Distributed Impulsive Consensus of the Multiagent System without Velocity Measurement

Abstract

This paper deals with the distributed consensus of the multiagent system. In particular, we consider the case where the velocity (second state) is unmeasurable and the communication among agents occurs at sampling instants. Based on the impulsive control theory, we propose an impulsive consensus algorithm that extends some of our previous work to account for the lack of velocity measurement. By using the stability theory of the impulsive system, some necessary and sufficient conditions are obtained to ensure the consensus of the controlled multiagent system. It is shown that the control gains, the sampled period and the eigenvalues of Laplacian matrix of communication graph play key roles in achieving consensus. Finally, a numerical simulation is provided to illustrate the effectiveness of the proposed algorithm.

1. Introduction

Recently, distributed consensus has received great interest in the control community, due to broad applications in formation [1], flocking [2, 3], synchronization in complex network [4, 5], distributed filtering [6], distributed optimization [7], and so forth. The main idea of distributed consensus is that each agent only communicates with its neighbors while the whole system of agents can converge to a common value, which by nature is a local distributed algorithm. Vicsek et al. [8] studied a simple discrete-time model of agents moving in the plane with the same speed but with different headings via simulations. The corresponding theoretical analysis was provided in [9]. Olfati-Saber and Murray presented the framework of the distributed consensus in [10], where the distributed consensus was studied in the multiagent system with fixed/switching topology and with/without delays. From then on, much progress has been made in the studies of the distributed consensus of the multiagent system in recent years [11–14]. There is a growing interest focusing on the consensus algorithms of the second-order multiagent system. Lin and Jia [15] studied the consensus problem of the multiagent system with nonuniform timedelays and dynamically changing topologies. In [16, 17], Su et al. investigated second-order consensus of the multiagent system with nonlinear dynamics and a virtual leader in a dynamic proximity network.

Due to the application of communication, the distributed consensus with sampled communication has received much attention in recent years. Many valuable algorithms have been proposed to deal with sampled communication [18–25], where distributed algorithms regulate the velocity of each agent continuously in the sampling period. On the other hand, most consensus algorithms for the multiagent system rely on the availability of the full state, only limited works [26–29] have been done when velocity information is unmeasurable.

The main contribution of this paper is to propose an impulsive consensus algorithm for the multiagent system without velocity measurements in the presence of sampled communication. The impulsive control strategy is effective when the state can be regulated instantaneously. This kind of algorithms are reasonable for many network systems. For example, in multi vrobot systems, the velocity of each robot can be changed very quickly, and the operating time of the actuator is much smaller than the sampling time. Impulsive control strategies for the multiagent system with nonlinear (linear) dynamics were considered in [30–32], where the impulsive controllers regulate all states of each agent in the system. We introduced impulsive algorithms for the multiagent system in [33–35], where only the velocity of each agent is regulated by the algorithms. In [33], some necessary and sufficient conditions are obtained for consensus/static consensus of the multiagent system. The consensus means that all the agents asymptotically tend to the zero-relative position (the agents may still change their positions) with a common velocity. The static consensus can ensure that all the agents tend to a common position. The leader-following case was studied in [35]. In [34], we proposed an impulsive consensus algorithm without velocity measurement for static consensus of multiagent system. How to achieve consensus without velocity measurement is still an open problem, which is the motivation of the study presented in this paper.

This paper is organized as follows. In Section 2, some necessary mathematical preliminaries are given, and the impulsive algorithm without using velocity information is also introduced. The main results of this paper, that is, the convergence of the proposed algorithm, are presented in Section 3. In Section 4, an illustrative numerical example is given. The concluding remarks are finally stated in Section 5.

Notation. Let ℕ and ℝ denote the natural numbers and the set of real numbers, respectively. 𝐼_𝑛 and 𝟎_𝑛×𝑚 are the identity matrixes of order 𝑛 (or simply 𝐼 if no confusion arises) and the 𝑛×𝑚 matrix with all elements equal to zero (or simply 𝟎 if no confusion arises), respectively. 𝜌(𝐴) denotes the spectral radius of squares matrix 𝐴. For 𝛾∈ℂ, Re(𝛾) and Im(𝛾) are the real part and the imaginary part of 𝛾.

