Consensus conditions of continuous-time multi-agent systems with time-delays and measurement noises

0.96 MB·

by  Xiaofeng Zong

Consensus conditions of continuous-time multi-agent systems with
(cid:73)
time-delays and measurement noises
Xiaofeng Zonga, Tao Lib,∗, Ji-Feng Zhanga
aKey Laboratory of Systems and Control, Academy of Mathematics and Systems Science, Chinese Academy
of Sciences, Beijing 100190, China.
bShanghai Key Laboratory of Power Automation Technology, School of Mechatronic Engineering and
Automation, Shanghai University, Shanghai 200072, China.
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2 Abstract

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This work is concerned with stochastic consensus conditions of multi-agent systems with
a
J both time-delays and measurement noises. For the case of additive noises, we develop some

0 necessaryconditionsandsufficientconditionsforthestochasticweakconsensusbyestimating
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the differential resolvent function for functional equations. By the martingale convergence

] theorem, we obtain the necessary conditions and the sufficient conditions for the stochastic
Y
strong consensus and prove that the small time-delay does not affect the agents to reach the
S
. stochastic strong consensus. For the case of multiplicative noises, we consider two kinds of
s
c time-delays, appeared in the measurement term and the noise term, respectively. A funda-
[
mental theorem is developed to show that the stochastic weak consensus with an exponential

1 pairwiseconvergencerateimpliesthestochasticstrongconsensus. ByconstructingLyapunov
v
functional, we show that for any bounded time-delay in the noise term, the stochastic con-
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6 sensus can be achieved by carefully choosing the control gain according to the time-delay in
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the measurement term.
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Keywords: Multi-agent system, time-delay, measurement noise, mean square consensus,
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almost sure consensus
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: 1. Introduction
v
i
X Theresearchonconsensusinmulti-agentsystems, whichinvolvescoordinationofmultiple
r entitieswithonlylimitedneighborhoodinformationtoreachaglobalgoalfortheentireteam,
a
has offered promising support for solutions in distributed systems, such as flocking behavior
and swarms (Liu, Passino & Polycarpou, 2003; Martin, Girard, Fazeli & Jadbabaie, 2014)
and sensor networks (Akyildiz, Su, Sankarasubramaniam & Cayirci, 2002; Ogren, Fiorelli &
Leonard, 2004). Due to that the time-delay appears in almost all practical systems and the
real communication processes are often disturbed by various random factors, each agent can
(cid:73)This paper was not presented at any IFAC meeting. This paper was submitted to Automatica on July
12, 2015
∗Corresponding author, Phone: [phone], Fax: [phone].
Email addresses: [e-mail] (Xiaofeng Zong), [e-mail] (Tao Li),
[e-mail] (Ji-Feng Zhang)
Preprint submitted to Automatica February 2, 2016
not measure its neighbors’ states timely and accurately. Hence, there has been substantial
and increasing interest in recent years in the consensus problem of the multi-agent systems
subject to the phenomenon of time-delay and stochasticity.
Sofarlotsofachievementshavebeenmadeintheresearchofconsensusproblemsofmulti-
agent systems with the time-delay. Olfati-Saber and Murray (2004) presented that the small
time-delaydoesnotaffecttheconsensuspropertyoftheprotocol. LinandRen(2014)studied
a constrained consensus problem for multi-agent systems in unbalanced networks in the
presence of time-delays. For the case of distributed time-delays, Munz, Papachristodoulou
and Allgower (2011) showed that the consensus for single integrator multi-agent systems can
be reached under the same conditions as the delay-free case. For the high-order linear multi-
agent systems, the exact time-delay bound was investigated in Cepeda-Gomez and Olgac
(2011) and Cepeda-Gomez (2015). The mentioned above works are for the continuous-time
models. For the discrete-time models, we refer to Liu, Li and Xie (2011), Hadjicostis and
Charalambous (2014), Park, Kwon, Choi and Cha (2014) and Sakurama and Nakano (2015).
