Consensus of Heterogeneous Multi-Agent Systems with Intermittent Communication

Journal of Systems Science and Information Aug., 2017, Vol. 5, No. 4, pp. 328 342 DOI: 10.21078/JSSI-2017-328-15 Consensus of Heterogeneous Multi-Agent Systems with Intermittent Communication Lü XU School of Automation, Nanjing University of Science and Technology, Nanjing 210094, China E-mail: zdmisslv@163.com Shuanghe MENG School of Automation, Nanjing University of Science and Technology, Nanjing 210094, China E-mail: h7spice@sina.cn Liang CHEN School of Automation, Nanjing University of Science and Technology, Nanjing 210094, China E-mail: 15651722230@163.com Abstract This paper studies consensus of a class of heterogeneous multi-agent systems composed of first-order and second-order agents with intermittent communication. For leaderless multi-agent systems, we propose a distributed consensus algorithm based on the intermittent information of neighboring agents. Some sufficient conditions are obtained to guarantee the consensus of heterogeneous multi-agent systems in terms of bilinear matrix inequalities (BMIs). Meanwhile, the relationship between communication duration and each control period is sought out. Moreover, the designed algorithm is extended to leader-following multi-agent systems without velocity measurements. Finally, the effectiveness of the main results is illustrated by numerical simulations. Keywords intermittent communication; multi-agent systems; consensus; bilinear matrix inequality (BMI); velocity measurements 1 Introduction Over the past several years, cooperative control of multi-agent systems has drawn much attention from the researchers in different research communities. It provides theoretical guidance for unmanned aerial vehicle formation, regulation and control of the traffic system, and so on. Based on the pioneering work of Reynolds [1] and Viseck [2], the distributed coordination control of multi-agent system has made significant progress. As the primary condition and common problem of realizing cooperative control, consensus problem has received considerable attention recently. The goal is to design a suitable protocol or algorithm to guarantee a group of agents to achieve agreement on a common value by exchanging information with their neighbors. Consensus problems for multi-agent systems in many situations have been investigated such as communication delay, packet dropouts, switching topology. Meanwhile, a great deal Received May 24, 2016, accepted December 19, 2016 Supported by the National Natural Science Foundation of China (612731200)

Consensus of Heterogeneous Multi-Agent Systems with Intermittent Communication 329 of results have been obtained. The consensus problems of multi-agent systems with first-order dynamics have been investigated from different perspectives in [2 6]. The consensus problems of multi-agent systems with second-order dynamics [7 12] and high-order dynamics [13 15] have been investigated as well. It should be noted that all the aforementioned work focused on homogeneous systems, i.e., all agents possess the same-order dynamics. However, in reality, the dynamics of the agents may be different because of various communication restrictions and external influence. Therefore, it is also important to study the consensus problems of heterogeneous multi-agent systems. Nowadays, more and more researchers throw themselves into the study of heterogeneous multi-agent systems. In [16], Tian and Zhang studied the high-order consensus problem for heterogeneous multi-agent systems with unknown communication delays, and got a necessary and sufficient consensus condition. Zheng and Wang [17] studied the finite-time consensus problem of heterogeneous multi-agent systems, and two classes of consensus protocols with and without velocity measurements were proposed by combining the homogeneous domination method with the addition of power integrator method. [18, 19] investigated consensus problem for discrete-time heterogeneous multi-agent systems composed of first-order and second-order agents. The consensus problem for a class of heterogeneous multi-agent systems composed of the first-order and second-order integrator agents together with the nonlinear Euler-Lagrange agents was considered in [20]. Most of the above-mentioned results were derived based on a common assumption that the information shared among agents is transmitted continuously. However, in practice, agents may only communicate with their neighbors at some disconnected time intervals because of the limitation of sensing ranges, the effect of obstacles and the failure of physical devices. In consideration of this situation, researchers have started to study multi-agent systems with intermittent communication. In [21], first-order consensus problem of multi-agent systems with non-linear dynamics and external disturbance was investigated under intermittent communication. [22, 23] investigated second-order consensus problems of multi-agent systems with and without non-linear dynamics under intermittent communication. Huang and Duan [24] studied leader-following consensus of second-order non-linear multi-agent systems with directed intermittent communication. [25] investigated consensus problems for multi-agent systems with second-order dynamics under delayed and intermittent communication. Qin and Liu [26] studied a consensus control problem for second-order nonlinear multi-agent systems with a leader under intermittent communication where the velocity of the active leader cannot be obtained in real time. The work on the intermittent communication is carried out in homogeneous systems, but heterogeneous systems has not been widely concerned in this field. Motivated by the above discussions, we consider consensus for heterogeneous multi-agent systems composed of first-order and second-order agents with intermittent communication in this paper. Firstly, a consensus protocol is proposed for leaderless multi-agent systems with intermittent communication. It is shown that when communication duration is larger than a value in each control period, multi-agent systems achieve consensus. Furthermore, the proposed protocol is extended to leader-following multi-agent systems, where the velocities of the follower agents are unknown. We design a consensus protocol with a distributed velocity filter such that

