# Average Consensus Analysis of Distributed Inference with Uncertain Markovian Transition Probability

Authors: Won Il Kim; Rong Xiong; Qiuguo Zhu; Jun Wu
Journal: Mathematical Problems in Engineering
 (2013)

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## Abstract

The average consensus problem of distributed inference
in a wireless sensor network under Markovian communication
topology of uncertain transition probability is studied. A sufficient
condition for average consensus of linear distributed
inference algorithm is presented. Based on linear matrix inequalities and
numerical optimization, a design method of fast
distributed inference is provided.


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## Body

## 1. Introduction

During the past few decades, consensus problems of multiagent systems by information exchange have been extensively studied by many researchers, due to their widespread applications in autonomous spacecraft, unmanned air vehicles, mobile robots, and distributed sensor networks. Olfati-Saber and Murray introduced in [1, 2] a theoretical framework for solving consensus problems. In [3, 4], consensus problems of first-order integrator systems were proposed based on algebra graph theory. In [5, 6], consensus problems of directed second-order systems were presented. In [5], the authors provided necessary and sufficient condition for reaching mean square consensus of discrete-time second order systems. Consensus conditions were studied in [6] of continuous-time second order systems by Linear Matrix Inequality (LMI) approach.Among consensus problems, the average consensus problem is challenging which requires distributed computation of the average of the initial state of a network [1, 2]. For a strongly connected network, [1] proved that the average consensus problem is solvable if and only if the network is balanced. The discrete-time average consensus plays a key role in distributed inference in sensor networks. In networks of fixed topology, [7] gave necessary and sufficient conditions for linear distributed inference to achieve average consensus. A design method was presented in [7] to implement linear distributed inference of fastest consensus. Because of noisy communication channels, link failures often occur in a real network. Therefore, it is meaningful to study distributed inference in networks of swing topology. Through a common Lyapunov function, a result of [1] stated that distributed inference reaches average consensus in a network of swing topology if the network holds strongly connected and balanced topology. Reference [8] modeled a network of swing topology using a Bernoulli process and established a necessary and sufficient condition for average consensus of distributed inference. The condition is related to a mean Laplacian matrix.The Bernoulli process in [8] means that the network link failure events are temporally independent. From the viewpoint of engineering, it is more reasonable to consider network link failures of temporal independence. The most famous and most tractable stochastic process of temporal independence is Markovian chain in which any future event is independent of the past events and depends only on the present event. This motivates us to model a network of swing topology using a Markovian chain and hence to study distributed inference using Markovian jump linear system method [9–12]. In practice, transition probabilities of a Markovian chain are not known precisely a priori, and only estimated values of transition probabilities are available. Hence, this paper thinks of networks with Markovian communication of uncertain transition probability.In fact, in the research on networked control systems, Markovian chain has been used by several authors to describe random communication in networks. Reference [13] provided packet-loss model by Markovian chain in H∞ networked control. Under network communication of update times driven by Markovian chain, [14] gave stability conditions of model-based networked control systems. Networked control systems with bounded packet losses and transmission delays are modeled through Markovian chain in [15]. The networked predictive control system in [16] adopted 2 Markovian chains to express date transmission in both the controller-actuator channel and the sensor-controller channel.In this paper,ℤ is used to denote the set of all nonnegative integers. The real identity matrix of n×n is denoted by In. Let 1 be the vector whose elements are all equal to 1. The Euclidean norm is denoted by ∥∙∥. If a matrix P is positive (negative) definite, it is denoted by P>0(<0). The notation ⋇ within a matrix represents the symmetric term of the matrix. The expected value is represented by E[∙].The paper is organized as follows. Section2 contains a description of the network and linear distributed inference. Section 3 presents average consensus conditions and a design method. Numerical simulation results are in Section 4. Finally, Section 5 draws conclusions.


