# Distributed Robust Kalman Filtering with Unknown and Noisy Parameters in Sensor Networks

Authors: Donghua Chen; Ya Zhang; Cheng-Lin Liu; Yangyang Chen
Journal: Discrete Dynamics in Nature and Society
 (2018)

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## Abstract

This paper investigates the distributed filtering for discrete-time-invariant systems in sensor networks where each sensor’s measuring system may not be observable, and each sensor can just obtain partial system parameters with unknown coefficients which are modeled by Gaussian white noises. A fully distributed robust Kalman filtering algorithm consisting of two parts is proposed. One is a consensus Kalman filter to estimate the system parameters. It is proved that the mean square estimation errors for the system parameters converge to zero if and only if, for any one system parameter, its accessible node subset is globally reachable. The other is a consensus robust Kalman filter to estimate the system state based on the system matrix estimations and covariances. It is proved that the mean square estimation error of each sensor is upper-bounded by the trace of its covariance. An explicit sufficient stability condition of the algorithm is further provided. A numerical simulation is given to illustrate the results.


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## Body

## 1. Introduction

Distributed estimation is a fundamental problem in sensor networks and has attracted broad attention of the researchers. Earlier works on distributed estimation often assume that the system model is precisely known to all sensors. However, in most practical applications, an exact model of the system may not be available to sensors and the robust performance of the distributed filter should be investigated.Consensus is a simple and feasible strategy for distributed estimation over networks. There have been many consensus estimation algorithms. Olfati-Saber ([1]) proposed consensus Kalman filtering algorithms based on the consensus protocol and standard Kalman filter. Stanković et al. ([2]) gave an algorithm which was composed of decentralized overlapping estimators and a consensus scheme for sensor networks where each sensor had prior knowledge of submatrices of the system matrices. Li et al. ([3]) developed optimal and suboptimal filters and gave sufficient conditions for the stochastic stability of the suboptimal filter. Cattivelli et al. ([4]) studied the networks in which each sensor had observable aggregated measurement matrix and proposed diffusion Kalman filtering algorithms. Yang et al. ([5]) studied sensor networks where each sensor could measure the target and was activated with a certain probability. Ugrinovskii ([6, 7]) proposed a suboptimal H∞ consensus-based estimation algorithm for continuous-time system with deterministic disturbance. The above-mentioned algorithms are called one time-scale method; i.e., the consensus iteration time interval is equal to the estimation/filtering one. References [8–13] designed consensus Kalman filtering algorithms with two time-scale strategy; i.e., during one estimation/filtering interval, multiple consensus iterations were processed to fuse the prior and novel information. The two time-scale algorithms are fully distributed and their stability just requires the collective detectability of the network. However, in this algorithm vast communication costs are required. In most existing works on the one time-scale algorithm, to design the observer gains or provide stability conditions, global knowledge of the network topology is needed. References [2–7, 14–18] gave stability conditions based on LMIs or the spectral radius of the whole system matrix, which requires global topology information and large computations.Earlier works on distributed estimation in sensor networks often suppose that an exact model of the target system is known. In most practical problems, it is difficult for all sensors to exactly know the model of the system, and each sensor may just have access to partial model information. The robust performance of the filter with respect to the prior unknown or noisy system parameters is an important issue. Although there have been many works solving the robust state estimation problem ([19–21]), there have been few works designing the consensus-based robust estimation algorithm and investigating the robust performance for sensor networks. Recently, Zhang et al. ([22]) proposed a distributed Kalman/H∞ filter for sensor networks with partial unknown system parameters. Han et al. ([23]) proposed a distributed H∞ consensus filtering problem for a class of discrete-time-varying systems with stochastic nonlinearities and gave sufficient stability conditions based on the recursive linear matrix inequalities. However, the linear matrix inequalities of sensors are correlated and their verification requires global knowledge of topology and large computations.In this paper, we mainly study the distributed filtering for discrete-time-invariant system whose system matrix parameters are previously unknown or not precisely known to the sensors in the network. Each sensor may not be able to obtain the measurement. Even if one sensor can obtain the measurement, its measurement may not be observable. We design a distributed robust Kalman filtering algorithm to estimate the system parameters and system state. To estimate the system parameters, due to noises in the available parameter information, we apply a consensus Kalman filtering algorithm. We prove that the estimation errors of the system parameters converge to zero in mean square sense, if and only if for any one of the system parameters its corresponding node subset in which each node can obtain this parameter information is globally reachable. To estimate the system state, due to parameter estimation errors, we apply a consensus robust Kalman filtering algorithm where the consensus weights are designed based on the covariances of the sensors. We provide a sufficient condition guaranteeing the boundedness of the mean square estimation errors of the sensors. The contributions of this paper include the following: (1) to propose a fully distributed robust Kalman filtering algorithm to deal with the noises in available system parameters; (2) to provide a sufficient stability condition based on some uncorrelated LMIs, whose computation is small and does not require global topology information.Notation.In this paper, Ip denotes a unit matrix of size p. tr(·), ·∗, and ·2 represent the trace, the trace norm, and the 2-norm of a matrix, respectively. x denotes the 2-norm of vector x. diag{G1,…,Gn} denotes a block-diagonal matrix with diagonal blocks G1,…,Gn. Matrix P is positively definite and is written as P>0. E(X) denotes the expectation of X. For a finite node set V, |V| denotes the number of nodes in this set.


