Wei LIU,Jie HUANG

Department of Mechanical and Automation Engineering,The Chinese University of Hong Kong,Shatin,N.T.,Hong Kong,China

1 Introduction

In the past few years,the cooperative control problem s for multi-agent system s have attracted extensive attention due to their applications in sensor networks,robotic team s,satellite clusters,unmanned air vehicle formations and so on.The consensus problem is one of the basic cooperative control problem s,w hose objective is to design a distributed control law for each agent such that the states (or outputs) of all agents synchronize to a common trajectory[1-4].Depending on whether or not a multi-agent system has a leader,the consensus problem can be divided into two classes:leaderless and leader-following.The leaderless consensus problem does not specify the common trajectory[2,3],while the leader-following consensus problem requires the states(or outputs)of all agents to track a desired trajectory generated by a so-called leader system[4-7].

An important class of multi-agent system s is the second-order nonlinear multi-agent system s.Recently,considerable efforts have been made to hand le the leader-following consensus problem for a class of second-order nonlinear multi-agent system s[8-13].For example,references[8-10]studied the leader-following consensus problem for some second-order nonlinear multiagent system s under the assumption that the nonlinear functions satisfy the global Lipschitz condition or global Lipschitz-like condition.The system studied in reference[11]contains disturbance but no uncertainty.The system s considered in[12,13]allow both disturbance and uncertainty,but the boundary of the uncertainty is known.

In this paper,we will further consider the leader following consensus problem for a class of secondorder nonlinear m ulti-agent system s subject to linearly parameterized uncertainty and disturbance.Com pared with[8-10],we do not impose the global Lipschitz condition or the global Lipschitz-like condition on the nonlinear functions.Com pared with[8-11],the nonlinear multi-agent system here contains both linearly parameterized uncertainty and disturbance.Finally,com pared with[12,13],our uncertainty can be arbitrarily large,and we do not assume the uncontrolled system has an equilibrium point at the origin.

Our distributed control law is based on a combination of the adaptive control technique and the adaptive distributed observer method developed in[14].It turns out that such a control law is quite effective in dealing with the problem studied in this paper.

The rest of this paper is organized as follow s.In Section 2,we will give our problem formulation and some preliminaries.In Sections 3,we will give our main result.In Section 4 we will provide an example to illustrate our design.Finally,in Section 5,we will finish the paper with some conclusions.

NotationFor any column vectorsai,i=1,...,s,denotedenotes the Kronecker product of matrices.‖x‖denotes the Euclidean norm of vectorx.‖A‖denotes the induced norm of matrixAby the Euclidean norm.

2 Problem formulation

Consider a class of second-order nonlinear multiagent system s as follows:

w hereqi,pi∈Rnare the states,ui∈Rnis the input,is a matrix with every elem ent being known continuous function,θi∈ Rmis an unknown constant parameter vector,denotes the disturbance withdi(·)being some C1function,andwis generated by the following linear exosystem

withw∈ RnwandSb∈ Rnw×nw.It is assumed that the reference signal is generated by the following linear exosystem

System(1)and the exosystem (4) together can be viewed be viewed as a multi-agent system of(N+1)agents with(4)as the leader and theNsubsystem s of(1)asNfollowers.

Next,we introduce some graph notation which can also be found in[15].A digraph G=(V,E)consists of a finite set of nodes V={1,...,N}and an edge set E?V×V.An edge of E from nodeito nodejis denoted by(i,j),where nodeiandjare called the parent node and the child node of each other.Define Ni={j|(j,i)∈E},which is called the neighbor set of nodei.The edge(i,j)is called undirected if(i,j)∈E implies(j,i)∈E.The digraph G iscalled undirected ifevery edge in E is undirected.If the digraph G contains a sequence of edges of the form(i1,i2),(i2,i3),...,(ik,ik+1),then the set{(i1,i2),(i2,i3),...,(ik,ik+1)}is called a path of G from nodei1to nodeik+1and nodeik+1is said to be reachable from nodei1.A digraph is called connected if there exists a nodeisuch that any other nodes are reachable from nodei.The weighted adjacency matrix of the digraph G is a nonnegative matrix A=[aij] ∈ RN×Nwhereaii=0 andaij＞0?(j,i)∈E,i,j=1,...,N.On the other hand,given a matrix A=[aij]∈ RN×Nsatisfyingaii=0 andaij≥0 fori,j=0,1,...,N,we can always define a digraph G such that A is the weighted ad jacencymatrix of thedigraph G.We call G thedigraph of A.

with respect to the plant(1)and the exosystem(4),we can define a digraphwith={0,1,...,N}andwhere the node 0 is associated with the leader system(4)and the nodei,i=1,...,N,is associated with theith subsystem of system(1).Fori=1,...,N,j=0,1,...,Nandif and only ifuican use the information of thejth subsystem for control.Letbe the weighted ad jacency matrix of.Letdenote the neighbor set of agenti.