## 2. Network Description

Consider distributed inference in a wireless sensor network consisting ofn sensor. Each sensor i∈N≜{1,2,…,n} collects a local measurement yi∈ℝ about the situation of environment. It is assumed that these local measurements y1,y2,…,yn are independent and identically distributed random variables. The goal of inference is for all sensors to reach the global measurement
(1)y¯=1n∑i=1n‍yi
such that the true situation of environment can be monitored convincingly.This paper studies iterative distributed inference. Define(2)U={(s,t)∣s<t,s∈N,t∈N}
which includes all realizable undirected links in the wireless sensor network. At the kth iteration k∈ℤ, the successful communication links in the wireless sensor network are described by the set
(3)E(k)⊂U.
A pair (s1,t1)∈E(k) means that the sensors s1 and t1 communicate with each other at k. A pair (s2,t2)∈U but (s2,t2)∉E(k) means that there is no communication link between the sensors s2 and t2 at k. Due to noisy communication channels and limited network power budget, E(k) is assumed to be random and to be modeled as follows. Given m distinct subsets F1,F2,…,Fm⊂U. Let θk be a stochastic process taking values in M={1,2,…,m} and driven by a Markov chain with a transition probability matrix Γ=[γhl]∈ℝm×m, where γhl=Pr(θk+1=l∣θk=h), for all h∈M, for all l∈M. However, these γhls are not known precisely. Each γhl is expressed as
(4)γhl=γ¯hl+Δγhl,
with a known γ¯hl and a unknown Δγhl whose absolute value is less than a given positive constant 2πhl. For any for all h∈M, ∑l=1m‍γ¯hl=1 and ∑l=1m‍Δγhl=0. This paper models
(5)E(k)=Fθk.
The neighborhood of sensor i at k is denoted by
(6)Ωi(k)={j∈N∣(i,j)∈Fθkor(j,i)∈Fθk},
and the element number of set Ωi(k) is denoted by di(k).For sensori, set its initial state xi(0)=yi. At the kth iteration, each sensor i obtains its neighbors’ states and updates its state using the following linear iteration law:
(7)xi(k+1)=xi(k)+α∑j∈Ωi(k)‍(xj(k)-xi(k)),i∈N,
where α>0 is the weight parameter which is assigned by designers. Our study on the above distributed inference has two objectives: one is to derive a condition on the convergence of xi(k) to y¯ in the sense of mean square; the other is how to find α such that a fast convergence is achieved.


