Distributed Fusion Filtering in Networked Systems with Random Measurement Matrices and Correlated Noises

Distributed Fusion Filtering in Networked Systems with Random Measurement Matrices and Correlated Noises Raquel Caballero-´ Aguila(a), Irene Garc´ıa-Garrido(b)and Josefa Linares-P´erez(b) 1 (a)Dpto. de Estad´ıstica. Universidad de Ja´en. Paraje Las Lagunillas. 23071. Ja´en. Spain. E-mail: r[email protected] (b)Dpto. de Estad´ıstica. Universidad de Granada. Avda. Fuentenueva. 18071. Granada. Spain. E-mail:{irenegarciag, jlinares}@ugr.es Abstract In this paper, the distributed fusion state estimation problem is addressed for sensor network systems with random state transition matrix and random measurement matrices, which provide a unified framework to consider some network-induced random phenomena. The process noise and all the sensor measurement noises are assumed to be one-step autocorrelated and different sensor noises are one-step cross-correlated; also, the process noise and each sensor measurement noise are two-step cross-correlated. These correlation assumptions cover many practical situations, were the classical independence hypothesis is not realistic. Using an innovation methodology, local least-squares linear filtering estimators are recursively obtained at each sensor. The distributed fusion method is then used to form the optimal matrix-weighted sum of these local filters according to the mean squared error criterion. A numerical simulation example shows the accuracy of the proposed distributed fusion filtering algorithm and illustrates some of the network-induced stochastic uncertainties that can be dealt with the current system model, such as sensor gain degradation, missing measurements and multiplicative noise. 1 Introduction In recent years, information and communication technologies have experienced a fast development, making the use of sensor networks become very popular for measurement acquisition and data processing, as they usually provide more information than traditional single-sensor communication systems. For this reason, the study of the estimation problem in sensor network stochastic systems has achieved great interest in many important research fields of engineering, computing and mathematics, mainly, by their broad scope of applications (target tracking, environment observation, habitat monitoring, animal tracking, communications, etc.). 1Corresponding author. E-mail: jl[email protected] 1 Although fusion algorithms have been proposed according to different methods (see e.g. [1]- [3]), most existing results do not consider the new problems that arise inevitably in sensor network systems due to the restrictions of the physical equipment, mainly the limitations of bandwidth channels and uncertainties in the external environment, in both the modeling process and the transmission of information. These situations can dramatically worsen the quality of fusion estimators designed. Multiplicative noise uncertainties, random delays, packet dropouts and missing measurements, are some of the most common problems that motivated the need to develop new estimation algorithms. Therefore, it is not surprising that, in the past few years, the study of the state estimation problem in network systems with only one or several of the aforementioned uncertainties has become an active research area (see e.g. [4] -[9] and references therein). Clearly, some of these situations with networked-induced phenomena are special cases of systems with transition and/or measurement random parameter matrices, which have important practical significance and arise in many application areas such as digital