Synchronization of discrete-time Markovian jump complex dynamical networks with random delays via non-fragile control

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Abstract

In this proposed article, a framework is presented for the analysis of the problem of synchronization of Markovian jumping discrete-time complex dynamical networks (CDNs) with probabilistic interval time-varying delay in the dynamical node and in the network coupling. The networks are expressed in terms of Kronecker product technique. The delay in time is taken to be unexpected and the probability distribution is known a prior. The synchronization is achieved by introducing a non-fragile procedure. This controller is subject to randomly occurring perturbation and is assumed to belong to the Binomial sequence. A suitable Lyapunov–Krasovskii functional (LKF) with triple summation terms is considered. By utilizing the reciprocal convex combination approach and Finsler׳s Lemma, conditions for the synchronization of networks are established in terms of linear matrix inequalities (LMIs). The effectiveness of the results obtained theoretically are illustrated through two numerical examples.

Introduction

In the past few years, complex dynamical networks (CDNs) have become a focal research topic. A CDN is a collection of dynamic systems, called as nodes, connected to one another with complex topological properties. Examples of complex networks include the Internet, the World Wide Web (WWW), food chain, electricity distribution networks, relationship networks, disease transmission networks, and so on. Many of these networks manifests complexity in the overall topological and dynamical properties of the network nodes and also in the coupled units. Among all the dynamical phenomena of complex networks, synchronization is an important property. Many advantageous methods have been proposed for the network synchronization, as can be seen in [1], [2], [3], [4], [5]-6].

The time-delay is caused due to traffic congestions, finite speeds of transmission, memory effects, etc. Even though the introduction of time-delays complicate the dynamical behavior of the network, time-delays have to be considered for simulating a more realistic complex network. The result for CDNs with various forms of time delays such as mixed or multiple time delays, time-varying delays, delays in coupling terms has been developed in [7], [8], [9], [10]-11].

As in [12], the role of discrete nature of network topology is vital in understanding the interaction topology of complex networks. Such a remarkable yet stimulating problems has reasons as the process of discretization of a continuous-time network cannot keep the dynamics of the continuous-time part even for small sampling periods. Discrete-time network models digitally transmitted signals and has been applied in different areas, such as time series analysis, image processing, system identification, and quadratic optimization problems. The study on the analysis of stability, H control and filtering of various models of discrete-time systems with time-varying delays have been established in [13], [14], [15]-16]. In [17], authors have obtained the synchronization criteria for both continuous and discrete time CDNs.

The abrupt phenomena such as random failures, sudden environmental changes, changes in the interconnections of subsystems and repairs of the components, can be modeled by a class of hybrid systems termed as Markovian jump systems. Complex networks with Markovian jumping parameters are of great significance in modeling complex networks with finite network modes. Stability analysis for discrete-time Markovian jump neural networks and synchronization analysis of complex networks with hybrid coupling by handling multitude Kronecker product terms have been investigated in [18], [19], [20], [21], [22]-23].

As is known well, there exist few systems which are stable with some nonzero delay and are unstable without delay. In such cases, if there is a time-varying perturbation on the nonzero delay, it is meaningful to study the problems of the stability analysis and controller design of the systems with interval time-varying delay [24]. The stochastic variable α(t)R is introduced to describe the phenomena of randomly occurring controller gain fluctuation. Synchronization of stochastic complex networks with probabilistic interval discrete time-varying delays has been achieved in [25]. Global synchronization results for CDN has been attained in [26], [27] with probabilistic time-varying delay and information exchange at discrete-time.

Different control techniques have been used to study the stability of CDNs. It has to be noted in practice that the designed controller should be capable of tolerating some uncertainty in its coefficients due to the fact that the uncertainty cannot be avoided which is caused by many reasons, such as finite word length in digital systems, the imprecision inherent in analog systems, and the need for additional tuning of parameters in the final controller implementation [28]. In recent years, synchronization of neural networks and complex networks have been derived in [29], [30]-31]. For stochastic CDN and neural network, the adaptive control technique has been used to analyze the exponential state estimation in [32] and synchronization in [33]. To overcome the abrupt changes in the control input, very recently the authors in [34] have designed the non-fragile control with randomly changing perturbation and have derived the conditions for the synchronization of the networks.

