# Mean Square Consensus for Uncertain Multiagent Systems with Noises and Delays

Authors: Yuangong Sun
Journal: Abstract and Applied Analysis
 (2012)

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## 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.


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## Body

## 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,AT 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 byG=(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=[aij] is an n×n-dimensional weighted adjacency matrix with aii=0. Say (i,j)∈E if aji>0. The set of neighbors of the ith agent is denoted by Ni={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.Then×n-dimensional Laplacian matrix L(G)=[lij] of digraph G is defined by lij=∑k=1naik for i=j and lij=-aij 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, where(2.1)E=[1-In-1],F=[0TIn-1].Consider a network of continuous-time first-order integrator agents with the dynamics(2.2)ẋi(t)=ui(t),i∈V,
where xi∈R is the state of the ith agent, ui∈R is the control input. When only taking measurement noises into consideration, the control input ui (or protocol) is designed to take the form [12]:(2.3)ui(t)=∑j∈Nia(t)aij[yij(t)-xi(t)],i∈V,
where the consensus-gain function a(t):[0,∞)→(0,∞) is piecewise continuous; yij(t)=xj(t)+σijηij(t) denotes the measurement of the jth agent’s state xj(t) by the ith agent; {ηij(t):i,j∈V} are independent standard white noises; σij≥0 is the noise intensity; σij=0 for j∉Ni. 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:(2.4)ui(t)=∑j∈Nia(t)(aij+Δaij(t))[zij(t)-xi(t-τij(t))],
where zij(t)=xj(t-τij(t))+σijηij(t); time delays τij(t):[0,∞)→[0,∞), i∈V, j∈Ni, are piecewise continuous and bounded functions; a(t) and σij are defined as above; Δaij(t), i,j∈V are parametric uncertainties satisfying |Δaij(t)|≤ϵij (ϵij>0) and Δaii(t)=0.For the sake of convenience, letτij(t)∈{τk(t):k=1,2,…,m}. Denote the ith row of the matrix A(t)=A+ΔA(t) by αi(t),(2.5)Σi=diag⁡{σi1,σi2,…,σin},Σt=diag⁡{α1(t)Σ1,α2(t)Σ2,…,αn(t)Σn},ηi(t)=[ηi1(t),ηi2(t),…,ηin(t)]T,η(t)=[η1T(t),η2T(t),…,ηnT(t)]T,
and x(t)=[x1(t),x2(t),…,xn(t)]T. Substituting the control (2.4) into the system (2.1) leads to(2.6)ẋ(t)=-a(t)∑k=1mLk(t)x(t-τk(t))+a(t)Σtη(t),
where the n×n-dimensional matrix Lk(t)=[lij(k)(t)] is defined as followings:(2.7)lij(k)(t)={-aij-Δaij(t),j≠i,τk(⋅)=τij(⋅)0,j≠i,τk(⋅)≠τij(⋅)-∑p=1nlip(k)(t),j=i.
It is easy to see(2.8)Lk(t)1=0,∑k=1mLk(t)=L(G)+ΔL(t),
where ΔL(t)=[Δlij(t)] with Δlij(t)=∑k=1nΔaik(t) for i=j and Δlij(t)=-Δaij(t) for i≠j.Lety=Ex, E, and F be defined by (2.1). From (2.6), we have the following reduced-order system:(2.9)ẏ(t)=a(t)∑k=1mELk(t)Fy(t-τk(t))+a(t)EΣtη(t).
It is a system driven by an n2-dimensional standard white noise, which can be written in the form of the Itô stochastic differential equation(2.10)dy(t)=a(t)∑k=1mELk(t)Fy(t-τk(t))dt+a(t)EΣtdw(t),
where w(t)=[w11(t),…,w1n(t),…,wnn(t)]T is an n2-dimensional standard Brownian motion. Without loss of generality, we let τk(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:(2.11)ui(t)=∑j∈Nia(t)(aij+Δaij(t))[yij(t-τ(t))-xi(t-τ(t))],
and the system (2.10) reduces to(2.12)dy(t)=a(t)E(L+ΔL(t))Fy(t-τ(t))dt+a(t)EΣtdw(t).In the sequel, we assume that the parametric uncertaintyΔL(t) to be of the form(2.13)ΔL(t)=DG(t)H,
where D and H are constant matrices with appropriate dimensions, and G(t) is an unknown matrix satisfying GT(t)G(t)≤I. Suppose also that there exists at least one edge (j,i)∈E such that σij>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)⋯xn(0)]T on [-h,0].We say the system (2.2) under protocol (2.11) asymptotically achieves mean square consensus if limt→∞E[(xi(t)-xj(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 graphG, which can be used to analyze the feasibility of the given consensus condition.Lemma 3.1.