## 2. Problem Formulation

Consider a target system with discrete-time linear dynamics(1)x0k+1=A0x0k+vk,where x0∈Rp denotes the state of the system and v(k)∈Rp is a system Gaussian noise with zero mean and covariance matrix W0>0. It is supposed that A0∈Rp×p is nonsingular.Suppose target (1) is monitored by a network of n sensors with the following measuring system:(2)yik=Cix0k+ωik,where, for sensor i∈{1,2,…,n}, yi∈Rqi is the measurement vector at sensor i, Ci∈Rqi×p is the measuring matrix, and ωi(k)∈Rqi is the Gaussian measurement white noise with zero mean and covariance matrix Wi>0. The measurement noises of different sensors and system noise are uncorrelated. Here (A0,Ci) may not be observable.In the network, some sensors may not be able to obtain the measurements (2) due to limited sensing range and energy saving. Let bi denote whether the measurement is available. If sensor i can obtain its measurement (2), then bi=1; otherwise bi=0. Let B={bi,1≤i≤n} (bi∈{0,1}) represent the sensing topology.In applications, it is a really strong assumption that all sensors in the network precisely know the system parameters of the target. In many cases, if one sensor can just obtain the measurement outputs with respect to partial system state, it is likely to obtain the corresponding partial parameters in system matrixA0. This paper discusses the case in which the parameters may not be previously known to some sensors and sensors’ available system information contains noises. For (s,l)∈N (N={(s,l),1≤s,l≤p}), we apply bisl to denote whether the sth rank, lth column element of matrix A0 is known to sensor i. If the information is unknown to i, bisl=0; otherwise, bisl=1 and sensor i’s corresponding available system information Gisl(k) is(3)Gislk=A0sl+δislk.where A0=[A0sl]p×p and δisl(k) is a zero mean Gaussian white noise with variance Xisl, which is independent of v(k), ωi(k), and δi1s1l1(k) for any one different i1, s1, or l1 (1≤i1≤n,1≤s1,l1≤p).In this network, even if one sensor can obtain the measurement of the target, it does not completely know the target’s system matrix and its measuring system may be also not observable, which makes it infeasible to estimate the state of the target without cooperation. The objective is to construct an estimation algorithm on the basis of the local system parameter information, local measurements (when available), and data received from all adjacent neighbors to estimate the state of the target.


