Synchronization criterion for Lur'e type complex dynamical networks with time-varying delay

Abstract

In this Letter, the synchronization problem for a class of complex dynamical networks in which every identical node is a Lur'e system with time-varying delay is considered. A delay-dependent synchronization criterion is derived for the synchronization of complex dynamical network that represented by Lur'e system with sector restricted nonlinearities. The derived criterion is a sufficient condition for absolute stability of error dynamics between the each nodes and the isolated node. Using a convex representation of the nonlinearity for error dynamics, the stability condition based on the discretized Lyapunov–Krasovskii functional is obtained via LMI formulation. The proposed delay-dependent synchronization criterion is less conservative than the existing ones. The effectiveness of our work is verified through numerical examples.

Introduction

Complex dynamical networks have received a great deal of attention since they are shown to widely exist in various fields of real world [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], [21], [22]. A complex network is a set of interconnected nodes, in which a node is a basic unit with specific contents or dynamics. Examples of complex networks include the Internet, the World Wide Web (WWW), food chain, electricity distribution networks, relationship networks and disease transmission networks, etc. Many of these networks exhibit complexity in the overall topological properties and dynamical properties of the network nodes and the coupled units. The complex nature of the networks has results in a series of important research problems. In particular, one significant and interesting phenomenon is the synchronization of all its dynamics. Therefore, many researchers have focused on this topic and have developed several efficient synchronization techniques for complex dynamical networks [3], [4], [5], [6], [7], [8], [9]. Wang and Chen introduced a uniform dynamical network model and investigated its synchronization and control [6]. Li and Chen [7] further extended the network model to include coupling delays among the network nodes and studied its synchronization. Gao et al. [8] considered the synchronization stability of both continuous- and discrete-time networks with coupling delays. Li et al. [9] proposed the synchronization criterion of both continuous- and discrete-time networks with time varying delays. However, the obtained results by using the linearization are not guaranteed the globally asymptotic stability.

Recently, many researches for the synchronization of Lur'e systems were presented, because various chaotic systems such as Chua's circuit [10] can be modeled as Lur'e systems. The Lur'e system is continuum of a linear system and a feedback nonlinearity satisfying sector bound constraints. The stability of the Lur'e system is called absolute stability that means global asymptotic stability [11]. For these reason, many researcher have studied the synchronization for Lur'e systems and applied to various applications. Suykens et al. [12], [13], [14] investigated extensively synchronization problem including ${\mathcal{H}}_{\infty }$ synchronization for Lur'e systems. More recently, practical issues of the synchronization such as propagation delay have been considered since Chen and Liu introduced the delay on the chaotic synchronization and showed that the delay may break synchronization [15]. There are several studies that considered the delay effect on the synchronization of Lur's systems [16], [17], [18], [19], [20], [21]. In those researches, synchronization criteria were derived for given gain matrices of the synchronization controller and time delay, which are sufficient conditions for absolute stability of the synchronization. However, the model transformation technique used in [16], [17] can lead to conservative conditions by inducing additional dynamics as addressed in [18]. In order to derive less conservative conditions for synchronization, synchronization criteria that does not use the model transformation were presented independently in [18], [19], [20]. Guo et al. [18] applied a free weighting matrix approach employed on the Libnitz–Newton formula and an equality constraint for the synchronization criterion. Xiang et al. [19] used the integral inequality and a free weighting matrix approach. In [20], a more general Lur'e–Postnikov Lyapunov functional was presented to derive a less conservative criterion. Furthermore, a synchronization method for the chaotic Lur'e system was extended for time-varying delay in [21]. Very recently, the synchronization problem is addressed for complex dynamical networks in which every identical node is a time-delayed Lur'e system [22]. However, it is worth noting that the time-delays of system dynamical states considered in these works are assumed to be constant. Time-varying delay case, which is more general than the constant one should be considered. Moreover, in much of the literature, time delays in the couplings are considered. However, the time delays in the dynamical nodes, which are more complex, are still relatively unexplored. Therefore, the synchronization problem of nonlinear Lur'e type complex dynamical networks with time-varying delay still remains challenging.

In this Letter, we will propose a delay dependent synchronization criterion for Lur'e type complex dynamical networks with a time varying delay. Convex representation of the nonlinearity of the Lur'e networks is introduced, and then, a sector bounded constraint of the nonlinearity is converted to an equality constraint. The projection lemma [23] is utilized for handling the equality constraint so that a less conservative delay-dependent criterion is obtained. Furthermore, the discretized Lyapunov–Krasovskii functional that employs redundant state of differential equations shifted in time by a fraction of the time delay is also applied to reduce conservatism in searching the maximum allowable delay such that the error dynamics of synchronization are absolutely stable. The derived criterion is formulated by LMIs that are easily solvable using various numerical methods [24].

Notations

The following notations are used in the Letter. ${\mathbb{R}}^n$ denotes the n-dimensional Euclidean space. ${\mathbb{R}}^{n\times m}$ is the set of all $n\times m$ real matrices. For a real matrix X, $X>0$ and $X<0$ mean that X is a positive/negative definite symmetric matrix, respectively. I is an identity matrix with appropriate dimension and 0 is a null matrix with appropriate dimension. For given matrix $A\in {\mathbb{R}}^{n\times m}$ such that $\mathrm{rank}(A)=r$, we define $A^{\perp }\in {\mathbb{R}}^{n\times (n-r)}$ as the right orthogonal complement of A by $A{A}^{\perp }=0$. $\mathrm{diag}(\cdots )$ represents a block diagonal matrix. $A\otimes B$ indicates the Kronecker product of a $n\times m$ matrix A and a $p\times q$ matrix B, i.e.,

$$A\otimes B=\left[\begin{array}{ccc}\hfill a_{11}B\hfill & \hfill \cdots \hfill & \hfill a_{1m}B\hfill \\ \hfill ⋮\hfill & \hfill \ddots \hfill & \hfill ⋮\hfill \\ \hfill a_{n1}B\hfill & \hfill \cdots \hfill & \hfill a_{nm}B\hfill \end{array}\right].$$