Adaptive decentralized NN control of large-scale stochastic nonlinear time-delay systems with unknown dead-zone inputs

Author's Accepted Manuscript

Adaptive decentralized NN control of largescale stochastic nonlinear time-delay systems with unknown dead-zone inputs Guozeng Cui, Zhen Wang, Guangming Zhuang, Ze Li, Yuming Chu

www.elsevier.com/locate/neucom

PII: DOI: Reference:

S0925-2312(15)00093-4 http://dx.doi.org/10.1016/j.neucom.2015.01.048 NEUCOM15086

To appear in:

Neurocomputing

Received date: 6 October 2014 Revised date: 19 December 2014 Accepted date: 25 January 2015 Cite this article as: Guozeng Cui, Zhen Wang, Guangming Zhuang, Ze Li, Yuming Chu, Adaptive decentralized NN control of large-scale stochastic nonlinear time-delay systems with unknown dead-zone inputs, Neurocomputing, http://dx.doi.org/10.1016/j.neucom.2015.01.048 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

Adaptive decentralized NN control of large-scale stochastic nonlinear time-delay systems with unknown dead-zone inputs Guozeng Cuia , Zhen Wangb , Guangming Zhuang a,c, Ze Lid , Yuming Chu e a School

of Automation, Nanjing University of Science and Technology, Nanjing 210094, Jiangsu, P.R. China Corresponding author. Email address: [email protected] b College of Mathematics and System Science, Shandong University of Science and Technology, Qingdao 266590, Shandong, P.R. China c School of Mathematical Sciences, Liaocheng University, Liaocheng 252059, Shandong, P.R. China d School of Mechanical and Electrical Engineering, Suzhou University of Science and Technology, Suzhou 215000, Jiangsu, P.R. China e School of Science, Huzhou Teachers College, Huzhou 313000, Zhejiang, P.R. China

Abstract In this paper, the problem of adaptive decentralized neural network (NN) control for a class of large-scale stochastic nonlinear time-delay systems with unknown dead-zone inputs is investigated. Neural networks are utilized to approximate unknown nonlinear functions, and an adaptive decentralized controller is constructed by incorporating the minimal learning parameters algorithm into backstepping design procedure. It is proved that the proposed control scheme guarantees that all the signals in the closed-loop system are semi-globally uniformly ultimately bounded in probability. Finally, a numerical example is provided to demonstrate the effectiveness of the present results. Keywords: Adaptive decentralized control, Stochastic nonlinear systems, Neural network (NN), Time-varying delays, Dead-zones.

1. Introduction It is well known that large-scale nonlinear systems are often constituted by a serial of lower order subsystems with interactions. Due to the complexity and limitation of information exchange among subsystems, one of the effective methods is to design a decentralized controller using only local measurements in each subsystem. In recent years, the decentralized control for large-scale nonlinear systems has received considered attention, and many remarkable results have been reported in some literatures and and references therein [1–3]. Reference [4] introduced the backstepping technique, and now it has been widely applied to design decentralized controllers for large-scale nonlinear systems without satisfying the so-called matching condition. Moreover, some approximation-based adaptive neural network (NN) or adaptive fuzzy control schemes have been proposed to handle the structured uncertainties for large-scale nonlinear systems [5–7]. However, these results only focused on deterministic nonlinear systems. Compared with the deterministic case, the design of adaptive decentralized controller for large-scale stochastic nonlinear systems is more difficult owing to the appearance of higher-order Hessian term in the infinitesimal generator, and till now, few results are studied large-scale stochastic nonlinear systems [8–13]. In [9], an adaptive fuzzy decentralized state feedback control scheme was proposed for large-scale stochastic nonlinear systems and it guaranteed all the signals of closedloop systems were bounded in probability. Without considering the measurements of the states, an adaptive fuzzy decentralized control for a class of large-scale stochastic nonlinear systems was investigated on the basis of dynamic surface control technique in [10]. It should be pointed out that the aforementioned results have not considered time delays. In fact, time delays are unavoidable in practical systems. The existence of time delays is the main cause of instability and poor performance of a system [14–17]. Hence, the stability analysis and control design for nonlinear time-delay systems have been paid more and more attention recently. In general, there exist two approaches for solving nonlinear systems with time delays. One is based on the Lyapunov-Krasovskii functionals. The problem of approximation-based adaptive control has been studied for deterministic or stochastic nonlinear systems with time delays [18–23]. The other is the Lyapunov-Razumikhin functionals. The stability analysis and controller design are more easier than the LyapunovKrasovskii functional approach. However, most of existing results on the approximation-based adaptive control for Preprint submitted to Neurocomputing

February 10, 2015

nonlinear time-delay systems are obtained by using the Lyapunov-Krasovskii functional, and few works are reported on Lyapunov-Razumikhin functional [24–28]. Combined Lyapunov-Razumikhin functional and the backstepping technique, the problem of adaptive NN control for a class of stochastic nonlinear systems with time-varying delays was considered and obtained the result that the tracking error was regulated to a small neighborhood of the origin in [26–28]. Although much progress has been made in the approximation-based adaptive control for nonlinear time-delays (time-delays free) systems, the aforementioned control schemes did not consider the problem of non-smooth nonlinear inputs. In practical application, the actuator of control system is often subject to nonsmooth characteristics, such as dead-zone, hysteresis, backlash. These nonsmooth input nonlinearities severely restrict the system’s performance and cause the instability of controlled systems. More recently, dead-zone nonlinearity has been attracting significant attention, and a great number of results have been reported in the literatures [29–37]. In [31, 32], the author studied the decentralized adaptive control for large-scale nonlinear systems with unknown dead-zone inputs, in which all the signals in the interconnected closed-loop system were proved to be semi-globally uniformly bounded. Subsequently, some approximation-based adaptive control schemes for a class of stochastic nonlinear systems with unknown deadzone were proposed in [33, 34]. However, the controlled systems considered in [31, 32] are deterministic nonlinear systems, and the results are not straightforward to be extended to stochastic cases. References [33, 34] are uncertain single input single output (SISO) stochastic nonlinear systems, and they are invalid for large-scale stochastic nonlinear systems in the presence of time-delay and unknown interconnection terms. So far, to the best of our knowledge, the problem of adaptive decentralized NN control for large-scale stochastic nonlinear systems with time-varying delays and dead-zone inputs has not been investigated in the literature, which motivates the present study. This paper deals with the problem of adaptive decentralized NN control for a class of large-scale stochastic nonlinear systems with time-varying delay and unknown dead-zones. By combining Lyapunov-Razumikhin functional approach, minimal learning parameters algorithm and backstepping design technique, all the signals in the closedloop system can be shown to be semi-globally uniformly ultimately bounded in probability. The contributions of this paper are as follows. First, a novel adaptive decentralized NN control scheme is successfully proposed based on the Lyapunov-Razumikhin functional approach, which solves time-varying delay and unknown dead-zone inputs simultaneously. Second, the designed controller and adaptation tuning law have a straightforward structure, which are not related to the nodes of neural network. Therefore, it is not necessary to estimate the centers and widths of basis functions. In addition, the presented adaptive decentralize NN controller contains only one adaptive law for each subsystem by estimating the maximum of the optimal weight vectors’ Euclidean norm instead of the vectors themselves. This alleviates the computation complexity and makes the algorithm be implemented more easily. A simulation example is given to illustrate the effectiveness of the proposed control scheme. The remainder of this paper is organized as follows. Section 2 offers problem formulation and some preliminary results. The adaptive decentralized NN controller is designed and analyzed in Section 3. After that, a numerical simulation example is presented to show the effectiveness of the results in Section 4. Section 5 concludes the work. Notations. The following standard notations are used throughout this paper. R n denotes the real n-dimensional space. For a given vector or matrix X, X T denotes its transpose; T r{X} denotes its trace when X is square; ||X|| denotes the Euclidean norm of a vector X; C i denotes the set of all functions with continuous ith partial derivative. K denotes the set of all functions: R + → R+ , which are continuous, strictly increasing and vanishing at zero; K ∞ denotes the set of all functions which are of class K and unbounded. 2. Problem Formulation and Preliminaries 2.1. Problem formulation Consider a class of large-scale stochastic nonlinear systems composed of N subsystems with time-varying delays and dead-zone inputs, the ith subsystem has the following form ⎧ ⎪ dxi,1 = (gi,1 ( x¯i,1 )xi,2 + fi,1 ( x¯i,1 ) + qi,1 ( x¯i,1 (t − τi,1 (t))) + hi,1 (¯y))dt + ψTi,1 (yi )dwi , ⎪ ⎪ ⎪ ⎪ ⎪ dxi,2 = (gi,2 ( x¯i,2 )xi,3 + fi,2 ( x¯i,2 ) + qi,2 ( x¯i,2 (t − τi,2 (t))) + hi,2 (¯y))dt + ψTi,2 (yi )dwi , ⎪ ⎪ ⎪ ⎪ ⎨ .. (1) ⎪ . ⎪ ⎪ ⎪ ⎪ T ⎪ ⎪ dxi,ni = (gi,ni ( x¯i,ni )ui + fi,ni ( x¯i,ni ) + qi,ni ( x¯i,ni (t − τi,ni (t))) + hi,ni (¯y))dt + ψi,ni (yi )dwi , ⎪ ⎪ ⎪ ⎩ yi = xi,1 , xi (t) = φi (t), −τ ≤ t ≤ 0, 2

