Abstract and Applied AnalysisVolume 2012 (2012), Article ID 621060, 18 pagesdoi:10.1155/2012/621060

Research Article

Mean Square Consensus for Uncertain Multiagent Systems with Noises and Delays

Yuangong Sun

School of Mathematical Sciences, University of Jinan, Shandong, Jinan 250022, China

Received 31 October 2011; Revised 12 January 2012; Accepted 12 January 2012

Academic Editor: Jinhu Lü

Copyright © 2012 Yuangong Sun. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

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.

1. Introduction

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 [1–9]. A simple but interesting model of multiple agents moving in the plane was proposed and discussed in [3]. A theoretical framework for consensus problems of continuous-time multi-agent systems was presented in [5]. Some recent progress on consensus of multi-agent systems was given in [6, 7]. When there exist time-varying delays between agents, a reduced-order system approach is used to consensus of multi-agent systems in [8, 9].

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 [10–15]. In [10, 11], the authors studied consensus problems when there exist noisy measurements of the states of neighbors, and a stochastic approximation approach was applied to obtain mean square and almost sure convergence in models with fixed network topologies or with independent communications failures. Necessary and/or sufficient conditions for stochastic consensus of multiagent systems were established for the case of fixed topology and time-varying topologies in [12, 13]. The distributed consensus problem for linear discrete-time multiagent systems with delays and noises was investigated in [14] by introducing a novel technique to overcome the difficulties induced by the delays and noises. In [15], a novel kind of cluster consensus of multiagents systems with several different subgroups was considered based on Markov chains and nonnegative matrix analysis.

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 [16–18]. Note that measurement noises, time delays, and parametric uncertainties may arise naturally in the process of information transmission between agents, for example, because of the congestion of the communication channels, the asymmetry of interactions, and the finite transmission speed due to the physical characteristics of the medium transmitting the information. Then, it is natural to consider the effect of measurement noises, time-varying delays, and parametric uncertainties on consensus problem of multi-agent systems.

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 [19], the authors proposed an algorithm which is robust against the bounded time-varying delays and bounded noises. It is natural to conjecture that consensus of multi-agent systems should also be robust to uncertainties. However, it leads to difficulties due to the existence of measurement noises, parametric uncertainties, and symmetric or asymmetric time-varying delays, since most of methods in the literature fail to apply.

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, means the transpose of the matrix . We say that if is positive definite, where and are symmetric matrices of same dimensions. refers to the Euclidean norm for vectors. is a column vector of appropriate dimension, where is a constant. means an identity matrix of appropriate dimension.

2. Preliminaries

We denote a weighted digraph by , where is the set of nodes with , node represents the th agent; is the set of edges, and an edge of is denoted by an order pair ; is an -dimensional weighted adjacency matrix with . Say if . The set of neighbors of the th agent is denoted by . If is an edge of , node is called the parent of node . A directed tree is a directed graph, where every node, except one special node without any parent, which is called the root, has exactly one parent, and the root can be connected to any other nodes through paths. A spanning tree of a digraph is a directed tree formed by graph edges that connect all the nodes of the graph.

The -dimensional Laplacian matrix of digraph is defined by for and for . It is easy to see that has at least one zero eigenvalue and . Below is an important property of Laplacian matrices shown in [9].

has a spanning tree if and only if the matrix is Hurwitz stable, where

Consider a network of continuous-time first-order integrator agents with the dynamics where is the state of the th agent, is the control input. When only taking measurement noises into consideration, the control input (or protocol) is designed to take the form [12]: where the consensus-gain function is piecewise continuous; denotes the measurement of the th agent’s state by the th agent; are independent standard white noises; is the noise intensity; for . It has been shown that the consensus-gain function plays a key role in the convergence analysis of the designed protocol.

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: where ; time delays , , , are piecewise continuous and bounded functions; and are defined as above; , are parametric uncertainties satisfying () and .

For the sake of convenience, let . Denote the th row of the matrix by , and . Substituting the control (2.4) into the system (2.1) leads to where the -dimensional matrix is defined as followings: It is easy to see where with for and for .

Let , , and be defined by (2.1). From (2.6), we have the following reduced-order system: It is a system driven by an -dimensional standard white noise, which can be written in the form of the Itô stochastic differential equation where is an -dimensional standard Brownian motion. Without loss of generality, we let for since the case of multiple delays can be similarly studied. In this case, the protocol (2.4) takes the following simple form: and the system (2.10) reduces to

In the sequel, we assume that the parametric uncertainty to be of the form where and are constant matrices with appropriate dimensions, and is an unknown matrix satisfying . Suppose also that there exists at least one edge such that , and is a piecewise continuous function on and , where is a constant. Here, the initial function of the system (2.12) is assumed to satisfy on .

We say the system (2.2) under protocol (2.11) asymptotically achieves mean square consensus if for all and .

3. Convergence Analysis of Protocol (2.11)

Before establishing the main result of this paper, we first show the relation between a linear matrix inequality and the collectivity of graph , which can be used to analyze the feasibility of the given consensus condition.

Lemma 3.1. If has a spanning tree, then there exist matrices , , , of compatible dimensions, constants , and such that where , , , and , , and is defined by (2.1).

Proof. By Lemma 3.1, we have that there exists a matrix such that if has a spanning tree. Based on the Finsler Lemma [20], there exist matrices and such that Let . Then, (3.2) implies (3.1) by choosing and sufficiently small.