## 3. Distributed Filtering Algorithm

To explain the principle of graph theory in distributed estimation, preliminaries about graph theory are firstly introduced.In the cooperative estimation in the sensor network, each sensor is treated as a node and the nodes communicate according to the communication graph. For a communication digraphG=(V,E,A), V={1,…,n} denotes the node set, E={(i,j),1≤i,j≤n} denotes the edge set, and A denotes the adjacency matrix. If node i can receive information from node j, then there is a corresponding edge in G; i.e., (i,j)∈E. The adjacency elements associated with the edges of the graph are defined as aij=1⇔(i,j)∈E, aij=0⇔(i,j)∉E. Since each node can always get its own information, for all i, (i,i)∈E. Neighbor set of sensor i is denoted as Ni={j:(i,j)∈E}.InG, a simple path of length l from i to j is such that there exists a sequence of nodes i1,i2,…,il-1 with subsequent edges (i,i1),(i1,i2),…,(il-1,j)∈E. For node i and a node subset V0, there exists at least one path from i to the node set V0 if there exists at least one node j∈V0 such that there is a path from node i to node j. A node subset V0 is said to be globally reachable in the communication topology if, for any node in in V, there exists at least one path from i to the node set V0.Consensus protocol is a simple and effective strategy in cooperation of sensors. In this distributed filtering algorithm, two consensus processes, aimed at estimating the system parameters and system state, respectively, are included. Let us assume that, at timek, each node i∈V in the sensor network can transmit its system matrix information, including parameter estimation error covariances {Yisl(k),(s,l)∈N} and parameter estimations {Aisl(k),(s,l)∈N}, and its system state estimation information, including measurement yi(k), measurement matrix Ci, measurement noise covariance Wi (when available), state estimation error covariance Pi(k), state covariance Qi(k), and estimated state xi(k), to its neighbors. Then the consensus-based distributed robust Kalman filtering algorithm is summarized by Algorithm 1.Algorithm 1
1.Initialization: Ai=Ai(0), Pi(0)=S1, Qi(0)=S2, xi=xi(0); 2.while new data is received, do 3.Locally compute the weighted average of estimations and covariances about system parameters: A ^ i = ∑ j ∈ N i m i j A j; Y ^ i = ∑ j ∈ N i m i j Y j . 4.Compute the Kalman estimate of the system parameters: A i s l = A ^ i s l + b i s l Y ^ i s l Y ^ i s l + X i s l - 1 G i s l - A ^ i s l, A i = A i s l 1 ≤ s , l ≤ p, Y i s l = Y ^ i s l - b i s l Y ^ i s l Y ^ i s l + X i s l - 1 Y ^ i s l ,   s , l ∈ N . 5.Locally aggregate data and compute the weighted average of estimations and covariances about state estimation: z i = y i 1 T y i 2 T ⋯ y i M i T T, C - i = C i 1 T C i 2 T ⋯ C i M i T T, W - i = d i a g W i 1 , W i 2 , ⋯ , W i M i, P ^ i = ∑ j ∈ N i w i j P j, Q ^ i = ∑ j ∈ N i w i j Q j, x ^ i = ∑ j ∈ N i w i j x j . where is∈Ni∗={j∈V:aij=1 and bj=1}, Mi=Ni∗ is the number of nodes in set Ni∗; if (Ai,C-i) is observable, wii=1 and for j≠i, wij=0, else w i j = 1 / P j ∗ ∑ s ∈ N i 1 / P s ∗ . 6.Compute the robust Kalman estimate of the target state: if there exists αi>0 such that α i I p - N i Q ^ i N i T > 0, then xi=A-ix^i+Fizi-C-ix^i; where, A - i = A i + α i I p + W 0 A i - T R i, F i = A - i P ^ i C - i T C - i P ^ i C - i T + W - i - 1, N i = max l ⁡ ∑ s = 1 p Y i s l 1 / 2 I p, R i = Q ^ i - 1 - α i - 1 N i N i T, P i and Qi are updated using the following equation: P i = A - i P ^ i A - i T - A - i P ^ i C - i T C - i P ^ i C - i T + W - i - 1 C - i P ^ i A - i T + α i I p + W 0 A i - T R i A i - 1 α i I p + W 0 + α i I p + W 0, Q i = A i R i - 1 A i T + α i I p + W 0 .For nodei, when new data is received, it firstly locally computes the weighted average of its neighbors’ estimations and covariances for the parameters in the system matrix:(4)A^ik=∑j∈NimijAjkY^ik=∑j∈NimijYjkwhere A^i and Y^i denote the fused estimation and covariance for system matrix, respectively, mij is the weight of edge (i,j) and can be any positive value satisfying ∑j∈Ni‍mij=1, and Ni denotes the neighbor set of sensor i. And then a local Kalman filtering is constructed to estimate the system parameters.In the second part, sensori locally aggregates the measurement vectors, measurement matrices, and measurement noise covariance of its neighbors i1,i2,…,iMi, respectively, and obtains zi(k)=yi1T(k)yi2T(k)⋯yiMiT(k)T, C¯i=Ci1TCi2T⋯CiMiTT, W¯i=diag{Wi1,Wi2,…,WiMi}. Then it computes the weights of in-edges wij(k) (j∈Ni) and obtains the weighted average of its neighbors’ covariance matrices and estimated states(5)P^ik=∑j∈NiwijkPjk,Q^ik=∑j∈NiwijkQjk,x^ik=∑j∈Niwijkxjk.In the third part, a local robust Kalman filter is constructed based on the aggregated measurements and weighted average estimate state for each sensor node to estimate the state of the target. For sensori, (6)xik+1=A-ikx^ik+Fikzik-C-ix^ik,(7)Pik+1=A-ikP^ikA-iTk-A-ikP^ikC-iTC-iP^ikC-iT+W-i-1C-iP^ikA-iTk+αiIp+W0Ai-TkRiAi-1kαiIp+W0+αiIp+W0,where A¯i(k) and Fi(k) are the filter gains to be determined. Assume xi(0)=0.The weights in the consensus estimation of the system matrix can be any positive values with row sum being equal to 1, while in the consensus estimation of the system state, due to the dynamical variation of the state, the weights cannot be arbitrary. When the weights are not properly designed, the estimation error of the overall network may be divergent. In [6, 7, 14–18] the weights are constants and concerning conditions are implied in the LMIs-based stability conditions, which is difficult to be computed. Here we apply a fully distributed weight design approach based on the traces of the estimation covariances of the sensors.At timek, for one node i, if (Ai(k),C¯i) is observable, then the weight of edge (i,j)∈E is designed as wii(k)=1 and wij(k)=0 for node j≠i, j∈Ni; otherwise(8)wijk=1/Pjk∗∑s∈Ni1/Psk∗,where Pi(k)∗ denotes the trace of covariance matrix Pi(k).Remark 1.
The weighted average consensus-based Kalman filter in Algorithm1 is a little similar to that proposed in [15] where Matei et al. studied sensor networks with completely known parameters and applied a distributed suboptimal filter with weighted average consensus protocol and standard Kalman filter. For sensor i,(9)P^ik=∑j∈NiwijPjk,x^ik=∑j∈Niwijxjk.and(10)x^ik=Axik-1+Fikyik-CiAxik-1,Pik=AP^ik-1AT+W0-AP^ik-1CiTCiP^ik-1CiT+Wi-1CiP^ik-1AT,Fik=P^ik-1CiTCiP^ik-1CiT+Wi-1.By Markovian jump linear system, a sufficient stability condition is given based on the feasibility of the following LMIs:(11)Qi>∑j∈NiwijAQjAT+W0-AQjCjTCjQjCjT+Wj-1CjQjAT.If the system matrix is not precisely known to sensors and just its estimation can be used, when estimating the system state, standard Kalman filter is not feasible and a robust Kalman filtering algorithm is designed in this paper.Remark 2.
In the network, when one sensor can obtain the measurement outputs of the process, it just obtains partial parameters of the system matrix with noises. Thus, Algorithm1 contains two consensus-based filtering processes. One is a consensus Kalman filtering for system parameters. The other is a distributed filtering for system state. Moreover, being different from existing algorithms ([6, 7, 14–18]), the weights in the consensus filtering for system state are adaptively designed. The correction step for filtering is done at sensor levels, and thus the filtering algorithm is completely distributed.


