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We apply the linear matrix inequality method to consensus and

In recent years, decentralized coordination of multiagent systems has received many researchers' attention in the areas of system control theory, biology, communication, applied mathematics, computer science, and so forth. In cooperative control of multiagent systems, a critical problem is to design appropriate protocols such that multiple agents in a group can reach consensus. So far, by using the matrix theory, the graph theory, the frequency-domain analysis method, the Lyapunov direct method, and so forth, consensus problems for various kinds of multiagent systems have been studied extensively [

In the field of systems and control theory, the pioneering work was done by Borkar and Varaiya [

For the case of single integrator multiagent systems, Olfati-Saber and Murray [

When considering communication delays in the feedback, three types of consensus protocols have been analyzed: (i) both the state of the agent and its neighbors are affected by identical delays [

We are here concerned with the third type of consensus protocols, where the heterogeneous delays are time varying, and the involved graph is directed. It seems to us that the frequency-domain analysis method in [

We also consider the

To the best of our knowledge, little has been known about the

This paper is structured as follows. The problem statement and the transformation of the multiagent system are summarized in Section

The notation used throughout this paper is fairly standard.

Throughout this paper, we denote a weighted digraph by

Consider the following multiagent system with heterogeneous delays:

Denote

Next, we make use of transformation (

Denote

When the system involves disturbance input, we consider the following multiagent system of the form

Denote the

In this paper, say that system (

The following two lemmas can be concluded from [

The digraph

For any continuous vector

Based on Lemmas

If the digraph

First, we see that (

On the other hand, we show that (

Next, by the definition of vector

The method used in Theorem

For given constants

Choose the Lyapunov function defined by (

Similar to the analysis in the proof of Theorem

Unlike most of consensus analysis for multiagent systems, it does not require that

The method used in this paper can also be extended to the case of switching topology. Consider the following multiagent system with switched topologies and heterogeneous delays:

Under the reduced-order transformation (

For given constants

Similarly, for the following switched multiagent system with disturbance input and heterogeneous delays

For given constants

Consider the following four digraphs with six nodes shown in Figure

Four digraphs: (a)

When

State trajectory of system (

State trajectory of system (

For the case of disturbance input, assume that

State trajectory of system (

For the case of switching topologies

In this paper, we first apply the linear matrix inequality method to consensus of the single integrator multiagent system with heterogeneous time-varying delays in directed networks. Unlike the case of identical delays, the multiagent system with heterogeneous delays usually cannot be transformed to a reduced-order system. To overcome such difficulty, we introduce a partially-reduced-order system and an integral system. As a result, the linear matrix inequality method becomes useful in the analysis of consensus and

This work was supported by National Natural Science Foundation of China (Grant nos. 61174217 and 61374074) and Natural Science Foundation of Shandong Province (Grant nos. ZR2010AL002 and JQ201119).