ˆi 2a 2a 2a i,k, 1 i,k, 2 i,k, 2 ∂ θ k= j k=1 k= j

ni N  

≤

ni N   i=1 j=2

 hi bi,m

∂αi, j−1 ˆ bi zi,3 j βi ∂ βˆi

+

∂αi, j−1 ˆ di zi,3 j θi ∂ θˆi

−

j−1  3 ∂αi, j−1 zi, j ∂ βˆi k=1

j

j−1  

3 ∂αi,k−1

Ki,1 6 T Ki,2 6

− z3 ∂αi, j−1 z + zi, j Si, j Si, j zi,k i,k i, j



ˆ ˆ 2ai, j,1 ∂ βi ∂ θi k=1 2ai,k,2 k=2

 j

Ki,2 6 

3 ∂αi,k−1

z + z 2ai, j,2 i, j k=2 i,k ∂ θˆi

Ki,1 6 T z S Si,k 2ai,k,1 i,k i,k

4

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0.1

Without disturbances With disturbances

0 -0.1

System State

12

0.2 -0.2 0 -0.3

-0.2

-0.4

-0.4

-0.5

-0.6

-0.6

-0.8 0

0.02

0.04

0.06

0.08

0.1

-0.7 -0.8 0

0.5

1

1.5

2

2.5

3

3.5

4

t/s Fig. 3. The comparison on state x1,2.

ni N  



∂αi, j−1 ˆ ∂αi, j−1 ˆ ∂αi, j−1  Ki,1 6 T ≤− −bi βi − di θi + zi,k Si,k Si,k ∂ βˆi ∂ θˆi ∂ βˆi k=1 2ai,k,1 i=1 j=2

 j−1 j

j

∂αi, j−1  Ki,2 6 Ki,1 3 2 

3 ∂αi,k−1

Ki,2 3 

3 ∂αi,k−1

z − z + zi,k − z s z 2ai, j,1 i, j k=2 i,k ∂ βˆi 2ai, j,2 i, j k=2 i,k ∂ θˆi

∂ θˆi k=1 2ai,k,2 =−

ni N  

j−1

hi bi,m zi,3 j

  hi bi,m zi,3 j ξi, j Zi, j .

i=1 j=2

According to Eq. (56), the auxiliary functions can be designed as follows. ∂αi, j−1 ˆ ∂αi, j−1  Ki,1 6 T = − bi βi + zi,k Si,k Si,k ∂ βˆi ∂ βˆi k=1 2ai,k,1

j

∂αi, j−1 ˆ Ki,1 3 2 

3 ∂αi,k−1

z − di − z s θi 2ai, j,1 i, j k=2 i,k ∂ βˆi

∂ θˆi

j−1 j

∂αi, j−1  Ki,2 6 Ki,2 3 

3 ∂αi,k−1

z , + zi,k − z 2ai, j,2 i, j k=2 i,k ∂ θˆi

∂ θˆi k=1 2ai,k,2 j−1

ξi, j

(56)