330 XU L, MENG S H, CHEN L. all the agents can achieve the same position with the leader and keep the same pace with the leader. The rest of this paper is organized as follows. In Section 2, we give some preliminaries and formulate the model. The main results of this paper are proposed in Section 3. Numerical simulations and conclusion are given in Sections 4 and 5, respectively. 2 Problem Formulation and Preliminaries 2.1 Graph Theory The interconnection topology of a multi-agent system is usually described by a graph. Let a weighted directed or undirected G(V,E,A) model the interaction communication between n agents, where V = {1, 2,,n} is a finite non empty set of nodes, E V V is a set of ordered pairs of nodes, called edges, and A =[a ij ] R n n is the weighted adjacency matrix with nonnegative adjacency elements a ij. In a directed graph, an edge (j, i) represents an information link from agent j to agent i, which means agent i can receive information from agent j. If (j, i) E, a ij > 0. Otherwise, a ij = 0. Moreover, self-loops are not allowed here, that is a ii =0fori =1, 2,,n. A graph is undirected, (j, i) E implies (i, j) E, thatisa ij = a ji and A is symmetric. The set of neighbors of agent i is denoted by N i = {j V :(i, j) E}. A directed path is a sequence of edges in a directed graph with the form (j, j k1 ), (j k1,j k2 ),,(j kl,i), where j km V,m =1, 2,,l. A directed graph has a directed spanning tree if thereexistsatleastonenodethathasadirected path to all the other nodes. We define an undirected graph is connected if there is a path between any two nodes of the graph. The Laplacian matrix L with respect to the graph is L =[l ij ]withl ii = j N i a ij and l ij = a ij, i j. ItiseasytoverifythatL has at least one zero eigenvalue with the corresponding eigenvector 1 n,where1 n =(1, 1,, 1) T. Obviously, for an undirected graph, L is a symmetric matrix, then 1 T nl =0 T n. In the multi-agent systems, the agent is defined as a leader which only sends information to other agents and cannot receive information from other agents. The connection weight between agent i and the leader is denoted by b i. If agent i is connected to the leader, b i > 0. Otherwise, b i =0. 2.2 Problem Formulation In this paper, consider a heterogeneous multi-agent system composed of first-order and second-order integrator agents. The number of agents is n. Assume that there are m first-order integrator agents (m <n). Each first-order agent dynamic is given as follows: ẋ i (t) =u i (t), i =1, 2,,m, (1) where x i R and u i R are the position and the control input of agent i, respectively. Each second-order agent dynamic is given as follows: ẋ i (t) =v i (t), i = m +1,m+2,,n, (2) v i (t) =u i (t), where x i R, v i R and u i R are the position, velocity and control input of agent i, respectively.

