傅 勤（蘇州科技大學 數(shù)學科學學院，江蘇 蘇州 215009）

1 Introduction

Over the past decades，there has been a rapid development in the control field of multi-agent systems（MASs）.Generally speaking，there are two main approaches to deal with the control problem of MASs.One of them is the coordination control[1-10]，the main concern of which is to study the coordination among different agents，such as the consensus control for leaderless MASs[1-3]，the consensus tracking for leader-follower MASs with one leader[4-7]，and the containment control for leader-follower MASs with multiple leaders[8-10].The other one is the stabilization control[11-14]，which considers how to design the feedback controllers so that the states（or outputs）of all agents can converge to zero as time tends to infinity.Paper[11]studied the stabilizability of a class of MASs under switching topologies，and the necessary and/or sufficient conditions were presented in terms of graph topology.Paper[12]considered the decentralized stabilizability for a class of MASs with general linear dynamics，and the stabilizability problem was formulated in a way that the protocol performance could be evaluated by means of the stabilizability region and the feedback gain.The distributed robust stabilization control problem of MASs with general linear dynamics was investigated in Ref[13]，where the topology of the network was directed and the dynamics of each agent were subject to unknown uncertainties.In Ref[14]，the distributed H∞r(nóng)obust control problem synthesized with transient performance was investigated for a group of autonomous agents governed by uncertain general linear node dynamics，and based on the relative information between neighboring agents and some information of other agents，distributed state-feedback and observer-type output-feedback control protocols were designed and analyzed，respectively.It is noticed that some assumptions about the communication topologies ofMASs，such as being strongly connected[11]，having a spanning tree[13]，and so on，must be given in the process of control design for MASs so that the coordination or stability of systems can be obtained[1-14].However，to the best of our knowledge，there seems no relevant literatures for how to apply feedback design approaches to the MASs with arbitrary communication topologies and to solve the stabilization problems of such MASs.

In this paper，the stabilization control problem for a class of heterogeneous MASs with general linear dynamics is studied，where the communication topologies of MASs are arbitrary and time-varying.The main contribution of this paper is that the stabilization control problem of MASs is considered under arbitrary communication topologies，and the asymptotic stability of the closed-loop systems is obtained only relying on an often used reachability condition of linear systems.

2 Problem statement

Notations：Imrepresents the m×m dimensional identity matrix.For a given vector or matrix X，XTis used to denote its transpose，||X||is used to denote its 2-norm，and X-1is used to denote its inverse matrix if X is nonsingular.

Consider the following linear heterogeneous MASs which consist of N agents

where xi（t）∈Rni，ui（t）∈R，and yi（t）∈R are the state variable，input variable，and output variable of agent i，respectively.

Assumption 1（Fi，Bi）is a reachable pair[15]，i=1，2，…，N.

Some basic concepts in graph theory，which are often used to describe the communication among agents in MASs，are given as follows.

A digraph with vertex set V＝{1，2，…，N}and edge set E?V×V is denoted by G=（V，E，A），where A=（aij）∈RN×Nis the adjacency matrix of G，which is defined as aii=0 and aij＞0 if and only if（j，i）∈E.The communication topology of a MAS which consists of N agents can be described by the digraph G as follows：ith vertex represents the ith agent，and a directed edge（i，j）∈E represents that agent j can directly receive information from agent i.

The adjacency matrix of（at time t）the communication topology of system（1）is denoted by A（t）=（aij（t））∈RN×N.

Assumption 2There exists a positive constant M such that aij（t）≤M holds for all t＞0.i=1，2，…，N，j=1，2，…，N，i≠j.

For system（1），it is assumed that the input of agent i consists of two parts：the control input uic（t）and the external input signal uie（t）from the other adjacency agents，ie.，

Furthermore，the external input signal uie（t）can be usually expressed as

Therefore，system（1）can be rewritten as follows

3 Main results

By Assumption 1，we know from the theory of linear systems（see Ref[15]）that there exist a nonsingular ni×nimatrix Tiand a 1×nivector Kisuch that

Construct the controller as follows

where uif（t）is feedback controller to be determined later，then system（2）can be transformed into

Denote

Denote

where li（i=1，2，…，N）are positive constants to be determined later（see Ref[16]）.Let，and notice that

then system（4）can be transformed into

It is well-known from the theory of linear systems（see Ref[17]）that there exist a symmetric positive-definite matrix Pi∈Rni×niand a 1×nivector kisuch that

Remark 1It is easy to see that the feedback stabilization problem of system（1）is equivalent to that of system（5）.

Theorem 1For the system（1）satisfying Assumptions 1-2，the asymptotic stability of the closed-loop system can be obtained under the action of the following feedback control law

Proof.Construct the following Lyapunov function

By（5）-（7）and calculating the derivative of V（t），we have

Combining with Assumption 2，it yields

Take lj≥1，then||Qlj－1||=1，j=1，2，…，N.Denote

thus

Take li≥1+2（N-1）c＞1，then

which implies

That is，the closed-loop system（5）is globally asymptotically stable.□

4 Illustrate example

Consider the following MAS which consists of three agents

where

Therefore

The initial conditions of system（8）are given as followsThe adjacency matrix of the communication topology of system（8）is as follows

where aij（t）is set randomly to 0 or 1 for each t∈（0，5]，i=1，2，3，j=1，2，3，i≠j.System（8）can be transformed into

Take li=42≥1+2（N-1）c=1+4c=1+4×1×1.309×2.828×2.732=41.453 8，then

5 Conclusions

In this paper，we consider the stabilization control problem for a class of linear heterogeneous MASs，where the communication topologies of the MASs are arbitrary and time-varying.Under the condition that（Fi，Bi）of agent i（i=1，2，…，N）are reachable pairs，the feedback control laws are obtained by constructing an appropriate state transformation.When the feedback control laws are applied to the systems，the states of all agents can converge to zero as time tends to infinity.The simulation result proves the validity of the theoretical analysis.

Fig.1 Trajectory of||x（t）||