Mean Square Consensus for Uncertain Multiagent Systems with Noises and Delays

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, 𝐴^𝑇 means the transpose of the matrix 𝐴. We say that 𝑋>𝑌 if 𝑋−𝑌 is positive definite, where 𝑋 and 𝑌 are symmetric matrices of same dimensions. ‖⋅‖ refers to the Euclidean norm for vectors. 𝐚=[𝑎⋯𝑎]^𝑇 is a column vector of appropriate dimension, where 𝑎 is a constant. 𝐼 means an identity matrix of appropriate dimension.

2. Preliminaries

We denote a weighted digraph by 𝐺=(𝑉,𝐸,𝐴), where 𝑉={1,2,…,𝑛} is the set of nodes with 𝑛≥2, node 𝑖 represents the 𝑖th agent; 𝐸⊆𝑉×𝑉 is the set of edges, and an edge of 𝐺 is denoted by an order pair (𝑖,𝑗); 𝐴=[𝑎_𝑖𝑗] is an 𝑛×𝑛-dimensional weighted adjacency matrix with 𝑎_𝑖𝑖=0. Say (𝑖,𝑗)∈𝐸 if 𝑎_𝑗𝑖>0. The set of neighbors of the 𝑖th agent is denoted by 𝑁_𝑖={𝑗∈𝑉∶(𝑗,𝑖)∈𝐸}. If (𝑖,𝑗) is an edge of 𝐺, node 𝑖 is called the parent of node 𝑗. 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 𝑛×𝑛-dimensional Laplacian matrix 𝐿(𝐺)=[𝑙_𝑖𝑗] of digraph 𝐺 is defined by 𝑙_𝑖𝑗=∑^𝑛_𝑘=1𝑎_𝑖𝑘 for 𝑖=𝑗 and 𝑙_𝑖𝑗=−𝑎_𝑖𝑗 for 𝑖≠𝑗. It is easy to see that 𝐿(𝐺) has at least one zero eigenvalue and 𝐿(𝐺)𝟏=0. Below is an important property of Laplacian matrices shown in [9].

𝐺=(𝑉,𝐸,𝐴) has a spanning tree if and only if the matrix 𝐸𝐿(𝐺)𝐹 is Hurwitz stable, where𝐸=𝟏−𝐼_𝑛−1𝟎,𝐹=^𝑇𝐼_𝑛−1. (2.1)

Consider a network of continuous-time first-order integrator agents with the dynamics𝑥̇_𝑖(𝑡)=𝑢_𝑖(𝑡),𝑖∈𝑉, (2.2) where 𝑥_𝑖∈𝑅 is the state of the 𝑖th agent, 𝑢_𝑖∈𝑅 is the control input. When only taking measurement noises into consideration, the control input 𝑢_𝑖 (or protocol) is designed to take the form [12]:𝑢_𝑖(𝑡)=_{𝑗∈𝑁𝑖}𝑎(𝑡)𝑎_𝑖𝑗𝑦_𝑖𝑗(𝑡)−𝑥_𝑖(𝑡),𝑖∈𝑉, (2.3) where the consensus-gain function 𝑎(𝑡)∶[0,∞)→(0,∞) is piecewise continuous; 𝑦_𝑖𝑗(𝑡)=𝑥_𝑗(𝑡)+𝜎_𝑖𝑗𝜂_𝑖𝑗(𝑡) denotes the measurement of the 𝑗th agent’s state 𝑥_𝑗(𝑡) by the 𝑖th agent; {𝜂_𝑖𝑗(𝑡)∶𝑖,𝑗∈𝑉} are independent standard white noises; 𝜎_𝑖𝑗≥0 is the noise intensity; 𝜎_𝑖𝑗=0 for 𝑗∉𝑁_𝑖. 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:𝑢_𝑖(𝑡)=_{𝑗∈𝑁𝑖}𝑎(𝑡)𝑎_𝑖𝑗+Δ𝑎_𝑖𝑗(𝑡)𝑧_𝑖𝑗(𝑡)−𝑥_𝑖𝑡−𝜏_𝑖𝑗(𝑡), (2.4) where 𝑧_𝑖𝑗(𝑡)=𝑥_𝑗(𝑡−𝜏_𝑖𝑗(𝑡))+𝜎_𝑖𝑗𝜂_𝑖𝑗(𝑡); time delays 𝜏_𝑖𝑗(𝑡)∶[0,∞)→[0,∞), 𝑖∈𝑉, 𝑗∈𝑁_𝑖, are piecewise continuous and bounded functions; 𝑎(𝑡) and 𝜎_𝑖𝑗 are defined as above; Δ𝑎_𝑖𝑗(𝑡), 𝑖,𝑗∈𝑉 are parametric uncertainties satisfying |Δ𝑎_𝑖𝑗(𝑡)|≤𝜖_𝑖𝑗 (𝜖_𝑖𝑗>0) and Δ𝑎_𝑖𝑖(𝑡)=0.