## 3. Average Consensus Analysis

### 3.1. Convergence Condition

Denote(8)x(k)=[x1(k)x2(k)⋮xn(k)]∈ℝn.
The system in Section 2 can be described as
(9)x(k+1)=W(α,θk)x(k),k∈ℤ,111.x(0)=[y1y2⋯yn]T,
where W(α,θk) is a n×n matrix with entries wij(α,θk). For i≠j,
(10)wij(α,θk)={α,whenj∈Ωi(k)0,whenj∉Ωi(k).
For i=j,
(11)wii(α,θk)=1-αdi(k).
From (2)~(6), it is seen that i∈Ωj(k) if and only if j∈Ωi(k), and hence
(12)W(α,θk)=WT(α,θk).Furthermore, (10)~(12) imply that for all k∈ℤ(13)1TW(α,θk)=1T,W(α,θk)1=1.
Then, we have
(14)W(α,θk)y¯1=y¯W(α,θk)1=y¯1,(15)1n11Tx(k+1)=1n11TW(α,θk)x(k)=1n11Tx(k),
which means that for all k∈ℤ(16)1n11Tx(k)=1n11Tx(k-1)=⋯=1n11Tx(0)=y¯1.
Denote e(k)=x(k)-y¯1. The iterative distributed inference is said to be average consensus in mean square sense if limk→∞E[∥e(k)∥2]=0 for any initial condition x(0)∈ℝn and θ0∈M.Theorem 1.
The linear distributed inference (7) reaches average consensus by choice of α, if there exist m positive definite matrices P1,P2,…,Pm∈ℝn×n such that for all h∈M(17)QT(α,h)(∑l=1m‍γ¯hlPl+∑l=1,l≠hm‍πhl2In)Q(α,h)+QT(α,h)×∑l=1m‍(Ph-Pl)2Q(α,h)-Ph<0
with Q(α,h)=W(α,h)-1/n11T.Proof.
Assumeθk=h∈M at time step k. From (9), (14), and (16), one has
(18)e(k+1)=x(k+1)-y¯1=W(α,h)x(k)-W(α,h)y¯1-1n11Tx(k)+y¯1=W(α,h)x(k)-W(α,h)y¯1-1n11Tx(k)+1n11Ty¯1=(W(α,h)-1n11T)(x(k)-y¯1)=Q(α,h)e(k).
We now consider the stochastic Lyapunov function(19)V(e(k),θk)=eT(k)Pθke(k).
Then for allθk=h∈M, we have
(20)E[V(e(k+1),θk+1∣e(k),θk)]-V(e(k),θk)=eT(k)(QT(α,h)∑l=1m‍γhlPlQ(α,h)-Ph)e(k)=eT(k)(QT(α,h)∑l=1m‍(γ¯hl+Δγhl)PlQ(α,h)-Ph)e(k)=eT(k)(QT(α,h)∑l=1m‍γ¯hlPlQ(α,h)+QT(α,h)×(∑l=1,l≠hm‍ΔγhlPl+ΔγhhPh)×Q(α,h)-Ph∑l=1,l≠hm‍)e(k)=eT(k)(QT(α,h)∑l=1m‍γ¯hlPlQ(α,h)+QT(α,h)(∑l=1,l≠hm‍ΔγhlPl-∑l=1,l≠hm‍ΔγhlPh)×Q(α,h)-Ph∑l=1,l≠hm‍)e(k)=eT(k)(QT(α,h)∑l=1m‍γ¯hlPlQ(α,h)+QT(α,h)(∑l=1,l≠hm‍Δγhl(Pl-Ph))×Q(α,h)-Ph∑l=1,l≠hm‍)e(k)≤eT(k)(QT(α,h)∑l=1m‍γ¯hlPlQ(α,h)+QT(α,h)∑l=1,l≠hm‍(14Δγhl2In+(Pl-Ph)2)×Q(α,h)-Ph∑l=1,l≠hm‍)e(k)≤eT(k)(QT(α,h)∑l=1m‍γ¯hlPlQ(α,h)+QT(α,h)∑l=1,l≠hm‍(πhl2In+(Pl-Ph)2)×Q(α,h)-Ph∑l=1,l≠hm‍)e(k)≤eT(k)(QT(α,h)(∑l=1m‍γ¯hlPl+∑l=1,l≠hm‍πhl2In)Q(α,h)+QT(α,h)∑l=1m‍(Pl-Ph)2Q(α,h)-Ph)e(k).
Denote(21)Θh=QT(α,h)(∑l=1m‍γ¯hlPl+∑l=1,l≠hm‍πhl2In)Q(α,h)+QT(α,h)∑l=1m‍(Ph-Pl)2Q(α,h)-Ph.
When the conditions in Theorem1 are satisfied, we have
(22)E[V(e(k+1),θk+1∣e(k),θk)]-V(e(k),θk)≤-λmin(-Θh)∥e(k)∥2≤-η∥e(k)∥2,
where λmin(-Θh) denotes the minimal eigenvalue of -Θh and
(23)η=inf{λmin(-Θh),h∈M}.
Therefore, for alle(0)∈ℝn, for all θ0∈M, for all T∈ℤ,
(24)∑k=0TE[∥e(k)∥2]≤1η∑k=0T‍(E[V(e(k),θk)]-E[V(e(k+1),θk+1)])≤1η(V(e(0),θ0)-E[V(e(T+1),θT+1)])≤1ηV(e(0),θ0).
This means limk→∞E[∥e(k)∥2]=0.


