This paper investigates the consensus problem in mean square for uncertain multiagent systems with stochastic measurement noises and symmetric or asymmetric time-varying delays. By combining the tools of stochastic analysis, algebraic graph theory, and matrix theory, we analyze the convergence of a class of distributed stochastic approximation type protocols with time-varying consensus gains. Numerical examples are also given to illustrate the theoretical results.

In recent years, more and more researchers in the control community have focused their attention on distributed coordination of multiagent systems due to its broad applications in many fields such as unmanned aerial vehicles, mobile robots, autonomous underwater vehicles, automated highway systems, and formation control of satellite clusters.

In the cooperative control, a key problem is to design distributed protocols such that group of agents can achieve consensus through local communications. So far, many consensus results have been established for both discrete-time and continuous-time multiagent systems [

For most of consensus results in the literature, it is usually assumed that each agent can obtain its neighbor’s information precisely. Since real networks are often in uncertain communication environments, it is necessary to consider consensus problems under measurement noises. Such consensus problems have been studied by several researchers [

Generally speaking, multiagent systems usually can be regarded as a special kind of complex networks. Complex networks have been intensively investigated over the last two decades [

To the best of our knowledge, little has been known about the consensus of uncertain multi-agent systems with measurement noises and time-varying delays. In [

In this paper, by taking measurement noises, symmetric or asymmetric time-varying delays, and parametric uncertainties into consideration, we will study the consensus problem for networks of continuous-time integrator agents under dynamically changing and directed topologies. Based on a reduced-order transformation and a new Lyapunov function, we establish two sufficient conditions in terms of linear matrix inequalities such that mean square consensus is achieved asymptotically for all admissible delays and uncertainties. The feasibility of the given linear matrix inequalities is also analyzed.

Throughout this paper,

We denote a weighted digraph by

The

Consider a network of continuous-time first-order integrator agents with the dynamics

Note that time delays and parametric uncertainties may arise naturally in the process of information transmission between agents. We consider the following protocol of the form:

For the sake of convenience, let

Let

In the sequel, we assume that the parametric uncertainty

We say the system (

Before establishing the main result of this paper, we first show the relation between a linear matrix inequality and the collectivity of graph

If

By Lemma

The following two lemmas will be used in the proof of the main result.

For any continuous vector

Let

Now, let us present the main result of this paper. We assume that the positive consensus-gain function

Assume that (A1) or (A2) holds and

For the reduced order system (

Let

On the other hand, integrating (

Note that

By Lemma

For the case of multiple delays, it is not difficult to conclude that the system (

The method used in this paper can also be applied to the case when delay only affects the state of neighbors. Assume that there exists at least one agent such that the information exchange between this agent and its neighbors is free of delay, stochastic noises, and parametric uncertainties. For example, among agents there exists a leader

Note that

Assume that (A1) or (A2) holds and

Let

By Lemma

Consider a digraph

State trajectories under protocol (

State trajectories under protocol (

Consider again the digraph defined above. Solving (

State trajectories under protocol (

State trajectories under protocol (

In this paper, we study the mean square consensus problem for continuous-time multi-agent systems with measurement noises, time-varying delays, and parametric uncertainties. By introducing a reduced-order transformation and a new Lyapunov function, we combine the tools of stochastic analysis, algebraic graph theory, and matrix theory to analyze the convergence of a class of distributed stochastic approximation type protocols with the time-varying consensus gain. When imposing appropriate conditions on the consensus gain, we show that mean square consensus will be achieved asymptotically for admissible delays and uncertainties if the digraph

The authors thank the reviewers for their helpful suggestions and valuable comments on this paper. This work was supported by the National Natural Science Foundations of China (60704039, 61174217) and the Natural Science Foundations of Shandong Province (ZR2010AL002, JQ201119).