For the sake of convenience, let 𝜏_𝑖𝑗(𝑡)∈{𝜏_𝑘(𝑡)∶𝑘=1,2,…,𝑚}. Denote the 𝑖th row of the matrix 𝐴(𝑡)=𝐴+Δ𝐴(𝑡) by 𝛼_𝑖(𝑡),Σ_𝑖=diag𝜎_𝑖1,𝜎_𝑖2,…,𝜎_𝑖𝑛,Σ_𝑡=diag𝛼₁(𝑡)Σ₁,𝛼₂(𝑡)Σ₂,…,𝛼_𝑛(𝑡)Σ_𝑛,𝜂_𝑖(𝑡)=𝜂_𝑖1(𝑡),𝜂_𝑖2(𝑡),…,𝜂_𝑖𝑛(𝑡)^𝑇,𝜂(𝑡)=𝜂^𝑇₁(𝑡),𝜂^𝑇₂(𝑡),…,𝜂^𝑇_𝑛(𝑡)^𝑇, (2.5) and 𝑥(𝑡)=[𝑥₁(𝑡),𝑥₂(𝑡),…,𝑥_𝑛(𝑡)]^𝑇. Substituting the control (2.4) into the system (2.1) leads to𝑥̇(𝑡)=−𝑎(𝑡)^𝑚_𝑘=1𝐿_𝑘(𝑡)𝑥𝑡−𝜏_𝑘(𝑡)+𝑎(𝑡)Σ_𝑡𝜂(𝑡), (2.6) where the 𝑛×𝑛-dimensional matrix 𝐿_𝑘(𝑡)=[𝑙^(𝑘)_𝑖𝑗(𝑡)] is defined as followings:𝑙^(𝑘)_𝑖𝑗(𝑡)=−𝑎_𝑖𝑗−Δ𝑎_𝑖𝑗(𝑡),𝑗≠𝑖,𝜏_𝑘(⋅)=𝜏_𝑖𝑗(⋅)0,𝑗≠𝑖,𝜏_𝑘(⋅)≠𝜏_𝑖𝑗(⋅)−^𝑛_𝑝=1𝑙^(𝑘)_𝑖𝑝(𝑡),𝑗=𝑖. (2.7) It is easy to see𝐿_𝑘(𝑡)𝟏=0,^𝑚_𝑘=1𝐿_𝑘(𝑡)=𝐿(𝐺)+Δ𝐿(𝑡), (2.8) where Δ𝐿(𝑡)=[Δ𝑙_𝑖𝑗(𝑡)] with Δ𝑙_𝑖𝑗(𝑡)=∑^𝑛_𝑘=1Δ𝑎_𝑖𝑘(𝑡) for 𝑖=𝑗 and Δ𝑙_𝑖𝑗(𝑡)=−Δ𝑎_𝑖𝑗(𝑡) for 𝑖≠𝑗.

Let 𝑦=𝐸𝑥, 𝐸, and 𝐹 be defined by (2.1). From (2.6), we have the following reduced-order system:𝑦̇(𝑡)=𝑎(𝑡)^𝑚_𝑘=1𝐸𝐿_𝑘(𝑡)𝐹𝑦𝑡−𝜏_𝑘(𝑡)+𝑎(𝑡)𝐸Σ_𝑡𝜂(𝑡). (2.9) It is a system driven by an 𝑛²-dimensional standard white noise, which can be written in the form of the Itô stochastic differential equation𝑑𝑦(𝑡)=𝑎(𝑡)^𝑚_𝑘=1𝐸𝐿_𝑘(𝑡)𝐹𝑦𝑡−𝜏_𝑘(𝑡)𝑑𝑡+𝑎(𝑡)𝐸Σ_𝑡𝑑𝑤(𝑡), (2.10) where 𝑤(𝑡)=[𝑤₁₁(𝑡),…,𝑤_1𝑛(𝑡),…,𝑤_𝑛𝑛(𝑡)]^𝑇 is an 𝑛²-dimensional standard Brownian motion. Without loss of generality, we let 𝜏_𝑘(𝑡)≡𝜏(𝑡) for 𝑘=1,2,…,𝑚 since the case of multiple delays can be similarly studied. In this case, the protocol (2.4) takes the following simple form:𝑢_𝑖(𝑡)=_{𝑗∈𝑁𝑖}𝑎(𝑡)𝑎_𝑖𝑗+Δ𝑎_𝑖𝑗(𝑡)𝑦_𝑖𝑗(𝑡−𝜏(𝑡))−𝑥_𝑖(𝑡−𝜏(𝑡)), (2.11) and the system (2.10) reduces to𝑑𝑦(𝑡)=𝑎(𝑡)𝐸(𝐿+Δ𝐿(𝑡))𝐹𝑦(𝑡−𝜏(𝑡))𝑑𝑡+𝑎(𝑡)𝐸Σ_𝑡𝑑𝑤(𝑡). (2.12)

In the sequel, we assume that the parametric uncertainty Δ𝐿(𝑡) to be of the formΔ𝐿(𝑡)=𝐷𝐺(𝑡)𝐻, (2.13) where 𝐷 and 𝐻 are constant matrices with appropriate dimensions, and 𝐺(𝑡) is an unknown matrix satisfying 𝐺^𝑇(𝑡)𝐺(𝑡)≤𝐼. Suppose also that there exists at least one edge (𝑗,𝑖)∈𝐸 such that 𝜎_𝑖𝑗>0, and 𝜏(𝑡) is a piecewise continuous function on [0,∞) and 0≤𝜏(𝑡)≤ℎ, where ℎ>0 is a constant. Here, the initial function of the system (2.12) is assumed to satisfy 𝜙(𝑡)≡𝑥(0)=[𝑥₁(0)⋯𝑥_𝑛(0)]^𝑇 on [−ℎ,0].

We say the system (2.2) under protocol (2.11) asymptotically achieves mean square consensus if lim_𝑡→∞𝐸[(𝑥_𝑖(𝑡)−𝑥_𝑗(𝑡))²]=0 for all 𝑖,𝑗∈𝑉 and 𝑖≠𝑗.

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 𝐺, which can be used to analyze the feasibility of the given consensus condition.

Lemma 3.1. If 𝐺 has a spanning tree, then there exist matrices 𝑃>0, 𝑄>0, 𝑃₁, 𝑃₂ of compatible dimensions, constants ℎ>0, and 𝛾>0 such that ΦΦ=¹¹Φ₁₂𝑃^𝑇₁𝐿𝑃^𝑇₁𝐿∗Φ₂₂𝑃^𝑇₂𝐿𝑃^𝑇₂𝐿∗∗−ℎ⁻¹𝑄0∗∗∗−𝛾𝐼_𝑛<0, (3.1) where Φ₁₁=𝐿^𝑇𝑃₁+𝑃^𝑇₁𝐿, Φ₁₂=𝑃−𝑃^𝑇₁+𝐿^𝑇𝑃₂, Φ₂₂=−𝑃^𝑇₂−𝑃₂+ℎ𝑄, and 𝐿=𝐸𝐿(𝐺)𝐹, 𝐸, and 𝐹 is defined by (2.1).