## 4. Stability Analysis

In this section, we analyze the stability properties of the distributed robust Kalman filtering algorithm in Algorithm1. In this paper, we assume the network is time-invariant.Firstly, we analyze the estimation of the system parameters. For sensori∈V and (s,l)∈N, from Algorithm 1 we have that(12)A^islk+1=∑j∈NimijAjslk,(13)Y^islk+1=∑j∈NimijYjslk,with(14)Aislk=A^islk+bislY^islkY^islk+Xisl-1Gislk-A^islk,(15)Yislk=Y^islk-bislY^islkY^islk+Xisl-1Y^islk,where bisl∈{0,1} (1≤s,l≤p). If sensor i has an access to the parameter A0sl with noise, bisl=1; otherwise bisl=0.DefineΔAisl(k)=Aisl(k)-A0sl, ΔA^isl(k)=A^isl(k)-A0sl as sensor i’s estimation error and fused estimation error of system parameter A0sl, respectively. From (12) and (14) we have that(16)ΔA^islk+1=∑j∈Nimij(ΔA^jslk-bjslY^jslkY^jslk+Xjsl-1ΔA^jslk-δjslk,and, obviously, E((ΔA^isl(k))2)≤Y^isl(k) holds for all 1≤i≤n,1≤s,l≤p, when E((ΔA^isl(0))2)=Y^isl(0).Proposition 3.
For any(s,l)∈N and any node i∈V, limk→∞Y^isl(k)=0 holds if and only if in the communication topology there exists at least one path from i to the node set Vsl, where Vsl={j:bjsl=1,j∈V}.Proof.
(Sufficiency). If in the graph there exists one node i0, bi0sl=1, and node i0 is globally reachable from any other node, then we partition the graph into strongly connected subgraphs (Vi,Ei,Ai), 1≤i≤m. Renumber the nodes such that M=[Mij]m×m is an upper block-diagonal matrix and the node subset V1 corresponding to M11 contains node i0. Define ξisl(k) as a vector stacking the values Y^jsl(k), j∈Vi, ξsl(k)=[(ξ1sl(k))T,…,(ξmsl(k))T]T, and define ηsl(k) as a vector stacking the values Y^jsl(k)(Y^jsl(k)+Xjsl)-1Y^jsl(k),1≤j≤n, Bsl=diag{b1sl,…,bnsl}. Then,(17)ξslk+1=Mξslk-MBslηslk.
For irreducible stochastic matrixM11, there exists a vector β1=[β11,…,β1n1]T with positive elements satisfying that β1TM11=β1T. Thus, ∑i∈V1‍β1iY^isl(k+1)=∑i∈V1‍β1iY^isl(k)-∑i∈V1‍β1i∑j∈Ni‍bjslY^jsl(k)(Y^jsl(k)+Xjsl)-1Y^jsl(k). Since the subgraph is strongly connected, if for one node i∈V1, Y^isl(k)=0, then from (13), we have that, for any other node j∈V1, Y^jsl(k)=0 holds. When Y^i0sl(k)>0, since i0∈V1, we have that ∑i∈V1‍β1i∑j∈Ni‍bjsl>0 and then ∑i∈V1‍β1iY^isl(k+1)<∑i∈V1‍β1iY^isl(k), which infers that limk→∞∑i∈V1‍β1iY^isl(k)=0, and thus for i∈V1, limk→∞Y^isl(k)=0.
For matrixM22, there exists a diagonal matrix D22, in which the diagonal elements are all larger than 1 and there exists at least one diagonal element being strictly larger than 1, such that D22M22 is an irreducible stochastic matrix. Define β2=[β21,…,β2n2]T with positive elements satisfying that β2TD22M22=β2T, and define β2TM22=[β^21,…,β^2n2]T; then for 1≤j≤n2, β2j≥β^2j, and there exists at least one j0,1≤j0≤n2; β2j0>β^2j0 holds. Thus, if for all i∈V2, Y^isl(k)>0, ∑i∈V2‍β2iY^isl(k+1)<∑i∈V2‍β2iY^isl(k)+∑i∈V2‍β2i∑j∈Ni∩V1‍Y^jsl(k)-∑i∈V2‍β2i∑j∈Ni‍bjslY^jsl(k)(Y^jsl(k)+Xjsl)-1Y^jsl(k). From the above analysis, for i∈V1, limk→∞Y^isl(k)=0; thus, for i∈V2, limk→∞Y^isl(k)=0 also holds.
Similarly, fori∈Vs(s>1), limk→∞Y^isl(k)=0. Therefore, if in the graph there exists one node i0, bi0sl=1, and node i0 is globally reachable from any other node, then limk→∞Y^isl(k)=0 holds for all i.
