Impulsive Cluster Anticonsensus of Discrete Multiagent Linear Dynamic Systems

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 𝑄-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 𝑄-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 𝑄-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 “𝑇” stand for the inverse and transpose of a matrix, respectively; ℝ^𝑛 denotes the 𝑛-dimensional Euclidean space; let ℝ₊=[0,∞), ℕ={0,1,2,…}, ℕ₊={1,2,…}; ℝ^𝑛×𝑚 is the set of all 𝑛×𝑚 real matrices; for real symmetric matrices 𝑋 and 𝑌, the notation 𝑋≥𝑌 (resp., 𝑋>𝑌) means that the matrix 𝑋-𝑌 is positive semidefinite (resp., positive definite); 𝐼_𝑛∈ℝ^𝑛×𝑛 is an identity matrix; 𝜆_min(𝑃) (𝜆_max(𝑃)) denotes the smallest (largest) eigenvalue of 𝑃. For a vector 𝑥∈ℝ^𝑛, let ‖𝑥‖ denote the Euclidean vector norm, that is, ‖𝑥‖=𝑥^𝑇𝑥, and for 𝐴∈𝑅^𝑛×𝑛, let ||𝐴|| indicate the norm of 𝐴 induced by the Euclidean vector norm, that is, ||𝐴||=𝜆_max(𝐴^𝑇𝐴). The Kronecker product of two matrices 𝐴=[𝑎_𝑖𝑗]∈ℝ^𝑚×𝑛 and 𝐵=[𝑏_𝑖𝑗]∈ℝ^𝑝×𝑞 is denoted by 𝐴⊗𝐵. For more properties of the Kronecker product the reader is referred to [20].

2. Preliminaries

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

An undirected graph 𝐺 of order 𝑁 consists of a vertex set 𝑉={1,2,…,𝑁} and an edge set 𝐸={(𝑖,𝑗)∶𝑖,𝑗∈𝑉}⊂𝑉×𝑉. The set of neighbors of vertex 𝑖 is denoted by 𝒩_𝑖={𝑗∈𝑉∶(𝑖,𝑗)∈𝐸,𝑗≠𝑖}. A path between each distinct vertices 𝑖 and 𝑗 is meant a sequence of distinct edges of 𝐺 of the form (𝑖,𝑘₁),(𝑘₁,𝑘₂),…,(𝑘_𝑙,𝑗). 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 𝐺, then 𝐺 is connected, otherwise disconnected. A graph 𝐺 is a bipartite graph if 𝑉(𝐺) can be partitioned into two disjoint subsets 𝑈 and 𝑊, called partite sets, such that every edge of 𝐺 joins a vertex of 𝑈 and a vertex of 𝑊. A graph is bipartite if and only if it does not contain an odd cycle.

A weighted adjacency matrix 𝐴=[𝑎_𝑖𝑗]∈ℝ^𝑁×𝑁, where 𝑎_𝑖𝑖=0 and 𝑎_𝑖𝑗=𝑎_𝑗𝑖≥0, 𝑖≠𝑗. 𝑎_𝑖𝑗>0 if and only if there is an edge between vertex 𝑖 and vertex 𝑗. For an unweighted graph 𝐺, 𝐴 is a 0-1 matrix. The out-degree of vertex 𝑖 is defined as follows deg_out(𝑖)=∑^𝑛_𝑗=1𝑎_𝑖𝑗. Let 𝐷 be the diagonal matrix with the out-degree of each vertex along the diagonal and call it the degree matrix of 𝐺. The signless Laplacian matrix of the weighted graph is defined as 𝑄_𝐺=𝐷+𝐴. For an unweighted graph 𝐺, 𝑄_𝐺=[𝑞_𝑖𝑗]_𝑁×𝑁, where 𝑞_𝑖𝑗=𝒩_𝑖,𝑖=𝑗,1,𝑗∈𝒩_𝑖,0,otherwise, (2.1) here |𝒩_𝑖| denotes the cardinality of the set 𝒩_𝑖.