IfG 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
(3.1)Φ=[Φ11Φ12P1TL̃P1TL̃*Φ22P2TL̃P2TL̃**-h-1Q0***-γIn]<0,
where Φ11=L̃TP1+P1TL̃, Φ12=P-P1T+L̃TP2, Φ22=-P2T-P2+hQ, and L̃=EL(G)F, E, and F is defined by (2.1).Proof.
By Lemma3.1, we have that there exists a matrix P>0 such that L̃TP+PL̃<0 if G has a spanning tree. Based on the Finsler Lemma [20], there exist matrices P1 and P2 such that
(3.2)[L̃TP1+P1TL̃P-P1T+L̃TP2*-P2T-P2]<0.
Let Q=In. 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 vectorz(t) on [t-τ(t),t] and matrix W>0, where t∈[0,∞) and 0≤τ(t)≤h, the following inequality holds:
(3.3)(∫t-τ(t)tz(u)du)TW∫t-τ(t)tz(u)du≤h∫t-τ(t)tzT(u)Wz(u)du.Lemma 3.3 (see [22]).
LetU, V, and F be real matrices of appropriate dimensions with FTF≤I, then for any scalar ϵ>0, one hasUFV+VTFTUT≤ϵ-1UUT+ϵVTV.Now, let us present the main result of this paper. We assume that the positive consensus-gain functiona(t) satisfies one of the following assumptions:(A1)
∫0∞a(t)dt=+∞, ∫0∞a2(t)dt<∞, and a(t)≤1 for sufficiently large t;(A2)
∫0∞a(t)dt=+∞, limt→∞a(t)=0.Theorem 3.4.
Assume that (A1) or (A2) holds and0≤τ(t)≤h. If there exist matrices P>0, Q>0, P1, P2 of compatible dimensions, constants γ>0 and ϵ>0 such that
(3.4)[Φ̃11Φ̃12Φ̃13Φ̃14P1TD̃*Φ̃22P2TL̃P2TL̃P2TD̃**Φ̃33Φ̃340***Φ̃440****-ϵI]<0,
where Φ̃11=Φ11+ϵFTHTHF, Φ̃12=Φ12, Φ̃13=P1TL̃+ϵFTHTHF, Φ̃14=P1TL̃+ϵFTHTHF, Φ̃22=Φ22, Φ̃33=-h-1Q+ϵFTHTHF, Φ̃34=ϵFTHTHF, Φ̃44=-γI+ϵFTHTHF, Φ11, Φ12, and Φ22 are defined as in Lemma 3.1, E and F is defined by (2.1), and D̃=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
(3.5)ξ(t)=E(L+ΔL(t))Fy(t-τ(t))=L̃ty(t-τ(t)).
Then, (2.12) reduces to
(3.6)dy(t)=a(t)ξ(t)dt+a(t)EΣdw(t).
Note that (3.4) holds. We can choose a constant λ>0 sufficiently small such that
(3.7)[Φ̃11+λPΦ̃12P1TL̃Φ̃14P1TD̃*Φ̃22P2TL̃P2TL̃P2TD̃**-h̃-1Q00***Φ̃440****-ϵI]<0,
where h̃=e-λhh-1.