where x¯ i, j = [xi,1 , . . . , xi, j ]T ∈ R j , j = 1, 2, . . . , ni , xi = [xi,1 , . . . , xi,ni ]T ∈ Rni and yi ∈ R are the state vector of the system and output of the system, respectively. x¯ i, j (t − τi, j (t)) = [xi,1 (t − τi,1 (t)), . . . , xi, j (t − τi, j (t))]T ∈ R j represents delayed state variables, 1 ≤ i ≤ N, 1 ≤ j ≤ n i ; τi, j (t) : R+ → [0, τ] is uncertain time-varying delay; φ i (t) is the initial condition of state variables. w i is an ri -dimensional standard Wiener process. q i, j (xi, j (t − τi, j (t))) is unknown smooth nonlinear time delay function with q i, j (0) = 0. fi, j (·), gi, j (·): R j → R are the unknown smooth nonlinear functions with fi, j (0) = 0. hi, j (¯y) is the interconnection term between the ith subsystem and other subsystems with h i, j (0) = 0, and y¯ = [y1 , y2 , . . . , yN ]T . ψi, j (·) is the unknown smooth function with ψ i, j (0) = 0. Moreover, u i ∈ R denotes the output of the ith dead zone, which is given as ⎧ ⎪ mi,r (vi ), if vi ≥ bi,r , ⎪ ⎪ ⎨ 0, if bi,l < vi < bi,r , (2) ui = Di (vi ) = ⎪ ⎪ ⎪ ⎩ mi,l (vi ), if vi ≤ bi,l , where vi is the input of the ith dead zone, b i,l < 0, and bi,r > 0 are unknown parameters of the ith dead zone. Throughout this paper, the following assumptions are needed for the system (1). Assumption 1. [35]. The growth of the ith dead-zone’s function, m i,l and mi,r , are smooth, and there exist unknown positive constants kil0 , kil1 , kir0 , and kir1 such that 0 < kil0 ≤ mi,l (vi ) ≤ kil1 ,



∀vi ∈ (−∞, bi,l],

(3)



∀vi ∈ [bi,r , +∞),

(4)

0 < kir0 ≤ mi,r (vi ) ≤ kir1 ,



and βi0 ≤ min{kil0 , kir0 } is an known positive constant, where m i,l (vi ) =

dmi,l (z) dz |z=vi ,



and mi,r (vi ) =

dmi,r (z) dz |z=vi .

Based on Assumption 1, the dead-zone (2) can be rewritten as ui = Di (vi ) = KiT (t)Φi (t)vi + di (vi ),

(5)

where Φi (t) =

[ϕi,r (t), ϕi,l (t)]T , 

if vi (t) > bi,l , if vi (t) ≤ bi,l ,

(7)

1, 0,

if vi (t) < bi,r , if vi (t) ≥ bi,r ,

(8)

[Ki,r (vi (t)), Ki,l (vi (t))]T , 

(9)

ϕi,r (t) =  ϕi,l (t) = Ki (t) = Ki,r (vi (t)) =  Ki,l (vi (t)) = di (vi ) =

(6)

1, 0,

⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩

0,  mi,r (ξi,r (vi (t))),

if vi (t) ≤ bi,l , if bi,l < vi (t) < +∞,

(10)

0,  mi,l (ξi,l (vi (t))),

if vi (t) ≥ bi,r , if − ∞ < vi (t) < bi,r ,

(11)



−mi,r (ξi,r (vi ))bi,r ,  −[mi,r (ξi,r (vi )) + mi,l (ξi,l (vi ))]vi ,  −mi,l (ξi,l (vi ))bi,l , 

if if if

vi ≥ bi,r , bi,l < vi < bi,r , vi < bi,l ,

(12)

where ξi,l (v) ∈ (vi , bi,l ), if vi < bi,l ; ξi,l (v) ∈ (bi,l , vi ), if bi,l ≤ vi < bi,r ; ξi,r (v) ∈ (bi,r , vi ), if bi,r < vi ; ξi,r (v) ∈ (vi , bi,r ), if bi,l < vi ≤ bi,r ; and |di (vi )| ≤ p∗i , p∗i is an unknown positive constant with p ∗i = (kir1 + kil1 ) max{bi,r , −bi,l }. Assumption 2. The sign of gi, j ( x¯i, j ) is known and there exist unknown constants b, c such that for 1 ≤ i ≤ N, 1 ≤ j ≤ ni 0 < b ≤ gi, j ( x¯i, j ) ≤ c < ∞, 3

∀ x¯i, j ∈ R j .

(13)

Assumption 3. For the time-delay terms q i, j ( x¯i, j (t − τi, j (t)), there exist K∞ functions Qi j (·) such that for 1 ≤ i ≤ N, 1 ≤ j ≤ ni , |qi, j ( x¯i, j (t − τi, j (t))| ≤ Qi j (|| x¯i, j (t − τi, j (t))||).

(14)

Assumption 4. [8]. There exist unknown smooth functions h i, j,l (yl ), ψ¯ i, j (yi ) such that for 1 ≤ i ≤ N, 1 ≤ j ≤ n i , |hi, j (¯y)|2 |ψi, j (yi )|2

N 

≤

h2i, j,l(yl ),

l=1 y2i ψ¯ 2i, j (yi ),

≤

(15) (16)

and assume that h i, j,l (yl ), ψ¯ i, j (yi ) are unknown smooth functions and h i, j,l (0) = 0, l = 1, 2, . . . , N. Remark 1. Based on Assumption 4, the following inequality can be obtained |hi, j (¯y)|2 ≤

N 

y2l h¯ 2i, j,l(yl ),

(17)

l=1

where h¯ i, j,l (yl ) is unknown smooth function. 2.2. Neural networks In this paper, the following radial basis function (RBF) NN is used to approximate unknown continuous function f (Z) : Rq → R, fnn (Z) = W T S (Z),

(18)

where Z ∈ ΩZ ⊂ Rq is input vector and q denotes NN input dimension. W = [w 1 , w2 , . . . , wl ]T ∈ Rl represents the weight vector; l > 1 denotes NN node number. S (Z) = [s 1 (Z), s2 (Z), . . . , sl (Z)]T ∈ Rl is the basis function vector. si (Z) is chosen as si (Z) = exp[−

(Z − μi )T (Z − μi ) ], i = 1, 2, . . . , l, η2i

(19)

where μi = [μi1 , μi2 , . . . , μiq ]T is the center of the receptive field and η i is the width of the Gaussian function. For any T unknown continuous nonlinear function f (Z) defined over a compact set Ω Z ∈ Rq , there exits NN W ∗ S (Z) such that f (Z) = W ∗ S (Z) + δ(Z), ∀Z ∈ ΩZ ⊂ Rq , T

(20)

where W ∗ is the ideal weight vector and δ(Z) is the approximation error. The ideal weight vector W ∗ is an artificial quantity required for analytical purposes. There exist ideal constant weights W ∗ defined as W ∗ := arg minl { sup | f (Z) − W T S (Z)|}, W∈R Z∈ΩZ

such that |δ(Z)| ≤ ε with ε > 0, for all Z ∈ Ω Z . 2.3. Preliminary results Consider the following stochastic nonlinear time-delay system dx =

f (t, x(t), x(t − τ(t)))dt + g(t, x(t))dw,

(21)

with initial condition {x(s) : −τ ≤ s ≤ 0} = ξ ∈ C Fb 0 × ([−τ, 0], Rn), where τ(t) : R + → [0, τ] is a Borel measurable function; x ∈ R n denotes the state variable and x(t − τ(t)) is the state vectors with time-delay; w is an r-dimensional standard Wiener process defined on the complete probability space (Ω, F, {F t }t≥0 , P) with Ω being a sample space, F being a σ-field, {F t }t≥0 being a filtration, and P being a probability measure. f (·), g(·) are locally lipschitz functions. 4

Definition 1. Define the infinitesimal generator L of function V(t, x) ∈ C 1,2 ([−τ, ∞] × Rn ) for stochastic nonlinear time-delay system (21) as follows LV(t, x) =

1 ∂V ∂V ∂2 V + f (t, x, x(t − τ(t))) + T r{gT (t, x) 2 g(t, x)}. ∂t ∂x 2 ∂x

(22)

Definition 2. [26]. Let p ≥ 1, the solution {x(t), t ≥ 0} of the stochastic nonlinear time-delay system (21) with initial condition ξ ∈ Ω 0 (Ω0 is some compact set which include the origin) is said to be p-Moment semi-globally uniformly ¯ the following equality holds ultimately bounded if there exists a constant d, ¯ ∀t ≥ T, E{||x(t, ξ)|| p } ≤ d,

for some T ≥ 0.