Based on the above discussions, the proposed work of this paper is to study the asymptotic mean square synchronization of discrete-time CDNs with Markov jump parameters and randomly occurring time-varying delays. By constructing new triple summation terms in the LKF and by utilizing some most updated techniques like second order reciprocally convex approach, the explicit conditions for the asymptotic mean square synchronization of discrete-time CDNs with randomly occurring perturbations in the control input is obtained and are established in terms of LMIs. The feasibility of the derived criteria can easily be checked by resorting to MATLAB LMI Toolbox. To the best of authors׳ knowledge, the asymptotic mean square synchronization of discrete-time CDNs with Markov jump parameters and probabilistic time-varying delay using non-fragile controller with randomly occurring perturbations has not yet been studied, which is the main work proposed in this paper.

The outline of this paper is organized as follows. Problem formulation and relevant preliminaries are given in Section 2. Kronecker product technique is used to represent the discrete-time CDNs. In Section 3, sufficient conditions for the considered discrete-time CDN with Markov jump parameters and probabilistic time-varying delay to be asymptotically mean-square stable are derived in terms of LMIs. The results thus obtained are in terms of LMIs which can be efficiently solved using standard convex optimization algorithms [35]. Numerical examples are given in Section 4 to exhibit the effectiveness of the proposed method.

Notations: Throughout this paper standard notations are presented. Rn is the n-dimensional Euclidean space, whereas Rm×n denotes the collection of all m×n real matrices. For real symmetric matrices X,Y,XY (or X>Y) means that the matrix XY is nonnegative (or positive) definite. The superscript T stands for transposition of the matrix. The basis of the null-space of X is denoted by X. In,0n and 0m×n respectively denotes n×n, identity matrix, n×n,m×n zero matrices. The mathematical expectation operator is denoted by E{·}. Euclidean vector norm is referred as ·. diag{} denotes the block diagonal matrix. For any vectors xiRm(i=1,2,,n),col(x1,x2,,xn)Rm×n. ⊗ denotes the Kronecker product.

Section snippets

Problem description and preliminaries

Consider the below discrete-time CDNs with interval time varying delays in the coupling termyi(k+1)=f(yi(k),yi(kτ(k)))+cj=1NwijΓyj(kτ(k)),i=1,2,,N,where yi(k)=[yi1(k),yi2(k),,yin(k)]TRn is the state vector of the ith node, N is the number of coupling nodes, the vector valued function f:RnRn describes the dynamics of the individual nodes, the constant c refers to the coupling strength and the time varying delay is denoted by τ(k) such that0τ1mτ(k)τ2M,where τ1m and τ2M are known

Main results

The LMI framework along with the Lyapunov method are to be used to obtain the synchronization criteria for the system (15). For simplicity, the block entry matrices are defined using ϑi=[0κ0κi1Iκ0κ0κ17i]R17κ×κ, where κ=(N1)n, the other matrices are defined asζT(k)={xT(k)xT(kτ1m)xT(kτ1k)xT(kτ1M)xT(kτ2m)xT(kτ2k)xT(kτ2M)ΔxT(k)ΔxT(kτ1m)ΔxT(kτ1M)ΔxT(kτ2m)ΔxT(kτ2M)i=kτ2mkτ1M1ΔxT(i)i=kτ1Mkτ1k1ΔxT(i)i=kτ1kkτ1mΔxT(i)i=kτ2Mkτ2k1ΔxT(i)i=kτ2kkτ2MΔxT(i)}T,ξ1,1=[ϑ1ϑ2ϑ4ϑ5ϑ7]

Numerical example

Two examples are to provided to exhibit the effectiveness of the method and the derived criteria. In Example 4.1, the effectiveness of the proposed method can be seen through the maximum of the upper bound attained for the probabilistic time-varying delay. The non-fragile controller designed with randomly occurring perturbation is considered in Example 4.2. The benefit of such a useful controller can be seen through the model given.

Example 4.1

A second order system as in [17] is considered. The system can

Conclusion

The discrete-time CDN model in our work not only includes the sudden changes in the network such as changes in the interconnection of subsystems, random occurrence of time-varying delay, but also random perturbation of the uncertainties in the control. Thus, the problem of synchronization of discrete-time CDNs with Markovian jump parameters and probabilistic interval time-varying delay in the dynamical node and network coupling is considered. Kronecker product technique has been used to express

Acknowledgements

The author R. Sasirekha is thankful to the Department of Science and Technology, Government of India, New Delhi, for providing the financial support under DST Women Scientist Scheme number SR/WOS-A/MS-03/2014(G), to carryout this research work.

References (38)

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