Let(3.8)V(t)=yT(t)Py(t)+∫t-hte-λ∫sta(u)du(s-t+h)a2(s)ξT(s)Qξ(s)ds+β∫t-hte-λ∫sta(u)du(s-t+h)a2(s)ds,
where β>0 is a constant to be determined. By the Itô formula, we have
(3.9)dV(t)=[2a(t)yT(t)Pξ(t)+c(t)a2(t)]dt+2a(t)yT(t)PEΣtdw(t)+[ha2(t)ξT(t)Qξ(t)-∫t-hte-λ∫sta(u)dua2(s)ξT(s)Qξ(s)ds-λa(t)∫t-hte-λ∫sta(u)du(s-t+h)a2(s)ξT(s)Qξ(s)dsdt-β∫t-hte-λ∫sta(u)dua2(s)ds-λβa(t)∫t-hte-λ∫sta(u)du(s-t+h)a2(s)ds]dt,
where c(t)=tr(ΣtTETPEΣt)+βh. It is not difficult to see that there exists a scalar m1>0 such that c(t)≤m1 due to the fact that |Δaij(t)|≤ϵij. Thus,
(3.10)dV(t)+λa(t)V(t)dt=LV(t)dt+2a(t)yT(t)PEΣtdw(t),
where
(3.11)LV(t)=λa(t)yT(t)Py(t)+2a(t)yT(t)Pξ(t)+c(t)a2(t)+ha2(t)ξT(t)Qξ(t)-∫t-hte-λ∫sta(u)dua2(s)ξT(s)Qξ(s)ds-β∫t-hte-λ∫sta(u)dua2(s)ds.
Without loss of generality, say a(t)≤1 for t≥0. Then, we have
(3.12)LV(t)≤λa(t)yT(t)Py(t)+2a(t)yT(t)Pξ(t)+m1a2(t)+ha(t)ξT(t)Qξ(t)-e-λha(t)∫t-τ(t)ta2(s)ξT(s)Qξ(s)ds-βe-λh∫t-τ(t)ta2(s)ds.
On the other hand, integrating (3.7) from t-τ(t) to t yields
(3.13)y(t)-y(t-τ(t))=∫t-τ(t)ta(s)ξ(s)ds+∫t-τ(t)ta(s)EΣsdw(s).
Therefore, by the definition of ξ(t), we have
(3.14)0=L̃ty(t-τ(t))-ξ(t)=L̃ty(t)-ξ(t)-L̃t∫t-τ(t)ta(s)ξ(s)ds-L̃t∫t-τ(t)ta(s)EΣsdw(s),
which implies that
(3.15)0=2a(t)[yT(t)P1T+ξT(t)P2T]×[L̃t(y(t)-∫t-τ(t)ta(s)ξ(s)ds-∫t-τ(t)ta(s)EΣdw(s))-ξ(t)].
By Lemma 3.2, we have
(3.16)∫t-τ(t)ta2(s)ξT(s)Qξ(s)ds≥h-1(∫t-τ(t)ta(s)ξ(s)ds)TQ∫t-τ(t)ta(s)ξ(s)ds.
Using the basic inequality -2uTv≤γuTu+γ-1vTv for any vector u and v, we have
(3.17)-2[yT(t)P1T+ξT(t)P2T]L̃t∫t-τ(t)ta(s)EΣsdw(s)=-2[yT(t)ξT(t)][P1TP2T]L̃t∫t-τ(t)ta(s)EΣsdw(s)≤γ-1[yT(t)ξT(t)][P1TP2T]L̃tL̃tT[P1P2][y(t)ξ(t)]+γ(∫t-τ(t)ta(s)EΣsdw(s))T∫t-τ(t)ta(s)EΣsdw(s).