Let 𝐺 be an undirected graph on 𝑁 vertices, having 𝑚 edges. Let 𝑅 be its vertex edge incidence matrix which is an 𝑁×𝑚 matrix such that 𝑟_𝑖𝑗=1 if the vertex 𝑖 and edge 𝑒_𝑗 are incident and 0 otherwise. The following relations are well-known: 𝑅𝑅^𝑇=𝐷+𝐴=𝑄. Thus the signless Laplacian is a positive semi-definite matrix, that is, all its eigenvalues are non-negative. Let 𝐺 be a graph with 𝑄-eigenvalues 𝑞₁,𝑞₂,…,𝑞_𝑁 (𝑞₁≤𝑞₂≤⋯≤𝑞_𝑁).

Lemma 2.1. Let 𝑄 be the singless Laplacian matrix of an undirected graph 𝐺 with 𝑁 vertices, and 𝑞₁≤𝑞₂≤⋯≤𝑞_𝑁 be the eigenvalues of 𝑄. Suppose that the graph 𝐺 is bipartite. Let 𝑈 and 𝑊 be two subsets of graph 𝐺. Define 𝜉∈ℝ^𝑁, 𝜉(𝑖)=1, 𝑖∈𝑈, 𝜉(𝑗)=−1, 𝑗∈𝑊 and 𝑒_𝑖∈ℝ^𝑛, 𝑒_𝑖(𝑖)=1, 𝑒_𝑖(𝑗)=0, 𝑗≠𝑖, 𝑖,𝑗=1,…,𝑛. Then(1)if 𝐺 is connected, then 𝑞₁=0 is the algebraically simple eigenvalue of 𝑄 and the corresponding eigenvector is 𝜉;(2) if 0 is the simple eigenvalue of 𝑄, then it is an 𝑛 multiplicity eigenvalue of 𝑄⊗𝐼_𝑛 and the corresponding eigenvectors are 𝜉⊗𝑒_𝑖, 𝑖=1,2,⋯,𝑛.

3. Problem Formulation

Here we consider a system consisting of 𝑁 agents indexed by 𝑖=1,2,…,𝑁. The dynamics of each agent is𝑥^𝑖(𝑡+1)=𝐴𝑥^𝑖(𝑡),𝑡≥𝑡₀≥0,𝑖=1,2,…,𝑁, (3.1) where 𝑥^𝑖(𝑡)=(𝑥^𝑖₁(𝑡),𝑥^𝑖₂(𝑡),…,𝑥^𝑖_𝑛(𝑡))^𝑇∈ℝ^𝑛 is the state of agent 𝑖 at time 𝑡, 𝑡∈ℕ, 𝐴∈ℝ^𝑛×𝑛 is a constant matrix. We assume that ‖𝐴‖≠0.

The following impulsive control protocol is applied to agent 𝑖𝑢^𝑖(𝑡)=Δ𝑥^𝑖(𝑡)=𝑥^𝑖𝑡⁺−𝑥^𝑖(𝑡⁻)=−𝐵_{𝑘𝑗∈𝒩𝑖(𝑡−)}𝑥^𝑗(𝑡⁻)+𝑥^𝑖(𝑡⁻),𝑡=𝑡_𝑘,𝑘∈ℕ₊, (3.2) where 𝐵_𝑘∈ℝ^𝑛×𝑛, 𝑘∈ℕ₊, are impulsive matrices to be determined later, 𝒩_𝑖(𝑡⁻) is the set of neighbors of agent 𝑖 at time 𝑡⁻, the discrete instants 𝑡_𝑘∈ℕ₊ satisfy 0≤𝑡₀<𝑡₁<𝑡₂<⋯<𝑡_𝑘−1<𝑡_𝑘<⋯, and lim_𝑘→+∞𝑡_𝑘=+∞. When 𝑡=𝑡_𝑘, 𝑘∈ℕ₊, 𝑡⁺ and 𝑡⁻ denote the instant just after 𝑡 and just before 𝑡, respectively, which is also considered in [8, 10]. Without loss of generality, we assume that 𝑥^𝑖(𝑡⁻)=𝑥^𝑖(𝑡), 𝑡=𝑡_𝑘, 𝑘∈ℕ₊.

