# Second-Order Consensus for Multiagent Systems under Directed and Switching Topologies

Authors: Lixin Gao; Jingjing Zhang; Wenhai Chen
Journal: Mathematical Problems in Engineering
 (2012)

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## Abstract

We consider multiagent consensus problems in a decentralized fashion. The interconnection topology among the agents is switching and directed. The agent dynamics is expressed in the form of a double-integrator model. Two different cases are considered: one is the leader-following case and the other is the leaderless case. Based on graph theory and the common Lyapunov function method, some sufficient conditions are established for the consensus stability of the considered systems with the neighbor-based feedback laws in both leader-following case and leaderless case, respectively. As special cases, the consensus conditions for balanced and undirected interconnection topology cases can be established directly. Finally, two numerical examples are given to illustrate the obtained results.


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## Body

## 1. Introduction

In recent years, the coordination problem of multiple autonomous agents has drawn an increasing attention. It is because there are many applications of multiagent systems in many areas including cooperative control of unmanned air vehicles, flocking of birds, schooling for underwater vehicles, distributed sensor networks, attitude alignment for cluster of satellites, collective motion of particles, and distributed computations [1–5].Although these practical backgrounds are different, the fundamental principles in the coordination of multiple spacecraft, robots, and even animals are very similar, that is, coordinating multiple agents to achieve a goal of the whole system by local information. As a result, a critical problem for coordinated control is to design appropriate protocols and algorithms such that the group of agents can reach consensus on the shared information in the presence of limited and unreliable information exchange and dynamically changing interaction topologies [6]. In [5], the author also pointed out that a key technology in cooperative control is the problem of consensus. The investigated feedback scheme associated with those problems is inspired by the aggregates of individuals in nature. Reynolds first proposed a computer animation model to simulate collective behaviors of multiple agents in [2]. In the fields of system and control, the development of consensus theory is primarily impelled by Vicsek’s particle swarm model mentioned in [3]. In [1], Jadbabaie et al. gave a theoretical explanation of the consensus behavior of Vicsek’s model and proved that the states of all the jointly connected agents converged to the same value or the value of a given leader. Ren and Beard [6] extended the work of Jadbabaie et al. [1] to the case of directed graphs and explored the minimum requirements to reach global consensus. Olfati-Saber and Murray studied the average-consensus problem with strongly connected and balanced directed topologies in [7], which is essentially the same problem as in [1, 6] for continuous-time consensus scheme. Similar or generalized consensus problems have also been studied in [8–12]. There are also many interesting results about the similar topic for stochastic complex networks [13–15].In real applications, the interacting topology between agents may change dynamically since the communication links between agents may be unreliable due to disturbances and/or subject to communication time delay. It is well known that switching of the communication topology and communication time delays may lower the system performance and even cause the network system to diverge or oscillate. Theoretically, the consensus in undirected switching topology is much easier than that of directed switching topology. Recently, some preliminary results have been reported to deal with the directed switching topology [16, 17].Furthermore, with many practical applications, especially involving mechanical systems, the dynamics of agents is usually modeled as a double integratorẍ=u. There are many interesting agent-related works, such as in [18–20], involving double-integrator dynamics under switching and undirected interconnection topology among the agents. Most existing results are obtained under undirected interconnection topology. In directed topology case, [21] discussed this topic under the simple fixed topology. In [9], the authors pointed out that the system matrix related to this multiagent system with double-integrator dynamics does not satisfy the properties related to stochastic matrix and the norm-based method proposed by [1] may fail for the double-integrator form agent dynamic. Therefore, the Lyapunov-based approach is often chosen to solve consensus problem of multi-agent systems with double integrator model.Motivated by the above works, we study a group of agents with the double-integrator dynamics. The main purpose of this paper is to develop a decentralized control strategy to reach the global consensus of the multi-agent systems under directed switching interconnection topology among the agents. Two different cases are considered: one is the leader-following case and the other is the leaderless case. First, we establish a sufficient consensus condition by constructing a parameter-dependent common Lyapunov function in leader-following formulation. The established consensus condition is expressed as a reduced-order Lyapunov matrix inequalities. As commonly known, it is not an easy task to construct a common Lyapunov function for a switching system, let alone the parameter-dependent common Lyapunov function. As for the leaderless case, the common Lyapunov method cannot be used directly. By using the Schur orthogonal transformation, we decomposes the group tracking dynamics into two subsystems: one represses the leader and the other represses a leader-following subsystem. Then, a sufficient condition is established by applying the obtained leader-following consensus condition to leader-following subsystem. As special cases, we can obtain consensus conditions directly for the multi-agent system with undirected and balanced switching interconnection topology. The jointly connected convergence condition can guarantee that the multi-agent with linear dynamical model achieves consensus. But for double-integrator model, we construct a counterexample to show that the jointly connected condition may not guarantee that the multi-agent system achieves consensus, which implies that the proposed globally reachable convergence condition may be moderatable and acceptable. By using similar analysis method, we also can probe consensus problems of [9, 18, 19] with switching and directed interconnection topology. Finally, the numerical examples also show that the established consensus condition is solvable.The rest of the paper is organized as follows. In Section2, we give a formulation of the coordination problem with help of graph theory. Then in Section 3, some results on the consensus stability are obtained for the multi-agent system with fixed and varying interconnection topologies in the leader-following case, while in Section 4, the leaderless case is studied. In Section 5, two simulation examples are presented to illustrate our theoretical results. Finally, concluding remarks with discussions of the future work are given in Section 6.The notation of this paper is standard.R is the real number set. I is an identity matrix with compatible dimension. AT denotes as transpose of a matrix A. For symmetric matrices A and B, A>(≥)B means A-B is positive (semi-)definite. λ(A) represents an eigenvalue of A. For symmetric matrices A, λmin(A) and λmax(A) represent the minimum and maximum eigenvalue of A, respectively. ∥•∥ denotes the Euclidean norm. ⊗ denotes the Kronecker product, which satisfies (1) (A⊗B)(C⊗D)=(AC)⊗(BD); (2) if A≥0 and B≥0, then A⊗B≥0.