Thereare alsomanyliteraturesonthe consensusproblemof multi-agentsystems withthe
measurement noises, including the additive and multiplicative cases. For the case of additive
noises, the distributed stochastic approximation method was introduced for discrete-time
models and the mean square and the almost sure consensus conditions were obtained (Huang
& Manton, 2009; Kar & Moura, 2009; Li & Zhang, 2010; Huang & Manton, 2010; Huang,
Dey, Nair & Manton, 2010; Xu, Zhang & Xie, 2012; Aysal & Barner, 2010). For continuous-
time models, the necessary and sufficient conditions of the mean square average-consensus
were obtained in Li and Zhang (2009) and Cheng, Hou, Tan and Wang (2011). And the
sufficient conditions of the almost sure strong consensus were stated in Wang and Zhang
(2009). For the case of the multiplicative noises, Ni and Li (2013) investigated the consensus
problems of the continuous-time systems with the noise intensities being proportional to
the absolute value of the relative states of an agent and its neighbor. Then this work was
extended to the discrete-time version in Long, Liu and Xie (2014). Li, Wu and Zhang
(2014) studied the distributed consensus with general multiplicative noises and developed
some necessary conditions and sufficient conditions for the mean square and the almost sure
average-consensus. Taking the two classes of measurement noises into consideration, Zong,
Li and Zhang (2015) gave the necessary and sufficient conditions of the mean square and the
almost sure weak and strong consensus for the continuous-time models.
When the time-delay and noise coexist in real multi-agent networks, these works above
are far from enough to deal with the consensus problem. Based on this phenomena, the
distributed consensus problem was addressed in Liu, Xie and Zhang (2011), and the ap-
proximate mean square consensus problem was examined in Amelina, Fradkov, Jiang and
Vergados (2015) for the discrete-time models. For the continuous-time models, Liu, Liu,
Xie and Zhang (2011) presented some sufficient conditions for the mean square average-
consensus. Note that the works stated above are for the additive measurement noise and
little is known about the consensus conditions for the case with the noises coupled with the
time-delay states. Moreover, the stochastic weak consensus problems, including the mean
square and the almost sure weak consensus, have not been taken into account.
Motivated by the above discussions and partly based on our recent works (Zong, Li &
Zhang,2015;Li, Wu&Zhang,2014;Li&Zhang,2009), thisworkinvestigatesthedistributed
consensus problem of continuous-time multi-agent systems with time-delays and measure-
2
ment noises, including the additive and multiplicative cases. Due to the presence of noises,
existing techniques for the case with time-delay (Olfati-Saber & Murray (2004); Xu, Zhang
& Xie (2013)) are no longer applicable to the analysis of stochastic consensus. Moreover,
the coexistence of time-delays and noises leads to the difficulty in finding the optimal delay
bound (as optimal as possible) for stochastic consensus problem. In this paper, the Lya-
punov functional and the stochastic analysis are used to overcome the difficulties induced by
the time-delays and noises.
We use the variable transformation to transform the closed-loop system into a stochastic
differential delay equation (SDDE) driven by the additive or multiplicative noises. Then the
keyistoanalysetheasymptoticstabilityofSDDEs. Ourconcernisnotonlyimportantinthe
consensus analysis mentioned above but also has its own mathematical interest because the
relevant stochastic stability theorem for this kind of SDDEs has not been well established.
The contribution of the current work can be concluded as the following two parts.
• Additive noises case: We first give the estimation of a differential resolvent function
for the delayed equation. Based on the estimation and the variation of constants
formula, we prove that if the graph is connected, then the mean square weak consen-
(cid:82)∞
sus can be achieved under the conditions λ τ sup c(t) < 1, c(t)dt = ∞ and
N 1 t≥0 0
limt→∞(cid:82)0te−(cid:37)0(cid:82)stc(u)du c2(s)ds = 0, where τ1 is the time-delay, λN is the spectral radius
of the Laplacian matrix, c(t) is the control gain and (cid:37) is a positive constant less than
0
(cid:82)∞
the algebraic connectivity λ . Moreover, if λ τ sup c(t) < 1 and c(t)ds = ∞,
2 N 1 t≥0 0
(cid:82)t
then the almost sure weak consensus can be achieved if lim c(t)log c(s) ds = 0,
t→∞ 0
(cid:82)t
and only if liminf c(t)log c(s)ds = 0. For the stochastic strong consensus, we
t→∞ 0
show that small time-delay does not affect the control gain to guarantee the stochastic
strong consensus.