### 3.2. Optimal Design

From the above proof, it is seen that the conditions in Theorem1 result in not only limk→∞E[∥e(k)∥2]=0 but also decreasing E[V(e(k),θk)]. Moreover, limk→∞E[∥e(k)∥2]=0 implies limk→∞E[V(e(k),θk)]=0; that is, E[V(e(k),θk)] also converges to zero. Therefore, the decrease rate of E[V(e(k),θk)] can express the convergence speed of distributed inference. The following theorem is about the decrease rate of E[V(e(k),θk)].Theorem 2.
Givenα>0 and ρ∈ℝ, if there exist m positive definite matrices P1,P2,…,Pm∈ℝn×n such that for all h∈M,(25)[QT(α,h)(∑l=1m‍γ¯hlPl+∑l=1,l≠hm‍πhl2In)Q(α,h)-(1+ρ)Ph⋇(Ph-P1)Q(α,h)⋮-Imn(Ph-Pm)Q(α,h)]<0,then in linear distributed inference (7), for any nonzero e(k)∈ℝn,
(26)E[V(e(k+1),θk+1)]-E[V(e(k),θk)]E[V(e(k),θk)]<ρ.Proof.
According to Schur complement [17], condition (25) can be rewritten as
(27)QT(α,h)(∑l=1m‍γ¯hlPl+∑l=1,l≠hm‍πhl2In+∑l=1m‍(Ph-Pl)2)Q(α,h)-(1+ρ)Ph<0.
From (27), for any nonzero e(k)∈ℝn, one has
(28)eT(k)(QT(α,h)(∑l=1m‍γ¯hlPl+∑l=1,l≠hm‍πhl2In+∑l=1m‍(Ph-Pl)2)×Q(α,h)-Ph∑l=1m‍)e(k)<ρeT(k)Phe(k),
for all θk=h∈M and for all θk+1=l∈M. From (28), (19), and (20), it is known that
(29)E[V(e(k+1),θk+1∣e(k),θk)]-V(e(k),θk)<ρV(e(k),θk),
and hence
(30)E[V(e(k+1),θk+1)]-E[V(e(k),θk)]E[V(e(k),θk)]<ρ.Condition (25) in Theorem 2 is a LMI. We denote condition (25) as Ξ(α,ρ)<0 and for any α>0 define
(31)ρ_(α)=inf{ρ∣ρ∈ℝ,Ξ(α,ρ)<0has  solutions}.
Using the LMI toolbox of MATLAB, ρ_(α) can be computed by Algorithm 1.Algorithm 1
ChooseTOL>0;
Choose enough largeρ1∈ℝ such that Ξ(α,ρ1) has solutions;
Choose enough smallρ2∈ℝ such that Ξ(α,ρ2) has no solution;
repeat until ρ1-ρ2<TOL
ρ0←(ρ1+ρ2)/2;
Solve Ξ(α,ρ0)<0
if  Ξ(α,ρ0)<0 has solutions
ρ1←ρ0;
else
ρ2←ρ0;
end (repeat)
Set  ρ_(α)←ρ1.For aα>0 with ρ_<0, it is known from Theorems 1 and 2 that linear distributed inference (7) is average consensus and that ρ_ is a bound of convergence speed. Since a less value of ρ_<0 gives a faster convergence speed, the fast distributed inference problem is addressed as
(32)μ=infα>0ρ_(α)
which is an unconstrainted optimization problem of only one variable. Many existing numerical optimization methods [18] can be utilized to solve this problem efficiently. When μ<0, the optimal parameter
(33)αopt=arginfα>0ρ_(α)
provides a fast linear distributed inference which reaches average consensus.