control of chemical processes, radar control, navigation systems or economic systems [10]. On the one hand, random state transition matrices arise in the context of systems with state-dependent multiplicative noise, of great interest for applications in aerospace systems, communication, processing images, etc. [11]. On the other hand, systems with observation multiplicative noises [12] clearly are special cases of systems with random parameter measurement matrices. Also, networked systems with stochastic sensor gain degradation as those considered in [13] or the systems with state and measurement multiplicative noises in [9], can be rewritten by transition and measurement random parameter matrices. It must be mentioned that in many papers, see e.g. [8], systems with random delays and packet dropouts are transformed into systems with random parameter matrices. Consequently, these kind of systems can model a great variety of real situations and, for this reason, the estimation problem in this type of systems has gained a considerable interest in recent years (see e.g. [14]-[18] and references therein). Furthermore, in the latest research on signal estimation, the fairly conservative assumption that the process and measurement noises are uncorrelated is commonly weakened as, in many practical situations, such noises are usually correlated. For example, when all the sensors operate in the same noisy environment, the sensor noises are usually correlated. Likewise when the process noise and the sensor noises are state dependent, there may be cross-correlation between them, as well as between different sensor noises. Also, the augmented systems used to describe random delays and measurement losses are systems with correlated noises, and discretized continuous-time systems have also inherently correlated noises. Hence, in both, systems with deterministic matrices 2 and systems with random parameter matrices, the estimation problem with correlated and crosscorrelated noises, has become a challenging research topic. In the first case, the optimal Kalman filtering fusion problem in systems with cross-correlated process noises and measurement noises at the same sampling time is addressed for example in [19], and at consecutive sampling times in [8]. Under different correlation assumptions of the noises, centralized and distributed fusion algorithms are obtained in [11], for systems with multiplicative noise in the state equation, in [9], when multiplicative noises exist in both the state and observation equations, and in [13], for systems where the measurements might have partial information about the signal. For systems with random parameter matrices and autocorrelated and cross-correlated noises, many research efforts have been devoted to the centralized fusion estimation problem ([15]-[18]). Centralized algorithms are based on a fusion centre able to receive all the measured data from sensors for being processed; they provide optimal estimators from the measurements of all the sensors and hence, when all the sensors work correctly, they have the best accuracy. Nevertheless, as it is known, the centralized approach has several drawbacks such as bad robustness, poor survivability, reliability, heavy communication and expensive computational cost, which can be overcome by using distributed approaches. In the distributed fusion method, each sensor estimates the state based on its own measurement data, and these local estimators are combined according to a certain information fusion criterion. To the best of the authors knowledge, the distributed fusion estimation problem in networked systems with both random parameter