Optimal Fusion Filtering in Multisensor Stochastic Systems with Missing Measurements and Correlated Noises

Abstract

The optimal least-squares linear estimation problem is addressed for a class of discrete-time multisensor linear stochastic systems with missing measurements and autocorrelated and cross-correlated noises. The stochastic uncertainties in the measurements coming from each sensor (missing measurements) are described by scalar random variables with arbitrary discrete probability distribution over the interval [0,1]; hence, at each single sensor the information might be partially missed and the different sensors may have different missing probabilities. The noise correlation assumptions considered are (i) the process noise and all the sensor noises are one-step autocorrelated; (ii) different sensor noises are one-step cross-correlated; and (iii) the process noise and each sensor noise are two-step cross-correlated. Under these assumptions and by an innovation approach, recursive algorithms for the optimal linear filter are derived by using the two basic estimation fusion structures; more specifically, both centralized and distributed fusion estimation algorithms are proposed. The accuracy of these estimators is measured by their error covariance matrices, which allow us to compare their performance in a numerical simulation example that illustrates the feasibility of the proposed filtering algorithms and shows a comparison with other existing filters.

1. Introduction

For a long time, the least-squares (LS) estimation problem in linear stochastic systems from measurements perturbed by additive noises has received considerable attention in the scientific community due to its wide applicability in many practical situations (e.g., video and laser tracking systems, satellite navigation, radar and meteorological applications, etc. [1]). As it is well known, one of the major contributions made to solve this problem is the Kalman filter, which provides a recursive algorithm for the optimal LS estimator when the additive white noises and the initial state are Gaussian and mutually independent (or, equivalently, uncorrelated due to the Gaussianity assumption) and, therefore, the optimal LS estimator is the optimal LS linear estimator. From the publication of the Kalman filter [2] in 1960, numerous results and several solution methods have been reported in the literature to address the state estimation problem from noisy observations, which depend on models representing possible relationships between the unknown state and the observable variables and also on the noise processes assumptions.

Specifically, during the past decades, there has been an increasing interest in the filtering problem in multisensor systems, where sensor networks are used to obtain the whole available information on the system state and its estimation must be carried out from the observations provided by all the sensors. A basic matter for this class of systems is how to fuse the measurement data from the different sensors to address the estimation problem. Commonly, two methods are used to process the measured data coming from multiple sensors: centralized and distributed fusion methods. In the centralized fusion method all the measured data from sensors are communicated to the fusion center for being processed; nevertheless, as is widely known, centralized estimators have many computational disadvantages, which motivate the research into other fusion methods. In the distributed fusion method, each sensor estimates the state based on its own single measurement data, and then it sends such estimate to the fusion center for fusion according to a certain information fusion criterion. Although the use of sensor networks offers several advantages, the unreliable network characteristics usually cause problems during data transmission from sensors to the fusion center, such as missing measurements, random communication packet losses and/or delays. Taking into account these network uncertainties, the models representing the relationships between the state and measurements do not allow to apply the Kalman filter, and modifications of conventional estimation algorithms have been proposed (see e.g., [3–9] and references therein).

As in the Kalman filter, independent white noises are considered in all the mentioned papers; however, this assumption may not be realistic and can be a limitation in many real-world problems in which noise correlation may be present. This problem arises, for example, when a target is taking an electronic countermeasure, for example, noise jamming [10], or if the process noise and the sensor measurement noises are dependent on the system state, then there may be cross-correlation between different sensor noises and cross-correlation between process noise and sensor noises. Also, if all the sensors are observed in the same noisy environment, the measurement noises of different sensors are usually correlated.

For these reasons, the estimation problem in systems with correlated noises has received significant research interest in recent years. For example, the optimal Kalman filtering fusion problem in systems with cross-correlated sensor noises is addressed in [10], while [11, 12] study the same problem in systems with cross-correlated process noises and measurement noises; in these papers correlated noises at the same sampling time are considered. In general, the assumption of correlation and cross-correlation of the noise process and measurement noises in different sampling times makes difficult the identification of optimal estimators; this limitation has encouraged a wider research into suboptimal Kalman-type estimation problems. In [13], a Kalman-type recursive filter is presented for systems with finite-step correlated process noises, and the filtering problem with multistep correlated process and measurement noises is investigated in [14]. The optimal robust nonfragile Kalman-type recursive filtering problem is studied in [15] for a class of uncertain systems with finite-step autocorrelated measurement noises and multiple packet dropouts. The problem of distributed weighted robust Kalman filter fusion is studied in [16] for a class of uncertain systems with autocorrelated and cross-correlated noises. In [17], a stochastic singular system with correlated noises at the same sampling time is transformed into an equivalent nonsingular system with correlated noises at the same and neighboring sampling times. Also, in [18], an augmented parameterized system with correlated noises at the same and neighboring sampling times is used to describe the sensor delay, packet dropout, and uncertain observation phenomenons.

On the other hand, as noted above, the use of communication networks for transmitting measured data motivates the need of considering stochastic uncertainties. Missing measurements have been widely treated due to its applicability to model a large class of real-world problems, such as fading phenomena in propagation channels, target tracking or, in general, situations where there exist intermittent failures in the observation mechanism, accidental loss of some measurements, or inaccessibility of the data during certain times. The state estimation problem from missing measurement transmitted by multiple sensors has been studied based on the assumption that all the sensors are identical (see, e.g., [19–22]); however, this assumption can be unreasonable since some real systems usually involve multiple sensors with different characteristics. Recently, the filtering problem using missing measurements whose statistical properties are assumed not to be the same in all the sensors has been addressed by several authors under different approaches and hypotheses on the processes involved (see, e.g., [23–27]). In all the above papers, Bernoulli random variables are used to model the missing measurements phenomenon, and hence, it is assumed that the measurement signal is either completely lost (if the corresponding Bernoulli variable takes the value zero) or successfully transferred (when the Bernoulli variable is equal to one). Recently, this missing measurement model has been generalized considering any discrete distribution on the interval [0,1], which allows to cover some practical applications where only partial information is missing (see [28, 29] and references therein).

Motivated by the above considerations, our attention is focused on investigating the optimal LS linear centralized and distributed fusion estimation problems in multisensor systems with missing measurements and autocorrelated and cross-correlated noises. In each sensor, the missing measurement phenomenon is governed by a scalar random variable with arbitrary discrete probability distribution over the interval [0,1], and the different sensors may have different missing probabilities. Assume that the process noise and all the sensor noises are one-step autocorrelated; different sensor noises are one-step cross-correlated; and the process noise and each sensor noise are two-step cross-correlated. This paper makes a twofold substantial novel contribution: (1)  unlike most previous results with correlated noises, in which suboptimal Kalman-type estimators are proposed, in this paper optimal LS linear estimators are obtained by using an innovation approach, which provides a simple derivation of the estimation algorithms due to the fact that the innovations constitute a white process; and (2)  our missing measurement model considers at each sensor the possibility of observations containing only partial information about the state, or even only noise.

The paper is organized as follows. In Section 2 the system model with autocorrelated and cross-correlated noises and missing measurements coming from multiple sensors is described. Also, the suitable properties on the state and noise processes are specified and a brief description of the innovation approach to the optimal LS linear estimation problem is included. In Section 3 a recursive algorithm for the centralized optimal linear filter is presented for the considered model (the derivation has been deferred to Appendix A). Next, in Section 4, the local LS linear filters and their corresponding error covariance matrices between any two local estimates are provided, and then the distributed optimal weighted fusion estimators and their error covariance matrices are obtained by applying the optimal information fusion criterion weighted by matrices in the linear minimum variance sense. Finally, in Section 5, a numerical simulation example is presented to show the effectiveness of the estimation algorithms proposed in the current paper, and some conclusions are drawn in Section 6.

Notation. The notation used throughout the paper is standard. For any matrix 𝐴, the notation symbols 𝐴^𝑇 and 𝐴⁻¹ represent its transpose and inverse, respectively; ℝ^𝑛 denotes the 𝑛-dimensional Euclidean space and ℝ^𝑚×𝑛 is the set of all real matrices of dimension 𝑚×𝑛. The shorthand Diag(𝑎₁,…,𝑎_𝑟) denotes a diagonal matrix whose diagonal entries are 𝑎₁,…,𝑎_𝑟. If the dimensions of matrices are not explicitly stated, they are assumed to be compatible for algebraic operations. 𝛿_𝑘−𝑠 is the Kronecker delta function, which is equal to one, if 𝑘=𝑠, and zero otherwise. Moreover, for arbitrary random vectors 𝛼 and 𝛽, we will denote Cov[𝛼,𝛽]=𝐸[(𝛼−𝐸[𝛼])(𝛽−𝐸[𝛽])^𝑇] and Cov[𝛼]=Cov[𝛼,𝛼], where 𝐸[⋅] stands for the mathematical expectation operator. Finally, 𝛼 denotes the estimator of 𝛼 and 𝛼=𝛼−𝛼 the estimation error.

2. Problem Formulation

Our aim is to obtain recursive algorithms for the optimal LS linear filtering problem in a class of discrete-time stochastic systems with missing measurements coming from multiple sensors, by using centralized and distributed fusion methods. In this section, firstly the system model and the assumptions about the state and noise processes are presented and, secondly, the optimal LS linear estimation problem is formulated using an innovation approach.

2.1. Stochastic System Model

Consider a discrete-time linear stochastic system with autocorrelated and cross-correlated noises and missing measurements coming from 𝑟 sensors. The phenomenon of missing measurements occurs randomly and, for each sensor, a different sequence of scalar random variables with discrete distribution over the interval [0,1] is used to model this phenomenon. Specifically, the following system is considered: 𝑥_𝑘=𝐹_𝑘−1𝑥_𝑘−1+𝑤_𝑘−1,𝑘≥1, (1) where 𝑥_𝑘∈ℝ^𝑛 is the state, {𝑤_𝑘;𝑘≥0} is the process noise, and 𝐹_𝑘, for 𝑘≥0, are known matrices with compatible dimensions.

Consider 𝑟 sensors which, at any time 𝑘, provide scalar measurements of the system state, perturbed by additive and multiplicative noises according to the following model: 𝑦^𝑖_𝑘=𝜃^𝑖_𝑘𝐻^𝑖_𝑘𝑥_𝑘+𝑣^𝑖_𝑘,𝑘≥1,𝑖=1,2,…,𝑟, (2) where {𝑦^𝑖_𝑘;𝑘≥1} are the measured data; {𝑣^𝑖_𝑘;𝑘≥1} are measurement noises; {𝜃^𝑖_𝑘;𝑘≥1} are scalar random variables sequences; 𝐻^𝑖_𝑘, for 𝑘≥1, are known time-varying matrices with compatible dimensions; superscript 𝑖 denotes the 𝑖th sensor, and 𝑟 is the number of sensors.