## 2. Problem Formulation

Stability analysis of the group of agents is based on several results of algebraic graph theory. In this section, we first introduce some basic concepts and notations in graph theory that will be used throughout this paper. More details are available in [22].Wen by 𝒢={𝒱,ε,A} with an index set ℐ={1,2,…,n}, where 𝒱={v1,v2,…,vn} is the set of vertices, ε⊂𝒱×𝒱 is the set of edges of the digraph, and each edge is denoted by (vi,vj)∈ε,i,j∈ℐ, and a weighted adjacency matrix A=[aij] has nonnegative adjacency elements aij. Throughout this paper, we assume that all the graphs have no edges from a node to itself. A weighted graph is called undirected if forall(vi,vj)∈ε⇒(vj,vi)∈ε and aij=aji. Otherwise, the graph is called a directed graph or digraph. The adjacency elements associated with the edges of the graph are positive, that is aij>0⇔(vi,vj)∈ε. Obviously, we also have aij=0⇔(vi,vj)∉ε. A path from vertex vi to vertex vj is a sequence of distinct vertices starting with vi and ending with vj such that consecutive pair of vertices make an edge of digraph. If there is a path from one node vi to another node vj, then vj is said to be reachable from vi. If a node vi is reachable from every other node of the digraph, then it is said to be globally reachable. Obviously, every node of an undirected connected graph is globally reachable.The relationships betweenn agents can be conveniently described by a simple digraph 𝒢. We denote the set of all neighbor vertices of vertex vi by 𝒩i={j∣(vi,vj)∈ε}. Neighboring relations reflect physical proximity between two agents, or the existence of a communication channel [6]. Its degree matrix D=diag{d1,…,dn}∈Rn×n is a diagonal matrix, where diagonal elements di=∑j∈𝒩iaij for i=1,…,n. Then the Laplacian matrix of 𝒢 is defined as(2.1)L=D-A.The following lemma relates the Laplacian matrix with its directed graph, as well as supplying an algebraic characterization. It is well known that the Laplacian matrix captures many interesting properties of the graphs. The result of Lemma 2.1 can be found in [6].Lemma 2.1.
The weighted graph Laplacian matrixL associated with graph 𝒢 has at least one zero eigenvalue, and all of the nonzero eigenvalues are located on the open right half plane. Furthermore, L has exactly one zero eigenvalue if and only if the directed graph 𝒢 has a globally reachable node.Moreover, for a digraph with a globally reachable node, then-dimensional eigenvector associated with the single zero eigenvalue is the vector of ones, 1:=(1,1,…,1)T. For undirected weight graph 𝒢, L is symmetric, positive and semidefinite, and all eigenvalues of L are real and nonnegative.A weighted graph𝒢=(𝒱,ε,A) is said to be balanced if(2.2)∑j=1naij=∑j=1naji,i=1,2,…,n,
Any undirected weighted graph is balanced. Furthermore, a weighted graph is balanced if and only if 1TL=0 (see [7]).Letε̃ be the set of reverse edges of 𝒢 obtained by reversing the order of all the pairs in ε. The mirror of 𝒢 is denoted by an undirected graph in the form 𝒢̂=(𝒱,ε̂,Â) with the same set of nodes of 𝒢, the set of edges ε̂=ε∪ε̃, and the symmetric adjacency matrix Â=[âij] with elements âij=(aij+aji)/2. The Laplacian matrix for mirror graph 𝒢̂ is L(𝒢̂)=(1/2)(L(𝒢)+LT(𝒢)) if and only if 𝒢 is balanced graph [7].In what follows, we mainly consider a graph𝒢¯ associated with the system with n agents (labeled by vi,i=1,2,…,n) and one leader (labeled by v0). A simple and directed graph 𝒢 describing the topology relation of these n followers, and 𝒢¯ contains graph 𝒢 and v0 with the directed edges from some agents to the leader describes the topology relation among all agents. In other words, we denote a weighted digraph of order n+1 by 𝒢¯={𝒱¯,ε¯,A¯}, where 𝒱¯={v0,v1,…,vn}, and the weighted matrix has the following form: (2.3)A¯=(00bA),b=(b1,b2,⋯,bn)T∈Rn,A=(aij)∈Rn×n.