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8069

0.4 Without disturbances With disturbances

0.2

System State

22

0 -0.2 0.5 -0.4 0 -0.6

-0.5

-0.8

-1

-1

-1.5 0

0.02

0.04

0.06

0.08

0.1

-1.2 -1.4 0

0.5

1

1.5

2

2.5

3

3.5

4

t/s Fig. 4. The comparison on state x2,2. n −1

i ∂αi,ni −1 ˆ ∂αi,ni −1  Ki,1 6 T βi + zi,k Si,k Si,k ∂ βˆi ∂ βˆi k=1 2ai,k,1

ni

3 ∂αi,k−1

Ki,1 3 2 

− di ∂αi,ni −1 θˆi z − z s 2ai,ni ,1 i,ni k=2 i,k ∂ βˆi

∂ θˆi

ni −1 ni

3 ∂αi,k−1

∂αi,ni −1  Ki,2 6 Ki,2 3 

z

, + zi,k − zi,ni

i,k

2ai,ni ,2 ∂ θˆi k=1 2ai,k,2 ∂ θˆi k=2

ξi,ni = − bi

where s has been defined in Lemma 5. From Eq. (56), the following inequality can be proved easily.   ni N     ∂αi, j−1 ˙ ∂αi, j−1 ˙ˆ 3 ˆ hi bi,m zi, j ξi, j Zi, j − βi − θi ≤ 0. (57) ∂ βˆi ∂ θˆi i=1 j=2 The following inequality can be obtained by using Eqs. (55) and (57).   ni N  N   1 1 2 4 2 ˜ ˆ ˜ ˆ H ≤− hi bi,m ci, j zi, j + hi bi,m bi βi βi + di θi θi Ki,1 Ki,2 i=1 j=1 i=1 ⎛ ⎞ ni ni  N  N 2      h 2 ⎝Mi + i + hi ni + 1 − k − hi γi2 i,k + ai, j,k ⎠. 2 j=1 k=1 i=1 k=1 i=1

(58)

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10 Without disturbances With disturbances

0 20

Control signal

1

-10

0 -20

-20 -40

-30

-60 0

-40

0.05

0.1

0.15

0.2

2

2.5

-50

-60 0

0.5

1

1.5

3

3.5

4

t/s Fig. 5. The comparison on control signal u1.

Substituting Eq. (13) into Eq. (55) yields ⎛ ⎞ ni ni  N  N 2    h i ⎝Mi + H ≤− hi b2i,m ci, j zi,4 j + ai, j,k ⎠ 2 i=1 j=1 i=1 j=1 k=1 +

N  i=1

 hi b2i,m

 1 1 bi β˜i βˆi + di θ˜i θˆi . Ki,1 Ki,2

It follows from Lemma 1 and Eq. (59) that   1 1 hi b2i,m bi β˜i βˆi = hi b2i,m bi β˜i βi − β˜i Ki,1 Ki,1 1 1 =− hi b2i,m bi β˜i2 + hi b2 bi β˜i βi Ki,1 Ki,1 i,m 1 1 1 ≤− hi b2i,m bi β˜i2 + hi b2i,m bi β˜i2 + hi b2 bi β 2 Ki,1 2Ki,1 2Ki,1 i,m i 1 1 =− hi b2 bi β˜ 2 + hi b2 bi β 2 . 2Ki,1 i,m i 2Ki,1 i,m i

(59)

(60)

Similarly, 1 1 1 hi b2 di θ˜i θˆi ≤ − hi b2 di θ˜2 + hi b2 di θ 2 . Ki,1 i,m 2Ki,2 i,m i 2Ki,2 i,m i

(61)

H. Liu, X. Li and X. Liu et al. / Journal of the Franklin Institute 356 (2019) 8049–8079

With the help of Eqs. (59)–(61), one has ⎛ ⎞   ni N N    1 1 ⎝ H ≤− hi b2i,m ci, j zi,4 j + hi b2i,m bi β˜i2 + hi b2i,m di θ˜i2 ⎠ + Bi , 2Ki,1 2Ki,2 i=1 j=1 i=1

8071

(62)

( i (2 where Bi = 2K1i,1 hi b2i,m bi βi2 + 2K1i,2 hi b2i,m di θi2 + Mi + h2i nj=1 k=1 ai, j,k . Combining Eq. (45) with Eqs. (62), (63) can be verified ⎛ ⎞   ni N N    1 1 2 4 2 2 2 2 ⎠ ˜ ˜ ⎝ LV ≤ − hi bi,m ci, j zi, j + hi b bi β + hi b di θ + Bi 2Ki,1 i,m i 2Ki,2 i,m i i=1 j=1 i=1 +

N 

hi2 γi2 i 2 −

i=1

N 

hi2 zi,2 1 .

(63)

i=1

From Eq. (63), we know ⎛ ⎞   ni N   1 1 ⎝ LV ≤ − hi b2i,m ci, j zi,4 j + hi b2 bi β˜ 2 + hi b2 di θ˜2 ⎠ 2Ki,1 i,m i 2Ki,2 i,m i i=1 j=1 +

N 

Bi +

i=1

N 

hi2 γi2 i 2 .

(64)

i=1

Considering the stability analysis of the stochastic nonlinear system (1), let i j (t ) = 0, for i = 1, 2, . . . , N, j = 1, 2, . . . , ni , which implies that i  = 0.