We describe our control law as follows:

wherehiandliare some nonlinear functions.A control law of the form(5)is called a distributed control law,sinceuionly dependson the information ofitsneighbors and itself.Our problem is described as follow s.

Prob lem 1Given the multi-agent system(1),the exosystem(4)and a digraphdesign a control law of the form(5),such that,for any initial statesqi(0),pi(0),ζi(0)andv(0),qi(t)andpi(t)exist for allt≥ 0,and satisfy

Rem ark 1Note that,like in[12,13],here we assume that the reference signal and the disturbance are generated by a linear exosystem(4)called the leader.Indeed,this formulation is more general than the case that the disturbancedi(w)is generated by an individual exosystem for each follower.

To solve our problem,we make two assumptions as follow s.

Assum p tion 1The exosystem(4)is neutrally stable,i.e.,all the eigenvalues ofSare sem i-sim p le with zero real parts.

Assum p tion 2Every nodei=1,...,Nis reachable from the node 0 in the diagraph.

Rem ark 2Assum ption 1 is standard and has been used in[12].Under Assumption 1,the exosystem(2)can generate arbitrarily large constant signals and some sinusoidal signals with arbitrary initial phases and am plitudes,and >with arbitrary initial phases with arbitrary initial phases and am plitudes.What’s more,under Assum ption 1,given any com pact setV0,thereexistsa com pactsetVsuch that,foranyv(0)∈V0,the trajectoryv(t)of the exosystem(4)remains inVfor allt≥0.

Rem ark 3Assum ption 2 is also a standard assumption and has been used in m any literatures on cooperative control problem s of multi-agent systems[12-14,16].Note that Assum ption 2 allows the network to be directed and thus is less restrictive than that in[11,17].

3 Main resu lt

In this section,we will consider the leader-following consensus problem for system(1)and exosystem(4).

We first recall the concept of the distributed observer for the leader system developed in[16]as follow s:

wherefori=1,...,N,μ0is any positive constant.By Theorem 1 and Rem ark 4 of[16],under assumptions 1 and 2,we havei=1,...,N.That is why w e call(6)the distributed observer for(4).

However,adrawback of(6)is thatthematrixSis used by every follower which may not be realistic in some applications.To overcome this drawback,an adaptive distributed observer was further proposed in[14]as follow s:

whereμ1and μ2are any positive constants.The adaptive distributed observer(7)is more realistic than the distributed observer(6),since heredepends onSat the timetiff the leader is the neighbor of theith follower at timet,while the matrixSis used by every follower in(6).

Letandfori=0,1,...,N.Then,fori=1,...,N,

Let,andThen(8)can be put into the following com pact form

wherewithforThen we introduce the following lemm a.

Lemm a 1(Lemma 2 of[14])Under assumptions 1 and 2,we have

exponentially and

exponentially.

To synthesize our control law,let

where α is a positive constant,and

Then,our control law is as follow s:

w herekiis some positive constant,and

The closed-loop system com posed of(1)and(17)is as follow s:

whereandIt is easy to see thatfor allw∈Rnw.Under Assum ption 1,by Rem ark 2,we know thatw∈W for allt≥0 with W being some com pact subset of Rnw.Then,by Lemma 7.8 of[18],there exists some smooth functionsuch that,for allw∈W,

Now we give our result as follows.

Theorem 1Under assumptions 1 and 2,the leaderfollowing consensus problem for the system com posed of(1)and(4)is solvable by the distributed control law(17).

ProofLet

Then the time derivative ofValong the trajectory of the closed-loop system(19)is given by

and thus

(25)can be view ed as a stable first order linear system inqiwith a bounded input sinceand ξiare all bounded,bothqiand˙qiare bounded.Therefore,from(15)and(18),priandare both bounded.From the second equation of(19),is bounded.Thusis also bounded.Note that

Sinceandare all bounded,we can conclude thatis bounded for allt≥0.Then,by Barbalat’s Lemm a,and thus,from(23),we haveNext,by(7),(15)and(16),we have

our proof is thus com p leted.