• Multiplicative noises case: We consider that the measurement by each agent from its
neighbors is composed of two terms: the measurement term with time-delay τ and
1
noisetermwithtime-delayτ . Wefirstdevelopafundamentaltheoremtoshowthatthe
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mean square (or the almost sure) weak consensus with an exponential convergence rate
implies the mean square (or the almost sure) strong consensus. Then by constructing
the Lyapunov functional, we prove that the control gain K(t) = kI satisfing 0 < k <
n
λ2 can ensure the mean square and the almost sure strong consensus, where
λ2Nτ1+NN−1σ¯2λ2
σ¯ is the bound of the noise intensities and N is the number of agents. It is shown
that for any bounded time-delay in the noise term, the stochastic strong consensus
can be achieved by carefully choosing the control gain according to the time-delay in
the measurement term. Especially, we show that 0 < k < N is necessary and
(N−1)σ2
sufficient for the mean square strong consensus if the noise intensity grows with the
rate σ and τ = 0.
1
The rest of the paper is organized as follows. Section 2 serves as an introduction to the
networked systems and consensus problems. Section 3 gives the necessary conditions and
the sufficient conditions of the stochastic weak and strong consensus for multi-agent systems
with the time-delay and additive noises. Section 4 aims to consider stochastic consensus
problem of multi-agent systems with time-delays and multiplicative noises. Section 5 gives
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thenumericalexperimentstoconfirmthetheoreticalresults. Section6givessomeconcluding
remarks and discusses the future research topics.
Notations: Throughout this paper, unless otherwise specified, we use the following
notations. 1 denotes a n-dimensional column vector with all ones. η denotes the N-
n N,i
dimensional column vector with the ith element being 1 and others being zero. J denotes
N
the matrix 11 1T . I denotes the N-dimensional identity matrix. For a given matrix or
N N N N
vector A, its transpose is denoted by AT, and its Euclidean norm is denoted by (cid:107)A(cid:107). For
two matrices A and B, A⊗B denotes their Kronecker product. For a,b ∈ R, a∨b represents
max{a,b} and a∧b denotes min{a,b}. For any given real symmetric matrix K, we denote its
maximum and minimum eigenvalues by λ (K) and λ (K), respectively. Let (Ω,F,P)
max min
be a complete probability space with a filtration {F } satisfying the usual conditions,
t t≥0
namely, it is right continuous and increasing while F contains all P-null sets. For a given
0
random variable or vector X, the mathematical expectation of X is denoted by EX. For a
continuous martingale M(t), its quadratic variation is denoted by (cid:104)M(cid:105)(t) (see Revuz & Yor
(1999)). For the fixed τ > 0, we use C([−τ,0];Rn) to denote the space of all continuous
Rn-valued functions ϕ defined on [−τ,0] with a norm (cid:107)ϕ(cid:107) = sup (cid:107)ϕ(t)(cid:107).
C t∈[−τ,0]
2. Problem formulation
Consider N agents distributed according to an undirected graph G = {V,E,A}, where
V = {1,2,...,N} is the set of nodes with i representing the ith agent, E denotes the set of
undirected edges and A=[a ]∈RN×N is the adjacency matrix of G with element a = 1 or
ij ij
0 indicating whether or not there is an information flow from agent j to agent i directly. N
i
denotes the set of the node i’s neighbors, that is, for j ∈ N , a = 1. Also, deg = (cid:80)N a
i ij i j=1 ij
is called the degree of i. The Laplacian matrix of G is defined as L = D − A, where
D = diag(deg ,...,deg ). Since the graph is undirected, A and L are symmetric.
1 N
For agent i, denote its state at time t by x (t) ∈ Rn. In real multi-agent networks, for
i
each agent, the information from its neighbors may have time-delays and noises. Hence, we
consider that the state of each agent is updated by the rule
N
(cid:88)
x˙ (t) = K(t) a z (t), i = 1,2,...,N,t > 0, (1)
i ij ji
j=1
with
z (t) = x (t−τ )−x (t−τ )+f (x (t−τ )−x (t−τ ))ξ (t) (2)
ji j 1 i 1 ji j 2 i 2 ji
denoting the measurements of relative states by agent i from its neighbor j ∈ N . Here,
i
K(t) ∈ Rn×n is the control gain matrix function to be designed, τ ≥ 0 and τ ≥ 0 are
1 2
the time-delays, ξ (t) ∈ R denotes the measurement noise and f : Rn (cid:55)→ Rn is the noise
ji ji
intensity function. Let τ = τ ∨ τ and the initial data x (t) = η (t) for t ∈ [−τ,0], i =
1 2 i i
1,2,··· ,N be deterministic continuous functions. Let x(t) = [xT(t),...,xT (t)]T and η(t) =
1 N
[ηT(t),...,ηT(t)]T.