(65)

It follows from Eqs. (64) and (65) that ⎛ ⎞   ni N N    1 1 ⎝ LV ≤ − hi b2i,m ci, j zi,4 j + hi b2i,m bi β˜i2 + hi b2i,m di θ˜i2 ⎠ + Bi 2Ki,1 2Ki,2 i=1 j=1 i=1 ≤ −aV + b.

(66) (N

where a = min{4bi,m ci, j , bi , di , i = 1, 2, . . . , N, j = 1, 2,. . . , ni }, b = i=1 (Bi + ami ). Therefore, from Lemma 3, all the signals in the closed-loop large-scale stochastic nonlinear system are bounded in the sense of probability. It is obvious that Assumption 1 is satisfied with ψi, j = 0. Therefore, Eq. (66) is true no matter if there are interconnections or not between the subsystems, which implies that the system (1) is connectively stable. Furthermore, according to the conclusion in [22], Eq. (66) can be written as dE [V (Z (t ) )] ≤ −aE [V (Z (t ) )] + b. dt It satisfies   b −at b e + , 0 ≤ E [V (Z (t ) )] ≤ E [V (Z (0 ) )] − a a

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100 Without disturbances With disturbances

Control signal

2

50

0 100 -50 0 -100

-100

-200

-150

0

0.05

0.1

-200 0

0.5

1

1.5

2

2.5

3

3.5

4

t/s Fig. 6. The comparison on control signal u2.

which implies that   b¯ 0 ≤ E V (Z (t ) ) ≤ E V (Z (0 ) ) e−at + , a (N (N where b¯ = i=1 Bi , V (Z (t ) ) = i=1 ( 2K1i,1 hi b2i,m β˜i2 + Furthermore, from Eq. (67), one has

(67) 1 h b2 θ˜2 2Ki,2 i i,m i

+

1 4

(n i j=1

hi bi,m zi,4 j ).

 b¯ 0 ≤ E V (Z (t ) ) ≤ , t → ∞. a Remark 4. According to Eq. (67), it is obvious that the error variables zi,1 , zi,2 , . . . , zi,ni , β˜i , θ˜i are bounded by a positive number, which is independent of the constant mi mentioned in Remark 1. At present, the stability proof of the stochastic system has been completed. Next, H∞ performance will be proved. Especially, formula (62) is obtained under the conditions of Eqs. (50)–(54). From Eq. (62), the following result can be obtained easily. H≤

N  i=1

Bi ≤

N 

ρiVi (Zi ).

i=1

which ρ i is an unknown positive constant.

(68)

H. Liu, X. Li and X. Liu et al. / Journal of the Franklin Institute 356 (2019) 8049–8079

8073

0.25 Without disturbances With disturbances

The estimate value of

1

0.2

0.15

10 -4

6 0.1

4 2

0.05 0 3

3.5

4

0 0

0.5

1

1.5

2

2.5

3

3.5

4

t/s Fig. 7. The comparison on the adaptive parameter βˆ1 .

Then, Eq. (69) can be obtained due to Eqs. (63) and (68). LV (Z ) ≤

N 

ρiVi (Zi ) +

i=1

N 

hi2 (γi2 i 2 − zi,2 1 ).

(69)

i=1

With the help of Eqs. (38) and (42), one has V (Z ) > 0.

(70)

Taking the integral over time from 0 to t and the mathematical expectation for both sides of the inequality (69), and with the help of Eq. (43), we can obtain  t   N N N   E [Vi (Zi (t ) )] ≤ E ρiVi (Zi (s ) )ds + E [ψi (t )], (71) i=1

where

0

i=1



t

ψi (t ) = 0

i=1

2 hi2 (γi2 i (s )2 − zi,1 (s ) )ds + [Vi (Zi (0 ) )].

According to Corollary 2, it can be deduced that N N  t N    ρi (t−s ) E [Vi (Zi (t ) )] ≤ ρi e E [ψi (s )]ds + E [ψi (t )]. i=1

i=1

0

i=1

(72)

(73)

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0.4 Without disturbances With disturbances

0.35

The estimate value of

2

0.3 0.25 0.2

10 -3

1

0.15 0.5 0.1 0.05

0 3

3.5

4

0 0

0.5

1

1.5

2

2.5

3

3.5

4

t/s Fig. 8. The comparison on the adaptive parameter βˆ2 .