Next, the statistical properties assumed about the initial state and noise processes involved in (1) and (2) are specified.(i)The initial state 𝑥₀ is a random vector with 𝐸[𝑥₀]=𝑥₀ and Cov[𝑥₀]=𝑃₀.(ii)The process noise, {𝑤_𝑘;𝑘≥0}, and the measurement noises, {𝑣^𝑖_𝑘;𝑘≥1}, 𝑖=1,2,…,𝑟, are zero-mean sequences with covariances and cross-covariances: Cov𝑤_𝑘,𝑤_𝑠=𝑄_𝑘,𝑘𝛿_𝑘−𝑠+𝑄_𝑘,𝑠𝛿_{𝑘−𝑠+1}+𝑄_𝑘,𝑠𝛿_{𝑘−𝑠−1},Cov𝑣^𝑖_𝑘,𝑣^𝑗_𝑠=𝑅^𝑖𝑗_𝑘,𝑘𝛿_𝑘−𝑠+𝑅^𝑖𝑗_𝑘,𝑠𝛿_{𝑘−𝑠+1}+𝑅^𝑖𝑗_𝑘,𝑠𝛿_{𝑘−𝑠−1},Cov𝑤_𝑘,𝑣^𝑖_𝑠=𝑆^𝑖_𝑘,𝑘𝛿_𝑘−𝑠+𝑆^𝑖_𝑘,𝑠𝛿_{𝑘−𝑠+1}+𝑆^𝑖_𝑘,𝑠𝛿_{𝑘−𝑠+2}. (3)(iii) The multiplicative noises {𝜃^𝑖_𝑘;𝑘≥1}, 𝑖=1,2,…,𝑟, are white sequences of scalar variables with discrete distribution over the interval [0,1], with 𝐸[𝜃^𝑖_𝑘]=𝜃^𝑖_𝑘 and Var[𝜃^𝑖_𝑘]=𝑉^𝜃𝑖_𝑘.(iv) The initial state 𝑥₀ and the multiplicative noises {𝜃^𝑖_𝑘;𝑘≥1}, for 𝑖=1,2,…,𝑟, are mutually independent, and they are independent of the additive noises {𝑤_𝑘;𝑘≥0} and {𝑣^𝑖_𝑘;𝑘≥1}, for 𝑖=1,2,…,𝑟.

Remark 1. From assumption (ii) the following correlation properties of the additive noises are easily deduced. (1)The noise vectors 𝑤_𝑘 and 𝑤_𝑠 are correlated at consecutive sampling times, |𝑘−𝑠|=1, and independent otherwise; the covariance matrices of 𝑤_𝑘 with 𝑤_𝑘−1, and 𝑤_𝑘+1 are 𝑄_{𝑘,𝑘−1}, and 𝑄_𝑘,𝑘+1, respectively. (2)For 𝑖,𝑗=1,2,…,𝑟, the measurement noises 𝑣^𝑖_𝑘 and 𝑣^𝑗_𝑠 are cross-correlated at the same sampling time and at consecutive sampling times, |𝑘−𝑠|=0,1, and independent otherwise; the cross-covariances of 𝑣^𝑖_𝑘 with 𝑣^𝑗_𝑘, 𝑣^𝑗_𝑘−1 and 𝑣^𝑗_𝑘+1 are 𝑅^𝑖𝑗_𝑘,𝑘, 𝑅^𝑖𝑗_{𝑘,𝑘−1} and 𝑅^𝑖𝑗_𝑘,𝑘+1, respectively. (3)For 𝑖=1,2,…,𝑟, the measurement noises 𝑣^𝑖_𝑘 are correlated with the noise vectors 𝑤_𝑠, for 𝑠=𝑘,𝑘−1,𝑘−2, and independent otherwise; the cross-covariance matrices of 𝑣^𝑖_𝑘 with 𝑤_𝑘, 𝑤_𝑘−1 and 𝑤_𝑘−2 are 𝑆^𝑖_𝑘,𝑘, 𝑆^𝑖_{𝑘−1,𝑘} and 𝑆^𝑖_{𝑘−2,𝑘}, respectively. The correlation conditions of the process noise and the measurement noises considered in this paper are the same as those in [16]. Systems with only finite-step correlated process noises or multistep correlated process and measurement noises are considered in [13–15], among others. The current study can be extended to more general systems involving finite-step autocorrelated and cross-correlated noises with no difficulty, except for a greater complexity in the mathematical derivations.

Remark 2. From the state equation (1) and assumptions (ii) and (iv), it is easy to deduce that 𝐷_𝑘=𝐸[𝑥_𝑘𝑥^𝑇_𝑘] is recursively calculated by 𝐷_𝑘=𝐹_𝑘−1𝐷_𝑘−1𝐹^𝑇_𝑘−1+𝑄_{𝑘−1,𝑘−1}+𝐹_𝑘−1𝑄_{𝑘−2,𝑘−1}+𝑄_{𝑘−1,𝑘−2}𝐹^𝑇_𝑘−1𝐷,𝑘≥2,₁=𝐹₀𝐷₀𝐹^𝑇₀+𝑄_0,0,𝐷₀=𝑃₀+𝑥₀𝑥^𝑇₀. (4)

Also, it is easy to see that the state 𝑥_𝑘 is correlated with the measurement noises 𝑣^𝑖_𝑘, for 𝑖=1,2,…,𝑟, and the expectations 𝐸^𝑖_𝑘=𝐸[𝑥_𝑘𝑣^𝑖_𝑘] satisfy 𝐸^𝑖_𝑘=𝐹_𝑘−1𝑆^𝑖_{𝑘−2,𝑘}+𝑆^𝑖_{𝑘−1,𝑘},𝑘≥2;𝐸^𝑖₁=𝑆^𝑖_0,1. (5)

Remark 3. According to assumption (iii), the scalar random variables 𝜃^𝑖_𝑘 take values over the interval [0,1] and they can satisfy any arbitrary discrete probability distribution over such interval, for instance, a Bernoulli distribution. Usually, Bernoulli random variables have been used to model the phenomenon of missing measurements (see, e.g., [25] and references therein), with 𝜃^𝑖_𝑘=1 meaning that the state 𝑥_𝑘 is present in the measurement 𝑦^𝑖_𝑘 coming from the 𝑖th sensor at time 𝑘, while 𝜃^𝑖_𝑘=0 means that the state is missing in the measured data at time 𝑘 or, equivalently, that such observation only contains additive noise 𝑣^𝑖_𝑘. However, in practice, the information transmitted at a sampling time can usually be neither completely missing nor completely successful, but only part of the information can go through; in such situations, only partial information is missing and the proportion of missed data at one moment is a fraction other than 0 or 1 (see, e.g., [28, 29] and references therein).

2.2. Stacked Measurement Equation

As noted above, our aim is to solve the optimal LS linear estimation problem of the state 𝑥_𝑘 based on the measurements {𝑦^𝑖₁,𝑦^𝑖₂,…,𝑦^𝑖_𝑘}, for 𝑖=1,2,…,𝑟, by using centralized and distributed fusion methods to process the measured sensor data. The centralized fusion method considers that all the measurement data coming from 𝑟 sensors are transmitted to a fusion center for being processed; for this purpose and to simplify the notation, the measurement equation (2) is rewritten in a stacked form as follows: 𝑦_𝑘=Θ_𝑘𝐻_𝑘𝑥_𝑘+𝑣_𝑘,𝑘≥1, (6) where 𝑦_𝑘=(𝑦¹_𝑘,…,𝑦^𝑟_𝑘)^𝑇,𝑣_𝑘=(𝑣¹_𝑘,…,𝑣^𝑟_𝑘)^𝑇, 𝐻_𝑘=(𝐻_𝑘^1𝑇,…,𝐻_𝑘^𝑟𝑇)^𝑇, and Θ_𝑘=Diag(𝜃¹_𝑘,…,𝜃^𝑟_𝑘).

The following properties of the noises in (6) are easily inferred from the model assumptions (ii)–(iv) previously stated.(i)The additive noise {𝑣_𝑘;𝑘≥1} is a zero-mean process satisfying: Cov𝑣_𝑘,𝑣_𝑠=𝑅_𝑘,𝑘𝛿_𝑘−𝑠+𝑅_𝑘,𝑠𝛿_{𝑘−𝑠+1}+𝑅_𝑘,𝑠𝛿_{𝑘−𝑠−1},Cov𝑤_𝑘,𝑣_𝑠=𝑆_𝑘,𝑘𝛿_𝑘−𝑠+𝑆_𝑘,𝑠𝛿_{𝑘−𝑠+1}+𝑆_𝑘,𝑠𝛿_{𝑘−𝑠+2}, (7) where 𝑅_𝑘,𝑠=(𝑅^𝑖𝑗_𝑘,𝑠)_{𝑖,𝑗=1,2,…,𝑟} and 𝑆_𝑘,𝑠=(𝑆¹_𝑘,𝑠,…,𝑆^𝑟_𝑘,𝑠).(ii)The state vector 𝑥_𝑘 and the measurement noise vector 𝑣_𝑘 are correlated with 𝐸_𝑘=𝐸[𝑥_𝑘𝑣^𝑇_𝑘] satisfying 𝐸_𝑘=𝐹_𝑘−1𝑆_{𝑘−2,𝑘}+𝑆_{𝑘−1,𝑘},𝑘≥2,𝐸₁=𝑆_0,1. (8)(iii)The random matrices {Θ_𝑘;𝑘≥1} satisfy 𝐸[Θ_𝑘]=Θ_𝑘=Diag(𝜃¹_𝑘,…,𝜃^𝑟_𝑘) and 𝐸[(Θ_𝑘−Θ_𝑘)²]=Diag(𝑉^𝜃1_𝑘,…,𝑉^𝜃𝑟_𝑘); also, denoting 𝜃_𝑘=(𝜃¹_𝑘,…,𝜃^𝑟_𝑘)^𝑇, it is clear that Cov[𝜃_𝑘]=Diag(𝑉^𝜃1_𝑘,…,𝑉^𝜃𝑟_𝑘).Moreover, for any random matrix 𝐺 independent of {Θ_𝑘;𝑘≥1}, it is easily deduced that 𝐸Θ_𝑘−Θ_𝑘𝐺Θ_𝑘−Θ_𝑘=Cov𝜃_𝑘∘𝐸[𝐺], (9) where ∘ denotes the Hadamard product [23].(iv)The initial state 𝑥₀ and {Θ_𝑘;𝑘≥1} are independent, and they are independent of {𝑤_𝑘;𝑘≥0} and {𝑣_𝑘;𝑘≥1}.

2.3. Innovation Approach to the Optimal LS Linear Estimation Problem

To address the optimal LS linear estimation problem of the state 𝑥_𝑘 based on the measurements {𝑦^𝑖₁,𝑦^𝑖₂,…,𝑦^𝑖_𝑘}, 𝑖=1,2,…,𝑟, the centralized and distributed fusion methods will be used. In both cases, recursive algorithms for the LS linear estimators will be established using an innovation approach and the orthogonal projection Lemma (OPL); more specifically we have the following.