Proof. By Lemma 3.1, we have that there exists a matrix 𝑃>0 such that 𝐿^𝑇𝑃+𝑃𝐿<0 if 𝐺 has a spanning tree. Based on the Finsler Lemma [20], there exist matrices 𝑃₁ and 𝑃₂ such that 𝐿^𝑇𝑃₁+𝑃^𝑇₁𝐿𝑃−𝑃^𝑇₁+𝐿^𝑇𝑃₂∗−𝑃^𝑇₂−𝑃₂<0. (3.2) Let 𝑄=𝐼_𝑛. Then, (3.2) implies (3.1) by choosing ℎ and 𝛾 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 𝑧(𝑡) on [𝑡−𝜏(𝑡),𝑡] and matrix 𝑊>0, where 𝑡∈[0,∞) and 0≤𝜏(𝑡)≤ℎ, the following inequality holds: ^𝑡_{𝑡−𝜏(𝑡)}𝑧(𝑢)𝑑𝑢^𝑇𝑊^𝑡_{𝑡−𝜏(𝑡)}𝑧(𝑢)𝑑𝑢≤ℎ^𝑡_{𝑡−𝜏(𝑡)}𝑧^𝑇(𝑢)𝑊𝑧(𝑢)𝑑𝑢. (3.3)

Lemma 3.3 (see [22]). Let 𝑈, 𝑉, and 𝐹 be real matrices of appropriate dimensions with 𝐹^𝑇𝐹≤𝐼, then for any scalar 𝜖>0, one has𝑈𝐹𝑉+𝑉^𝑇𝐹^𝑇𝑈^𝑇≤𝜖⁻¹𝑈𝑈^𝑇+𝜖𝑉^𝑇𝑉.

Now, let us present the main result of this paper. We assume that the positive consensus-gain function 𝑎(𝑡) satisfies one of the following assumptions:(A1)∫^∞₀𝑎(𝑡)𝑑𝑡=+∞, ∫^∞₀𝑎²(𝑡)𝑑𝑡<∞, and 𝑎(𝑡)≤1 for sufficiently large 𝑡;(A2)∫^∞₀𝑎(𝑡)𝑑𝑡=+∞, lim_𝑡→∞𝑎(𝑡)=0.

