Discrete Dynamics in Nature and Society

Volume 2012 (2012), Article ID 857561, 11 pages

http://dx.doi.org/10.1155/2012/857561

## Impulsive Cluster Anticonsensus of Discrete Multiagent Linear Dynamic Systems

School of Mathematics, Yancheng Teachers University, Yancheng 224002, China

Received 8 September 2011; Accepted 9 December 2011

Academic Editor: Piyapong Niamsup

Copyright © 2012 Liping Zhang and Haibo Jiang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Cluster anticonsensus is another important type consensus of multiagent systems. In this paper, we investigate the problem of impulsive cluster anticonsensus of discrete multiagent linear dynamic systems. Firstly, an impulsive protocol is designed to achieve the cluster anticonsensus. Then sufficient conditions are given to guarantee the cluster anticonsensus of the discrete multiagent linear dynamic system based on the -theory. Numerical simulation shows the effectiveness of our theoretical results.

#### 1. Introduction

Recently, the consensus problem of multiagent systems has been intensively studied in the literature [1–4]. In [1, 2], a systematical framework of consensus problem in networks of agents was investigated. The problem of information consensus among multiple agents in the presence of limited and unreliable information exchange with dynamically changing interaction topologies was considered in [3]. In [4], the authors considered the consensus problem for multiagent systems, in which all agents have an identical linear dynamic mode that can be of any order. On the other hand, cluster anticonsensus is another important type consensus of multiagent systems. When the multiagent systems achieve cluster anticonsensus, the nodes in the same group achieve consensus with each other, but there is no consensus between nodes in different groups. Very recently the signless Laplacian has attracted the attention of researchers. Several papers on the signless Laplacian spectrum have been reported since 2005 and a new spectral theory of graphs which is called the -theory is developing by many researchers [5, 6]. To the best of our knowledge, however, there are very few results on cluster anticonsensus of multiagent systems, which motivates this study.

Impulsive control is widely used in various applications, such as ecosystems, financial systems, mechanical systems with impacts, and orbital transfer of satellite [7–12]. Very recently, impulsive control protocol for multiagent systems has received much attention [13–19]. In [14], the authors introduced impulsive control protocols for multiagent linear continuous dynamic systems. The convergence analysis of the impulsive control protocol for the networks with fixed and switching topologies is presented, respectively. The proposed impulsive control protocol is only applied to the multiagent system at certain discrete instants, which is different from continuous control protocol [1, 2, 4]. In [15], the authors investigated the problem of impulsive synchronization of networked multiagent systems, where each agent can be modeled as an identical nonlinear dynamical system.

In this paper, we investigate the problem of impulsive cluster anticonsensus of discrete multiagent linear dynamic systems. The main contribution of our paper includes (1) an impulsive control protocol is introduced to seek the cluster anticonsensus of discrete multiagent linear dynamic systems and (2) a new type consensus, that is, cluster anticonsensus is studied.

This paper is organized as follows. In Section 2, we provide some results in the -theory. In Section 3, we formulate the cluster anticonsensus problem for discrete multiagent linear dynamic systems and introduce the impulsive control protocol. The convergence analysis of the cluster anticonsensus problem is discussed in Section 4. In Section 5, numerical simulation is included to show the effectiveness of our theoretical results. Some conclusions are drawn in Section 6.

*Notation 1. *Throughout this paper, the superscripts “−1” and “” stand for the inverse and transpose of a matrix, respectively; denotes the -dimensional Euclidean space; let , , ; is the set of all real matrices; for real symmetric matrices and , the notation (resp., ) means that the matrix - is positive semidefinite (resp., positive definite); is an identity matrix; () denotes the smallest (largest) eigenvalue of . For a vector , let denote the Euclidean vector norm, that is, , and for , let indicate the norm of induced by the Euclidean vector norm, that is, . The Kronecker product of two matrices and is denoted by . For more properties of the Kronecker product the reader is referred to [20].

#### 2. Preliminaries

In this section, we provide some results in the -theory [4–6, 21].

An undirected graph of order consists of a vertex set and an edge set . The set of neighbors of vertex is denoted by . A path between each distinct vertices and is meant a sequence of distinct edges of of the form . A cycle is a path such that the start vertex and end vertex are the same. If there is a path between any two vertices of a graph , then is connected, otherwise disconnected. A graph is a bipartite graph if can be partitioned into two disjoint subsets and , called partite sets, such that every edge of joins a vertex of and a vertex of . A graph is bipartite if and only if it does not contain an odd cycle.

A weighted adjacency matrix , where and , . if and only if there is an edge between vertex and vertex . For an unweighted graph , is a 0-1 matrix. The out-degree of vertex is defined as follows . Let be the diagonal matrix with the out-degree of each vertex along the diagonal and call it the degree matrix of . The signless Laplacian matrix of the weighted graph is defined as . For an unweighted graph , , where here denotes the cardinality of the set .

Let be an undirected graph on vertices, having edges. Let be its vertex edge incidence matrix which is an matrix such that if the vertex and edge are incident and 0 otherwise. The following relations are well-known: . Thus the signless Laplacian is a positive semi-definite matrix, that is, all its eigenvalues are non-negative. Let be a graph with -eigenvalues ().