Centralized Fusion Estimation Problem. Our aim is to obtain the optimal LS linear filter, 𝑥_𝑘/𝑘, of the state 𝑥_𝑘 based on the measurements {𝑦₁,𝑦₂,…,𝑦_𝑘}, given in (6), by recursive algorithms.

As known, the LS linear filter 𝑥_𝑘/𝑘 is the orthogonal projection of the state 𝑥_𝑘 over the linear space spanned by {𝑦₁,𝑦₂,…,𝑦_𝑘}. These observations are generally nonorthogonal vectors, but the Gram-Schmidt orthogonalization procedure allows us to substitute them by a set of orthogonal vectors, called innovations, defined as the difference between each observation and its one-stage predictor. Due to the orthogonality property of the innovations and since the innovation process is uniquely determined by the observations, the LS linear filter, 𝑥_𝑘/𝑘, can be calculated as linear combination of the innovations; namely, 𝑥_𝑘/𝑘=^𝑘_𝑠=1𝒳_𝑘,𝑠Π⁻¹_𝑠,𝑠𝜇_𝑠,𝑘≥1, (10) where 𝜇_𝑠=𝑦_𝑠−𝑦_{𝑠/𝑠−1} are the innovation vectors, with 𝑦_{𝑠/𝑠−1} the one-stage observation predictor, Π_𝑠,𝑠=𝐸[𝜇_𝑠𝜇^𝑇_𝑠], and 𝒳_𝑘,𝑠=𝐸[𝑥_𝑘𝜇^𝑇_𝑠].

Distributed Fusion Estimation Problem. To address the distributed fusion estimation problem, firstly, recursive algorithms to obtain local LS linear filters, 𝑥^𝑖_𝑘/𝑘, for 𝑖=1,2,…,𝑟, and the error cross-covariance matrices between any two local estimates, are derived. Secondly, the distributed fusion filter, 𝑥^𝐷_𝑘/𝑘, is established by applying the optimal information fusion criterion weighted by matrices in the linear minimum variance sense [30].

Analogously to (10), denoting 𝜇^𝑖_𝑠=𝑦^𝑖_𝑠−𝑦^𝑖_{𝑠/𝑠−1}, Π^𝑖𝑖_𝑠,𝑠=𝐸[𝜇^𝑖_𝑠𝜇^𝑖_𝑠], and 𝒳^𝑖_𝑘,𝑠=𝐸[𝑥_𝑘𝜇^𝑖_𝑠], the local filter 𝑥^𝑖_𝑘/𝑘 is expressed as 𝑥^𝑖_𝑘/𝑘=^𝑘_𝑠=1𝒳^𝑖_𝑘,𝑠Π^𝑖𝑖_𝑠,𝑠⁻¹𝜇^𝑖_𝑠,𝑘≥1. (11)

3. Optimal LS Linear Centralized Fusion Estimation

In this section a recursive algorithm for the centralized optimal (under the LS criterion) linear filter, 𝑥_𝑘/𝑘 is derived. Such algorithm is deduced using (10) and the OPL, and it is presented in Theorem 5. Firstly, in order to simplify the proof of Theorem 5, the following lemma is established.

Lemma 4. Under assumptions (i)–(iv), the following results hold: 𝒲_𝑘,𝑘=𝐸𝑤_𝑘𝜇^𝑇_𝑘=𝑄_{𝑘,𝑘−1}𝐻^𝑇_𝑘Θ_𝑘+𝑆_𝑘,𝑘𝒱,𝑘≥1,_{𝑘,𝑘−1}=𝐸𝑣_𝑘𝜇^𝑇_𝑘−1=𝑆^𝑇_{𝑘−2,𝑘}𝐻^𝑇_𝑘−1Θ_𝑘−1+𝑅_{𝑘,𝑘−1},𝑘≥2. (12)(13)

Proof. Since 𝑤_𝑘 is independent of 𝑦₁,…,𝑦_𝑘−1, 𝐸[𝑤_𝑘𝑦^𝑇_{𝑘/𝑘−1}]=0 and hence 𝒲_𝑘,𝑘=𝐸[𝑤_𝑘𝑦^𝑇_𝑘]. Now, using (1) and (6), 𝒲_𝑘,𝑘 can be calculated as follows: 𝒲_𝑘,𝑘=𝐸𝑤_𝑘Θ_𝑘𝐻_𝑘𝑥_𝑘+𝑣_𝑘^𝑇=𝐸𝑤_𝑘𝑥^𝑇_𝑘𝐻^𝑇_𝑘Θ_𝑘+𝑆_𝑘,𝑘=𝐸𝑤_𝑘𝐹_𝑘−1𝑥_𝑘−1+𝑤_𝑘−1^𝑇𝐻^𝑇_𝑘Θ_𝑘+𝑆_𝑘,𝑘=𝑄_{𝑘,𝑘−1}𝐻^𝑇_𝑘Θ_𝑘+𝑆_𝑘,𝑘. (14) Taking into account that 𝑣_𝑘 is independent of 𝑦₁,…,𝑦_𝑘−2, the calculation of 𝒱_{𝑘,𝑘−1} is similar to that of 𝒲_𝑘,𝑘, and hence the proof is omitted.

Theorem 5. For the system model (1) and measurement model (6), under assumptions (i)–(iv), the optimal LS linear filter 𝑥_𝑘/𝑘 is obtained as 𝑥_𝑘/𝑘=𝑥_{𝑘/𝑘−1}+𝒳_𝑘,𝑘Π⁻¹_𝑘,𝑘𝜇_𝑘,𝑘≥1,𝑥_0/0=𝑥₀, (15) where the state predictor, 𝑥_{𝑘/𝑘−1}, satisfies 𝑥_{𝑘/𝑘−1}=𝐹_𝑘−1𝑥_{𝑘−1/𝑘−1}+𝒲_{𝑘−1,𝑘−1}Π⁻¹_{𝑘−1,𝑘−1}𝜇_𝑘−1𝑥,𝑘≥2,_1/0=𝐹₀𝑥_0/0. (16)
The innovation, 𝜇_𝑘, is given by 𝜇_𝑘=𝑦_𝑘−Θ_𝑘𝐻_𝑘𝑥_{𝑘/𝑘−1}−𝒱_{𝑘,𝑘−1}Π⁻¹_{𝑘−1,𝑘−1}𝜇_𝑘−1𝜇,𝑘≥2,₁=𝑦₁−Θ₁𝐻₁𝑥_1/0. (17) The matrix 𝒳_𝑘,𝑘=𝐸[𝑥_𝑘𝜇^𝑇_𝑘] is calculated by 𝒳_𝑘,𝑘=𝑃_{𝑘/𝑘−1}𝐻^𝑇_𝑘Θ_𝑘+𝐸_𝑘−𝒳_{𝑘,𝑘−1}Π⁻¹_{𝑘−1,𝑘−1}𝒱^𝑇_{𝑘,𝑘−1}𝒳,𝑘≥2,_1,1=𝑃_1/0𝐻^𝑇₁Θ₁+𝐸₁, (18) where 𝒳_{𝑘,𝑘−1}=𝐸[𝑥_𝑘𝜇^𝑇_𝑘−1] satisfies 𝒳_{𝑘,𝑘−1}=𝐹_𝑘−1𝒳_{𝑘−1,𝑘−1}+𝒲_{𝑘−1,𝑘−1},𝑘≥2. (19) The prediction error covariance matrix, 𝑃_{𝑘/𝑘−1}, is obtained by 𝑃_{𝑘/𝑘−1}=𝐹_𝑘−1𝑃_{𝑘−1/𝑘−1}𝐹^𝑇_𝑘−1+𝑄_{𝑘−1,𝑘−1}+𝐹_𝑘−1𝒥_𝑘−1+𝒥^𝑇_𝑘−1𝐹^𝑇_𝑘−1−𝒲_{𝑘−1,𝑘−1}Π⁻¹_{𝑘−1,𝑘−1}𝒲^𝑇_{𝑘−1,𝑘−1}𝑃,𝑘≥2,_1/0=𝐹₀𝑃_0/0𝐹^𝑇₀+𝑄_0,0, (20) where 𝒥_𝑘=𝐸[𝑥_𝑘/𝑘𝑤^𝑇_𝑘] is calculated by 𝒥_𝑘=𝑄_{𝑘−1,𝑘}−𝒳_𝑘,𝑘Π⁻¹_𝑘,𝑘𝒲^𝑇_𝑘,𝑘,𝑘≥1. (21) The filtering error covariance matrix, 𝑃_𝑘/𝑘, is given by 𝑃_𝑘/𝑘=𝑃_{𝑘/𝑘−1}−𝒳_𝑘,𝑘Π⁻¹_𝑘,𝑘𝒳^𝑇_𝑘,𝑘,𝑘≥1,𝑃_0/0=𝑃₀. (22) The innovation covariance matrix, Π_𝑘,𝑘, satisfies Π_𝑘,𝑘=Cov𝜃_𝑘∘𝐻_𝑘𝐷_𝑘𝐻^𝑇_𝑘+𝑅_𝑘,𝑘+Θ_𝑘𝐻_𝑘𝒳_𝑘,𝑘+𝒳^𝑇_𝑘,𝑘𝐻^𝑇_𝑘Θ_𝑘−Θ_𝑘𝐻_𝑘𝑃_{𝑘/𝑘−1}𝐻^𝑇_𝑘Θ_𝑘−𝒱_{𝑘,𝑘−1}Π⁻¹_{𝑘−1,𝑘−1}𝒱^𝑇_{𝑘,𝑘−1}Π,𝑘≥2,_1,1=Cov𝜃₁∘𝐻₁𝐷₁𝐻^𝑇₁+𝑅_1,1+Θ₁𝐻₁𝒳_1,1+𝒳^𝑇_1,1𝐻^𝑇₁Θ₁−Θ₁𝐻₁𝑃_1/0𝐻^𝑇₁Θ₁. (23) The matrices 𝐷_𝑘, 𝐸_𝑘, 𝒲_𝑘,𝑘, and 𝒱_{𝑘,𝑘−1} are given in (4), (8), (12), and (13), respectively.

Proof. See Appendix A.

Remark 6. In conventional estimation problems in systems with missing measurements and uncorrelated additive white noises, the one-stage state and observation predictors are calculated as 𝑥_{𝑘/𝑘−1}=𝐹_𝑘−1𝑥_{𝑘−1/𝑘−1} and 𝑦_{𝑘/𝑘−1}=Θ_𝑘𝐻_𝑘𝑥_{𝑘/𝑘−1}, respectively. However, this is not true for the problem at hand since, due to the correlation assumption (ii), the noise estimators 𝑤_{𝑘−1/𝑘−1} and 𝑣̂_{𝑘/𝑘−1} must be taken into account for the derivation of the predictors. Besides the fact of considering missing measurements, this is the main difference between the optimal estimators proposed in the current paper and the suboptimal Kalman-type ones proposed in [16], where the noise estimators are considered to be equal to zero.

