# Impulsive Consensus for Leader-Following Multiagent Systems with Fixed and Switching Topology

Authors: Zhi-Wei Liu; Zhi-Hong Guan; Hong Zhou
Journal: Mathematical Problems in Engineering
 (2013)

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## Abstract

This paper studied the consensus problem of the leader-following multiagent system. It is assumed that the state information of the leader is only available to a subset of followers, while the communication among agents occurs at sampling instant. To achieve leader-following consensus, a class of distributed impulsive control based on sampling information is proposed. By using the stability theory of impulsive systems, algebraic graph theory, and stochastic matrices theory, a necessary and sufficient condition for fixed topology and sufficient condition for switching topology are obtained to guarantee the leader-following consensus of the multiagent system. It is found that leader-following consensus is critically dependent on the sampling period, control gains, and interaction graph. Finally, two numerical examples are given to illustrate the effectiveness of the proposed approach and the correctness of theoretical analysis.


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## Body

## 1. Introduction

During the past several decades, the consensus problem of the multiagent system has drawn a great deal of attentions because of its broad applications in many domains, including distributed coordination [1], synchronization of dynamical networks [2], distributed filtering [3], and load balancing [4]. The basic idea of consensus is to design a distributed control such that the team of agents can achieve a state agreement only by locally available information without central control stations. Consensus problem has been addressed in various situations, such as time delay [5], switching topology [6], asynchronous algorithms [4, 7], nonlinear algorithms [8, 9], quantized data [4, 10], noisy communication channel [11], and second-order model [12, 13].Inspired by some biological systems and engineering applications, the leader-following consensus problem has received a lot of interest. The leader is a special agent whose motion is independent of all other agents and thus is followed by all other agents. It has been widely used in many applications [14, 15]. For the first-order multiagent systems, Jadbabaie et al. [16] considered a leader-following consensus problem and discussed the convergence properties of the leader-follower systems. Cao and Ren [17] studied a leader-following consensus problem with reduced interaction for both first- and second-order multiagent systems. Su et al. [18] studied a flocking algorithm with a virtual leader. Zhu and Cheng [19] considered leader-following consensus of second-order agents with multiple time-varying delays. Meng et al. [20] studied the leaderless and leader-following consensus algorithms with communication and input delays under a directed network by the Lyapunov theorems and the Nyquist stability criterion.In recent years, owing to the development of digital sensors and the constraints of transmission bandwidth of networks, many control systems can be modeled by continuous-time systems together with discrete sampling. Therefore, it is significant to design the distributed control for continuous-time multiagent systems based on sampled information. There are a few reports [20–25] dealing with this problem, where the control inputs regulate the velocity of each agent continuously over the sampling period.On the other hand, impulsive dynamical systems exhibit continuous evolutions typically described by ordinary differential equations and instantaneous state jumps or impulses. It is also well known that the impulsive control is more efficient than one of continuous control in many situations. The examples include ecosystems management [26], orbital transfer of satellite [27], and optimal control of economic systems [28]. The main idea of impulsive control is to instantaneously change the state of a system when some conditions are satisfied. During the last few decades, it has been widely applied into the synchronization problems of complex dynamical networks [29–31], which can be regarded as first-order multiagent systems with nonlinear dynamics. In many real-world system, agents are governed by both position and velocity dynamics. The impulsive control for second-order multiagent system was studied in [32, 33], where both velocity and position are instantaneously changed by impulsive control, but position cannot change quickly in many situation. Therefore, it is more reasonable to only regulate the velocity of each agent to reach consensus [34, 35]. In [34], we designed impulsive velocity-control for multiagent systems with fixed topology to achieve consensus. In [35], an impulsive control was proposed in which the current position data of its neighbours and the past position data of its own state were utilised to regulate the velocity of agents.This paper aims to investigate the consensus problem of leader-following multiagent systems by using impulsive control which only regulates the velocity of agents. Our main contributions are summarised as follows. First, a necessary and sufficient condition under fixed topology is derived, and it is found that the leader-following consensus in multiagent systems with sampling information can be reached if and only if the sampled period is bounded by critical values which depend on control gains and the interaction graph. Second, a sufficient condition under switching topology is obtained, and it is shown that the impulsive interval is restricted by an upper bound which depends on control gains, the diagonal element of the Laplacian matrix, and the connections between agents and leader. The two key difference between this paper and our earlier work [34] are that the leader-following case is taken into account and that this paper considers multiagent systems under switching topology.The remainingpart of the paper is organized as follows. In Section2, some necessary mathematical preliminaries are given. Main results of this paper, that is, the convergence of the distributed impulsive control under fixed and switching topology, are presented in Sections 3 and 4. In Section 5, some illustrative numerical examples are given. Concluding remarks are finally stated in Section 6.