If there is no globally reachable nodei0 with bi0sl=1, then the graph can be partitioned into subgraphs; in each subgraph there exists one node denoted as i0, bi0sl=1, and node i0 is globally reachable from any other node in the subgraph. From the above analysis, it is not difficult to prove that limk→∞Y^isl(k)=0 holds for all i. The sufficiency part has been proved.
(Necessity).If there exists a node having no path from itself to the node set Vsl, then there exists a node subset in which each node has no neighbor in other node subsets and is not be able to obtain the information of A0sl. Obviously, the prediction error covariances of the nodes in this subset cannot converge to 0. The necessity part has been proved.In the following, we analyze the estimation for the system state. To begin with, we give a lemma about robust Kalman filtering for a system with unknown random uncertainty modeled by white noise.Lemma 4.
Consider a target with discrete-time system:(18)ζk+1=Aζk+vk,(19)yk=Cζk+ωk,where x(k)∈Rp is the state; y(k)∈Rq is the measured output with observable measurement matrix C; v(k) and ω(k) are uncorrelated zero mean Gaussian white noises with variance matrix W0 and W, respectively. The observer cannot obtain the precise system matrix and just obtains the value with noise; i.e., A(k)=A+ΔA(k), where ΔA(k)=[ΔAsl]p×p represents the unknown parameter noise and satisfies E(ΔA(k))=0, E(ΔAs1l1(k)ΔAs2l2(k))=0 for any s1≠s2 or l1≠l2, E((ΔAsl(k))2)≤Ysl(k). If there exist α(k)>0 such that(20)αkIp-NkQ2kNTk>0,where Q1(k), Q2(k) satisfy the following two Riccati equations(21)Q1k+1=A¯kQ1kA¯Tk-A¯kQ1kCTCQ1kCT+W-1CQ1kA¯Tk+αkIp+W0A-TkRkA-1kαkIp+W0+αkIp+W0(22)Q2k+1=AkR-1kATk+αkIp+W0and(23)Nk=maxl⁡∑s=1pYslk1/2Ip,Rk=Q2-1k-α-1kNTkNk,A¯k=Ak+αkIp+W0A-TkRk,then, the filter(24)ζ^k+1=A¯kζ^k+Fkyk-Cζ^k,with F(k)=A¯(k)Q1(k)CT(CQ1(k)CT+W)-1 satisfies(25)Eζk-ζ^kζk-ζ^kT≤trQ1kif E((ζ(0)-ζ^(0))T(ζ(0)-ζ^(0)))<Q1(0)00Q2(0).Proof.
Definee(k)=ζ(k)-ζ^(k), ez(k)=z(k)-z^(k); the augmented system of (19) and (24) can be expressed by(26)ζek+1=Aek+ΔAekζek+Bekw¯k,where(27)ζek=eTk,ζTkTw¯k=vTk,ωTkT,Aek=A¯k-FkCAk-A¯k0Ak,ΔAek=0ΔAk0ΔAk,Bek=Ip-FkIp0.
From the properties of the parameter noises, we haveE(ΔAT(k)ΔA(k))=diag{∑s=1p‍Ys1(k),…,∑s=1p‍Ysp(k)}. Define N(k)=maxl{(∑s=1p‍Ysl(k))1/2}Ip; it is obvious that E(ΔAT(k)ΔA(k))≤NT(k)N(k).
DefineΣ(k)=E(ζe(k)ζeT(k)), then from (26),(28)Σk+1=EAek+ΔAekΣkAek+ΔAekT+BekW~kBeTkwhere(29)ΔAek=HΔAkN-1kN¯k,H=IpIp,N¯k=0Nk,W~k=W000W,and satisfy E((ΔA(k)N-1(k))T(ΔA(k)N-1(k)))≤Ip.
From [19], if there exists α(k) such that α(k)Ip-N¯(k)Σ(k)N¯T(k)>0, then(30)Aek+ΔAekΣkAek+ΔAekT≤AekΣ-1k-α-1kN¯TkN¯k-1AeTk+αkHHT.Therefore,(31)Σk+1≤AekΣ-1k-α-1kN¯TkN¯k-1AeTk+αkHHT+BekW~kBeTk.Define an equation(32)Σ¯k+1=AekΣ¯-1k-α-1kN¯TkN¯k-1AeTk+2αkHHT+BekW~kBeTkwith Σ¯(0)=Σ(0). Obviously, Σ¯(k)≥Σ(k) holds for all k. Thus, α(k)Ip-N¯(k)Σ¯(k)N¯T(k)>0 implies α(k)Ip-N¯(k)Σ(k)N¯T(k)>0.
IfΣ¯(k) is assumed to have the form Q1(k)00Q2(k), from (32) we have Σ¯(k+1)=Σ¯11(k+1)Σ¯12(k+1)Σ¯21(k+1)Σ¯22(k+1), where(33)Σ¯11k+1=A¯k-FkCQ1kA¯k-FkCT+Ak-A¯kR-1kAk-A¯kT+αkIp+W0+FkWFTk,Σ¯12k+1=Ak-A¯kR-1kATk+W0+αkIp,Σ¯21k+1=Σ¯12Tk+1,Σ¯22k+1=AkR-1kATk+αkIp+W0,Rk=Q2-1k-α-1kNTkNk.
WhenA¯(k)=A(k)+(α(k)Ip+W0)A-T(k)R(k), there holds Σ¯12(k+1)=0, and thus Σ¯(k) is a block-diagonal positive definite matrix with diagonal blocks.(34)Q1k+1=A¯k-FkCQ1kA¯k-FkCT+Ak-A¯kR-1kAk-A¯kT+αkIp+W0+FkWFTk(35)Q2k+1=AkR-1kATk+αkIp+W0