Next, the method of reduction-to-absurdity is adopted to prove make an assumption N 

E [ψi (t )] ≤ 0,

(N i=1

E [ψi (t )] > 0. First, let’s

(74)

i=1

then, the following inequality holds N N  t   E [Vi (Zi (t ) )] ≤ ρi eρi (t−s ) E [ψi (s )]ds ≤ 0. i=1

i=1

(75)

0

However, there is a contradiction between Eqs. (70) and (75), so the assumption made in ( Eq. (74) is unreasonable. Therefore, Ni=1 E [ψi (t )] > 0. Then, the following inequality can be obtained according to Eq. (72).  t    t   N N N  2 2 2 2 2 E hi zi (s ) ds < E hi γi i (s ) ds + E [Vi (Zi (0 ) )]. (76) i=1

0

i=1

0

i=1

According to Eq. (13), it is obvious to know hi > 1. So the following result can be obtained.  t    t     N N N  1 2 2 2 zi (s ) ds < E E γi i (s ) ds + E 2 V (Z (0 ) ) , hi 0 0 i=1 i=1 i=1

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8075

0.25 Without disturbances With disturbances

The estimate value of

1

0.2

0.15 10 -4

6 0.1

4

2 0.05 0 3

3.5

4

0 0

0.5

1

1.5

2

2.5

3

3.5

4

t/s Fig. 9. The comparison on the adaptive parameter θˆ1 .

then,  t     t   N N N  2 2 2 zi (s ) ds < i (s ) ds + E E γi E [Vi (x (0 ) )], i=1

0

i=1

0

(77)

i=1

Eq. (77) implies that the large-scale system satisfies H∞ performance in Definition 2. Therefore, the following theorem can be obtained. Theorem 1. Consider the large-scale stochastic nonlinear system (1), which meets Assumptions 1 and 2. If the virtual control laws, the actual control law and the adaptive laws of every subsystem are chosen according to Eqs. (50)–(54), then all the signals in the closed-loop large-scale stochastic system are uniformly ultimately bounded in probability. The designed decentralized bounded-H∞ adaptive neural controllers are able to attenuate the effect of external disturbances. The large-scale stochastic system has bounded-H∞ performance.

4. Simulation study In this section, a numerical example is given to show the effectiveness of the proposed control scheme. The following large-scale stochastic strict-feedback nonlinear system in [21] is considered. Additionally, the external disturbances are added to the model. It is described as

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0.2 Without disturbances With disturbances

0.18

2

0.16

The estimate value of

0.14 0.12

10 -4

6 0.1 4

0.08 0.06

2

0.04 0 3

0.02

3.5

4

0 0

0.5

1

1.5

2

2.5

3

3.5

4

t/s Fig. 10. The comparison on the adaptive parameter θˆ2 .

follows.         ⎧ y1 y2 dx1,1 = 1 + sin x1,1 x1,2 + x12,1 sin x1,1 + y22 + y1 y2 + x1,1 sin x1,1 1,1 d t + 1+ d w1 , ⎪ y22 ⎪        ⎪ 2 2 ⎪ ⎪ 2+ x u x  d 1 dx = cos x + x x + y y + x x sin t +y ln + y 1,2 1,1 1,2 1 1,1 1,2 1 2 1,1 1,2 1,2 1,2 2 ⎪ 1 d w1 , ⎪ ⎨ y1 = x1,1 ,      2 2 2 dx2,1 =  3 + sin (x2,1 ) x2,2 + 2x2, t + y1 cos (y2 )d w2 , 2,1 d ⎪ 1 + y1 + y1 y2 + x 2,1 cos x2,1   ⎪ 2     ⎪ x2,1 ⎪ −x2,1 x2,2 2 ⎪ 1 2  u x + dx = + e + x sin y y + + 2 +x 2 2 2,2 2,1 1 2 2,2 d t + y2 sin (y1 )d w2 , ⎪ x2, ⎪ 2,2 1 2,2 ⎩ y2 = x2,1 . (78) It is obvious that the interconnections in the two subsystems satisfy the condition in Assumption 1. The H∞ control laws and adaptive laws can be designed according to Theorem 1. In the simulation, the initial conditions are given by [x1,1 (0), x1,2 (0), x2,1 (0), x2,2 (0)]T = [0.02, −0.05, 0.05, 0.04]T , and [β1 (0), β2 (0), θ1 (0), θ2 (0)]T = [0.2, 0.2, 0.2, 0.2]T . The external disturbances are chosen as 1,1 = 0.1 sin (t )e−t + 0.02 cos(2t )e−2t , 1,2 = 0.04 cos(t )e−0.5t , 2,1 = 0.1 sin (t )e−1.5t , 2,2 = 0.02 cos(2t )e−2t − 0.3 sin (3t )e−t . The design parameters are taken as follows: K1,1 = K1,2 = K2,1 = K2,2 = a1,1,1 = a1,1,2 = a1,2,1 = a1,2,2 = a2,1,1 = a2,1,2 = a2,2,1 = a2,2,2 = 1, c1,1 = c1,2 = 40, c2,1 = c2,2 = 50 , √ 2 b1 = b2 = d1 = d2 = 2. The disturbance attenuation constants can be chosen as γ1 = 2 (h1 = 5)and γ2 = 0.5 (h2 = 9). The four RBF neural networks are chosen as follows. W1T,1 S1,1 (Z1,1 ) contains 7 nodes, which are evenly distributed in interval [−3, 3] with the