1 N
In this work, x (t−τ )−x (t−τ ) is called the measurement term and f (x (t−τ )−
j 1 i 1 ji j 2
x (t − τ ))ξ (t) is called the noise term. We also assume that the measurement noises are
i 2 ji
independent Gaussian white noises.
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Assumption 2.1. The noise process ξ (t) ∈ R satisfies (cid:82)tξ (s)ds = w (t), t ≥ 0,j,i =
ji 0 ji ji
1,2,...,N, where {w (t),i,j = 1,2,...,N} are independent Brownian motions.
ji
Note that the measurement noise in (2) includes the two cases: First, the noises in (2) are
additive, that is, each intensity f is independent of the agents’ states; Second, the noises
ji
are multiplicative, that is, the intensity f depends on the relative states. And both the two
ji
cases will be under consideration. Then the key in stochastic consensus problems is to find
an appropriate control gain function K(t) such that the agents reach mean square or almost
sure consensus under the two types of noises. Here, the definitions on the mean square and
the almost sure consensus for the multi-agent system (1) are given as follows.
Definition 2.1. The agents are said to reach the mean square weak consensus if the system
(1) and (2) have the property that for any initial data η ∈ C([−τ,0],RNn) and all distinct
i,j ∈ V, lim E(cid:107)x (t) − x (t)(cid:107)2 = 0. If, in addition, there is a random vector x∗ ∈ Rn,
t→∞ i j
such that E(cid:107)x∗(cid:107)2 < ∞ and lim E(cid:107)x (t) − x∗(cid:107)2 = 0, i = 1,2,...,N, then the agents
t→∞ i
are said to reach the mean square strong consensus. Particularly, if Ex∗ = 1 (cid:80)N x (0),
N j=1 j
then the agents are said to reach the asymptotically unbiased mean square average-consensus
(AUMSAC) and E(cid:107)x∗ − 1 (cid:80)N x (0)(cid:107)2 is called the mean square steady-state error.
N j=1 j
Definition 2.2. The agents are said to reach the almost sure weak consensus if the system
(1) and (2) have the property that for any initial data η ∈ C([−τ,0],RNn) and all distinct
i,j ∈ V, lim (cid:107)x (t)−x (t)(cid:107) = 0, a.s. If, in addition, there is a random vector x∗ ∈ Rn,
t→∞ i j
such that P{(cid:107)x∗(cid:107) < ∞} = 1 and lim (cid:107)x (t)−x∗(cid:107) = 0, a.s. i = 1,2,...,N, then the agents
t→∞ i
are said to reach the almost sure strong consensus. Particularly, if Ex∗ = 1 (cid:80)N x (0),
N j=1 j
then the agents are said to reach the asymptotically unbiased almost sure average-consensus
(AUASAC).
Denote δ(t) = [(I −J )⊗I ]x(t). Let δ(t) = [δT(t),...,δT(t)]T, where δ (t) ∈ Rn, i =
N N n 1 N i
1,2,...,N. Define the unitary matrix T = [√1N ,φ ,...,φ ], where φ is the unit eigenvector
L 2 N i
N
of L associated with the eigenvalue λ = λ (L), that is, φTL = λ φT, (cid:107)φ (cid:107) = 1, i = 2,...,N.
i i i i i i
Denote φ = [φ ,...,φ ] and Λ = diag(λ ,λ ,··· ,λ ). Let δ(t) = (T ⊗I )δ(cid:101)(t) and δ(cid:101)(t) =
2 N 2 3 N L n
[δ(cid:101)T(t),..., δ(cid:101)T(t)]T, then it can be verified that δ(cid:101) (t) ≡ 0. Denote δ(t) = [δ(cid:101)T(t),...,δ(cid:101)T(t)]T,
1 N 1 2 N
which is an (N −1)n dimensional column vector. After the variable transformation above,
the initial data η(t) is transformed to a new function, denoted by δ(t) = ζ(t) with ζ(t) ∈
C([−τ,0];R(N−1)n). Denote (cid:107)ζ(cid:107)2 = sup (cid:107)ζ(θ)(cid:107)2.
