Delay-dependent H∞ synchronization for chaotic neural networks with network-induced delays and packet dropouts

Author’s Accepted Manuscript Delay-dependent H∞ synchronization for chaotic neural networks with network-induced delays and packet dropouts Yichun Niu, Li Sheng, Weibo Wang www.elsevier.com/locate/neucom

PII: DOI: Reference:

S0925-2312(16)30364-2 http://dx.doi.org/10.1016/j.neucom.2016.05.026 NEUCOM17042

To appear in: Neurocomputing Received date: 23 January 2016 Revised date: 31 March 2016 Accepted date: 9 May 2016 Cite this article as: Yichun Niu, Li Sheng and Weibo Wang, Delay-dependent H∞ synchronization for chaotic neural networks with network-induced delays and packet dropouts, Neurocomputing, http://dx.doi.org/10.1016/j.neucom.2016.05.026 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.

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Delay-dependent H∞ synchronization for chaotic neural networks with network-induced delays and packet dropouts Yichun Niu, Li Sheng∗ , Weibo Wang

Abstract This paper investigates the problem of H∞ synchronization for chaotic neural networks with network-induced delays and packet dropouts. A novel master-slave synchronization scheme is established where the networkinduced delays and data packet dropouts are taken into consideration. By constructing the Lyapunov functional and employing the Wirtinger-based integral inequality, several delay-dependent conditions are obtained to guarantee that the error system is globally asymptotically stable and satisfies a prescribed H∞ performance constraint. Finally, two numerical examples are presented to validate the feasibility and effectiveness of the results derived.

Chaotic neural networks, H∞ synchronization, network-induced delays, packet dropouts. I. I NTRODUCTION In the past several decades, successful applications of recurrent neural networks have been witnessed in many information processing systems, such as signal processing, optimization algorithm and intelligent control [1]–[4]. It has been found that if the parameters are appropriately chosen, the neural network could exhibit some complicated dynamics and even chaotic behaviors. Recently, the synchronization of chaotic neural networks has been extensively studied due to its potential applications in many different areas including secure communication, chemical and biological systems, information science, optics and so on [5]–[10]. In [5], a multiple chaotic neural network algorithm has been proposed to find a valley involving globally optimal solutions by embedding an annealing self-feedback signal and a nonlinear delay self-feedback connection. Based on the dual-stage impulsive control method, the robust masterslave synchronization problem has been studied in [8] for chaotic delayed neural networks with different parametric uncertainties. In [9], a fuzzy disturbance observer has been proposed to make two uncertain chaotic neural networks with mixed time delays synchronous. In [10], the quantitative analysis of error dynamics has been studied by finding the exact analytical error bound for the synchronization of delayed neural networks. It is well known that robust H∞ control is one of the most effective ways to attenuate influence of the exogenous disturbances below a given level [12], [13]. In order to reduce the effects of noises or disturbances in the synchronization processes for chaotic neural networks, the problem of H∞ synchronization has attracted an ever-increasing research interest [14]–[16]. In [14], an H∞ master-slave synchronization This work is supported by National Natural Science Foundation of China (Nos. 61403420, 61573377) and Fundamental Research Fund for the Central Universities (Nos. 12CX02010A, 13CX02098A, 15CX08014A). Y. Niu, L. Sheng and W. Wang are with the College of Information and Control Engineering, China University of Petroleum (East China), Qingdao, 266580, China. (∗ Corresponding author. Email: [email protected])

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method has been established for a class of uncertain chaotic neural networks with mixed time delays. By constructing a suitable Lyapunov-Krasovskii functional and using a reciprocally convex approach, an H∞ synchronization criterion has been derived in [15] for chaotic neural networks with time-varying delays. In [16], the state feedback controller has been designed for the exponential H∞ synchronization of the discrete-time chaotic neural networks with or without time-delays. In all the above-mentioned works, it is assumed that there exists the ideal communication channel in the remote master-slave synchronization schemes. However, the devices in the master-slave systems are mutually connected via communication cables that are of limited capacity [17], [18]. The H∞ synchronization problem for chaotic neural networks has not been properly investigated in the background of networked control systems, and the purpose of this paper is to shorten such a gap. Over the past few years, networked control systems have received much attention owing to their advantages in many aspects such as the simple installation, the large flexibility and the low cost [19][22]. Nevertheless, because of the limited bandwidth of the communication channels, some networkinduced challenging problems have inevitably emerged which could degrade the system performances [23]–[26]. Among others, two frequently studied network-induced phenomena are communication delays and packet dropouts [27]–[37]. By using a sampled-data approach, the problem of H∞ output tracking has been solved in [27] for networked control systems with network-induced delays and data packet dropouts. In [28], the necessary and sufficient stabilizing condition has been developed for networked control systems, and the maximum packet dropout rate and the maximum allowable delay bound have been derived explicitly. In [29] and [35], the H∞ control problem has been studied for a class of T-S fuzzy systems with interval time-varying delays and successive packet dropouts. From the hardware point of view, the dual communication channels sharing-based compensation method has been adopted in [31] to reduce the effects of network-induced delays and packet dropouts. However, to the best of author’s knowledge, the corresponding results on the synchronization problem for chaotic neural networks in the networked environment have been really scattered, not to mention the H∞ synchronization of chaotic neural networks with external disturbances. Motivated by the above discussions, the aim of this paper is to investigate the H∞ synchronization problem for chaotic neural networks with network-induced delays and packet dropouts. The main contributions of this paper are twofold. 1) The novel master-slave synchronization scheme is introduced, and the network-induced delays and data packet dropouts are taken into consideration in this scheme. 2) This paper represents the first of few attempts to study the H∞ synchronization problem for chaotic neural networks with external disturbances, network-induced delays and packet dropouts. The rest of this paper is organized as follows: Section II gives the problem description and presents several necessary lemmas. In Section III, the feedback controllers are designed to synchronize the master and slave systems with a guaranteed H∞ performance. Two numerical examples are provided to show the feasibility of the proposed results in Section IV. Finally, the conclusion is summarized in Section V. Notations. The notation in this paper is fairly standard. Rn and Rm×n stand for the n-dimensional Euclidean space and the set of all m × n real matrices, respectively. The superscript T denotes matrix transposition. A > 0(A ≥ 0) means that A is a real symmetric positive definite (positive semidefinite) matrix. The asterisk ∗ within a symmetric block matrix represents the symmetric term of the matrix. diag{· · · } denotes a diagonal matrix. The space of square-integrable vector functions over [0, ∞) with