Theorem 3.4. Assume that (A1) or (A2) holds and 0≤𝜏(𝑡)≤ℎ. If there exist matrices 𝑃>0, 𝑄>0, 𝑃₁, 𝑃₂ of compatible dimensions, constants 𝛾>0 and 𝜖>0 such that Φ₁₁Φ₁₂Φ₁₃Φ₁₄𝑃^𝑇₁𝐷∗Φ₂₂𝑃^𝑇₂𝐿𝑃^𝑇₂𝐿𝑃^𝑇₂𝐷∗∗Φ₃₃Φ₃₄0∗∗∗Φ₄₄0∗∗∗∗−𝜖𝐼<0, (3.4) where Φ₁₁=Φ₁₁+𝜖𝐹^𝑇𝐻^𝑇𝐻𝐹, Φ₁₂=Φ₁₂, Φ₁₃=𝑃^𝑇₁𝐿+𝜖𝐹^𝑇𝐻^𝑇𝐻𝐹, Φ₁₄=𝑃^𝑇₁𝐿+𝜖𝐹^𝑇𝐻^𝑇𝐻𝐹, Φ₂₂=Φ₂₂, Φ₃₃=−ℎ⁻¹𝑄+𝜖𝐹^𝑇𝐻^𝑇𝐻𝐹, Φ₃₄=𝜖𝐹^𝑇𝐻^𝑇𝐻𝐹, Φ₄₄=−𝛾𝐼+𝜖𝐹^𝑇𝐻^𝑇𝐻𝐹, Φ₁₁, Φ₁₂, and Φ₂₂ are defined as in Lemma 3.1, 𝐸 and 𝐹 is defined by (2.1), and 𝐷=𝐸𝐷, 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 𝜉(𝑡)=𝐸(𝐿+Δ𝐿(𝑡))𝐹𝑦(𝑡−𝜏(𝑡))=𝐿_𝑡𝑦(𝑡−𝜏(𝑡)). (3.5) Then, (2.12) reduces to 𝑑𝑦(𝑡)=𝑎(𝑡)𝜉(𝑡)𝑑𝑡+𝑎(𝑡)𝐸Σ𝑑𝑤(𝑡). (3.6) Note that (3.4) holds. We can choose a constant 𝜆>0 sufficiently small such that Φ₁₁+𝜆𝑃Φ₁₂𝑃^𝑇₁𝐿Φ₁₄𝑃^𝑇₁𝐷∗Φ₂₂𝑃^𝑇₂𝐿𝑃^𝑇₂𝐿𝑃^𝑇₂𝐷∗∗−ℎ⁻¹Φ𝑄00∗∗∗₄₄0∗∗∗∗−𝜖𝐼<0, (3.7) where ℎ=𝑒^−𝜆ℎℎ⁻¹.
Let 𝑉(𝑡)=𝑦^𝑇(𝑡)𝑃𝑦(𝑡)+^𝑡_𝑡−ℎ𝑒^{−𝜆∫𝑡𝑠𝑎(𝑢)𝑑𝑢}(𝑠−𝑡+ℎ)𝑎²(𝑠)𝜉^𝑇(𝑠)𝑄𝜉(𝑠)𝑑𝑠+𝛽^𝑡_𝑡−ℎ𝑒^{−𝜆∫t𝑠𝑎(𝑢)𝑑𝑢}(𝑠−𝑡+ℎ)𝑎²(𝑠)𝑑𝑠, (3.8) where 𝛽>0 is a constant to be determined. By the Itô formula, we have 𝑑𝑉(𝑡)=2𝑎(𝑡)𝑦^𝑇(𝑡)𝑃𝜉(𝑡)+𝑐(𝑡)𝑎²(𝑡)𝑑𝑡+2𝑎(𝑡)𝑦^𝑇(𝑡)𝑃𝐸Σ_𝑡(𝑡)+𝑑𝑤ℎ𝑎²(𝑡)𝜉^𝑇(𝑡)𝑄𝜉(𝑡)−^𝑡_𝑡−ℎ𝑒^{−𝜆∫𝑡𝑠𝑎(𝑢)𝑑𝑢}𝑎²(𝑠)𝜉^𝑇(𝑠)(𝑠)(𝑡)𝑄𝜉𝑑𝑠−𝜆𝑎^𝑡_𝑡−ℎ𝑒^{−𝜆∫𝑡𝑠𝑎(𝑢)𝑑𝑢}(𝑠−𝑡+ℎ)𝑎²(𝑠)𝜉^𝑇(𝑠)𝑄𝜉(𝑠)𝑑𝑠𝑑𝑡−𝛽^𝑡_𝑡−ℎ𝑒^{−𝜆∫𝑡𝑠𝑎(𝑢)𝑑𝑢}𝑎²(𝑠)𝑑𝑠−𝜆𝛽𝑎(𝑡)^𝑡_𝑡−ℎ𝑒^{−𝜆∫𝑡𝑠𝑎(𝑢)𝑑𝑢}(𝑠−𝑡+ℎ)𝑎²(𝑠)𝑑𝑠𝑑𝑡, (3.9) where 𝑐(𝑡)=tr(Σ^𝑇_𝑡𝐸^𝑇𝑃𝐸Σ_𝑡)+𝛽ℎ. It is not difficult to see that there exists a scalar 𝑚₁>0 such that 𝑐(𝑡)≤𝑚₁ due to the fact that |Δ𝑎_𝑖𝑗(𝑡)|≤𝜖_𝑖𝑗. Thus, 𝑑𝑉(𝑡)+𝜆𝑎(𝑡)𝑉(𝑡)𝑑𝑡=𝐿𝑉(𝑡)𝑑𝑡+2𝑎(𝑡)𝑦^𝑇(𝑡)𝑃𝐸Σ_𝑡𝑑𝑤(𝑡), (3.10) where 𝐿𝑉(𝑡)=𝜆𝑎(𝑡)𝑦^𝑇(𝑡)𝑃𝑦(𝑡)+2𝑎(𝑡)𝑦^𝑇(𝑡)𝑃𝜉(𝑡)+𝑐(𝑡)𝑎²(𝑡)+ℎ𝑎²(𝑡)𝜉^𝑇(𝑡)𝑄𝜉(𝑡)−^𝑡_𝑡−ℎ𝑒^{−𝜆∫𝑡𝑠𝑎(𝑢)𝑑𝑢}𝑎²(𝑠)𝜉^𝑇(𝑠)𝑄𝜉(𝑠)𝑑𝑠−𝛽^𝑡_𝑡−ℎ𝑒^{−𝜆∫𝑡𝑠𝑎(𝑢)𝑑𝑢}𝑎²(𝑠)𝑑𝑠. (3.11) Without loss of generality, say 𝑎(𝑡)≤1 for 𝑡≥0. Then, we have 𝐿𝑉(𝑡)≤𝜆𝑎(𝑡)𝑦^𝑇(𝑡)𝑃𝑦(𝑡)+2𝑎(𝑡)𝑦^𝑇(𝑡)𝑃𝜉(𝑡)+𝑚₁𝑎²(𝑡)+ℎ𝑎(𝑡)𝜉^𝑇(𝑡)𝑄𝜉(𝑡)−𝑒^−𝜆ℎ𝑎(𝑡)^𝑡_{𝑡−𝜏(𝑡)}𝑎²(𝑠)𝜉^𝑇(𝑠)𝑄𝜉(𝑠)𝑑𝑠−𝛽𝑒^{−𝜆ℎ𝑡}_{𝑡−𝜏(𝑡)}𝑎²(𝑠)𝑑𝑠. (3.12)
On the other hand, integrating (3.7) from 𝑡−𝜏(𝑡) to 𝑡 yields 𝑦(𝑡)−𝑦(𝑡−𝜏(𝑡))=^𝑡_{𝑡−𝜏(𝑡)}𝑎(𝑠)𝜉(𝑠)𝑑𝑠+^𝑡_{𝑡−𝜏(𝑡)}𝑎(𝑠)𝐸Σ_𝑠𝑑𝑤(𝑠). (3.13) Therefore, by the definition of 𝜉(𝑡), we have 0=𝐿_𝑡𝑦((𝑡(𝑡)=𝐿𝑡−𝜏))−𝜉_𝑡𝑦(𝑡)−𝜉(𝑡)−𝐿_𝑡^𝑡_{𝑡−𝜏(𝑡)}𝑎(𝑠)𝜉(𝑠)−𝐿𝑑𝑠_{𝑡𝑡𝑡−𝜏(𝑡)}𝑎(𝑠)𝐸Σ_𝑠𝑑𝑤(𝑠), (3.14) which implies that 0=2𝑎(𝑡)𝑦^𝑇(𝑡)𝑃^𝑇₁+𝜉^𝑇(𝑡)𝑃^𝑇₂×𝐿_𝑡𝑦(𝑡)−^𝑡_{𝑡−𝜏(𝑡)}𝑎(𝑠)𝜉(𝑠)𝑑𝑠−^𝑡_{𝑡−𝜏(𝑡)}𝑎(𝑠)𝐸Σ𝑑𝑤(𝑠)−𝜉(𝑡). (3.15) By Lemma 3.2, we have ^𝑡_{𝑡−𝜏(𝑡)}𝑎²(𝑠)𝜉^𝑇(𝑠)𝑄𝜉(𝑠)𝑑𝑠≥ℎ^−1𝑡_{𝑡−𝜏(𝑡)}𝑎(𝑠)𝜉(𝑠)𝑑𝑠^𝑇𝑄^𝑡_{𝑡−𝜏(𝑡)}𝑎(𝑠)𝜉(𝑠)𝑑𝑠. (3.16) Using the basic inequality −2𝑢^𝑇𝑣≤𝛾𝑢^𝑇𝑢+𝛾⁻¹𝑣^𝑇𝑣 for any vector 𝑢 and 𝑣, we have −2𝑦^𝑇(𝑡)𝑃^𝑇₁+𝜉^𝑇(𝑡)𝑃^𝑇₂𝐿_𝑡^𝑡_{𝑡−𝜏(𝑡)}𝑎(𝑠)𝐸Σ_𝑠(𝑠)𝑦𝑑𝑤=−2^𝑇(𝑡)𝜉^𝑇(𝑡)𝑃^𝑇₁𝑃^𝑇₂𝐿_𝑡^𝑡_{𝑡−𝜏(𝑡)}𝑎(𝑠)𝐸Σ_𝑠𝑑𝑤(𝑠)≤𝛾⁻¹𝑦^𝑇(𝑡)𝜉^𝑇(𝑡)𝑃^𝑇₁𝑃^𝑇₂𝐿_𝑡𝐿^𝑇_𝑡𝑃₁𝑃₂𝑦(𝑡)𝜉(𝑡)+𝛾^𝑡_{(𝑡)𝑡−𝜏}𝑎(𝑠)𝐸Σ_𝑠𝑑𝑤(𝑠)^𝑇𝑡_{𝑡−𝜏(𝑡)}𝑎(𝑠)𝐸Σ_𝑠𝑑𝑤(𝑠). (3.17) Substituting (3.15)–(3.17) into (3.12) gives 𝐿𝑉(𝑡)≤𝑎(𝑡)𝜔^𝑇(𝑡)Ω(𝑡)𝜔(𝑡)+𝑚₁𝑎²(𝑡)−𝛽𝑒^{−𝜆ℎ𝑡}_{𝑡−𝜏(𝑡)}𝑎²(𝑠)𝑑𝑠+𝛾^𝑡_{𝑡−𝜏(𝑡)}𝑎(𝑠)𝐸Σ𝑑𝑤(𝑠)^𝑇𝑡_{𝑡−𝜏(𝑡)}𝑎(𝑠)𝐸Σ𝑑𝑤(𝑠), (3.18) where 𝜔(𝑡)=[𝑦^𝑇(𝑡)𝜉^𝑇(𝑡)−∫^𝑡_{𝑡−𝜏(𝑡)}𝑎(𝑠)𝜉^𝑇(𝑠)𝑑𝑠]^𝑇, and Ω(𝑡)=Ω₁₁Ω₁₂𝑃^𝑇₁𝐿_𝑡∗Ω₂₂𝑃^𝑇₂𝐿_𝑡∗∗−ℎ⁻¹𝑄+𝛾⁻¹𝑃^𝑇₁𝑃^𝑇₂𝐿_𝑡𝐿^𝑇_𝑡𝑃₁𝑃₂, (3.19) with Ω₁₁=𝐿^𝑇_𝑡𝑃₁+𝑃^𝑇₁𝐿_𝑡+𝜆𝑃, Ω₁₂=𝑃−𝑃^𝑇₁+𝐿^𝑇_𝑡𝑃₂, Ω₂₂=Φ₂₂. It is easy to see that Ω(𝑡)<0 if and only if 𝑃Φ+^𝑇1𝐷𝑃^𝑇₂𝐷00𝐺(𝑡𝐻𝐻+𝐻)𝐻0^𝑇0𝐻^𝑇𝐻^𝑇𝐺(𝑡)𝐷^𝑇𝑃₁𝐷^𝑇𝑃₂00<0, (3.20) where Φ is defined by (3.1) and 𝐻=𝐻𝐹. By Lemma 3.3, we have that (3.20) is implied by (3.7). Thus, (3.7) yields that Ω(𝑡)<0 for 𝑡≥0.
Note that Ω(𝑡)<0 for 𝑡≥0, and 𝐸^𝑡_{𝑡−𝜏(𝑡)}𝑎(𝑠)𝐸Σ_𝑠𝑑𝑤(𝑠)^𝑇𝑡_{𝑡−𝜏(𝑡)}𝑎(𝑠)𝐸Σ_𝑠𝑑𝑤(𝑠)=^𝑡_{𝑡−𝜏(𝑡)}𝑑(𝑠)𝑎²(𝑠)𝑑𝑠, (3.21) where 𝑑(𝑡)=tr(Σ^𝑇_𝑡𝐸^𝑇𝐸Σ_𝑡) satisfying 𝑑(𝑡)≤𝑚₂ for some 𝑚₂>0. By choosing 𝛽=𝛾𝑚₂𝑒^𝜆ℎ, we get from (3.10) and (3.18) that 𝐸[𝑉(𝑡)]≤𝐸[𝑉(0)]−𝜆^𝑡₀𝑎(𝑠)𝐸[𝑉(𝑠)]𝑑𝑠+𝑚₁^𝑡₀𝑎²(𝑠)𝑑𝑠. (3.22) By the comparison theorem [23], we have 𝐸[𝑉(𝑡)]≤𝐸[𝑉(0)]𝑒^{−𝜆∫t0𝑎(𝑠)𝑑𝑠}+𝑚₁^𝑡₀𝑒^{−𝜆∫𝑡𝑠𝑎(𝑢)𝑑𝑢}𝑎²(𝑠)𝑑𝑠. (3.23) If (A1) holds, that is, ∫^∞₀𝑎(𝑠)𝑑𝑠=∞ and ∫^∞₀𝑎²(𝑠)𝑑𝑠<∞, then, similar to the proof of Theorem  3.2 in [12], we can conclude from (3.23) that lim_𝑡→∞𝐸[𝑉(𝑡)]=0. If (A2) holds, by the L’Hopital rule, we have lim_𝑡→∞^𝑡₀𝑒^{−𝜆∫𝑡𝑠𝑎(𝑢)𝑑𝑢}𝑎²(𝑠)𝑑𝑠=lim_𝑡→∞𝑎(𝑡)𝜆=0, (3.24) which also implies lim_𝑡→∞𝐸[𝑉(𝑡)]=0. Based on the construction of 𝑉(𝑡) and the transformation 𝑦=𝐸𝑥, we have lim_𝑡→∞𝐸𝑥₁(𝑡)−𝑥_𝑖+1(𝑡)²=lim_𝑡→∞𝐸𝑦²_𝑖(𝑡)=0,𝑖=1,2,…,𝑛−1. (3.25) It implies that lim_𝑡→∞𝐸[(𝑥_𝑖(𝑡)−𝑥_𝑗(𝑡))²]=0 for 𝑖,𝑗∈𝑉 and 𝑖≠𝑗. The proof is complete.