4. Distributed Fusion Estimation

One of the main disadvantages of the centralized fusion estimators derived in Section 3 is that they may have a high computational cost due to augmentation. Moreover, as is widely known, the centralized approach has several other drawbacks, such as fault detection, isolation, poor reliability, and so forth. To overcome these disadvantages, our aim in this section is to address the optimal distributed fusion estimation problem, in which each single sensor provides its local LS linear estimator and their estimation error covariance matrices, and then these local estimators along with the covariances and cross-covariance matrices of the estimation errors between any two sensors are sent to the fusion center for fusion based on the matrices-weighted fusion estimation criterion in the linear minimum variance sense [30].

4.1. Local LS Linear Filtering Algorithms

For each single sensor subsystem of systems (1) and (2), the following theorem provides recursive formulas for the local LS linear filters, 𝑥^𝑖_𝑘/𝑘, and their corresponding error covariance matrices, 𝑃^𝑖𝑖_𝑘/𝑘.

Theorem 7. For the 𝑖th sensor subsystem of systems (1) and (2) under assumptions (i)–(iv), the local LS linear filter, 𝑥^𝑖_𝑘/𝑘, is calculated by 𝑥^𝑖_𝑘/𝑘=𝑥^𝑖_{𝑘/𝑘−1}+𝒳^𝑖_𝑘,𝑘Π^𝑖𝑖_𝑘,𝑘⁻¹𝜇^𝑖_𝑘,𝑘≥1,𝑥^𝑖_0/0=𝑥₀, (24) where the local LS linear predictor, 𝑥^𝑖_{𝑘/𝑘−1}, satisfies 𝑥^𝑖_{𝑘/𝑘−1}=𝐹_𝑘−1𝑥^𝑖_{𝑘−1/𝑘−1}+𝒲^𝑖_{𝑘−1,𝑘−1}Π^𝑖𝑖_{𝑘−1,𝑘−1}⁻¹𝜇^𝑖_𝑘−1𝑥,𝑘≥2,_𝑖1/0=𝐹₀𝑥^𝑖_0/0, (25) with 𝒲^𝑖_𝑘,𝑘=𝜃^𝑖_𝑘𝑄_{𝑘,𝑘−1}𝐻_𝑘^𝑖𝑇+𝑆^𝑖_𝑘,𝑘,𝑘≥1.
The innovation, 𝜇^𝑖_𝑘, is given by 𝜇^𝑖_𝑘=𝑦^𝑖_𝑘−𝜃^𝑖_𝑘𝐻^𝑖_𝑘𝑥^𝑖_{𝑘/𝑘−1}−𝒱^𝑖𝑖_{𝑘,𝑘−1}Π^𝑖𝑖_{𝑘−1,𝑘−1}⁻¹𝜇^𝑖_𝑘−1𝜇,𝑘≥2,_𝑖1=𝑦^𝑖₁−𝜃^𝑖₁𝐻^𝑖₁𝑥^𝑖_1/0, (26) with 𝒱^𝑖𝑖_{𝑘,𝑘−1}=𝜃^𝑖_𝑘−1𝑆^𝑖𝑇_{𝑘−2,𝑘}𝐻^𝑖𝑇_𝑘−1+𝑅^𝑖𝑖_{𝑘,𝑘−1},𝑘≥2.
The vector 𝒳^𝑖_𝑘,𝑘=𝐸[𝑥_𝑘𝜇^𝑖_𝑘] is calculated from the following expression 𝒳^𝑖_𝑘,𝑘=𝜃^𝑖_𝑘𝑃^𝑖𝑖_{𝑘/𝑘−1}𝐻_𝑘^𝑖𝑇+𝐸^𝑖_𝑘−𝒳^𝑖_{𝑘,𝑘−1}Π^𝑖𝑖_{𝑘−1,𝑘−1}⁻¹𝒱^𝑖𝑖_{𝑘,𝑘−1},𝒳𝑘≥2,_𝑖1,1=𝜃^𝑖₁𝑃^𝑖𝑖_1/0𝐻₁^𝑖𝑇+𝐸^𝑖₁, (27) where 𝒳^𝑖_{𝑘,𝑘−1}=𝐹_𝑘−1𝒳^𝑖_{𝑘−1,𝑘−1}+𝒲^𝑖_{𝑘−1,𝑘−1},𝑘≥2.
The local prediction error covariance matrix, 𝑃^𝑖𝑖_{𝑘/𝑘−1}, is obtained by 𝑃^𝑖𝑖_{𝑘/𝑘−1}=𝐹_𝑘−1𝑃^𝑖𝑖_{𝑘−1/𝑘−1}𝐹^𝑇_𝑘−1+𝑄_{𝑘−1,𝑘−1}+𝐹_𝑘−1𝒥^𝑖_𝑘−1+𝒥^𝑖𝑇_𝑘−1𝐹^𝑇_𝑘−1−𝒲^𝑖_{𝑘−1,𝑘−1}Π^𝑖𝑖_{𝑘−1,𝑘−1}⁻¹𝒲^𝑖𝑇_{𝑘−1,𝑘−1}𝑃,𝑘≥2,_𝑖𝑖1/0=𝐹₀𝑃^𝑖𝑖_0/0𝐹^𝑇₀+𝑄_0,0, (28) where 𝒥^𝑖_𝑘=𝑄_{𝑘−1,𝑘}−𝒳^𝑖_𝑘,𝑘(Π^𝑖𝑖_𝑘,𝑘)⁻¹𝒲^𝑖𝑇_𝑘,𝑘,𝑘≥1, and 𝑃^𝑖𝑖_𝑘/𝑘, the filtering error covariance matrix, is given by 𝑃^𝑖𝑖_𝑘/𝑘=𝑃^𝑖𝑖_{𝑘/𝑘−1}−𝒳^𝑖_𝑘,𝑘Π^𝑖𝑖_𝑘,𝑘⁻¹𝒳^𝑖𝑇_𝑘,𝑘,𝑘≥1,𝑃^𝑖𝑖_0/0=𝑃₀. (29)
The innovation variance, Π^𝑖𝑖_𝑘,𝑘, satisfies Π^𝑖𝑖_𝑘,𝑘=𝑉^𝜃𝑖_𝑘𝐻^𝑖_𝑘𝐷_𝑘𝐻_𝑘^𝑖𝑇+𝑅^𝑖𝑖_𝑘,𝑘+𝜃^𝑖_𝑘𝐻^𝑖_𝑘𝒳^𝑖_𝑘,𝑘+𝜃^𝑖_𝑘𝒳^𝑖𝑇_𝑘,𝑘𝐻_𝑘^𝑖𝑇−𝜃^𝑖_𝑘²𝐻^𝑖_𝑘𝑃^𝑖𝑖_{𝑘/𝑘−1}𝐻_𝑘^𝑖𝑇−𝒱^𝑖𝑖_{𝑘,𝑘−1}²Π^𝑖𝑖_{𝑘−1,𝑘−1}⁻¹,Π𝑘≥2,_𝑖𝑖1,1=𝑉^𝜃𝑖₁𝐻^𝑖₁𝐷₁𝐻₁^𝑖𝑇+𝑅^𝑖𝑖_1,1+𝜃^𝑖₁𝐻^𝑖₁𝒳^𝑖_1,1+𝜃^𝑖₁𝒳^𝑖𝑇_1,1𝐻₁^𝑖𝑇−𝜃^𝑖₁²𝐻^𝑖₁𝑃^𝑖𝑖_1/0𝐻₁^𝑖𝑇. (30) The matrix 𝐷_𝑘 and the vector 𝐸^𝑖_𝑘 are given in (4) and (5), respectively.

Proof. The proof, based on the innovation approach and the OPL, is omitted for being analogous to that of Theorem 5. Nevertheless, it should be indicated that, in this proof, the Hadamard product is not used since, instead of the diagonal stochastic matrix Θ_𝑘, the scalar variable 𝜃^𝑖_𝑘 is now involved in the derivation of the estimators.

Remark 8. As indicated in Remark 6 for the centralized estimators, it must be noted that, due to the correlation assumption (ii) of the additive noises {𝑤_𝑘} and {𝑣^𝑖_𝑘}, the estimators 𝑤^𝑖_{𝑘−1/𝑘−1}=𝒲^𝑖_{𝑘−1,𝑘−1}(Π^𝑖𝑖_{𝑘−1,𝑘−1})⁻¹𝜇^𝑖_𝑘−1 and 𝑣̂^𝑖_{𝑘/𝑘−1}=𝒱^𝑖𝑖_{𝑘,𝑘−1}(Π^𝑖𝑖_{𝑘−1,𝑘−1})⁻¹𝜇^𝑖_𝑘−1 are not equal to zero, and hence the optimal local state predictor, 𝑥^𝑖_{𝑘/𝑘−1}=𝐹_𝑘−1𝑥^𝑖_{𝑘−1/𝑘−1}+𝑤^𝑖_{𝑘−1/𝑘−1}, and the observation predictor, 𝑦^𝑖_{𝑘/𝑘−1}=𝜃^𝑖_𝑘𝐻^𝑖_𝑘𝑥^𝑖_{𝑘/𝑘−1}+𝑣̂^𝑖_{𝑘/𝑘−1}, are quite different from conventional filtering algorithms with uncorrelated white noises. This issue, along with the consideration of missing measurements at each single sensor, constitutes the main difference between the current optimal local estimators and the suboptimal local estimators proposed in [16].

4.2. Cross-Covariance Matrices of Local Estimation Errors

To apply the optimal fusion criterion weighted by matrices in the linear minimum variance sense, the filtering, 𝑃^𝑖𝑗_𝑘/𝑘, and prediction, 𝑃^𝑖𝑗_{𝑘/𝑘−1}, error cross-covariance matrices between local estimators of any two subsystems must be calculated.

For simplicity, besides the notation of Theorem 7, for 𝑖≠𝑗, 𝑖,𝑗=1,2…,𝑟, we introduce the following notation: 𝐿_𝑘^𝑖𝑗=𝐸𝑥^𝑖_{𝑘/𝑘−1}𝜇^𝑗_𝑘,Π^𝑖𝑗_𝑘,𝑠=𝐸𝜇^𝑖_𝑘𝜇^𝑗_𝑠,𝒱^𝑖𝑗_{𝑘,𝑘−1}=𝐸𝑣^𝑖_𝑘𝜇^𝑗_𝑘−1. (31) Also, in order to simplify the calculation of the error cross-covariance matrices, the following lemmas are given.