Lemma 1. (Razumikhin Lemma [26]) Let p ≥ 1, consider the stochastic nonlinear time-delay system (21), if there exist function V(t, x) ∈ C 1,2 ([−τ, ∞] × Rn ) and some positive constants c 1 , c2 , μ1 , μ2 , q > 1 satisfying the following inequalities c1 |x| p ≤ V(t, x) ≤ c2 |x| p ,

t ≥ −τ,

EV(t + s, x(t + s)) < qEV(t, x(t)),

x ∈ Rn ,

(23)

∀s ∈ [−τ, 0],

(24)

for all t ≥ 0 such that ELV(t, x) ≤ −μ1 V(t, x) + μ2 .

(25)

Then the solution x(t, ξ) of system (1) is p-Moment uniformly ultimately bounded. Lemma 2. [38]. For any η ∈ R and σ > 0, the following inequality holds η 0 ≤ |η| − η tanh( ) ≤ δσ, σ

(26)

with δ being a constant which satisfies δ = e −(δ+1) ; i.e., δ = 0.2785. Lemma 3. (Young’s inequality) For ∀(x, y) ∈ R 2 , the following inequality holds xy ≤

1 εp p |x| + q |y|q , p qε

(27)

where ε > 0, p > 1, q > 1, and (p − 1)(q − 1) = 1. The following coordinate transformation is defined to develop adaptive decentralized NN control scheme zi,1 = xi,1 , zi, j = xi, j − αi, j−1 (Zi, j−1 ),

i = 2, 3, . . . , N,

j = 2, 3, . . . , ni .

To begin with the backstepping design procedure, constants are defined as   Ni, j ∗ ||Wi, j ||, i = 1, 2, . . . , N, j = 2, 3, . . . , ni , θi = max b

(28)

(29)

where Ni, j is the dimension of S i, j , and ||Wi,∗ j || is the Euclidean norm of the optimal weight vector W i,∗ j . The unknown constant θi is estimated by θˆi in the following design procedure, and θ˜i = θi − θˆi is the parameter error. The intermediate control function α i, j , the true control law v i , and the adaption law θˆi for stochastic nonlinear time-delay systems with dead-zone inputs (1) will be designed as z3i, j 3 −(ki, j + )zi, j − θˆi tanh( ), 4 i, j z3i,n

1 3 vi (Zi,ni ) = − (ki,ni + )zi,ni + θˆi tanh( i ) , βi0 4 i,ni ni  z3i, j λi z3i, j tanh( ) − γi θˆi , θ˙ˆi = i, j j=1

αi, j (Zi, j ) =

5

(30) (31) (32)

where ki, j , i, j , λi and γi are positive design parameters, Z i,1 = xi,1 ∈ ΩZi,1 ⊂ R1 , Zi, j = [ x¯Ti, j , θˆi ] ∈ ΩZi ⊂ R j+1 , (i = 1, 2, . . . , N, j = 2, 3, . . . , n i ). Lemma 4. Based on the coordinate transformations (28), the following inequality holds || x¯i, j || ≤ φi (||Zi (t)||) + i ,

(33)

where Zi (t) = [zi,1 , zi,2 , . . . , zi,ni , |θ˜i |1/2 ]T , and i is a positive constant; φ i (s) = s(ai0 s + bi0 ) is an unknown class K ∞ function with ai0 and bi0 being positive constants. Proof. See Appendix.  3. Controller Design and Stability Analysis 3.1. Controller design In this section, an adaptive decentralized NN controller is constructed based on the backstepping technique, minimal learning parameters algorithm, and Lyapunov-Razumikhin functional approach. In each step, RBF NN is used to approximate the unknown nonlinear function f¯i, j , and an intermediate control function α i, j will be obtained to stabilize subsystem while the true control law v i will be designed for ith subsystem in the final step. The detailed backstepping design procedures are given as follows. Step 1: Based on the first equation in (1) and noting z i,1 = yi = xi,1 , we have dzi,1

= (gi,1 xi,2 + fi,1 + qi,1 + hi,1 (¯y))dt + ψTi,1 (yi )dwi .

(34)

Consider the Lyapunov function candidate V i,1 as =

Vi,1

1 4 b ˜2 z + θ . 4 i,1 2λi i

From the definition 1, we get LVi,1

=

3 b y3i (gi,1 xi,2 + fi,1 + qi,1 + hi,1 (¯y)) + y2i ψTi,1 ψi,1 − θ˜i θ˙ˆi . 2 λi

(35)

Based on Young’s inequality and Assumption 4, the following inequalities can be obtained 3 2 T y ψ ψi,1 2 i i,1

≤

y3i hi,1 (¯y) ≤ ≤

3 4 y + 4 i 3 4 y + 4 i

3 4 4 y ψ¯ (yi ), 4 i i,1 1 4 h (¯y) 4 i,1 N 3 4 1  4¯4 y h (yl ). yi + N 4 4 l=1 l i,1,l

(36)

(37)

By applying Assumption 3 and Lemma 4 to the time-delay term q i,1 , it implies that y3i qi,1 ( x¯i,1 (t − τi,1 (t)) ≤ ≤ ≤

|y3i |Qi,1 (|| x¯i,1 (t − τi,1 (t))||) |y3i |Qi,1 (φi ||Zi (t − τi,1 (t))|| + i ) |y3i |(Q¯ i,1 (||Zi (t − τi,1 (t))||) + Qi,1 (2i )),

where Q¯ i,1 (s) = Qi,1 (2φi (s)). Q¯ i,1 (s) is still a class K∞ function, and it can be written as Q¯ i,1 (s) = sφi,1 (s) with φi,1 (s) ni  being a continuous function. By utilizing ||Z i (t)|| ≤ ||Z¯ i,1 || + |zi,k | and Lemma 3, we have k=2

y3i qi,1 ( x¯i,1 (t − τi,1 (t)) ≤

|y3i |(Q¯ i,1 (qi ||Zi (t)||) + Qi,1 (2i )) ni ni  y3 Fi,1 3 43 4  1 4 4 ≤ li,1 yi + zi,k φi,1 (li,1 |zi,k |) + y3i Fi,1 tanh( i ) + δσi,1 , 4 4 σi,1 k=2 k=2 6

(38)

where li,1 = qi ni , and F i,1 = (Q¯ i,1 (li,1 ||Z¯ i,1 ||) + Qi,1 (2i )). Thus, in view of inequalities (36)-(38), it follows that i  3 b 1 4 4 z φ (li,1 |zi,k |) y3i (gi,1 xi,2 + f¯i,1 ) − y4i + δσi,1 − θ˜i θ˙ˆi + 4 λi 4 i,k i,1 k=2

n

LVi,1

≤

ni  ni ni s−1 s s  N   1 3 1  y4l h¯ i,m,l (yl ) − (s − 1) y4i ψ¯ 4i,p (yi ) − s y4i ψ¯ 4i,m (yi ), − N 4 s=2 m=1 l=1 4 s=2 4 p=1 s=2 m=1

(39)

where f¯i,1

=

fi,1 + 3 + yi 4

ni  3 i=2 ni  s=1

4 s

4

3 li,1 yi + Fi,1 tanh(

ni −1  nl  s N  s  y3i Fi,1 1 1  ) + Nyi s ψ¯ 4i,p (yi ) h¯ 4l,m,i (yi ) + yi σi,1 4 4 l=1 s=1 m=1 s=1 p=1

s 

9 ψ¯ 4i,m (yi ) + yi . 4 m=1

Obviously, f¯i,1 is an unknown nonlinear function as it contains unknown function f i,1 , h¯ l,m,i(yi ), ψ¯ i,p (yi ), ψ¯ i,m (yi ), which ∗T cannot be implemented in practice. Hence, there exists NN W i,1 S i,1 (Zi,1 ), Zi,1 = xi,1 ∈ ΩZi,1 ⊂ R1 such that f¯i,1