Remark 3.5. By Lemma 3.1, we can easily see that (3.4) holds for appropriate constant ℎ>0 and admissible uncertainties (e.g., ‖𝐻‖ is sufficiently small) if 𝐺 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 𝐷 and 𝐻, 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 𝐺 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 𝑥_𝑖0 satisfying 𝑥̇_𝑖0=0. Without loss of generality, consider the following protocol:𝑢₁(𝑡)=_𝑗∈𝑁1𝑎(𝑡)𝑎_1𝑗𝑥_𝑗(𝑡)−𝑥₁(𝑡,𝑢_𝑖)(𝑡)=_{𝑗∈𝑁𝑖}𝑎(𝑡)𝑎_𝑖𝑗+Δ𝑎_𝑖𝑗(𝑡)𝑧_𝑖𝑗(𝑡)−𝑥_𝑖(𝑡),𝑖=2,…,𝑛, (4.1) where 𝑧_𝑖𝑗(𝑡)=𝑥_𝑗(𝑡−𝜏(𝑡))+𝜎_𝑖𝑗𝜂_𝑖𝑗, 𝜏, 𝑎, 𝑎_𝑖𝑗, Δ𝑎_𝑖𝑗, 𝜎_𝑖𝑗 and 𝜂_𝑖𝑗 are defined as above. The case of multiple time-varying delays can be similarly discussed.