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the norm ω(t)2 = dimensions.

 ∞ 0

ω T (t)ω(t) dt is represented by L2 [0, ∞). I is a unit matrix with appropriate

II. P ROBLEM F ORMULATION AND P RELIMINARIES Consider a networked synchronization system of chaotic neural networks shown in Fig. 1. Suppose the master system is given by the following n-neuron delayed neural network:  x(t) ˙ = Cx(t) + Af (x(t)) + Bf (x(t − τ )) + J (1) θ(t) = M x(t) where x(t) = [x1 (t), x2 (t), · · · , xn (t)]T ∈ Rn is the state vector associated with n neurons and θ(t) ∈ Rm is the output vector. C = diag{c1 , c2 , · · · , cn } < 0 is a diagonal matrix. A = (aij )n×n and B = (bij )n×n denote the connection weights matrix and the delayed connection weights matrix, respectively. M ∈ Rm×n is a known constant matrix. J = [J1 , J2 , · · · , Jn ]T is an external input vector. τ ≥ 0 is a constant time delay. f (·) = [f1 (·), f2 (·), · · · , fn (·)]T is the neuron activation function that satisfies the following Lipschitz condition: |fi (x1 ) − fi (x2 )| ≤ Li |x1 − x2 |,

∀x1 , x2 ∈ R, i = 1, 2, · · · , n

(2)

where Li > 0 is the Lipschitz constant and denote L = diag{L1 , L2 , · · · , Ln }. For the purpose of synchronization, we introduce the following slave system that is driven by (1) via signal u(t)  y(t) ˙ = Cy(t) + Af (y(t)) + Bf (y(t − τ )) + u(t) + J + W ω(t) (3) β(t) = M y(t) where y(t) ∈ Rn and β(t) ∈ Rm are the state vector and the output vector of the slave system, respectively. u(t) ∈ Rn is the networked control input. Constant matrices A, B, C ∈ Rn×n , M ∈ Rm×n , the activation function f (·) and the external input J are the same as defined in system (1). ω(t) ∈ Rnv is the disturbance input and it belongs to L2 [0, ∞), and W ∈ Rn×nv is a given coefficient matrix of the disturbance. ω(t) Master System

ZOH

u(t)

Slave System

u(tk) θ(t)

θ(tk-sk) Sampler I

β(t)

Controller

ε(iδ+sk)

ε(iδ) &RPPXQLFDWLRQ&KDQQHO

β(tk-sk)

DPP

ι(iδ)

Fig. 1. Networked synchronization system of chaotic neural networks.

Sampler II

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In Fig. 1, it is assumed that the outputs of the master system and the slave system, θ(t) and β(t), are measurable variables. The sensor is clock-driven while the controller and the zero order holder (ZOH) are event-driven. In addition, the sampling period is supposed to be δ and the updating instant of the ZOH is tk . ε(iδ) is a data packet containing the output vector of the master system and the sampling time, and it will suffer a long-distance network transmission and arrive at the DPP at the instant iδ + sk . Similarly, ι(iδ) contains the output vector of the slave system and the sampling time. Sampling Time (i+r)δ

Sampling Time iδ

Dropout Number r

Sampler I

DPP Delay s k

Delay s k+1

updating instant tk

updating instant tk+1

Fig. 2. The schematic of network transmission.

The schematic of the network transmission is shown in Fig. 2. The signal transmission delay time at the instant tk is sk . Between the instant tk and tk+1 , the number of accumulated data packet dropouts is rk . Denote sM as the upper boundary of the network-induced delay sk , and rM is the maximum number of allowed data packet dropouts. Then, the longest updating interval of the controller is M = max{tk+1 − tk } = max{sk+1 − sk + (r + 1)δ} = sM + (rM + 1)δ, k = 1, 2, · · · .

