This paper investigates the robust leaderless consensus problem of uncertain multiagent systems with directed fast switching topologies. The topologies are assumed to jointly contain a directed spanning tree. Based on a special property of the graph Laplacian matrix, the consensus problem is converted into a stabilization problem by performing a proper variable transformation. Averaging method is employed for analysis. It is proved that if the topologies switch sufficiently fast and the controllers are properly designed, the robust leaderless consensus can still be achieved even when all the possible topologies are unconnected in the switching time intervals. Finally, a numerical simulation is provided to illustrate the effectiveness of the theoretical results.

In the past few years, the consensus problem of multiagent systems has drawn great attention for its broad potential applications in many areas such as cooperative control of vehicle, unmanned air vehicle formation, and flocking control [

In many applications, the interaction topology among agents may change dynamically. This may happen when the communication links among agents may be unreliable due to disturbance or subject to communication range limitations [

In contrast with the leader-following consensus problem, in the homogeneous multiagent systems, the leaderless consensus problem which can include the leader-following consensus problem as a special case is more complex and challenging especially with directed communication topologies. Since the leader-following consensus problem can be conveniently converted into a stabilization problem by constructing tracking error variables, the presence of the leader in the multiagent systems facilitates the derivation [

Motivated by this, the objective of this paper is to solve the robust leaderless consensus problem of uncertain multiagent systems with directed fast switching topologies. It is assumed that the switching topologies jointly contain a directed spanning tree. The analysis process is of two steps. Firstly, based on the property that the graph Laplacian matrix can be factored into the product of two specific matrices, the consensus problem with switching topologies is converted into a stabilization problem of a switched system by constructing a proper disagreement vector. Secondly, by using averaging method which is widely used for stability analysis of fast switching systems, sufficient conditions for achieving the leaderless consensus are obtained. It is shown that if the topologies switch sufficiently fast and the feedback gain matrices in the consensus controllers are properly designed, consensus can still be achieved even when the topologies are not connected in the switching time intervals. The main contribution of our paper is that the leaderless consensus problem of high-order dimension multiagent systems is solved under fast switching topologies which jointly contain a directed spanning tree.

The remainder of this paper is organized as follows. In Section

Throughout this paper, the following notations will be used.

A directed graph

In this paper, the communication topology is molded by a directed graph and we assume that the communication topology is time-varying. Denote

A union graph of a collection of graphs

Zero is a simple eigenvalue of

Consider a multiagent system composed of

The consensus of system (

In order to achieve consensus, the following distributed consensus controller based on local relative states information of neighbor agents is proposed:

The closed-loop system dynamics of (

Without loss of generality, consider an infinite sequence of nonempty, bounded, nonoverlapping, and contiguous time intervals

In this paper, we assume that across each time interval

For any given

For a Laplacian matrix

From the definition of union graph

If the graph fulfills Assumption

Let

Note that

Suppose there exists a constant

Let

Suppose that Assumption

According to Schur complement lemma [

Consider the following Lyapunov candidate of time-average system (

Using Lemma

According to Lemma

Commonly, the leader-following consensus problem can be conveniently converted into a stabilization problem by constructing the tracking error variables [

Most of existing works about the consensus problem of higher order systems with jointly connected topologies are restricted to be undirected topologies [

In this section, we provide an example to illustrate the effectiveness of the above theoretical results. Consider a multiagent system consisting of four agents in the form of position-speed model of moving plant with uncertainties

The directed switching communication topologies

Communication topologies

Solving LMI (

Then, the feedback matrix can be chosen as

Switching signal.

Trajectories of

Trajectories of

This paper has used the averaging method to solve the robust leaderless consensus problem of uncertain multiagent systems with fast switching topologies. The communication topologies are assumed to jointly contain a directed spanning tree. It has been proved that if the topologies switch sufficiently fast and the feedback matrix is properly designed, the consensus can be achieved even when the topologies are not connected in the switching time intervals.

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