Synchronization of decentralized event-triggered uncertain switched neural networks with two additive time-varying delays

This paper addresses the problem of synchronization for decentralized event-triggered uncertain switched neural networks with two additive time-varying delays. A decentralized eventtriggered scheme is employed to determine the time instants of communication from the sensors to the central controller based on narrow possible information only. In addition, a class of switched neural networks is analyzed based on the Lyapunov–Krasovskii functional method and a combined linear matrix inequality (LMI) technique and average dwell time approach. Some sufficient conditions are derived to guarantee the exponential stability of neural networks under consideration in the presence of admissible parametric uncertainties. Numerical examples are provided to illustrate the effectiveness of the obtained results. 


Introduction
In recent years, neural networks (NNs) have become the active research field, and their successive application used in various areas, such as image processing and optimization problems [3,6,26]. Time delays, which always cause instability and degrade performance, are ubiquitously present in many NNs due to the signal transmission (see [1,12] and references therein). Switched systems, which provide a unified framework for mathematical modeling of many physical or man-made systems displaying switching features, such as power electronics, flight control systems, network control systems, have been widely studied recently [5,22,23]. The switched system consists of a collection of subsystems and a switching signal governing the switching among them. In addition, the average dwell time method introduced in [7] has been recognized to be flexible and efficient in finding a suitable switching signal to guarantee the stability of switched systems or improve the system performance [4,20].
In the recent years, event-triggered control has received increasing attention in real time control systems. Especially to the case of battery-powered wireless devices, reducing the number of network transmissions has an important effect on the battery lifespan [14,15]. Therefore, how to saving limited network resource is a significant and challenging task. The main task to design a decentralized event-triggered scheme for saving the limited communication resources while guaranteeing that the drive response system is synchronous. As a result, event-triggered scheme particularly decentralized event-triggered scheme has received a lot of research interest, and some important results have been published [9,16]. In [18], network-based event-triggered filtering for Markovian jump systems is studied. In this paper, the model of decentralized event-triggered is for saving the limited communication resources while assure that the switched neural networks with additive time-varying delay is exponentially stable.
Recently, a new type of systems with two additive time-varying delays were proposed in [8], and the stability problem was further discussed in [11]. In networked systems, signals are communicated from one point to another may experience two network segments, which can possibly produce two time-varying delays with different properties by cause of variable network transmission conditions [13,23,28]. Moreover, synchronization has been extensively studied due to its strong potential applications in engineering, such as secure communication, robot queue, and chemical reaction [10,27]. Synchronization strategies can have communication between nodes, which cause the network congestion and waste the network resources. In order to overcome the conservativeness of synchronization strategies, the event-triggered strategy is proposed. Many results have been reported in the literature for synchronization-based event-triggered problem [17,19,25]. In addition, to the best of our knowledge, synchronization of uncertain switched neural networks with two additive delay components via decentralized event-triggered scheme has not been completely investigated, which motivates the study of this paper.
Based on the above discussions, in this paper, the problem of synchronization of decentralized event-triggered uncertain switched neural networks with two additive delay components is considered. By utilizing a novel Lyapunov-Krasovskii functional, integral techniques, some sufficient conditions are expressed in terms of linear matrix inequalities http://www.journals.vu.lt/nonlinear-analysis (LMIs), which can be easily checked by using MATLAB LMI Control toolbox. Finally, numerical examples are given to verify the effectiveness of the obtained criteria. The main contributions of this paper are listed as follows: (i) a sufficient condition of exponential stability with two additive time-varying delays for uncertain switched neural networks is established by using average dwell time method, Lyapunov functional method, and some mathematical techniques; and (ii) A decentralized event-triggered scheme is proposed to synchronize the drive-response uncertain switched neural networks.
Notations. Throughout this manuscript, R n and R n×n denote the n-dimensional Euclidean space and the set of all n×n real matrices, respectively. The superscript T denotes the transposition, and the notation P > 0 means that P is real, symmetric and positive definite; diag denotes the block-diagonal matrix; · refers to the Euclidean vector norm. The notation * always denotes the symmetric block in one symmetric matrix. λ min (A) or λ max (A) denotes the maximum eigenvalue or the minimum eigenvalue of matrix A, respectively. Matrices, if their dimensions are not explicitly stated, are assumed to be compatible for algebraic operations.