Lemma 2.1. *Let be the singless Laplacian matrix of an undirected graph with vertices, and be the eigenvalues of . Suppose that the graph is bipartite. Let and be two subsets of graph . Define , , , , and , , , , . Then*(1)*if is connected, then is the algebraically simple eigenvalue of and the corresponding eigenvector is ;*(2)* if 0 is the simple eigenvalue of , then it is an multiplicity eigenvalue of and the corresponding eigenvectors are , .*

#### 3. Problem Formulation

Here we consider a system consisting of agents indexed by . The dynamics of each agent is where is the state of agent at time , , is a constant matrix. We assume that .

The following impulsive control protocol is applied to agent where , , are impulsive matrices to be determined later, is the set of neighbors of agent at time , the discrete instants satisfy , and . When , , and denote the instant just after and just before , respectively, which is also considered in [8, 10]. Without loss of generality, we assume that , , .

Under the impulsive control protocol (3.2), the dynamics of agent satisfies the following equations

*Definition 3.1. *For the system (3.1), the cluster anticonsensus is said to be achieved under the impulsive control protocol (3.2) if
where and are two nonempty subsets of and , .

Consider the following discrete impulsive system where , , . Then the solution satisfies where , .

The representation (3.6) implies the following.

Lemma 3.2. *All solutions of the system (3.5) are asymptotically stable if the conditions (H1) and (H2) hold,*(H1)*, , *(H2)*, ,** where , .*

#### 4. Main Results

In this section, we provide convergence analysis of the cluster anticonsensus problem of discrete multiagent linear dynamic systems.

Let , then the system (3.3) can be described as

Since is symmetric, there is an orthogonal matrix such that

where is the spectrum of .

Let Using the properties of Kronecker product, we have when , ,

and when , ,

Thus (4.1) becomes Therefore

Theorem 4.1. *Consider the system (3.1). Assume that the graph of the network is connected and bipartite. If there exist discrete instants and impulsive matrices such that the conditions (H3) and (H4) hold, then the cluster anticonsensus is achieved under the impulsive control protocol (3.2),*(H3)*, ,*(H4)*, , ,** where , , .*

*Proof. *Since the graph is connected and bipartite, by Lemma 2.1, is the algebraically simple eigenvalue of . All the other eigenvalues of are positive. Then we have .

By Lemma 3.2, it follows that if there exist discrete instants and impulsive matrices such that the conditions (H3) and (H4) hold, then the system (4.7) is asymptotically stable, that is, , , .

It can be verified that
Hence , . Since the graph is connected and bipartite, by Lemma 2.1, 0 is the eigenvalue of with multiplicity . The linearly independent eigenvectors associated with the eigenvalue 0 of are , . Therefore , , where , , . Thus the system (3.1) achieves the cluster anticonsensus under the impulsive control protocol (3.2). This completes the proof.

*Remark 4.2. *For simplicity, we usually choose the equidistant impulsive interval , . The impulsive matrices , , are chosen as , . If , , then the conditions (H3) and (H4) are satisfied.

#### 5. Simulations

Consider the following multiagent discrete linear dynamic system where . Simulation results for the system (5.1) are shown in Figures 1, 2, and 3. From Figures 1–3, we know that the system cannot achieve the cluster anticonsensus without the impulsive control protocols.

The control input of agent is designed as (3.2). The graph is considered as a simple path on four vertices and the singless Laplacian matrix of graph is . For simplicity, the impulsive matrices , , are chosen as . Choose the equidistant impulsive interval , . It is easy to check that , , , where , , . The initial values are chosen as , , , . Simulation results for are shown in Figures 4, 5, and 6. The simulation results show that the cluster anticonsensus of the multiagent discrete linear dynamic system is achieved by the impulsive control protocol. Besides, let , then , and , .

If the graph is considered as a simple circle on four vertices, then the singless Laplacian matrix of graph is . The other parameters are chosen as above. Then the conditions (H3) and (H4) in Theorem 4.1 are also satisfied. Simulation results for are shown in Figures 7, 8, and 9. From Figures 4–9, it can be seen that the speed of convergence of cluster anticonsensus algorithms is closely related with the network structure.

#### 6. Conclusions

In this paper, we have introduced impulsive control protocols for discrete multiagent linear dynamic systems. The convergence analysis of the cluster anticonsensus is presented. When the multiagent systems achieve cluster anticonsensus, the nodes in the same group achieve consensus with each other, but there is no consensus between nodes in different groups. In our future, we will consider the impulsive cluster anticonsensus problem of multiagent nonlinear dynamic systems.

#### Acknowledgments

The authors are grateful to the anonymous reviewers for their valuable comments and suggestions that have helped to improve the presentation of the paper. They also thank Professor Shuguang Guo and Mr. Zhongfeng Hu for helpful discussion. This paper is partially supported by the National Natural Science Foundation of China (Grant no. 11171290), the Natural Science Foundation of Jiangsu Province of China (Grant nos. BK2010292, BK2010293), the Natural Science Foundation of the Jiangsu Higher Education Institutions of China (Grant nos. 09KJB510018, 10KJB510026), and the Qin Lan Project of the Jiangsu Higher Education Institutions of China.

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