Average Consensus Analysis of Distributed Inference with Uncertain Markovian Transition Probability

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.

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 𝐼_𝑛. 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. Section 2 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 of 𝑛 sensor. Each sensor 𝑖∈𝑁≜{1,2,…,𝑛} collects a local measurement 𝑦_𝑖∈ℝ about the situation of environment. It is assumed that these local measurements 𝑦₁,𝑦₂,…,𝑦_𝑛 are independent and identically distributed random variables. The goal of inference is for all sensors to reach the global measurement 1𝑦=𝑛^𝑛_𝑖=1𝑦_𝑖 (1) such that the true situation of environment can be monitored convincingly.

This paper studies iterative distributed inference. Define 𝑈={(𝑠,𝑡)∣𝑠<𝑡,𝑠∈𝑁,𝑡∈𝑁} (2) 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 𝐸(𝑘)⊂𝑈. (3) 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,…,𝐹_𝑚⊂𝑈. Let 𝜃_𝑘 be a stochastic process taking values in 𝑀={1,2,…,𝑚} and driven by a Markov chain with a transition probability matrix Γ=[𝛾_ℎ𝑙]∈ℝ^𝑚×𝑚, where 𝛾_ℎ𝑙=Pr(𝜃_𝑘+1=𝑙∣𝜃_𝑘=ℎ), for all ℎ∈𝑀, for all 𝑙∈𝑀. However, these 𝛾_ℎ𝑙s are not known precisely. Each 𝛾_ℎ𝑙 is expressed as 𝛾_ℎ𝑙=𝛾_ℎ𝑙+Δ𝛾_ℎ𝑙, (4) with a known 𝛾_ℎ𝑙 and a unknown Δ𝛾_ℎ𝑙 whose absolute value is less than a given positive constant 2𝜋_ℎ𝑙. For any for all ℎ∈𝑀, ∑^𝑚_𝑙=1𝛾_ℎ𝑙=1 and ∑^𝑚_𝑙=1Δ𝛾_ℎ𝑙=0. This paper models 𝐸(𝑘)=𝐹_𝜃𝑘. (5) The neighborhood of sensor 𝑖 at 𝑘 is denoted by Ω_𝑖(𝑘)=𝑗∈𝑁∣𝑖,𝑗∈𝐹_𝜃𝑘or𝑗,𝑖∈𝐹_𝜃𝑘, (6) and the element number of set Ω_𝑖(𝑘) is denoted by 𝑑_𝑖(𝑘).

For sensor 𝑖, set its initial state 𝑥_𝑖(0)=𝑦_𝑖. At the 𝑘th iteration, each sensor 𝑖 obtains its neighbors’ states and updates its state using the following linear iteration law: 𝑥_𝑖(𝑘+1)=𝑥_𝑖(𝑘)+𝛼_{𝑗∈Ω𝑖(𝑘)}𝑥_𝑗(𝑘)−𝑥_𝑖(𝑘),𝑖∈𝑁, (7) 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 𝑥_𝑖(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 𝑥(𝑘)=𝑥₁(𝑘)𝑥₂(𝑘)⋮𝑥_𝑛(𝑘)∈ℝ^𝑛. (8) The system in Section 2 can be described as 𝑥(𝑘+1)=𝑊𝛼,𝜃_𝑘𝑥(𝑘)𝑥(0)=𝑦,𝑘∈ℤ,₁𝑦₂⋯𝑦_𝑛^𝑇, (9) where 𝑊(𝛼,𝜃_𝑘) is a 𝑛×𝑛 matrix with entries 𝑤_𝑖𝑗(𝛼,𝜃_𝑘). For 𝑖≠𝑗, 𝑤_𝑖𝑗𝛼,𝜃_𝑘=𝛼,when𝑗∈Ω_𝑖(𝑘)0,when𝑗∉Ω_𝑖(𝑘). (10) For 𝑖=𝑗, 𝑤_𝑖𝑖𝛼,𝜃_𝑘=1−𝛼𝑑_𝑖(𝑘). (11) From (2)~(6), it is seen that 𝑖∈Ω_𝑗(𝑘) if and only if 𝑗∈Ω_𝑖(𝑘), and hence 𝑊𝛼,𝜃_𝑘=𝑊^𝑇𝛼,𝜃_𝑘. (12)