Under the impulsive control protocol (3.2), the dynamics of agent 𝑖 satisfies the following equations 𝑥^𝑖(𝑡+1)=𝐴𝑥^𝑖(𝑡),𝑡≠𝑡_𝑘,Δ𝑥^𝑖(𝑡)=−𝐵_{𝑘𝑗∈𝒩𝑖(𝑡)}𝑥^𝑗(𝑡)+𝑥^𝑖(𝑡),𝑡=𝑡_𝑘,𝑘∈ℕ₊. (3.3)

Definition 3.1. For the system (3.1), the cluster anticonsensus is said to be achieved under the impulsive control protocol (3.2) if lim_{𝑖1,𝑖2∈𝑈,𝑡→+∞}𝑥^𝑖1(𝑡)−𝑥^𝑖2(𝑡)=0,lim_{𝑗1,𝑗2∈𝑊,𝑡→+∞}𝑥^𝑗1(𝑡)−𝑥^𝑗2(𝑡)=0,lim_{𝑖∈𝑈,𝑗∈𝑊,𝑡→+∞}𝑥^𝑖(𝑡)+𝑥^𝑗(𝑡)=0, (3.4) where 𝑈 and 𝑊 are two nonempty subsets of 𝑉(𝐺) and 𝑈∩𝑊=∅, 𝑈∪𝑊=𝑉(𝐺).

Consider the following discrete impulsive system𝑥(𝑡+1)=𝐴𝑥(𝑡),𝑡≠𝑡_𝑘,Δ𝑥(𝑡)=𝑥𝑡⁺−𝑥(𝑡⁻)=𝐵_𝑘𝑥(𝑡),𝑡=𝑡_𝑘,𝑘∈ℕ₊, (3.5) where 𝑥∈ℝ^𝑛, 𝐴∈ℝ^𝑛×𝑛, 𝐵_𝑘∈ℝ^𝑛×𝑛. Then the solution 𝑥(𝑡;𝑡₀,𝑥₀) satisfies𝑥𝑡;𝑡₀,𝑥₀=𝐴^{(𝑡−𝑡𝑘)𝑘}_𝑖=1𝐼_𝑛+𝐵_𝑖𝐴^{(𝑡𝑖−𝑡𝑖−1)}𝑥₀, (3.6) where 𝑡_𝑘<𝑡≤𝑡_𝑘+1, 𝑘∈ℕ₊.

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)0<𝜃₁≤𝑡_𝑘−𝑡_𝑘−1≤𝜃₂<∞, 𝑘∈ℕ₊, (H2)||Ξ_𝑘||≤𝑟<1, 𝑘∈ℕ₊, where Ξ_𝑘=(𝐼_𝑛+𝐵_𝑘)𝐴^{(𝑡𝑘−𝑡𝑘−1)}, 𝑘∈ℕ₊.

4. Main Results

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

Let 𝑥(𝑡)=(𝑥¹(𝑡),𝑥²(𝑡),…,𝑥^𝑁(𝑡))^𝑇, then the system (3.3) can be described as𝑥(𝑡+1)=𝐼_𝑁⊗𝐴𝑥(𝑡),𝑡≠𝑡_𝑘,Δ𝑥(𝑡)=𝐼_𝑁⊗𝐵_𝑘−𝑄⊗𝐼_𝑛𝑥(𝑡),𝑡=𝑡_𝑘,𝑘∈ℕ₊. (4.1)

Since 𝑄 is symmetric, there is an orthogonal matrix 𝑌∈ℝ^𝑁×𝑁 such that𝑌𝑄𝑌^𝑇=Λ=diag𝑞₁,𝑞₂,…,𝑞_𝑁, (4.2)

where {𝑞₁,𝑞₂,…,𝑞_𝑁}=𝜒(𝑄) is the spectrum of 𝑄.

