Impulsive cluster synchronization for complex dynamical networks with packet loss and parameters mismatch

Highlights

• An impulsive control strategy with multiple control gain is proposed.

• Sufficient conditions are derived to guarantee that CDNs can be synchronized to their cluster sets.

• The relationship among rate coefficients, impulsive control law, and iPLR is revealed.

Abstract

This paper investigates the cluster synchronization (CS) for a class of complex dynamical networks (CDNs) with parameters mismatch to the selected cluster pattern. An impulsive control strategy with multiple control gains is proposed. Particularly, the possible packet loss phenomenon in control input is fully considered, and it is characterized by a series of impulse packet loss rate (iPLR) conditions. In terms of the reverse average dwell time (rADT) condition and the average impulse gain method, some sufficient conditions guaranteing that all the nodes of CDNs can be synchronized to their cluster sets are derived, from which the constraint relationships among rate coefficients, impulsive control law, and iPLR can be revealed. Finally, an example is given to illustrate the correctiveness of our results.

Introduction

During the last decade, the study of complex dynamical networks (CDNs) has become a focal research topic in various fields, e.g., mathematics, biology, physics, sociology and so on [1], [2], [3], [4]. CDNs usually consist of a large of sets of coupled interconnected nodes, in which each node is an independent dynamical system. Synchronization phenomenon, which is firstly studied by Huygens in the 17th century [5], can be defined as a process wherein two (or many) dynamical systems adjust a given property of their motion to a common behavior as time goes to infinity, due to coupling or forcing. Up to now, the problems of synchronization on CDNs have been attracting a lot of attention [6], [7], [8], [9], [10]. As a special kind of synchronization phenomenon, cluster synchronization (CS) means that the CDNs can be divided into several subgroups called clusters, and the coupled nodes belonging to the same group can achieve synchronization, whereas the synchronous states of these groups are mutually different [11]. In recent years, CS gradually reveals its potential application in the field of network engineering, and many excellent and outstanding results have been derived [12], [13], [14], [15]. Specially, Wu et al. studied the problem of how to drive a general network to a selected cluster synchronization pattern by means of a single control strategy [12], and the method of adapting the coupling strength was provided to refine the results; The issue on CS of CDNs with parameters mismatch and time-varying delay was investigated by Wang et al. [15], where the proposed control strategy was dependent not only on the current state and the history state but also on the number of nodes to be controlled.

In engineering environment, some abrupt changes may interfere system behaviour at certain instants, which is hard to be considered continuously. These abrupt changes are usually called impulsive phenomenon [16], [17], [18], [19], [20], [21]. Interestingly, sometimes even only impulse can be used for control purpose instead of continuous feedback control. Hence, the control method based on impulse, named impulsive control, with simple characteristic is activated at artificial impulse sequence and has excellence in installing, implementing, and maintaining, which could reduce the consumption of control costs and the resource loss of information interaction significantly. As an effective control method, impulsive control has been receiving increasing attention in the recent years [8], [22], [23], [24], [25], [26], [27], especially for CDNs. For instance, Lu et al. proposed the concept of average impulsive interval (AII) to characterize impulse behaviour in directed impulsive dynamical networks [8], and presented the corresponding unified synchronization criterion; The pinning impulsive synchronization problem for a class of CDNs with time-varying delay was studied by Wang et al., and a set of novel sufficient conditions were constructed to relax the restrictions on the size of time-delay and guarantee the synchronization of concerned networks with large delay [27].

In networks engineering practice, the information packet may be lost because of some internal or external factors, such as communication attack or congestion, insufficient power supply, etc [28], [29]. When the information packet is lost, the control input cannot be implemented to help achieve some properties of networks, such as CS. As consequence, it is more meaningful to consider the control problem of CDNs under packet loss, and many meaningful works focusing on packet loss have been proposed, see [30], [31] and so on. However, most of existing results are only focused on the case that packet loss occurs at continuous-time dynamical behaviour. Considering the randomness of packet loss phenomenon, it is natural to think what if packet loss occurs at some discontinuous dynamical behaviour of networks. More seriously, when those discontinuous dynamical behaviours play positive effects on some dynamical performances of networks, whether the expected performances can be achieved remains unanswered. Based on this basic thinking, we fully consider the possible packet loss phenomenon in impulsive control strategy, which is proposed to achieve the CS for CDNs with parameters mismatch to a selected cluster pattern. By characterizing the packet loss phenomenon via the designed impulse packet loss rate (iPLR) condition, some sufficient criteria, under which all the nodes of CDNs can be synchronized to their cluster sets, and the constraint relationships among rate coefficients, impulsive control law, and iPLR can be revealed, are derived in terms of reverse average dwell time (rADT) condition and average impulse gain method. Accordingly, it is shown that the robustness against packet loss can be improved as required by adjusting the corresponding parameters.

The rest of this paper is organized as follows: Section 2 introduces some notations and research problem. The main results are presented in Section 3. An example is provided in Section 4. Section 5 gives the conclusion.

Notations. Let ${\mathbb{Z}}_{+}$ denote the set of positive integer numbers, $\mathbb{R}$ the set of real numbers, ${\mathbb{R}}_{+}$ the set of nonnegative real numbers, and ${\mathbb{R}}^n$ and ${\mathbb{R}}^{n\times m}$ the $n$-dimensional Euclidean space and the set of all $n\times m$ real matrices, respectively. $I$ represents the identity matrix with compatible dimension. The notation $P>(\ge ,<,\le )0$ is used to denote a real symmetric positive-definite (respectively, positive-semidefinite, negative, and negative-semidefinite) matrix. ${\lambda }_{max}\left(A\right)$ and ${\lambda }_{min}\left(A\right)$ represent the maximum and minimum eigenvalues of symmetric real matrix $A,$ respectively. For vector $x\in {\mathbb{R}}^n,$ $\parallel x\parallel$ is the Euclidean norm defined as $\parallel x\parallel ={\left({\Sigma }_{i=1}^n{x}_i^2\right)}^{\frac{1}{2}}$. For matrix $A\in {\mathbb{R}}^{n\times n},\parallel A\parallel =\sqrt{{\lambda }_{max}\left(A^{T}A\right)}$. The symbol $★$ indicates a symmetric structure in matrix expressions. $A^T$ and $A^{-1}$ represent the transpose and inverse matrix of $A$, respectively. $a\vee b$ and $a\wedge b$ are the maximum and minimum of $a$ and $b$, respectively.