Note that𝑥_𝑗(𝑡−𝜏(𝑡))−𝑥_𝑖(𝑡)=𝑥_𝑗(𝑡−𝜏(𝑡))−𝑥₁(𝑡−𝜏(𝑡))+𝑥₁(𝑡)−𝑥_𝑖(𝑡−𝑥₁)(𝑡)−𝑥₁((𝑡𝑥𝑡−𝜏)),𝑖,𝑗=2,3,…,𝑛,₁(𝑡)−𝑥₁(𝑡−𝜏(𝑡))=^𝑡_{𝑡−𝜏(𝑡)}𝑥̇₁(𝑠)𝑑𝑠. (4.2) If we set 𝑦=𝐸𝑥, where 𝐸 is defined as above, then we have the following reduced-order system:𝑦̇(𝑡)=𝑎(𝑡)𝐿₁(𝑡)𝑦(𝑡)+𝐿₂(𝑡)𝑦(𝑡−𝜏(𝑡))+𝐿₃(𝑡)^𝑡_{𝑡−𝜏(𝑡)}𝑎(𝑠)𝑦(𝑠)𝑑𝑠+𝑎(𝑡)Σ_𝑡𝜂(𝑡), (4.3) where 𝐿_𝑖(𝑡)=𝐿_𝑖+Δ𝐿_𝑖(𝑡) for 𝑖=1,2,3,𝐿₁=−diag^𝑛_𝑗=1𝑎_2𝑗,^𝑛_𝑗=1𝑎_3𝑗,…,^𝑛_𝑗=1𝑎_𝑛𝑗−𝟏𝑎₁₂𝑎₁₃⋯𝑎_1𝑛,Δ𝐿₁(𝑡)=−diag^𝑛_𝑗=1Δ𝑎_2𝑗(𝑡),^𝑛_𝑗=1Δ𝑎_3𝑗(𝑡),…,^𝑛_𝑗=1Δ𝑎_𝑛𝑗(𝑡),𝐿₂=𝑎₂₂𝑎₂₃⋯𝑎_2𝑛𝑎₃₂𝑎₃₃⋯𝑎_3𝑛𝑎⋮⋮⋱⋮_𝑛2𝑎_𝑛3⋯𝑎_𝑛𝑛,Δ𝐿₂(𝑡)=Δ𝑎₂₂(𝑡)Δ𝑎₂₃(𝑡)⋯Δ𝑎_2𝑛(𝑡)Δ𝑎₃₂(𝑡)Δ𝑎₃₃(𝑡)⋯Δ𝑎_3𝑛(𝑡)⋮⋮⋱⋮Δ𝑎_𝑛2(𝑡)Δ𝑎_𝑛3(𝑡)⋯Δ𝑎_𝑛𝑛(𝑡),𝐿₃=−diag^𝑛_𝑗=1𝑎_2𝑗,^𝑛_𝑗=1𝑎_3𝑗,…,^𝑛_𝑗=1𝑎_𝑛𝑗𝟏𝑎₁₂𝑎₁₃⋯𝑎_1𝑛,Δ𝐿₃(𝑡)=−diag^𝑛_𝑗=1Δ𝑎_2𝑗(𝑡),^𝑛_𝑗=1Δ𝑎_3𝑗(𝑡),…,^𝑛_𝑗=1Δ𝑎_𝑛𝑗(𝑡)𝟏𝑎₁₂𝑎₁₃⋯𝑎_1𝑛, (4.4)Σ_𝑡=−diag{𝛼₂(𝑡)Σ₂,…,𝛼_𝑛(𝑡)Σ_𝑛}, 𝜂(𝑡)=[𝜂^𝑇₂(𝑡)𝜂^𝑇₃(𝑡)⋯𝜂^𝑇_𝑛(𝑡)]^𝑇, 𝛼_𝑖(𝑡), Σ_𝑖 and 𝜂_𝑖 are defined as above. Therefore, (4.3) can be written in the form of the Itô stochastic differential equation.𝑑𝑦(𝑡)=𝑎(𝑡)𝐿₁(𝑡)𝑦(𝑡)+𝐿₂(𝑡)𝑦(𝑡−𝜏(𝑡))+𝐿₃(𝑡)^𝑡_{𝑡−𝜏(𝑡)}𝑎(𝑠)𝑦(𝑠)𝑑𝑠𝑑𝑡+𝑎(𝑡)Σ_𝑡𝑑𝑤(𝑡). (4.5) It is not difficult to verify that 𝐿₁+𝐿₂=𝐸𝐿(𝐺)𝐹. In the following, we assume that the uncertainties Δ𝐿_𝑖(𝑡) to be of the formΔ𝐿₁(𝑡)+Δ𝐿₂(𝑡)=𝐷₁𝐹(𝑡)𝐻₁,Δ𝐿_𝑖(𝑡)=𝐷_𝑖𝐹(𝑡)𝐻_𝑖,𝑖=2,3, (4.6) where 𝐷_𝑖, 𝐻_𝑖 are constant matrices of appropriate dimensions and 𝐹(𝑡) satisfies (2.13).