C θ∈[−τ,0]
3. Networks with the time-delay and additive noises
In this section, we consider the case with additive noises and the matrix function K(t) =
c(t)I , which is concluded as the following assumption.
n
Assumption 3.1. For any x ∈ Rn, f (x) = σ 1 with σ > 0, i,j = 1,...,N, and
ji ji n ji
K(t) = c(t)I , where c(t) ∈ C((0,∞);[0,∞)).
n
Note that under Assumption 3.1, time-delay τ vanishes in the network system. For the
2
mean square and the almost sure strong consensus, the following conditions on the control
gain function c(t) were addressed before:
5
(cid:82)∞
(C1) c(s)ds = ∞;
0
(C2) (cid:82)∞c2(s)ds < ∞;
0
(C3) lim c(t) = 0.
t→∞
Note that (C2) does not imply (C3), since c(t) may not be uniformly continuous.
For the systems with additive noise, the necessary and sufficient conditions of the mean
square and the almost sure strong and weak consensus seems to be clear now in view of Zong,
Li and Zhang (2015). When the time-delay appears, the sufficient conditions involving (C1)
and (C2) were obtained for mean square strong consensus in Liu, Liu, Xie and Zhang (2011).
But little is known about the necessary conditions of the stochastic strong consensus and
the necessary and sufficient conditions of the stochastic weak consensus. This section will
fill in this gap for the case of undirected graphs.
Here, we first consider the linear scalar equation
˙
X(t) = −λc(t)X(t−τ ),t > 0, (3)
1
X(t) = ξ(t) for t ∈ [−τ ,0], where λ > 0, τ ≥ 0 and ξ ∈ C([−τ ,0],R). The solution to (3)
1 1 1
has the form (Gripenberg, Londen & Staffans, 1990, p. 295)
X(t) = r(t,s)X(s), ∀ t ≥ s ≥ 0, (4)
where r(t,s) is the differential resolvent function, satisfying r(t,t) = 1 for t > 0, r(t,s) = 0
for t < s and
∂
r(t,s) = −λc(t)r(t−τ ,s). (5)
1
∂t
Let c¯ := sup c(t). We now give the following lemma to reveal the estimation of the
t≥0
differential resolvent function r(t,s) for 0 < s < t.
Lemma 3.1. For each λ > 0 satisfying λτ c¯< 1, the solution to (5) satisfies
1
(cid:107)r(t,s)(cid:107)2 ≤ b(λ)e(cid:37)(λ)τ1c¯e−(cid:37)(λ)(cid:82)stc(u)du, t > s > 0, (6)
where b(λ) and (cid:37)(λ) < λ(1−λc¯τ ) are two positive constants depending on λ. Moreover, if
1
lim |r(t,0)| = 0 for sufficiently small τ > 0, then condition (C1) holds.
t→∞ 1
The proof of this lemma is given in Appendix A. By using Lemma 3.1, we now examine
the mean square and the almost sure consensus.
3.1. Mean square consensus
Let (cid:37)(λ) be defined in Lemma 3.1 and introduce another condition on c(t):
(C4) limt→∞(cid:82)0te−(cid:37)0(cid:82)stc(u)duc2(s)ds = 0, where (cid:37)0 = min2≤j≤N (cid:37)(λj).
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Theorem 3.2. Suppose that Assumptions 2.1 and 3.1 hold and τ λ c¯< 1. Then the agents
1 N
reach the mean square weak consensus if the network topology graph G is connected and con-
ditions (C1) and (C4) hold, and only if the network topology graph G is connected. Moreover,
the agents reach the mean square strong consensus if the network topology graph G is con-
nected and conditions (C1)-(C2) hold, and only if the network topology graph G is connected
and condition (C2) holds.
Proof. Substituting (2) into the system (1) and using Assumption 3.1 yield
N
(cid:88)
dx(t) = −c(t)(L⊗I )x(t−τ )dt+c(t) a σ (η ⊗1 )dw (t). (7)
n 1 ij ji N,i n ji
i,j=1
By the definition of δ(t), we have dδ(t) = −c(t)(L ⊗ I )δ(t)dt + c(t)(cid:80)N a σ [((I −
n i,j=1 ij ji N
¯
J )η ) ⊗ 1 )]dw (t), which together with the definition of δ(t) produces the following
N N,i n ji
SDDE:
dδ(t) = −c(t)(Λ⊗I )δ(t−τ )dt+dM(t), (8)
n 1
where M(t) = (cid:80)N a σ [φT(I −J )η ⊗1 ](cid:82)tc(s)dw (s). By (8), we get
i,j=1 ij ji N N N,i n 0 ji
dδ (t) = −λ c(t)δ (t−τ )dt+dM (t), j = 2,...,N, (9)
j j j 1 j
where M (t) = (cid:80)N a σ [φT(I − J )η ⊗ 1 ](cid:82)tc(s) dw (s). Hence, the mean square
j i,l=1 il li j N N N,i n 0 li
weak consensus is equivalent to that for any initial data η(t), lim E(cid:107)δ (t)(cid:107)2 = 0, j =
t→∞ j
2,...,N. Note that the network topology graph G is connected, then λ > 0 for j =
j
2,...,N. By means of a variation of constants formula for equation (9) considered pathwise
(Gripenberg, Londen & Staffans, 1990, p. 295),
(cid:90) t
δ (t) = r (t,0)δ (0)+ r (t,s)dM (s), (10)
j j j j j
0
where r (t,s) is the differential resolvent function defined by (5) with λ replaced by λ .
