Synchronization for general complex dynamical networks with sampled-data

doi:10.1016/j.neucom.2010.11.007

Neurocomputing

Volume 74, Issue 5, February 2011, Pages 805-811

https://doi.org/10.1016/j.neucom.2010.11.007 Get rights and content

Abstract

In this paper, the sampled-data synchronization control problem is investigated for a class of general complex networks with time-varying coupling delays. A rather general sector-like nonlinear function is used to describe the nonlinearities existing in the network. By using the method of converting the sampling period into a bounded time-varying delay, the addressed problem is first transformed to the problem of stability analysis for a differential equation with multiple time-varying delays. Then, by constructing a Lyapunov functional and using Jensen's inequality, a sufficient condition is derived to ensure the exponential stability of the resulting delayed differential equation. Based on that, the desired sampled-data feedback controllers are designed in terms of the solution to certain linear matrix inequalities (LMIs) that can be solved effectively by using available software. Finally, a numerical simulation example is exploited to demonstrate the effectiveness of the proposed sampled-data control scheme.

Introduction

Complex dynamical networks are comprised a large set of nodes evolving according to their respective dynamical equations. Some of these nodes are usually coupled according to the network topology. In the real world, a large number of practical systems can be represented by models of complex networks, such as Internet, World Wide Web, food webs, electric power grids, cellular and metabolic networks, scientific citation networks, social networks, etc. So far, the complex dynamical network has already been one of the most popular topics in the areas of scientific research. As we all know, the synchronization of all dynamical nodes is one of the most significant and interesting properties in a complex network. Therefore, the synchronization problem for complex dynamical networks has received increasing research attention, and a great deal of results have been available in the literature [1], [2], [3], [7], [9], [11], [12], [13], [14], [15], [18], [19], [21], [22], [26], [29]. For example, the dynamical network model in terms of a differential equation with a coupling term has been proposed in [22], where the synchronization phenomenon of all dynamical nodes has been investigated for the considered networks. Subsequently, some synchronization criteria have been established for a class of complex dynamical networks in [7], [13], where the coupling delays have been taken into account. In [15], the synchronization problem has been investigated for an array of coupled complex discrete-time networks with the simultaneous presence of both the discrete and distributed time delays. In [14], the exponential synchronization problem has been studied for a class of stochastic delayed discrete-time complex networks and the corresponding synchronization criteria have been derived. Very recently, in [26], the global synchronization problem has been fully investigated for a class of discrete-time stochastic complex networks, where both randomly occurred nonlinearities and mixed time delays have been taken into account.

Note that it is often the case that, for a complex network, all the behaviors of the dynamical nodes do not evolve synchronously. As such, it would be more interesting to study the condition that can guarantee the synchronization of all the nodes in complex networks and how to achieve synchronization for an asynchronous complex network. A natural idea is to introduce a set of controllers to adjust the behaviors of the network nodes and accordingly achieve the network synchronization. This is referred to as the synchronization control problem. Some efforts have been made on this topic and various control schemes have been proposed. For example, in [31], an impulsive control scheme has been proposed to achieve synchronization for complex dynamical networks with unknown coupling, and the synchronization strategy considers the influence of all nodes in the dynamical network. In [28], the synchronization problem via pinning control has been studied and the derived pinning condition with controllers given in a high-dimensional setting can be reduced to a low-dimensional condition without the pinning controllers involved. In [33], the locally and globally adaptive synchronization control problems have been considered for an uncertain complex dynamical network and a set of adaptive controllers has been designed for the network synchronization.

On the other hand, as the rapid development of computer hardware, the sampled-data control technology has shown more and more superiority over other control approaches. The sampled-data control theory also has rapidly developed. It is worthwhile to mention that, in [5], [6], a new approach to dealing with the sample-data control problems has been proposed by converting the sampling period into a time-varying but bounded delay. By following this idea, in [30], the exponential synchronization sampled-data control problem has been studied for neural networks with time-varying mixed delays, while the robust sampled-data $H_{\infty}$ control problem for vehicle active suspension systems has been investigated in [8]. To the best of our knowledge, so far, the sampled-data synchronization control problem has not been considered for the complex networks yet. Therefore, in this paper, we aim to design a set of sampled-data feedback controllers to achieve the synchronization of a class of general asynchronous complex networks with time-varying coupling delays. The contributions of this work can be summarized as follows: (1) we make the first attempt to address the sampled-data synchronization control problem for a class of general complex networks with time-varying coupling delays; (2) a synchronization criterion depending on the sampling period is obtained by constructing an appropriate Lyapunov functional and using Jensen's inequality; and (3) a set of sampled-data feedback controllers is designed that can achieve the synchronization of the considered complex network.

The remainder of this paper is organized as follows. The exponential sampled-data synchronization control problem for complex networks with time-varying coupling delays is formulated in Section 2. In Section 3, a exponential synchronization criterion is derived and a set of sampled-data feedback controllers is designed for the considered complex network. An illustrative example is provided to show the effectiveness of the proposed sampled-data control scheme in Section 4. Finally, conclusions are given in Section 5.