2. Preliminary and Problem Formulation

The communication structure of the multiagent system is described by an undirected graph 𝒢=(𝒱,ℰ) with a set of agents 𝒱={1,2,…,𝑁} and a set of edges ℰ⊆𝒱×𝒱 (𝒢 has no self-loops or repeated edges). An edge {𝑖,𝑗} in 𝒢 means that node 𝑖 can receive information from node 𝑗. 𝒩_𝑖 denotes the set of neighbors of agents 𝑖, that is, 𝒩_𝑖={𝑗∈𝒱∣(𝑖,𝑗)∈ℰ}. The Laplacian matrix ℒ of the graph 𝒢 is defined as 𝑙ℒ=^𝑖𝑗−<0,if𝑖,𝑗∈ℰ,_𝑁𝑗=1∑_,𝑗≠𝑖𝑙_𝑖𝑗,𝑖=𝑗. (1)

A directed path in a digraph 𝒢 is an ordered sequence 𝑣₁,𝑣₂,…,𝑣_𝑘 of agents such that any ordered pair of vertices appearing consecutively in the sequence is an edge of the digraph, that is, (𝑣_𝑖,𝑣_𝑖+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. 𝒯_𝒢={𝒱_𝒯,ℰ_𝒯} is a directed spanning tree of 𝒢, if 𝒯_𝒢 is a directed tree and 𝒱_𝒯=𝒱.

We consider a multiagent system with 𝑁 identical agents: 𝑝̇_𝑖(𝑡)=𝑣_𝑖(𝑡),𝑣̇_𝑖(𝑡)=𝑢_𝑖(𝑡), (2) where 𝑖∈𝒩, 𝑝_𝑖∈ℝ and 𝑣_𝑖∈ℝ are the position and velocity of agent 𝑖, respectively, 𝑢_𝑖∈ℝ is a control input. All results in this paper still hold for 𝑝_𝑖, 𝑣_𝑖, 𝑢_𝑖∈ℝ^𝑛 by using the Kronecker product operations.

Definition 1. Consensus in the multiagent system (2) is said to be achieved, if, for any initial state, lim_𝑡→∞‖𝑝_𝑖(𝑡)−𝑝_𝑗(𝑡)‖=0 and lim_𝑡→∞‖𝑣_𝑖(𝑡)−𝑣_𝑗(𝑡)‖=0, where 𝑖,𝑗∈𝒱.
In this paper, we assume that both the absolute and relative velocities are unmeasurable, and the communication among agents occurs at sampling instants. The sampled sequence is given by {𝑡_𝑘∣^∞_𝑘=1}, which satisfies 0<𝑡₁<𝑡₂<⋯<𝑡_𝑘<⋯, lim_𝑘→∞𝑡_𝑘=∞, and 𝑡_𝑘+1−𝑡_𝑘=ℎ, where sampling period ℎ is positive constant. The following impulsive algorithm without using any velocity information is proposed and described by the following impulsive differential equations: 𝑝̇_𝑖(𝑡)=𝑣_𝑖(𝑡),𝑣̇_𝑖(𝑡)=0,Δ𝑣_𝑖𝑡_𝑘=−𝛽_{1𝑗∈𝒩𝑖}𝑙_𝑖𝑗𝑝_𝑗𝑡_𝑘−𝑝_𝑖𝑡_𝑘−𝛽_{2𝑗∈𝒩𝑖}𝑙_𝑖𝑗𝑦_𝑗(𝑘)−𝑦_𝑖(𝑘),𝑦_𝑖(𝑘+1)=−𝛼_{𝑗∈𝒩𝑖}𝑙_𝑖𝑗𝑝_𝑗𝑡_𝑘−𝑝_𝑖𝑡_𝑘, (3) where Δ𝑣_𝑖(𝑡_𝑘)=𝑣_𝑖(𝑡⁺_𝑘)−𝑣_𝑖(𝑡_𝑘), 𝑣_𝑖(𝑡⁺_𝑘)=lim_𝑡→𝑡⁺_𝑘𝑣_𝑖(𝑡), 𝑖∈𝒱. We assumed that 𝑣_𝑖(𝑡) is left-hand continuous at 𝑡=𝑡_𝑘, 𝑘∈ℕ, and 𝑣(𝑡) is continuous at 𝑡₀=0.

Remark 2. The proposed algorithm only uses sampled information of relative position (i.e. 𝑥_𝑖(𝑡)−𝑥_𝑗(𝑡)) which is different from [26–29], where the continuous position information is required. It is also different from our previous work [34] which requires the sampled information of relative position to itself in previous sampling instant (i.e., 𝑥_𝑖(𝑡_𝑘)−𝑥_𝑖(𝑡_𝑘−1)).
The following lemmas are needed in the proof of the theorem.

