Prescribed-Time Leader-Follower Consensus for Nonlinear Multi-Agent Systems∗

LU Wangming,YU Zhiyong,JIANG Haijun,XU Xuening

(School of Mathematics and System Sciences,Xinjiang University,Urumqi Xinjiang 830017,China)

Abstract: The prescribed-time leader-follower consensus problems are investigated for nonlinear multi-agent systems(MASs)with undirected and directed network topologies.First of all,a novel distributed control protocol is proposed for MASs with an undirected network topology.Based on Lyapunov stability theory,some sufficient conditions are derived for MASs to achieve leader-follower consensus within a prescribed-time.Moreover,we consider the directed network topology and give some sufficient conditions to ensure the prescribed-time leader-follower consensus.Finally,two examples are given to demonstrate the effectiveness of the main results.

Key words: prescribed-time;multi-agent systems;leader-follower consensus

0 Introduction

During the past decades,due to its wide applications in sensor networks,electrical power grids,and multi-unmanned air vehicle formation[1-3],the distributed cooperative control of MASs has witnessed substantial progress.In comparison with traditional centralized control,it has stronger adaptivity,higher robustness,and flexibility.As a fundamental and crucial issue of distributed cooperative control,the consensus of MASs has received extensive attention[4].

The existing consensus problems can be classified into leaderless consensus and leader-follower consensus.Compared with leaderless consensus,the leader-follower consensus can conserves resources and boosts efficiency.Therefore,the leaderfollower consensus problems were investigated in [5-6].However,the aforementioned consensus protocols are asymptotic convergent.That means the convergence time approaches infinity,which may not be verifiable in many practical situations.In view of this phenomenon,some researchers have developed finite-time algorithms[7-9],which can provide a faster convergence speed and better disturbance rejection performance.In[7-8],the finite-time consensus problems were put forward for firstorder and second-order integrator MASs,respectively.Further,the finite-time consensus control algorithms for higher-order integrator MASs were proposed in [9].However,the convergence time of the aforementioned results [7-9] depend on the initial conditions,which are unprocurable in real applications since the initial state is usually unknown or unavailable.

To resolve the above limitation of finite-time control,lots of results about fixed-time consensus problems have been reported[10-13].For instance,some fixed-time consensus algorithms were proposed for first-order integrator systems[10],second-order integrator systems[11-12],and high-order integrator systems[13].However,for the fixed-time consensus algorithms,the designer still cannot arbitrarily assign the settling time since it always depends on design parameters.In order to solve the shortcomings of the finite-time/fixed-time consensus algorithms.Wang et al.[14]proposed a novel scaling function to study the prescribed-time consensus problem of MASs.Further,a prescribed-time consensus problem was researched with second-order integrator MASs in[15].It is worth noticing that the systems considered in[10-15]are integral dynamics.Nevertheless,in many practical applications,the nonlinear dynamics are usually used to describe system behaviors.Thus,the prescribed-time consensus for nonlinear MASs is worth considering.

Motivated by the above observations,we will study the prescribed-time leader-follower consensus problems for nonlinear MASs under undirected and directed network topologies in this paper.The major contributions of this research are as follows:

(i)In contrast to[14],the model we considered has a nonlinear function term,and the state of the leader is time-varying.

(ii)Different from the finite-time/fixed-time consensus convergence conditions,the settling time of the prescribed-time algorithms is independent of the initial state and control parameters.

NotationsIn this paper,R and Rndenote the set of real number and the n-dimensional Euclidean space,respectively.In∈Rn×nrepresents n×n identity matrix.1n(or 0n)denotes an n-dimensional column vector whose all entries being 1(or 0).For a matrix A,AT,λmax(A),λmin(A)represent the transpose,maximum and minimum eigenvalues of A,respectively.A>0 means that A is a positive definite matrix.‖·‖and ⊗represent the Euclidean norm and the Kronecker product,respectively.

1 Preliminaries

1.1 Graph theory

The network of MASs is usually modeled by a graph G=(V,E,A),where V={v1,v2,···,vN} is the vertex set and E={(vi,vj)|vi,vj∈V} is the edge set.(vi,vj) ∈E indicates a directed edge from vito vj.The adjacency matrix of G is A=[aij]∈RN×N,aii=0,aij≠0 ⇔(vj,vi)∈E and aij=0,otherwise.The in-degree matrix associated with G is defined as D=diag(d1,d2,···,dN),where di=aij.The Laplacian matrix is defined as L=[lij]N×Nwith lij=-aijfor i ≠j,and lii=

In this paper,the interaction graph among N followers and the leader is denoted asThe communication subgraph among followers is defined as G.Let matrix B=diag(b1,b2,···,bN)with bi∈{0,1},where bi=1 indicates that agent i receives the information from the leader,otherwise bi=0.Moreover,the matrix H is defined by H=L+B.