Notation: The notation used here is fairly standard except where otherwise stated. $R^{n}$ denotes the n-dimensional Euclidean space. $∥ A ∥$ refers to the norm of a matrix A defined by $∥ A ∥ = \sqrt{trace (A^{T} A)}$ . The notation $X \geq Y$ (respectively, $X > Y$ ), where X and Y are real symmetric matrices, means that X−Y is positive semi-definite (respectively, positive definite). M^T represents the transpose of the matrix M. I denotes the identity matrix of compatible dimension. $diag {\dots}$ stands for a block-diagonal matrix and the notation ${diag}_{n} {⋄}$ is employed to stand for $diag {\overset{n}{\overset{︷}{⋄, \dots, ⋄}}}$ . Moreover, the asterisk $⁎$ in a matrix is used to denote term that is induced by symmetry. Matrices, if they are not explicitly specified, are assumed to have compatible dimensions.

Section snippets

Problem formulation and preliminaries

Consider a complex dynamical network consisting of N identical coupled nodes as follows: ${\dot{x}}_{i} (t) = f (x_{i} (t)) + c \sum_{j = 1}^{N} G_{ij} {Ax}_{j} (t - τ (t)) + u_{i} (t), i = 1, 2, \dots, N,$ where $x_{i} (t) \in R^{n}$ and $u_{i} (t) \in R^{n}$ are, respectively, the state variable and the control input of the node i. $f : R^{n} \to R^{n}$ is a continuous vector-valued function. The scalar $τ (t)$ denotes the time-varying delay satisfying $0 \leq τ (t) \leq μ, \dot{τ} (t) \leq ν,$ where $μ$ and $ν$ are known positive constants. $c > 0$ is the coupling strength, $A = (a_{ij})_{n \times n} \in R^{n \times n}$ is a constant inner-coupling matrix of the

Main results

In this section, we first analyze the stability and deriving the corresponding exponential stability condition for the error system (7). Then, the sampled-data feedback controllers are designed to achieve the exponential synchronization of the complex network (1).

The following lemma will be used to derive our main results.

Lemma 1

Jensen's inequality [10]

For any constant matrix $R \in R^{n \times n}$ , $R = R^{T} > 0$ , scalar $b > 0$ , and vector function $x : [0, b] \to R^{n}$ , one has $b \int_{0}^{b} x^{T} (s) Rx (s) ds \geq {(\int_{0}^{b} x (s) ds)}^{T} R (\int_{0}^{b} x (s) ds)$ provided that the above integrals are well

An illustrative example

In this section, a simulation example is presented to demonstrate the effectiveness of the proposed synchronization control scheme for the complex network (1).

Consider a complex network (1) with three nodes. The outer-coupling matrix is assumed to be $G = (G_{ij})_{3 \times 3}$ with $G_{ij} = [\begin{matrix} - 1 & 0 & 1 \\ 0 & - 1 & 1 \\ 1 & 1 & - 2 \end{matrix}] .$ The inner-coupling matrix and coupling strength are given as A=diag{1,1,1} and c=0.5, respectively. The time-varying delay is chosen as $τ (t) = 0.2 + 0.05 \sin (10 t)$ . Accordingly, we have $μ = 0.25$ and $ν = 0.5$ .

The nonlinear

Conclusions

In this paper, we have studied the sampled-data synchronization control problem for a class of general complex networks with time-varying coupling delays. The addressed problem has first been transformed to the problem of stability analysis for a differential equation with multiple time-varying delays by converting the sampling period into a bounded time-varying delay. Then, by constructing an appropriate Lyapunov functional and using Jensen's inequality, an exponential stability criterion has

Nan Li received her master degree in Control Theory and Control Engineering from Zhejiang University of Technology in 2009. Now she is a lecturer at Zhejiang Ocean University. Her research interests include neural networks, complex networks, nonlinear systems and bioinformatics.

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Cited by (0)

Yulian Zhang received her master degree in Engineering Mechanics from Xi’an Jiaotong University in 2002. Now she is a professor at Zhejiang Ocean University. Her research interests include complex networks, fluid dynamics and structural optimization.

Jiawen Hu received his bachelor degree in Mechanical Design, Manufacturing and Automation from Zhejiang Ocean University in 2003. Now he is a technician at Zhejiang Ocean University. His research interests include microcontroller technology and electronic technology.

Zhenyu Nie received his master degree in Power Electronics from Nanchang University in 2005. Now he is a lecturer at Zhejiang Ocean University. His research interests include the development of power electronics device of shipping.

^☆: This work was supported by the Research Project of Zhejiang Ocean University under Grant X09Z4.

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Synchronization for general complex dynamical networks with sampled-data☆

Abstract

Introduction

Section snippets

Problem formulation and preliminaries

Main results

Jensen's inequality [10]

An illustrative example

Conclusions

Physica D-Nonlinear Phenomena

Neural Networks

Automatica

Systems & Control Letters

Physics Letters A

Physica A—Statistical Mechanics and its Applications

Neural Networks

Neurocomputing

Automatica

Automatica

Neurocomputing

Automatica

Automatica

Neurocomputing

Neurocomputing

Graph operations and synchronization of complex networks

Physical Review E