j j
Combining (10) and Lemma 3.1 gives
(cid:90) t
E(cid:107)δ (t)(cid:107)2 = (cid:107)r (t,0)(cid:107)2(cid:107)δ (0)(cid:107)2 +E(cid:107) r (t,s)dM (s)(cid:107)2
j j j j j
0
(cid:90) t
≤ b0e−(cid:37)0(cid:82)0tc(u)du(cid:107)δj(0)(cid:107)2 +b1 e−(cid:37)0(cid:82)stc(u)duc2(s)ds, (11)
0
where b = max b(λ )e(cid:37)(λj)τ1c¯ and b = N−1nb (cid:80)N a σ2. By (C1) and (C4), we
0 2≤j≤N j 1 N 0 i,j=1 ij ji
immediately obtain lim E(cid:107)δ (t)(cid:107)2 = 0 for j = 2,...,N. Therefore, the agents under (C1)
t→∞ j
and (C4) reach the mean square weak consensus. Next, we assume that the agents reach the
mean square weak consensus and show the necessity for the graph to be connected. If the
network topology graph G is disconnected, then λ = 0. By (9) and the definition of δ(t),
2
we would have
δ (t) = δ (0)+M (t), (12)
2 2 2
7
which means liminf E(cid:107)δ(t)(cid:107)2 = liminf E(cid:107)δ(t)(cid:107)2 ≥ (cid:107)δ (0)(cid:107)2 > 0 for any given x(0)
t→∞ t→∞ 2
with δ (0) (cid:54)= 0. This is in contradiction with the definition of mean square weak consensus.
2
So the network topology graph G must be connected. Above all, the first assertion holds.
Now, we prove the second assertion. By the definitions of the mean square weak and
strong consensus, we can see that the mean square strong consensus is equivalent to the
mean square weak consensus plus that the average 1 (cid:80)N x (t) is convergent in the sense of
N j=1 j
mean square. It is proved in Li and Zhang (2009) that (C2) implies (C4), then from the first
assertion, conditions (C1), (C2) and the connectivity give the mean square weak consensus.
Note the time-delay τ does not change the average of the states of agents, that is,
1
N N
1 (cid:88) 1 (cid:88)
¯
x (t) = x (0)+M(t), (13)
j j
N N
j=1 j=1
where M¯(t) = 1 1 (cid:80)N a σ (cid:82)tc(s)dw (s). It is easy to see that
nN i,j=1 ij ji 0 ji
n(cid:80)N a2 σ2 (cid:90) t
E(cid:107)M¯(t)(cid:107)2 = i,j=1 ij ji c2(s)ds, (14)
N2
0
and then the mean square convergence of 1 (cid:80)N x (t) is equivalent to the mean square
N j=1 j
¯
boundedness of M(t) (Lipster & Shiryayev, 1989, Theorem 1, p.20), which is also equivalent
to (C2). Hence, the second assertion holds with the consensus limit x∗ = 1 (cid:80)N x (0) +
N j=1 j
¯ ¯ ¯
M(∞), where M(∞) := lim M(t) is a common Gaussian random variable.
t→∞
Remark 3.1. Based on the proof above, the connected graph and conditions (C1)-(C2)
produce the consensus limit x∗ = 1 (cid:80)N x (0) + M¯(∞) satisfying Ex∗ = 1 (cid:80)N x (0).
N j=1 j N j=1 j
Hence, under the connected undirected graph and conditions (C1)-(C2), the agents reach
not only the mean square strong consensus, but also the AUMSAC.
Remark 3.2. Note that the time-delay τ does not change the average 1 (cid:80)N x (t). If the
1 N j=1 j
AUMSAC is achieved, then the mean square steady-state error is n(cid:80)Ni,j=1a2ijσj2i (cid:82)∞c2(s)ds,
N2 0
which equals to that of the delay-free case. That is, the time-delay does not change the value
of the consensus limit and the mean square steady-state error.