## 3.1. Convergence Condition

Denote(8)x(k)=[x1(k)x2(k)⋮xn(k)]∈ℝn.
The system in Section 2 can be described as
(9)x(k+1)=W(α,θk)x(k),k∈ℤ,111.x(0)=[y1y2⋯yn]T,
where W(α,θk) is a n×n matrix with entries wij(α,θk). For i≠j,
(10)wij(α,θk)={α,whenj∈Ωi(k)0,whenj∉Ωi(k).
For i=j,
(11)wii(α,θk)=1-αdi(k).
From (2)~(6), it is seen that i∈Ωj(k) if and only if j∈Ωi(k), and hence
(12)W(α,θk)=WT(α,θk).Furthermore, (10)~(12) imply that for all k∈ℤ(13)1TW(α,θk)=1T,W(α,θk)1=1.
Then, we have
(14)W(α,θk)y¯1=y¯W(α,θk)1=y¯1,(15)1n11Tx(k+1)=1n11TW(α,θk)x(k)=1n11Tx(k),
which means that for all k∈ℤ(16)1n11Tx(k)=1n11Tx(k-1)=⋯=1n11Tx(0)=y¯1.
Denote e(k)=x(k)-y¯1. The iterative distributed inference is said to be average consensus in mean square sense if limk→∞E[∥e(k)∥2]=0 for any initial condition x(0)∈ℝn and θ0∈M.Theorem 1.
The linear distributed inference (7) reaches average consensus by choice of α, if there exist m positive definite matrices P1,P2,…,Pm∈ℝn×n such that for all h∈M(17)QT(α,h)(∑l=1m‍γ¯hlPl+∑l=1,l≠hm‍πhl2In)Q(α,h)+QT(α,h)×∑l=1m‍(Ph-Pl)2Q(α,h)-Ph<0
with Q(α,h)=W(α,h)-1/n11T.Proof.
Assumeθk=h∈M at time step k. From (9), (14), and (16), one has
(18)e(k+1)=x(k+1)-y¯1=W(α,h)x(k)-W(α,h)y¯1-1n11Tx(k)+y¯1=W(α,h)x(k)-W(α,h)y¯1-1n11Tx(k)+1n11Ty¯1=(W(α,h)-1n11T)(x(k)-y¯1)=Q(α,h)e(k).
We now consider the stochastic Lyapunov function(19)V(e(k),θk)=eT(k)Pθke(k).
Then for allθk=h∈M, we have
(20)E[V(e(k+1),θk+1∣e(k),θk)]-V(e(k),θk)=eT(k)(QT(α,h)∑l=1m‍γhlPlQ(α,h)-Ph)e(k)=eT(k)(QT(α,h)∑l=1m‍(γ¯hl+Δγhl)PlQ(α,h)-Ph)e(k)=eT(k)(QT(α,h)∑l=1m‍γ¯hlPlQ(α,h)+QT(α,h)×(∑l=1,l≠hm‍ΔγhlPl+ΔγhhPh)×Q(α,h)-Ph∑l=1,l≠hm‍)e(k)=eT(k)(QT(α,h)∑l=1m‍γ¯hlPlQ(α,h)+QT(α,h)(∑l=1,l≠hm‍ΔγhlPl-∑l=1,l≠hm‍ΔγhlPh)×Q(α,h)-Ph∑l=1,l≠hm‍)e(k)=eT(k)(QT(α,h)∑l=1m‍γ¯hlPlQ(α,h)+QT(α,h)(∑l=1,l≠hm‍Δγhl(Pl-Ph))×Q(α,h)-Ph∑l=1,l≠hm‍)e(k)≤eT(k)(QT(α,h)∑l=1m‍γ¯hlPlQ(α,h)+QT(α,h)∑l=1,l≠hm‍(14Δγhl2In+(Pl-Ph)2)×Q(α,h)-Ph∑l=1,l≠hm‍)e(k)≤eT(k)(QT(α,h)∑l=1m‍γ¯hlPlQ(α,h)+QT(α,h)∑l=1,l≠hm‍(πhl2In+(Pl-Ph)2)×Q(α,h)-Ph∑l=1,l≠hm‍)e(k)≤eT(k)(QT(α,h)(∑l=1m‍γ¯hlPl+∑l=1,l≠hm‍πhl2In)Q(α,h)+QT(α,h)∑l=1m‍(Pl-Ph)2Q(α,h)-Ph)e(k).
Denote(21)Θh=QT(α,h)(∑l=1m‍γ¯hlPl+∑l=1,l≠hm‍πhl2In)Q(α,h)+QT(α,h)∑l=1m‍(Ph-Pl)2Q(α,h)-Ph.
When the conditions in Theorem1 are satisfied, we have
(22)E[V(e(k+1),θk+1∣e(k),θk)]-V(e(k),θk)≤-λmin(-Θh)∥e(k)∥2≤-η∥e(k)∥2,
where λmin(-Θh) denotes the minimal eigenvalue of -Θh and
(23)η=inf{λmin(-Θh),h∈M}.
Therefore, for alle(0)∈ℝn, for all θ0∈M, for all T∈ℤ,
(24)∑k=0TE[∥e(k)∥2]≤1η∑k=0T‍(E[V(e(k),θk)]-E[V(e(k+1),θk+1)])≤1η(V(e(0),θ0)-E[V(e(T+1),θT+1)])≤1ηV(e(0),θ0).
This means limk→∞E[∥e(k)∥2]=0.