Substituting (3.15)–(3.17) into (3.12) gives
(3.18)LV(t)≤a(t)ωT(t)Ω(t)ω(t)+m1a2(t)-βe-λh∫t-τ(t)ta2(s)ds+γ(∫t-τ(t)ta(s)EΣdw(s))T∫t-τ(t)ta(s)EΣdw(s),
where ω(t)=[yT(t)ξT(t)-∫t-τ(t)ta(s)ξT(s)ds]T, and
(3.19)Ω(t)=[Ω11Ω12P1TL̃t*Ω22P2TL̃t**-h̃-1Q]+γ-1[P1TP2T]L̃tL̃tT[P1P2],
with Ω11=L̃tTP1+P1TL̃t+λP, Ω12=P-P1T+L̃tTP2, Ω22=Φ22. It is easy to see that Ω(t)<0 if and only if
(3.20)Φ+[P1TD̃P2TD̃00]G(t)[H̃0H̃H̃]+[H̃T0H̃TH̃T]G(t)[D̃TP1D̃TP200]<0,
where Φ is defined by (3.1) and H̃=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 thatΩ(t)<0 for t≥0, and
(3.21)E[(∫t-τ(t)ta(s)EΣsdw(s))T∫t-τ(t)ta(s)EΣsdw(s)]=∫t-τ(t)td(s)a2(s)ds,
where d(t)=tr(ΣtTETEΣt) satisfying d(t)≤m2 for some m2>0. By choosing β=γm2eλh, we get from (3.10) and (3.18) that
(3.22)E[V(t)]≤E[V(0)]-λ∫0ta(s)E[V(s)]ds+m1∫0ta2(s)ds.
By the comparison theorem [23], we have
(3.23)E[V(t)]≤E[V(0)]e-λ∫0ta(s)ds+m1∫0te-λ∫sta(u)dua2(s)ds.
If (A1) holds, that is, ∫0∞a(s)ds=∞ and ∫0∞a2(s)ds<∞, then, similar to the proof of Theorem  3.2 in [12], we can conclude from (3.23) that limt→∞E[V(t)]=0. If (A2) holds, by the L’Hopital rule, we have
(3.24)lim⁡t→∞∫0te-λ∫sta(u)dua2(s)ds=lim⁡t→∞a(t)λ=0,
which also implies limt→∞E[V(t)]=0. Based on the construction of V(t) and the transformation y=Ex, we have
(3.25)lim⁡t→∞E[(x1(t)-xi+1(t))2]=lim⁡t→∞E[yi2(t)]=0,i=1,2,…,n-1.
It implies that limt→∞E[(xi(t)-xj(t))2]=0 for i,j∈V and i≠j. The proof is complete.Remark 3.5.
By Lemma3.1, 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 leaderxi0 satisfying ẋi0=0. Without loss of generality, consider the following protocol:(4.1)u1(t)=∑j∈N1a(t)a1j[xj(t)-x1(t)],ui(t)=∑j∈Nia(t)(aij+Δaij(t))[z̃ij(t)-xi(t)],i=2,…,n,
where z̃ij(t)=xj(t-τ(t))+σijηij, τ, a, aij, Δaij, σij and ηij are defined as above. The case of multiple time-varying delays can be similarly discussed.Note that(4.2)xj(t-τ(t))-xi(t)=[xj(t-τ(t))-x1(t-τ(t))]+[x1(t)-xi(t)]-[x1(t)-x1(t-τ(t))],i,j=2,3,…,n,x1(t)-x1(t-τ(t))=∫t-τ(t)tẋ1(s)ds.
If we set y=Ex, where E is defined as above, then we have the following reduced-order system:(4.3)ẏ(t)=a(t)[L1(t)y(t)+L2(t)y(t-τ(t))+L3(t)∫t-τ(t)ta(s)y(s)ds]+a(t)Σ̂tη̂(t),
where Li(t)=Li+ΔLi(t) for i=1,2,3,(4.4)L1=-diag⁡{∑j=1na2j,∑j=1na3j,…,∑j=1nanj}-1[a12a13⋯a1n],ΔL1(t)=-diag⁡{∑j=1nΔa2j(t),∑j=1nΔa3j(t),…,∑j=1nΔanj(t)},L2=[a22a23⋯a2na32a33⋯a3n⋮⋮⋱⋮an2an3⋯ann],ΔL2(t)=[Δa22(t)Δa23(t)⋯Δa2n(t)Δa32(t)Δa33(t)⋯Δa3n(t)⋮⋮⋱⋮Δan2(t)Δan3(t)⋯Δann(t)],L3=-diag⁡{∑j=1na2j,∑j=1na3j,…,∑j=1nanj}1[a12a13⋯a1n],ΔL3(t)=-diag⁡{∑j=1nΔa2j(t),∑j=1nΔa3j(t),…,∑j=1nΔanj(t)}1[a12a13⋯a1n],Σ̂t=-diag{α2(t)Σ2,…,αn(t)Σn}, η̂(t)=[η2T(t)η3T(t)⋯ηnT(t)]T, αi(t), Σi and ηi are defined as above. Therefore, (4.3) can be written in the form of the Itô stochastic differential equation.(4.5)dy(t)=a(t)[L1(t)y(t)+L2(t)y(t-τ(t))+L3(t)∫t-τ(t)ta(s)y(s)ds]dt+a(t)Σ̂tdw(t).