In order to find the optimal filter gainsF(k) minimizing Q1(k), we take the first variation to (34) with respect to F(k) and equal it to zero.(36)∂Q1k+1∂Fk=A¯k-FkCQ1k-CT+FW=0From (36) we obtain the optimal gains F(k)=A¯(k)Q1(k)CT(CQ1(k)CT+W)-1. This completes the proof.In the following we analyze the stability of the distributed robust Kalman filtering algorithm.For sensori, i∈V, it can only use its own estimated system matrix Ai(k), and then the target dynamics can be depicted by(37)x0k+1=Aik-ΔAikx0k+ω0k.Here ΔAik=Aik-A and satisfies E(ΔAis1l1(k)ΔAis2l2(k))=0 for any s1≠s2 or l1≠l2, E((ΔAisl(k))2)≤Yisl(k).From Algorithm1 we have that(38)x^ik=∑j∈Niwijkxjk,(39)xjk+1=A¯jkx^jk+Fjkzjk-C¯jx^jk,(40)P^ik=∑j∈NiwijkPjk,(41)Q^ik=∑j∈NiwijkQjk,where wij(k)≥0 with ∑j∈Niwij(k)=1 being the consensus weight of edge (i,j) in the target of state estimation. Fi(k) and A¯i(k) are filter gains given by Algorithm 1; Pi(k) and Qi(k) satisfy the following equations:(42)Pik+1=A¯ik-FikC¯iP^ikA¯ik-FikC¯iT+Aik-A¯ikRi-1kAik-A¯ikT+αikIp+W0+FikW¯iFiTk(43)Qik+1=AikRi-1kAiTk+αikIp+W0.Defineei(k)=xi(k)-x0(k), e^i(k)=x^i(k)-x0(k) as sensor i’s estimation error and collective estimation error, respectively. Then, from (38) we have that(44)e^ik+1=∑j∈Niwijkejk+1,(45)ejk+1=A¯jk-FjkC¯je^jk+A¯jk-Ajkx0k+ΔAjkx0k+Fjkω¯jk-vk.Letξi(k)=[eiT(k),x0T(k)]T and ξ^i(k)=[e^iT(k),x0T(k)]T. Then E(ξ^i(k)ξ^iT(k))≤∑j∈Ni‍wij(k)E(ξj(k)ξjT(k)). From the proof of Lemma 4, if there exists αi(k)>0 such that αi(k)Ip-N¯i(k)Σ^i(k)N¯iT(k)>0, then(46)Eξik+1ξiTk+1≤Σik+1where(47)Σik+1=ΠikΣ^i-1k-αi-1kN¯iTkN¯ik-1ΠiTk+αikI2p+BikW~ikBiTk(48)Σ^ik=∑j∈NiwijkΣjk,(49)Πik=A¯ik-FikC¯iAik-A¯ik0Aik,Bik=Ip-FikIp0,W~ik=W000W¯i,N¯ik=0Nik.Furthermore, if E(ξi(0)ξiT(0))≤Pi(0)00Qi(0), then Σi(k) and Σ^i(k) are both block-diagonal matrices with diagonal blocks Pi(k), Qi(k) and P^i(k), Q^i(k), respectively. The inequality αi(k)Ip-N¯i(k)Σ^i(k)N¯iT(k)>0 is equivalent to(50)αikIp-NikQ^ikNiTk>0.Moreover, E(eiT(k)ei(k))≤Pik∗. Thus, we can obtain the following result.Theorem 5.
Consider a system (1) being monitored by sensor networks (2). Then, under Algorithm 1 the mean square estimation error of sensor i is upper-bounded by Pi(k)∗, i∈V.In the following, we focus on discussing the boundedness of the sequence{Pi(k),k≥0}. Firstly we definecollaboratively observable as follows. A node i is said to becollaboratively observable if (A0,C¯i) is observable.Suppose the sensor network satisfies the following assumption.(A1)
In the network, for any given(s,l)∈N, for any node j∈V, there exists at least one path from j to the node set Vsl, where Vsl={i:bisl=1,i∈V}.(A2)
In the network, there exists at least one collaboratively observable node. LetV0 denote the set of collaboratively observable nodes; i.e., V0={i∈V:(A0,C¯i)iscompletelyobservable}. The node subset V0 is globally reachable in the communication topology G.Notice that in (A2) each node’s measurement matrix is not required to be observable. Just one node’s aggregated measurement matrix is required to be observable. Although this assumption is a little more conservative than cooperative observability of all sensors, it is easy to be satisfied by properly placing sensors.Theorem 6.
Consider a system (1) being monitored by sensor networks (2) satisfying (A1) and (A2). For i∈V0, suppose there exist a constant αi>0 such that the following three matrix inequalities (51)-(53) have positive definite solutions Pi and Qi:(51)Pi>A¯iPiA¯iT-A¯iPiC¯iTC¯iPiC¯iT+W¯i-1C¯iPiA¯iT+αiIp+W0A0-TRiA0-1αiIp+W0+αiIp+W0,(52)Qi>A0Ri-1A0T+αiIp+W0,(53)αiIp-NiQiNiT>0where (54)A¯i=A0+αiIp+W0A0-TRi,Fi=A¯iPiC¯iTC¯iPiC¯iT+W¯i-1,Ni=limk→∞⁡sup⁡maxl⁡∑s=1pYislk1/2Ip,Ri=Qi-1-αi-1NiNiT.Then, under Algorithm 1 the mean square estimation errors of all sensors are bounded.Proof.
