Distributed Estimation in Periodically Switching Sensor Networks

This paper studies the distributed estimation problem of sensor networks, in which each node is periodically sensing and broadcasting in order. A consensus estimation algorithm is applied, and a weight design approach is proposed. The weights are designed based on an adjusting parameter and the nodes’ lengths of their shortest paths to the target node. By introducing a (T + 2)-partite graph of the time-varying networks over a time period [0, T] and studying the relationships between the product of the time-sequence estimation error system matrices and the sequences of edges in the (T + 2)-partite graph, a sufficient condition in terms of the observer gain and the adjusting parameter for the stability of the estimation error system is proposed. A simulation example is given to illustrate the results.


Introduction
Distributed estimation is an important problem in applications of sensor networks such as health monitoring of bridges, collaborative tracking and positioning, and intelligent transportation.In many cases, although partial sensors may be unable to get the measurement of the target, cooperation between neighboring sensors in a network makes it possible that all sensors can participate in a distributed estimation process and reach an estimation of the target's varying state.
Consensus, which means the states of all agents achieve agreement, is a simple and feasible protocol for cooperation of sensors [1][2][3].Consensus protocol has been commonly applied in developing distributed estimation algorithms.For first-order data processing, consensus algorithm was applied in [4,5], and topology and weight design were discussed to optimize the estimators.For general dynamical systems, distributed estimation algorithms are composed of both consensus protocols and observers.There have been many works addressing this consensus estimation problem.The literature includes consensus Kalman filtering [6][7][8][9], Luenberger-like consensus estimation [10][11][12][13][14][15][16], and consensus H ∞ estimation [17,18].The stability conditions given in the above works are mainly based on multiple linear matrix inequalities (LMIs).
Matei and Baras [14] combined Luenberger-like observers with consensus protocol to present distributed estimation techniques for linear time-invariant systems.They showed that the weights in consensus algorithms were important for distributed detectability of networks.However, by the existing literature, it is not clear how to derive useful guidelines for weight design from LMIs-based conditions.
The weights play an essential role in the cooperation.In recent years, the works concerning designing the weights for the network are limited.Xiao et al. [5] proposed an optimal weight design for first-order data processing.Jafarizadeh [19] designed weights to optimize the second largest eigenvalue modulus of the weighted stochastic matrix.Park et al. [20] designed two weighted consensus schemes based on the edge betweenness centrality and the eigenvector centrality of the topology.Wei et al. [21] allocated the weights to minimize the  ∞ norm of the network by solving a semidefinite program problem.To the best of our knowledge there is no work providing an explicit weight design approach for consensus of time-varying networks.This paper will design a distributed estimator for sensors over periodically sensing and broadcasting networks and propose a weight design approach in the consensus protocol.Firstly, a consensus based estimation algorithm, where the 2 Mathematical Problems in Engineering weights are designed based on the length of the shortest path from each sensor to the target and an adjusting parameter, is proposed.Secondly, by introducing the ( + 2)-partite graph of the time-varying network over a time period [0, ] to depict the sequences of edges in time-sequence graphs as paths, and by developing the relationships between the product of time-sequence network stochastic matrices and the paths, a lower bound of certain value in the multiplications of the stochastic matrices in one switching period is provided.And then, based on the properties of the stochastic matrices in one switching period, a sufficient condition of the parameter and estimator gain for the stability of the networked estimation error system is further given.The main contributions of this paper lie in that we provide an explicit condition on the weight's parameter and estimation gain for periodically switching networks, and the condition requires limited topology information.
Notation.In this paper,   denotes a unit matrix of size , and 0 denotes a zero matrix with proper dimension.Re(⋅) and | ⋅ | denote the real part and modulus of one value, respectively.For a finite set V, |V| denotes the number of nodes in this set.(⋅) represents the spectral radius of a matrix.The norm ‖‖ 2   is defined as max  ̸ =0 (    /  ).diag{ 1 , . . .,   } denotes a block diagonal matrix with diagonal blocks  1 , . . .,   .⊗ denotes the Kronecker product of matrices.
[]  denotes the element in the th row and th column of matrix .