The element aij>0 represents that agent i is connected to agent j, and bi>0 represents that agent i is connected to leader. Meanwhile, aij=0 and bi=0 represents that agent i is not connected to agent j and leader, respectively. The induced subgraph 𝒢={𝒱,ε,A} of 𝒢¯ represses the interconnection topology among n following agents.For the multi-agent system under consideration, the relationships between neighbors can change over time and the interconnection topology may be dynamically changing. Lett1=0,t2,t3,… be an infinite time sequence at which the interconnection graph of the considered multi-agent system switches. Usually, it is assumed that chattering does not occur; that is, there is a constant τ0>0, often called dwell time, with ti+1-ti≥τ0,foralli. Moreover, we assume that only finite possible interconnection topologies can be switched. In the leaderless case, denote 𝒮={𝒢1,𝒢2,…,𝒢𝒩} as a set of the graphs of all possible topologies, while in the leader-following case, denote 𝒮¯={𝒢¯1,𝒢¯2,…,𝒢¯N0} as a set of the graphs of all possible topologies. Denote 𝒫={1,2,…,N} and 𝒫0={1,2,…,N0} as index sets, respectively.In this paper, we intend to coordinaten mobile agents with each agent expressed in the form of a double integrator:(2.4)ṙi=pi,ṗi=ui,i=1,…,n,
where ri∈Rm is the position of agent i,pi∈Rm its velocity and ui∈Rm its control inputs. In the leader-following case, the dynamics of the leader is taken as(2.5)ṙ0=p0ṗ0=a0(t),
where r0∈Rm and p0∈Rm are the position and velocity vectors of the leader, respectively, and a0(t)∈Rm is a known input, which may be regarded as some given policy known to all the agents. Particularly, a0(t)≡0 expresses that the leader moves in constant velocity.Then mobile agents are said to achieve global consensus if for any given initial values ri(0) and pi(0) (i=1,2,…,n), limt→+∞(ri(t)-rj(t))=0 and limt→+∞(pi(t)-pj(t))=0 for any i,j=1,2,…,n. Thus, in the leader-following case, the multi-agent system achieves global consensus if and only if limt→+∞(ri(t)-r0(t))=0 and limt→+∞(pi(t)-p0(t))=0 for any initial values.The control law is said to beneighbor control law or local control law if all the control inputsui only depend on the states of agent i and its neighbors. We are interested in using neighbor control law to solve the consensus problem. To this end, the controller ui of agent i, regarded as node i in a graph, requires state information from a subset of the agent’s flockmates, called the neighbor set 𝒩i and defined as above. There are two cases considered for multi-agent system as follows.Leader-Following Case
For anyi∈ℐ, take the local control law, which is neighbor-based feedback law as follows:(2.6)ui=a0(t)-κ(∑j∈Ni(t)aij(t)(ri-rj)+bi(t)(ri-r0))-κγ(∑j∈Ni(t)aij(t)(pi-pj)+bi(t)(pi-p0)),
where κ is considered as a “control” parameter, which is positive constant and will be determined later, and γ is a positive weighted parameter. The set of switching interconnection topology graph is assumed to be 𝒢¯l={𝒱¯,ε¯l,A¯l},l∈𝒫0, where the A¯l has the following form: (2.7)A¯l=(00blAl),bl=(b1(l),b2(l),…,bn(l))T∈Rn,Al=(aij(l))∈Rn×n.
In time t, the interconnection topology graph is assumed to be switched to 𝒢¯l,l∈𝒫0, so aij(t) and bi(t) are chosen by aij(t)=aij(l) and bi(t)=bi(l).Denotesj=rj-r0 and qj=pj-p0. Take s:=(s1T,…,snT)T∈Rmn, q:=(q1T,…,qnT)T∈Rmn, and u:=(u1T-a0T,u2T-a0T,…,unT-a0T)T∈Rmn. Then error dynamics of closed system (2.4)–(2.6) can be rewritten as(2.8)ṡ=q,q̇=uσ=-κ(Hσ⊗Im)s-κγ(Hσ⊗Im)q,
where Hσ=Lσ+Bσ, σ:[0,+∞)→𝒫0={1,2,…,N0} is a piecewise constant switching signal with successive times when the topology of graph 𝒢¯ switches. Lσ is the Laplacian matrix of the switching graph 𝒢 consisting of n vertices (presenting agents 1,…,n), and Bσ is a diagonal matrix whose i diagonal entry is bi(t) at time t.Leaderless Case
Because there is no leader dynamics (2.5), take the local control law for any i∈ℐ as follows:(2.9)ui=-κ∑j∈Ni(t)aij(t)(ri-rj)-κγ∑j∈Ni(t)aij(t)(pi-pj).
Let r:=(r1T,…,rnT)T and p:=(p1T,…,pnT)T. Then we can rewrite the closed-loop system (2.4) and (2.9) as(2.10)ṙ=p,ṗ=uσ=-κ(Lσ⊗Im)r-κγ(Lσ⊗Im)p,
where σ:[0,+∞)→𝒫={1,2,…,N} is a piecewise constant switching signal to describe the switch of the interconnection graph 𝒢. The aij(t) is chosen by aij(t)=aij(l),l∈𝒫.The next lemma shows well-known results and will be used later, which is given in [23].Lemma 2.2.
Suppose that a symmetric matrix is partitioned as(2.11)(C1C2C2TC3),
where C1 and C3 are square. This matrix is positive definite if and only if C1 is positive definite and C3-C2TC1-1C2 is positive definite.In what follows, we first focus on the analysis of the leader-following case and then study the leaderless case in a similar trace.