## 2. Problem Formulation

Letℝ and ℂ denote the set of real numbers and complex numbers, respectively. For A=(aij)m×m∈ℝm×m,  λ1(A),  λ2(A),…,λm(A) are the eigenvalues of A, ρ(A) represent the spectral radius of A. The identity matrix of order n is denoted as In (or simply I if no confusion arises). For γ∈ℂ, Re(γ) and Im(γ) are the real and imaginary part of γ, respectively. 1n=(1,1,…,1)T is the column vector. 0n×m denotes the n×m matrix with all elements equal to zero.Let𝒢={𝒱,ℰ,𝒜} be a directed graph (digraph) with the set of nodes 𝒱={1,2,…N}, the set of edges ℰ∈𝒱×𝒱, and the weighted adjacency matrix 𝒜=(aij)N×N. In the digraph 𝒢, node i represents the agent i, and an edge in 𝒢 is denoted by an ordered pair {j,i}. {j,i}∈ℰ if and only if the agent i can directly receive information from the jth agent. In this case, the jth agent is the neighbor of the ith agent. The set of neighbors of the ith agent is denoted by 𝒩i={j∈𝒱∣(j,i)∈ℰ}. All elements of adjacency matrix are nonnegative. For i,j∈𝒱,  j∈𝒩i⇔aij>0, and assume that aii=0, i∈𝒱. A directed path in a digraph 𝒢 is an ordered sequence v1,v2,…,vk of agents such that any ordered pair of vertices appearing consecutively in the sequence is an edge of the digraph, that is, (vi,vi+1)∈ℰ, for any i=1,2,…,k-1. A directed tree is a digraph, where there exists an agent, called the root, such that any other agent of the digraph can be reached by one and only one path starting at the root. 𝒯𝒢={𝒱𝒯,ℰ𝒯} is a directed spanning tree of 𝒢, if 𝒯𝒢 is a directed tree and 𝒱𝒯=𝒱. The Laplacian matrix L(𝒢)=(lij)N×N of 𝒢 is defined as
(1)lij={-aij,i≠j,∑k=1,k≠iNaik,i=j.Given a matrixP=(pij)∈ℝN×N, the digraph (without self-link) of P denotes by 𝒢(P), which is the digraph with node set 𝒱={1,2,…,N} such that there is an edge in 𝒢(P) from j to i if and only if pij≠0. The matrix A is nonnegative, that is, A≥0, if all element of A is non-negative. The matrix A,  B∈ℝN×N, A≥B denote A-B≥0. The non-negative matrix A is row stochastic if all of its row sum are equal to 1. The row stochastic matrix A∈ℝN×N is called indecomposable and aperiodic (SIA) if limk→∞Ak=1NyT, where y is some N×1 column vector.Consider that a multiagent system consists ofN identical agents indexed by 1,2,…,N, which is described by
(2)x˙i(t)=vi(t),v˙i(t)=ui(t),
where i=1,2,…,N, xi(t)∈ℝ,  vi(t)∈ℝ are the position and velocity states of the agent i, respectively. ui(t)∈ℝn is a control input for i=1,2,…,N. The static leader for the system (2) is a static agent represented by x0(t)=x0, where x0∈ℝ. The edges between the agents and the leader is unidirectional; namely, there are only partial agents that can obtain information from the leader. It is also assumed that each agent can only obtain information from other agents or the leader at sampling times.This paper focuses on the problem of designingui(t), i=1,2,…,N based on sampling information to make all N agents converge to a static leader.Definition 1.
The leader-following consensus of the multiagent system (2) with static leader is said to be achieved if
(3)limt→+∞xi(t)=x0,limt→+∞vi(t)=0,i∈𝒱,
for any initial state.