Lemma 3 (see [36]). Zero is a simple eigenvalue of ℒ, and all the other eigenvalues have positive real parts if and only if 𝒢 contains a spanning tree.

Define 𝑀=1−10⋯010−10⋯0⋮⋱⋮10⋯0−1010⋯0−1_{(𝑁−1)×𝑁},𝐺=000⋯0−10000−1⋱0⋱⋱⋱0⋮⋱−1000⋯0−1_{𝑁×(𝑁−1)}. (4)(5)

From [4, 37], we can get the following lemma.

Lemma 4. Let ℒ be the Laplacian matrix of the graph 𝒢. Then the (𝑁−1)×(𝑁−1) matrix ℒ defined by ℒ=𝑀ℒ𝐺 satisfies 𝑀ℒ=ℒ𝑀. Furthermore, 𝑙𝐿=²²−𝑙₁₂𝑙₂₃−𝑙₁₃⋯𝑙_2𝑁−𝑙_1𝑁𝑙₃₂−𝑙₁₂𝑙₃₃−𝑙₁₃⋯𝑙_3𝑁−𝑙_1𝑁𝑙⋮⋮⋱⋮_𝑁2−𝑙₁₂𝑙_𝑁2−𝑙₁₃⋯𝑙_𝑁2−𝑙_{𝑁𝑁(𝑁−1)×(𝑁−1)}. (6)

Lemma 5 (see [29]). The complex polynomial R(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)Re(𝑏)−Im²(𝑏)>0.

3. Consensus in Multi-Agent System

Denote the eigenvalues of 𝐿, respectively, by 𝜆₁,𝜆₂,...,𝜆_𝑟, where 𝜆₁=0. According to Lemma 3, 𝜆₁=0 is a simple eigenvalue if 𝒢 contains a spanning tree. Note that when 𝒢 is a directed graph, 𝜆_𝑖, for 𝑖=1,2,…,𝑟, may be complex numbers.

Theorem 6. The controlled multiagent system (3) can achieve consensus if and only if the graph G contains a spanning tree and 𝜌(𝑀_𝑙)<1, where 𝜆_𝑙 are the nonzero eigenvalues of ℒ, 𝑙=2,…,𝑟, 𝑀_𝑙=1−𝜆_𝑙ℎ𝛽₁ℎ−𝜆_𝑙ℎ𝛽₂−𝜆_𝑙𝛽₁1−𝜆_𝑙𝛽₂−𝜆_𝑙𝛼00. (7)