## 3. Leader-Following Case

In this section, we concentrate on the analysis of the multi-agent systems with a leader. If the information of the inputa0(t) can be used in local control design, then we can prove that although the leader keeps changing, the agents can follow the leader that is, consensus is achieved.Using neighbor-based feedback law (2.6), the error dynamics system (2.8) will be expressed in a compact form as follows:(3.1)ẋ=(Fσ⊗Im)x,
where (3.2)x=(sq),Fσ=(0I-κHσ-κγHσ).The interconnection topology graph is𝒢¯σ(t), and the interconnection graphs associated with all followers are 𝒢σ(t).Note thatFσ is not directly related to a stochastic matrix, and therefore, the method reported in [1] cannot be applied directly. In what follows, we will propose an approach based on common Lyapunov function for the system (3.1) to demonstrate the convergence of the dynamics system (3.1).Now we give the main result in the leader-following case as follows.Theorem 3.1.
Suppose that the interconnection graph is connected for any interval[tj,tj+1). If there exist a positive definite matrix P¯1 and a positive constant μ such that
(3.3)HlTP¯1+P¯1Hl≥μI,l∈P0,
and taking a constant
(3.4)κ>2μγ2λmax⁡(P¯1),
then the local control law (2.6) can guarantee that multi-agent system (2.4), (2.5) achieves consensus for any given initial condition r(0) and p(0).Proof.
To prove the theorem, we consider the dynamics in each interval at first. Note that, in any interval (say[tj,tj+1)), the interconnection topology does not change. Therefore, Fσ(t) is a constant matrix for t∈[tj,tj+1) for any j≥0, and then the solution to (3.1) is well defined. Thus, we can assume that σ(t)=l,l∈𝒫0,t∈[tj,tj+1). Choose matrix P̃1 as
(3.5)P̃1=(2P¯1γP¯1γP¯1γ2P¯1).
We can verify that P̃1 is positive definite by applying Lemma 2.2. Consider a common Lyapunov function V(x)=xT(P̃1⊗Im)x, where P̃1 is defined in (3.5). Then, for any l∈𝒫0 which corresponds to interval [tj,tj+1), we have
(3.6)V̇(x)∣(3.1)=xT[(FlT⊗Im)(P̃1⊗Im)+(P̃1⊗Im)(Fl⊗Im)]x=xT[(FlTP̃1+P̃1Fl)⊗Im]x:=-xT(Ql⊗Im)x,
where
(3.7)Ql=(κγ(HlTP¯1+P¯1Hl)κγ2(HlTP¯1+P¯1Hl)-2P¯1κγ2(HlTP¯1+P¯1Hl)-2P¯1κγ3(HlTP¯1+P¯1Hl)-2γP¯1).
Note that
(3.8)(κγκγ2κγ2κγ3)⊗(HlTP¯1+P¯1Hl-μI)≥0.
Then we have
(3.9)Ql≥Q:=(κγμIκγ2μI-2P¯1κγ2μI-2P¯1κγ3μI-2γP¯1).
By using Lemma 2.2, (3.4) guarantees that matrix Q is positive definite.
In addition, the maximum eigenvalue ofP̃1 is(3.10)λmax⁡(P̃1)=2+γ2+21+γ42λmax⁡(P¯1),min⁡xT(Ql⊗I)xxT(P̃1⊗I)x≥λmin⁡(Q)λmax⁡(P̃1).
Let β:=λmin(Q)/2λmax(P̃2). Then, we have V̇(x)≤-2βV(x), which implies V(x)≤V(x(0))e-2βt. Moreover, we have
(3.11)‖x(t)‖≤V(x)λmin⁡(P̃1)≤V(0)λmin⁡(P̃1)e-βt≤λmax⁡(P̃1)λmin⁡(P̃1)‖x(0)‖e-βt.
Thus we have limt→+∞x(t)=0 with at least an exponent rate β. The proof is now completed.Remark 3.2.
Obviously, the consensus problem of closed-loop multi-agent system (2.4), (2.5) and (2.6) is equivalent to stability problem of error system (3.1). The closed-loop multi-agent system and the error system are both 2n-order systems, and the consensus condition (3.3) given in Theorem 3.1 is n-order Lyapunov matrix inequalities. Thus, the established consensus condition is a reduced order condition. In many applications, the system normally switches in finite model. The assumption that the system switches only in finite model is often used. Because the interconnection topology is modeled by weighted digraph, different weight will lead different interconnection topology. Although the set of nodes and the set of possible edges are finite, the set of possible interconnection topology may be infinite, but from the point of mathematics, the assumption that the index set 𝒫 is finite in Theorem 3.1 may be not necessary. From the proof of Theorem 3.1, we can also obtain the result of Theorem 3.1 by the assumption that condition (3.3) is satisfied for any element of an infinite index set 𝒫.Note that the matrixH=L+B plays a key role in the convergence analysis of system (3.1). A matrix is said to be a positive stable matrix if all its eigenvalues have positive real parts. The matrix H=L+B is positive stable if and only if node v0 is globally reachable in 𝒢¯ (see [18]).Of course, the result of Theorem3.1 can be applied to fixed topology case directly. Because H is positive stable, there exists a positive definite matrix P¯2∈Rn×n such that (3.12)HTP¯2+P¯2H=I.
Therefore, we can obtain the following corollary for fixed topology case.Corollary 3.3.
Suppose the interconnection topology𝒢¯ is fixed. If node v0 is globally reachable in 𝒢¯ and the constant κ satisfies
(3.13)κ>2γ2λmax⁡(P¯2),
