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The consensus of the multiagent system with directed topology and a leader is investigated, in which the leader is dynamic. Based on Laplace transform method, the accurate upper error bound between the leader and the followers can be obtained. It is also proved that all agents of the system will aggregate and eventually form a cohesive cluster following the leader if the leader is globally reachable. Finally, some simulation examples are given to illustrate the theoretical results.

In recent years, the consensus problem of multiagent systems has become a hot topic due to their broad applications, such as cooperative unmanned air vehicles, automated highway systems, air traffic control, and autonomous underwater vehicles [

Various algorithms and models about multiagent systems have been discussed based on different tasks or interests. The leader-following system is one of the most interesting topics in the motion control of the multiagent systems. Vicsek et al. [

The coupling topology plays an important role in the studies of multiagent systems. Because of the complexity in the consensus analysis with directed topology, most researchers focus on the undirected topology or balance topology to simplify the problem [

This paper is organized as follows. Section

Consider a multiagent system of

To discuss the coordinated control among the agents, graph theory is a very effective tool. Regard the agent as a node and the connection link between any two agents as an edge; the coupling topology is conveniently described by a directed graph. Let

A diagonal matrix

It is easy to see that

A digraph

The nonzero eigenvalues of

The proof is similar to that of [

The eigenvalues of

If node

Notice that the block matrix of

In this section, we focus on the coordinated control problem of model (

If leader

To solve this problem, we introduce the Laplace transform, succinctly denoted by

If the fractional expressions in (

This completes the proof.

In Theorem

In order to verify the above theoretical analysis, we present some numerical simulations to illustrate the systems. These simulations are performed with ten followers and one leader, and the initial positions of the agents are chosen randomly. The coupling matrix

Figures

Errors between leader and followers about system (

Errors between leader and followers about system (

Errors between leader and followers about system (

Trajectories of the agents about system (

Trajectories of the agents about system (

Trajectories of the agents about system (

Trajectories of the agents about system (

In this paper, we have investigated the coordinated control of leader-following multiagent systems. It is proved that the agents of the systems will aggregate and form a cluster following the leader asymptotically. Meanwhile, we have studied the coupling topology among the agents in the general case. The systems considered in this paper can better reflect the collective behavior in practice. It is clear that the ideas and approaches about graph theory and linear dynamical system theory will play an important role in the analysis of the multiagent system. Finally, simulations give an effective demonstration of the leader-following systems.

The authors declare that there is no conflict of interests regarding the publication of this paper.

This work was supported by the National Natural Science Foundation of China under Grant no. 61304049, the Science and Technology Development Plan Project of Beijing Education Commission (no. KM201310009011), and the Plan training project of excellent young teacher of North China University of Technology.