Proof. Note that 𝑝_𝑖(𝑡) is continuous at 𝑡=𝑡_𝑘. From (3), one has 𝑝_𝑖𝑡_𝑘+1=𝑝_𝑖𝑡_𝑘+ℎ𝑣_𝑖𝑡⁺_𝑘,𝑣_𝑖𝑡⁻_𝑘+1=𝑣_𝑖𝑡⁺_𝑘,𝑣_𝑖𝑡⁺_𝑘=𝑣_𝑖𝑡⁻_𝑘−𝛽_{1𝑗∈𝒩𝑖}𝑙_𝑖𝑗𝑝_𝑗𝑡_𝑘−𝑝_𝑖𝑡_𝑘−𝛽_{2𝑗∈𝒩𝑖}𝑙_𝑖𝑗𝑦_𝑗(𝑘)−𝑦_𝑖(𝑘),𝑦_𝑖(𝑘+1)=−𝛼_{𝑗∈𝒩𝑖}𝑙_𝑖𝑗𝑝_𝑗𝑡_𝑘−𝑝_𝑖𝑡_𝑘. (8)
Then, one has 𝑝_𝑖𝑡_𝑘+1=𝑝_𝑖𝑡_𝑘+ℎ𝑣_𝑖𝑡⁻_𝑘−𝛽_{1𝑗∈𝒩𝑖}𝑙_𝑖𝑗𝑝_𝑗𝑡_𝑘−𝑝_𝑖𝑡_𝑘−𝛽_{2𝑗∈𝒩𝑖}𝑙_𝑖𝑗𝑦_𝑗(𝑘)−𝑦_𝑖(𝑘),𝑣_𝑖𝑡⁻_𝑘+1=𝑣_𝑖𝑡⁻_𝑘−𝛽_{1𝑗∈𝒩𝑖}𝑙_𝑖𝑗𝑝_𝑗𝑡_𝑘−𝑝_𝑖𝑡_𝑘−𝛽_{2𝑗∈𝒩𝑖}𝑙_𝑖𝑗𝑦_𝑗(𝑘)−𝑦_𝑖(𝑘),𝑦_𝑖(𝑘+1)=−𝛼_{𝑗∈𝒩𝑖}𝑙_𝑖𝑗𝑝_𝑗𝑡_𝑘−𝑝_𝑖𝑡_𝑘. (9)
Let 𝑌_𝑖(𝑘)=(𝑝_𝑖(𝑡_𝑘),𝑣_𝑖(𝑡_𝑘),𝑦(𝑘))^𝑇 and 𝑌(𝑘)=(𝑌₁(𝑘),𝑌₂(𝑘),…,𝑌_𝑁(𝑘))^𝑇; then, 𝑌(𝑘+1)=𝐴𝑌(𝑘), (10) where 𝐴=𝐼_𝑁⊗𝐵−ℒ⊗𝐶. Let 𝑋(𝑘+1)=(𝑀⊗𝐼₃)𝑌(𝑘+1), where 𝑀 is defined in (4). From (10), one has 𝑋(𝑘+1)=𝑀⊗𝐼₃𝐴𝑌(𝑘),𝑀⊗𝐼₃𝐴=𝑀⊗𝐵−𝑀ℒ⊗𝐶=𝐼_𝑁−1=𝐼𝑀⊗𝐵−ℒ𝑀⊗𝐶_𝑁−1⊗𝐵−ℒ⊗𝐶𝑀⊗𝐼₃, (11) where 𝐿 is defined in (6). Then, 𝑋(𝑘+1)=𝐼_𝑁−1⊗𝐵−𝐿⊗𝐶𝑋(𝑘). (12)
Note that 𝐸⁻¹𝟎𝐿𝐸=0𝑏_(𝑁−1)ℒ, (13) where 𝑏=(𝑙₁₂,𝑙₁₃,...,𝑙_1𝑁) and 𝐸=1𝟎^𝑇_𝑁−1𝟏_𝑁−1𝐼_𝑁−1, (14)
is an invertible matrix. According to Lemma 3, 𝜆₁=0 is a simple eigenvalue of ℒ if the 𝒢 contains a spanning tree (it is well known that 𝒢 contains a spanning tree which is a necessary condition for consensus). Then, ℒ do not have zero eigenvalue. This implies that the eigenvalues of ℒ are 𝜆₂,𝜆₃,…,𝜆_𝑟. Then, there exists a nonsingular matrix 𝑃∈ℝ^{(𝑛−1)×(𝑛−1)}, such that 𝑃ℒ𝑃⁻¹=𝐽, (15) where 𝐽=diag(𝐽₂,𝐽₃,…,𝐽_𝑟), 𝐽_𝑙=𝜆_𝑙0001⋱000⋱⋱0001𝜆_{𝑙𝑁𝑙×𝑁𝑙}, (16)
𝑁_𝑙 is multiplicity of eigenvalue 𝜆_𝑙 and 𝑁₂+⋯+𝑁_𝑟=𝑁−1.
Let 𝑋(𝑘)=(𝑃⊗𝐼₃)𝑋(𝑘)=(𝑥₂,𝑥₃,…,𝑥_𝑟)^𝑇, where 𝑥_𝑙∈ℝ^3𝑁𝑙. Then, from (12), 𝑋(𝑘+1)=𝐼_𝑁−1⊗𝐵−𝑃ℒ𝑃⁻¹⊗𝐶𝑋(𝑘)=𝐼_𝑁−1⊗𝐵−𝐽⊗𝐶𝑋(𝑘), (17) where 𝐽=diag{𝐽₂,𝐽₃,…,𝐽_𝑟}. 𝑋(𝑘+1) is asymptotically stable if and only if 𝑥̇‌=(𝐼_𝑁−1⊗𝐵−𝐽_𝑙⊗𝑋)𝑥. Similar to analysis in [24, 29], 𝑋(𝑘+1) is asymptotically stable if and only if 𝑧(𝑘+1)=(𝐵−𝜆_𝑙)𝑧(𝑘) is stable. Note that 𝑀_𝑙=(𝐵−𝜆_𝑙) which immediately leads to the conclusion.