Consensus of Heterogeneous Multi-Agent Systems with Intermittent Communication 331 Definition 1 (see [25]) The heterogeneous multi-agent system (1) (2) is said to reach consensus if for any initial condition, we have lim x i(t) x j (t) =0, t lim v i(t) v j (t) =0, t i,j =1, 2,,n, i,j = m +1,m+2,,n. Definition 2 (see [25]) The communication topology among all agents is called completely periodical intermittence, if all agents periodically lose contact with their neighboring agents as time evolves. In the previous work, it is commonly assumed that all the information is transmitted continuously among all the agents. However, in practice, the agents may lose contact with their neighbors due to limitations of sensing ranges, unreliability of information channels and so on. They may only communicate with their neighbors over some disconnected time intervals. Therefore this study intends to design a consensus protocol with intermittent communication for heterogeneous multi-agent systems in the following. 3 Main Results 3.1 Leaderless Consensus with Intermittent Communication In this subsection, in consideration of the above-mentioned communication limitations existing in information exchange of neighboring agents, we investigate the leaderless consensus problem for heterogeneous multi-agent system (1) (2) with completely intermittent communication. For each agent, a consensus protocol with completely intermittent communication is designed as follows: u i (t) = a ij (x j (t) x i (t)), i =1, 2,,m, t [lt, lt + δ), (4a) u i (t) = 2kv i + k a ij (x j (t) x i (t)), i = m +1,m+2,,n, u i (t) =0, i =1, 2,,m, u i (t) = 2kv i (t), i = m +1,m+2,,n, t [lt + δ, (l +1)T ), where k>0andt>0arethe coupling strength and control period, respectively. [lt, lt +δ),l N, represents the time intervals over which each agent could communicate with its neighbors; [lt + δ, (l +1)T ),l N, represents the time intervals over which the communication between each pair of neighboring agents is no longer in force. Obviously, 0 <δ<t. The system (1) (2) with the protocol (4) can be rewritten as follows: When t [lt, lt +δ), ẋ i (t) = a ij (x j (t) x i (t)), i =1, 2,,m, (5a) ẋ i (t) =v i (t), v i (t) = 2kv i + k a ij (x j (t) x i (t)), i = m +1,m+2,,n; (3) (4b) (5b)

332 XU L, MENG S H, CHEN L. When t [lt + δ, (l +1)T ), ẋ i (t) =0, i =1, 2,,m, (6a) ẋ i (t) =v i (t), i = m +1,m+2,,n. (6b) v i (t) = 2kv i (t), Let v i (t) =v i (t)/k +x i (t), y(t) =[x 1 (t) x m (t),x m+1 (t) x n (t), v m+1 (t) v n (t)] T,then system (5) (6) can be transformed into the closed-loop system as follows: ẏ(t) = Gy(t), t [lt, lt + δ), (7) ẏ(t) =Hy(t), t [lt + δ, (l +1)T ), where L 11 L 12 0 G = 0 ki n m ki n m, H = 0 0 0 0 ki n m ki n m. L 21 L 22 ki n m ki n m 0 ki n m ki n m L 11 R m m, L 12 R m (n m), L 21 R (n m) m and L 22 R (n m) (n m) are parts of the Laplacian matrix L = L 11 L 12 L 21 L 22. Suppose that an undirected graph G(V,E,A) is connected, note that L1 n =0 n, according to the symmetry of matrix L, we can get 1 T n L =0T n,then 1 T 2n m ( Gy(t)) = 0 1T2n mẏ(t) =0. (8) Denote e(t) =y(t) ξ1 2n m,wheree =(e 1,e 2,,e 2n m ) T R 2n m and ξ =(1/(2n m)) 2n m i=1 y i. From (8), it is clear that ξ is a constant, and i e i(t) =0. Based on the above analysis, system (7) can be rewritten as ė(t) = Ge(t), t [lt, lt + δ), (9) ė(t) =He(t), t [lt + δ, (l +1)T ). If lim t e(t) = 0 2n m, then lim t y(t) = ξ1 2n m, that is lim t x i (t) = ξ, i = 1, 2,,n, lim t v i (t) =ξ, i = m +1,m+2,,n. Due to v i (t) =v i (t)/k + x i (t), we can get lim t v i (t) = 0, i = m +1,m +2,,n. Therefore, the consensus problem of heterogeneous multi-agent system (7) can be transformed into the stability issue of system (9). Now, we have the following result. Theorem 1 Suppose that the undirected communication topology G is connected. If there exist a symmetric positive definite matrix P R (2n m) (2n m), and constants k>0, γ>0 and η>0 such that (G 1 + kg 2 ) T P + P (G 1 + kg 2 ) γp, (10)