## 3. Leader-Following Consensus under Fixed Topology

In this section, the leader-following consensus problem under fixed topology is considered. The interaction between agents in this part is described by a fixed digraph𝒢={𝒱,ℰ,𝒜}, and the connections between agents and leader are described by bi∈ℝ,  bi>0 if and only if the agent i can obtain information from the leader, otherwise, bi=0.In order to achieve the leader-following consensus of the multiagent system (2) with sampled information under fixed topology, the impulsive control for the agent i, is designed as
(4)ui(t)=-∑k=1+∞‍p1[(∑j∈Ni‍lij(xj(t)-xi(t))--------+bi(xi(t)-x0)∑j∈Ni‍lij(xj(t)-xi(t)))+p2vi(t)]δ(t-tk),
where i∈𝒱, the sampling time sequence {tk}k=1∞ satisfies tk+1-tk=h (h∈ℝ is sampled period) and limk→∞tk=∞, p1,p2>0 are the control gain to be determined, and δ(·) is the Dirac impulsive function.Equivalently, the multiagent system (2) with impulsive controller (4) can be rewritten as follows:
(5)x˙i(t)=vi(t),v˙i(t)=0,t∈(tk,tk+1],Δvi(tk)=-p1(∑j∈𝒱‍lijxj(tk)+bi(xi(tk)-x0))-p2vi(tk),
where Δvi(tk)=vi(tk+)-vi(tk), vi(tk+)=limt→tk+vi(tk). For simplicity, it is assume that vi(t) is left continuous at tk.Remark 2.
From (5), the control input of each agent only uses the information from its neighbors at sampling instants and are only applied at sampling instants. This is quite different from the previously mentioned works, where the control inputs are applied continuously. The velocity of the agent is instantaneously changed at sampling times. This is feasible when the operating time of the impulsive controller is much smaller than the sampled period.Lemma 3.
The multiagent system (2) with impulsive control (4) achieves leader-following consensus asymptotically if and only if ρ(P)<1, where
(6)P=(INhIN-p1(L+B)(1-p2)IN-p1h(L+B)),B=(b1b2⋱bN).Proof.
Letx^i(t)=xi(t)-x0(t), for i∈𝒱 and note that ∑j∈𝒱‍lijx0=0, system (5) can be rewritten as follows:
(7)x^˙i(t)=vi(t),v˙i(t)=0,t∈(tk,tk+1],Δvi(tk)=-p1(∑j∈𝒱‍lijx^j(tk)+bix^i(tk))-p2vi.
From (7), one has
(8)x^i(tk+1)=x^i(tk)+hvi(tk+),vi(tk+1)=vi(tk+),vi(tk+1+)=(1-p2)vi(tk+1)-p1(∑j∈𝒱lijx^j(tk+1)+bix^i(tk+1)).
From (8), one has
(9)vi(tk+1+)=(1-p2)vi(tk+)-p1h∑j∈𝒱‍lijvj(tk+)-p1bihvi(tk+)-p1(∑j∈𝒱‍lij(x^j(tk))+bix^i(tk)).
Then, the evolution of x^i(tk), vi(tk) under impulsive control (4) can be described as follows:
(10)x^i(tk+1)=x^i(tk)+hvi(tk+),vi(tk+1+)=(1-p2)vi(tk+)-p1h∑j∈𝒱‍lijvj(tk+)-p1bihvi(tk+)-p1(∑j∈𝒱‍lij(x^j(tk))+bix^i(tk)).
Let x^(k)=(x^1(tk),…,x^N(tk))T and v(k)=(v1(tk+),…,vN(tk+))T. Then, the multiagent system (2) achieves leader-following consensus, if and only if limk→+∞x^(k)=0,  limk→+∞v(k)=0.
Equivalently, (10) can be rewritten as follows:
(11)(x^(k+1)v(k+1))=P×(x^(k)v(k)).