(4)

The data packet processor (DPP) is introduced to choose and transmit the matched data packet to the controller. The DPP contains a register and a logical comparator, and the mechanism of the DPP can be simply described as follow: 1) ι(iδ) will be stored in the register in the form of array. (The capacity of the register is M .) δ 2) When ε(iδ + sk ) arrives at the comparator, the sampling time in this data packet will be used to compare with that in the array in the register. Then, the matched output vectors θ(tk − sk ) and β(tk − sk ) are transmitted to the controller. Remark 1: In most master-slave synchronization schemes, the controllers are very close to the slave systems and far away from the master systems. Thus, it is natural to assume that there does not exist communication delays and packet dropouts between Sampler II and the DPP. The output vectors θ(t) and β(t) are measured at the same instant by Sampler I and Sampler II, respectively. However, the data packets ε(iδ) and ι(iδ) would arrive at the DPP at the different instants and ε(iδ) may suffer dropouts because of the long-distance communication channel. Therefore, it is necessary for the DPP to choose and transmit the matched data packet to the controller. Considering the disturbances in communication channel, the feedback controller is generated by the following form u(tk ) = K(β(tk − sk ) − θ(tk − sk )) = KM (y(tk − sk ) − x(tk − sk ))

(5)

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where K is the feedback gain matrix. Under the action of the ZOH, we have u(t) = KM (y(tk − sk ) − x(tk − sk )),

tk ≤ t < tk+1

(6)

where tk+1 is the next updating instant after tk . In order to analysis the dynamic behavior of the continuous system with a discontinuous controller, we transform the controller (6) into (7) u(t) = KM (y(t − η(t)) − x(t − η(t)))

(7)

where η(t) is a non-differentiable time-varying delay with upper limit ηM η(t) = t − tk + sk < ηM = M + sM = 2sM + (rM + 1)δ,

tk ≤ t < tk+1 .

(8)

Define e(t) = y(t) − x(t) as the synchronization error. The error system between the master system (1) and the slave system (3) can be obtained as follow:  e(t) ˙ = Ce(t) + Ah(e(t)) + Bh(e(t − τ )) + KM e(t − η(t)) + W ω(t) (9) p(t) = M e(t) where p(t) = β(t) − θ(t) and h(e(t)) = f (y(t)) − f (x(t)). Remark 2: In many existing references on H∞ synchronization for chaotic neural networks, it has been assumed that there exists an ideal communication channel between the transmitter and receiver [14]–[16]. However, the perfect communication is not always possible in many engineering systems especially in a networked environment due to the limited bandwidth and network congestion. Therefore, it makes more practical sense to study the synchronization problem of chaotic neural networks subject to network-induced delays and packet dropouts. The main difficulty of the investigation on the networked synchronization problem is that the controller cannot receive the sampling data timely in the network communication. The main purpose of this paper is to design an H∞ controller such that the following requirements are satisfied simultaneously: (a) The zero-solution of the error system (9) with zero disturbance (ω(t) = 0) is globally asymptotically stable. (b) Under the zero initial condition, the following condition holds:  ∞ p(t)2 [pT (t)p(t) − γ 2 ω T (t)ω(t)] dt < 0, i.e., sup <γ (10) J= ω=0,ω∈L2 [0,∞) ω(t)2 0 where γ > 0 is a given disturbance attenuation level. To give our main results, the following lemmas are necessary. Lemma 1: [38] For a real matrix S > 0, the following integral inequality is always hold  b 1 3 e˙ T (s)S e(s) ˙ ds ≥ (e(b) − e(a))T S(e(b) − e(a)) + ΩT SΩ b−a b−a a  b 2 e(s) ds. where Ω = e(b) + e(a) − b−a a Rn×m , Lemma 2: [38] For a real symmetric positive definite matrix S ∈ Rn×n , two matrices H1 ,H2 ∈  S R a vector α ∈ Rm and a given scalar η ∈ (0, 1), if there exists a matrix R ∈ Rn×n such that > 0, ∗ S the following inequality is always hold     S R H 1 1 T T 1 α. α H1 SH1 α + αT H2T SH2 α ≥ αT H1T H2T η 1−η ∗ S H2

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III. M AIN R ESULT In this section, by resorting to the Lyapunov functional method, several delay-dependent synchronization criterions are derived for the H∞ synchronization of chaotic neural networks with network-induced delays and packet dropouts. Theorem 1: For given scalars λ and γ > 0, the error system (9) is globally asymptotically stable while achieving the performance constraint (10), if there exist positive definite matrices P > 0, Q > 0, Z > 0, U > 0, S > 0, T1 > 0, T2 > 0 and matrices N, X, R11 , R12 , R21 , R22 such that the following LMIs hold ⎤ ⎡ 6 6 Z S ζ −λN A −λN B ζ −λN W ψ1 − τ2 Z ζ1 ζ2 3 4 τ ηM ⎥ ⎢ 6 ⎥ ⎢ ∗ 0 0 Z 0 0 0 0 0 0 ψ2 τ ⎥ ⎢ T T ⎥ ⎢ ∗ ∗ ψ ζ 0 ζ ζ 0 0 −M X 0 3 5 6 7 ⎥ ⎢ ⎥ ⎢ ∗ 6 0 ζ8 S 0 0 0 0 ∗ ∗ ψ4 ⎥ ⎢ ηM ⎥ ⎢ 12 ⎥ ⎢ ∗ 0 0 0 0 0 0 ∗ ∗ ∗ −τZ ⎥ ⎢ 12 4 ⎥ < 0,(11) ⎢ Π2 = ⎢ ∗ 0 0 0 0 ∗ ∗ ∗ ∗ − ηM S − ηM R22 ⎥ ⎥ ⎢ 12 ∗ ∗ ∗ ∗ ∗ − ηM S 0 0 0 0 ⎥ ⎢ ∗ ⎥ ⎢ T T ⎥ ⎢ ∗ 0 −A N 0 ∗ ∗ ∗ ∗ ∗ ∗ −T1 ⎥ ⎢ ⎥ ⎢ ∗ T T −B N 0 ∗ ∗ ∗ ∗ ∗ ∗ ∗ −T 2 ⎥ ⎢ ⎥ ⎢ −N W ⎦ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ψ5 ⎣ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ −γ 2 I ⎡