1.2 Problem statement

We consider a system with one leader and N followers.The dynamics of the ith follower is given by

where xi(t) ∈Rn,ui(t) ∈Rnand f(t,xi(t)) denote the state,control input and the nonlinear function of the ith follower,respectively.W ∈Rn×nis a constant matrix.The leader has the following dynamics

where x0(t)∈Rn,f(t,x0(t))represent the state and nonlinear function of the leader,respectively.Denote x(t)=(xT1(t),xT2(t),···,xTN(t))Tand F(t,x(t))=(fT(t,x1(t)),fT(t,x2(t)),···,fT(t,xN(t)))T.

Consider the system

where ξ(t)∈Rnis the state vector,g:Rn×R+→Rnis a nonlinear vector valued function and g(0,t)=0 for all t>0.

Definition 1[16]The origin of the system(3)is said to be globally prescribed-time stable if limt→Tξ(t)=0,ξ(t)≡0(t ≥T)and the settling time T is a user-assignable finite constant,i.e.,0

We introduce a time-varying scaling function[14]:

where h > 0 and T > 0.Note that µ(t)-r(r > 0) is monotonically decreasing during the interval [0,T),µ(0)-r=1 and limt→T-µ(t)-r=0.

Lemma 1Let V(ξ(t)):Rn→R is a continuous and non-negative function.If there exist α>0,β>0 such that

then the origin of the system(3)is globally prescribed-time stable with the settling time T.Additionally,

and V(ξ(t))≡0 for any t ∈[T,∞).

ProofWhen t ∈[0,T),by multiplyingµα(t)on both sides of(5),one has

Then,it follows that

Solving the differential inequality(8),one obtains

µα(t)V(ξ(t))≤µα(0)V(ξ(0))exp(-βt).

Hence,it yields that V(ξ(t))≤µ-α(t)exp(-βt)V(ξ(0)).When t ∈[T,∞),due to limt→T-µ-α(t)=0 and combination with the continuity of V(ξ(t)),we get that V(ξ(T))=limt→T-V(ξ(t))=0.Based on(ξ(t))≤0 for t ≥T,it has V(ξ(t))≡0,t ∈[T,∞).

Assumption 1For any y1(t),y2(t)∈Rn,the nonlinear function f(·,·)satisfies

‖f(t,y1(t))-f(t,y2(t))‖≤ρ‖y1(t)-y2(t)‖,

where ρ is a nonnegative constant.

Assumption 2The graphcontains a directed spanning tree with the leader as the root node,and the subgraph G is undirected.

Assumption 3The graphhas a directed spanning tree with the leader as the root node,and the subgraph G is directed.

Lemma 2[17]If Assumption 3 holds,then the matrix-H is Hurwitz stable.Moreover,there exists a positive diagonal matrix P=diag(φ1,φ2,···,φN)such that PH+HTP is positive definite.

2 Main Results

This section analyzes the prescribed-time consensus problem of MASs(1)∼(2).With the above preparation,the control protocol for agent i is designed as

where k and c are positive constants.

Define the tracking error as=xi(t)-x0(t).The controller(9)can be rewritten in the following compact form

where u(t)=[uT1(t),uT2(t),···,uTN(t)]Tand

Theorem 1Suppose that Assumptions 1 and 2 hold,then the MASs (1)∼(2) can achieve prescribed-time leaderfollower consensus under the control protocol(10)within the settling time T,if the positive constants k and c satisfies

ProofConsider the Lyapunov function asThe time derivative of V(t)along the systems(1)∼(2)are given as

For simplicity,we define F(t,x(t))-1N⊗f(t,x0(t))=(t,x(t),x0(t)),then the above equation can be written as

According to Lemma 1 and condition(11),it yields

which further implies that

From the control protocol(10),we can see that(t)/µ(t)will tend to infinity as time approaches T,which will lead the controller (10) to be unbounded.In practical applications,the control protocol u(t) needs to remain bounded.Hence,the boundedness of the controller(10)should be proved over the whole time interval[0,∞).In light of(15),one gets that

and

Combining(16)and(17),we have

Next,we will discuss that the consensus is kept and the control protocol u(t)≡0 over[T,∞).Choose the same Lyapunov function candidates as in Theorem 1,and following the same procedure from(12)to(13),it can be learned that

Note that V(t)is continuous at t=T,then it follows

Combining(19)with(20),it yields

We can easily get V(t)≡0,t ≥T.Furthermore,we obtain u(t)≡0 on[T,∞),which imply

In other words,the prescribed-time leader-follower consensus of the systems (1)∼(2) are proved,and the control law u(t)is always bounded for t ∈[0,∞).

Remark 1It should be pointed out that the control law(10)is significant for achieving the prescribed-time consensus.However,the functionµ1/h(t)→∞as t →T.We need to demonstrate that the control law(10)is bounded.According to the preceding analysis,the controller u(t)is always bounded over the whole time interval[0,∞),so our design is reasonable.