matrices and autocorrelated and cross-correlated noises has not been investigated. Motivated by the above considerations, this paper deals with the distributed fusion estimation problem in sensor network systems including simultaneously random parameter matrices and correlated noises in the state-space model. The main contributions of our study can be highlighted as follows: (1) The network system model with random parameter matrices considered provides a unified framework to treat some network-induced phenomena, such as multiplicative noise uncertainties, missing measurements or sensor gain degradation, and, hence, the proposed distributed fusion filter has a wide applicability. (2) One-step autocorrelation of the noises and, also, two-step cross-correlation between the process noise and different sensor noises are considered. (3) The innovation technique is used to obtain algorithms for the local least-squares linear filtering estimators which are recursive and computationally simple. (4) The proposed distributed fusion filter is generated by a matrix-weighted linear combination of the local filtering estimators using the mean squared error as optimality criterion, requiring the cross-covariance matrices between any two local filters, but not the error cross-covariance matrices, as in [1]. 3 The rest of the paper is organized as follows. The system model with multiple sensors and random parameter matrices is presented in Section 2, including a brief description of the traditional centralized and distributed fusion estimation methods. The local least-squares linear filtering algorithms are derived in Section 3, using an innovation approach. In Section 4, the proposed distributed fusion filter is obtained by a matrix-weighted linear combination of the local filtering estimators using the mean squared error as optimality criterion. A simulation example is given in Section 5 to show the performance of the proposed estimation algorithms, and some conclusions are drawn in Section 6. Notation: The notation used throughout the paper is standard. Rndenotes the n-dimensional Euclidean space. ATand A−1denote the transpose and inverse of a matrix A, respectively. The shorthand (A1,...,Am) denotes a partitioned matrix into sub-matrices A1,...,Am. If a matrix dimension is not explicitly stated, it is assumed to be compatible for algebraic operations. Moreover, for any function Gk,s, depending on the time instants kand s, we write Gk=Gk,k for simplicity. Analogously, we write K(i)=K(ii)for any function K(ij), depending on the sensors iand j. δk−srepresents the Kronecker delta function, which is equal to one if k=s, and zero otherwise. Orthogonal Projection Lemma is abbreviated as OPL. Finally, all the random vectors will be defined on the probabilistic space (Ω,A, P), and for arbitrary random vectors Xand Y, we denote Cov[X, Y ] = E[(X−E[X]) (Y−E[Y])T] and Cov[X] = Cov[X, X], where E[·] stands for the mathematical expectation operator. 2 System Formulation and Problem Statement Consider the following discrete-time linear stochastic system with mdifferent sensors: xk+1 =Fkxk+wk, k ≥0,(1) y(i) k=H(i) kxk+v(i) k, k ≥1, i = 1,2,...,m, (2) where xk∈Rnxis the state vector and y(i) k∈Rnyis the output measured by sensor i, both at time k.