⎤ S 0 R11 R12 ⎢ ∗ 3S R ⎥ ⎢ 21 R22 ⎥ F =⎢ ⎥>0 ⎣∗ ∗ S 0 ⎦ ∗ ∗ ∗ 3S

(12)

where 4 4 4 S + M T M, ψ2 = LT2 L − Q − Z, ψ1 = LT1 L + Q + U − λN C − λC T N T − Z − τ ηM τ 8 1 1 T 1 1 T 1 1 T 1 1 T R11 + R11 + R12 + R12 − R21 − R21 − R22 − R , ψ3 = − S + ηM ηM ηM ηM ηM ηM ηM ηM ηM 22 4 S, ψ5 = N + N T + τ Z + ηM S, ψ4 = −U − ηM 2 1 1 1 1 ζ1 = −λXM − S− R11 − R12 − R21 − R22 , ηM ηM ηM ηM ηM 1 1 1 1 2 2 R11 − R12 + R21 − R22 , ζ3 = R12 + R22 , ζ2 = ηM ηM ηM ηM ηM ηM 2 1 1 1 1 R11 + R12 + R21 − R22 , ζ4 = λN − C T N T + P, ζ5 = − S − ηM ηM ηM ηM ηM 6 2 T 2 T 6 2 2 2 T 2 T ζ6 = S+ R21 + R22 , ζ7 = S− R12 + R22 , ζ8 = − R21 + R . ηM ηM ηM ηM ηM ηM ηM ηM 22 Moreover, if LMIs (11) and (12) are feasible, the feedback gain matrix K can be obtained by K = N −1 X. We first prove the asymptotic stability of the error system (9) with zero disturbance. Construct the Lyapunov functional as V (t) = V1 (t) + V2 (t) + V3 (t)

(13)

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where V1 (t) = eT (t)P e(t),  t  t T e (α)Qe(α) dα + eT (α)U e(α) dα, V2 (t) = t−τ t−ηM  0 t  0  t T V3 (t) = e˙ (α)Z e(α) ˙ dα dβ + e˙ T (α)S e(α) ˙ dα dβ. −τ

t+β

−ηM

t+β

By computing the time derivative of V (t), one has V˙ 1 (t) = 2eT (t)P e(t), ˙ T V˙ 2 (t) = e (t)Qe(t) − eT (t − τ )Qe(t − τ ) + eT (t)U e(t) − eT (t − ηM )U e(t − ηM ),  t  t T T T ˙ V3 (t) = τ e˙ (t)Z e(t) ˙ − e˙ (α)Z e(α) ˙ dα + ηM e˙ (t)S e(t) ˙ − e˙ T (α)S e(α) ˙ dα. t−τ

t−ηM

≤−

t−η(t)

1 [e(t − η(t)) − e(t − ηM )]T S[e(t − η(t)) − e(t − ηM )] − η(t)

where 2 Ω1 = e(t) + e(t − τ ) − τ



(18)

t t−τ

e(α) dα, 2 − η(t)



t−η(t)

e(α) dα, Ω2 = e(t − η(t)) + e(t − ηM ) − ηM t−ηM  t 2 Ω3 = e(t) + e(t − η(t)) − e(α) dα. η(t) t−η(t) T

T

(16)

(17)

ηM 1 − [e(t) − e(t − η(t))]T S[e(t) − e(t − η(t))] η(t) 3 T 3 ΩT2 SΩ2 − Ω SΩ3 − ηM − η(t) η(t) 3



(15)

t−ηM

According to Lemma 1, we have  t 1 3 e˙ T (α)Z e(α) ˙ dα ≤ − [e(t) − e(t − τ )]T Z[e(t) − e(t − τ )] − ΩT1 ZΩ1 , − τ τ t−τ  t−η(t)  t  t e˙ T (α)S e(α) ˙ dα = − e˙ T (α)S e(α) ˙ dα − e˙ T (α)S e(α) ˙ dα − t−ηM

(14)

1 η(t)

T

t

T

1 ηM −η(t)

 t−η(t)