It is not difficult to verify that L1+L2=EL(G)F. In the following, we assume that the uncertainties ΔLi(t) to be of the form(4.6)ΔL1(t)+ΔL2(t)=D1F(t)H1,ΔLi(t)=DiF(t)Hi,i=2,3,
where Di, Hi are constant matrices of appropriate dimensions and F(t) satisfies (2.13).Theorem 4.1.
Assume that (A1) or (A2) holds and0≤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 ϵi>0 for i=1,2,3 such that
(4.7)[Ψ11Ψ12P1TL2Ψ14P1TL2P1TD1P1TD2P1TD3*Ψ22P2TL2Ψ24P2TL2P2TD1P2TD2P2TD3**Ψ33L2TP3ϵ2E2TE2000***Ψ44P3TL2P3TD1P3TD20****Ψ55000*****-ϵ1I00******-ϵ2I0*******-ϵ3I]<0,
where Ψ11=(L1+L2)TP1+P1T(L1+L2)+hS+ϵ1E1TE1, Ψ12=P-P1T+(L1+L2)TP2, Ψ14=P1TL3+(L1+L2)TP3, Ψ22=-P2T-P2+hQ, and Ψ24=P2TL3-P3, Ψ33=-h-1Q+ϵ2E2TE2, Ψ44=-h-1S+ϵE3TE3, Ψ55=-γI+ϵ2E2TE2, then the system (2.2) under protocol (4.1) asymptotically achieves mean square consensus for all admissible uncertainties satisfying (2.13).Proof.
Let(4.8)ξ(t)=L1(t)y(t)+L2(t)y(t-τ(t))+L3(t)∫t-τ(t)ta(s)y(s)ds.
Then, (4.5) reduces to
(4.9)dy(t)=a(t)ξ(t)+a(t)Σ̂tdw(t).
Choose the Lyapunov function as the following:
(4.10)V(t)=yT(t)Py(t)+∫t-hte-λ∫sta(u)du(s-t+h)a2(s)ξT(s)Qξ(s)ds+∫t-hte-λ∫sta(u)du(s-t+h)a2(s)yT(s)Sy(s)ds×β∫t-hte-λ∫sta(u)du(s-t+h)a2(s)ds,
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
(4.11)dV(t)+λa(t)V(t)dt=LV(t)dt+2a(t)yT(t)PΣ̂tdw(t),
where
(4.12)LV(t)≤λa(t)yT(t)Py(t)+2a(t)yT(t)Pξ(t)+c̃(t)a2(t)+ha(t)ξT(t)Qξ(t)+ha(t)yT(t)Sy(t)-e-λha(t)∫t-τ(t)ta2(s)ξT(s)Qξ(s)ds-e-λha(t)∫t-τ(t)ta2(s)yT(s)Sy(s)ds-βe-λh∫t-τ(t)ta2(s),
and c̃(t)=tr(Σ̂tTPΣt)+βh. On the other hand, by the definition of ξ(t), we have
(4.13)0=L1(t)y(t)+L2(t)y(t-τ(t))+L3(t)∫t-τ(t)ta(s)y(s)ds-ξ(t)=(L1(t)+L2(t))y(t)+L3(t)∫t-τ(t)ta(s)y(s)ds-ξ(t)-L2(t)∫t-τ(t)ta(s)ξ(s)ds-L2(t)∫t-τ(t)ta(s)Σ̂dw(s).
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 Lemma3.1, 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 Di and Hi(i=1,2,3), the tolerable upper bound of delay can be derived from (4.7).


## 5. Simulation Results

Consider a digraphG=(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 digraphG has a spanning tree.



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*Source: 621060-2012-04-01.xml*