Theorem 4.1. Assume that (A1) or (A2) holds and 0≤𝜏(𝑡)≤ℎ. If there exist matrices 𝑃>0, 𝑄>0, 𝑆>0, 𝑃₁, 𝑃₂, and 𝑃₃ of compatible dimensions, constants ℎ>0, 𝛾>0 and 𝜖_𝑖>0 for 𝑖=1,2,3 such that Ψ₁₁Ψ₁₂𝑃^𝑇₁𝐿₂Ψ₁₄𝑃^𝑇₁𝐿₂𝑃^𝑇₁𝐷₁𝑃^𝑇₁𝐷₂𝑃^𝑇₁𝐷₃∗Ψ₂₂𝑃^𝑇₂𝐿₂Ψ₂₄𝑃^𝑇₂𝐿₂𝑃^𝑇₂𝐷₁𝑃^𝑇₂𝐷₂𝑃^𝑇₂𝐷₃∗∗Ψ₃₃𝐿^𝑇₂𝑃₃𝜖₂𝐸^𝑇₂𝐸₂000∗∗∗Ψ₄₄𝑃^𝑇₃𝐿₂𝑃^𝑇₃𝐷₁𝑃^𝑇₃𝐷₂0∗∗∗∗Ψ₅₅000∗∗∗∗∗−𝜖₁𝐼00∗∗∗∗∗∗−𝜖₂𝐼0∗∗∗∗∗∗∗−𝜖₃𝐼<0, (4.7) where Ψ₁₁=(𝐿₁+𝐿₂)^𝑇𝑃₁+𝑃^𝑇₁(𝐿₁+𝐿₂)+ℎ𝑆+𝜖₁𝐸^𝑇₁𝐸₁, Ψ₁₂=𝑃−𝑃^𝑇₁+(𝐿₁+𝐿₂)^𝑇𝑃₂, Ψ₁₄=𝑃^𝑇₁𝐿₃+(𝐿₁+𝐿₂)^𝑇𝑃₃, Ψ₂₂=−𝑃^𝑇₂−𝑃₂+ℎ𝑄, and Ψ₂₄=𝑃^𝑇₂𝐿₃−𝑃₃, Ψ₃₃=−ℎ⁻¹𝑄+𝜖₂𝐸^𝑇₂𝐸₂, Ψ₄₄=−ℎ⁻¹𝑆+𝜖𝐸^𝑇₃𝐸₃, Ψ₅₅=−𝛾𝐼+𝜖₂𝐸^𝑇₂𝐸₂, then the system (2.2) under protocol (4.1) asymptotically achieves mean square consensus for all admissible uncertainties satisfying (2.13).