Consensus of Heterogeneous Multi-Agent Systems with Intermittent Communication 333 and the communication duration satisfies H T P + PH ηp, (11) η γ + η < δ T < 1, (12) then, under control protocol (4), heterogeneous multi-agent system (1) (2) can achieve consensus, where L 11 L 12 0 G 1 = 0 0 0 L 21 L 22 0, G 2 = 0 0 0 0 I n m I n m 0 I n m I n m. Proof For system (9), construct the following Lyapunov function candidate V (t) =e T (t)pe(t), (13) where P is a positive definite matrix. For t [lt, lt + δ), l =0, 1,, taking the time derivative of V (t) along the trajectory of (9) gives where Q 1 = G T P + PG, G = G 1 + kg 2. From (10), we get V (t) =ė T (t)pe(t)+e T (t)p ė(t) = e T (t)g T Pe(t)+e T (t)p ( Ge(t)) = e T (t)(g T P + PG)e(t) = e T (t)q 1 e(t), V (t) = e T (t)q 1 e(t) γe T (t)pe(t) = γv (t). (14) Similarly, we can obtain from (11) that, for t [lt + δ, (l +1)T ), l =0, 1,, V (t) =e T (t)(h T P + PH)e(t) ηe T (t)pe(t) =ηv (t). (15) Combining (14) and (15) leads to V (lt)e γ(t lt ), t [lt, lt + δ), V (t) V (lt + δ)e η(t lt δ), t [lt + δ, (l +1)T ). (16) In what follows, the estimation of V (t) for any t is carried out from (16). Setting β = γδ η(t δ), from (12), we obtain β>0. For t [lt, lt + δ), l =0, 1,,wehave V (t) V (lt)e γ(t lt ) V ((l 1)T + δ)e η(t δ) e γ(t lt )

334 XU L, MENG S H, CHEN L. V ((l 1)T )e γδ e η(t δ) e γ(t lt ).. V (0)e lγδ e lη(t δ) e γ(t lt ) = V (0)e lβ e γ(t lt ) V (0)e lβ Similarly, for t [lt + δ, (l +1)T ), l =0, 1,,weget It follows from (17) (18) that where M = V (0)e β. Then V (0)e β e t T β. (17) η(t lt δ) V (t) V (lt + δ)e V (lt)e γδ η(t lt δ) e V ((l 1)T + δ)e η(t δ) e γδ η(t lt δ) e V ((l 1)T )e 2γδ e η(t δ) η(t lt δ) e. V (0)e (l+1)γδ e lη(t δ) η(t lt δ) e = V (0)e lβ e γδ η(t lt δ) e V (0)e lβ e γδ η(t δ) e = V (0)e lβ e β V (0)e β e t T β. (18) V (t) Me β T t, t >0, (19) λ min (P ) e(t) 2 V (t) λ max (P ) e(0) 2 e β e β T t e(t) 2 λ max(p ) λ min (P ) e(0) 2 e β e β T t, (20) which indicates that the states of system (9) are exponentially convergent. In other words, under the control protocol (4), multi-agent system (1) (2) achieves consensus. This completes the proof. Remark 1 It is noted that (10) (11) are bilinear matrix inequalities, which can not be directly solved via LMI Toolbox in Matlab. However, we can solve (10) (11) according to the algorithm proposed in [27]. Algorithm 1 Step 1 Let i = 0, constant N>0, initial value k = k (0), t (0) min =0,t(0) min 1 =0. Step 2 Let i = i +1,k = k (i 1), solve the following optimization problem: min s.t. t (G 1 + kg 2 ) T P + P (G 1 + kg 2 ) γp ti,

Consensus of Heterogeneous Multi-Agent Systems with Intermittent Communication 335 H T P + PH ηp ti. Step 3 If t (i) min <t(i 1) min 1, we can get the feasible solutions P. Otherwise, stop. Step 4 Let P (i) = P, solve the optimization problem as well. If t (i) min 1 <t(i) min, we can get feasible solution k. Otherwise, stop. Step 5 Let k (i) = k, ifi N, return to Step 2. Otherwise, stop. Remark 2 In Theorem 1, condition (12) provides a definite relationship between communication duration δ and each control period T. As for the applicability of Theorem 1, it is easy to see that: a smaller γ and a larger η will be favorable to the feasibility of bilinear matrix inequalities (10) and (11), while a larger γ and a smaller η are more expectable to decrease the lower bound of the communication duration δ. In view of these, we put forward the following iterative algorithm to obtain a lower bound of δ. Algorithm 2 Step 1 Give initial values k 0, γ 0 and η 0. (For the first time, we can choose asmallγ and a large η.) Let i =1,j =1. Step 2 Increase γ at a small step length Δγ, letγ = γ 0 + iδγ, i = i +1, if i<n 1, continue. Otherwise, stop. Step 3 Decrease η at a small step length Δη, letη = η 0 j Δη, solve bilinear matrix inequalities (10) (11). Step 4 Let j = j +1,ifj N 2, return to step 3. Otherwise, let j =1,returntoStep2. 3.2 Leader-Following Consensus with Intermittent Communication In this subsection, we consider the leader-following consensus problem for heterogeneous multi-agent system (1) (2) with completely intermittent communication. The dynamic of the leader is described as follows: x 0 (t) =v 0, (21) where x 0 R is the position of the leader, v 0 is the desired constant velocity. Definition 3 (see [26]) The heterogeneous multi-agent system (1) (2) with a leader (21) is said to reach consensus if for any initial condition, we have lim t x i (t) x 0 (t) =0, i =1, 2,,n, lim t v i (t) v 0 =0, i = m +1, 2,,n. (22) For the interconnection topology, we make the following assumption. Assumption 1 An undirected graph G(V,E,A) is connected and at least one agent is connected with the leader. A control protocol for heterogeneous multi-agent system (1) (2) with an active leader is designed as follows: u i (t) = a ij (x j (t) x i (t))+b i (x 0 (t) x i (t)) + v 0, i =1, 2,,m (23) u i (t) = α(v i (t) v 0 )+ a ij (x j (t) x i (t))+b i (x 0 (t) x i (t)), i= m +1,m+2,,n, where α>0.

