We consider consensus of a class of third-order continuous-time multiagent systems with time delay in undirected networks. By using matrix analysis and a frequency domain approach, a necessary and sufficient condition for consensus is established. A simulation result is also given to illustrate the main theoretical result.

In recent years, more and more people are interested in multiagent systems due to their extensive applications in real world, such as cooperative control of unmanned aerial vehicle, autonomous underwater submarine, flocking, and network control in information. Among the corresponding theories for multiagent systems, consensus problem is a critical issue which has attracted interdisciplinary researchers from the fields of control, mathematics, biology, physics, computer, robot, communication and artificial intelligence, and so forth.

Most work on consensus focuses on protocols taking the form of first-order dynamics [

It is well known that the dynamic equation given by second-order consensus protocols contains the derivatives of position and velocity. In practice, the dynamics of agents may involve the derivative of accelerated speed. The derivative of accelerated speed is usually called Jerk in engineering. Jerk is a physical quantity to describe the changing rate of acceleration. A Jerk is often required in engineering, especially in transportation design and materials, and so forth. Therefore, it is necessary and significant to design a class of third-order consensus protocols by using the information of position, velocity, and accelerated speed of agents, which is the main purpose of this paper.

Recently, some people have studied some higher-order consensus protocols [

An outline of this paper is as follows. In Section

Let

The Laplacian matrix of the graph

The following lemma is well known.

If the undirected graph

Suppose the dynamics of each node is described by the following third-order system:

In this paper we consider the following third-order protocol:

Say system (

We first have the following lemma for system (

If system (

(i) Since the graph is undirected, by (

(ii) By (

(iii) By (

In the sequel, set

By Lemma

By Lemma

System (

Based on the above analysis, consensus of system (

For the particular case when

System (

In this section, we will give a simulation to illustrate the main result. The topology graph in the simulation is shown in Figure

An undirected graph.

It is easy to see the corresponding Laplacian matrix

If we consider

Trajectories of system (

Trajectories of system (

In this paper, we study the consensus problem of the third-order dynamic multiagent system with time delay in undirected graphs. A necessary and sufficient condition for consensus of the system has been established. A simulation result illustrates the effectiveness of the theoretical result. Consensus of third-order multiagent systems with time delay in directed graphs will be further studied in the future.

The authors declare that they have no competing interests.

This work was supported by the Natural Science Foundation of Shandong Province under Grant no. JQ201119 and the National Natural Science Foundation of China under Grant nos. 61174217, 61374074, and 61473133.