Lemma 9. Under assumptions (i)–(iv), the following results hold. (a)The expectation 𝐸[𝑥^𝑖_{𝑘/𝑘−1}𝜇^𝑗_𝑘−1] satisfies 𝐸𝑥^𝑖_{𝑘/𝑘−1}𝜇^𝑗_𝑘−1=𝐹_𝑘−1𝐿^𝑖𝑗_𝑘−1+𝒳^𝑖_{𝑘,𝑘−1}Π^𝑖𝑖_{𝑘−1,𝑘−1}⁻¹Π^𝑖𝑗_{𝑘−1,𝑘−1},𝑘≥2. (32)(b)The expectation 𝐸[𝑥^𝑖_{𝑘/𝑘−1}𝑣^𝑗_𝑘] satisfies 𝐸𝑥^𝑖_{𝑘/𝑘−1}𝑣^𝑗_𝑘=𝒳^𝑖_{𝑘,𝑘−1}Π^𝑖𝑖_{𝑘−1,𝑘−1}⁻¹𝒱^𝑗𝑖_{𝑘,𝑘−1},𝑘≥2, (33) where 𝒱^𝑗𝑖_{𝑘,𝑘−1}=𝜃^𝑖_𝑘−1𝐻^𝑖_𝑘−1𝑆^𝑗_{𝑘−2,𝑘}+𝑅^𝑖𝑗_{𝑘−1,𝑘}.(c)The expectation 𝐸[𝑣^𝑖_𝑘𝜇^𝑗_𝑘] satisfies 𝐸𝑣^𝑖_𝑘𝜇^𝑗_𝑘=𝜃^𝑗_𝑘𝐸_𝑘^𝑖𝑇𝐻_𝑘^𝑗𝑇+𝑅^𝑖𝑗_𝑘,𝑘−𝒱^𝑖𝑗_{𝑘,𝑘−1}Π^𝑗𝑗_{𝑘−1,𝑘−1}⁻¹𝜃^𝑗_𝑘𝐻^𝑗_𝑘𝒳^𝑗_{𝑘,𝑘−1}+𝒱^𝑗𝑗_{𝑘,𝑘−1}^𝑇,𝑘≥2. (34)

Proof. (a) From (25) for 𝑥^𝑖_{𝑘/𝑘−1} and (24) for 𝑥^𝑖_{𝑘−1/𝑘−1}, we have 𝐸𝑥^𝑖_{𝑘/𝑘−1}𝜇^𝑗_𝑘−1=𝐹_𝑘−1𝐸𝑥^𝑖_{𝑘−1/𝑘−1}𝜇^𝑗_𝑘−1+𝒲^𝑖_{𝑘−1,𝑘−1}Π^𝑖𝑖_{𝑘−1,𝑘−1}⁻¹Π^𝑖𝑗_{𝑘−1,𝑘−1}=𝐹_𝑘−1𝐸𝑥^𝑖_{𝑘−1/𝑘−2}𝜇^𝑗_𝑘−1+𝐹_𝑘−1𝒳^𝑖_{𝑘−1,𝑘−1}×Π^𝑖𝑖_{𝑘−1,𝑘−1}⁻¹Π^𝑖𝑗_{𝑘−1,𝑘−1}+𝒲^𝑖_{𝑘−1,𝑘−1}Π^𝑖𝑖_{𝑘−1,𝑘−1}⁻¹Π^𝑖𝑗_{𝑘−1,𝑘−1}=𝐹_𝑘−1𝐿^𝑖𝑗_𝑘−1+𝐹_𝑘−1𝒳^𝑖_{𝑘−1,𝑘−1}+𝒲^𝑖_{𝑘−1,𝑘−1}×Π^𝑖𝑖_{𝑘−1,𝑘−1}⁻¹Π^𝑖𝑗_{𝑘−1,𝑘−1}, (35) and since 𝒳^𝑖_{𝑘,𝑘−1}=𝐹_𝑘−1𝒳^𝑖_{𝑘−1,𝑘−1}+𝒲^𝑖_{𝑘−1,𝑘−1}, expression (32) is proved.
(b) Analogously, taking into account that 𝐸[𝑥^𝑖_{𝑘−1/𝑘−2}𝑣^𝑗_𝑘]=0, we have 𝐸𝑥^𝑖_{𝑘/𝑘−1}𝑣^𝑗_𝑘=𝐹_𝑘−1𝐸𝑥^𝑖_{𝑘−1/𝑘−1}𝑣^𝑗_𝑘+𝒲^𝑖_{𝑘−1,𝑘−1}Π^𝑖𝑖_{𝑘−1,𝑘−1}⁻¹𝐸𝜇^𝑖_𝑘−1𝑣^𝑗_𝑘=𝐹_𝑘−1𝒳^𝑖_{𝑘−1,𝑘−1}+𝒲^𝑖_{𝑘−1,𝑘−1}Π^𝑖𝑖_{𝑘−1,𝑘−1}⁻¹𝒱^𝑗𝑖_{𝑘,𝑘−1}, (36) and expression (33) is immediately obtained. Finally, the derivation of expression 𝒱^𝑗𝑖_{𝑘,𝑘−1}=𝜃^𝑖_𝑘−1𝐻^𝑖_𝑘−1𝑆^𝑗_{𝑘−2,𝑘}+𝑅^𝑖𝑗_{𝑘−1,𝑘} is similar to that of (13) and hence it is omitted.
(c) Taking into account expression (26) for 𝜇^𝑗_𝑘, with (2) for 𝑦^𝑗_𝑘, we have 𝐸𝑣^𝑖_𝑘𝜇^𝑗_𝑘=𝜃^𝑗_𝑘𝐸_𝑘^𝑖𝑇𝐻_𝑘^𝑗𝑇+𝑅^𝑖𝑗_𝑘,𝑘−𝜃^𝑗_𝑘𝐸𝑣^𝑖_𝑘𝑥^𝑗𝑇_{𝑘/𝑘−1}𝐻_𝑘^𝑗𝑇−𝒱^𝑖𝑗_{𝑘,𝑘−1}Π^𝑗𝑗_{𝑘−1,𝑘−1}⁻¹𝒱^𝑗𝑗_{𝑘,𝑘−1},𝑘≥2, (37) and using (33) for 𝐸[𝑣^𝑖_𝑘𝑥^𝑗𝑇_{𝑘/𝑘−1}], expression (34) is obtained.

Lemma 10. Under assumptions (i)–(iv), for 𝑖≠𝑗, 𝑖,𝑗=1,2…,𝑟, the expectations 𝐿_𝑘^𝑖𝑗=𝐸[𝑥^𝑖_{𝑘/𝑘−1}𝜇^𝑗_𝑘] are recursively obtained by 𝐿_𝑘^𝑖𝑗=𝜃^𝑗_𝑘𝑃^𝑗𝑗_{𝑘/𝑘−1}−𝑃^𝑖𝑗_{𝑘/𝑘−1}𝐻_𝑘^𝑗𝑇−𝐹_𝑘−1𝐿^𝑖𝑗_𝑘−1Π^𝑗𝑗_{𝑘−1,𝑘−1}⁻¹𝒱^𝑗𝑗_{𝑘,𝑘−1}+𝒳^𝑖_{𝑘,𝑘−1}Π^𝑖𝑖_{𝑘−1,𝑘−1}⁻¹×𝒱^𝑗𝑖_{𝑘,𝑘−1}−𝒱^𝑗𝑗_{𝑘,𝑘−1}Π^𝑗𝑗_{𝑘−1,𝑘−1}⁻¹Π^𝑖𝑗_{𝑘−1,𝑘−1},𝑘≥2, (38) with initial condition 𝐿₁^𝑖𝑗=0.

Proof. Taking into account expression (26) for 𝜇^𝑗_𝑘, with (2) for 𝑦^𝑗_𝑘, we have 𝐿_𝑘^𝑖𝑗=𝜃^𝑗_𝑘𝐸𝑥^𝑖_{𝑘/𝑘−1}𝑥^𝑇_𝑘𝐻_𝑘^𝑗𝑇+𝐸𝑥^𝑖_{𝑘/𝑘−1}𝑣^𝑗_𝑘−𝜃^𝑗_𝑘𝐸𝑥^𝑖_{𝑘/𝑘−1}𝑥^𝑗𝑇_{𝑘/𝑘−1}𝐻_𝑘^𝑗𝑇−𝐸𝑥^𝑖_{𝑘/𝑘−1}𝜇^𝑗_𝑘−1Π^𝑗𝑗_{𝑘−1,𝑘−1}⁻¹𝒱^𝑗𝑗_{𝑘,𝑘−1},𝑘≥2. (39) From the OPL, 𝐸[𝑥^𝑖_{𝑘/𝑘−1}𝑥^𝑇_𝑘]=𝐸[𝑥^𝑖_{𝑘/𝑘−1}𝑥^𝑖𝑇_{𝑘/𝑘−1}]; then, taking into account (32) for 𝐸[𝑥^𝑖_{𝑘/𝑘−1}𝜇^𝑗_𝑘−1], and (33) for 𝐸[𝑥^𝑖_{𝑘/𝑘−1}𝑣^𝑗_𝑘], it is enough to prove that 𝐸𝑥^𝑖_{𝑘/𝑘−1}𝑥^𝑖𝑇_{𝑘/𝑘−1}−𝐸𝑥^𝑖_{𝑘/𝑘−1}𝑥^𝑗𝑇_{𝑘/𝑘−1}=𝑃^𝑗𝑗_{𝑘/𝑘−1}−𝑃^𝑖𝑗_{𝑘/𝑘−1}, (40) which is easily deduced since 𝐸𝑥^𝑖_{𝑘/𝑘−1}𝑥^𝑗𝑇_{𝑘/𝑘−1}=𝑃^𝑖𝑗_{𝑘/𝑘−1}−𝐷_𝑘+𝐸𝑥^𝑖_{𝑘/𝑘−1}𝑥^𝑖𝑇_{𝑘/𝑘−1}+𝐸𝑥^𝑗_{𝑘/𝑘−1}𝑥^𝑗𝑇_{𝑘/𝑘−1},𝐸𝑥^𝑗_{𝑘/𝑘−1}𝑥^𝑗𝑇_{𝑘/𝑘−1}=𝐷_𝑘−𝑃^𝑗𝑗_{𝑘/𝑘−1}. (41)