Furthermore, (10)~(12) imply that for all 𝑘∈ℤ𝟏^𝑇𝑊𝛼,𝜃_𝑘=𝟏^𝑇,𝑊𝛼,𝜃_𝑘𝟏=𝟏. (13) Then, we have 𝑊𝛼,𝜃_𝑘𝑦𝟏=𝑦𝑊𝛼,𝜃_𝑘𝟏=1𝑦𝟏,𝑛𝟏𝟏^𝑇𝑥()=1𝑘+1𝑛𝟏𝟏^𝑇𝑊𝛼,𝜃_𝑘𝑥(𝑘)=1𝑛𝟏𝟏^𝑇𝑥(𝑘), (14)(15) which means that for all 𝑘∈ℤ1𝑛𝟏𝟏^𝑇𝑥(𝑘)=1𝑛𝟏𝟏^𝑇𝑥()1𝑘−1=⋯=𝑛𝟏𝟏^𝑇𝑥(0)=𝑦𝟏. (16) Denote 𝑒(𝑘)=𝑥(𝑘)−𝑦𝟏. The iterative distributed inference is said to be average consensus in mean square sense if lim_𝑘→∞𝐸[‖𝑒(𝑘)‖²]=0 for any initial condition 𝑥(0)∈ℝ^𝑛 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,…,𝑃_𝑚∈ℝ^𝑛×𝑛 such that for all ℎ∈𝑀𝑄^𝑇(𝛼,ℎ)^𝑚_𝑙=1𝛾_ℎ𝑙𝑃_𝑙+^𝑚_{𝑙=1,𝑙≠ℎ}𝜋²_ℎ𝑙𝐼_𝑛𝑄(𝛼,ℎ)+𝑄^𝑇()×𝛼,ℎ_𝑚𝑙=1𝑃_ℎ−𝑃_𝑙²𝑄(𝛼,ℎ)−𝑃_ℎ<0 (17) with 𝑄(𝛼,ℎ)=𝑊(𝛼,ℎ)−1/𝑛𝟏𝟏^𝑇.