then the local control law (2.6) can guarantee that system (2.4), (2.5) achieves consensus for any given initial condition r(0) and p(0).Moreover, letλi(H)=αi+jβi(i=1,2,…,n) be ith eigenvalue of H. A necessary and sufficient condition for fixed topology case obtained by [21] is given as follows. The local control law (2.6) can guarantee that the multi-agent system in leader-following case achieves consensus for any given initial condition r(0) and p(0) if and only if node v0 is globally reachable in 𝒢¯ and the constants κ and γ satisfy(3.14)κ>1γ2max⁡iβi2αi[αi2+βi2].The assumption that nodev0 is globally reachable in 𝒢¯ can guarantee that αi>0. For any given γ>0,αi>0, and βi, it is not difficult to see (3.14) can be satisfied for larger enough κ. Obviously, (3.14) always holds for any γ>0,κ>0 if all βi=0. When the interconnection graphs 𝒢 associated with all followers are undirected and node v0 is globally reachable in 𝒢¯, all eigenvalues of H are real and positive (see [9]). Thus, in this fixed undirected case, multi-agent system (2.4)–(2.6) achieves consensus for any given initial condition r(0) and p(0) if and only if node v0 is globally reachable in 𝒢¯.Consider the special case that the graph𝒢σ associated with all followers is balanced. The matrix Lσ+LσT is positive semidefinite. Moreover, we have (see [18]).Lemma 3.4.
Suppose𝒢l is balanced. Then HlT+Hl is positive definite if and only if node v0 is globally reachable in 𝒢¯l.Based on Lemma3.4 and the fact that the set 𝒫0 is finite, define(3.15)λ¯1:=12min⁡l∈P0{λmin⁡(HlT+Hl)∣node0isgloballyreachableofG¯landGlisbalanced}>0
which is fixed and depends directly on the constants aij(l) and βi(l) for i,j=1,…,n and l=1,2,…,N. Obviously, if all the interconnection graphs 𝒢σ associated with followers are undirected and node v0 is globally reachable in 𝒢¯σ, λ¯1 can be expressed as λ¯1=minp∈𝒫0{λmin(Hl)}. Now we propose following the corollary for the balanced graph case.Corollary 3.5.
In any interval[tj,tj+1), suppose that node v0 is globally reachable in the interconnection graph 𝒢¯l, and 𝒢l associated with all followers is balanced. Taking a constant
(3.16)κ>1γ2λ¯1,
then multi-agent system (2.4)–(2.6) achieves consensus for any given initial condition r(0) and p(0).Proof.
According to Lemma3.4 and the definition of λ¯1, we have
(3.17)HlTI+IHl≥2λ¯1I.
Then we can obtain the corollary directly by using the result of Theorem 3.1.Because the interconnection graph considered in this paper is weighted graph, two graphs with same edge set and different weighted adjacency matrices are different interconnection graphs. Although the node set and edge are finite, the set of possible interconnection graph may be infinite. For the case that all the graphs𝒢l,l∈𝒫0 are associated with all followers are balanced and the switching index set 𝒫0 is infinite, we also can prove that multi-agent system achieves consensus by only assumption that there exists a positive constant amin such that all nonzero weighted factors aij(l),bi(l) are equal or greater than amin.To probe the consensus stability condition in this case, letΓ0={𝒢¯p,p∈𝒫0} denote the class of all possible interconnection graphs with the following properties: (1) v0 is globally reachable node; (2) the interconnection graph 𝒢p related to all followers is balanced; (3) all constant weighted factors aij,bi are equal or greater than a positive constant amin. 𝒫0 is index set of Γ0, which is infinite set. Let Γ̃0 denote the class of all possible nonweighted interconnection graphs with the properties: (1) v0 is globally reachable node; (2) the interconnection graph 𝒢 related to all followers is undirected; (3) all nonzero weights are 1.For any graph𝒢¯∈Γ0, we construct a related graph 𝒢̃ by replacing all directed edges between all follower agents with undirected edges and all nonzero weights with 1. Because v0 is globally reachable node in the interconnection graph 𝒢¯, v0 is also globally reachable node in 𝒢̃. Thus, we have 𝒢̃∈Γ̃0. Moreover, H(𝒢̃)=B(𝒢̃)+L(𝒢̃) is symmetric and positive definite by Lemma 3.4. Because of the finiteness of node set and edge set, Γ̃0 is a finite set. Define (3.18)λ̃1≔min⁡G̃∈Γ̃0{λmin⁡(H̃(G̃))},
which is fixed, and λ̃1>0.On the other hand, consider a weighted graph𝒢̂ with same node set and edge set as 𝒢̃l. In addition, its weight of edge (i,j) is taken as (aij+aji)/2-amin≥0 and weight of edge (0,i) is taken as 2bi-amin≥0. It is not difficult to see that(3.19)H(Ĝ)=B(Ĝ)+L(Ĝ)=2B(G¯)+L(G¯)+LT(G¯)-amin⁡[B(G̃)+L(G̃)]=H(G¯)+HT(G¯)-amin⁡H(G̃).
Noticing that H(𝒢̂) is symmetric and positive semidefinite matrix, we have (3.20)H(G¯)+HT(G¯)≥amin⁡H(G̃)≥amin⁡λ̃1I.Similarly, we can get the following result.Corollary 3.6.
Suppose that the interconnection graph𝒢¯σ(t) of any interval [tj,tj+1) belongs in Γ0. Take a constant
(3.21)κ>2amin⁡γ2λ̃1.
Then the local control law (2.6) can guarantee that the multi-agent system in leader-following case achieves consensus for any given initial condition r(0) and p(0).