Therefore, it is easy to obtain the result by the stability theory of discrete-time systems.The following lemmas and definition are needed for the subsequent development.Lemma 4 (bilinear transformation theorem [36]).
PolynomialR(z) (of degree d) is Schur stable if and only if Q(z) is Hurwitz stable, where
(12)R(z)=(z-1)dQ(z+1z-1).For complex polynomialR(z), let
(13)R(iω)=m(ω)+in(ω),
where m(ω),  n(ω)∈ℝ and i is the imaginary unit.Lemma 5 (see [37, 38]).
The complex polynomialR(z)=z2+az+b, where α∈ℂ and β∈ℂ, is Hurwitz stable if and only if Re(a)>0 and Re(a)Im(a)Im(b)+Re2(a)Re2(b)-Im2(b)>0.Next theorem will show what kind of interaction topology can reach leader-following consensus and how to determine the control gainsp1,  p2 and sampling period h.Theorem 6.
The multiagent system (2) with impulsive control (4) under fixed topology achieves the leader-following consensus asymptotically if and only if
(14)h<min1<i<N2p22(2-p2)Re(λi)Im2(λi)p1(p2-2)2+Re2(λi)p1p22,
where λi,  i=1,2,…,N are the eigenvalues of L+B.Proof.
Let theγ be an eigenvalue of matrix P. Then,
(15)det(γI2N-P)=det((γ-1)IN-hINp1(L+B)γIN-(1-p2)IN+p1h(L+B))=∏i=1N(γ2+γ(p2-2+p1hλi)+1-p2).
Let(16)Qi(γ)=γ2+γ(p2-2+p1hλi)+1-p2,i∈𝒱.
Then, we only need to prove that polynomialsQi(γ) for i∈𝒱 are Schur stable.
Let(17)Ri(σ)=(σ-1)2Qi(σ+1σ-1)=p1hλiσ2+2p2σ+4-2p2-p1hλi.
Let
(18)R′(σ)=σ2+2p2p1hλi′σ+(4p1h-2p2p1h)λi′-1,
where λi′=1/λi′. Then, according to Lemma 4, polynomials Ri′(σ), for i=1,2,3,…,N, are Hurwitz stable if and only if polynomials Qi(γ) for i=1,2,3,…,N are Schur stable.
It can be proved by Lemma5 that Ri(σ) is Hurwitz stable if and only if (14) is satisfied. Therefore, ρ(P)<1 if and only if (14) is satisfied. The proof is thus completed.Remark 7.
It can be observed from the inequality (14) that the real and imaginary part of the eigenvalues of L+B, the sampling period h, and two control gains p1 and p2 play important roles in achieving consensus. p2<2 and Re(λi(L+B))>0, for i∈𝒱, are necessary conditions for leader-following consensus. It is easy to see that the critical value of h increases as p1 decreases.Remark 8.
Let𝒢~={𝒱~,ℰ~,𝒜~} with 𝒱~={0,1,2,…,N}, and the Laplace matrix is
(19)L~=(000⋯0-b1l11+b1l12⋯l1N-b2l21l22+b2⋯l2N⋮⋮⋮⋱⋮-bNlN1lN2⋯lNN+bN).
Note that
(20)E-1L~E=(00NT0NL+B),
where
(21)E=(10NT1NIN)
is an invertible matrix. Re(λi)>0, for i∈𝒱 imply that L~ has a simple eigenvalue 0, and all the other eigenvalues have positive real parts. This implies that the graph 𝒢~ contains a spanning tree. The root of the spanning tree is the leader.Remark 9.
How to choose a suitable control gainp1 and p2 when the sampling period h is given. According to Theorem 6, p2<2 is a necessary condition for consensus. Therefore, one can choose p2 from (0,2], and then compute
(22)Θ=mini∈𝒱2p22(2-p2)Re(λi)(Im(λi)(p2-2))2h+hp22Re2(λi).
Then, one can choose p1 from (0,Θ).