Proof. Assume 𝜃_𝑘=ℎ∈𝑀 at time step 𝑘. From (9), (14), and (16), one has 𝑒(𝑘+1)=𝑥(𝑘+1)−()𝑥(𝑘)()𝑦𝟏=𝑊𝛼,ℎ−𝑊𝛼,ℎ1𝑦𝟏−𝑛𝟏𝟏^𝑇𝑥(𝑘)+()𝑥(𝑘)()𝑦𝟏=𝑊𝛼,ℎ−𝑊𝛼,ℎ1𝑦𝟏−𝑛𝟏𝟏^𝑇𝑥(𝑘)+1𝑛𝟏𝟏^𝑇=𝑊()−1𝑦𝟏𝛼,ℎ𝑛𝟏𝟏^𝑇𝑥(𝑘)−()𝑒(𝑘).𝑦𝟏=𝑄𝛼,ℎ (18)
We now consider the stochastic Lyapunov function 𝑉𝑒(𝑘),𝜃_𝑘=𝑒^𝑇(𝑘)𝑃_𝜃𝑘𝑒(𝑘). (19)
Then for all 𝜃_𝑘=ℎ∈𝑀, we have 𝐸𝑉𝑒(𝑘+1),𝜃_𝑘+1∣𝑒(𝑘),𝜃_𝑘−𝑉𝑒(𝑘),𝜃_𝑘=𝑒^𝑇(𝑘)𝑄^𝑇(𝛼,ℎ)^𝑚_𝑙=1𝛾_ℎ𝑙𝑃_𝑙𝑄(𝛼,ℎ)−𝑃_ℎ𝑒(𝑘)=𝑒^𝑇(𝑘)𝑄^𝑇(𝛼,ℎ)^𝑚_𝑙=1𝛾_ℎ𝑙+Δ𝛾_ℎ𝑙𝑃_𝑙𝑄(𝛼,ℎ)−𝑃_ℎ𝑒(𝑘)=𝑒^𝑇(𝑘)𝑄^𝑇(𝛼,ℎ)^𝑚_𝑙=1𝛾_ℎ𝑙𝑃_𝑙𝑄(𝛼,ℎ)+𝑄^𝑇()×𝛼,ℎ_{𝑚𝑙=1,𝑙≠ℎ}Δ𝛾_ℎ𝑙𝑃_𝑙+Δ𝛾_ℎℎ𝑃_ℎ×𝑄(𝛼,ℎ)−𝑃_ℎ𝑒(𝑘)=𝑒^𝑇(𝑘)𝑄^𝑇(𝛼,ℎ)^𝑚_𝑙=1𝛾_ℎ𝑙𝑃_𝑙𝑄(𝛼,ℎ)+𝑄^𝑇(𝛼,ℎ)^𝑚_{𝑙=1,𝑙≠ℎ}Δ𝛾_ℎ𝑙𝑃_𝑙−^𝑚_{𝑙=1,𝑙≠ℎ}Δ𝛾_ℎ𝑙𝑃_ℎ×𝑄(𝛼,ℎ)−𝑃_ℎ𝑒(𝑘)=𝑒^𝑇(𝑘)𝑄^𝑇(𝛼,ℎ)^𝑚_𝑙=1𝛾_ℎ𝑙𝑃_𝑙𝑄(𝛼,ℎ)+𝑄^𝑇(𝛼,ℎ)^𝑚_{𝑙=1,𝑙≠ℎ}Δ𝛾_ℎ𝑙𝑃_𝑙−𝑃_ℎ×𝑄(𝛼,ℎ)−𝑃_ℎ𝑒(𝑘)≤𝑒^𝑇(𝑘)𝑄^𝑇(𝛼,ℎ)^𝑚_𝑙=1𝛾_ℎ𝑙𝑃_𝑙𝑄(𝛼,ℎ)+𝑄^𝑇(𝛼,ℎ)^𝑚_{𝑙=1,𝑙≠ℎ}14Δ𝛾²_ℎ𝑙𝐼_𝑛+𝑃_𝑙−𝑃_ℎ²×𝑄(𝛼,ℎ)−𝑃_ℎ𝑒(𝑘)≤𝑒^𝑇(𝑘)𝑄^𝑇(𝛼,ℎ)^𝑚_𝑙=1𝛾_ℎ𝑙𝑃_𝑙𝑄(𝛼,ℎ)+𝑄^𝑇(𝛼,ℎ)^𝑚_{𝑙=1,𝑙≠ℎ}𝜋²_ℎ𝑙𝐼_𝑛+𝑃_𝑙−𝑃_ℎ²×𝑄(𝛼,ℎ)−𝑃_ℎ𝑒(𝑘)≤𝑒^𝑇(𝑘)𝑄^𝑇(𝛼,ℎ)^𝑚_𝑙=1𝛾_ℎ𝑙𝑃_𝑙+^𝑚_{𝑙=1,𝑙≠ℎ}𝜋²_ℎ𝑙𝐼_𝑛𝑄(𝛼,ℎ)+𝑄^𝑇(𝛼,ℎ)^𝑚_𝑙=1𝑃_𝑙−𝑃_ℎ²𝑄(𝛼,ℎ)−𝑃_ℎ𝑒(𝑘). (20)
DenoteΘ_ℎ=𝑄^𝑇(𝛼,ℎ)^𝑚_𝑙=1𝛾_ℎ𝑙𝑃_𝑙+^𝑚_{𝑙=1,𝑙≠ℎ}𝜋²_ℎ𝑙𝐼_𝑛𝑄(𝛼,ℎ)+𝑄^𝑇(𝛼,ℎ)^𝑚_𝑙=1𝑃_ℎ−𝑃_𝑙²𝑄(𝛼,ℎ)−𝑃_ℎ. (21)
When the conditions in Theorem 1 are satisfied, we have 𝐸𝑉𝑒(𝑘+1),𝜃_𝑘+1∣𝑒(𝑘),𝜃_𝑘−𝑉𝑒(𝑘),𝜃_𝑘≤−𝜆_min−Θ_ℎ‖𝑒(𝑘)‖²≤−𝜂‖𝑒(𝑘)‖², (22) where 𝜆_min(−Θ_ℎ) denotes the minimal eigenvalue of −Θ_ℎ and 𝜂=inf𝜆_min−Θ_ℎ,ℎ∈𝑀. (23)
Therefore, for all 𝑒(0)∈ℝ^𝑛, for all 𝜃₀∈𝑀, for all 𝑇∈ℤ, ^𝑇_𝑘=0𝐸‖𝑒(𝑘)‖²≤1𝜂^𝑇_𝑘=0𝐸𝑉𝑒(𝑘),𝜃_𝑘−𝐸𝑉𝑒(𝑘+1),𝜃_𝑘+1≤1𝜂𝑉𝑒(0),𝜃₀−𝐸𝑉𝑒(𝑇+1),𝜃_𝑇+1≤1𝜂𝑉𝑒(0),𝜃₀. (24) This means lim_𝑘→∞𝐸[‖𝑒(𝑘)‖²]=0.