Proof. Let 𝜉(𝑡)=𝐿₁(𝑡)𝑦(𝑡)+𝐿₂(𝑡)𝑦(𝑡−𝜏(𝑡))+𝐿₃(𝑡)^𝑡_{𝑡−𝜏(𝑡)}𝑎(𝑠)𝑦(𝑠)𝑑𝑠. (4.8) Then, (4.5) reduces to 𝑑𝑦(𝑡)=𝑎(𝑡)𝜉(𝑡)+𝑎(𝑡)Σ_𝑡𝑑𝑤(𝑡). (4.9) Choose the Lyapunov function as the following: 𝑉(𝑡)=𝑦^𝑇(𝑡)𝑃𝑦(𝑡)+^𝑡_𝑡−ℎ𝑒^{−𝜆∫𝑡𝑠𝑎(𝑢)𝑑𝑢}(𝑠−𝑡+ℎ)𝑎²(𝑠)𝜉^𝑇(𝑠)(𝑠)+𝑄𝜉𝑑𝑠_{𝑡𝑡−ℎ}𝑒^{−𝜆∫𝑡𝑠𝑎(𝑢)𝑑𝑢}(𝑠−𝑡+ℎ)𝑎²(𝑠)𝑦^𝑇(𝑠)𝑆𝑦(𝑠)𝑑𝑠×𝛽^𝑡_𝑡−ℎ𝑒^{−𝜆∫𝑡𝑠𝑎(𝑢)𝑑𝑢}(𝑠−𝑡+ℎ)𝑎²(𝑠)𝑑𝑠, (4.10) where 𝛽>0 is an appropriate constant to be determined. Without loss of generality, say 𝑎(𝑡)≤1 for 𝑡≥0. By (4.9) and the Itô formula, we have 𝑑𝑉(𝑡)+𝜆𝑎(𝑡)𝑉(𝑡)𝑑𝑡=𝐿𝑉(𝑡)𝑑𝑡+2𝑎(𝑡)𝑦^𝑇(𝑡)𝑃Σ_𝑡𝑑𝑤(𝑡), (4.11) where 𝐿𝑉(𝑡)≤𝜆𝑎(𝑡)𝑦^𝑇(𝑡)𝑃𝑦(𝑡)+2𝑎(𝑡)𝑦^𝑇(𝑡)𝑃𝜉(𝑡)+𝑐̃(𝑡)𝑎²(𝑡)+ℎ𝑎(𝑡)𝜉^𝑇(𝑡)𝑄𝜉(𝑡)+ℎ𝑎(𝑡)𝑦^𝑇(𝑡)𝑆𝑦(𝑡)−𝑒^−𝜆ℎ𝑎(𝑡)^𝑡_{𝑡−𝜏(𝑡)}𝑎²(𝑠)𝜉^𝑇(𝑠)𝑄𝜉(𝑠)𝑑𝑠−𝑒^−𝜆ℎ𝑎(𝑡)^𝑡_{𝑡−𝜏(𝑡)}𝑎²(𝑠)𝑦^𝑇(𝑠)𝑆𝑦(𝑠)𝑑𝑠−𝛽𝑒^{−𝜆ℎ𝑡}_{𝑡−𝜏(𝑡)}𝑎²(𝑠), (4.12) and 𝑐(̃𝑡)=tr(Σ^𝑇_𝑡𝑃Σ_𝑡)+𝛽ℎ. On the other hand, by the definition of 𝜉(𝑡), we have 0=𝐿₁(𝑡)𝑦(𝑡)+𝐿₂(𝑡)𝑦(𝑡−𝜏(𝑡))+𝐿₃(𝑡)^𝑡_{𝑡−𝜏(𝑡)}𝑎(𝑠)𝑦(𝑠)(𝑡)=𝐿𝑑𝑠−𝜉₁(𝑡)+𝐿₂(𝑡)𝑦(𝑡)+𝐿₃(𝑡)^𝑡_{𝑡−𝜏(𝑡)}𝑎(𝑠)𝑦(𝑠)𝑑𝑠−𝜉(𝑡)−𝐿₂(𝑡)^𝑡_{𝑡−𝜏(𝑡)}𝑎(𝑠)𝜉(𝑠)𝑑𝑠−𝐿₂(𝑡)^𝑡_{𝑡−𝜏(𝑡)}𝑎(𝑠)Σ𝑑𝑤(𝑠). (4.13) 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 ℎ>0 and admissible uncertainties if 𝐺 has a spanning tree. For given matrices 𝐷_𝑖 and 𝐻_𝑖(𝑖=1,2,3), the tolerable upper bound of delay can be derived from (4.7).

5. Simulation Results

Consider a digraph 𝐺=(𝑉,𝐸,𝐴) with six nodes and 0-1 weights, where 𝑎₂₁=𝑎₃₂=𝑎₄₃=𝑎₅₄=𝑎₆₅=𝑎₁₆=1. It is evident that 𝐺 has a spanning tree. Let 𝑎(𝑡)=1/(𝑡+1). Using Matlab to solve (3.4) without uncertainties yields that ℎ≤0.4995. Thus, by Theorem 3.4, we have that protocol (2.11) asymptotically solves mean square consensus for any time-varying delay 𝜏(𝑡) satisfying 0≤𝜏(𝑡)≤0.4995 if (A1) or (A2) holds. Let 𝑎(𝑡)=1/(𝑡+1) and the intensity of the measurement noises 𝜎₂₁=𝜎₃₂=𝜎₄₃=𝜎₅₄=𝜎₁₆=1. The state trajectories of the system under protocol (2.11) and a random initial state 𝑥(0) are shown in Figure 1 when 𝑎(𝑡)=1/(𝑡+1). Figure 2 shows that the system is divergent when 𝑎(𝑡)=1.

Figure 1 

State trajectories under protocol (2.11) and 𝑎(𝑡)=1/(𝑡+1).

Figure 2 

State trajectories under protocol (2.11) and 𝑎(𝑡)=1.

Consider again the digraph defined above. Solving (4.7) without uncertainties gives ℎ≤0.5315. By Theorem 4.1, we have that protocol (4.1) asymptotically solves mean square consensus for any time-varying delay 𝜏(𝑡) satisfying 0≤𝜏(𝑡)≤0.5315 if (A1) or (A2) holds. Let 𝑎(𝑡)=1/(𝑡+1) and the intensity of the measurement noises 𝜎₂₁=𝜎₃₂=𝜎₄₃=𝜎₅₄=𝜎₁₆=1. Under protocol (4.1) and a stochastic initial state 𝑥(0), the state trajectories of the system are shown in Figures 3 and 4 for 𝑎(𝑡)=1/(𝑡+1) and 𝑎(𝑡)=1, respectively. We see that the system is divergent for the case of 𝑎(𝑡)=1.

Figure 3 

State trajectories under protocol (4.1) and 𝑎(𝑡)=1/(𝑡+1).

Figure 4 

State trajectories under protocol (4.1) and 𝑎(𝑡)=1.

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 𝐺 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).