## 4. Leaderless Case

In this section, similar to the analysis given in the last section, we probe the consensus problem for leaderless case. The involved interconnection graph is𝒢 instead of 𝒢¯. The closed-loop system (2.10) will be expressed in a compact form as follows:(4.1)ẋ=(Fσ⊗Im)x
with (4.2)x=(rp),Fσ=(0I-κLσ-κγLσ).Noticing that vector1 is the eigenvector of the Laplacian matrix Lσ corresponding to its zero eigenvalue by Lemma 2.1, we choose an orthogonal matrix with form(4.3)U=(1n*⋯*1n*⋯*⋮⋮⋮⋮1n*⋯*),
where * denotes the other entries in this matrix. We have(4.4)UTLσU=(0B¯σ0L¯σ),
where B¯σ∈R1×(n-1),L¯σ∈R(n-1)×(n-1). Taking s¯=UTr and q¯=UTp, system (4.1) is equivalent to be described by(4.5)s¯̇=q¯q¯̇=[(0-κB¯σ0-κL¯σ)⊗Im]s¯+[(0-κγB¯σ0-κγL¯σ)⊗Im]q¯.
For convenience, set (4.6)ξ0=s¯1,ξ1=(s¯2T,s¯3T,…,s¯nT)T,ψ0=q¯1,ψ1=(q¯2T,q¯3T,…,q¯nT)T.Then the whole system decomposes into two subsystems as follows: one is(4.7)ξ̇1=ψ1,ψ̇1=-κ(L¯σ⊗Im)ξ1-κγ(L¯σ⊗Im)ψ1,
and the other one is(4.8)ξ̇0=ψ0,ψ̇0=-κ(B¯σ⊗Im)ξ1-κγ(B¯σ⊗Im)ψ1.
Notice that all eigenvalues of the matrix L¯σ are eigenvalues of Lσ from (4.4), which means all of the nonzero eigenvalues of L¯σ are located on the open right half plane by Lemma 2.1. If the graph has a globally reachable node, the rank of L¯σ is n-1, and therefore all the eigenvalues of matrix L¯σ are located on the open right half plane, which implies matrix L¯σ is positive stable matrix. Now we give the main result in leaderless case as follows.Theorem 4.1.
Suppose that the interconnection graph has a globally reachable node for any interval[tj,tj+1). If there exist a matrix U1∈Rn×(n-1), a positive definite P¯3∈R(n-1)×(n-1), and a positive constant μ such that
(4.9)U1T1=0,U1TU1=In-1,(U1TLlU1)TP¯3+P¯3(U1TLlU1)≥μI,l∈P,
and taking a constant
(4.10)κ>2μγ2λmax⁡(P¯3),
then the local control law (2.9) can guarantee that the multi-agent system in leaderless case achieves consensus for any given initial condition r(0) and p(0).Proof.
TakeU=((1/n)1,U1). Due to U1T1=0 and U1TU1=In-1, it is easy to know that U is an orthogonal matrix and satisfies (4.4). Because the interconnection graph 𝒢l has a globally reachable node, L¯l=U1TLlU1 is positive stable matrix. Applying the result of Theorem 3.1 to system (4.7), we have
(4.11)lim⁡t→∞(ξ1ψ1)=0.
Noticing thatUT1=(n,0,…,0)T, we have(4.12)r(t)-1nξ0(t)1=U(s¯-1nξ0(t)UT1)=U(0ξ1(t))⟶0(t⟶+∞),p(t)-1nψ0(t)1=U(q¯-1nψ0(t)UT1)=U(0ψ1(t))⟶0(t⟶+∞),
which imply limt→+∞(ri(t)-rj(t))=0 and limt→+∞(pi(t)-pj(t))=0 for any i,j=1,2,…,n. This completes the proof.Similarly, consider the special case that all the interconnection graphs𝒢σ are balanced. Due to 1TLσ=Lσ1=0 for balanced graph 𝒢σ, we have(4.13)UTLσU=(000L¯σ),L¯σ∈R(n-1)×(n-1),
where U is orthogonal matrix having form (4.3). The next lemma is given for L¯σT+L¯σ.Lemma 4.2.
If𝒢σ is balanced and has a globally reachable node, then L¯σT+L¯σ is positive definite.Proof.
1/2(LσT+Lσ) is the Laplacian matrix of the mirror graph of 𝒢σ. Because 𝒢σ is balanced and has a globally reachable node, the mirror graph is undirected and connected. Therefor, 1/2(LσT+Lσ) is positive semidefinite and its rank is n-1. From (4.13), we have L¯σT+L¯σ is positive definite.Based on Lemma4.2 and the fact that the set 𝒫 is finite, define
(4.14)λ¯2∶=min⁡l∈P{12λmin⁡(L¯lT+L¯l)∣Glisbalancedandhasagloballyreachablenode}>0.λ¯2 is also well defined. From (4.13), λ¯2 is equivalent to be expressed as(4.15)λ¯2=min⁡l∈P{12λ2(LlT+Ll)∣Glisbalancedandhasagloballyreachablenode}>0,
where λ2(LlT+Ll) is the second small eigenvalue of LlT+Ll. Moreover, if all the interconnection graphs 𝒢σ associated with any time interval are undirected and connected, then we have λ¯2=minl∈𝒫{λ2(Ll)}. For balanced graph case, the following corollary is similarly obtained as in Corollary 3.5.Corollary 4.3.
In any interval[tj,tj+1), suppose that the interconnection graph 𝒢σ(t) is balanced and has a globally reachable node. Taking a constant
(4.16)κ>1γ2λ¯2,