T

T

Let Φ(t) = e (t) e (t − η(t)) e (t − ηM ) e (α) dα e (α) dα . By applyt−η(t) t−ηM ing Lemma 2 and condition (12), it follows from (18) that        t 1 1 0 0 T T T S T S e˙ (α)S e(α) ˙ dα ≤ − Φ (t) H1 + H2 Φ(t) H H − η(t) 1 ∗ 3S ηM − η(t) 2 ∗ 3S t−ηM 1 T T ≤− Φ (t)H12 F H12 Φ(t) (19) ηM where



H12 = [H1T , H2T ]T ,

 I −I 0 0 0 H1 = , I I 0 −2I 0



 0 I −I 0 0 H2 = . 0 I I 0 −2I

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From the condition (2), it can be found that there exist matrices T1 > 0 and T2 > 0 such that G1 = eT (t)LT1 Le(t) − hT (e(t))T1 h(e(t)) ≥ 0,

(20)

G2 = eT (t − τ )LT2 Le(t − τ ) − hT (e(t − τ ))T2 h(e(t − τ )) ≥ 0.

(21)

In addition, for any real matrix N and a given constant λ, the following equality is obviously true G3 = 2[eT (t)λN + e˙ T (t)N ][e(t) ˙ − Ce(t) − Ah(e(t)) − Bh(e(t − τ )) − KM e(t − η(t))] = 0

(22)

and denote X = KN in the following analysis. Then, from (14)-(22), one obtains V˙ (t) ≤ V˙ 1 (t) + V˙ 2 (t) + V˙ 3 (t) + G1 + G2 + G3 = ΞT1 (t)Π1 Ξ1 (t) where

(23)

⎤ 6 6 Z S ζ3 −λN A −λN B ζ4 ρ1 − τ2 Z ζ1 ζ2 τ ηM ⎥ ⎢ ∗ 6 ψ2 0 0 Z 0 0 0 0 0 ⎥ ⎢ τ ⎢ T T ⎥ ⎢ ∗ 0 ζ6 ζ7 0 0 −M X ⎥ ∗ ψ 3 ζ5 ⎥ ⎢ 6 ⎥ ⎢ ∗ ∗ ∗ ψ 0 ζ S 0 0 0 4 8 ηM ⎥ ⎢ ⎥ ⎢ ∗ Z 0 0 0 0 0 ∗ ∗ ∗ − 12 ⎥ ⎢ τ Π1 = ⎢ ⎥, 12 4 ⎥ ⎢ ∗ 0 0 0 ∗ ∗ ∗ ∗ − ηM S − ηM R22 ⎥ ⎢ 12 ⎥ ⎢ ∗ ∗ ∗ ∗ ∗ ∗ − ηM S 0 0 0 ⎥ ⎢ ⎢ T T ⎥ 0 −A N ⎥ ∗ ∗ ∗ ∗ ∗ ∗ −T1 ⎢ ∗ ⎥ ⎢ ⎣ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ −T2 −B T N T ⎦ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ψ5   t  t 1 1 eT (α) dα eT (α) dα Ξ1 (t) = eT (t) eT (t − τ ) eT (t − η(t)) eT (t − ηM ) τ t−τ η(t) t−η(t)  T  t−η(t) 1 eT (α) dα hT (e(t)) hT (e(t − τ )) e˙ T (t) , ηM − η(t) t−ηM 4 4 ρ1 = LT1 L + Q + U − λN C − λC T N T − Z − S τ ηM and the other corresponding matrices are defined in (11). From the condition (11), it is not difficult to see that Π1 < 0 which implies V˙ (t) < 0 for all Ξ1 (t) = 0. According to the Lyapunov theory, the error system (9) with zero disturbance is globally asymptotically stable. Next, we will prove that (10) is satisfied under the zero initial condition for all nonzero ω(t) ∈ L2 [0, ∞). Considering V (t) in (13), we have V (0) = 0 and V (t) ≥ 0, ∀t > 0 under zero initial condition. Therefore,  ∞ J= [pT (t)p(t) − γ 2 ω T (t)ω(t)] dt 0 ∞ [pT (t)p(t) − γ 2 ω T (t)ω(t) + V˙ (t)] dt − V (∞) = 0 ∞ [pT (t)p(t) − γ 2 ω T (t)ω(t) + V˙ (t)] dt. (24) ≤ ⎡

0

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Taking the time derivative of V (t) along the error system (9) and similar to the proof of (23), we can obtain that pT (t)p(t) − γ 2 ω T (t)ω(t) + V˙ (t) ≤ ΞT2 (t)Π2 Ξ2 (t) where

(25)



  t 1 t T 1 Ξ2 (t) = e (t) e (t − τ ) e (t − η(t)) e (t − ηM ) e (α) dα eT (α) dα τ t−τ η(t) t−η(t) T  t−η(t) 1 eT (α) dα hT (e(t)) hT (e(t − τ )) e˙ T (t) ω T (t) . ηM − η(t) t−ηM T

