Distributed Impulsive Consensus of the Multiagent System without Velocity Measurement

and Applied Analysis 3 Lemma4. LetL be the Laplacianmatrix of the graphG.Then the (N−1)×(N−1)matrix L̂ defined by L̂ = MLG satisfies ML = L̂M. Furthermore, ?̂? = ( l 22 − l 12 l 23 − l 13 ⋅ ⋅ ⋅ l 2N − l 1N l 32 − l 12 l 33 − l 13 ⋅ ⋅ ⋅ l 3N − l 1N .. .. d .. l N2 − l 12 l N2 − l 13 ⋅ ⋅ ⋅ l N2 − l NN )


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][12][13][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][19][20][21][22][23][24][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][27][28][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][31][32], where the impulsive controllers regulate all states of each agent in the system.We introduced impulsive algorithms for the multiagent system in [33][34][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.  and 0 × are the identity matrixes of order  (or simply  if no confusion arises) and the  ×  matrix with all elements equal to zero (or simply 0 if no confusion arises), respectively.() denotes the spectral radius of squares matrix .For  ∈ C, Re() and Im() are the real part and the imaginary part of .

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, . . ., } and a set of edges E ⊆ V × V (G has no self-loops or repeated edges).An edge {, } in G means that node  can receive information from node .N  denotes the set of neighbors of agents , that is, N  = { ∈ V | (, ) ∈ E}.The Laplacian matrix L of the graph G is defined as A directed path in a digraph G is an ordered sequence V 1 , V 2 , . . ., V  of agents such that any ordered pair of vertices appearing consecutively in the sequence is an edge of the digraph, that is, (V  , V +1 ) ∈ E, for any  = 1, 2, . . .,  − 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.
We consider a multiagent system with  identical agents: where  ∈ N,   ∈ R and V  ∈ R are the position and velocity of agent , respectively,   ∈ R is a control input.All results in this paper still hold for   , V  ,   ∈ R  by using the Kronecker product operations.
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 and  +1 −   = 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: where Remark 2. The proposed algorithm only uses sampled information of relative position (i.e.  ()−  ()) which is different from [26][27][28][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.,   (  ) −   ( −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.

Consensus in Multi-Agent System
Denote the eigenvalues of , respectively, by  1 ,  ) .
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
where  = 2, 3, . . ., .
Proof.It is well known that L contains  − 1 positive real eigenvalues if G is a connected undirected graph.Then, one has  = 0 and  = 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 which implies that So, we can choose the control gains  and  2 from (28) and choose  1 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.
Remark 12.When  1 >    2 is not satisfied, the consensus will fail.The upper bound of sampled period increases as  max , ,  1 , and  2 decrease.The sampled period ℎ does not have the lower bound, which is different from [34].

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  When the sampled period ℎ = 1 is given, from (28), choose  = 1 and  2 = 0.05 which satisfy From Corollary 9, the consensus can be achieved if and only if 0.2596 <  1 < 0.5110.Figures 2 and 3 show that consensus cannot be reached when  1 = 0.2 and  1 = 0.55 but can be achieved when  1 = 0.3 (shown in Figure 4).

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.