Lemma 11. Under assumptions (i)–(iv), for 𝑖≠𝑗, 𝑖,𝑗=1,2…,𝑟, the innovation cross-covariance Π^𝑖𝑗_𝑘,𝑘=𝐸[𝜇^𝑖_𝑘𝜇^𝑗_𝑘] satisfies Π^𝑖𝑗_𝑘,𝑘=𝜃^𝑖_𝑘𝐻^𝑖_𝑘𝒳^𝑗_𝑘,𝑘−𝐿_𝑘^𝑖𝑗+𝜃^𝑗_𝑘𝐸_𝑘^𝑖𝑇𝐻_𝑘^𝑗𝑇+𝑅^𝑖𝑗_𝑘,𝑘−𝒱^𝑖𝑖_{𝑘,𝑘−1}Π^𝑖𝑖_{𝑘−1,𝑘−1}⁻¹Π^𝑖𝑗_{𝑘−1,𝑘}−𝒱^𝑖𝑗_{𝑘,𝑘−1}Π^𝑗𝑗_{𝑘−1,𝑘−1}⁻¹𝜃^𝑗_𝑘𝐻^𝑗_𝑘𝒳^𝑗_{𝑘,𝑘−1}+𝒱^𝑗𝑗_{𝑘,𝑘−1}^𝑇,Π𝑘≥2,_𝑖𝑗1,1=𝜃^𝑖₁𝐻^𝑖₁𝒳^𝑗_1,1+𝜃^𝑗₁𝐸^𝑖₁𝐻₁^𝑗𝑇+𝑅^𝑖𝑗_1,1, (42) where Π^𝑖𝑗_{𝑘−1,𝑘}=𝐸[𝜇^𝑖_𝑘−1𝜇^𝑗_𝑘] is given by Π^𝑖𝑗_{𝑘−1,𝑘}=𝜃^𝑗_𝑘𝒳^𝑖_{𝑘,𝑘−1}−𝐹_𝑘−1𝐿^𝑗𝑖_𝑘−1−𝒳^𝑗_{𝑘,𝑘−1}Π^𝑗𝑗_{𝑘−1,𝑘−1}⁻¹Π^𝑗𝑖_{𝑘−1,𝑘−1}^𝑇𝐻_𝑘^𝑗𝑇+𝒱^𝑗𝑖_{𝑘,𝑘−1}−Π^𝑖𝑗_{𝑘−1,𝑘−1}Π^𝑗𝑗_{𝑘−1,𝑘−1}⁻¹𝒱^𝑗𝑗_{𝑘,𝑘−1},𝑘≥2. (43)

Proof. Taking into account expression (26) for 𝜇^𝑖_𝑘, with (2) for 𝑦^𝑖_𝑘, we have Π^𝑖𝑗_𝑘,𝑘=𝜃^𝑖_𝑘𝐻^𝑖_𝑘𝐸𝑥_𝑘𝜇^𝑗_𝑘+𝐸𝑣^𝑖_𝑘𝜇^𝑗_𝑘−𝜃^𝑖_𝑘𝐻^𝑖_𝑘𝐸𝑥^𝑖_{𝑘/𝑘−1}𝜇^𝑗_𝑘−𝒱^𝑖𝑖_{𝑘,𝑘−1}Π^𝑖𝑖_{𝑘−1,𝑘−1}⁻¹𝐸𝜇^𝑖_𝑘−1𝜇^𝑗_𝑘=𝜃^𝑖_𝑘𝐻^𝑖_𝑘𝒳^𝑗_𝑘,𝑘−𝐿_𝑘^𝑖𝑗+𝐸𝑣^𝑖_𝑘𝜇^𝑗_𝑘−𝒱^𝑖𝑖_{𝑘,𝑘−1}Π^𝑖𝑖_{𝑘−1,𝑘−1}⁻¹Π^𝑖𝑗_{𝑘−1,𝑘},𝑘≥2, (44) and, from (34) for 𝐸[𝑣^𝑖_𝑘𝜇^𝑗_𝑘], expression for Π^𝑖𝑗_𝑘,𝑘 is clear.
Analogously, taking into account expression (26) for 𝜇^𝑗_𝑘, with (2) for 𝑦^𝑗_𝑘, we have Π^𝑖𝑗_{𝑘−1,𝑘}=𝜃^𝑗_𝑘𝐸𝜇^𝑖_𝑘−1𝑥^𝑇_𝑘𝐻_𝑘^𝑗𝑇+𝐸𝜇^𝑖_𝑘−1𝑣^𝑗_𝑘−𝜃^𝑗_𝑘𝐸𝜇^𝑖_𝑘−1𝑥^𝑗𝑇_{𝑘/𝑘−1}𝐻_𝑘^𝑗𝑇−𝐸𝜇^𝑖_𝑘−1𝜇^𝑗_𝑘−1Π^𝑗𝑗_{𝑘−1,𝑘−1}⁻¹𝒱^𝑗𝑗_{𝑘,𝑘−1},=𝜃^𝑗_𝑘𝒳^𝑖_{𝑘,𝑘−1}−𝐸𝜇^𝑖_𝑘−1𝑥^𝑗𝑇_{𝑘/𝑘−1}𝐻_𝑘^𝑗𝑇+𝒱^𝑗𝑖_{𝑘,𝑘−1}−Π^𝑖𝑗_{𝑘−1,𝑘−1}Π^𝑗𝑗_{𝑘−1,𝑘−1}⁻¹𝒱^𝑗𝑗_{𝑘,𝑘−1},𝑘≥2, (45) and, from (32) for 𝐸[𝜇^𝑖_𝑘−1𝑥^𝑗𝑇_{𝑘/𝑘−1}], expression for Π^𝑖𝑗_{𝑘−1,𝑘} is immediately derived.

In the following theorem, recursive formulas to calculate the filtering and prediction error cross-covariance matrices, 𝑃^𝑖𝑗_𝑘/𝑘 and 𝑃^𝑖𝑗_{𝑘/𝑘−1}, respectively, are derived.

Theorem 12. Under assumptions (i)–(iv), the cross-covariance matrices, 𝑃^𝑖𝑗_𝑘/𝑘, of the filtering errors between the 𝑖th and the 𝑗th sensor subsystems are recursively computed by 𝑃^𝑖𝑗_𝑘/𝑘=𝑃^𝑖𝑗_{𝑘/𝑘−1}+𝒳^𝑖_𝑘,𝑘Π^𝑖𝑖_𝑘,𝑘⁻¹Π^𝑖𝑗_𝑘,𝑘Π^𝑗𝑗_𝑘,𝑘⁻¹𝒳^𝑗𝑇_𝑘,𝑘−𝒳^𝑗_𝑘,𝑘−𝐿_𝑘^𝑖𝑗Π^𝑗𝑗_𝑘,𝑘⁻¹𝒳^𝑗𝑇_𝑘,𝑘−𝒳^𝑖_𝑘,𝑘Π^𝑖𝑖_𝑘,𝑘⁻¹𝒳^𝑖_𝑘,𝑘−𝐿_𝑘^𝑗𝑖𝑇𝑃,𝑘≥1,_𝑖𝑗0/0=𝑃₀, (46) where 𝑃^𝑖𝑗_{𝑘/𝑘−1}, the cross-covariance matrix of the prediction error between the 𝑖th and the 𝑗th sensor subsystems, satisfies 𝑃^𝑖𝑗_{𝑘/𝑘−1}=𝐹_𝑘−1𝑃^𝑖𝑗_{𝑘−1/𝑘−1}𝐹^𝑇_𝑘−1+𝑄_{𝑘−1,𝑘−1}+𝐹_𝑘−1𝒥^𝑖_𝑘−1+𝒥^𝑗𝑇_𝑘−1𝐹^𝑇_𝑘−1+𝒲^𝑖_{𝑘−1,𝑘−1}Π^𝑖𝑖_{𝑘−1,𝑘−1}⁻¹Π^𝑖𝑗_{𝑘−1,𝑘−1}×Π^𝑗𝑗_{𝑘−1,𝑘−1}⁻¹𝒲^𝑗𝑇_{𝑘−1,𝑘−1}−𝒢^𝑖𝑗_𝑘−1Π^𝑗𝑗_{𝑘−1,𝑘−1}⁻¹𝒲^𝑗𝑇_{𝑘−1,𝑘−1}−𝒲^𝑖_{𝑘−1,𝑘−1}Π^𝑖𝑖_{𝑘−1,𝑘−1}⁻¹𝒢^𝑗𝑖𝑇_𝑘−1𝑃,𝑘≥2,_𝑖𝑗1/0=𝐹₀𝑃^𝑖𝑗_0/0𝐹^𝑇₀+𝑄_0,0, (47) where 𝒢_𝑘^𝑖𝑗=𝒲^𝑗_𝑘,𝑘+𝐹_𝑘(𝒳^𝑗_𝑘,𝑘−𝐿_𝑘^𝑖𝑗−𝒳^𝑖_𝑘,𝑘(Π^𝑖𝑖_𝑘,𝑘)⁻¹Π^𝑖𝑗_𝑘,𝑘), 𝑘≥1. The vectors 𝐿_𝑘^𝑖𝑗 and the innovation cross-covariances Π^𝑖𝑗_𝑘,𝑘 are given in Lemmas 10 and 11, respectively.

Proof. By using (24) for 𝑥^𝑖_𝑘/𝑘 and 𝑥^𝑗_𝑘/𝑘, we have 𝑃^𝑖𝑗_𝑘/𝑘=𝑃^𝑖𝑗_{𝑘/𝑘−1}+𝒳^𝑖_𝑘,𝑘Π^𝑖𝑖_𝑘,𝑘⁻¹Π^𝑖𝑗_𝑘,𝑘Π^𝑗𝑗_𝑘,𝑘⁻¹𝒳^𝑗𝑇_𝑘,𝑘−𝐸𝑥_𝑘−𝑥^𝑖_{𝑘/𝑘−1}𝜇_𝑘^𝑗𝑇Π^𝑗𝑗_𝑘,𝑘⁻¹𝒳^𝑗𝑇_𝑘,𝑘−𝒳^𝑖_𝑘,𝑘Π^𝑖𝑖_𝑘,𝑘⁻¹𝐸𝜇^𝑖_𝑘(𝑥_𝑘−𝑥^𝑗_{𝑘/𝑘−1})^𝑇. (48) Taking into account that 𝐸[𝑥_𝑘𝜇^𝑗_𝑘]=𝒳^𝑗_𝑘,𝑘 and 𝐸[𝑥^𝑖_{𝑘/𝑘−1}𝜇^𝑗_𝑘]=𝐿_𝑘^𝑖𝑗, the recursive expression for the cross-covariance matrices of the local filtering errors is immediately deduced.
Following an analogous reasoning, using now (25) and taking into account that 𝐸[(𝑥_𝑘−𝑥^𝑖_𝑘/𝑘)𝑤^𝑇_𝑘]=𝒥^𝑖_𝑘 and 𝐸[𝜇^𝑖_𝑘𝑤^𝑇_𝑘]=𝒲^𝑖𝑇_𝑘,𝑘, it is easy to see that𝑃^𝑖𝑗_{𝑘/𝑘−1}=𝐹_𝑘−1𝑃^𝑖𝑗_{𝑘−1/𝑘−1}𝐹^𝑇_𝑘−1+𝑄_{𝑘−1,𝑘−1}+𝐹_𝑘−1𝒥^𝑖_𝑘−1+𝒥^𝑗𝑇_𝑘−1𝐹^𝑇_𝑘−1+𝒲^𝑖_{𝑘−1,𝑘−1}Π^𝑖𝑖_{𝑘−1,𝑘−1}⁻¹Π^𝑖𝑗_{𝑘−1,𝑘−1}×Π^𝑗𝑗_{𝑘−1,𝑘−1}⁻¹𝒲^𝑗𝑇_{𝑘−1,𝑘−1}−𝒲^𝑗_{𝑘−1,𝑘−1}+𝐹_𝑘−1𝐸𝑥^𝑖_{𝑘−1/𝑘−1}𝜇^𝑗_𝑘−1×Π^𝑗𝑗_{𝑘−1,𝑘−1}⁻¹𝒲^𝑗𝑇_{𝑘−1,𝑘−1}−𝒲^𝑖_{𝑘−1,𝑘−1}Π^𝑖𝑖_{𝑘−1,𝑘−1}⁻¹×𝒲^𝑖_{𝑘−1,𝑘−1}+𝐹_𝑘−1𝐸[𝑥^𝑗_𝑘/𝑘𝜇^𝑖_𝑘−1]^𝑇. (49) Finally, using again (24) for 𝑥^𝑖_{𝑘−1/𝑘−1}, and since 𝐸[𝑥^𝑖_{𝑘−1/𝑘−2}𝜇^𝑗_𝑘−1]=𝐿^𝑖𝑗_𝑘−1, we have 𝐸𝑥^𝑖_{𝑘−1/𝑘−1}𝜇^𝑗_𝑘−1=𝒳^𝑗_{𝑘−1,𝑘−1}−𝐿^𝑖𝑗_𝑘−1−𝒳^𝑖_{𝑘−1,𝑘−1}Π^𝑖𝑖_{𝑘−1,𝑘−1}⁻¹Π^𝑖𝑗_{𝑘−1,𝑘−1} (50) and the expression for the cross-covariance matrices of the local prediction errors is easily obtained.