Problem statement and preliminaries
Consider the following uncertain drive-response neural networks with two additive timevarying delay components: where y(t) = [y 1 (t), y 2 (t), . . . , y n (t)] T ∈ R n , z(t) = [z 1 (t), z 2 (t), . . . , z n (t)] T ∈ R n is the neuron state vector in the drive system (1) and the response system (2), respectively; W β(t) = diag{w β(t)i } with w β(t)i > 0 (i = 1, 2, . . . , n); A β(t) , B β(t) , C β(t) ∈ R n×n are the connection weight matrix, discretely delayed connection weight matrix and the distributed delayed connection weight matrix, respectively; ν(t) = [ν 1 (t), ν 2 (t), . . . , ν n (t) ∈ R n ] is an external input vector; u(t) ∈ R n is the control input of the response system (2); f (·) = [f 1 (·), . . . , f n (·)] T ∈ R n is the neuron activation function; ∆W β(t) (t), ∆A β(t) (t), ∆B β(t) (t), ∆C β(t) (t) are time-varying parametric uncertainties, where β(t) denotes the switching signal and takes the values in the finite set M = {1, 2, . . . , m}, which means that the matrices W β(t) , A β(t) , B β(t) , C β(t) are allowed to take values in the finite set Corresponding to the switching signal β(t), we have the switching sequence {x t0 ; (l 0 , t 0 ), . . . (l k , t k ), . . . , | l k ∈ M, k = 0, 1, 2 . . . }, it means that lth subsystem is active when t ∈ [t k , t k+1 ). In the drive-response neural networks,ρ(t) represents the distributed delay, andĥ 1 (t) andĥ 2 (t) are additive time-varying delays that satisfy whereĥ 1 ,ĥ 2 ,ρ, µ 1 , µ 2 and µ 3 are known constants. Also, we denoteĥ(t) =ĥ 1 (t) + h 2 (t),ĥ =ĥ 1 +ĥ 2 and µ = µ 1 + µ 2 . We assume that the matrices ∆W l (t), ∆A l (t), ∆B l (t), ∆C l (t) are norm bounded and satisfy where G l (t), X 1l , X 2l , X 3l , X 4l are known real constant matrices with appropriate dimensions. The uncertain matrix F l (t) satisfies F T l (t)F l (t) I, t 0. In order to simplify the equations, we write Assumption 1. Each neuron activation function f z (·) is bounded and there exist constants For simplicity of presentation, we denote H 1 = diag{l − 1 l + 1 , l − 2 l + 2 , . . . , l − n l + n }, H 2 = diag{(l − 1 + l + 1 )/2, (l − 2 + l + 2 )/2, . . . , (l − n + l + n )/2}. Combining (1) and (2) with r(t) = y(t) − z(t), the synchronization error system can be obtained aṡ where g(r(t)) = f (z(t)) − f (y(t)), and it can be checked that the function g z (·) satisfies the following condition: The decentralized event-triggered scheme is initiate to decrease the communication burden. In this design, the (m) entries and the measurement errors r(t) are collected intô v nodes, therefore the signals corresponding to node j ∈ {1, 2, . . . ,v} are denoted by k j (t) ∈ R nj for v j=1 n j = n. A decentralized event-triggered condition embedded in the event generators (EGs) is used to decide whether the sampled data should be released to the controller or not. We denote the release instants of the j th event generator by [t j kjh ] ∞ Kj = 0, and the next release instant t j kj +1h of event generator j is determined by where κ j , φ j > 0, j ∈ N, and also defined the error between the current sampling vector and the latest transmission is We will consider a decentralized event-triggered scheme in this paper, which is one of the most important components for designing every control framework and reducing the communication burden in the network. Generally, from (4), the set of release instants , all the sampled signals are transmitted to the controller. The decentralized event-triggered communication scheme is designed to reduce some unwanted data transmissions. Therefore, the real-time detection hardware is no longer needed. In this paper, we are interested in designing the following controller: where K β(t kh ) is the gain matrix to be determined. When β(t kh ) = β(t k+1h ) = l, the lth subsystem is activated on [t kh , t k+1h ) and Therefore, the threshold error ω j (t kh + jh) can be rewritten as Then the control input u l (t) can be obtained as Substituting (5) in system (3), which giveṡ The following definitions and lemmas will play an important role in the derivation of our result. (3) is said to be robustly exponentially stable under switching signal β(t) if there exist some scalars k 0 and η 0 such that [12].) For any constant matrix N > 0, the following inequality holds for all continuously differentiable functionθ in [3].) For scalarsâ andb satisfyingâ b and a matrix R ∈ S n + , the following inequality holds: [26].) For a scalar α ∈ (0, 1), matrices R 1 , R 2 ∈ S n + and Y 1 , Y 2 ∈ R n×n , the following matrix inequality