{Fk;k≥0}and {H(i) k;k≥1}are sequences of random parameter matrices with compatible dimensions. {wk;k≥0}is the process noise and {v(i) k;k≥1}is the measurement noise of the i-th sensor. Model assumptions. The assumptions about the initial state, the random parameter matrices and the noises involved in the system model (1)-(2), under which the fusion filtering problem will be addressed, are: 4 (i) The initial state x0is a random vector with E[x0] = x0and Cov[x0] = Σ0. (ii) {Fk;k≥0}and {H(i) k;k≥1}are sequences of independent random parameter matrices with known means, E[Fk] = Fk,E[H(i) k] = H(i) k,i= 1,2,...,m, and the covariances of their entries, Cov[fpq (k), fp′q′(k)], Cov[h(i) pa (k), h(i) qb (k)], are also assumed to be known. fpq (k) denotes the (p, q)-th entry of matrix Fk, for p, q = 1,2,...,nx, and h(i) pq (k) denotes the (p, q)-th entry of H(i) k, for p= 1,2,...,nxand q= 1,2,...,ny. (iii) The noises {wk;k≥0}and {v(i) k;k≥1},i= 1,2,...,m, are zero-mean sequences with known covariances and cross-covariances: Cov[wk, ws] = Qk,s (δk−s+δk−s+1 +δk−s−1), Cov[v(i) k, v(j) s] = R(ij) k,s (δk−s+δk−s+1 +δk−s−1), Cov[wk, v(i) s] = S(i) k,s (δk−s+δk−s+1 +δk−s+2). (iv) For i= 1,2,...,m, the initial state x0and the processes {Fk;k≥0}and {H(i) k;k≥1} are mutually independent and they are independent of the additive noises {wk;k≥0}and {v(i) k;k≥1}. Remark 1. By denoting e Fk=Fk−Fk,e H(i) k=H(i) k−H(i) k, i = 1,2,...,m, and Dan arbitrary deterministic matrix, the following identities hold for the (p, q)-th entries of the matrices E[e FkDe FT k] and E[e H(i) kDe H(i)T k]: E[e FkDe FT k]pq = nx X a=1 nx X b=1 Cov[fpa (k), fqb (k)]Dab, p, q = 1,2,...,nx. E[e H(i) kDe H(i)T k]pq = nx X a=1 nx X b=1 Cov[h(i) pa (k), h(i) qb (k)]Dab, p, q = 1,2,...,ny. Remark 2. Assumptions (i)-(iii) lead to the following recursive formula for Dk≡E[xkxT k], the correlation matrix of the state vector xk(see, e.g. [15]): Dk+1 =FkDkFT k+E[e FkDke FT k] + Qk+FkQk−1,k +Qk,k−1FT k, k ≥1; D1=F0D0FT 0+E[e F0D0e FT 0] + Q0, D0= Σ0+x0xT 0. (3) Remark 3. The following correlation properties of the vector noises wkand v(i) kare easily inferred from the assumptions (iii)-(iv): •For i= 1,2,...,m, the noise vector wkis uncorrelated with the observations y(i) 1,...,y(i) k−1, and correlated with y(i) k, with W(i) k≡E[wky(i)T k] = Qk,k−1H(i)T k+S(i) k, k ≥1.(4) 5 •For i, j = 1,2,...,m, the noise vector v(i) kis uncorrelated with the observations y(j) 1,...,y(j) k−2, and correlated with y(j) k−1, with V(ij) k,k−1≡E[v(i) ky(j)T k−1] = S(i)T k−2,kH(j)T k−1+R(ij) k,k−1, k ≥2.(5) Our aim is to address the optimal least-squares (LS) linear filtering problem of state xkby fusing effectively the observations y(i) 1,...,y(i) k,i= 1,2,...,m; specifically, we use the traditional centralized and distributed fusion methods. As is known, the main drawbacks of the first are the expensive computational cost, poor robustness and flexibility. The latter overcomes these disadvantages and provides greater accuracy than local estimators. The centralized fusion method use all measurement data coming from msensors directly in the fusion center for state estimation, while in the distributed fusion method the observations in the fusion center are replaced by estimates that have been locally computed. Centralized fusion algorithm. Combining the mmeasurement equations given by (2) and setting yk=y(1)T k,...,y(m)T kT, the discrete-time multi-sensor system with random parameter matrices and correlated additive noises (1)-(2) considered in this paper, is a special case of the discretetime stochastic system with random parameter matrices and correlated additive noises considered in [17]. Hence, the optimal centralized fusion filter could be obtained by the optimal LS lineal filtering algorithm in [17]. Distributed fusion algorithm. The distributed fusion method computes, at each sensor, a local optimal LS linear state filter using its own measurement data, and, subsequently, the fusion center computes the LS matrix-weighted linear combination of the local filtering estimators. Hence, the distributed fusion filtering algorithm is performed in two steps. In the first one (Section 3), for each i= 1,2,...,m, a local LS linear estimator of the signal xk, denoted by bx(i) k/k, is produced using the measurements y(i) 1,...,y(i) k, by a recursive algorithm. In the second step (Section 4), a fusion distributed estimator, bx(D) k/k, is generated by a matrix-weighted linear combination of the local estimators, bx(i) k/k, i = 1,2,...,m, using the mean squared error as optimality criterion. 