336 XU L, MENG S H, CHEN L. However, for the second-order integrator agents, the velocity information sometimes is unmeasurable because of technology limitations or environmental disturbances. To this end, we construct the following distributed velocity filter to estimate the velocity of the ith follower agent: ˆv i (t) = 2α(ˆv i (t) v 0 )+ a ij (x j (t) x i (t))+b i (x 0 (t) x i (t)), i = m+1,m+2,,n, (24) where ˆv i R is the estimation of the velocity v i. The initial value of ˆv i can be chosen arbitrarily. Due to the existence of intermittent communication, a consensus protocol for system (1), (2) and (21) is designed as follows: u i (t) = a ij (x j (t) x i (t))+b i (x 0 (t) x i (t)) + v 0,i=1, 2,,m, t [lt, lt + δ), u i (t) = α(ˆv i (t) v 0 )+ a ij (x j (t) x i (t)) +b i (x 0 (t) x i (t)), i = m +1,m+2,,n, u i (t) =v 0, i =1, 2,,m u i (t) = α(ˆv i (t) v 0 ), i = m +1,m+2,,n, t [lt+δ, (l+1)t ). (25a) (25b) Meanwhile, the velocity filter with intermittent information is given as follows: ˆv i (t) = 2α(ˆv i (t) v 0 )+ a ij (x j (t) x i (t)) i = m +1,m+2,,n. (26) +b i (x 0 (t) x i (t)), t [lt, lt + δ), ˆv i (t) = 2α(ˆv i (t) v 0 ), t [lt + δ, (l +1)T ), Let x i (t) =x i (t) x 0 (t), ṽ i (t) =v i (t) v 0 (t), ẽ i (t) =ˆv i (t) v i (t), the heterogeneous multi-agent system (1), (2) and (21) with the protocol (25) and (26) can be written as follows: When t [lt, lt + δ), x i (t) = a ij ( x j (t) x i (t)) b i x i (t), i =1, 2,,m, (27a) x i (t) =ṽ i (t), ṽ i (t) = α(ẽ i (t)+ṽ i (t)) + a ij ( x j (t) x i (t)) b i x i (t), i = m +1,m+2,,n, (27b) ẽ i (t) = α(ẽ i (t)+ṽ i (t)), i =1, 2,,m; (27c) When t [lt + δ, (l +1)T ), x i (t) =0, i =1, 2,,m, (28a)