Remark 3.3. If c¯ = 1, Liu, Liu, Xie and Zhang (2011) showed that λ2 τ < λ , (C1) and
N 1 2
(C2) are the sufficient conditions for the AUMSAC by using the Gronwall-Bellman-Halanay
type inequality. In this work, we relax the restriction λ2 τ < λ to λ τ < 1, give the
N 1 2 N 1
necessary conditions and the sufficient conditions for the mean square weak consensus and
show that the time-delay does not affect the mean square steady-state error. Here, the main
skills are the variation of constants formula and the Lyapunov functional method.
By L’Hˆopital’s rule, we have the following corollary, which reveals the relationship be-
tween (C1), (C3) and the mean square weak consensus and can be proved similarly to that
in Zong, Li and Zhang (2015).
Corollary 3.3. Suppose that Assumptions 2.1 and 3.1 hold and τ λ c¯ < 1. Then the
1 N
agents reach the mean square weak consensus if the network topology graph G is connected
and conditions (C1) and (C3) hold.
8
3.2. Almost sure consensus
Here, we give some necessary conditions and sufficient conditions for the almost sure
weak and strong consensus. In order to examine the almost sure weak consensus, we need
two more conditions:
(cid:82)t
(C5) lim c(t)log c(s)ds = 0,
t→∞ 0
(C5(cid:48)) liminf c(t)log(cid:82)tc(s)ds = 0.
t→∞ 0
We first recall the almost sure consensus results for the case without time-delay (Zong, Li &
Zhang, 2015).
Lemma 3.4. Suppose that Assumptions 2.1, 3.1 hold and τ = 0. Then the agents reach the
1
almost sure weak consensus if the network topology graph G is connected and conditions (C1)
and (C5) hold, and only if the network topology graph G is connected and conditions (C1)
and (C5(cid:48)) hold. Moreover, the agents reach the almost sure strong consensus if and only if
the digraph G is connected and conditions (C1) and (C2) hold.
With the aid of Lemma 3.4, we now establish the almost sure consensus for the multi-
agent systems with the time-delay and additive noises.
Theorem 3.5. Suppose that Assumptions 2.1, 3.1 and condition (C1) holds and τ λ c¯< 1.
1 N
Then the agents reach the almost sure weak consensus if the network topology graph G is
connected and condition (C5) holds, and only if the network topology graph G is connected
and condition (C5(cid:48)) holds. Moreover, the agents reach the almost sure strong consensus if
and only if the network topology graph G is connected and condition (C2) holds.
Proof. Note that the almost sure weak consensus is equivalent to that for any initial data
η(t), lim (cid:107)δ (t)(cid:107) = 0, a.s., j = 2,...,N. It is enough to consider the jth component
t→∞ j
given by (9). Let λ = λ and denote the solution of (9) with τ = 0 by z (t) and the solution
j 1 1
of (9) with τ (cid:54)= 0 by z (t).
1 2
We now prove the first assertion. By Lemma 3.4, we have lim z (t) = 0 a.s., under
t→∞ 1
the connected graph and conditions (C1) and (C5). Let z(t) = z (t)−z (t), then z(0) = 0,
1 2
and we have
z˙(t) = −λc(t)z(t−τ )+c(t)g(t),t > 0, (15)
1
whereg(t) = λ(z (t−τ )−z (t))iscontinuous. ByLemma3.4, wehavethatlim (cid:107)g(t)(cid:107) =
1 1 1 t→∞
0, a.s. By means of a variation of constants formula for equation (15), we have
(cid:90) t
z(t) = r(t,s)c(s)g(s)ds,
0
where r(t,s) is the differential resolvent function, satisfying (5). By (6), we get
(cid:90) t (cid:90) t
(cid:107)z(t)(cid:107) ≤ (cid:107)r(t,s)(cid:107)c(s)(cid:107)g(s)(cid:107)ds ≤ (cid:112)b0 e−0.5(cid:37)0(cid:82)stc(u)duc(s)(cid:107)g(s)(cid:107)ds.