3.2. Optimal Design

From the above proof, it is seen that the conditions in Theorem 1 result in not only lim_𝑘→∞𝐸[‖𝑒(𝑘)‖²]=0 but also decreasing 𝐸[𝑉(𝑒(𝑘),𝜃_𝑘)]. Moreover, lim_𝑘→∞𝐸[‖𝑒(𝑘)‖²]=0 implies lim_𝑘→∞𝐸[𝑉(𝑒(𝑘),𝜃_𝑘)]=0; that is, 𝐸[𝑉(𝑒(𝑘),𝜃_𝑘)] also converges to zero. Therefore, the decrease rate of 𝐸[𝑉(𝑒(𝑘),𝜃_𝑘)] can express the convergence speed of distributed inference. The following theorem is about the decrease rate of 𝐸[𝑉(𝑒(𝑘),𝜃_𝑘)].

Theorem 2. Given 𝛼>0 and 𝜌∈ℝ, if there exist m positive definite matrices 𝑃₁,𝑃₂,…,𝑃_𝑚∈ℝ^𝑛×𝑛 such that for all ℎ∈𝑀,𝑄^𝑇(𝛼,ℎ)^𝑚_𝑙=1𝛾_ℎ𝑙𝑃_𝑙+^𝑚_{𝑙=1,𝑙≠ℎ}𝜋²_ℎ𝑙𝐼_𝑛𝑄(𝛼,ℎ)−1+𝜌𝑃_ℎ⋇𝑃_ℎ−𝑃₁𝑄(𝛼,ℎ)⋮−𝐼_𝑚𝑛𝑃_ℎ−𝑃_𝑚𝑄(𝛼,ℎ)<0, (25)then in linear distributed inference (7), for any nonzero 𝑒(𝑘)∈ℝ^𝑛, 𝐸𝑉𝑒(𝑘+1),𝜃_𝑘+1−𝐸𝑉𝑒(𝑘),𝜃_𝑘𝐸𝑉𝑒(𝑘),𝜃_𝑘<𝜌. (26)