∗ Wi,1 S i,1 (Zi,1 ) + δi,1 (Zi,1 ), T

=

|δi,1 (Zi,1 )| ≤ εi,1 ,

(40)

where δi,1 (Zi,1 ) denotes the approximation error, and ε i,1 is a positive constant. From Lemma 3 and the definition of θi , we have y3i f¯i,1

≤ ≤

3 1 ∗ |y3i |||Wi,1 ||||S i,1(Zi,1 )|| + y4i + ε4i,1 4 4 3 y 3 1 i bθi y3i tanh( ) + bθi δi,1 + y4i + ε4i,1 . i,1 4 4

(41)

Substituting inequality (41) into (39) results in ≤

LVi,1

y3i gi,1 xi,2 + bθi y3i tanh(

ni  y3i 1 4 4 1 b zi,k φi,1 (li,1 |zi,k |) − θ˜i θ˙ˆi ) + δ(σi,1 + bθi,1 ) + ε4i,1 + i,1 4 4 λ k=2

ni  ni ni s−1 s s  N   1  1 3 − N y4l h¯ i,m,l (yl ) − (s − 1) y4i ψ¯ 4i,p (yi ) − s y4i ψ¯ 4i,m (yi ). 4 s=2 m=1 l=1 4 s=2 4 p=1 s=2 m=1

(42)

By adding and subtracting α i,1 in (42), we have LVi,1

≤

y3i gi,1 zi,2 + z3i,1 gi,1 αi,1 + bθˆi y3i tanh(

ni  y3i 1 4 4 1 z φ (li,1 |zi,k |) ) + δ(σi,1 + bθi,1 ) + ε4i,1 + i,1 4 4 i,k i,1 k=2

+

ni  ni s−1 s  N  y3 1  1 b ˜ y4l h¯ i,m,l (yl ) − (s − 1) y4i ψ¯ 4i,p (yi ) θi (λi y3i tanh( i ) − θ˙ˆ i ) − N λi i,1 4 s=2 m=1 l=1 4 s=2 p=1

−

ni s  3 s y4 ψ¯ 4 (yi ). 4 s=2 m=1 i i,m

(43)

Considering the intermediate control function in (30) with i = 1 and Young’s inequality gives y3i gi,1 αi,1

≤

y3i gi,1 zi,2

≤

y3 3 ˆ 3i tanh( i ), −ki,1 by4i − gi,1 y4i − bθy 4 i,1 3 1 gi,1 y4i + gi,1 z4i,2 . 4 4 7

(44) (45)

Therefore, it follows from (43) that LVi,1

≤ −ki,1 bz4i,1 + −

ni ni  s  N  z3i,1 1 4 4 b˜ 1  zi,k φi,1 (li,1 |zi,k |) − N y4l h¯ i,m,l (yl ) θi (λi z3i,1 tanh( ) − θ˙ˆi ) + λi i,1 4 4 k=2 s=2 m=1 l=1

ni ni s−1 s   3 1 1 (s − 1) y4i ψ¯ 4i,p (yi ) − s y4i ψ¯ 4i,m (yi ) + ρi,1 + gi,1 z4i,2 , 4 s=2 4 4 p=1 s=2 m=1

(46)

where ρi,1 = δ(σi,1 + bθi,1 ) + 14 ε4i,1 . Step j (2 ≤ j ≤ ni − 1): From zi, j = xi, j − αi, j−1 , the error dynamic system can be written as dzi, j

=

(gi, j xi, j+1 + fi, j + qi, j + hi, j (¯y) − Lαi, j−1 )dt + (ψi, j −

j−1  ∂αi, j−1 m=1

∂xi,m

ψi,m )T dwi ,

(47)

where Lαi, j−1

=

j−1  ∂αi, j−1 m=1

∂xi,m

(gi,m xi,m+1 + fi,m + qi,m + hi,m (¯y)) +

j−1 ∂αi, j−1 ˙ 1  ∂2 αi, j−1 T ψ ψi,q . θˆ i + 2 p,q=1 ∂xi,p ∂xi,q i,p ∂θˆi

(48)

Choose the following Lyapunov candidate V i, j Vi, j

=

1 Vi, j−1 + z4i, j . 4

(49)

From (49), the infinitesimal generator satisfies LVi, j

=

LVi, j−1 + z3i, j (gi, j xi, j+1 + fi, j + qi, j + hi, j (¯y) − Lαi, j−1 ) j−1 j−1   ∂αi, j−1 ∂αi, j−1 3 + z2i, j (ψi, j − ψi,m )T (ψi, j − ψi,m ). 2 ∂xi,m ∂xi,m m=1 m=1

(50)

Similarly the aforementioned Step 1, it obtains that LVi, j−1

≤ −

j−1 

ki,m bz4i,m +

m=1

+

j−1 

j−1 z3i,m b ˜  3 ) − θ˙ˆi ) θi ( λi zi,m tanh( λi m=1 i,m

(z3i,m ϕi,m (Zi,m ) − z3i,m

m=2

∂αi,m−1 ˙ θˆi − δκi,m ) ∂θˆi

+

i i  1  1  z4i,k φ4i,m (li,s |zi,k |) − N y4 h¯ i,m,l (yl ) 4 s=1 m=1 k= j 4 s= j m=1 l=1 l

−

ni ni s−1 s   1 3 (s − 1) y4i ψ¯ 4i,p (yi ) − s y4i ψ¯ 4i,m (yi ) 4 s= j 4 s= j m=1 p=1

j−1

+

s

n

n

j−1 

1 ρi,m + gi, j−1 z4i, j , 4 m=1

s

N

(51)

where ρi,1 = δ(σi,1 + bθi i,1 ) + 14 ε4i,1 , and ρi,m = δ(σi,m + κi,m + bθi i,m ) + 14 ε4i,m , m = 2, 3, . . . , j − 1. On the basis of Assumption 3 and Lemma 2, we have z3i, j qi, j − z3i, j

j−1  ∂αi, j−1 m=1

∂xi,m

qi,m 8

≤

j−1 ni j ni ni  3 43 4   3 43 ∂αi, j−1 4 4 1  li, j zi, j + li, j | | 3 zi, j + z4i,k φ4i,m (li, j |zi,k |) 4 4 ∂x 4 i,m m=1 k= j+1 m=1 k= j+1 k= j+1

+z3i, j Fi, j tanh( where F i, j = (Q¯ i, j (li, j ||Z¯ i, j ||)+Qi, j(2i ))+

z3i, j Fi, j σi, j

j−1  m=1

|

) + δσi, j ,

(52)

∂αi, j−1 ¯ ¯ ∂xi,m |( Qi,m (li, j ||Zi, j ||)+Qi,m (2i )), li, j

= qi ((ni − j)+1), Z¯i, j = [zi,1 , zi,2 , . . . , zi, j , |θ˜i |1/2 ]T

and Q¯ i j (s) = sφi j (s). By Young’s inequality and Assumption 4, we get the following inequalities z3i, j hi, j (¯y) − z3i, j

j−1  ∂αi, j−1 m=1

∂xi,m

hi,m (¯y)

j j−1 N 3 4 3  ∂αi, j−1 43 1   4 ¯ zi, j + z4i, j + N y hi,m,l (yl ), 4 4 m=1 ∂xi,m 4 m=1 l=1 l

≤

(53)

1   ∂2 αi, j−1 T − z3i, j ψ (yi )ψi,q (yi ) 2 p=1 q=1 ∂xi,p xi.q i,p j−1 j−1

j−1 j−1 j−1  1 6   ∂2 αi, j−1 2 1 zi, j + ( j − 1) y4i ψ¯ 4i,p (yi ), 4 ∂x x 4 i,p i.q p=1 q=1 p=1

≤

(54)

j−1  ∂αi, j−1 3 2 z ||ψi, j (yi ) − ψi,m (yi )||2 2 i, j ∂xi,m m=1 j j−1  ∂αi, j−1 4 3  4 4 3 4 3 jzi, j + jz4i, j ( ) + j y ψ¯ (yi ). 4 4 ∂xi,m 4 m=1 i i,m m=1

≤

(55)