0 0
9
Let p(t) = (cid:82)0te0.5(cid:37)0(cid:82)0sc(u)duc(s)(cid:107)g(s)(cid:107)ds and Y(t) = e−0.5(cid:37)0(cid:82)0tc(u)dup(t), then limt→∞p(t) < ∞
or lim p(t) = ∞ since p(t) is increasing. It is easy to see from (C1) that lim (cid:107)Y(t)(cid:107) =
t→∞ t→∞
0 if lim p(t) < ∞. If lim p(t) = ∞, by L’Hˆopital’s rule, we also have
t→∞ t→∞
(cid:82)te0.5(cid:37)0(cid:82)0sc(u)duc(s)(cid:107)g(s)(cid:107)ds 2
lim (cid:107)Y(t)(cid:107) = lim 0 = lim (cid:107)g(t)(cid:107) = 0,a.s. (16)
t→∞ t→∞ e0.5(cid:37)0(cid:82)0tc(u)du (cid:37)0 t→∞
Hence, lim (cid:107)z(t)(cid:107) = 0, a.s. This together with lim (cid:107)z (t)(cid:107) = 0 gives lim (cid:107)z (t)(cid:107) =
t→∞ t→∞ 1 t→∞ 2
0, a.s., and then the ”if” part follows.
For the ”only if” part, we let lim (cid:107)z (t)(cid:107) = 0 a.s. If the graph G is disconnected, then
t→∞ 2
λ = 0. From (12), we must require M (t) to converge almost surely to any −δ (0), which
2 2 2
is impossible. That is, the graph G is connected. Now, we show lim z (t) = 0. Define
t→∞ 1
G(t) = λ(z (t) − z (t − τ )), then lim (cid:107)G(t)(cid:107) = 0, a.s., and dz (t) = −λc(t)z (t)dt +
2 2 1 t→∞ 2 2
G(t)c(t)dt+dM (t), which together with the variation of constants formula implies
j
(cid:90) t
z2(t) = z1(t)+e−λ(cid:82)0tc(u)du eλ(cid:82)0sc(u)duG(s)c(s)ds.
0
This also implies
(cid:90) t
(cid:107)z1(t)(cid:107) ≤ (cid:107)z2(t)(cid:107)+ e−λ(cid:82)stc(u)du(cid:107)G(s)(cid:107)c(s)ds. (17)
0
If (cid:82)teλ(cid:82)sc(u)du(cid:107)G(s)(cid:107)c(s)ds < ∞, then
0
0
(cid:90) t
lim e−λ(cid:82)0tc(u)du eλ(cid:82)0sc(u)du(cid:107)G(s)(cid:107)c(s)ds = 0
t→∞
0
and then we have from (17) that lim z (t) = 0, a.s. If (cid:82)teλ(cid:82)sc(u)du(cid:107)G(s)(cid:107) c(s)ds = ∞,
t→∞ 1 0 0
then, by L’Hˆopital’s rule, we also have
(cid:82)teλ(cid:82)0sc(u)du(cid:107)G(s)(cid:107)c(s)ds 1
lim 0 = lim (cid:107)G(t)(cid:107) = 0, a.s.
t→∞ eλ(cid:82)tc(u)du λ t→∞
0
This together with (17) also produces lim z (t) = 0, a.s. and thereby, the ”only if” part
t→∞ 1
follows from Lemma 3.4.
Wenowprovethesecondassertion. ByLemma3.4, wehavethattheconnectedgraphand
conditions (C1), (C2) guarantee that lim z (t) = 0 a.s. Then the proof of the ”if” part
t→∞ 1
above gives that the connected graph and conditions (C1), (C2) can guarantee the almost
sure weak consensus. By the property of the matrix L and (7), we have (13). Then the
almost sure strong consensus requires and also implies that 1 (cid:80)N x (t) converges almost
N j=1 j
surely to a random variable. Note that 1 (cid:80)N x (t) converges almost surely to a random
N j=1 j
¯
variable if and only if the limit of the continuous local martingale M(t) exists, denoted by
¯ ¯
M(∞). But this is also equivalent to lim (cid:104)M(cid:105)(t) < ∞ (Revuz & Yor, 1999, Proposition
t→∞
1.8, p. 183). Note that (cid:104)M¯(cid:105)(t) = n (cid:80)N a2 σ2 (cid:82)tc2(s)ds (Revuz & Yor, 1999, Theorem
N2 i,j=1 ij ji 0
1.8, p. 124). Hence, 1 (cid:80)N x (t) converges almost surely to a random variable if and only
N j=1 j
if (C2) holds. Moreover, the limit is x∗ = 1 (cid:80)N x (0)+M¯(∞).
N j=1 j
10