Consensus of Heterogeneous Multi-Agent Systems with Intermittent Communication 337 x i (t) =ṽ i (t), ṽ i (t) = α(ẽ i (t)+ṽ i (t)), i = m +1,m+2,,n, (28b) ẽ i (t) = α(ẽ i (t)+ṽ i (t)), i =1, 2,,m. (28c) Setting z(t) =[ x 1 (t) x m (t), x m+1 (t) x n (t), ṽ m+1 (t) ṽ n (t), ẽ m+1 (t) ẽ n (t)] T,wecan obtain the following error dynamical system: ż(t) = Uz(t), t [lt, lt + δ), (29) ż(t) =Wz(t), t [lt + δ, (l +1)T ), where L 11 + B f L 12 0 0 00 0 0 0 0 I n m 0 00 I n m 0 U =, W =, L 21 L 22 + B s αi n m αi n m 00 αi n m αi n m 0 0 αi n m αi n m 00 αi n m αi n m B f =diag{b 1,b 2,,b m }, B s =diag{b m+1,b m+2,,b n }, and L 11 R m m,l 12 R m (n m),l 21 R (n m) m,l 22 R (n m) (n m) are parts of the Laplacian matrix L = L 11 L 12. L 21 L 22 Therefore, the leader-following consensus problem of heterogeneous multi-agent system (1) (2) can be transformed into the stability issue of error system (29). Theorem 2 Under Assumption 1, if there exist a symmetric positive definite matrix Q R (3n 2m) (3n 2m), constants α>0, μ>0 and ϑ>0 such that and the communication duration satisfies (U 1 + αu 2 ) T Q + Q(U 1 + αu 2 ) μq, (30) (W 1 + αw 2 ) T Q + Q(W 1 + αw 2 ) ϑq, (31) ϑ μ + ϑ < δ T < 1, (32) then under the control protocol (25), heterogeneous multi-agent system (1), (2) and (21) can achieve consensus tracking, where L 11 + B f L 12 0 0 00 0 0 0 0 I n m 0 00 0 0 U 1 =, U 2 =. L 21 L 22 + B s 0 0 00I n m I n m 0 0 0 0 00I n m I n m

338 XU L, MENG S H, CHEN L. 00 0 0 00I n m 0 W 1 =, W 2 = U 2. 00 0 0 00 0 0 Proof For system (29), we construct the following Lyapunov function candidate V (t) =z T (t)qz(t). The remainder of the proof is similar to that of Theorem 1, and it is omitted here. Remark 3 It should be noted that (30) (32) are similar to (10) (12), we can solve (30) (32) according to Algorithms 1 and 2. 4 Numerical Simulations In this section, numerical simulations will be given to verify the effectiveness of the proposed results. Example 1 Figure 1 shows a heterogeneous multi-agent system which consists of firstorder agents and second-order agents. The communication topology is connected. 1 and 2 denote the first-order agents. 3, 4, 5 and 6 denote the second-order agents. 4 3 1 6 5 2 Figure 1 Heterogeneous multi-agent system The Laplacian matrix is 2 1 1 0 0 0 1 1 0 0 0 0 1 0 2 0 1 0 L =. 0 0 0 2 1 1 0 0 1 1 2 0 0 0 0 1 0 1 In case of consensus with completely intermittent communication,it is assumed that there exists an infinite time sequence [lt, (l +1)T ),l N with control period T = 20s. Let initial values k 0 =3,γ 0 =0.2, η 0 =1.2, one obtains γ =0.35,η =0.2,k =6.8968 by Algorithms 1 and 2. Choose δ =0.37T =7.4s, which satisfies the condition (12) of Theorem 1.

Consensus of Heterogeneous Multi-Agent Systems with Intermittent Communication 339 The initial positions are randomly chosen within [ 10, 10], i =1, 2,, 6, and the initial velocities of agents are randomly chosen within [0, 0.5], i =3, 4, 5, 6. The simulation results are presented in Figures 2 4. Figure 2 shows that the positions x i, i =1, 2,, 6 asymptotically converge to their average value. From Figure 3, we can see that the velocities v i, i =3, 4, 5, 6 asymptotically converge to zero. Figure 4 shows intermittent communication between agents. 10 8 6 4 2 8 6 4 2 v 3 v 4 v 5 v 6 x i 0 v i 0 2 4 6 8 x 1 x 2 x 3 x 4 x 5 x 6 2 4 6 10 0 20 40 60 80 t/s 8 0 20 40 60 80 t/s Figure 2 The trajectories of position x i Figure 3 The trajectories of velocity v i 3 1:with communication 2:without communication 2 1 0 0 20 40 60 80 t/s Figure 4 Periodical intermittent communication Example 2 Figure 5 shows a heterogeneous multi-agent system with an active leader (labelled as 0). 1 and 2 denote the first-order agents. 3, 4, 5 and 6 denote the second-order agents. The control period T = 20s. Let initial values α 0 =0.2, μ 0 =0.05, ϑ 0 =2,oneobtains μ =0.13,ϑ=1.7,α=0.1921. Choose δ =0.93T =18.6s, which satisfies the condition (30) in Theorem 2.