Substituting (51)-(55) into (50) leads to LVi, j

≤

−

j−1 

ki,m bz4i,m +

m=1

j−1 j  z3i,m ∂αi,m−1 ˙ˆ b ˜  3 ) − θ˙ˆi ) + (z3i,m ϕi,m (Zi,m ) − z3i,m θi − δκi,m ) θi ( λi zi,m tanh( λi m=1 i,m ∂θˆi m=2

+

j ni  ni ni s−1 s s  N  1   1  1  z4i,k φ4i,m (li,s |zi,k |) − N y4l h¯ i,m,l (yl ) − (s − 1) y4i ψ¯ 4i,p (yi ) 4 s=1 m=1 k= j+1 4 s= j+1 m=1 l=1 4 s= j+1 p=1

−

j−1 ni s   3  3 s y4i ψ¯ 4i,m (yi ) + ρi,m + z3i, j (gi, j xi, j+1 + f¯i, j (Zi, j )) + δ(σi, j + κi, j ) − z4i, j , 4 s= j+1 m=1 4 m=1

where f¯i, j (Zi, j ) =

fi, j −

j−1  ∂αi, j−1 m=1

∂xi,m

+Fi, j tanh(

(gi,m xi,m+1 + fi,m ) +

z3i, j Fi, j σi, j

j−1  ni ni   3 43 3 43 ∂αi, j−1 4 li, j zi, j + li, j | | 3 zi, j 4 4 ∂x i,m m=1 k= j+1 k= j+1

j−1 j−1 j−1 3 3  ∂αi, j−1 43 1 3   ∂2 αi, j−1 2 ) + zi, j + zi, j + zi, j 2 4 m=1 ∂xi,m 4 p=1 q=1 ∂xi,p xi.q

j−1 s j−1  ∂αi, j−1 4 1   3 3 1 + jzi, j + jzi, j ( ) + zi, j φ4i,m (li,s |zi,k |) + gi, j−1 zi, j − ϕi, j (Zi, j ). 4 4 ∂xi,m 4 s=1 m=1 4 m=1

9

(56)

T The function ϕ i, j (Zi, j ) will be specified later. Hence, f¯i, j (Zi, j ) can be approximated by NN W i,∗ j S i, j (Zi, j ), Zi, j = [ x¯i, j , θˆi ]T ∈ ΩZi, j ⊂ Ri, j+1 such that

Wi,∗ j S i, j (Zi, j ) + δi, j (Zi, j ), T

=

f¯i, j

|δi, j (Zi, j )| ≤ εi, j .

(57)

It implies that z3i, j f¯i, j

≤ ≤

3 1 |z3i, j |||Wi,∗ j ||||S i, j(Zi, j )|| + z4i, j + ε4i, j 4 4 z3i, j 3 1 bθi z3i, j tanh( ) + bθi δi, j + z4i, j + ε4i, j . i, j 4 4

(58)

With the help of the same method as used in step 1, we have LVi, j

≤

−

j−1 

ki,m bz4i,m +

m=1

+

j j  z3i,m ∂αi,m−1 ˙ˆ b ˜  3 ) − θ˙ˆi ) + (z3i,m ϕi,m (Zi,m ) − z3i,m θi − δκi,m ) θi ( λi zi,m tanh( λi m=1 i,m ∂θˆi m=2

j ni  ni ni s−1 s s  N  1   1  1  z4i,k φ4i,m (li,s |zi,k |) − N y4l h¯ i,m,l (yl ) − (s − 1) y4i ψ¯ 4i,p (yi ) 4 s=1 m=1 k= j+1 4 s= j+1 m=1 l=1 4 s= j+1 p=1

j−1 ni s   z3i, j 3  1 4 ¯4 3 3 3 ˆ − s y ψ (yi ) + ρi,m + zi, j gi, j zi, j+1 + zi, j gi, j αi, j + bθi zi, j tanh( ) + ε4i, j 4 s= j+1 m=1 i i,m  4 i, j m=1

+δ(σi, j + κi, j + bθi i, j ).

(59)

According to the intermediate control function α i, j in (30) and Young’s inequality, we have z3i, j gi, j αi, j z3i, j gi, j zi, j+1

z3i, j 3 ≤ −ki, j bz4i, j − gi, j z4i, j − bθˆi z3i, j tanh( ), 4 i, j 3 1 gi, j z4i, j + gi, j z4i, j+1 . ≤ 4 4

(60) (61)

From inequalities (59)-(61), it follows that LVi, j

≤

−

j  m=1

ki,m bz4i,m +

j j  z3i,m ∂αi,m−1 ˙ b ˜  3 θi ( λi zi,m tanh( ) − θ˙ˆi ) + (z3i,m ϕi,m (Zi,m ) − z3i,m θˆ i − δκi,m ) λi m=1 i,m ∂θˆi m=2

+

j ni  ni ni s−1 s s  N  1   1  1  z4i,k φ4i,m (li,s |zi,k |) − N y4l h¯ i,m,l (yl ) − (s − 1) y4i ψ¯ 4i,p (yi ) 4 s=1 m=1 k= j+1 4 s= j+1 m=1 l=1 4 s= j+1 p=1

−

j ni s   3  1 s y4i ψ¯ 4i,m (yi ) + ρi,m + gi, j z4i, j+1 4 s= j+1 m=1 4 m=1

(62)

where ρi,1 = δ(σi,1 + bθi i,1 ) + 14 ε4i,1 , and ρi,m = δ(σi,m + κi,m + bθi,m ) + 14 ε4i,m , (m = 2, 3, . . . , j). Step ni : The true controller v i will be developed in the final step. From z i,ni = xi,ni − αi,ni −1 , we have

dzi,ni = gi,ni (KiT (t)Φi (t)vi + di (vi )) + fi,ni + qi,ni + hi,ni (¯y) − Lαi,ni −1 dt +(ψi,ni −

n i −1 m=1

∂αi,ni −1 ψi,m )T dwi . ∂xi,m

(63)

Consider the Lyapunov function V i,ni Vi,ni

=

1 Vi,ni −1 + z4i,ni . 4 10

(64)

From the definition 1, it yields LVi,ni

=

LVi,ni −1 + z3i,ni (gi,ni (KiT (t)Φi (t)vi + di (vi )) + fi,ni + qi,ni + hi,ni (¯y) − Lαi,ni −1 i i   ∂αi,ni −1 ∂αi,ni −1 3 + z2i,ni (ψi,ni − ψi,m )T (ψi,ni − ψi,m ), 2 ∂x ∂xi,m i,m m=1 m=1

n −1



n −1

(65)

where Lαi,ni −1 is given in (48) with j = n i , and LVi,ni −1 denotes (62) with j = n i − 1. Via the similar method to deal with the time-delay terms, we can obtain the following inequality z3i,ni qi,ni − z3i,ni

ni −1  ∂αi,n −1 i

m=1

∂xi,m

qi,m

≤

ni −1 

∂αi,ni −1 z3i,ni (Q¯ i,ni (li,ni ||Z¯ i,ni ||) + Qi,ni (2i )) + | |(Q¯ i,m (li,ni ||Z¯ i,ni ||) + Qi,m (2i )) ∂xi,m m=1

≤

z3i,ni Fi,ni tanh(

z3i,ni Fi,ni σi,ni

) + δσi,ni ,

(66)

n i −1 ∂α where F i,ni = Q¯ i,ni (li,ni ||Z¯ i,ni ||) + Qi,ni (2i ) + | ∂xi,ni,mi −1 |(Q¯ i,m (li,ni ||Z¯ i,ni ||) + Qi,m (2i )). Based on Young’s inequality and m=1

Assumption 4, it follows that z3i,ni hi,ni (¯y) − z3i,ni

ni −1  ∂αi,n −1 i

m=1

≤

∂xi,m

hi,m (¯y)

n ni  N i −1 ∂αi,ni −1 43 1  3 4 3 zi,ni + z4i,ni + N y4l h¯ i,m,l (yl ), 4 4 ∂x 4 i,m m=1 m=1 l=1

(67)

i i   ∂2 αi,ni −1 T 1 ψ (yi )ψi,q (yi ) − z3i,ni 2 ∂xi,p xi.q i,p p=1 q=1

n −1 n −1

≤

ni −1  ni −1 ni −1 2  ∂ αi,ni −1 2 1 1 6  zi,ni + (ni − 1) y4i ψ¯ 4i,p (yi ), 4 ∂x x 4 i,p i.q p=1 q=1 p=1