## 4. Leader-Following Consensus under Switching Topology

In this section, the leader-following consensus under switching topology is considered. The interaction between agents at sampling timetk is described by time-varying digraph 𝒢(t)={𝒱,ℰ(t),𝒜(t)}, where 𝒜(t)=(aij(t))N×N and the connections between agents and leader at time t are described by bi(t),  bi(t)>0 if and only if the agent i can obtain information from the leader at time t; otherwise,  bi(t)=0.In order to achieve leader-following consensus under switching topology, the impulsive control input is designed as(23)ui(t)=-∑k=1+∞‍p1[(∑j∈Ni‍lij(t)(xj(t)-xi(t))-------+bi(t)(xi(t)-x0)∑j∈Ni‍lij(t)(xj(t)-xi(t)))+p2vi]δ(t-tk),
where i=1,2,…,N. Let 𝒢~(t)={𝒱~,ℰ~(t),𝒜~(t)} with 𝒱~(t)={0,1,2,…,N} and
(24)𝒜~(t)=(000⋯0b1(t)0a12(t)⋯a1N(t)b2(t)a21(t)0⋯a2N(t)⋮⋮⋮⋱⋮bN(t)aN1(t)aN2(t)⋯0).
Let L~(t)=(l~ij)(N+1)×(N+1) denotes the Laplace matrix of 𝒢~. Equivalently, the multiagent system (2) with the impulsive controller (23) can be rewritten as follows:
(25)x˙i(t)=vi(t),v˙i(t)=0,t∈(tk,tk+1],Δvi(tk)=-p1∑j∈𝒱‍l~ij(t)xj(tk)-p2vi(tk),
where i=0,1,…,N.Remark 10.
Note that the communication among agents only occurs at sampling times. This implies that interation graph does not contain any edges𝒢(t)=0 where t≠tk.Similar to the discussion in Section4, one has
(26)xi(tk+1)=xi(tk)+hvi(tk+),vi(tk+1+)=(1-p2)vi(tk+)-p1∑j=0N‍l~ij(tk)xj(tk)-p1h∑j=0N‍l~ij(tk)vj(tk+).
Let x~i(k)=xi(tk),  v~i(k)=xi(tk)+αvi(tk+), where α=2h/p2.It is easy to know that the network (2) achieves leader-following consensus, if x~i(k)→β and v~i(k)→β, for some β∈ℝ,  i∈𝒱~.From (26), one has
(27)x~i(k+1)=(1-p22)x~i(k)+p22v~i(k),v~i(k+1)=p22x~i(k)+(1-p22)v~i(k)-(2p2-1)p1h∑j=0N‍lij(x~j(tk))-p1h∑j=0N‍lijv~j(k).Letx~(k)=(x~0T(k),x~1T(k),x~2T(k),…,x~NT(k))T and v~(k)=(v~0T(k),v~1T(k),v~2T(k),…,v~NT(k))T; then,
(28)(x~(k+1)v~(k+1))=P(k)×(x~(k)v~(k)),
where
(29)P(k)=((1-p22)Ip22Ip22I-(2p2-1)p1hL~(k)(1-p22)I-p1hL~(k)).Before moving on, the following lemmas are needed.Lemma 11 (see [16]).
Letm≥2 be a positive integer, and let P1,P2,…,Pm be non-negative N×N matrices with positive diagonal entries; then, P1P2⋯Pm≥ε(P1+P2+⋯+Pm), where ε>0 can be specified from matrices Pi,  i=1,2,…,m.Lemma 12 (see [39]).