T

T

T

According to the LMI (11), we have J < 0 which is equivalent to (10). The proof is complete. Remark 3: In [14], the integral terms in (16) have been dealt with by using the Jensen inequality which may introduce an undesirable conservatism. In order to reduce the conservatism of the Jensen inequality, an alternative inequality has been proposed in [38] based on the Wirtinger inequality. By using the recently developed Wirtinger-based integral inequality in [38] and introducing a slack variable, a less conservative delay-dependent criterion has been derived in Theorem 1 in terms of LMIs. Remark 4: There exist two different kinds of time-delays in the error systems (9), where τ is the inherent delay in chaotic neural networks and η(t) results from the network-induced delays and packet dropouts. Theorem 1 provides a delay-dependent condition and the delays τ and η(t) both have strong influences on the solvability of LMIs (11) and (12). It will be shown in Examples 1 and 2 that, the upper bound of the non-differentiable time-varying delay η(t) in the error system (9) with τ > 0 is smaller than that in (9) with τ = 0. Considering chaotic neural networks without time-delays, i.e., τ = 0 in systems (1) and (3), we have the following corollary. Corollary 1: For given scalars λ and γ > 0, the error system (9) with τ = 0 is globally asymptotically stable while achieving the performance constraint (10) if there exist positive definite matrices P > 0, U > 0, S > 0, T > 0, and matrices N, X, R11 , R12 , R21 , R22 such that the following LMIs hold ⎤ ⎡ κ3 −λN A κ4 −λN W χ1 κ1 κ2 η6M S ⎥ ⎢ ⎥ ⎢ ∗ χ2 κ 5 κ6 κ7 0 −M T X T 0 ⎥ ⎢ 6 ⎥ ⎢ ∗ ∗ χ3 κ S 0 0 0 8 ηM ⎥ ⎢ ⎥ ⎢ ∗ ∗ ∗ − 12 S − 4 R 0 0 0 ⎥ ⎢ 22 ηM ηM Π3 = ⎢ (26) ⎥ < 0, 12 ⎥ ⎢ ∗ ∗ ∗ ∗ − S 0 0 0 η M ⎥ ⎢ ⎥ ⎢ ∗ ∗ ∗ 0 ∗ ∗ −T −AT N T ⎥ ⎢ ⎥ ⎢ −N W ⎦ ∗ ∗ ∗ χ4 ⎣ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ −γ 2 I ⎤ S 0 R11 R12 ⎥ ⎢ ∗ 3S R ⎢ 21 R22 ⎥ F =⎢ ⎥ > 0, ⎣∗ ∗ S 0 ⎦ ∗ ∗ ∗ 3S ⎡

(27)

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where 4 S + M T M, ηM 8 1 1 T 1 1 T 1 1 T 1 1 T χ2 = − S + R11 + R11 + R12 + R12 − R21 − R21 − R22 − R , ηM ηM ηM ηM ηM ηM ηM ηM ηM 22 4 S, χ4 = N + N T + ηM S, χ3 = −U − ηM 2 1 1 1 1 κ1 = −λXM − S− R11 − R12 − R21 − R22 , ηM ηM ηM ηM ηM 1 1 1 1 2 2 R11 − R12 + R21 − R22 , κ3 = R12 + R22 , κ2 = ηM ηM ηM ηM ηM ηM 2 1 1 1 1 κ5 = − S − R11 + R12 + R21 − R22 , κ4 = λN − C T N T + P, ηM ηM ηM ηM ηM 6 2 T 2 T 6 2 2 2 T 2 T κ6 = S+ R21 + R22 , κ7 = S− R12 + R22 , κ8 = − R21 + R . ηM ηM ηM ηM ηM ηM ηM ηM 22

χ1 = LT L + U − λN C − λC T N T −

Furthermore, if LMIs (26) and (27) are feasible, the feedback gain matrix K can be derived by K = N −1 X. Choose the following the following Lyapunov functional  t  0  t T T V (t) = e (t)P e(t) + e (α)U e(α) dα + e˙ T (α)S e(α) ˙ dα dβ. t−ηM

−ηM

t+β

Following the same procedure in Theorem 1, this corollary can be proved and the details are omitted here. IV. N UMERICAL E XAMPLES In this section, two numerical examples are presented to illustrate the effectiveness of the obtained results. Example 1: Consider the following delayed chaotic neural network  x(t) ˙ = Cx(t) + Af (x(t)) + Bf (x(t − τ )) + J (28) θ(t) = M x(t) with parameters     −1 0 1 + π/4 20 C= , A= , 0 −1 0.1 1 + π/4   0.3 0 M= , τ = 1, J = 0. 0 0.5

 B=

 √ −1.3π 2/4 0.1 √ , 0.1 −1.3π 2/4

The activation function is chosen as fi (x) = |x+1|−|x−1| , (i = 1, 2) which satisfies (2) with L = diag{1, 1}. 2 Under the initial conditions x1 (s) = 1.0, x2 (s) = 0.5, ∀s ∈ [−1.0 0], the model (28) has a chaotic attractor shown in Fig. 3. Take (28) as the master system, and the following slave system is designed for synchronization:  y(t) ˙ = Cy(t) + Af (y(t)) + Bf (y(t − τ )) + u(t) + J + W ω(t) (29) β(t) = M y(t)

SUBMITTED

11 t

t

− T 5 sin(t)e 5 ] where u(t) is the controller in the form of (7), ω(t) = [n(t)e−  is the external disturbance 0.2 0 with n(t) is uniformly distributed over [0 1], W = , and the initial conditions of the slave 0 0.1 system are y1 (s) = −0.5, y2 (s) = −1.5, ∀s ∈ [−1.0 0].