(68)

ni −1  ∂αi,ni −1 3 2 zi,ni ||ψi,ni (yi ) − ψi,m (yi )||2 2 ∂x i,m m=1

≤

n ni i −1 ∂αi,ni −1 4 3  3 4 3 ni zi,ni + ni z4i,ni ( ) + ni y4 ψ¯ 4 (yi ). 4 4 ∂xi,m 4 m=1 i i,m m=1

(69)

Substituting (66)-(69) into (65) gives LVi,ni

≤

−

ni −1  m=1

+

ni 

ki,m bz4i,m +

ni −1 z3i,m b˜  ) − θ˙ˆi ) θi ( λi z3i,m tanh( λi m=1 i,m

(z3i,m ϕi,m (Zi,m ) − z3i,m

m=2

∂αi,m−1 ˙ˆ θi − δκi,m ) ∂θˆi

+z3i,ni gi,ni (KiT (t)Φi (t)vi + di (vi )) + f¯i,ni (Zi,ni ) +

n i −1

3 ρi,m + δ(σi,ni + κi,ni ) − z4i,ni , 4 m=1 11

(70)

where f¯i,ni (Zi,ni ) =

fi,ni −

ni −1  ∂αi,n −1 i

∂xi,m

m=1

(gi,m xi,m+1 + fi,m ) + F i,ni tanh(

z3i,ni Fi,ni σi,ni

3 ) + zi,ni 2

n ni −1 n i −1 i −1 2 ∂αi,ni −1 43 1 3  ∂ αi,ni −1 2 3 3 + zi,ni + zi,ni + ni zi,ni 4 ∂xi,m 4 ∂xi,p xi.q 4 m=1 p=1 q=1 n ni −1  s i −1 ∂αi,ni−1 4 1  3 + ni zi,ni ( ) + zi,ni φ4i,m (li,s |zi,ni |) 4 ∂x 4 i,m m=1 s=1 m=1

1 + gi,ni −1 zi,ni − ϕi,ni (Zi,ni ). 4 ∗T S i,ni (Zi,ni ), Zi,ni = [ x¯i,ni , θˆi ]T ∈ ΩZi,ni ⊂ Rni +1 such that Thus, there exists NN Wi,n i

z3i,ni f¯i,ni

3 1 ∗ |z3i,ni |||Wi,n ||||S i,ni (Zi,ni )|| + z4i,ni + ε4i,ni i 4 4 3 z 3 1 i,n bθz3i,ni tanh( i ) + bθi δi,ni + z4i,ni + εi,ni . i,ni 4 4

≤ ≤

(71)

From the inequalities (70)-(71), we get LVi,ni

≤

−

ni −1 

ki,m bz4i,m +

m=1

ni z3i,m b˜  θi ( λi z3i,m tanh( ) − θ˙ˆi ) λi m=1 i,m

ni  ∂αi,m−1 ˙ + (z3i,m ϕi,m (Zi,m ) − z3i,m θˆi − δκi,m ) ∂θˆi m=2

+z3i,ni gi,ni +



KiT (t)Φi (t)vi



+ di (vi ) + bθˆi z3i,ni tanh(

z3i,ni i,ni

)

n i −1

1 ρi,m + δ(σi,ni + κi,ni + bθi i,ni ) + ε4i,ni . 4 m=1

(72)

Now, by constructing the true control controller v i as (31), it is straightforward to show that z3i,n 3 −ki,ni bz4i,ni − gi,ni z4i,ni − bθˆi z3i,ni tanh( i ), 4 i,ni 4 3 1 |z3i,ni |gi,ni |di (vi )| ≤ gi,ni z4i,ni + cp∗i . 4 4

≤

z3i,ni gi,ni KiT (t)Φi (t)vi

z3i,ni gi,ni di (vi ) ≤

(73) (74)

Based on inequalities (73) and (74), it implies that LVi,ni

≤

−

ni 

ki,m bz4i,m +

m=1

+

ni 

ni z3i,m b˜  ) − θ˙ˆi ) θi ( λi z3i,m tanh( λi m=1 i,m

(z3i,m ϕi,m (Zi,m ) − z3i,m

m=2

+

∂αi,m−1 ˙ˆ θi − δκi,m ) ∂θˆi

n i −1

1 1 4 ρi,m + δ(σi,ni + κi,ni + bθi i,ni ) + ε4i,ni + cp∗i . 4 4 m=1

(75)

Furthermore, we choose the adaptation law as θ˙ˆi

=

ni 

λi z3i,m tanh(

m=1

12

z3i,m i,m

) − γi θˆi .

(76)

From Young’s inequality, we have bγi ˜ bγi ˜2 bγi 2 bγi ˜ ˆ θ . θi θi = θi (θi − θ˜i ) ≤ − θ + λi λi 2λi i 2λi i

(77)

Then, substituting (76) and (77) into (75) results in LVi,ni

≤

−

ni 

ki,m bz4i,m −

m=1

ni ni  ∂αi,m−1 ˙ˆ bγi ˜2  ρi,m + (z3i,m ϕi,m (Zi,m ) − z3i,m θi − δκi,m ) θi + 2λi ∂θˆi m=1 m=2

(78)

where ρi,1 = δ(σi,1 + bθi,1 ) + 14 ε41 , ρi,m = δ(σi, j + κi, j + bθi, j ) + 14 ε4j , (m = 2, . . . , ni − 1), ρi,ni = δ(σi,ni + κi,ni + bθi i,ni ) + 1 4 4 εi,ni

+ 14 cp∗i + 4

bγi 2 2λi θi .

Remark 2. Noted that the adaptive decentralized NN controller (31) and adaptive law (32) have a simple structure, which aren’t related to RBF NN. In addition, by estimating the maximum of the optimal weight vectors’ Euclidean norm instead of the vectors themselves, the number of adaptive tuning law is reduced to one for each subsystem. 3.2. Stability analysis The following theorem is based on the backstepping design procedure just given and is our main result. Now, the main result is summarized. Theorem 1. Consider the large-scale stochastic nonlinear time-delay systems in (1) subject to dead-zone inputs (2) under Assumptions 1-4. For bounded initial conditions with θˆi ≥ 0 and Zi, j ∈ ΩZi, j ⊂ Ri, j+1 (Ωi, j is a sufficiently large compact set), if the intermediate control function α i, j , the true control law vi and the adaptive law θˆi are defined as (30)-(32), then all the signals in the closed-loop system are semi-globally uniformly ultimately bounded in probability. Proof: Choose the Lyapunov functional candidate for the whole system V

N 

=

Vi,ni .

(79)

i=1

According to Definition 1, we have LV

≤ −

ni N  

ki,m bz4i,m −

i=1 m=1

+

ni N  

N N ni  bγi ˜2   ρi,m θi + 2λi i=1 i=1 m=1

(z3i,m ϕi,m (Zi,m ) − z3i,m

i=1 m=2

∂αi,m−1 ˙ˆ θi − δκi,m ). ∂θˆi

(80)

By the definition of θˆi , we have −

ni N   i=1 m=2

z3i,m

∂αi,m−1 ˙ˆ θi ∂θˆi

=

ni N  

z3i,m

i=1 m=2

=

ni N  

i=1 m=2

z3i,m

i=1 m=2

−

ni N ni z3i, j ∂αi,m−1 ˆ   ∂αi,m−1  γi θi − z3i,m λi z3i, j tanh( ) i, j ∂θˆi ∂θˆi

ni N   i=1 m=2

∂αi,m−1 ˆ γi θi − ∂θˆi

z3i,m

ni  j=m

ni N   i=1 m=2

j=1

z3i,m

m−1  j=1

z3i, j

∂αi,m−1 3 λi zi, j tanh( ). i, j ∂θˆi

13

z3i, j ∂αi,m−1 3 λi zi, j tanh( ) i, j ∂θˆi (81)

Applying Lemma 2 to the last term in (81) results in −

ni N  

z3i,m

ni N  

|z3i,m |

=

ni 

λi |z3i,m |

i=1 m=2

≤

ni N  

|z3i, j

j=m

i=1 m=2 ni N  

∂θˆi

j=m

i=1 m=2

≤

ni  ∂αi,m−1

m 

m  j=2

|z3i, j

∂αi, j−1 |. ∂θˆi

z3i, j i, j

)

∂αi,m−1 |λi ∂θˆi

|z3i, j

j=2

(z3i,m Θi,m tanh(

∂αi, j−1 | ∂θˆi z3i,m Θi,m κi,m

i=1 m=2

where Θi,m = λi

λi z3i, j tanh(

) + δκi,m ),

(82)

It implies that

ϕi,m (Zi,m ) =

−

m−1 z3i, j ∂αi,m−1 ˆ  ∂αi,m−1 3 γi θi + λi zi, j tanh( ) i, j ∂θˆi ∂θˆi j=1

−Θi,m tanh(

z3i,m Θi,m κi,m

), m = 2, 3, . . . , ni − 1.