Proof. According to Schur complement [17], condition (25) can be rewritten as 𝑄^𝑇(𝛼,ℎ)^𝑚_𝑙=1𝛾_ℎ𝑙𝑃_𝑙+^𝑚_{𝑙=1,𝑙≠ℎ}𝜋²_ℎ𝑙𝐼_𝑛+^𝑚_𝑙=1𝑃_ℎ−𝑃_𝑙²𝑄(𝛼,ℎ)−1+𝜌𝑃_ℎ<0. (27)
From (27), for any nonzero 𝑒(𝑘)∈ℝ^𝑛, one has 𝑒^𝑇(𝑘)𝑄^𝑇(𝛼,ℎ)^𝑚_𝑙=1𝛾_ℎ𝑙𝑃_𝑙+^𝑚_{𝑙=1,𝑙≠ℎ}𝜋²_ℎ𝑙𝐼_𝑛+^𝑚_𝑙=1𝑃_ℎ−𝑃_𝑙²×𝑄(𝛼,ℎ)−𝑃_ℎ𝑒(𝑘)<𝜌𝑒^𝑇(𝑘)𝑃_ℎ𝑒(𝑘), (28) for all 𝜃_𝑘=ℎ∈𝑀 and for all 𝜃_𝑘+1=𝑙∈𝑀. From (28), (19), and (20), it is known that 𝐸𝑉𝑒(𝑘+1),𝜃_𝑘+1∣𝑒(𝑘),𝜃_𝑘−𝑉𝑒(𝑘),𝜃_𝑘<𝜌𝑉𝑒(𝑘),𝜃_𝑘, (29) and hence 𝐸𝑉𝑒(𝑘+1),𝜃_𝑘+1−𝐸𝑉𝑒(𝑘),𝜃_𝑘𝐸𝑉𝑒(𝑘),𝜃_𝑘<𝜌. (30)

Condition (25) in Theorem 2 is a LMI. We denote condition (25) as Ξ(𝛼,𝜌)<0 and for any 𝛼>0 define 𝜌(𝛼)=inf𝜌∣𝜌∈ℝ,Ξ𝛼,𝜌<0hassolutions. (31) Using the LMI toolbox of MATLAB, 𝜌(𝛼) can be computed by Algorithm 1.

Choose TOL>0; Choose enough large 𝜌1∈ℝ such that Ξ(𝛼,𝜌1) has solutions; Choose enough small 𝜌2∈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.

Algorithm 1 

For a 𝛼>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 𝜇=inf_𝛼>0𝜌(𝛼) (32) 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 𝛼_opt=arginf_𝛼>0𝜌(𝛼) (33) 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. Figure 1 illustrates 3 communication situations. The estimated transition probabilities of 𝜃_𝑘 is Γ=𝛾_ℎ𝑙=0.70.20.10.450.30.250.50.10.4. (34) The estimate error Δ𝛾_ℎ𝑙 satisfies Δ𝛾_ℎ𝑙≤2𝜋_ℎ𝑙=0.1,∀ℎ∈{1,2,3},∀𝑙∈{1,2,3}. (35) 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 𝛼_opt=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 of 𝜌(𝛼).

In order to compare our method with that in [7], the initial states of each sensor node is selected as𝑥(0)=𝑦₁𝑦₂𝑦₃𝑦₄𝑦₅𝑦₆𝑦₇𝑦₈𝑦₉𝑦₁₀^𝑇=15.214.614.315.615.314.515.714.915.114.8^𝑇. (36)

Thus from (1), we have 𝑦=15. The real transition probability matrix is set as Γ=𝛾_ℎ𝑙=0.680.240.080.40.270.330.530.110.36. (37) Figures 3, 4, and 5 show state curves of all sensor nodes under 𝛼_opt, 𝛼₁, and 𝛼₂, respectively. It can be seen that all sensor states convergence to 𝑦=15 and that our 𝛼_opt 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 when 𝛼opt=0.4812.

Figure 4 

State curve when 𝛼1=0.5528.

Figure 5 

State curve when 𝛼2=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.

Acknowledgment

This work was supported by 973 Program of China (Grant 2009CB320603).