This paper addresses the average consensus problem of neutral multiagent systems in undirected networks with fixed and switching topologies. For the case of fixed topology, necessary and sufficient conditions to average consensus are established by decoupling the neutral multiagent system in terms of the eigenvalues of the Laplacian matrix. For the case of switching topology, sufficient conditions to average consensus are given in terms of linear matrix inequalities to determine the allowable upper bound of the timevarying communication delay. Finally, two examples are worked out to explain the effectiveness of the theoretical results.
In the last few years, more and more researchers have focused their attention on the study of multiagent systems due to many applications in unmanned aerial vehicles, satellite cluster, automated highways, and mobile robots. In all cases, the aim is to control a group of agents connected through a wireless network. Many profound theoretical results have been established for cooperative control of multiagent systems. In cooperative control of multiagent systems, an important issue is to design appropriate control strategies or protocols based on local information such that the group of dynamic agents can reach an agreement on certain quantities of interest. Such a problem is usually called a consensus problem, which has attracted many people in recent years.
Consensus problem has a long history in the field of computer science. In the field of control theory, a discretetime model of
In recent years, more and more consensus results with single integrator models were established. Xiao and Wang studied the consensus problem of discretetime multiagent systems with time delay [
Unlike the single integrator case, a spanning tree is a necessary rather than a sufficient condition for consensus seeking with double integrator dynamics. However, the extension of consensus algorithms from the firstorder case to the secondorder case is nontrivial [
It is well known that neutral systems play an important role in engineering area. If the agent’s dynamical equation takes the neutral form, what about the consensus of multiple agents in networks? To the best of our knowledge, consensus problem of neutral multiagent systems is not resolved by now. Therefore, it is necessary to study the consensus of neutral multiagent systems, which is the main purpose of this paper. We make the following contributions. Firstly, average consensus for neutral multiagent systems in undirected networks is proposed. Secondly, the convergence of the given proposed protocols is analyzed, and sufficient and necessary or sufficient conditions to average consensus of neutral multiagent systems have been established.
The remainder of this paper is organized as follows. In Section
As usual, the communication network is modeled through an undirected weighted graph
It is easy to see that
(a) The operator
Lemma
As a result of the supposed piecewise continuity of the delay, the Cauchy problem associated with (
The average consensus for the neutral multiagent system (
The neutral multiagent system (
For the neutral multiagent system (
By Lemma
The function
By Lemma
In this section, we will consider the average consensus of neutral multiagent systems with fixed and switching topologies, respectively.
Let
Let
For the particular case
The following scalar neutral differential equation
Based on Lemma
Let
By Theorem
Note that system (
We are here interested in discussing such a problem of whether a network with switching topology can still solve average consensus. In this case, the following switched system is considered:
The neutral multiagent system (
Let us first take the following reducedorder transformation:
Before giving the main result, we need the following lemma.
For any real differentiable vector function
Assume that (
In spite of the existence of switching topology, similar to the discussion given in Remark
When
Assume that
In Theorem
Theorem
We now work out two examples to illustrate the theoretical results in this paper.
Consider the neutral multiagent system (
The Laplacian matrix
An undirected connected weighted graph.
Consider an undirected network with the following switching topologies shown in Figure
Solving (
When

0  0.1  0.2  0.3  0.4  0.5  Nothing 



0.3431  0.3337  0.3249  0.3162  0.3081  0.2975  0.2612 

0.1  0.2  0.3  0.4  0.5  0.6 



0.3337  0.2798  0.2294  0.1816  0.1348  0.0874 
Two examples of undirected connected graphs.
The average consensus problem of neutral multiagent systems in undirected networks with fixed and switching topologies is studied in this paper. Firstly, for the case of fixed topology, we establish sufficient and necessary conditions to average consensus by decoupling the neutral multiagent system in terms of the eigenvalues of the Laplacian matrix. Secondly, by using a linear matrix inequality method, average consensus criteria in terms of linear matrix inequality are given to determine the allowable upper bound of the timevarying communication delay for the case of switching topology. Finally, two examples are worked out to illustrate the theoretical results.
The authors declare that there is no conflict of interests regarding the publication of this paper.
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.