3 Local LS Linear Filtering Algorithm This section is concerned with the problem of obtaining a recursive algorithm for the local LS linear filter at each sensor i, for i= 1,...,m, by using an innovation approach. Theorem 1 For system (1)-(2), under assumptions (i)-(iv), the local LS linear filter, bx(i) k/k, is given 6 by bx(i) k/k =bx(i) k/k−1+X(i) kΠ(i)−1 kµ(i) k, k ≥1; bx(i) 0/0=x0,(6) where the one-stage state predictor, bx(i) k/k−1, satisfies bx(i) k/k−1=Fk−1bx(i) k−1/k−1+W(i) k−1Π(i)−1 k−1µ(i) k−1, k ≥2; bx(i) 1/0=F0x0.(7) The filtering error covariance matrix, Σ(i) k/k, is given by Σ(i) k/k = Σ(i) k/k−1− X (i) kΠ(i)−1 kX(i)T k, k ≥1; Σ(i) 0/0= Σ0,(8) where the prediction error covariance matrix, Σ(i) k/k−1, is calculated by Σ(i) k/k−1=Fk−1Σ(i) k−1/k−1FT k−1+E[e Fk−1Dk−1e FT k−1] + Qk−1 +Fk−1J(i) k−1+J(i)T k−1FT k−1− W(i) k−1Π(i)−1 k−1W(i)T k−1, k ≥2; Σ(i) 1/0=F0Σ0/0FT 0+E[e F0D0e FT 0] + Q0, (9) with J(i) k=Qk−1,k − X(i) kΠ(i)−1 kW(i)T k, k ≥1.(10) The matrix X(i) kis obtained by X(i) k= Σ(i) k/k−1H(i)T k+M(i) k, k ≥1,(11) where M(i) kis given by M(i) k=Fk−1 S(i) k−2,k +S(i) k−1,k− X(i) k,k−1Π(i)−1 k−1V(i)T k,k−1, k ≥2; M(i) 1=S(i) 0,1,(12) with X(i) k,k−1=Fk−1X(i) k−1+W(i) k−1, k ≥2.(13) The innovation, µ(i) k, is given by µ(i) k=y(i) k−H(i) kbx(i) k/k−1− V(i) k,k−1Π(i)−1 k−1µ(i) k−1, k ≥2; µ(i) 1=y(i) 1−H(i) 1bx(i) 1/0, (14) and the innovation covariance matrix, Π(i) k, satisfies Π(i) k=E[e H(i) kDke H(i)T k] + H(i) kX(i) k+M(i)T kH(i)T k+R(i) k −V(i) k,k−1Π(i)−1 k−1V(i)T k,k−1, k ≥2; Π(i) 1=E[e H(i) 1D1e H(i)T 1] + H(i) 1X(i) 1+M(i)T 1H(i)T 1+R(i) 1. (15) The matrices Dk,W(i) kand V(i) k,k−1are given in (3), (4) and (5), respectively. 7 Proof. See Appendix.  The computational procedure of the proposed local LS linear filtering algorithm can be summarized as follows: The matrices W(i) kand V(i) k,k−1are computed by expressions (4) and (5), respectively. We obtain Dkrecursively by (3), where the matrix E[e Fk−1Dk−1e FT k−1], necessary to compute Dkand Σ(i) k/k−1, is obtained as indicated in Remark 1. The matrix E[e HkDke HT k] is also computed in order to obtain the innovation covariance matrix Π(i) k. Note that all these matrices depend only on the system model information and can be obtained before the observations are available. At the sampling time k, once the (k−1)-th iteration is finished and the new observation y(i) k is available, starting with the prior knowledge including X(i) k−1,µ(i) k−1, Π(i) k−1,bx(i) k/k−1, Σ(i) k/k−1the proposed filtering algorithm operates as follows: Step 1: From (13), X(i) k,k−1is computed and, from it, M(i) kis provided by (12), and then X(i) kis obtained by (11). Step 2: The innovation µ(i) kand its covariance matrix Π(i) kare computed by (14) and (15), respectively. Step 3: The filter, bx(i) k/k, and the filtering error covariance matrix, Σ(i) k/k, are computed by (6) and (8), respectively. Step 4: To implement the above steps at time k+ 1, we must: 4.1) Compute the state predictor, bx(i) k+1/k, by (7). 4.2) Compute J(i) kby (10), and from this, the prediction error covariance matrix, Σ(i) k+1/k, by (9). 