LetP1,P2,…,Pk∈ℝN×N be a finite set of SIA matrices with the property that for each sequence Pi1,Pi2,…,Pij of positive length, the matrix product Pi1Pi2⋯Pij is SIA. Then, for each infinite sequence Pi1,Pi2,…,Pij, there exists a column vector y such that
(30)limj→∞Pi1×Pi2×⋯Pij=1NyT.Lemma 13 (see [40]).
Suppose thatP∈ℝN×N is a row stochastic matrix with positive diagonal elements. If the digraph 𝒢(P) has a directed spanning tree, then P is SIA.Lemma 14.
Let(31)P=((1-α)IαIβI-μ1L(1-β)I-μ2L)
be non-negative matrix, where α,μ1,μ2>0. If L is a Laplace matrix of a digraph 𝒢, which has a directed spanning tree, then P is a row stochastic matrix and the digraph of P contains a directed spanning tree.Proof.
It is easy to check the non-negative matrixP12N=12N. Then, P is a row stochastic matrix. Let 𝒢(P) denote the digraph of P. Then, the Laplace matrix of 𝒢(P) is
(32)L(P)=(αI-αI-βI+μ1LβI+μ2L).
Let γi,  i=1,2,…,N denote the eigenvalues of L.
Letλ be an eigenvalue of matrix P; then, one has
(33)det(λI-αIαIβI-μ1LλI-βI-μ2L)=∏i=1N((λ-α)(λ-β-μ2γi)-α(β-μ1γi)).
Let Q(λ)=(λ-α)(λ-β-μ2γi)-α(β-μ1γi). Then,
(34)Q(0)=αμ1γi+αμ2γi.
Therefore, from (34), λ=0 only if γi=0.
Whenγi=0,
(35)Q(λ)=λ2-λ(α+β).
Thus, when γi(L)=0, the solutions of Q(λ)=0 are λ=0 and λ=α+β. On the other hand, if 𝒢 contains a spanning tree, L only has one simple eigenvalue equal to zero. Therefore, L(P) only has one simple eigenvalue equal to zero, which implies that the digraph of P has a spanning tree. The proof is completed.Theorem 15.
If there exists a positive integerl, the union of 𝒢~(tk) across k∈[k0,k0+l] contains a directed spanning tree, for any non-negative integer k0, and
(36)h<p222p1(2-p2)Δ,p2≤1,h<2-p22p1Δ,1<p2<2,
where Δ=maxi∈𝒱,k∈ℕ{lii(k)+bi(k)}; then, the multiagent system (2) achieves the leader-following consensus.Proof.
Let(37)P(k)=(P11(k)P12(k)P21(k)P22(k)),
where P(k) is defined in (29). From (1), one has lij<0, for i≠j. Then, the following statements are satisfied: (i)
P11(k) is nonnegative if and only if p2<2;(ii)
P12(k) is nonnegative if and only if p2>0;(iii)
