Wei LIU,Jie HUANG

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

Adaptive leader-following rendezvous and flocking for a class of uncertain second-order nonlinear multi-agent systems

Wei LIU,Jie HUANG?

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

In this paper,we study the leader-following rendezvous and flocking problems for a class of second-order nonlinear multiagent systems,which contain both external disturbances and plant uncertainties.What differs our problems from the conventional leader-following consensus problem is that we need to preserve the connectivity of the communication graph instead of assuming the connectivity of the communication graph.By integrating the adaptive control technique,the distributed observer method and the potential function method,the two problems are both solved.Finally,we apply our results to a group of van der Pol oscillators.

Adaptive control,connectivity preservation,multi-agent systems,nonlinear systems

1 Introduction

Over the past few years,the study of cooperative control problems for multi-agent systems has attracted extensive attention.In many cooperative control problems such as the consensus problem,the communication graph is predefined and has to satisfy certain connectivity assumption[1–5].However,in some real applications such as rendezvous problem and flocking problem,the communication graph is defined by the distance of various agents,and is thus state-dependent.It is more practical to enable a control law to not only achieve consensus but also preserve the connectivity of the graph instead of assuming the connectivity of the graph.Such a problem is called rendezvous with connectivity preservation problem.If the objective of collision avoidance is also imposed,then the problem can be further called flocking.

Depending on whether or not a multi-agent system has a leader,the rendezvous/flocking problem can be further divided into two classes:leaderless and leader-following.The leaderless rendezvous/flocking problem aims to make the state(or partial state)of all agents approach a same location,while the leader-following rendezvous/flocking problem further requires the state(or partial state)of all agents to track a desired trajectory generated by some leader system.The leaderless rendezvous/flocking problem has been studied for single-integrator multi-agent systems in[6–8]and double-integrator multi-agent systems in[9–11]while the leader-following rendezvous/flocking problem has also been studied for single-integrator multi-agent systems in[12,13]and double-integrator multi-agent systems in[10,13–16].

More recently,the leader-following rendezvous/flocking problem has been further studied for some second-order nonlinear multi-agent systems under various assumptions in [17–20].Specifically,in[17],the connectivity preserving leader-following consensus problem for uncertain Euler-Lagrange multi-agent systems is studied.In[18],the differences between the nonlinear functions of all agents are assumed to be bounded for all time.In[19,20],the nonlinear functions are assumed to satisfy global Lipschitz-like condition and all followers know the information of the virtual leader.

In this paper,we will study both the leader-following rendezvous problem and the leader-following flocking problem for a class of second-order nonlinear multiagent systems by a distributed state feedback control law with different potential functions.Our problems differ from existing works in at least two aspects.First,our system as given in next section is subject to not only external disturbances but also plant uncertainties.Second,the nonlinear functions in our system do not have to satisfy some bounded condition or global Lipschitz-like condition.To overcome these difficulties,we need to combine the adaptive control technique,the distributed observer method and the potential function method to solve our problems.

The rest of this paper is organized as follows.In Section 2,we give two problem formulations and some preliminaries.In Sections 3 and 4,we give the main results.In Section 5,we provide an example to illustrate our design.Finally,in Section 6,we conclude the paper with some remarks.It is noted that the preliminary version of this paper without any proof was presented in[21].

NotationFor any column vectors ai,i=1,...,s,denote col(a1,...,as)=[aT1,...,aTs]T.?denotes the Kronecker product of matrices.?x?denotes the Euclidean norm of vector x.?A?denotes the induced norm of matrix A by the Euclidean norm.For any real symmetric matrix A, λmin(A)and λmax(A)denote the minimum and maximum eigenvalues of A,respectively.For any two symmetric matrices A and B,the symbol A≥B means the matrix A?B is positive semi-definite.

2 Problem formulation

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

where qi,pi∈Rnare the states,ui∈Rnis the input,fi(qi,pi)∈ Rm×nis a known matrix with every element being continuous function,θi∈ Rmis an unknown constant parameter vector,di(w)∈Rndenotes the disturbance with di(·)being some C1function,and w is generated by the linear exosystem as follows:

with w ∈ Rnwand Sb∈ Rnw×nw.It is assumed that the reference signal is generated by the following linear exosystem

The plant(1)and the exosystem(4)together can be viewed as a multi-agent system of(N+1)agents with(4)as the leader and the N subsystems of(1)as N followers.As in[15,17],define a time-varying graph(t)=(V,E(t))with respect to(1)and(4),whereV={0,1,...,N}with 0 associated with the leader system and with i=1,...,N associated with the N followers,respectively,andE(t)?V×V is defined by different rules for rendezvous and flocking problem.The graphis said to be connected at time t if there is a directed path from node 0 to every other node.