Theorem 7. The controlled multiagent system (3) achieves consensus asymptotically if and only if the communication graph 𝒢 contains a spanning tree and Re𝜆_𝑙𝛽₁−𝛼𝛽₂Re𝜆_𝑙−Im²𝜆_𝑙𝛼𝛽₂>0,𝑎𝑏𝑑+𝑎²𝑐−𝑑²>0, (18)(19) where 𝑙=2,3,…,𝑟, 𝑎=Re2𝜆²_𝑙𝛼ℎ𝛽₂𝜆_𝑙ℎ𝛽₁−𝜆_𝑙𝛼𝛽₂,𝑏=Im2𝜆²_𝑙𝛼ℎ𝛽₂𝜆_𝑙ℎ𝛽₁−𝜆_𝑙𝛼𝛽₂,𝑐=Re4−𝜆²_𝑙𝛼ℎ𝛽₂−𝜆_𝑙ℎ𝛽₁𝜆_𝑙ℎ𝛽₁−𝜆_𝑙𝛼𝛽₂,𝑑=Im4−𝜆²_𝑙𝛼ℎ𝛽₂−𝜆_𝑙ℎ𝛽₁𝜆_𝑙ℎ𝛽₁−𝜆_𝑙𝛼𝛽₂. (20)

Proof. Let 𝜇 be an eigenvalue of matrix 𝑀_𝑙. Then, det𝜇𝐼₃−𝑀_𝑖=𝜇³−2−𝜆_𝑙ℎ𝛽₁𝜇²−𝜆²_𝑙𝛼ℎ𝛽₂−1𝜇. (21)
Let 𝑃_𝑙𝜇=𝜇²−2−𝜆_𝑙ℎ𝛽₁𝜇−𝜆²_𝑙𝛼ℎ𝛽₂−1. (22) It is easy to know that polynomials 𝑃_𝑙(𝜇), for 𝑙=2,3,…,𝑟, are Schur stable if and only if 𝜌(𝑀_𝑙)<1. (𝑠−1)𝑃_𝑖𝑠+1=𝜆𝑠−1_𝑙ℎ𝛽₁−𝜆²_𝑙𝛼ℎ𝛽₂𝑠²+2𝜆²_𝑙𝛼ℎ𝛽₂𝑠+4−𝜆²_𝑙𝛼ℎ𝛽₂−𝜆_𝑙ℎ𝛽₁. (23)
If 𝛽₁−𝜆_𝑖𝛼𝛽₂=0, 1 is a root of 𝑃_𝑙(𝜇)=1. Therefore, 𝜆_𝑖ℎ(𝛽₁−𝜆_𝑖𝛼𝛽₂)≠0, if the consensus can be achieved. Then, the consensus can be achieved if and only if the polynomials 𝑃_𝑙(𝑠), for 𝑙=2,…,𝑟, are Hurwitz stable, where 𝑃_𝑙(𝑠)=𝑠²+2𝜆²_𝑙𝛼ℎ𝛽₂𝜆_𝑙ℎ𝛽₁−𝜆_𝑙𝛼𝛽₂𝑠+4−𝜆²_𝑙𝛼ℎ𝛽₂−𝜆_𝑙ℎ𝛽₁𝜆_𝑙ℎ𝛽₁−𝜆_𝑙𝛼𝛽₂. (24)
It is easy to check Re2𝜆²_𝑙𝛼ℎ𝛽₂𝜆_𝑙ℎ𝛽₁−𝜆_𝑙𝛼𝛽₂>0, (25) if and only if (18) holds. By Lemma 5, the polynomials 𝑃_𝑖(𝑠), for 𝑖=1,2,…,𝑁, are Hurwitz stable if and only if (18) and (19) hold. The proof is thus completed.

Remark 8. According to the previous discussion, both the real and imaginary parts of the eigenvalues of the Laplacian matrix play key roles in achieving consensus. The necessary and sufficient conditions in Theorems 6 and 7 are too complicated to directly display the relationship among consensus, control gains, and sampled period.
When it comes to undirected graph, the results will be more simple.

Corollary 9. The controlled multiagent system (3) achieves consensus asymptotically if and only if the undirected communication graph 𝒢 is connected and 𝜆_𝑖𝛼𝛽₂<𝛽₁<4−𝜆²_𝑖𝛼ℎ𝛽₂𝜆_𝑖ℎ, (26) where 𝑖=2,3,…,𝑁.