P21(k) is nonnegative if and only if
(38)p22-(2p2-1)p1h(lii(k)+bi(k))>0,fori∈𝒱;(iv)
P22(k) is nonnegative if and only if
(39)(1-p22)-p1h(lii(k)+bi(k))>0,fori∈𝒱.If 0<p2<2, then 2/p2-1>0 and 1-p2/2>0. Note that lij+bi(k)≥0 and p1>0. Then, the following four statements are satisfied when p2<2.  (i)
Iflij+bi(k)=0, then we have
(40)p22-(2p2-1)p1h(lii(k)+bi(k))>0,(1-p22)-p1h(lii(k)+bi(k))>0,(ii)
Iflij+bi(k)>0, and
(41)h<p2/2(2/p2-1)p1(lii(k)+bi(k)),then we have(42)p22-(2p2-1)p1h(lii(k)+bi(k))>0,(iii)
Iflij+bi(k)>0, and
(43)h<1-p2/2p1(lii(k)+bi(k)),then we have(44)(1-p22)-p1h(lii(k)+bi(k))>0.According to the previous discussion, (38) is satisfied, if p2<2, and
(45)h<p2/2(2/p2-1)p1Δ.
Equation (39) is satisfied, if p2<2, and(46)h<1-p2/2p1Δ.
Note that
(47)p2/2(2/p2-1)p1Δ-1-p2/2p1Δ=2-2/p2(2/p2-1)p1Δ.
Then, (38) and (39) are satisfied if (36) holds. This implies that P11(k), P12(k), P21(k), and P22(k) are nonnegative. Then, P(k) is also nonnegative. Note that
(48)P(k)12N+2=(1N+11N+1-2p2p1hL~(k)1N+1),
and L~(k)1N+1=0. Then, P(k)12N+2=12N+2, P(k) is a row stochastic matrix. Then, ∑k=k0k0+l‍P(k) and ∏k=k0k0+l‍P(k) are also a row stochastic matrix.
Note that(49)∑k=k0k0+l‍P(k)=(l+1)×((1-p22)Ip22Ip22I-(2p2-1)p1hSk0l(1-p22)I-p1hSk0l),
where Sk0l=(1/(l+1))∑k=k0k0+l‍L~(k).
The union of𝒢(tk) across k∈[k0,k0+l], for any non-negative integer k0 contains a directed spanning tree. This implies that the digraph with the Laplace matrix Sk0l also contains a directed spanning tree.
By Lemma14, from (49), the digraph of ∑k=k0k0+l‍P(k) contains a spanning tree.
According to Lemma11, one has
(50)∏i=k0k0+lP(i)≥ε∑i=k0k0+l‍P(i),
for some ε. This implies that the digraph of ∏k=k0k0+l‍P(k) also contains a spanning tree. It follows from Lemma 13 that ∏k=k0k0+l‍P(k) is SIA. By Lemma 12,
(51)limk→∞P(k)P(k-1)⋯P(0)x(0)=12(N+1)yTx(0).
The proof is thus completed.Remark 16.
In this remark, we also show how to choose a suitable control gainp1 and p2 when the sampling period h is given. According to Theorem 15, p2<2 is also required. Similar to Remark 9, one can choose p2 from (0,2], and then compute
(52)Θ={p222h(2-p2)Δ,p2≤12-p22hΔ,1<p2<2.
Then, one can choose p1 from (0,Θ).