4.3. Distributed Fusion Filtering Estimators

Once the local LS linear filtering estimators 𝑥^𝑖_𝑘/𝑘 and their error covariance matrices 𝑃^𝑖𝑖_𝑘/𝑘, given in Theorem 7, along with the error cross-covariance matrices, 𝑃^𝑖𝑗_𝑘/𝑘, given in Theorem 12, are available, the distributed optimal weighted fusion estimators and their error covariance matrices are obtained by applying the optimal information fusion criterion weighted by matrices in the linear minimum variance sense [30].

Theorem 13. For the system model (1) and measurement model (2), under assumptions (i)–(iv), the distributed optimal fusion filter, 𝑥^𝐷_𝑘/𝑘, is given by 𝑥^𝐷_𝑘/𝑘=𝐴¹_𝑘𝑥¹_𝑘/𝑘+⋯+𝐴^𝑟_𝑘𝑥^𝑟_𝑘/𝑘,𝑘≥0, (51) where the local estimators 𝑥^𝑖_𝑘/𝑘,𝑘≥0(𝑖=1,2…,𝑟) are calculated by the recursive algorithm established in Theorem 7.
The optimal matrix weights 𝐴^𝑖_𝑘(𝑖=1,2,…,𝑟) are computed by 𝐴_𝑘=Σ⁻¹_𝑘/𝑘𝑒𝑒^𝑇Σ⁻¹_𝑘/𝑘𝑒⁻¹, (52) where the matrices 𝐴_𝑘=[𝐴¹_𝑘,…,𝐴^𝑟_𝑘]^𝑇 and 𝑒=[𝐼,…,𝐼]^𝑇 are both 𝑛𝑟×𝑛 matrices, and Σ_𝑘/𝑘=𝐸𝑥¹_𝑘/𝑘,…,𝑥^𝑟_𝑘/𝑘𝑥¹_𝑘/𝑘,…,𝑥^𝑟_𝑘/𝑘^𝑇=𝑃^𝑖𝑗_{𝑘/𝑘𝑖,𝑗=1,2,…,𝑟} (53) is an 𝑛𝑟×𝑛𝑟 positive definite symmetric block matrix, whose 𝑛×𝑛 matrix entries 𝑃^𝑖𝑗_𝑘/𝑘 are given in Theorems 7 and 12.
The error covariance matrices of the distributed weighted fusion filtering estimators are computed by 𝑃^𝐷_𝑘/𝑘=𝑒^𝑇Σ⁻¹_𝑘/𝑘𝑒⁻¹,𝑘≥0, (54) and the following inequality holds: 𝑃^𝐷_𝑘/𝑘≤𝑃^𝑖𝑖_𝑘/𝑘,𝑖=1,2,…,𝑟.

Proof. The proof is omitted because it follows directly from the optimal information criterion weighted by matrices in the linear minimum variance sense [30].

Remark 14. The proposed distributed optimal LS linear fusion filter requires the computation of an 𝑛𝑟×𝑛𝑟 inverse matrix, with 𝑛 the dimension of the system state and 𝑟 the number of sensors. Consequently, the proposed distributed fusion method has a computational complexity of 𝑂[(𝑛𝑟)³], equal to that of the distributed Kalman-type filter in [16] and less than that of the distributed fusion filters based on the state augmentation approach. Hence, our distributed fusion method is superior to the filter proposed in [16] (since it has the same computation burden but better accuracy) and also to the distributed fusion filters based on state augmentation (since it has less computational complexity).

5. Numerical Simulation Example

In this section, a numerical simulation example is presented to illustrate the effectiveness of the centralized and distributed filtering algorithms proposed in this paper. Consider a scalar first-order autoregressive model with missing measurements coming from two sensors with autocorrelated and cross-correlated noises. According to the proposed observation model, two different independent sequences of random variables with a certain probability distribution over the interval [0,1] are used to model the missing phenomenon. Specifically, the following model is considered as follows: 𝑥_𝑘=0.95𝑥_𝑘−1+𝑤_𝑘−1𝑦,𝑘≥1_𝑖𝑘=𝜃^𝑖_𝑘𝑥_𝑘+𝑣^𝑖_𝑘,𝑘≥1,𝑖=1,2, (55) where the initial state 𝑥₀ is a zero-mean Gaussian variable with variance 𝑃₀=1. The noise processes {𝑤_𝑘;𝑘≥0} and {𝑣^𝑖_𝑘;𝑘≥1}, 𝑖=1,2, are defined by 𝑤_𝑘=0.6𝜂_𝑘+1+𝜂_𝑘+2,𝑣^𝑖_𝑘=𝑐_𝑖𝜂_𝑘+𝜂_𝑘+1,𝑖=1,2, (56) where the sequence of variables {𝜂_𝑘;𝑘≥1} is a zero-mean Gaussian white process with variance 0.5. Clearly, according to assumption (ii), the additive noises {𝑤_𝑘} and {𝑣^𝑖_𝑘} are one-step autocorrelated and two-step cross-correlated with 𝑄_𝑘,𝑘=0.36,𝑄_𝑘,𝑘+1𝑆=0.18,_{𝑖𝑘,𝑘}=0.3𝑐_𝑖,𝑆^𝑖_{𝑘−1,𝑘}=0.6𝑐_𝑖,𝑆^𝑖_{𝑘−2,𝑘}=0.3𝑐_𝑖,𝑅^𝑖𝑖_𝑘,𝑘=𝑐²_𝑖,𝑅^𝑖𝑖_𝑘,𝑘+1=0.5𝑐²_𝑖,𝑅^𝑖𝑗_𝑘,𝑘=𝑐_𝑖𝑐_𝑗,𝑅^𝑖𝑗_𝑘,𝑘+1=0.5𝑐_𝑖𝑐_𝑗. (57) The phenomenon of missing measurements for each sensor is described as follows. (1)In the first sensor, a sequence of independent and identically distributed (i.i.d.) random variables, {𝜃¹_𝑘;𝑘≥1}, is considered, with probability distribution given by 𝑃𝜃¹_𝑘=0=0.1,𝑃𝜃¹_𝑘𝑃𝜃=0.5=0.5,_1𝑘=1=0.4. (58) If 𝜃¹_𝑘=0, which occurs with probability 0.1, the state 𝑥_𝑘 is missing and the observation 𝑦¹_𝑘 contains only noise 𝑣¹_𝑘; if 𝜃¹_𝑘=0.5, only partial information of the state 𝑥_𝑘 is missing in such observation, which happens with probability 0.5; and, finally, the state is present in the observation 𝑦¹_𝑘 with probability 0.4 when 𝜃¹_𝑘=1. The mean and variance of these variables are easily calculated, being 𝜃¹_𝑘=0.65 and 𝑉^𝜃1_𝑘=0.1025, for all 𝑘.(2)In the second sensor, a sequence of i.i.d. Bernoulli random variables, {𝜃²_𝑘;𝑘≥1}, is considered, with 𝑃[𝜃²_𝑘=1]=𝑝; in this case, if 𝜃²_𝑘=1 the state 𝑥_𝑘 is present in the measurement 𝑦²_𝑘 with probability 𝑝, whereas if 𝜃²_𝑘=0 such observation only contains additive noise, 𝑣²_𝑘, with probability 1−𝑝. So, no partial missing information is considered in this sensor. Clearly, for all 𝑘, 𝜃²_𝑘=𝑝 and 𝑉^𝜃2_𝑘=𝑝(1−𝑝).

To illustrate the feasibility and effectiveness of the proposed estimators we ran a program in MATLAB, in which fifty iterations of the proposed algorithms have been performed considering different values of 𝑐_𝑖 and 𝑝. Using simulated values of the state and the corresponding observations, both distributed and centralized filtering estimates of the state are calculated, as well as the corresponding error variances, which provide a measure of the estimation accuracy.

Firstly, for 𝑝=0.8, the local, centralized, and distributed filtering error variances are displayed in Figure 1 considering the values 𝑐₁=1 and 𝑐₂=0.5. According to Theorem 13, this figure corroborates that the optimal fusion distributed filter performs quite better than each local filter, but lightly worse than the centralized filter. Nevertheless, although the distributed fusion filter has a bit lower accuracy than the centralized one, both filters perform similarly and provide good estimations. Moreover, this slight difference is compensated because the distributed fusion structure is in general more robust, reduces the computational cost, and improves the reliability due to its parallel structure. For these reasons, the distributed filter is generally preferred in practice.

Figure 1 

Local, centralized, and distributed fusion filtering error variances.

Next, to analyze the performance of the proposed estimators versus the probability that the state 𝑥_𝑘 is present in the measurements of the second sensor, the centralized and distributed filtering error variances have been calculated for 𝑐₁=1, 𝑐₂=0.5 and different values of the probability 𝑝=0.2,0.6 and 0.8. The results are displayed in Figure 2; analysis of this figure reveals that as 𝑝 increases (or, equivalently, the probability 1−𝑝 that the state is missing in the observations from the second sensor decreases), the filtering error variances become smaller and, hence, better estimations are obtained. Also, this figure shows that, for all the considered probability values, the error variances corresponding to the centralized filter are always less than those of the distributed filter. Analogous results are obtained for other values of 𝑐₁, 𝑐₂ and the probability 𝑝.

Figure 2 

Centralized and distributed fusion filtering error variances for 𝑝=0.2,0.6,0.8, when 𝑐1=1, 𝑐2=0.5.