Problem Formulation
Consider a target with linear discrete-time dynamical system where  0 ∈   denotes the state of the target. ∈  × is the system matrix and not necessarily Schur stable.The target (1) is monitored by a network of  homogeneous sensors.Not all of the sensors could successfully measure the target simultaneously.If sensor  has access to the target at time , it measures the target with measuring equation where   ∈   is the measurement vector at sensor , 1 ≤  ≤ ;  ∈  × is the measurement matrix, and (, ) is assumed to be completely observable.
In practical applications, to save sensors' power and avoid congestions of communication networks, the sensors work intermittently and asynchronously.In this paper, we consider a periodically switching network satisfying the following assumptions.
Assumption 1.The available communication topology of the sensors is directed and prior given as Ĝ = (V, Ê, Â), Â = [â  ] × .Due to limited sensing range, some sensors cannot obtain the target's measurements and the available sensing vector of the sensors is prior given as B = { b1 , . . ., b }.Each sensor is periodically activated in order; i.e., if node  is activated at time , then it will be activated again at time +.At each time instant, just one node is activated and other nodes just receive information from their neighbors.If node  is activated at time , it measures the target and broadcasts its information to its neighbors, and correspondingly   () = b and   () = 0 ( ̸ = ); for all  ∈ {1, . . ., },  ̸ = , , there hold   () = â ,   () = 0, and   () = 1.
In G, a simple path of length  from  to  is a sequence of nodes  1 ,  2 , . . .,   with   =  and each subsequent edge (,  1 ), ( 1 ,  2 ), . . ., ( −1 ,   ) ∈ E. If there exists a path in G from node  to another node , then  is said to be reachable from .If a node  is reachable from every other node in V, then it is globally reachable.Assumption 2. For the given available communication topology Ĝ and sensing vector B, the target node +1 in topology G is globally reachable.Each node knows the length of its shortest path to node  + 1 in topology G.
This paper focuses on designing a distributed estimator for the time-varying network satisfying Assumptions 1-3 such that each sensor can asymptotically estimate the state of the target.

Main Results
In this section, we firstly propose a consensus based estimation algorithm.Then, we analyse the property of the stochastic matrices over the periodically switching network.Finally, we investigate the stability conditions of the estimation error system.

A Consensus Based Estimation Algorithm.
In this paper we apply a distributed estimation algorithm based on the consensus strategy.
From Assumptions 1-3, when sensor  ∈ {1, . . ., } is activated to measure the target and broadcast its information to its neighbor sensors at time , it computes its estimation by using its own available measurement information: and for other sensor , it computes its estimation using the network information based on the weighted average consensus protocol: where   ∈   denotes the estimation state made by sensor , and  ∈  × is the estimation gain to be designed.From (4), if sensor  is not sensor 's out-neighbor, i.e., â = 0, it updates its estimation based on its own information   ( + 1) =   () . ( It is well known that the weights of the network play an essential role in the stability of the estimation errors [10][11][12][13][14][15][16][17][18].In the following, we will propose a weight design approach based on the nodes' lengths of their shortest paths to node  + 1 in G, i.e.,   , 1 ≤  ≤ , and an adjusting parameter .If sensor  ∈ {1, . . ., } is activated at time , for  ̸ =  satisfying â = 1, where  ≥ 0 is an adjusting parameter to be designed,   = min{  ,   }.If  = 0,  0 = 1, and Define   () =   () −  0 () as the estimation error of sensor , 1 ≤  ≤ .Then if sensor  ∈ {1, . . ., } is activated at time , For other sensors, the estimation errors are given by Here we use   () and   ()  () to uniformly depict the time-varying sensing topology and communication topology, respectively.They are defined as follows: if sensor  ∈ {1, . . ., } is activated at time , then (1)   () = b and   () = 0 , and for  ̸ = , ,   ()  () = 0.Then, the estimation error system (11) of each sensor  can be formulated as a uniform equation , the estimation error system is where If the periodically switching system ( 13) is asymptotically stable, the estimation errors of sensors converge to zero.In the following, we discuss under what conditions system (13) in periodically switching networks satisfying Assumptions 1-3 is asymptotically stable.

