Event-driven control scheduling strategies for multiagent systems play a key role in future use of embedded microprocessors of limited resources that gather information and actuate the agent control updates. In this paper, a distributed event-driven consensus problem is considered for a multi-agent system with second-order dynamics. Firstly, two kinds of event-driven control laws are, respectively, designed for both leaderless and leader-follower systems. Then, the input-to-state stability of the closed-loop multi-agent system with the proposed event-driven consensus control is analyzed and the bound of the inter-event times is ensured. Finally, some numerical examples are presented to validate the proposed event-driven consensus control.

Recently, synthesis and analysis of multi-agent systems have drawn great attention in many disciplines, such as mathematics, physics, computer science, systems biology, engineering, and social science. Roughly speaking, multi-agent systems are a class of networked dynamic systems consisting of a group of autonomous agents, which interact with each other locally and achieve an emergence behavior over a communication network. The controlled multi-agent systems have a broad range of applications including flocking and swarming in animal groups, vehicle formation, satellite reconfiguration, and unmanned aerial vehicles for rescue and surveillance.

Consensus problems have a long history originated from management science and statistics in 1960s [

One potential application of multi-agent control is to equip each autonomous agent with a small embedded micro-processor to collect information from neighboring agents for actuating the controller updates. However, micro-processors are generally resource- and energy-limited [

In this paper, we consider an event-driven consensus problem of a second-order leaderless and leader-follower multi-agent system with a fixed directed communication network. Firstly, the event-driven consensus problem is formulated. Secondly, an event-driven consensus control is designed for each agent to achieve consensus. Then the closed-loop multi-agent system is proven to be input-to-state stable with respect to the measurement error and, simultaneously, a positive lower bound is found for the event-time between two consecutive actuation updates.

Throughout this paper, the following notations are used.

Let

A path from vertex

Vertex

According to Definitions

A diagonal matrix

The next lemma shows an important property of Laplacian matrix

Laplacian matrix

In the leader-follower consensus literature, it is always assumed that the leader-agent is self-active, that is, the leader does not need information feedback from other agents and thus, the adjacency coefficients

If vertex

A Schur-complement lemma will be used in the stability analysis of the close-loop multi-agent systems and is given to end this subsection.

Consider a symmetric matrix

In a leaderless consensus problem, a group of

In a leader-follower consensus problem, the dynamics of follower-agents are given as (

When agents are equipped with resource-limited micro-processors, it is preferable to design an event-driven consensus controls for all agents such that the consensus controls need no update in continuous-time. For agent

We say that the event-driven consensus problem of leaderless multi-agent system (

We assume that the consensus laws

An event-driven consensus control of agent

Then a compact form of (

Denote

Now we will analyze the convergence of the closed-loop multi-agent system (

For Laplacian matrix

From Lemma

Take a coordinate transformation

Essentially, system (

Let

Now a main result is obtained for system (

Assume that the interconnection topology

For system (

Differentiating

Therefore,

On the other hand, for system (

Since the system (

The gain

Next, we show that the inter-event times

Assume that

Then a result is stated as follows.

When the event-driven consensus control (

In the leader-following problem, we assume that the state information, that is,

For the leader-following problem (

Still denote the measurement errors

Assume that the vertex

For system (

Differentiating

Therefore,

Similar to Theorem

Consider four agents whose dynamics is described by (

The interconnection topology

Assume that the weighted adjacency matrix

Then the submatrices

The initial conditions of system (

The evolution of position states

The evolution of velocity state

Evolution of the measurement error norm

Evolution of the consensus controllers

Consider four followers and one leader whose dynamics are, respectively, described by (

It is not difficult to obtain the Laplacian matrix

The acceleration of the active leader is assumed to be

The evolution of position states

The evolution of velocity states

Evolution of the measurement error norm

An event-driven consensus problem of second-order multi-agent systems with/without a self-active leader was considered in this paper. The consensus controllers have been proposed for all autonomous mobile agents based on an event-driven control update strategy. The input-to-state stability of the closed-loop multi-agent system has been analyzed by employing an ISS Lyapunov function. Some numerical examples have been presented to validate the proposed event-driven controls. However, it is noted that the event-driven condition depends on the states of the whole multi-agent group and all agents have identical event-times. The result is somewhat preliminary due to the centralized information gathering, so further work will be devoted to designing a decentralized event-driven consensus control for a second-order multi-agent system in the future.

This work is supported by the National Natural Science Foundation of China under Grant 61104104 and the Scientific Research Foundation for the Returned Overseas Chinese Scholars, State Education Ministry of China.