## 5. Illustrative Examples

In this section, two illustrative numerical examples will be given to demonstrate the correctness of theoretical analysis.


### 5.1. Fix Topology

The communication topology is described as in Figure1. The Laplacian matrix L and matrix B are given as follows:
(53)L=(10-10-120-1000000-11),B=diag(0,0,1,1).
The eigenvalues of L+B are λ1(L+B)=λ2(L+B)=1, λ3(L+B)=λ4(L+B)=2. Let p2=1, h=2;  according to Theorem 6, the network can achieve leader-following consensus, if and only if
(54)p1<mini∈𝒱2p22(2-p2)Re(λi)(Im(λi)(p2-2))2h+hp22Re2(λi)=0.5.
Figure 2 shows that the leader-following consensus can be achieved when p1=0.49. But it cannot be achieved when p1=0.51 (as shown in Figure 3).Figure 1
Fixed topology.Trajectory of the multiagent system (2) under fixed topology, when p1=0.49. Evolution of (a) xi, and (b) vi.
(a)
(b)Trajectory of the multiagent system (2) under fixed topology, when p1=0.51. Evolution of (a) xi, and (b) vi.
(a)
(b)


### 5.2. Switching Topology

In this subsection, the network topology switches from a set{𝒢~1,𝒢~2,𝒢~3,𝒢~4} as shown in Figure 4. The corresponding Laplacian matrices of 𝒢1,𝒢2,𝒢3,𝒢4 and matrices B1,B2,B3,B4 are
(55)L1=(0000-110000000000),L2=(0000010-1000000-11),L3=(10-10000000000000),L4=(0000000000000000),B1=diag(0010), B2=diag(0000), B3=diag(0001), and B4=diag(0000). Assume that 𝒢(t0)=𝒢1, 𝒢(t1)=𝒢2, 𝒢(t2)=𝒢3, 𝒢(t3)=𝒢4, 𝒢(t4)=𝒢1,… and B(t0)=B1,  B(t1)=B2,  B(t2)=B3,  B(t3)=B4,  B(t4)=B1,…. Note that the union graph of 𝒢~1,  𝒢~2,  𝒢~3, and 𝒢~4, and maxi∈𝒩,k∈ℕ(lii(k)+bi(k))=1.Figure 4
Switching topology:𝒢~i, i=1,2,3,4.Letp2=1, h=0.5, according to Theorem 15, if
(56)p1<p222h(2-p2)maxi∈𝒩,k∈ℕ{lii(k)+bi(k)}=1.
Figure 5 shows that the leader-following consensus can be achieved when p1=0.95.Trajectory of the multiagent system (2) under switched topology, when p1=0.95. Evolution of (a) xi, and (b) vi.
(a)
(b)


## 5.1. Fix Topology

The communication topology is described as in Figure1. The Laplacian matrix L and matrix B are given as follows:
(53)L=(10-10-120-1000000-11),B=diag(0,0,1,1).
The eigenvalues of L+B are λ1(L+B)=λ2(L+B)=1, λ3(L+B)=λ4(L+B)=2. Let p2=1, h=2;  according to Theorem 6, the network can achieve leader-following consensus, if and only if
(54)p1<mini∈𝒱2p22(2-p2)Re(λi)(Im(λi)(p2-2))2h+hp22Re2(λi)=0.5.
Figure 2 shows that the leader-following consensus can be achieved when p1=0.49. But it cannot be achieved when p1=0.51 (as shown in Figure 3).Figure 1
Fixed topology.Trajectory of the multiagent system (2) under fixed topology, when p1=0.49. Evolution of (a) xi, and (b) vi.
(a)
(b)Trajectory of the multiagent system (2) under fixed topology, when p1=0.51. Evolution of (a) xi, and (b) vi.
(a)
(b)


## 5.2. Switching Topology

In this subsection, the network topology switches from a set{𝒢~1,𝒢~2,𝒢~3,𝒢~4} as shown in Figure 4. The corresponding Laplacian matrices of 𝒢1,𝒢2,𝒢3,𝒢4 and matrices B1,B2,B3,B4 are
(55)L1=(0000-110000000000),L2=(0000010-1000000-11),L3=(10-10000000000000),L4=(0000000000000000),B1=diag(0010), B2=diag(0000), B3=diag(0001), and B4=diag(0000). Assume that 𝒢(t0)=𝒢1, 𝒢(t1)=𝒢2, 𝒢(t2)=𝒢3, 𝒢(t3)=𝒢4, 𝒢(t4)=𝒢1,… and B(t0)=B1,  B(t1)=B2,  B(t2)=B3,  B(t3)=B4,  B(t4)=B1,…. Note that the union graph of 𝒢~1,  𝒢~2,  𝒢~3, and 𝒢~4, and maxi∈𝒩,k∈ℕ(lii(k)+bi(k))=1.Figure 4
Switching topology:𝒢~i, i=1,2,3,4.Letp2=1, h=0.5, according to Theorem 15, if
(56)p1<p222h(2-p2)maxi∈𝒩,k∈ℕ{lii(k)+bi(k)}=1.
Figure 5 shows that the leader-following consensus can be achieved when p1=0.95.Trajectory of the multiagent system (2) under switched topology, when p1=0.95. Evolution of (a) xi, and (b) vi.
(a)
(b)


## 6. Conclusions

In this paper, the leader-following consensus problem of the multiagent system is considered. The impulsive control, which only needs sampled information and regulates the velocity of each agent at sampling times, is proposed for the leader-following consensus. Several new criteria are established for the leader-following consensus of the system under both fixed and switching topology. Illustrated examples have been given to show the effectiveness of the proposed impulsive control.



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*Source: 762861-2013-06-18.xml*