1.5

1

x

2

0.5

0

−0.5

−1

−1.5 −20

−15

−10

−5

0 x1

5

10

15

20

Fig. 3. The chaotic attractor of the master system (28).

In this example, assume that the sampling period δ = 0.01s, the upper boundary of the network-induced delay sM = 0.05s and the maximum number of packet dropouts rM = 3. Then, we have ηM = 0.14s from (8). Let γ = 0.6 and λ = 5. By using the MATLAB software, a set of solutions to LMIs (11) and (12) in Theorem 1 are obtained as follows       0.0899 −0.0239 0.1903 −2.6715 0.0000 −0.0007 P = , Q= , Z= , −0.0239 21.8139 −2.6715 75.4251 −0.0007 0.0115       0.2268 2.5633 0.0012 −0.0163 0.1113 −0.0281 , T1 = U= , , S= 2.5633 47.3464 −0.0163 0.2774 −0.0281 27.0194       0.1897 −2.6640 −0.0168 0.0048 0.2632 −0.0398 , N= , X= . T2 = −2.6640 75.2978 0.0048 −4.0781 −0.0663 38.3084 Then, the controller gain matrix can be derived as follows   −15.6593 −0.3073 K = N −1 X = . −0.0021 −9.3940 By applying the controller (7), the state trajectories of variable xi (t), yi (t), (i = 1, 2) are shown in Fig. 4, and the synchronization error between the master system and the slave system is depicted in Fig. 5. The simulation results have confirmed that the designed H∞ controller performs very well.

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20 x (t) 1

x1(t),y1(t)

10

y (t) 1

0 −10 −20

0

20

40

60

80

100

t(s) 1 y2(t)

0

2

2

x (t),y (t)

x2(t)

−1

−2

100

80

60

40

20

0

t(s)

Fig. 4. State trajectories of the master system (28) and the slave system (29).

2 e (t) 1

e (t) 2

1

e(t)

0

−1

−2

−3

−4

0

5

10

15

t(s)

Fig. 5. The synchronization error between the master system (28) and the slave system (29).

Example 2: Consider a three-dimensional chaotic neural network model without time-delay as follows  x(t) ˙ = Cx(t) + Af (x(t)) + J (30) θ(t) = M x(t) where ⎡

⎤

−1 0 0 ⎢ ⎥ C = ⎣ 0 −1 0 ⎦ , 0 0 −2

⎡

⎤

2 −1 0 ⎢ ⎥ A = ⎣ 1.7 1.71 1.1 ⎦ , −2.5 −2.9 0.56

⎡ ⎢ ⎢ M =⎢ ⎣

0.4 0 0 0 0.5 0 0 0 0.3 0.1 0 0

⎤ ⎥ ⎥ ⎥, ⎦

J = 0.

The activation function is chosen as fi (x) = tanh(x), (i = 1, 2, 3) with L = diag{1, 1, 1}. Under the initial condition [x1 (0) x2 (0) x3 (0)]T = [1 1 1]T , the chaotic attractor of system (30) is shown in Fig. 6 and 7.

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3 2

x3

1 0 −1 −2 −3 2 1

1 0.5

0

0

−1 x

−0.5 −2

2

−1

x1

Fig. 6. The chaotic attractor of the master system (30) in phase space.

2

2.5 2

1.5 1.5 1

1 0.5 3

x

x2

0.5

0

0 −0.5 −1

−0.5

−1.5 −1 −2 −1.5 −1

−0.5

0 x1

0.5

−2.5 −1

1

−0.5

0 x1

(a) x1 -x2

0.5

1

(b) x1 -x3 2.5 2 1.5 1

x3

0.5 0 −0.5 −1 −1.5 −2 −2.5 −1.5

−1

−0.5

0

0.5

1

1.5

2

x2

(c) x2 -x3 Fig. 7. The chaotic attractor of the master system (30) in phase plane.

Taking (30) as the master system and considering the controller (7), the response system is  y(t) ˙ = Cy(t) + Af (y(t)) + u(t) + J + W ω(t) β(t) = M y(t)

(31)

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with the initial condition [y1 (0) y2 (0) y3 (0)]T ⎡ 0.3 0 ⎢ W = ⎣ 0 0.2 0 0

= [−0.5 − 1.5 − 2.5]T , and ⎤ ⎡ ⎤ t 0 n(t)e− 5 t ⎥ ⎥ ⎢ 0 ⎦ , ω(t) = ⎣ sin(t)e− 5 ⎦ , t 0.1 cos(t)e− 5