4 Distributed Fusion Filtering Estimators Once the local LS linear filters, bx(i) k/k for each sensor i= 1,2,...,m, have been obtained, our objective in this section is to design a distributed fusion filter, bx(D) k/k, by a matrix-weighted linear combination of such estimators, which minimizes the mean squared estimation error. To simplify the derivation of the proposed fusion estimators, we previously present some useful lemmas that provide the expectations K(ij) k/k−1=E[bx(i) k/k−1bx(j)T k/k−1], L(ij) k=E[bx(i) k/k−1µ(j)T k] and Π(ij) k=E[µ(i) kµ(j) k], necessary for subsequent calculations. The assumptions and notation in these lemmas are those of Section 3. 8 4.1 Preliminary Results Lemma 1 For i, j = 1,2,...,m,the cross-covariance matrix between any two local state predictors, K(ij) k/k−1=E[bx(i) k/k−1bx(j)T k/k−1], satisfies K(ij) k/k−1=Fk−1K(ij) k−1/k−2FT k−1+Fk−1L(ij) k−1Π(j)−1 k−1X(j)T k,k−1 +X(i) k,k−1Π(i)−1 k−1Π(ij) k−1Π(j)−1 k−1X(j)T k,k−1 +X(i) k,k−1Π(i)−1 k−1L(ji)T k−1FT k−1, k ≥2; i6=j, K(ij) 1/0=F0x0xT 0FT 0, K(i) k/k−1=Dk−Σ(i) k/k−1, k ≥1. Proof. Using (25) and that L(ij) k=E[bx(i) k/k−1µ(j)T k] and Π(ij) k=E[µ(i) kµ(j) k], the proof of this lemma is immediately clear.  Lemma 2 For i, j = 1,2,...,m and i6=j, the expectation L(ij) k=E[bx(i) k/k−1µ(j)T k]satisfies L(ij) k=K(i) k/k−1−K(ij) k/k−1H(j)T k+X(i) k,k−1Π(i)−1 k−1V(ji)T k,k−1 −L(ij) k,k−1Π(j)−1 k−1V(j)T k,k−1, k ≥2; L(ij) 1= 0, where L(ij) k,k−1=E[bx(i) k/k−1µ(j)T k−1]is given by L(ij) k,k−1=Fk−1L(ij) k−1+X(i) k,k−1Π(i)−1 k−1Π(ij) k−1, k ≥2. Proof. Taking into account expression (14) for µ(j) k, we have that L(ij) k=E[bx(i) k/k−1y(j)T k]−K(ij) k/k−1H(j)T k−L(ij) k,k−1Π(j)−1 k−1V(j)T k,k−1. Now, using (2) for y(j) k, (25) for bx(i) k/k−1, and the OPL, we obtain E[bx(i) k/k−1y(j)T k] = K(i) k/k−1H(j)T k+X(i) k,k−1Π(i)−1 k−1V(ji)T k,k−1. From the above relations, the expression for L(ij) kis immediately derived. Using again (25) for bx(i) k/k−1, expression for L(ij) k,k−1is also immediately clear, and the proof is completed.  Lemma 3 For i, j = 1,2,...,m and i6=j, the innovation cross-covariance matrix, Π(ij) k= E[µ(i) kµ(j)T k], satisfies Π(ij) k=H(i) kX(j) k−L(ij) k+M(ji)T kH(j)T k+R(ij) k −V(ij) k,k−1Π(j)−1 k−1V(j)T k,k−1−V(i) k,k−1Π(i)−1 k−1Π(ji)T k,k−1 , k ≥2; Π(ij) 1=H(i) 1X(j) 1−L(ij) 1+M(ji)T 1H(j)T 1+R(ij) 1, 9 the centralized filter over the distributed filter is compensated by better robustness and fault-tolerance abilities of the latter. This example has also highlighted the applicability of the proposed algorithm for a great variety of multi-sensor systems featuring networkinduced stochastic uncertainties, such as sensor gain degradation, missing measurements or multiplicative observation noises, which can be dealt with the observation model considered in this paper. A challenging further research topic is to address the estimation problem for this kind of systems with random parameter matrices, considering a sensor network whose nodes are distributed according to a given topology, characterized by a directed graph. Also, an interesting future research topic is to consider other kinds of stochastic uncertainties which often appear in networked systems, such as random delays and packet dropouts. Conflict of Interests The authors declare that there is no conflict of interests regarding the publication of this paper. Acknowledgments This research is supported by Ministerio de Econom´ıa y Competitividad (Grant No. MTM201452291-P and FPU programme). Appendix: Proof of Theorem 1 This appendix provides the proof of Theorem 1 by an innovation approach. For the i-th sensor, the innovation at time kis defined as µ(i) k=y(i) k−by(i) k/k−1, where by(i) k/k−1is the LS linear estimator of y(i) kbased on measurements y(i) s, s ≤k−1. Replacing the observation process by the innovation one, the LS linear estimator, bz(i) k/L, of a random vector zkbased on the observations y(i) 1,...,y(i) L, can be calculated as linear combination of the innovations µ(i) 1,...,µ(i) L; namely, bz(i) k/L = L X s=1 E[zkµ(i)T s](E[µ(i) sµ(i)T s])−1µ(i) s, k ≥1.