Let𝑥(𝑡)=𝑌⊗𝐼_𝑛𝑥(𝑡). (4.3) Using the properties of Kronecker product, we have when 𝑡≠𝑡_𝑘, 𝑘∈ℕ₊,𝑥(𝑡+1)=𝑌⊗𝐼_𝑛𝑥(𝑡+1)=𝑌⊗𝐼_𝑛𝐼_𝑁⊗𝐴𝑌⊗𝐼_𝑛⁻¹𝑥(𝑡)=𝐼_𝑁⊗𝐴𝑥(𝑡), (4.4)

and when 𝑡=𝑡_𝑘, 𝑘∈ℕ₊,Δ𝑥(𝑡)=𝑌⊗𝐼_𝑛Δ𝑥(𝑡)=𝑌⊗𝐼_𝑛𝐼_𝑁⊗𝐵_𝑘−𝑄⊗𝐼_𝑛𝑌⊗𝐼_𝑛⁻¹𝑥(𝑡)=−Λ⊗𝐵_𝑘𝑥(𝑡). (4.5)

Thus (4.1) becomes 𝑥(𝑡+1)=𝐼_𝑁⊗𝐴𝑥(𝑡),𝑡≠𝑡_𝑘,Δ𝑥(𝑡)=−Λ⊗𝐵_𝑘𝑥(𝑡),𝑡=𝑡_𝑘,𝑘∈ℕ₊. (4.6) Therefore𝑥^𝑖(𝑡+1)=𝐴𝑥^𝑖(𝑡),𝑡≠𝑡_𝑘,Δ𝑥^𝑖(𝑡)=−𝑞_𝑖𝐵_𝑘𝑥^𝑖(𝑡),𝑡=𝑡_𝑘,𝑘∈ℕ₊,𝑖=1,2,…,𝑁. (4.7)

Theorem 4.1. Consider the system (3.1). Assume that the graph 𝐺 of the network is connected and bipartite. If there exist discrete instants 𝑡_𝑘 and impulsive matrices 𝐵_𝑘 such that the conditions (H3) and (H4) hold, then the cluster anticonsensus is achieved under the impulsive control protocol (3.2),(H3)0<𝜃₁≤𝑡_𝑘−𝑡_𝑘−1≤𝜃₂<∞, 𝑘∈ℕ₊,(H4)‖𝑉_𝑖𝑘‖≤𝑟<1, 𝑖=2,…,𝑁, 𝑘∈ℕ₊, where 𝑉_𝑖𝑘=(𝐼_𝑛−𝑞_𝑖𝐵_𝑘)𝐴^{(𝑡𝑘−𝑡𝑘−1)}, 𝑖=2,…,𝑁, 𝑘∈ℕ₊.

Proof. Since the graph 𝐺 is connected and bipartite, by Lemma 2.1, 𝑞₁=0 is the algebraically simple eigenvalue of 𝑄. All the other eigenvalues of 𝑄 are positive. Then we have 0=𝑞₁<𝑞₂≤⋯≤𝑞_𝑁.
By Lemma 3.2, it follows that if there exist discrete instants 𝑡_𝑘 and impulsive matrices 𝐵_𝑘 such that the conditions (H3) and (H4) hold, then the system (4.7) is asymptotically stable, that is, 𝑥^𝑖(𝑡)→0, 𝑡→+∞, 𝑖=2,…,𝑁.
It can be verified that 𝑄⊗𝐼_𝑛𝑥(𝑡)=𝑌⊗𝐼_𝑛⁻¹𝑌⊗𝐼_𝑛𝑄⊗𝐼_𝑛𝑌⁻¹⊗𝐼_𝑛(𝑡)=𝑥𝑌⊗𝐼_𝑛⁻¹Λ⊗𝐼_𝑛𝑥(𝑡)=(𝑌⊗𝐼_𝑛)⁻¹0𝑞₂𝑥²(𝑡)⋮𝑞_𝑁𝑥^𝑁(𝑡). (4.8) Hence (𝑄⊗𝐼_𝑛)𝑥(𝑡)→0, 𝑡→+∞. Since the graph 𝐺 is connected and bipartite, by Lemma 2.1, 0 is the eigenvalue of 𝑄⊗𝐼_𝑛 with multiplicity 𝑛. The 𝑛 linearly independent eigenvectors associated with the eigenvalue 0 of 𝑄⊗𝐼_𝑛 are 𝜉⊗𝑒_𝑖, 𝑖=1,2,⋯,𝑛. Therefore 𝑥→𝜉⊗𝑠, 𝑡→+∞, where 𝑠=∑^𝑛_𝑖=1𝛾_𝑖𝑒_𝑖∈ℝ^𝑛, 𝛾_𝑖∈ℝ, 𝑖=1,2,…,𝑛. 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 Δ𝑡_𝑘=𝑡_𝑘+1−𝑡_𝑘=Δ, 𝑘∈ℕ₊. The impulsive matrices 𝐵_𝑘, 𝑘∈ℕ₊, are chosen as 𝑝𝐼_𝑛, 𝑝∈ℝ. If ||(1−𝑞_𝑖𝑝)𝐴^Δ||≤𝑟<1, 𝑖=2,…,𝑁, then the conditions (H3) and (H4) are satisfied.

