^{1,2}

^{2}

^{1}

^{2}

This paper considers the problem of the convergence of the consensus algorithm for multiple agents in a directed network where each agent is governed by double-integrator dynamics and coupling time delay. The advantage of this protocol is that almost all the existing linear local interaction consensus protocols can be considered as special cases of the present paper. By combining algebraic graph theory and matrix theory and studying the distribution of the eigenvalues of the associated characteristic equation, some necessary and sufficient conditions are derived for reaching the second-order consensus. Finally, an illustrative example is also given to support the theoretical results.

Recently, coordinated control of multiple agents has attracted a great deal of attention in many fields such as biology, physics, robotics, and control engineering [

In the references on the consensus problem, multiagent systems with double-integrator dynamics have been paid great attentions because of their importance in practice. In [

In this paper, we consider the convergence of the second-order consensus of multiagent systems composed of coupled double-integrators dynamics and coupling time delay. For this protocol, all the agents in the fixed directed network topology are governed by double-integrator dynamics. The advantage of this protocol is that almost all the existing linear local interaction consensus protocols can be considered as special cases of the present paper [

The rest of this paper is outlined as follows. In Section

In this subsection, some basic concepts and results about algebraic grapy theory are introduced. For more details about algebraic graph theory, please refer to [

If there is a sequence of edges of the form

Some mathematical notations are used for simplicity throughout this paper.

Suppose that the

Equivalently, system (

Let

Second-order consensus in the multiagent system (

In delayed systems, the time delays can be regarded as the bifurcation parameters. In [

Let

The characteristic equation for system (

In the following, one considers the case that

(1) If

(2) If

(3) If

Suppose that (

Define

Consider the exponential polynomial

Let

Suppose that the network contains a directed spanning tree, and all the roots of (

For

Also note that

Summarizing the previous discussions, we have the following theorem.

Suppose that the network contains a directed spanning tree; we have the following results.

If

If

For

For

In this section, some numerical results of simulating system (

The directed interaction topology of five agents.

Consider

Velocity and position states of five agents in a network under linear consensus protocols where

Consider

Velocity and position states of five agents in a network under periodic consensus protocols where

In this paper, the convergence of the second-order consensus of multiagent systems composed of coupled double-integrators dynamics and coupling time delay has been studied. The advantage of the considered protocol is that it can be treated as the extensions of most linear local interaction protocols in the existing literatures. By combining algebraic graph theory and matrix theory and studying the distribution of the eigenvalues of the associated characteristic equation, some necessary and sufficient conditions are derived for reaching the second-order consensus. Several simulation results further validate the effectiveness of theoretical analysis.

This work was supported in part by Project no. CDJXS12180002 supported by the Fundamental Research Funds for the Central Universities, in part by the National Natural Science Foundation of China under Grant 60973114 and Grant 61170249, in part by the Research Fund of Preferential Development Domain for the Doctoral Program of Ministry of Education of China under Grant 201101911130005, in part by the State Key Laboratory of Power Transmission Equipment & System Security and New Technology, Chongqing University, under Grant 2007DA10512711206, and in part by the Program for Changjiang Scholars.