# Impulsive Cluster Anticonsensus of Discrete Multiagent Linear Dynamic Systems

Authors: Liping Zhang; Haibo Jiang
Journal: Discrete Dynamics in Nature and Society
 (2012)

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## 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 theQ-theory. Numerical simulation shows the effectiveness of our theoretical results.


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## Body

## 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 Section2, 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; ℝn denotes the n-dimensional Euclidean space; let ℝ+=[0,∞), ℕ={0,1,2,…}, ℕ+={1,2,…}; ℝn×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); In∈ℝn×n is an identity matrix; λmin(P) (λmax(P)) denotes the smallest (largest) eigenvalue of P. For a vector x∈ℝn, let ∥x∥ denote the Euclidean vector norm, that is, ∥x∥=xTx, and for A∈Rn×n, let ||A|| indicate the norm of A induced by the Euclidean vector norm, that is, ||A||=λmax(ATA). The Kronecker product of two matrices A=[aij]∈ℝm×n and B=[bij]∈ℝp×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 theQ-theory [4–6, 21].An undirected graphG 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 𝒩i={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),…,(kl,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 matrixA=[aij]∈ℝN×N, where aii=0 and aij=aji≥0, i≠j. aij>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 degout(i)=∑j=1naij. 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 QG=D+A. For an unweighted graph G, QG=[qij]N×N, where
(2.1)qij={|Ni|,i=j,1,j∈Ni,0,otherwise,
here |𝒩i| denotes the cardinality of the set 𝒩i.LetG 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 rij=1 if the vertex i and edge ej are incident and 0 otherwise. The following relations are well-known: RRT=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,…,qN (q1≤q2≤⋯≤qN).Lemma 2.1.
LetQ be the singless Laplacian matrix of an undirected graph  G with  N vertices, and  q1≤q2≤⋯≤qN be the eigenvalues of  Q. Suppose that the graph  G is bipartite. Let  U and  W be two subsets of graph  G. Define  ξ∈ℝN,  ξ(i)=1,  i∈U,  ξ(j)=-1,  j∈W and  ei∈ℝn,  ei(i)=1,  ei(j)=0,  j≠i,  i,j=1,…,n.  Then(1)
ifG 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 ofQ, then it is an n multiplicity eigenvalue of Q⊗In and the corresponding eigenvectors are ξ⊗ei, i=1,2,⋯,n.