## 3.2. Optimal Design

From the above proof, it is seen that the conditions in Theorem1 result in not only limk→∞E[∥e(k)∥2]=0 but also decreasing E[V(e(k),θk)]. Moreover, limk→∞E[∥e(k)∥2]=0 implies limk→∞E[V(e(k),θk)]=0; that is, E[V(e(k),θk)] also converges to zero. Therefore, the decrease rate of E[V(e(k),θk)] can express the convergence speed of distributed inference. The following theorem is about the decrease rate of E[V(e(k),θk)].Theorem 2.
Givenα>0 and ρ∈ℝ, if there exist m positive definite matrices P1,P2,…,Pm∈ℝn×n such that for all h∈M,(25)[QT(α,h)(∑l=1m‍γ¯hlPl+∑l=1,l≠hm‍πhl2In)Q(α,h)-(1+ρ)Ph⋇(Ph-P1)Q(α,h)⋮-Imn(Ph-Pm)Q(α,h)]<0,then in linear distributed inference (7), for any nonzero e(k)∈ℝn,
(26)E[V(e(k+1),θk+1)]-E[V(e(k),θk)]E[V(e(k),θk)]<ρ.Proof.
According to Schur complement [17], condition (25) can be rewritten as
(27)QT(α,h)(∑l=1m‍γ¯hlPl+∑l=1,l≠hm‍πhl2In+∑l=1m‍(Ph-Pl)2)Q(α,h)-(1+ρ)Ph<0.
From (27), for any nonzero e(k)∈ℝn, one has
(28)eT(k)(QT(α,h)(∑l=1m‍γ¯hlPl+∑l=1,l≠hm‍πhl2In+∑l=1m‍(Ph-Pl)2)×Q(α,h)-Ph∑l=1m‍)e(k)<ρeT(k)Phe(k),
for all θk=h∈M and for all θk+1=l∈M. From (28), (19), and (20), it is known that
(29)E[V(e(k+1),θk+1∣e(k),θk)]-V(e(k),θk)<ρV(e(k),θk),
and hence
(30)E[V(e(k+1),θk+1)]-E[V(e(k),θk)]E[V(e(k),θk)]<ρ.Condition (25) in Theorem 2 is a LMI. We denote condition (25) as Ξ(α,ρ)<0 and for any α>0 define
(31)ρ_(α)=inf{ρ∣ρ∈ℝ,Ξ(α,ρ)<0has  solutions}.
Using the LMI toolbox of MATLAB, ρ_(α) can be computed by Algorithm 1.Algorithm 1
ChooseTOL>0;
Choose enough largeρ1∈ℝ such that Ξ(α,ρ1) has solutions;
Choose enough smallρ2∈ℝ such that Ξ(α,ρ2) has no solution;
repeat until ρ1-ρ2<TOL
ρ0←(ρ1+ρ2)/2;
Solve Ξ(α,ρ0)<0
if  Ξ(α,ρ0)<0 has solutions
ρ1←ρ0;
else
ρ2←ρ0;
end (repeat)
Set  ρ_(α)←ρ1.For aα>0 with ρ_<0, it is known from Theorems 1 and 2 that linear distributed inference (7) is average consensus and that ρ_ is a bound of convergence speed. Since a less value of ρ_<0 gives a faster convergence speed, the fast distributed inference problem is addressed as
(32)μ=infα>0ρ_(α)
which is an unconstrainted optimization problem of only one variable. Many existing numerical optimization methods [18] can be utilized to solve this problem efficiently. When μ<0, the optimal parameter
(33)αopt=arginfα>0ρ_(α)
provides a fast linear distributed inference which reaches average consensus.


## 4. Numerical Example

In this section, we present simulation results for average consensus of distributed inference in a simple sensor network. The network has 10 sensor nodes and switched in three possible communication situations. Figure1 illustrates 3 communication situations. The estimated transition probabilities of θk is
(34)Γ¯=[γ¯hl]=[0.70.20.10.450.30.250.50.10.4].
The estimate error Δγhl satisfies
(35)|Δγhl|≤2πhl=0.1,∀h∈{1,2,3},∀l∈{1,2,3}.
Using the computation procedure in Section 3, the optimization problem (32) is solved. Graph of ρ_(α) is displayed in Figure 2. The result is μ=-0.2407<0 and αopt=0.4812. For the communication situation in Figure 1(1), we use the design method in [7] of minimizing asymptotic convergence factor and obtain α1=0.5528. The design method in [7] is also applied to the other 2 situations in Figure 1 and get α2=α3=0.5359.Figure 1
Three communication situations.Figure 2
Graph ofρ_(α).In order to compare our method with that in [7], the initial states of each sensor node is selected as(36)x(0)=[y1y2y3y4y5y6y7y8y9y10]T=[15.214.614.315.615.314.515.714.915.114.8]T.Thus from (1), we have y¯=15. The real transition probability matrix is set as
(37)Γ=[γhl]=[0.680.240.080.40.270.330.530.110.36].
Figures 3, 4, and 5 show state curves of all sensor nodes under αopt, α1, and α2, respectively. It can be seen that all sensor states convergence to y¯=15 and that our αopt has faster convergence rate than α1 or α2 has. Achieving faster convergence is because our method considers the random switching among the 3 communication situations while [7]’s method considers only 1 communication situation.Figure 3
State curve whenαopt=0.4812.Figure 4
State curve whenα1=0.5528.Figure 5
State curve whenα2=0.5359.


## 5. Conclusion

The distributed average consensus problem in sensor networks has been studied under a Markovian switching communication topology of uncertain transition probabilities. Stochastic Lyapunov functions have been employed to investigate average consensus of linear distributed inference. A sufficient condition of average consensus has been proposed based on feasibility of a set of coupled LMIs. The design problem of fast distributed inference has been solved by numerical optimization techniques.



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*Source: 505848-2013-11-20.xml*