Main results
In this section, our aim is to study the exponential stability of switched neural networks with additive time-varying delays using the decentralized event-triggered design. First, we consider the following nominal system: In order to make the presentation more sententious, we definẽ x(s m ) ds m · · · ds 2 ds 1 , 3.1 Exponential stability analysis using decentralized event-triggered scheme Theorem 1. For given scalarsĥ u (u = 1, 2), ρ,ĥ, µ,h, µ j (j = 1, 2, 3), the equilibrium point of system (7) is exponentially stable if there exist symmetric positive-definite such that the following linear matrix inequalities hold: and Then system (7) is exponentially stable for any switching signal with average dwell time T a > T * a = lnμ α .
Based on Theorem 1, we presents the exponential stability for uncertain switched neural networks using decentralized event-triggered scheme.
, respectively, and following the similar line in the proof of Theorem 1, we obtainΠ where Ψ a = col[L 1 G l , 0, 0, 0 By Lemma in [1], the necessary and sufficient condition satisfy inequality (26), and there exists a scalar > 0 such that Now, by applying the Schur complement lemma in (27), we get (25), which guarantees that the drive system (1) and the response system (2) are synchronous. This completes the proof.
Remark 1. In the absence of switching signal, control input and distributed delays the nominal system (7) is reduced to the following neural networks: Then by Theorem 1, it is easy to have the following corollary.
Remark 2. It will be mentioned that condition (9) in Theorem 1 is dependent on the additive time-varying delayĥ 1 (t),ĥ 2 (t), which cannot be solved directly by LMI tool. Noted thatΠ is a linear function on the variableĥ 1 (t),ĥ 2 (t) it is easy to show that condition (9) is satisfied for all 0 ĥ 1 (t) ĥ 1 , 0 ĥ 2 (t) ĥ 2 ifΠĥ 1(t),ĥ2(t)=0 < 0 andΠĥ 1(t)=ĥ1 ,ĥ2(t)=ĥ2 < 0. Remark 3. In the available existing literature, the stability problem has been discussed for various NNs through different techniques. In [5], the authors studied the stability problem of switched Hopfield NNs of neutral type with additive time-varying delay using Jensen integral inequality and Finser lemma. In [11], the authors discussed the problem of generalized neural networks with additive time-varying delay by using integral inequality technique (IIT), Writinger double integral inequality (WDII). Furthermore, some pioneering works have been done to transform the event-triggered control for synchronization of switched neural networks in [19], and event-triggered synchronization strategy for complex dynamical networks was studied in [17]. The model considered in the present study is more practical than that proposed by [5,19], because in this paper we consider event-triggered switched neural networks with additive time-varying delay. The scheme of event-triggered sampling is an effective way within the electronic chips with limited capacity and energy for their great abilities to reduce the data transmission and power consumption. which is the main contribution and motivation of our work. Hence, the results presented in this paper are essentially new.

Remark 4.
In general, if size of the LMIs increase then the computation burden will also increase. However, large size of LMIs yield better system performances. In this paper, the proposed criteria are employed by the several integral inequalities; as a result, some computational complexity can occur in the proposed criterion. To avoid the computational burden, in future the Finsler's lemma was applied in the proof of the main results, which in turn to reduce the computational burden. Moreover, in the future work, we will focus on lower computational complexity of the stability problems while maintaining the desired system performances.

Conclusion
The problem of synchronization of decentralized event-triggered has been addressed for uncertain switched neural networks with two additive time-varying delays. To reduce the communication burden in the network, the novel decentralized event-triggered communication scheme has been proposed. By using the Lyapunov functional method, convex combination techniques and Wirtinger-based integral inequality some sufficient conditions in terms of linear matrix inequalities have been derived to guarantee the exponential stability of the neural networks under consideration. Numerical examples are provided to demonstrate the effectiveness of the obtained results. In future work, we will utilize the proposed method to deal with stochastic neural networks, fuzzy neural networks, or other types of neural networks with time-varying delays.