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8077

width being taken as η1,1 = 2, W1T,2 S1,2 (Z1,2 ) contains 2401 nodes, which are evenly distributed in interval [−3, 3] × [−3, 3] × [−3, 3] × [−3, 3] with the width being taken as η1,2 = 2, W2,T1 S2,1 (Z2,1 ) contains 7 nodes, which are evenly distributed in interval [−3, 3] with the width being taken as η2,1 = 2, W2,T2 S2,2 (Z2,2 ) contains 2401 nodes, which are evenly distributed in interval [−3, 3] × [−3, 3] × [−3, 3] × [−3, 3] with the width being taken as η2,2 = 2. All simulation results are depicted in Figs. 1–10. In these figures, the simulation results with or without external disturbances for each subsystem are given at the same time in order to compare the disturbance attenuation effect. In the simulation results, Figs. 1–4 denote the states (including the outputs y1 , y2 ) of the first subsystem and the second subsystem respectively. They show that all the state signals of each closed-loop stochastic subsystem can converge to a small area around the origin in probability, and each subsystem has a better stabilization performance no matter whether the external disturbances exist. Figs. 5–10 give the curves of the control signals u1 , u2 , the estimations β 1 , β 2 and the estimations θ 1 , θ 2 in the same way, respectively. They illustrate that all the signals in the closed-loop system are bounded in probability, and the external disturbances are attenuated by H∞ performance. Therefore, the effectiveness of the proposed method is verified. 5. Conclusion In this paper, a novel adaptive neural control design scheme has been proposed to design decentralized bounded-H∞ adaptive neural controllers for a class of strict-feedback large-scale stochastic nonlinear systems with Itô-type, in which each subsystem is influenced by external disturbances. The decentralized bounded-H∞ adaptive neural controller of each subsystem is designed by combining backstepping technique, Lyapunov theory, decentralized control method, RBF neural networks with H∞ performance criterion. The designed controllers guarantee that all the signals in the closed-loop stochastic system are uniformly ultimately bounded in probability. The effect of external disturbances in each closed-loop stochastic subsystem can be attenuated by H∞ performance. Specially, the assumption that the neural network approximation error is square-integrable has been eliminated in the design process. A numerical example is given to illustrate the effectiveness and feasibility of the proposed control scheme. The research in this paper is of significant theoretical value for robust control design of large-scale stochastic nonlinear systems. Acknowledgment This work is supported by the National Natural Science Foundation of China (no. 61403177), Natural Science Foundation of Liaoning province (no. 20180550319), and Natural Sciences and Engineering Research Council of Canada, as well as Graduate Education Reform and Science Technology Innovation Entrepreneurship Project of University of Science and Technology Liaoning (no. LKDYC201701). References [1] D.W.C. Ho, Y. Niu, Robust fuzzy design for nonlinear uncertain stochastic systems via sliding-mode control, IEEE Trans. Fuzzy Syst. 15 (3) (2007) 350–358. [2] H. Wang, Q. Zhu, Finite-time stabilization of high-order stochastic nonlinear systems in strict-feedback form, Automatica 54 (2015) 284–291.

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