Remark 1Compared with the second-order nonlinear systems studied in[18–20],our system contains not only the external disturbances but also the parameter uncertainties,and the boundaries of the uncertainties are allowed to be arbitrarily large,while the systems in[18,20]contain neither external disturbances nor plant uncertainties,and the system in[19]contains only plant uncertainties but no external disturbances.Moreover,the nonlinear function fiin(1)does not need to be bounded as assumed in[18],or satisfy the global Lipschitz-like condition as assumed in[19,20].

2.1 Leader-Following Rendezvous Problem

For leader-following rendezvous problem,E(t)is defined by the following rules:Given any r＞0 and∈∈(0,r),for any t≥0,E(t)={(i,j)|i,j∈V,i≠j}is defined such that

1)E(0)={(i,j)|?qi(0)?qj(0)?＜r?∈,i=0,1,...,N,j=1,...,N};

2)for i=0,1,...,N,j=1,...,N,if?qi(t)?qj(t)?≥ r,then(i,j)?E(t);

3)for i=0,1,...,N,(i,0)?E(t);

4)for i=0,1,...,N,j=1,...,N,if(i,j)?E(t?)and?qi(t)?qj(t)?＜r?∈,then(i,j)∈(t);

5)for i=0,1,...,N,j=1,...,N,if(i,j)∈E(t?)and?qi(t)?qj(t)?＜r,then(i,j)∈(t).

Note that the above rules are similar to those in[15].We denote the neighbor set of the ith agent at time t byi(t)={j|(j,i)∈(t)}.Then,we consider a control law of the following form:

where hiand liare some nonlinear functions,and ζi∈ Rnζiwith nζito be defined later.A control law of the form(5)is called a dynamic distributed state feedback control law,since uionly depends on the state information of its neighbors and itself.Then,we define the leader-following rendezvous problem for system(1)as follows.

Problem 1Given the plant(1),the exosystem(4),any r＞0 and∈∈(0,r),find a distributed control law of the form(5),such that,for any w∈W with W being some compact subset of Rnwand any initial condition qi(0),i=0,1,...,N,making(0)connected,theclosedloop system composed of(1)and(5)has the following properties:

2.2 Leader-following f l ocking problem

For leader-following flocking problem,E(t)is defined by the following rules:Given any r＞ 0,∈∈(0,r)and R∈[0,r?∈),for any t≥0,E(t)={(i,j)|i,j∈V,i≠j}is defined such that

2)for i=0,1,...,N,j=1,...,N,if?qi(t)?qj(t)?≥r,then(i,j)?E(t);

4)for i=0,1,...,N,j=1,...,N,if(i,j)?and R＜?qi(t)?qj(t)?＜r?∈,then(i,j)∈E(t);

5)for i=0,1,...,N,j=1,...,N,if(i,j)∈and R＜?qi(t)?qj(t)?＜r,then(i,j)∈(t).

Note that the above rules are similar to those in Section IV of[17].Then,we define the leader-following flocking problem for system(1)as follows.

Problem 2Given the plant(1),the exosystem(4),any r＞ 0,∈∈(0,r)and R ∈[0,r? ∈),find a distributed control law of the form(5),such that,for any w∈W with W being some compact subset of Rnwand any initial condition qi(0),i=0,1,...,N,satisfying?qi(0)?qj(0)?＞ R for i≠ j,i,j=0,1,...,N,and making(0)connected,the closed-loop system composed of(1)and(5)has the following properties:

3)Collision can be avoided among all agents,that is?qi(t)?qj(t)?＞ R for i,j=0,1,...,N,i≠ j and all t≥ 0.

2.3 One assumption

To solve the above two problems,we need one assumption as follows.

Assumption 1The exosystem(4)is neutrally stable,i.e.,all the eigenvalues of S are semi-simple with zero real parts.

Remark 2Under Assumption 1,the exosystem(4)can generate some fundamental types of reference signals and disturbance signals such as step signals,sinusoidal signals and their finite combinations.Moreover,under Assumption 1,given any compact set V0,there exists a compact set V such that,for any v(0)∈V0,the trajectory v(t)of the exosystem(4)remains in V for all t≥0.