340 XU L, MENG S H, CHEN L. The initial positions of the leader and followers are randomly chosen within [ 10, 10], i = 0, 1,, 6. The velocity of the leader is v 0 = 2. The initial velocities of followers and the initial estimations of velocities are randomly chosen within [0, 0.5], i =3, 4, 5, 6. 1 2 0 3 4 5 6 Figure 5 Heterogeneous multi-agent system 250 10 8 200 6 150 4 2 x i 100 x 1 v i 0 50 x 2 x 3 x 4 2 4 v 3 v 4 0 x 5 x 6 x 0 6 8 v 5 v 6 v 0 50 0 20 40 60 80 100 t/s 10 0 20 40 60 80 100 t/s Figure 6 The trajectories of position x i Figure 7 The trajectories of velocity v i 3 2.5 2 ẽ3 ẽ4 ẽ5 ẽ6 1.5 1 ẽi 0.5 0 0.5 1 1.5 2 0 20 40 60 80 100 t/s Figure 8 The errors between the real velocities and estimations

Consensus of Heterogeneous Multi-Agent Systems with Intermittent Communication 341 The simulation results are presented in Figures 6 8. Figure 6 shows that the positions x i, i =1, 2,, 6 asymptotically converge to the position of the leader. From Figure 7, we can see that the velocities v i, i =3, 4, 5, 6 asymptotically converge. Figure 8 shows the errors between estimations of the velocities and the real velocities, we can see that the errors asymptotically converge. 5 Conclusions This paper has dealt with the leaderless consensus and leader-following consensus of heterogeneous multi-agent system with periodical intermittent communication under the assumption that communication topology is undirected and connected. Some sufficient conditions for consensus have been obtained in terms of bilinear matrix inequalities (BMIs) based on Lyapunov function method. For the leaderless multi-agent systems, we have presented a control protocol based on the intermittent information. It has been proved that when communication duration is larger than a value in each control period, the multi-agent systems converge to a common position. For the leader-following multi-agent systems without velocity measurements, the control protocol with a distributed velocity filter has been designed to produce consensus between the following agents and to converge to the leader agent, along with communication duration is larger than a value. Two examples have been provided to verify the effectiveness of the proposed results. Our future work will focus on consensus for heterogeneous multi-agent systems under intermittent communication with communication delay. References [1] Reynolds C. Flocks, herds, and schools: A distributed behavioral model. Computers & Graphics, 1987, 21(4): 25 34. [2] Viseck T, Czirók A, Ben J E, et al. Novel type of phase transition in a system of self-driven particles. Physical Review Letters, 1995, 75(6): 1226 1229. [3] Jadbabaie A, Lin J, Morse A S. Coordination of groups of mobile autonomous agents using nearest neighbor rules. IEEE Transactions on Automatic Control, 2003, 48(6): 988 1001. [4] Olfati-Saber R, Murray R M. Consensus problems in networks of agents with switching topology and time-delays. IEEE Transactions on Automatic Control, 2004, 49(9): 1520 1533. [5] Sun Y G, Wang L. Consensus of multi-agent systems in directed networks with nonuniform time-varying delays. IEEE Transactions on Automatic Control, 2009, 54(7): 1607 1613. [6] Tian Y P, Liu C L. Consensus of multi-agent systems with diverse input and communication delays. IEEE Transactions on Automatic Control, 2008, 53(9): 2122 2128. [7] Ren W, Cao Y C. Convergence of sampled-data consensus algorithms for double-integrator dynamics. IEEE Conference on Decision and Control, 2008, 16(5): 3965 3970. [8] Zhang Y, Tian Y P. Consensus of data-sampled multi-agent systems with random communication delay and packet loss. IEEE Transactions on Automatic Control, 2010, 55(4): 939 943. [9] Zhu W, Cheng D Z. Leader-following consensus of second-order agents with multiple time-varying delays. Automatica, 2010, 46(12): 1994 1999. [10] Qin J, Gao H, Zheng W. Second-order consensus for multi-agent systems with switching topology and communication delay. Systems & Control Letters, 2011, 60(6): 390 397. [11] Liu K, Xie G, Wang L. Consensus for multi-agent systems under double integrator dynamics with timevarying communication delays. International Journal of Robust and Nonlinear Control, 2012, 22(17): 1881 1898. [12] Gao Y, Ma J, Zuo M, et al. Consensus of discrete-time second-order agents with time-varying topology and time-varying delays. Journal of the Franklin Institute, 2012, 349(8): 2598 2608.

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