5. Simulations

Consider the following multiagent discrete linear dynamic system𝑥^𝑖(𝑡+1)=𝐴𝑥^𝑖(𝑡),𝑥^𝑖(𝑡)∈ℝ³,𝑖=1,2,3,4, (5.1) where 𝐴=^√1/2_−√^3/20_3/21/20001. 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.

Figure 1 

The time histories of 𝑥𝑖1(𝑡), 𝑖=1,2,3,4.

Figure 2 

The time histories of 𝑥𝑖2(𝑡), 𝑖=1,2,3,4.

Figure 3 

The time histories of 𝑥𝑖3(𝑡), 𝑖=1,2,3,4.

The control input of agent 𝑖 is designed as (3.2). The graph is considered as a simple path on four vertices and the singless Laplacian matrix of graph 𝐺₁ is 𝑄₁=^11001210_01210011. For simplicity, the impulsive matrices 𝐵_𝑘, 𝑘∈ℕ₊, are chosen as 0.43∗𝐼₃. Choose the equidistant impulsive interval Δ𝑡_𝑘=𝑡_𝑘−𝑡_𝑘−1=2, 𝑘∈ℕ₊. It is easy to check that ‖𝑉_2𝑘‖=0.7481<1, ‖𝑉_3𝑘‖=0.1400<1, ‖𝑉_4𝑘‖=0.4681<1, where 𝑞₂=0.5858, 𝑞₃=2, 𝑞₄=4. The initial values are chosen as 𝑥¹(0)=[41−1]^𝑇, 𝑥²(0)=[−46−3]^𝑇, 𝑥³(0)=[−527]^𝑇, 𝑥⁴(0)=[5−72]^𝑇. Simulation results for 𝐺₁ 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 𝑥(𝑡)=(𝑥¹(𝑡)−𝑥²(𝑡)+𝑥³(𝑡)−𝑥⁴(𝑡))/4, then lim_𝑡→+∞‖𝑥^𝑖(𝑡)−𝑥(𝑡)‖=0, 𝑖=1,3 and lim_𝑡→+∞‖𝑥^𝑖(𝑡)+𝑥(𝑡)‖=0, 𝑖=2,4.

Figure 4 

The time histories of 𝑥𝑖1(𝑡), 𝑖=1,2,3,4 for 𝐺1.

Figure 5 

The time histories of 𝑥𝑖2(𝑡), 𝑖=1,2,3,4 for 𝐺1.

Figure 6 

The time histories of 𝑥𝑖3(𝑡), 𝑖=1,2,3,4 for 𝐺1.

If the graph is considered as a simple circle on four vertices, then the singless Laplacian matrix of graph 𝐺₂ is 𝑄₂=^21011210_01211012. The other parameters are chosen as above. Then the conditions (H3) and (H4) in Theorem 4.1 are also satisfied. Simulation results for 𝐺₂ 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.

Figure 7 

The time histories of 𝑥𝑖1(𝑡), 𝑖=1,2,3,4 for 𝐺2.

Figure 8 

The time histories of 𝑥𝑖2(𝑡), 𝑖=1,2,3,4 for 𝐺2.

Figure 9 

The time histories of 𝑥𝑖3(𝑡), 𝑖=1,2,3,4 for 𝐺2.

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.