3 Leader-following rendezvous

In this section,we will consider the leader-following rendezvous problem.We first recall the concept of the distributed observer for the leader system(4)proposed in[22]as follows:

By Theorem 1 and Remark 4 of[22],under Assumption 1 and the condition that the graph(t)is fixed and connected,we haveexponentially.That is why(6)is called the distributed observer for(4).

To achieve connectivity preservation,we will adopt the same potential function used in[17]as follows:

Now we propose our distributed dynamic control law as follows:

where kiis some positive constant,and

with C1=[0nw×2nInw]and C2=[0n×nIn0n×nw].

Letqi=qi?q0andpi=pi?p0for i=0,1,...,N.Note thatqi?qj=qi?qjandpi?pj=pi?pj.Thus,for i=1,...,N,we have

which implies

The closed-loop system composed of(1)and(10)is as follows:

Under Assumption 1,by Remark 2,w∈W for all t≥0 with W being some compact subset of Rnw.Together withwe can conclude that there exists some smooth function?d(?v)≥0 such that,for all w∈W,

Now we give our result as follows.

Theorem1Under Assumption1,the leaderfollowing rendezvous problem for the multi-agent system composed of(1)and(4)is solvable by the distributed state feedback control law(10)with the potential function(9).

ProofBy the continuity of the solution of the closedloop system(14),there exists 0＜t1≤+∞such that(t)=(0)for all 0≤t＜t1.Thus,ij(t)=ij(0)and H(t)=H(0)for all 0≤t＜t1.Let

Then,from(15)and(16),along the trajectory of the closed-loop system(14),for 0≤t＜t1,we have

Since the number of agents is finite,the number of connected graphs associated with these N+1 agents is also finite.Denote all connected graphs by{1,...,n0}and denote the H matrix associated with these connected graphs by{H1,...,Hn0}which are all symmetric and positive definite.Let

Then,along the trajectory of the distributed observer(8),for 0≤t＜t1,we have

where λ1= λmax(ST+S)and λ2=min{λmin(H1),...,λmin(Hn0)}.Choose(λ1+1).Then,for 0 ≤ t＜t1,we have

Choosesomesmoothfunctionρ(??v?2)≥?C2?2+?d(?v)+1.Let

Then,from(23)and(24),for 0≤t＜t1,we have

Finally,let

Then,it can be seen that for all initial condition qi(0),i=0,1,...,N,that makes(0)connected,

If t1=+∞,thenfor all t≥0,and thus(28)holds for all t≥0.

for any t∈[ti,ti+1)with i=0,1,...,k,t0=0 and tk+1=+∞.

Since V(t)≥0 is lower bounded,by(30),exists and for i=1,...,N,j∈i(tk)are bounded.Since the graphis connected for all t≥tk,qi?qjwith j∈are bounded and q0is bounded by Remark 2,we can easily obtain that qiis bounded for i=0,1,...,N.By Remark 2,v is bounded,thus?vi=v+?viis also bounded.From the second equation of(11),priis bounded for i=0,1,...,N.Then,fromthethirdequationof(11),piisalsobounded.By Remark 2,p0is bounded and thuspi=pi?p0is bounded.

Next,we will show that¨V is bounded for all t≥tkwhich implies that˙V is uniformly continuous for all t≥ tk.Note that,for t≥ tk

Now,for i=1,2,...,N,j=0,1,...,N and j≠i,and t≥0,let

which can be further put into the following form:

4 Leader-following f l ocking

In this section,we will consider the leader-following flocking problem.The technique is similar to that used in Section 3.However,what makes the flocking problem different from rendezvous problem is that we need to avoid collision among agents.For this purpose,we need to use a different potential function as follows:

which is similar to that in[9].Then,we give the result as follows.

Theorem2Under Assumption1,the leaderfollowing flocking problem for the multi-agent system composed of(1)and(4)is solvable by the distributed state feedback control law(10)with the potential function(40).

ProofThe proof is similar to the proof of Theorem1,the only difference is that we need to show that the collision can be avoided in the sense that?qi(t)?qj(t)?＞R,i,j=0,1,...,N and i≠ j for all t≥ 0.