then the local control law (2.9) can guarantee the multi-agent system in leaderless case achieves consensus for any given initial condition r(0) and p(0).Similarly, denote byΓ={𝒢p,p∈𝒫} the class of all possible interconnection graphs with the following properties: (1) the interconnection graph 𝒢p is balanced and has a globally reachable node; (2) all constant weighted factors aij are equal or greater than a positive constant amin. 𝒫 is index set of Γ, which is infinite set. Let Γ̃ denote the class of all possible nonweighted interconnection graphs with the properties that the graph is balanced and connected. Thus Γ̃ is a finite set. Define (4.17)λ̃2:=min⁡G̃∈Γ̃{λ2[LT(G̃)+L(G̃)]},
which is fixed, and λ̃2>0. By applying similar analysis of leader-following case, we propose the following result directly.Corollary 4.4.
Suppose that any interconnection graph𝒢σ(t) associated with interval [tj,tj+1) belongs in Γ. Take a constant
(4.18)κ>2amin⁡γ2λ̃2.
Then the local control law (2.6) can guarantee that the multi-agent system in leader-following case achieves consensus for any given initial condition r(0) and p(0).Remark 4.5.
We define the center of the multi-agent system asr¯=(1/n)∑i=1nri. The velocity of the center is expressed as p¯=(1/n)∑i=1npi. If all the interconnection graphs 𝒢σ associated with any time interval are balanced, the dynamics of center is given by
(4.19)r¯̇=1n∑i=1nẋi=1n∑i=1nvi=p¯,p¯̇=1n∑i=1nṗi=1n∑i=1nui,=1n∑i=1n[-κ∑j∈Niaij(ri-rj)-κ∑j∈Niaij(pi-pj)]=0.
Then, we can get that
(4.20)p¯(t)=p¯(0)=1n∑i=1npi(0)=1nψ0(t),r¯(t)=r¯0(0)+p¯(0)t=1nξ0(t).
The center r¯ of the closed-loop system (2.10) will move at a constant speed (1/n)∑i=1npi(0), which is the average initial velocity of all agents. The position and velocity of every agent will tend to the position and velocity of center, respectively, which means that the multi-agent network achieves consensus.Remark 4.6.
If the dynamics of the agent is linear, that is,ṙ=ui, and the interconnection graph is jointly connected, as pointed out in the local control law can guarantee the multi-agent system achieves consensus [1, 8]. But for double-integrator model, the jointly connected condition may not guarantee that the multi-agent system achieves consensus. We propose a counterexample as follows. Thus the assumption that the interconnection graph has a globally reachable node in many references and also in this paper may be moderatable and acceptable.Counterexample 1
The multi-agent system contains two agents, labeled by agent 1 and agent 2. Without loss of generality, we assumem=1. For any positive constant α, let t0=0,t1,t2,… be an infinite time sequence, which satisfies t2k+1=t2k+1 and t2k+2=t2k+1+(1/2α2)+(1/4α2)(e2α+e-2α), k=0,1,2,…. Suppose that the two agents are connected in time interval [t2k,t2k+1) and not connected in time interval [t2k+1,t2k+2). The dynamics of the two agents can be expressed as(4.21)ṙi=pi,ṗi=ui,i=1,2.
Take κ=2α,γ=α. All nonzero weighted elements of graph are taken as 1. Applying the local law (2.9) to the multi-agent system, the control input ui has the following form:(4.22)u1(t)={-2α(r1-r2)-2α2(p1-p2),t∈[t2k,t2k+1),0,t∈[t2k+1,t2k+2),u2(t)={-2α(r2-r1)-2α2(p2-p1),t∈[t2k,t2k+1),0,t∈[t2k+1,t2k+2).
The initial position and velocity of agents are taken as r1(0)=-r2(0)=(e2α-2α+1)/4α2 and p1(0)=-p2(0)=1. From the symmetry of the input and initial values, we can know that r1=-r2 and p1=-p2. Then, we can obtain(4.23)r1(t)=-r2(t)={[e2α-2α+14α2+e2α+12α(t-t4k)]e-2α(t-t4k),t∈[t4k,t4k+1),1+2α4α2+14α2e-2α2-(t-t4k+1),t∈[t4k+1,t4k+2),-[e2α-2α+14α2+e2α+12α(t-t4k)]e-2α(t-t4k),t∈[t4k+2,t4k+3),-1+2α4α2-14α2e-2α2+(t-t4k+3),t∈[t4k+3,t4k+4),p1(t)=-p2(t)={e-2α(t-t4k)-(e2α+1)(t-t4k)e-2α(t-t4k),t∈[t4k,t4k+1),-1,t∈[t4k+1,t4k+2),-e-2α(t-t4k)+(e2α+1)(t-t4k+2)e-2α(t-t4k+2),t∈[t4k+2,t4k+3),1,t∈[t4k+3,t4k+4),
for k=0,1,2,…. Although the interconnection graph related with any time interval [t2k+1,t2k+2) is not connected, the jointed graph related to the multi-agent system is jointly connected in any interval [t2k,t2(k+1)). Thus, the interconnection topology of the multi-agent system satisfies jointly connected condition proposed by [1]. On the other hand, it is easy to see that the multi-agent system cannot achieve consensus for any positive constant α in this case.