## 3. Problem Formulation

Here we consider a system consisting ofN agents indexed by i=1,2,…,N. The dynamics of each agent is(3.1)xi(t+1)=Axi(t),t≥t0≥0,i=1,2,…,N,
where xi(t)=(x1i(t),x2i(t),…,xni(t))T∈ℝn is the state of agent i at time t, t∈ℕ, A∈ℝn×n is a constant matrix. We assume that ∥A∥≠0.The following impulsive control protocol is applied to agenti(3.2)ui(t)=Δxi(t)=xi(t+)-xi(t-)=-Bk∑j∈Ni(t-)(xj(t-)+xi(t-)),t=tk,k∈N+,
where Bk∈ℝn×n, k∈ℕ+, are impulsive matrices to be determined later, 𝒩i(t-) is the set of neighbors of agent i at time t-, the discrete instants tk∈ℕ+ satisfy 0≤t0<t1<t2<⋯<tk-1<tk<⋯, and limk→+∞tk=+∞. When t=tk, 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 xi(t-)=xi(t), t=tk, k∈ℕ+.Under the impulsive control protocol (3.2), the dynamics of agent i satisfies the following equations
(3.3)xi(t+1)=Axi(t),t≠tk,Δxi(t)=-Bk∑j∈Ni(t)(xj(t)+xi(t)),t=tk,k∈N+.Definition 3.1.
For the system (3.1), the cluster anticonsensus is said to be achieved under the impulsive control protocol (3.2) if
(3.4)lim⁡i1,i2∈U,t→+∞‖xi1(t)-xi2(t)‖=0,lim⁡j1,j2∈W,t→+∞‖xj1(t)-xj2(t)‖=0,lim⁡i∈U,j∈W,t→+∞‖xi(t)+xj(t)‖=0,
where U and W are two nonempty subsets of V(G) and U∩W=∅, U∪W=V(G).Consider the following discrete impulsive system(3.5)x(t+1)=Ax(t),t≠tk,Δx(t)=x(t+)-x(t-)=Bkx(t),t=tk,k∈N+,
where x∈ℝn, A∈ℝn×n, Bk∈ℝn×n. Then the solution x(t;t0,x0) satisfies(3.6)x(t;t0,x0)=A(t-tk)∏i=1k(In+Bi)A(ti-ti-1)x0,
where tk<t≤tk+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≤tk-tk-1≤θ2<∞, k∈ℕ+,(H2)
||Ξk||≤r<1, k∈ℕ+, where Ξk=(In+Bk)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.Letx(t)=(x1(t),x2(t),…,xN(t))T, then the system (3.3) can be described as(4.1)x(t+1)=(IN⊗A)x(t),t≠tk,Δx(t)=(IN⊗Bk)(-Q⊗In)x(t),t=tk,k∈N+.SinceQ is symmetric, there is an orthogonal matrix Y∈ℝN×N such that(4.2)YQYT=Λ=diag⁡(q1,q2,…,qN),where{q1,q2,…,qN}=χ(Q) is the spectrum of Q.Let(4.3)x̃(t)=(Y⊗In)x(t).
Using the properties of Kronecker product, we have when t≠tk, k∈ℕ+,(4.4)x̃(t+1)=(Y⊗In)x(t+1)=(Y⊗In)(IN⊗A)(Y⊗In)-1x̃(t)=(IN⊗A)x̃(t),and whent=tk, k∈ℕ+,(4.5)Δx̃(t)=(Y⊗In)Δx(t)=(Y⊗In)(IN⊗Bk)(-Q⊗In)(Y⊗In)-1x̃(t)=(-Λ⊗Bk)x̃(t).Thus (4.1) becomes (4.6)x̃(t+1)=(IN⊗A)x̃(t),t≠tk,Δx̃(t)=(-Λ⊗Bk)x̃(t),t=tk,k∈N+.
Therefore(4.7)x̃i(t+1)=Ax̃i(t),t≠tk,Δx̃i(t)=-qiBkx̃i(t),t=tk,k∈N+,i=1,2,…,N.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 tk and impulsive matrices Bk such that the conditions (H3) and (H4) hold, then the cluster anticonsensus is achieved under the impulsive control protocol (3.2),(H3)
0<θ1≤tk-tk-1≤θ2<∞, k∈ℕ+,(H4)
∥Vik∥≤r<1, i=2,…,N, k∈ℕ+, where Vik=(In-qiBk)A(tk-tk-1), i=2,…,N, k∈ℕ+.Proof.
Since the graphG 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≤⋯≤qN.
By Lemma3.2, it follows that if there exist discrete instants tk and impulsive matrices Bk such that the conditions (H3) and (H4) hold, then the system (4.7) is asymptotically stable, that is, x̃i(t)→0, t→+∞, i=2,…,N.
It can be verified that(4.8)(Q⊗In)x(t)=(Y⊗In)-1(Y⊗In)(Q⊗In)(Y-1⊗In)x̃(t)=(Y⊗In)-1(Λ⊗In)x̃(t)=(Y⊗In)-1[0q2x̃2(t)⋮qNx̃N(t)].
Hence (Q⊗In)x(t)→0, t→+∞. Since the graph G is connected and bipartite, by Lemma 2.1, 0 is the eigenvalue of Q⊗In with multiplicity n. The n linearly independent eigenvectors associated with the eigenvalue 0 of Q⊗In are ξ⊗ei, i=1,2,⋯,n. Therefore x→ξ⊗s, t→+∞, where s=∑i=1nγiei∈ℝn, γi∈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 intervalΔtk=tk+1-tk=Δ, k∈ℕ+. The impulsive matrices Bk, k∈ℕ+, are chosen as pIn, p∈ℝ. If ||(1-qip)AΔ||≤r<1, i=2,…,N, then the conditions (H3) and (H4) are satisfied.


## 5. Simulations

Consider the following multiagent discrete linear dynamic system(5.1)xi(t+1)=Axi(t),xi(t)∈R3,i=1,2,3,4,
where A=[1/23/20-3/21/20001]. 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 ofx1i(t), i=1,2,3,4.Figure 2
The time histories ofx2i(t), i=1,2,3,4.Figure 3
The time histories ofx3i(t), i=1,2,3,4.The control input of agenti 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=[1100121001210011]. For simplicity, the impulsive matrices Bk, k∈ℕ+, are chosen as 0.43*I3. Choose the equidistant impulsive interval Δtk=tk-tk-1=2, k∈ℕ+. It is easy to check that ∥V2k∥=0.7481<1, ∥V3k∥=0.1400<1, ∥V4k∥=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 x̅(t)=(x1(t)-x2(t)+x3(t)-x4(t))/4, then limt→+∞∥xi(t)-x̅(t)∥=0, i=1,3 and limt→+∞∥xi(t)+x̅(t)∥=0, i=2,4.Figure 4
The time histories ofx1i(t), i=1,2,3,4 for G1.Figure 5
The time histories ofx2i(t), i=1,2,3,4 for G1.Figure 6
The time histories ofx3i(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 graphG2 is Q2=[2101121001211012]. 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 ofx1i(t), i=1,2,3,4 for G2.Figure 8
The time histories ofx2i(t), i=1,2,3,4 for G2.Figure 9
The time histories ofx3i(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.



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