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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.

Recently, distributed consensus has received great interest in the control community, due to broad applications in formation [

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 [

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 [

This paper is organized as follows. In Section

The communication structure of the multiagent system is described by an undirected graph

A directed path in a digraph

We consider a multiagent system with

Consensus in the multiagent system (

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

The proposed algorithm only uses sampled information of relative position (i.e.

The following lemmas are needed in the proof of the theorem.

Zero is a simple eigenvalue of

Define

From [

Let

The complex polynomial

Denote the eigenvalues of

The controlled multiagent system (

Note that

Then, one has

Let

Note that

is an invertible matrix. According to Lemma

Let

The controlled multiagent system (

Let

Let

If

It is easy to check

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

When it comes to undirected graph, the results will be more simple.

The controlled multiagent system (

It is well known that

Equation (

So, we can choose the control gains

The following corollary will show, when the control gains are given, how to determine suitable control gains

The controlled multiagent system (

When

In this section, an illustrative example is given to demonstrate the correctness of the theoretical analysis. We consider the controlled multiagent system (

Communication graph.

By calculation, one has

When the sampled period

From Corollary

Trajectory of controlled multiagent system (

Trajectory of controlled multiagent system (

Trajectory of controlled multiagent system (

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.

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.