From Proposition3, we have that under (A1), for any node i, limk→∞E(ΔAiT(k)ΔAi(k))=0. Thus, from (A2) there exists a finite time k0≥0, for any node i∈V0, when k≥k0, (Ai(k),C¯i) is observable. From Algorithm 1, in the consensus-based robust Kalman filtering part, if sensor i∈V0, then x^i(k)=xi(k), P^i(k)=Pi(k), Q^i(k)=Qi(k), and (55)xik+1=A¯ikxik+Fikzik-C¯ixik,(56)Pik+1=A¯ik-FikC¯iPikA¯ik-FikC¯iT+Aik-A¯ikRi-1kAik-A¯ikT+αikIp+W0+FikWiFiTk(57)Qik+1=AikRi-1kAiTk+αikIp+W0.And the LMI is as follows.(58)αikIp-NikQikNiTk>0The above filter is the one discussed in Lemma 4. From Proposition 3, Ni(k)NiT(k)≤NiNiT, Ni=limk→∞supmaxl{(∑s=1p‍Yisl)1/2}Ip must be bounded, and Ai converges to A0 in mean square sense, by setting the initial conditions to zero and replacing the Riccati difference equations by the corresponding algebraic Riccati equations ([21]), the stability of the Riccati difference equations can be converted to the feasibility of inequalities (51)-(53). If for i∈V0, inequalities (51)-(53) are feasible, P^i(k) must be bounded.
In the following, we discuss the other nodes.
Firstly, we prove that if fori∈V0, (52)-(53) are feasible, for any other node i, there exists αi(k)>0 such that (50) holds with Qi(k) satisfying (43).
DefineNmax=maxi{Ni}; then by replacing Ni with Nmax, the LMIs (52)-(53) are still feasible, which means there exists α¯(k)>0 such that α¯(k)Ip-NmaxQ¯(k)NmaxT>0 and Q¯(k+1)=A0(Q¯-1(k)-α¯-1(k)NmaxNmaxT)-1A0T+α¯(k)Ip+W0. When Q¯(0)≥Qi(0), since Ni(k)≤Nmax, there holds Q¯(k)≥Qi(k) and α¯Ip-Ni(k)Qi(k)NiT(k)>0 with Qi(k) satisfying Qi(k+1)=A0(Qi-1(k)-α¯-1(k)Ni(k)NiT(k))-1A0T+α¯(k)Ip+W0.
DefineQ^i(k)=∑j∈Ni‍wij(k)Qj(k). It is obvious that Q¯(k)≥Q^i(k), Q¯(k)≥Qi(k), and α¯(k)Ip-NmaxQ^i(k)NmaxT>0 with Qi(k) satisfying Qi(k+1)=A0(Q^i-1(k)-α¯-1(k)Ni(k)NiT(k))-1(k)A0T+α¯(k)Ip+W0. Since Ai converges to A0 in mean square sense, there also exists αi(k)>0 such that (58) holds with Qi(k) satisfying (43).
Secondly, we investigate the boundedness of the estimation errors. Fori∈V, define ϕi(k)=tr(Pi(k)), ϕ^i(k)=tr(P^i(k)). From (40) we obtain that(59)ϕ^ik+1=∑j∈Niwijkϕjk=∑j∈Ni1∑s∈Ni1/ϕsk≤diϕjk.where di=∑j∈Ni‍aij, j is any one of sensor i’s neighbors.
Since fori∈V0,  ϕ^i(k) and ϕi(k) are bounded, then for any one of sensor i’s out neighbors io, ϕ^io(k) must be bounded by dioϕi(k), which means that the mean square estimation error of node io must be also bounded. For node io’s out neighbor, its mean square estimation error must be also bounded.
Since from (A2),V0 is globally reachable, for any node i, there exists at least one path from i to some node in V0. So by the same way, for any node i,  ϕ^i(k) must be bounded. This completes the proof.Remark 7.
Compared with the stability results in [6, 7, 14–18], the condition in Theorem 6 is based on some uncorrelated LMIs, which is easy to be verified and does not depend on the global topology information.Remark 8.
The feasibility of (51)-(53) does not depend on the value of Ni as long as Ni is bounded. Since from Proposition 3, Yisl(k) must be bounded, the Ni in LMIs (51)-(53) can be replaced by the identity matrix.