where n(t) is uniformly distributed over [0 1]. In this example, suppose that the sampling period δ = 0.01s, the upper boundary of the network-induced delay sM = 0.08s and the maximum number of packet dropouts rM = 9. Then we have ηM = 0.26s according to (8). Let λ = 4 and γ = 0.3. By using the MATLAB software, a set of feasible solutions to LMIs (26) and (27) in Corollary 1 are derived as follows ⎡ ⎤ ⎡ ⎤ 0.9100 −0.2746 −0.0336 0.0860 −0.0700 −0.0475 ⎢ ⎥ ⎢ ⎥ P = ⎣−0.2746 0.9617 0.3065 ⎦ , U = ⎣−0.0700 0.0956 0.0605 ⎦ , −0.0336 0.3065 0.3233 −0.0475 0.0605 0.0521 ⎤ ⎡ ⎡ ⎤ 2.7861 −0.8738 −0.0200 1.0071 −0.2948 −0.0442 ⎥ ⎢ ⎢ ⎥ S = ⎣−0.2948 1.0157 0.2969 ⎦ , T = ⎣−0.8738 3.0732 1.1320 ⎦ , −0.0442 0.2969 0.3012 −0.0200 1.1320 1.1601 ⎤ ⎡ ⎤ ⎡ −0.2431 0.0721 0.0103 −32.5115 −0.3621 −0.0208 135.8916 ⎥ ⎢ ⎥ ⎢ N = ⎣ 0.0739 −0.2497 −0.0777⎦ , X = ⎣ 6.5450 1.2907 0.4978 −28.0311⎦ . −28.8815 0.4445 0.5553 115.3360 0.0113 −0.0762 −0.0808 Therefore, the feedback gain matrix can be obtained as ⎡ ⎤ 109.6 0 −0.2 −462.4 ⎢ ⎥ K = N −1 X = ⎣−155.2 −4.9 0.1 621.8 ⎦ 518.9 −0.9 −7 −2077.4 Simulation results are shown in Figs. 8 and 9, where Fig. 8 plots the state trajectories of xi (t), yi (t), (i = 1, 2, 3), and Fig. 9 depicts the synchronization error between the master and slave systems.

x1(t)

1

x (t),y (t)

1 0

1

y1(t)

−1

0

20

40

60

80

100

t(s) x2(t)

2

x (t),y (t)

2 0

2

y2(t)

−2

0

20

40

60

80

100

t(s) x3(t)

3

x (t),y (t)

5 0

3

y3(t)

−5

0

20

40

60

80

100

t(s)

Fig. 8. State trajectories of the master system (30) and the slave system (31)

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1 e (t) 1

0.5

e2(t) e3(t)

0 −0.5

e(t)

−1 −1.5 −2 −2.5 −3 −3.5

0

5

10

15 t(s)

20

25

30

Fig. 9. The synchronization error between the master system (30) and the slave system (31)

V. C ONCLUSIONS In this paper, the H∞ synchronization problem has been addressed for chaotic neural networks with network-induced delays and packet dropouts. By utilizing the Lyapunov functional method and applying inequality techniques, several sufficient delay-dependent conditions which ensure the stability and H∞ performance constraint of the error system have been derived. Moreover, it has been shown that the controller for the H∞ synchronization of chaotic neural networks can be designed by solving a set of LMIs. Two numerical examples have been provided to illustrate the effectiveness of the obtained results. Some other network-induced phenomena, such as uniform quantization, fading channels, and saturation will be considered for the synchronization of chaotic neural networks in the future. R EFERENCES [1] X. Le, and J. Wang, “Robust pole assignment for synthesizing feedback control systems using recurrent neural networks,” IEEE Transactions on Neural Networks and Learning Systems, vol. 25, no. 2, pp. 383-393, 2014. [2] Y. Feng, W. Barr, and W. F. Harper, “Neural network processing of microbial fuel cell signals for the identification of chemicals present in water,” Journal of Environmental Management, vol. 120, pp. 84-92, 2013. [3] A. K. Manshad, H. Rostami, H. Rezaei, and S. M. Hosseini, “Application of artificial neural network-particle swarm optimization algorithm for prediction of asphaltene precipitation during gas injection process and comparison with gaussian process algorithm,” Journal of Energy Resources Technology-Transactions of the ASME, vol. 137, no. 6, 062904, 2015 [4] C. F. Hsu, T. T. Lee, and K. Tanaka, “Intelligent nonsingular terminal sliding-mode control via perturbed fuzzy neural network,” Intelligent Nonsingular Terminal Sliding-mode Control via Perturbed Fuzzy Neural Network, vol. 45, pp. 339-349, 2015. [5] G. Yang, J. Yi, “Dynamic characteristic of a multiple chaotic neural network and its application,” Soft Computing, vol. 17, no. 5, pp. 783-792, 2013. [6] C. D. Zheng, H. Zhang, and Z. Wang, “Exponential synchronization of stochastic chaotic neural networks with mixed time delays and Markovian switching,” Neural Computing & Applications, vol. 25, no. 2, pp. 429-442, 2014. [7] M. Jiang, J. Mei, and J. Hu, “New results on exponential synchronization of memristor-based chaotic neural networks,” Neurocomputing, vol. 156, pp. 60-67, 2015. [8] H. Zhang, T. Ma, G. B. Huang, and Z. Wang, “Robust global exponential synchronization of uncertain chaotic delayed neural networks via dual-stage impulsive control,” IEEE Transactions on Systems Man and Cybernetics Part B-Cybernetics, vol. 40, no. 3, pp. 831-844, 2010. [9] S. C. Jeong, D. H. Ji, J. H. Park, and S. C. Won, “Adaptive synchronization for uncertain chaotic neural networks with mixed time delays using fuzzy disturbance observer,” Applied Mathematics and Computation, vol. 219, no. 11, pp. 5984-5995, 2013.

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