## 5. Simulation Examples

In this section, to illustrate our theoretical results derived in the above sections, we will provide two numerical simulations. Without loss of generality, we takem=1 in numerical simulation. In leader-follower case, consider a multi-agent system with one leader and six followers. The interconnection topology is arbitrarily switched with switching period 1 among four graphs 𝒢¯i(i=1,2,3,4). The Laplacian matrices Li(i=1,2,3,4) for the four subgraphs 𝒢i(i=1,2,3,4) are (5.1)L1=(3.5-1.5000-2-15.5-2.50-200-12-10000-25-1-20-20-15-2-100-1-24),L2=(5-1-20-20-1.54.5-100-2-2-140-1000020-2-10-10200-20-204),L3=(300-20-1020-1-1000200-2-2-105-200-10-230-20-2004),L4=(200-1-1004-20-200-25-20-1-20-2400-1-2005-200-10-23),
and the diagonal matrices for the interconnection relationship between the leader and the followers are (5.2)B1=diag⁡(1,0,0,1,0,0),B2=diag⁡(0,1,0,1,0,0),B3=diag⁡(1,0,0,0,1,1),B4=diag⁡(0,0,1,1,0,0).
The positive definite (5.3)P¯2=(4.600.580.180.650.780.670.584.010.950.410.920.850.180.956.120.740.801.380.650.410.743.510.590.610.780.920.800.595.081.050.670.851.380.611.054.92)
and μ=1 satisfy condition (3.3). Take γ=2.5 and κ=3>(2/μγ2)λmax(P¯2), where λmax(P¯2)=8.81.The initial positions and velocities of the all agents are randomly produced. The position and velocity errors in Figure1 are defined as ri-r0 and pi-p0, respectively. Figure 1 shows that the follower agents can track the leader.Position and velocity tracking errors of six followers.
(a)(b)In leaderless case, consider a multi-agent system with six agents. The interconnection topology is also arbitrarily switched with switching period 1 among four graphs𝒢i(i=1,2,3,4). The Laplacian matrices Li(i=1,2,3,4) of graphs 𝒢i are also defined as above. The matrix (5.4)U1=(-0.46950.49260.34340.3596-0.3507-0.5804-0.5372-0.0859-0.4456-0.04330.35980.3469-0.5239-0.3718-0.41330.5426-0.29340.6559-0.0941-0.11740.01090.35770.0222-0.15580.82490.1366-0.3666-0.41180.70770.0998),
the positive definite (5.5)P¯3=(0.7577-0.0131-0.03770.0021-0.0559-0.01310.68630.02620.0109-0.0595-0.03770.02620.81840.08130.00270.00210.01090.08130.82990.0194-0.0559-0.05950.00270.01940.8698)
and μ=1 such that they satisfy condition (4.9). Take γ=2.5 and κ=1>(2/μγ2)λmax(P¯3), where λmax(P¯3)=0.93. The position and velocity errors in Figure 2 are defined as ri-r¯ and pi-p¯, respectively. Figure 2 shows the multi-agent system in leaderless case can achieve consensus.Position and velocity tracking errors of six followers.
(a)(b)


## 6. Conclusion

In this paper, we have investigated the control of multi-agent systems with varying directed interconnection topologies using graph theory and stability theory. The proposed consensus strategy is neighbor-based law that is, each agent regulated its position and velocity based on its neighbor agents. There are two different cases considered in this study. One is the leader-following case and the other the leaderless case. To make this less conservative, a parameter-dependent common Lyapunov function (CLF) was constructed to analyze the stability of the closed system. Some sufficient conditions are given to achieve consensus of these mobile agents with the proposed local control strategies in both leader-following and leaderless cases, respectively. Of course, we can obtain consensus condition directly for the multi-agent system with the switching and undirected interconnection topology or directed balanced interconnection topology. Due to conservativeness of the common Lyapunov function method, we will probe less conservative method in our future work. We also will probe multi-agentH∞ consensus control problems with external disturbance under time delay directed switching topologies in our future work. Moreover, several numerical simulations were shown to verify the theoretical analysis.



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*Source: 273140-2011-11-20.xml*