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

Lemma 3.2 (see [21]). For any continuous vector on and matrix , where and , the following inequality holds:

Lemma 3.3 (see [22]). Let , , and be real matrices of appropriate dimensions with , then for any scalar , one has.

Now, let us present the main result of this paper. We assume that the positive consensus-gain function satisfies one of the following assumptions:(A1), , and for sufficiently large ;(A2), .

Theorem 3.4. Assume that (A1) or (A2) holds and . If there exist matrices , , , of compatible dimensions, constants and such that where , , , , , , , , , , and are defined as in Lemma 3.1, and is defined by (2.1), and , then the system (2.2) under protocol (2.11) asymptotically achieves mean square consensus for all admissible uncertainties satisfying (2.13).

Proof. For the reduced order system (2.12), let Then, (2.12) reduces to Note that (3.4) holds. We can choose a constant sufficiently small such that where .Let where is a constant to be determined. By the Itô formula, we have where . It is not difficult to see that there exists a scalar such that due to the fact that . Thus, where Without loss of generality, say for . Then, we have On the other hand, integrating (3.7) from to yields Therefore, by the definition of , we have which implies that By Lemma 3.2, we have Using the basic inequality for any vector and , we have Substituting (3.15)–(3.17) into (3.12) gives where , and with , , . It is easy to see that if and only if where is defined by (3.1) and . By Lemma 3.3, we have that (3.20) is implied by (3.7). Thus, (3.7) yields that for .Note that for , and where satisfying for some . By choosing , we get from (3.10) and (3.18) that By the comparison theorem [23], we have If (A1) holds, that is, and , then, similar to the proof of Theorem  3.2 in [12], we can conclude from (3.23) that . If (A2) holds, by the L’Hopital rule, we have which also implies . Based on the construction of and the transformation , we have It implies that for and . The proof is complete.

Remark 3.5. By Lemma 3.1, we can easily see that (3.4) holds for appropriate constant and admissible uncertainties (e.g., is sufficiently small) if has a spanning tree. Therefore, Theorem 3.4 shows that mean square consensus of the system (2.2) under protocol (2.11) is robust to delays and uncertainties if (A1) or (A2) holds. For given matrices and , the tolerable upper bound of delay can be derived from (3.4) by using the Matlab’s LMI Toolbox.

Remark 3.6. For the case of multiple delays, it is not difficult to conclude that the system (2.2) under protocol (2.4) asymptotically achieves mean square consensus for admissible delays and parametric uncertainties if has a spanning tree and (A1) or (A2) holds. Since the analysis procedure is similar to the above, we omit it here and leave it to the interested readers.

4. Convergence Analysis for the Case of Asymmetric Delays

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 satisfying . Without loss of generality, consider the following protocol: where , , , , , and are defined as above. The case of multiple time-varying delays can be similarly discussed.

Note that If we set , where is defined as above, then we have the following reduced-order system: where for ,, , , and are defined as above. Therefore, (4.3) can be written in the form of the Itô stochastic differential equation. It is not difficult to verify that . In the following, we assume that the uncertainties to be of the form where , are constant matrices of appropriate dimensions and satisfies (2.13).

Theorem 4.1. Assume that (A1) or (A2) holds and . If there exist matrices , , , , , and of compatible dimensions, constants , and for such that where , , , , and , , , , then the system (2.2) under protocol (4.1) asymptotically achieves mean square consensus for all admissible uncertainties satisfying (2.13).

Proof. Let Then, (4.5) reduces to Choose the Lyapunov function as the following: where is an appropriate constant to be determined. Without loss of generality, say for . By (4.9) and the Itô formula, we have where and . On the other hand, by the definition of , we have Then, proceeding as in the proof of Theorem 3.4, we can get a desired result. This completes the proof of Theorem 4.1.

Remark 4.2. By Lemma 3.1, we can also show that (4.7) holds for appropriate constant and admissible uncertainties if has a spanning tree. For given matrices and , the tolerable upper bound of delay can be derived from (4.7).

5. Simulation Results

Consider a digraph with six nodes and 0-1 weights, where . It is evident that has a spanning tree. Let . Using Matlab to solve (3.4) without uncertainties yields that . Thus, by Theorem 3.4, we have that protocol (2.11) asymptotically solves mean square consensus for any time-varying delay satisfying if (A1) or (A2) holds. Let and the intensity of the measurement noises . The state trajectories of the system under protocol (2.11) and a random initial state are shown in Figure 1 when . Figure 2 shows that the system is divergent when .

Figure 1: State trajectories under protocol (2.11) and .

Figure 2: State trajectories under protocol (2.11) and .

Consider again the digraph defined above. Solving (4.7) without uncertainties gives . By Theorem 4.1, we have that protocol (4.1) asymptotically solves mean square consensus for any time-varying delay satisfying if (A1) or (A2) holds. Let and the intensity of the measurement noises . Under protocol (4.1) and a stochastic initial state , the state trajectories of the system are shown in Figures 3 and 4 for and , respectively. We see that the system is divergent for the case of .

Figure 3: State trajectories under protocol (4.1) and .

Figure 4: State trajectories under protocol (4.1) and .

6. Conclusions

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 has a spanning tree.

Acknowledgments

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).

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