(83)

It is obvious that the last term of (80) are negative on the basis of (81)-(83). Thus, LV

≤

−

ni N  

ki,m bz4i,m −

i=1 m=1

≤

N  bγi i=1

2λi

θ˜i2 +

ni N  

ρi,m

i=1 m=1

−μ1 V + μ2 ,

(84)

where μ1 = min{4ki,m b, γi , i = 1, 2, . . . , N, m = 1, 2, . . . , ni }, and μ2 =

ni N   i=1 m=1

ρi,m . Hence, from (84) and Razumikhin

Lemma, it is easy to obtain that the error variables z i,m are semi-globally uniformly ultimately bounded in the sense of 4-Moment, and θ˜i is bounded in probability. Since θ i is a constant, θˆi is bounded in probability. α i,m is a function of z i,m and θˆi , so αi,m is also bounded in probability. Furthermore, all the signals in the closed-loop system are semi-globally uniformly ultimately bounded in probability. This completes the proof of the Theorem 1.  Remark 3. It should be mentioned that all the signals in the closed-loop system can be regulated to a small neighborhood of the origin by appropriately choosing design parameters, for example, choosing  i, j , γi sufficiently small and ki, j , λi sufficiently large. However, this makes control energy more larger. Therefore, the design parameters are chosen carefully to achieve control action. 4. Simulation Example In this section, a simulation example is provided to demonstrate the feasibility of the proposed method in this paper. Consider the following second-order large-scale stochastic nonlinear time-delay system with dead-zone inputs

⎧ ⎪ dx11 = (2.3 + cos(x11 ))x12 + x211 cos(x11 ) + 0.5x211 (t − τ11 (t)) + y1 y2 + y22 dt + 0.5y21dw1 , ⎪ ⎪ ⎪

⎪ ⎪ 2 ⎪ ⎪ dx = (1.5 − sin(x x ))u + x sin(x ) + x (t − τ (t))x (t − τ (t)) + y y dt + (1 + y21 )dw1 , 12 11 12 1 11 11 12 12 12 1 2 ⎪ 12 ⎪ ⎪ ⎪ ⎪ ⎨ y1 = x11 ,

⎪ ⎪ ⎪ dx21 = (3 − cos(x321 ))x22 + 2x321 + x21 (t − τ21 (t)) + y1 + y1 y22 dt + y2 cos(y2 )dw2 , ⎪ ⎪

⎪ ⎪ ⎪ ⎪ ⎪ dx22 = (1.2 + sin(x21 x222 ))u2 + x22 cos(x21 ) + x21 (t − τ22 (t))x22 (t − τ22 (t)) + y1 y2 dt + y22 sin(y2 )dw2 , ⎪ ⎪ ⎪ ⎩ y =x , 2 21 14

where τ11 (t) = 1 + sin(t), τ12 (t) = 2 + cos(t), τ21 (t) = 1 − sin(t), and τ 22 (t) = 2 − cos(t). u 1 and u2 are defined as ⎧ ⎪ (1 − 0.2 sin v1 )(v1 − 3), v1 ≥ 3, ⎪ ⎪ ⎨ 0, −2 < v1 < 3, u1 = D1 (v1 ) = ⎪ ⎪ ⎪ ⎩ (0.8 − 0.1 cos v1 )(v1 + 2), v1 ≤ −2, ⎧ ⎪ (1 − 0.2 sin v2 )(v2 − 2.5), v2 ≥ 2.5, ⎪ ⎪ ⎨ 0, −1.5 < v2 < 2.5, u2 = D2 (v2 ) = ⎪ ⎪ ⎪ ⎩ (0.8 − 0.1 cos v2 )(v2 + 1.5), v2 ≤ −1.5. Based on Theorem 1, the intermediate control functions α 1,1 , α2,1 and the true control laws v 1 , v2 are chosen respectively as αi,1 (Zi,1 ) = v(Zi,2 ) =

z3i,1 3 −(ki,1 + )zi,1 − θˆi tanh( ), i = 1, 2, 4 i,1 z3i,2

1 3 − (ki,2 + )zi,2 + θˆi tanh( ) , i = 1, 2, βi0 4 i,2

where zi,1 = xi,1 , zi,2 = xi,2 − αi,1 , Zi,1 = zi,1 ∈ R1 , Zi,2 = [zi,1 , zi,2 , θˆi ] ∈ R3 , (i = 1, 2). The adaptive laws are given as θ˙ˆ i

=

2  j=1

λi z3i, j tanh(

z3i, j i, j

) − γi θˆi ,

i = 1, 2.

In the simulation, the design parameters are chosen as k 11 = k12 = 20, k21 = k22 = 30, 11 = 12 = 2, 21 = 22 = 4, λ1 = 2, λ2 = 4, γ1 = 2, γ2 = 4 and β10 = 0.8, β20 = 0.8. The simulation results are shown in Figures 1-4 with the initial condition φ 1 (t) = [0.1, 0] T , φ2 (t) = [−0.1, 0] T , t ∈ [−τ, 0], [θˆ1 (0), θˆ2 (0)]T = [0, 0]T . Figure 1 gives the response of the state variable x i,1 and xi,2 , (i = 1, 2). Figure 2 illustrates the trajectory of adaptive laws θˆ1 , θˆ2 . Figure 3 depicts the trajectory of dead-zone function output signals u 1 , u2 . Figure 4 shows the control input signals v 1 , v2 . 5. Conclusions In this paper, an adaptive decentralized NN controller has been proposed for a class of large-scale stochastic nonlinear systems with time-varying delay and unknown dead-zone inputs by combining the backstepping technique, Lyapunov-Razumikhin functional and the minimal learning parameters algorithm. The main advantage of proposed approach lies in that NN terms are canceled in controller and adaptive tuning law and only one adaptive parameter needs to be estimated for each subsystem. It has been shown that all the signals in the closed-loop system semi-globally uniformly ultimately bounded in probability can be guaranteed. A simulation example has further demonstrated the effectiveness of the proposed approach. Acknowledgments The authors would like to thank the anonymous reviewers for their helpful comments that improve the quality of the paper. This work was supported in part by the National Natural Science Foundation of China under Grants 61403178, 61403199, 61473178, 61203048, 61374153, 61374086 and the Natural Science Foundation of Jiangsu Province under Grant BK20140770 and the Graduate Innovation and Creativity Foundation of Jiangsu Province under Grants KYLX 0373, KYLX 0376. Appendix. The proof of lemma 5 With the help of αi, j in (30), it easily obtains 3 |αi, j | ≤ (ki, j + )|zi, j | + |θˆi |. 4 15

(85)

Substituting (85) into (28), we have || x¯i, j || ≤

||¯zi, j || + ||α¯ i, j−1 || ≤ ||Zi (t)|| +

j−1 

|αi,m |

m=1 j−1 

3 ((ki,m + )|zi, j | + |θˆi |) 4 m=1

≤

||Zi (t)|| +

≤

||Zi (t)|| +

≤

φi (||Zi (t)||) + i ,

ni 

3 ((ki,m + )||Zi (t)|| + (||Zi (t)||2 + |θi |)) 4 m=1

where x¯ i, j = [xi,1 , . . . , xi, j ]T , z¯i, j = [zi,1 , . . . , zi, j ]T , α¯ i, j−1 = [αi,1 , . . . , αi, j−1 ]T , φi (s) = s(ai0 s + bi0 ), ai0 = bi0 =

ni 

(ki,m + 34 ) + 1, and i =

m=1

ni  m=1

ni 

1,

m=1

|θi |.