Stochastic Matrices for Periodically Switching Networks.
To analyse the stability of the estimation error system, in this subsection, we will investigate the properties of the stochastic matrices over the periodically switching networks satisfying Assumptions 1-3 and give important lemmas.
To begin with, we introduce some important notions.
Example 6.Consider a network with 4 nodes.The available communication topology is prior given by Figure 1 and its bipartite graph is given in Figure 2.
For analysis simplicity, renumber the nodes such that for all nodes in V +1 their numbers are larger than those in V  , 1 ≤  ≤  − 1.By Assumptions 1-3, without loss of generality we can assume that node (1 ≤  ≤ ) is activated in order at times  +  − 1,  ≥ 0.Here we introduce the ( + 1)-partite graph of the time-varying networks.The ( + 1)-partite graph of the time-varying networks during time 0 to time and when  > 0, V} ∪ {(V (+1) , V  ) |  ∈ V}, where Ê is the edge set of the prior given communication topology.
For  ∈ V 1 , from the weight design approach we have that, for all  and for 1 ≤  ≤ , (V  , V (−1) ) ∈ E −1 , and then there exists a sequence satisfying the conditions in Lemma 8 as long as V  −1 (−1) = V (−1) for all .
In the following, we use a simple example to illustrate the result in Lemma  0,  > 1.The available communication topology is prior given by Figure 1 and, for all , (, ) ∈ E. The network is periodically switching and satisfies Assumptions 1-3.We know that V 1 = {1}, V 2 = {2, 3}, and V 3 = {4}.The order of working sensor in one period is 1,2,3,4.
When  in weights is 0, the 5-partite graphs of the network in one period are, respectively, given in Figure 4.
Back to Example 11, we have 3 .These values satisfy the condition in Lemma 12.

Stability Condition.
In this subsection, we discuss how to select  and observer gain such that in periodically working sensor networks the estimation error system (13) can asymptotically converge to zero.

Numerical Examples
To verify the validity of the proposed consensus based estimation algorithm, a case where there is one mobile target to be monitored by the network is applied and MATLAB is employed for numerical simulation.
The target is moving with second-order system  0 ( + 1) =  0 () + V 0 () , V 0 ( + 1) = √ 2V 0 () . (31) There are 4 sensors in the network trying to measure the target's position; i.e., the sensing matrix is  = [1 0].Due to limited sensing capacity, just sensor 1 can get the measurement of the target.The available communication topology is given by Figure 1 as illustrated in Example 6.
According to the topology we have that  = 2.By applying Theorem 14 we choose  = [1.54320.768]  ,  = [ 2.747 −4.8433 −4.8433 8.9409 ]; then as long as  < 0.0131, the condition in Theorem 14 holds.Selecting  = 0.01, the estimation error of the network ∑ 4 =1 ‖  ()‖ 2 2 is given in Figure 5.It is shown that the estimation error of the network converges to 0, and thus the design approach is feasible.

Conclusion
In this paper we propose a distributed estimation algorithm with a path length based weighted consensus protocol for sensor networks with periodically sensing and broadcasting scheme.( + 2)-partite graph of the time-varying networks over a time period [0, ] is introduced and three lemmas specifying the properties of the multiplications of the stochastic matrices under the periodically switching networks are given.Based on the lemmas, a sufficient condition of the stability of the estimation error is provided.The sensing models considered in this paper are all observable.Individually unobservable while collaboratively observable case is of our interest in future.