Due to the fact that information sharing between MASs is not always bidirectional,studying the consensus of MASs on directed network topology is more meaningful.Therefore,the above results are extended to general directed network topology.

Theorem 2If Assumptions 1 and 3 hold and there exist positive constants k and c such that

then the MASs(1)∼(2)with the control protocol(10)can achieve the prescribed-time consensus.

ProofConstruct Lyapunov function

where ħi(t)=aij(xi(t)-xj(t))+bi(xi(t)-x0(t)),ħ(t)=[ħT1(t),ħT2(t),···,ħTN(t)]Twith ħi(t)=[ħi1(t),ħi2(t),···,ħin(t)]T.

Taking the time derivative of the Lyapunov function(24),it yields

According to Lemma 2,we get=λmin(PH+HTP)>0.By utilizing Assumption 1,it has

Substituting(26)into(25),one can get that

From the definition of V(t),one has

where φmax=max{φ1,φ2,···,φN}.By using condition(23),one can obtain

According to Lemma 1,we obtain V(t) ≤µ-α(t)exp(-βt)V(0),t ∈[0,T),and V(t) ≡0,t ∈[T,∞).This means that=x0(t),and xi(t)=x0(t)for t ≥T,i.e.,the prescribed-time consensus problem is solved under the directed network topology.

Taking a similar process as Theorem 1,we can derive that u(t)is bounded over the whole time interval[0,∞).

Remark 2It should be pointed out that the settling time of finite-time algorithms is related to the initial state,i.e.,a large initial state will increase the settling time.In addition,for fixed-time consensus protocol,the settling time is determined by the parameters of the system.However,in our proposed algorithms,the settling time T is not only regardless with the initial values of the system,but also does not depend on control parameters.

3 Numerical Simulations

In this section,two examples are presented to verify the effectiveness of the obtained results.

Example 1Consider MASs(1)∼(2)with one leader and four followers,in which W=the nonlinear function is f(t,xi(t))=[sin(xi1(t)),cos(xi2(t))]T,i=0,1,2,3,4,and ρ=1.The communication graph is given as in Fig 1.The corresponding Laplacian matrix among the followers is L=and B=diag(1,1,0,0).Choosing the initial states [x01(0),x11(0),x21(0),x31(0),x41(0)]=[-3,-4.3,2,-7,3],[x02(0),x12(0),x22(0),x32(0),x42(0)]=[5.5,7.3,2,5,11].According to Theorem 1,choose the other parameters c=3,k=6.54 and T=0.5.Figs 2∼3 depict the state trajectories of xi(t),i=0,···,4.Figs 4∼5 depict the state error of,i=1,···,4.It can be seen that the system achieves prescribed-time leader-follower consensus within T=0.5.

Fig 1 Communication topology

Fig 2 The state trajectories of xi1

Fig 3 The state trajectories of xi2

Fig 4 The state error of

Fig 5 The state error of

Example 2We consider MASs(1)∼(2)consist of five agents,which one leader agent and the remainder as followers.The communication graph is given in Fig 6.The nonlinear function is f(t,xi(t))=[sin(xi1(t)),cos(xi2(t))]T,i=0,1,2,3,4.It can be seen that f(t,xi(t)) satisfies Assumption 1 with ρ=1.Based on Lemma 2,we have P=diag(2,3,1,2).The system matrices are given as W=Meanwhile,the initial conditions of the systems (1) ∼(2) are given as[x01(0),x11(0),x21(0),x31(0),x41(0)]=[-2,-5.3,1,3,-4],[x02(0),x12(0),x22(0),x32(0),x42(0)]=[1,2,3,1.5,-1].In addition,set c=3.66,k=28.861,T=0.5.Figs 7∼8 show the state trajectories of xi(t),i=0,···,4.Figs 9∼10 show the state error of,i=1,···,4.Obviously,the prescribed-time control protocol ensures that the MASs achieves leader-follower consensus within T=0.5.

Fig 6 Communication topology

Fig 7 The state trajectories of xi1

Fig 8 The state trajectories of xi2

Fig 9 The state error of

Fig 10 The state error of

4 Conclusion

In this paper,a new control scheme was constructed to achieve prescribed-time leader-follower consensus for firstorder nonlinear MASs under undirected and directed network topologies.Based on Lyapunov stability theory,some sufficient conditions are derived to achieve leader-follower consensus.The proposed control scheme is bounded over the whole time interval.The prescribed-time consensus algorithm can ensure that the settling time is arbitrarily preallocated and is independent of the initial state and other parameters.This paper mainly focuses on only one leader prescribed-time consensus problem of first-order MASs.In future work,we will focus our research topic on prescribed-time consensus problems with second-order and high-order dynamics.