4 An exam p le

Consider the leader-following consensus problem for a group of Vol del Pol system s as follow s:

whereClearly,system(29)is in the form of(1)withand

The communication graph is described by Fig.1 where the node 0 is associated with the leader and the other nodes are associated with the followers.Clearly,every nodei=1,2,3,4 is reachable from the node 0 in the diagraphand thus Assumption 2 is satisfied.From Fig.1,we obtain that the ad jacency matrix ofis

Then,by Theorem 1,we can design a distributed control law as follow s:

where

siand˙priare defined as in(16)and(18)withD=[1 0 0 0]and α=1.

Fig.1 Communication graph

Sim ulation is perform ed with

and the following initial conditions:

Fig.2 shows the states of the leader system which are bounded for all timet≥0.Figs.3-6 show the estimation errors of the observer for each follower.It can be seen that all four estimations of leader’s states converge to the leader’s states ast→ +∞.

Fig.2 States of leader system:

Fig.3 Estimation errors:

Fig.4 Estimation errors:

Fig.6 Estimation errors:

Figs.7 and 8 further show the tracking performance ofqiandpi.As expected,the states of all followers approach the states of the leader asymptotically.

Fig.7 Tracking errors:

Fig.8 Tracking errors:pi-p0.

5 Conclusions

In this paper,we have studied the leader-following consensus problem for a class of second-order nonlinear multi-agent system s subject to linearly parameterized uncertainty and disturbance.We have solved the problem by integrating the adaptive control technique and the adaptive distributed observer method.It is interesting to further consider the case where the network topology is switching and satisfies the jointly connected condition.

References

[1] A.Jadbabaie,J.Lin,A.S.Morse.Coordination of groups of mobile agents using nearest neighbor rules.IEEE Transactions on Automatic Control,2003,48(6):988-1001.

[2]R.O lfati-Saber,R.M.Murray.Consensus problem s in networks of agents with switching topology and time-delays.IEEE Transactions on Automatic Control,2004,49(9):1520-1533.

[3]W.Ren.On consensus algorithm s for double-integrator dynamics.IEEE Transactions on Automatic Control,2008,53(6):1503-1509.

[4] Y.Su,J.Huang.Stability of a class of linear switching system s with applications to two consensus problem s.IEEE Transactions on Automatic Control,2012,57(6):1420-1430.

[5]Y.Hong,G.Chen,L.Bushnell.Distributed observers design for leader-following control of multi-agent networks.Automatica,2008,44(3):846-850.

[6]J.Hu,Y.Hong.Leader-following coordination of m ulti-agent system s with coupling time delays.Physica A:Statistical Mechanics and its Applications,2007,374(2):853-863.

[7]W.Ni,D.Cheng.Leader-following consensus of multi-agent system s under fixed and switching topologies.System s&Control Letters,2010,59(3/4):209-217.

[8]J.Mei,W.Ren,G.Ma.Distributed coordination for second-order m ulti-agent system s with nonlinear dynamics using only relative position measurem ents.Automatica,2013,49(7):2107-2115.

[9]Q.Song,J.Cao,W.Yu.Second-order leader-following consensus of nonlinear multi-agents via pinning control.System s&Control Letters,2010,59(9):553-562.

[10]W.Yu,W.Ren,W.Zheng,et al.Distributed control gains design for consensus in m ulti-agent system s with second-order nonlinear dynamics.Automatica,2013,49(5):1419-1427.

[11]C.Wang,H.Ji.Robust consensus tracking for a class of heterogeneous second-order nonlinear m ulti-agent system s.International Journal of Robust and Nonlinear Control,2015,25(17):3367-3383.

[12]Y.Su,J.Huang.Cooperative global output regulation of heterogenous second-order nonlinear uncertain m ulti-agent system s.Automatica,2012,49(11):3345-3350.

[13]X.Wang,D.Xu,Y.Hong.Consensus control of nonlinear leader-follower m ulti-agent systems with actuating disturbances.System s&Control Letters,2014,73:58-66.

[14]H.Cai,F.L.Lew is,G.Hu,et al.The adaptive distributed observer approach to the cooperative output regulation of linear m ulti-agent system s.Automatica,2016:DOI 10.1016/j.automatica.2016.09.038.

[15]C.Godsil,G.Royle.Algebraic Graph Theory.New York:Springer,2001.

[16]Y.Su,J.Huang.Cooperative output regulation of linear m ultiagent system s.IEEE Transactions on Automatic Control,2012,57(4):1062-1066.

[17]J.Wang,K.Chen,Q.Ma.Adaptive leader-following consensus of m ulti-agent system s with unknown nonlinear dynamics.Entropy,2014,16(9):5020-5031.

[18]J.Huang.Non linear Output Regu lation:Theory and Applications.Phildelphia:SIAM,2004.