## 5. Numerical Examples

Consider a system with matrix(60)A0=A0100A02,and system noise covariance W0=I4 in a network composed of 5 sensors, where(61)A01=A02=0.80.200.8.The measurement matrices of sensors are C1=C3=C5=1100, C2=C4=0011, and the measurement noises have covariances Wi=0.01 for all i. In the network, sensor 1, sensor 2, and sensor 4 can obtain the measurements of the target and partial parameters of system matrix A0 with noises. The covariance of the noises is equal to 0.01. The adjacency matrix of the communication topology is given by(62)A=1110011001101001001001001.Sensor 1 and sensor 3 have access toA01 with noises and sensor 2 has an access to A02 with noises. Other sensors cannot obtain the system information. Since node 1 and 2 are reachable in the communication topology, from Proposition 3, the mean square estimation errors for the system parameters converge to 0. Figure 1 gives the trajectory of sensor 1’s estimation covariance for system parameter A033 and shows its convergence.Figure 1
Parameter estimation covariance of sensor 1.Nodes 1, 2, and 4 are collectively observable, and the LMIs in Theorem6 are feasible. Set Pi(0)=Qi(0)=I4 and αi(k)=1. Under Algorithm 1, 50 Monte Carlo simulations are carried out and the estimation error and covariance of each sensor are shown in Figures 2 and 3. Obviously, each sensor’s estimation error is less than its trace of covariance and bounded.Figure 2
State estimation errors and covariances of sensors 1-3.Figure 3
State estimation errors and covariances of sensors 4-5.


## 6. Conclusion

This paper mainly proposes a fully distributed robust Kalman filtering algorithm for discrete-time-invariant systems over sensor networks in which each sensor can just obtain partial system parameters and measurement outputs with noises. In this algorithm, each sensor just uses its neighbors’ information to update filtering gains and weights. If in the network for any one of system parameters its accessible node subset is globally reachable, and there exists at least one globally reachable collaboratively observable node subset, then as long as a set of uncorrelated LMIs are feasible, the mean square estimation errors of the sensors must be bounded. The distributed filtering problem of time-varying networks is of our research interest in future.



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*Source: 7954263-2018-12-02.xml*