Proof. It is well known that ℒ contains 𝑁−1 positive real eigenvalues if 𝒢 is a connected undirected graph. Then, one has 𝑏=0 and 𝑑=0. From Theorem 7, (18) and (19) hold if and only if (26) is satisfied. The proof is thus completed.

Remark 10. Equation (26) is nonempty, when 4−𝜆²_𝑖𝛼ℎ𝛽₂𝜆_𝑖ℎ−𝜆_𝑖𝛼𝛽₂>0, (27) which implies that 𝛼𝛽₂<2𝜆²_𝑖ℎ. (28)
So, we can choose the control gains 𝛼 and 𝛽₂ from (28) and choose 𝛽₁ from (26). Therefore, it is quite easy to find suitable control gains for any connected graph 𝒢 and sampled period ℎ.
The following corollary will show, when the control gains are given, how to determine suitable control gains ℎ.

Corollary 11. The controlled multiagent system (3) can achieve consensus if and only if the undirected communication graph G is connected, 4ℎ<𝜆_max𝜆_max𝛼ℎ𝛽₂+𝛽₁,𝛽₁>𝜆_𝑖𝛼𝛽₂, (29) where 𝜆_max=max{𝜆₂,𝜆₃,…,𝜆_𝑁}.

Remark 12. When 𝛽₁>𝜆_𝑖𝛼𝛽₂ is not satisfied, the consensus will fail. The upper bound of sampled period increases as 𝜆_max, 𝛼, 𝛽₁, and 𝛽₂ decrease. The sampled period ℎ does not have the lower bound, which is different from [34].

4. Illustrative Examples

In this section, an illustrative example is given to demonstrate the correctness of the theoretical analysis. We consider the controlled multiagent system (3) with 8 agents. The communication graph is shown in Figure 1. The Laplacian matrix is ℒ=3−1−1−10000−1300−1−100−102000−10−1003−100−10−10−130−100−100020−100−100010000−10−102. (30)

Figure 1 

Communication graph.

By calculation, one has 𝜆₁=0, 𝜆₂=0.4965, 𝜆₃=1.7356, 𝜆₄=𝜆₅=2, 𝜆₆=3.5767, 𝜆₇=4, and 𝜆₈=5.1912.

When the sampled period ℎ=1 is given, from (28), choose 𝛼=1 and 𝛽₂=0.05 which satisfy 𝛼𝛽₂<2𝜆²_maxℎ. (31)

From Corollary 9, the consensus can be achieved if and only if 0.2596<𝛽₁<0.5110. Figures 2 and 3 show that consensus cannot be reached when 𝛽₁=0.2 and 𝛽₁=0.55 but can be achieved when 𝛽₁=0.3 (shown in Figure 4).

(a)

(b)

Figure 2 

Trajectory of controlled multiagent system (3) with communication graph shown in Figure 1, where 𝛽1=0.2, 𝛽2=0.05, 𝛼=1. Evolution of (a) 𝑝𝑖 and (b) 𝑣𝑖.

(a)

(b)

Figure 3 

Trajectory of controlled multiagent system (3) with communication graph shown in Figure 1, where 𝛽1=0.55, 𝛽2=0.05, 𝛼=1. Evolution of (a) 𝑝𝑖 and (b) 𝑣𝑖.

(a)

(b)

Figure 4 

Trajectory of controlled multiagent system (3) with communication graph shown in Figure 1, where 𝛽1=0.3, 𝛽2=0.05, 𝛼=1. Evolution of (a) 𝑝𝑖 and (b) 𝑣𝑖.

5. Conclusions

In this paper, the distributed consensus problem has been considered for the continuous-time multiagent system under intermittent communication. Motivated by impulsive control strategy, an impulsive consensus algorithm has been proposed, where the local algorithm of each agent is only based on the position information. Based on the stability theory of impulsive systems and the property of graph Laplacian matrix, some necessary and sufficient conditions for consensus have been obtained. From the results, we can easily find out suitable control gains for consensus. Finally, a numerical example is given to verify the theoretical analysis. It would be interesting to further investigate the multiagent system with switching topology via impulsive control to realize consensus.

Acknowledgments

This work was supported in part by the China Postdoctoral Science Foundation funded project 2012M511258 and the National Natural Science Foundation of China under Grants 61073026, 61170031, 61272069, and 61004030.