References [1] Z.P. Jiang, D.W. Repperger, D.J. Hill, Decentralized nonlinear output feedback stabilization with disturbance attenuation, IEEE Transactions on Automatic Control. 46 (2001) 1623-1629. [2] H.G. Zhang, L. Cai, Decentralized nonlinear adaptive control of an HVAC system, IEEE Transactions on Systems, Man and Cybernetics Part C: Applications and Reviews. 32 (2002) 493-498. [3] J. Zhou, C.Y. Wen, Decentralized backstepping adaptive output tracking of interconnected nonlinear systems, IEEE Transactions on Automatic Control. 53 (2008) 2378-2384. [4] M. Krsti´c, I. Kanellakopoulos, P.V. Kokotovic, Nonlinear and Adaptive Control Design. Wiley, New York, 1995. [5] W.S. Chen, J.M. Li, Decentralized output-feedback neural control for systems with unknown interconnections, IEEE Transactions on Systems, Man, and Cybernetics, Part B Cybernetics. 38 (2008) 258-266. [6] S. Mehraeen, S. Jagannathan, M.L. Crow, Decentralized dynamic surface control of large-scale interconnected systems in strict-feedback form using neural networks with asymptotic stabilization, IEEE Transactions on Neural Networks. 22 (2011) 1709-1722. [7] S.C. Tong, C.L. Liu, Y.M. Li, Fuzzy adaptive decentralized control for large-scale nonlinear systems with dynamical uncertainties, IEEE Transactions on Fuzzy Systems. 18 (2010) 845-861. [8] J. Li, W.S. Chen, J.M. Li, Adaptive NN output-feedback decentralized stabilization for a class of large-scale stochastic nonlinear strictfeedback systems, International Journal of Robust and Nonlinear control. 21 (2011) 452-472. [9] H.Q. Wang, B. Chen, C. Lin, Adaptive fuzzy decentralized control for a class of large-scale stochastic nonlinear systems, Neurocomputing. 103 (2013) 155-163. [10] S.C. Tong, Y.M. Li, T. Wang, Adaptive fuzzy decentralized output feedback control for stochastic nonlinear large-scale systems using DSC technique, International Journal of Robust and Nonlinear control. 23 (2013) 381-399. [11] T. Wang, S.C. Tong, Y.M. Li, Robust adaptive decentralized fuzzy control for stochastic large-scale nonlinear systems with dynamical uncertainties, Neurocomputing. 97 (2012) 33-43. [12] S. Sui, S.C. Tong, Y.M. Li, Adaptive fuzzy decentralised output feedback control of pure-feedback large-scale stochastic non-linear systems with unknown dead zone, IET Control Theory & Applications. 8 (2014) 488-502. [13] Q. Zhou, P. Shi, H.H. Liu, S.Y. Xu, Neural- Network-Based decentralized adaptive output-feedback control for large-scale stochastic nonlinear systems, IEEE Transactions on Systems, Man, and Cybernetics, Part B Cybernetics. 42 (2013) 1608-1609. [14] R.Q. Lu, Y. Xu, A.K. Xue, H∞ filtering for singular systems with communication delays, Signal Processing. 90 (2010) 1240-1248. [15] R.Q. Lu, H.Y. Wu, J.J. Bai, New delay-dependent robust stability criteria for uncertain neutral systems with mixed delays, Journal of the Franklin Institute. 351 (2014) 1386-1399. [16] R.Q. Lu, H.Y. Su, J. Chu, A.K. Xue, A simple approach to robust D-stability analysis for uncertain singular delay systems, Asian Journal of Control. 11 (2009) 411-419. [17] R.Q. Lu, H. Li, Y.P. Zhu, Quantized H∞ filtering for singular time-varying delay systems with unreliable communication channel, Circuits, Systems, and Signal Processing. 31 (2012) 521-538. [18] B. Chen, X.P. Liu, K.F. Liu, C. Lin, Adaptive neural control design for nonlinear MIMO time-delay systems, Automatica. 45 (2009) 15541560. [19] W.S. Chen, L.C. Jiao, J.M. Li, R.H. Li, Adaptive NN backstepping output-feedback control for stochastic nonlinear strict-feedback systems with time-varying delays, IEEE Transactions on Systems, Man, and Cybernetics, Part B: Cybernetics. 40 (2010) 939-950. [20] H.Q. Wang, B. Chen, C. Lin, Adaptive neural control for strict-feedback stochastic nonlinear systems with time-delay, Neurocomputing. 77 (2012) 267-274. [21] Z.X. Yu, S.G. Li, Neural-network-based output-feedback adaptive dynamic surface control for a class of stochastic nonlinear time-delay systems with unknown control directions, Neurocomputing. 129 (2014) 540-547. [22] Q. Zhou, P. Shi, S.Y. Xu, H.Y. Li, Adaptive output feedback control for nonlinear time-delay systems by fuzzy approximation approach, IEEE Transactions on Fuzzy Systems. 21 (2013) 301-313.

16

[23] Q. Zhou, P. Shi, S.Y. Xu, H.Y. Li, Observer-based adaptive neural network control for nonlinear stochastic systems with time delay, IEEE Transactions on Neural Networks and Learning Systems, 24 (2013) 71-80. [24] B. Chen, X.P. Liu, K.F. Liu, C. Lin, Direct adaptive fuzzy control for nonlinear systems with time-varying delays, Information Science. 180, (2010) 776-792. [25] G.Z. Cui, T.C. Jiao, Y.L. Wei, G.F. Song, Y.M. Chu, Adaptive neural control of stochastic nonlinear systems with multiple time-varying delays and input saturation, Neural Computing and Applications. 25 (2014) 779-791. [26] Z.X. Yu, H.B. Du, Adaptive neural control for uncertain stochastic nonlinear strict-feedback systems with time-varying delays: a Razumikhin functional method, Neurocomputing. 74 (2011) 2072-2082. [27] Z.X. Yu, Z.H. Jin, H.B. Du, Adaptive neural control for a class of non-affine stochastic nonlinear systems with time-varying delay: a Razumikhin-Nussbaum method, IET Control Theory & Applications. 6 (2012) 14-23. [28] Z.X. Yu, S.G. Li, H.B. Du, Razumikhin-Nussbaum-Lemma-Based adaptive neural control for uncertain stochastic pure-feedback nonlinear systems with time-varying delays, International Journal of Robust and Nonlinear Control. 23 (2013) 1214-1239. [29] J. Na, Adaptive prescribed performance control of nonlinear systems with unknown dead zone, International Journal of Adaptive Control Signal Processing. 27 (2013) 426-446. [30] S.C. Tong, Y.M. Li, Robust adaptive fuzzy outputfeedback tracking backstepping control of strict-feedback nonlinear systems with unknown dead zones, IEEE Transactions on Fuzzy Systems. 20 (2012) 168-180. [31] S.J. Yoo, J.B. Park, Y.H. Choi, Decentralized adaptive stabilization of interconnected nonlinear systems with unknown non-symmetric deadzone inputs, Automatica. 45 (2009) 436-443. [32] S.J. Yoo, Decentralized adaptive control of a class of interconnected non-linear systems with unknown time delays and dead-zone inputs. IET Control Theory & Applications. 4 (2010) 2639-2650. [33] H.Q. Wang, B. Chen, C. Lin, Direct adaptive neural tracking control for a class of stochastic pure-feedback nonlinear systems with unknown dead-zone, International Journal of Adaptive Control and Signal Processing. 27 (2013) 302-322. [34] H.Q. Wang, B. Chen, C. Lin, Adaptive fuzzy control for pure-feedback stochastic nonlinear systems with unknown dead-zone input, International Journal of Systems and Science. 45 (2014) 2552-2564. [35] T.P. Zhang, S.S. Ge, Adaptive neural control of MIMO nonlinear state time-varying delay systems with unknown dead-zones and gain signs, Automatica. 43 (2007) 1021-1033. [36] T.P. Zhang, S.S. Ge, Adaptive dynamic surface control of nonlinear systems with unknown dead zone in pure feedback form, Automatica. 44 (2008) 1895-1903. [37] J. Zhou, C.Y. Wen, Y. Zhang, Adaptive output control of nonlinear systems with uncertain dead-zone nonlinearity, IEEE Transactions on Automatic Control. 51 (2006) 504-511. [38] M.M. Polycapou, Stable adaptive neural control scheme for nonlinear systems, IEEE Transactions on Automatic Control. 41 (1996) 447-451.

1.5 x11 x12 x21

1

x22

0.5

0

−0.5

−1

0

0.5

1 Time(Sec)

1.5

Figure 1. States of closed-loop system x i,1 , xi,2 , (i = 1, 2).

17

2

0.7 θˆ1 θˆ2

0.6

0.5

0.4

0.3

0.2

0.1

0

0

0.5

1 Time(Sec)

1.5

2

Figure 2. Adaptive laws θˆ1 , θˆ2 .

140 u1

120

u2

100 80 60 40 20 0 −20 −40 −60

0

0.5

1 Time(Sec)

1.5

Figure 3. Dead-zone function output signals u 1 , u2 .

18

2

120 v1 100

v2

80 60 40 20 0 −20 −40 −60

0

0.5

1 Time(Sec)

1.5

Figure 4. Control input signals v 1 , v2 .

19

2