Average Consensus Analysis of Distributed Inference with Uncertain Markovian Transition Probability

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.


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 continuoustime 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][10][11][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  ∞ 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 × is denoted by   .Let 1 be the vector whose elements are all equal to 1.The Euclidean norm is denoted by ‖ • ‖.If a matrix  is positive (negative) definite, it is denoted by  > 0(<0).The notation ⋇ within a matrix represents the symmetric term of the matrix.The expected value is represented by 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.

Network Description
Consider distributed inference in a wireless sensor network consisting of  sensor.Each sensor  ∈  ≜ {1, 2, . . ., } collects a local measurement   ∈ R about the situation of environment.It is assumed that these local measurements  1 ,  2 , . . .,   are independent and identically distributed random variables.The goal of inference is for all sensors to reach the global measurement such that the true situation of environment can be monitored convincingly.This paper studies iterative distributed inference.Define which includes all realizable undirected links in the wireless sensor network.At the th iteration  ∈ Z, the successful communication links in the wireless sensor network are described by the set A pair ( with a known  ℎ and a unknown Δ ℎ whose absolute value is less than a given positive constant 2 ℎ .For any for all ℎ ∈ , ∑  =1  ℎ = 1 and ∑  =1 Δ ℎ = 0.This paper models The neighborhood of sensor  at  is denoted by and the element number of set Ω  () is denoted by   ().For sensor , set its initial state   (0) =   .At the th iteration, each sensor  obtains its neighbors' states and updates its state using the following linear iteration law: where  > 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   () to  in the sense of mean square; the other is how to find  such that a fast convergence is achieved.

Optimal Design.
From the above proof, it is seen that the conditions in Theorem 1 result in not only lim that is, [((),   )] also converges to zero.Therefore, the decrease rate of [((),   )] can express the convergence speed of distributed inference.The following theorem is about the decrease rate of [((),   )].
Theorem 2. Given  > 0 and  ∈ R, if there exist m positive definite matrices  1 ,  2 , . . .,   ∈ R × such that for all ℎ ∈ , then in linear distributed inference (7), for any nonzero Proof.According to Schur complement [17], condition (25) can be rewritten as From ( 27), for any nonzero () ∈ R  , one has for all   = ℎ ∈  and for all  +1 =  ∈ .From ( 28 For a  > 0 with  < 0, it is known from Theorems 1 and 2 that linear distributed inference (7) is average consensus and that  is a bound of convergence speed.Since a less value of  < 0 gives a faster convergence speed, the fast distributed inference problem is addressed as 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  < 0, the optimal parameter provides a fast linear distributed inference which reaches average consensus.

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 The estimate error Δ ℎ satisfies Using the computation procedure in Section 3, the optimization problem (32) is solved.Graph of () is displayed in Figure 2. The result is  = −0.2407< 0 and  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  1 = 0.5528.The design method in [7] is also applied to the other 2 situations in Figure 1 and get  2 =  3 = 0.5359.
In order to compare our method with that in [7], the initial states of each sensor node is selected as

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.