If the collision happens at a finite time tl,which impliesV(t)=+∞.However,by(30),we have V(t)≤V(0)＜+∞ for all t≥0,which makes the contradiction.Thus the collision can be avoided in the sense that?qi(t)?qj(t)?＞R,i,j=0,1,...,N and i≠ j for all t≥ 0.

Thus the proof is completed.

5 An example

In this section,we will apply our results to the leaderfollowing rendezvous/flocking problem for a group of van der Pol systems as follows:

where qi=[q1i,q2i]T∈R2and pi=[p1i,p2i]T∈R2for i=1,...,4,w=[w1,w2]T,and

Clearly,system(41)is in the form(1)with

The exosystem is in the form(4)with

Clearly,Assumption 1 is satisfied.

The initial communication graph(0)is described by Fig.1 where node 0 is associated with the leader and other nodes are associated with the followers.

Fig.1 The initial communication graph.

5.1 Leader-following rendezvous

By Theorem 1,we design a distributed state feedback control law of the form(10)with the potential function given by(9),r=3,∈=0.2,μ0=10 and ki=4 for i=1,2,3,4.

Simulation is performed with

and the following initial conditions:

It is easy to see that the initial diagraph(0)is connected under the first five rules defined in Section 2.

Figs.2,3 and 4 show that all followers approach the position of the leader asymptotically with the same velocity of the leader while preserving the connectivity,that is to say,the leader-following rendezvous problem for system(41)is solved by the distributed state feedback control law of the form(10)with the potential function given by(9).

Fig.2 Distances between leader and all followers.

Fig.3 Distances between all followers.

Fig.4 Velocity errors between leader and all followers.

5.2 Leader-following f l ocking

By Theorem 2,we design a distributed state feedback control law of the form(10)with the potential function given by(40),r=3,R=1,∈=0.2,μ0=10 and ki=4 for i=1,2,3,4.

Simulation is performed with the same θi,i=1,2,3,4,and initial conditions as given in the simulation for the leader-following rendezvous problem.It is also easy to see that the initial diagraph(0)is connected under the second five rules defined in Section2.

Figs.5 and 6 show that the connectivity is preserved and the collision is avoided.Fig.7 further shows that the velocities of all followers approach the velocity of the leader asymptotically.That is to say,the leader-following flocking problem for system(41)is solved by the distributed state feedback control law of the form(10)with the potential function given by(40).

Fig.5 Distances between leader and all followers.

Fig.6 Distances between all followers.

Fig.7 Velocity errors between leader and all followers.

6 Conclusions

In this paper,we have studied both the leaderfollowing rendezvous problem and flocking problem for a class of second-order nonlinear multi-agent systems.Compared with the existing results,our systems contain not only external disturbances but also parameter uncertainties,and the parameter uncertainties are allowed to be arbitrarily large.By combining the adaptive control technique,the distributed observer method and the potential function method,we have solved the two problems by the distributed state feedback control law.

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27 June 2017;revised 1 September 2017;accepted 1 September 2017

DOIhttps://doi.org/10.1007/s11768-017-7083-0

?Corresponding author.

E-mail:jhuang@mae.cuhk.edu.hk.Tel.:+852-39438473;fax:+852-26036002.

This paper is dedicated to Professor T.J.Tarn on the occasion of his 80th birthday.

This work was supported by the Research Grants Council of the Hong Kong Special Administration Region(No.14200515).

?2017 South China University of Technology,Academy of Mathematics and Systems Science,CAS,and Springer-Verlag GmbH Germany

Wei LIUreceived the B.Eng.degree in 2009 from Southeast University,Nanjing,China,the M.Eng.degree in 2012 from University of Science and Technology of China,Hefei,China,and the Ph.D.degree in 2016 from The Chinese University of Hong Kong,Hong Kong,China.He is currently a Postdoctoral Fellow at The Chinese University of Hong Kong.His research interests include output regulation,event-triggered control,nonlinear control,multi-agent systems,and switched systems.E-mail:wliu@mae.cuhk.edu.hk.

Jie HUANGis Choh-Ming Li professor and chairman of the Department of Mechanical and Automation Engineering,The Chinese University of Hong Kong,Hong Kong,China.His research interests include nonlinear control theory and applications,multi-agent systems,and flight guidance and control.Dr.Huang is a Fellow of IEEE,IFAC,CAA,and HKIE.E-mail:jhuang@mae.cuhk.edu.hk.