(20) From the system equations (1)-(2) and the OPL, the state predictor bx(i) k/k−1and the observation predictor by(i) k/k−1, verify: bx(i) k/k−1=Fk−1bx(i) k−1/k−1+bw(i) k−1/k−1, k ≥1,(21) 16 by(i) k/k−1=H(i) kbx(i) k/k−1+bv(i) k/k−1, k ≥1.(22) Because of the correlation assumption (iii), the noise filter bw(i) k/k and the one-stage noise predictor bv(i) k/k−1are not equal to zero and, hence, expressions for such estimators must be calculated. LS linear noise estimators bw(i) k/k and bv(i) k/k−1.From the general expression for the estimators (20), taking into account that by Remark 3, E[wkµ(i)T k] = E[wky(i)T k] = W(i) kand E[v(i) kµ(i)T k−1] = E[v(i) ky(i)T k−1] = V(i) k,k−1, we have that the noise filter, bw(i) k/k, and the one-stage noise predictor, bv(i) k/k−1, satisfy: bw(i) k/k =W(i) kΠ(i)−1 kµ(i) k, k ≥1; bw(i) 0/0= 0.(23) bv(i) k/k−1=V(i) k,k−1Π(i)−1 k−1µ(i) k−1, k ≥2; bv(i) 1/0= 0.(24) Now, we will prove the local filtering algorithm by several steps: (I) Derivation of the filter (6), one-stage state predictor (7) and their error covariance matrices (8) and (9), respectively. From (20), by denoting X(i) k=E[xkµ(i)T k] and Π(i) k=E[µ(i) kµ(i)T k], the expression (6) for the filter is obvious, and from this relation and the OPL, we obtain (8) for the filtering error covariance. From (21) and (23), the expression (7), with W(i) ksatisfying (4), for the state predictor is immediately obtained. This expression together with the state equation (1) lead to (9) for the prediction error covariance matrix, where Dkis given by (3), and J(i) k=E[(xk−bx(i) k/k)wT k] clearly satisfies (10). (II) Derivation of (11) for the matrix X(i) k=E[xkµ(i)T k]. From (6) and (7), the following recursive expression for the state predictor is obtained: bx(i) k/k−1=Fk−1bx(i) k−1/k−2+X(i) k,k−1Π(i)−1 k−1µ(i) k−1, k ≥2,(25) where, from (1), the matrix X(i) k,k−1=E[xkµ(i)T k−1] clearly verifies (13). Using relation (25) and again (1), we obtain that the correlation between the prediction error and the noise, M(i) k≡E[(xk−bx(i) k/k−1)v(i)T k], satisfies (12). This correlation property allows us to obtain easily expression (11) for the matrix X(i) k=E[xkµ(i)T k]. Indeed, by the OPL, we have X(i) k=E[(xk−bx(i) k/k−1)y(i)T k],and, from (2), expression (11) for X(i) kis obtained. (III) Derivation of the innovation (14) and its covariance matrix (15). From (22) and (24), is clear that the innovation is given by (14), with V(i) k,k−1satisfying (5). Next, an expression (15) for Π(i) k=E[µ(i) kµ(i)T k] is derived. From the OPL and (2), we have that Π(i) k=E[y(i) kµ(i)T k] = H(i) kE[xkµ(i)T k] + E[e H(i) kxkµ(i)T k] + E[v(i) kµ(i)T k].(26) 17 −Again, from the OPL and (2), and the conditional expectation properties, E[e H(i) kxkµ(i)T k] = E[e H(i) kxkxT ke H(i)T k] = E[e H(i) kDke H(i)T k], k ≥1. −Using (14) for µ(i) k, with (2) for y(i) k, we obtain that E[v(i) kµ(i)T k] = M(i)T kH(i)T k+R(i) k− V(i) k,k−1Π(i)−1 k−1V(i)T k,k−1, k ≥2; E[v(i) 1µ(i)T 1] = M(i)T 1H(i)T 1+R(i) 1. Substituting the above expectations into (26), we easily obtain (15), and the proof is completed. References [1] S. Sun, Multi-sensor optimal information fusion Kalman filters with applications, Aerospace Science and Technology, 8 (2004) 57–62. [2] X. Sun and G. 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