On the other hand, to compare the performance of the estimators for different degrees of correlation between the state and the observation noises, the centralized and distributed filtering error variances have been calculated considering 𝑐₂=0.5, 𝑝=0.8 and different values of 𝑐₁, specifically, 𝑐₁=0.25,0.5,0.75, and 1. These values provide different correlations between the noise process {𝑤_𝑘} and the first sensor observation noise {𝑣¹_𝑘} and, consequently, different correlations, 𝐸¹_𝑘, between the state and the first sensor observation noise. The error variances are displayed in Figure 3, from which it is inferred that the error variances are smaller (and, consequently, the performance of the estimators is better) as the value 𝑐₁ is greater; these results were expected, since the correlation between the state and observations increases with 𝑐₁. Analogous results are obtained for different values of 𝑐₂ and other values of the probability 𝑝.

Figure 3 

Centralized and distributed fusion filtering error variances for 𝑐1=0.25,0.5,0.75,1, when 𝑐2=0.5 and 𝑝=0.8.

Now, completing the results of the two previous figures, the performance of the filters is analyzed when 𝑐₂=0.5, the probability 𝑝 is varied from 0.1 to 0.9, and the values 𝑐₁=0.25,0.5,0.75,1,1.25, and 1.5 are considered. It must be noted that in all the cases examined, the error variances present insignificant variation from a certain iteration on and, consequently, only the values at a specific iteration (viz., 𝑘=50) are shown. The results are presented in Figure 4 which, for the sake of clarity, only displays the distributed filtering error variances. Agreeing with the comments about Figures 2 and 3, this figure shows that, for a fixed value of 𝑐₁, the performance of the estimators improves as 𝑝 becomes greater, and for a fixed value of 𝑝, also more accurate estimations are obtained as 𝑐₁ increases. Hence, from this figure it is gathered that, as 𝑐₁ or 𝑝 decreases (which means that the correlation between the state and the first sensor observation noise decreases or the probability that the state is present in the second sensor measurements decreases, resp.), the filtering error variances become greater and, consequently, worse estimations are obtained.

Figure 4 

Distributed fusion filtering error variances versus 𝑝, with 𝑐1=0.25,0.5,0.75,1,1.25,1.5, when 𝑐2=0.5.

Finally, a comparative analysis is presented between the classical Kalman filter [2], the Kalman-type filter with correlated and cross-correlated noises given in [16], the filter proposed in [23] for systems with different failure rates in multisensor networks, and the centralized and distributed filters proposed in this paper. For the comparison, the same parameter values as in Figure 1 are considered (𝑐₁=1, 𝑐₂=0.5, and 𝑝=0.8).

On the basis of one thousand independent simulations of the mentioned algorithms, a comparison between the different filtering estimates is performed using the mean square error (MSE) criteria. For 𝑠=1,…,1000, let {𝑥_𝑘^(𝑠),𝑘=1,…,50} denote the 𝑠th set of artificially simulated data (which is taken as the 𝑠th set of true values of the state), and 𝑥^(𝑠)_𝑘/𝑘 the filtering estimate at the sampling time 𝑘 in the 𝑠th simulation run. For each algorithm, the filtering MSE at time 𝑘 is calculated by MSE_𝑘=(1/1000)∑¹⁰⁰⁰_𝑠=1(𝑥_𝑘^(𝑠)−𝑥^(𝑠)_𝑘/𝑘)².

The values MSE_𝑘, for 𝑘=1,…,50, are displayed in Figure 5 which shows that, for all 𝑘, the proposed centralized and distributed filters have approximately the same MSE_𝑘 values, which in turn are smaller than the MSE_𝑘 values of the filter in [23] and considerably less than those of the filters [2, 16]. Hence, we can conclude that, according to the MSE criterion, the proposed filtering estimates perform significantly better than other filters in the literature.

Figure 5 

Comparison of MSE for different filters.

6. Conclusions

The LS linear estimation problem from missing measurements has been investigated for multisensor linear discrete-time systems with autocorrelated and cross-correlated noises. The main contributions are summarized as follows.(1)Using both centralized and distributed fusion methods to process the measurement data from the different sensors, recursive optimal LS linear filtering algorithms are derived by an innovation approach.(2)At each sensor, the possibility of missing measurements (i.e., observations containing only partial information about the state or even only noise) is modelled by a sequence of independent random variables taking discrete values over the interval [0,1].(3)The multisensor system model considered in the current paper covers those situations where the sensor and process noises are one-step autocorrelated and two-step cross-correlated. Also, one-step cross-correlations between different sensor noises is considered. This correlation assumption is valid in a wide spectrum of applications, for example, in target tracking systems with process and measurement noises dependent on the system state, or situations where a target is observed by multiple sensors and all of them operate in the same noisy environment. Nevertheless, the current study can be extended to more general systems involving finite-step autocorrelated and cross-correlated noises with no difficulty, except for a greater complexity in the mathematical expressions.(4)The applicability of the proposed centralized and distributed filtering algorithms is illustrated by a numerical simulation example, where a scalar state process generated by a first-order autoregressive model is estimated from missing measurements coming from two sensors with autocorrelated and cross-correlated noises. The results confirm that centralized and distributed fusion estimators have approximately the same error variances, with a slight inferiority of the distributed one which is compensated by a reduced computational burden and reduced communication demands for the sensor networks. Also, compared with some existing estimation methods, the proposed algorithms provide better estimations in the mean square error sense.

Appendix

Proof of Theorem 5

From (10), expression (15) for the state filter 𝑥_𝑘/𝑘 in terms of the one-stage predictor 𝑥_{𝑘/𝑘−1} is immediately clear.

Expression (16) for the state predictor 𝑥_{𝑘/𝑘−1} is obtained as follows:𝑥_{𝑘/𝑘−1}=^𝑘−1_𝑠=1𝐸𝑥_𝑘𝜇^𝑇_𝑠Π⁻¹_𝑠,𝑠𝜇_𝑠=^𝑘−1_𝑠=1𝐸𝐹_𝑘−1𝑥_𝑘−1+𝑤_𝑘−1𝜇^𝑇_𝑠Π⁻¹_𝑠,𝑠𝜇_𝑠=^𝑘−1_𝑠=1𝐹_𝑘−1𝐸𝑥_𝑘−1𝜇^𝑇_𝑠Π⁻¹_𝑠,𝑠𝜇_𝑠+𝐸𝑤_𝑘−1𝜇^𝑇_𝑘−1Π⁻¹_{𝑘−1,𝑘−1}𝜇_𝑘−1=𝐹_𝑘−1𝑥_{𝑘−1/𝑘−1}+𝒲_{𝑘−1,𝑘−1}Π⁻¹_{𝑘−1,𝑘−1}𝜇_𝑘−1,𝑘≥2, (A.1) and clearly, 𝑥_1/0=𝐸[𝑥₁]=𝐹₀𝐸[𝑥₀]=𝐹₀𝑥_0/0.

Now we show expression (17) for the innovation, 𝜇_𝑘=𝑦_𝑘−𝑦_{𝑘/𝑘−1}, for which it is enough to obtain an expression for 𝑦_{𝑘/𝑘−1}. A similar reasoning to that used to prove (16) leads to 𝑦_{𝑘/𝑘−1}=^𝑘−1_𝑠=1𝐸𝑦_𝑘𝜇^𝑇_𝑠Π⁻¹_𝑠,𝑠𝜇_𝑠=^𝑘−1_𝑠=1𝐸Θ_𝑘𝐻_𝑘𝑥_𝑘+𝑣_𝑘𝜇^𝑇_𝑠Π⁻¹_𝑠,𝑠𝜇_𝑠=Θ_𝑘𝐻_𝑘^𝑘−1_𝑠=1𝐸𝑥_𝑘𝜇^𝑇_𝑠Π⁻¹_𝑠,𝑠𝜇_𝑠+𝐸𝑣_𝑘𝜇^𝑇_𝑘−1Π⁻¹_{𝑘−1,𝑘−1}𝜇_𝑘−1=Θ_𝑘𝐻_𝑘𝑥_{𝑘/𝑘−1}+𝒱_{𝑘,𝑘−1}Π⁻¹_{𝑘−1,𝑘−1}𝜇_𝑘−1,𝑘≥2, (A.2) with 𝑦_1/0=𝐸[𝑦₁]=Θ₁𝐻₁𝐸[𝑥₁]=Θ₁𝐻₁𝑥_1/0. Hence, 𝑦_{𝑘/𝑘−1}=Θ_𝑘𝐻_𝑘𝑥_{𝑘/𝑘−1}+𝒱_{𝑘,𝑘−1}Π⁻¹_{𝑘−1,𝑘−1}𝜇_𝑘−1𝑦,𝑘≥2,_1/0=Θ₁𝐻₁𝑥_1/0 (A.3) and expression (17) for the innovation is clear.

Next, expression (18) for the matrix 𝒳_𝑘,𝑘=𝐸[𝑥_𝑘𝑦^𝑇_𝑘]−𝐸[𝑥_𝑘𝑦^𝑇_{𝑘/𝑘−1}] is derived. From (6) and the independence assumption, it is clear that 𝐸𝑥_𝑘𝑦^𝑇_𝑘=𝐷_𝑘𝐻^𝑇_𝑘Θ_𝑘+𝐸_𝑘,𝑘≥1. (A.4) From expression (A.3) for 𝑦_{𝑘/𝑘−1} and the OPL, 𝐸[𝑥_𝑘𝑦^𝑇_{𝑘/𝑘−1}] is calculated as follows: 𝐸𝑥_𝑘𝑦^𝑇_{𝑘/𝑘−1}=𝐸𝑥_𝑘𝑥^𝑇_{𝑘/𝑘−1}𝐻^𝑇_𝑘Θ_𝑘+𝐸𝑥_𝑘𝜇^𝑇_𝑘−1Π⁻¹_{𝑘−1,𝑘−1}𝒱^𝑇_{𝑘,𝑘−1}=𝐸𝑥_{𝑘/𝑘−1}𝑥^𝑇_{𝑘/𝑘−1}𝐻^𝑇_𝑘Θ_𝑘+𝒳_{𝑘,𝑘−1}Π⁻¹_{𝑘−1,𝑘−1}𝒱^𝑇_{𝑘,𝑘−1}=𝐷_𝑘−𝑃_{𝑘/𝑘−1}𝐻^𝑇_𝑘Θ_𝑘+𝒳_{𝑘,𝑘−1}Π⁻¹_{𝑘−1,𝑘−1}𝒱^𝑇_{𝑘,𝑘−1},𝐸𝑥𝑘≥2,₁𝑦^𝑇_1/0=𝐷₁−𝑃_1/0𝐻^𝑇₁Θ₁. (A.5) By substraction of the above expectations, expression (18) for 𝒳_𝑘,𝑘=𝐸[𝑥_𝑘𝑦^𝑇_𝑘]−𝐸[𝑥_𝑘𝑦^𝑇_{𝑘/𝑘−1}] is obtained. From (1), expression (19) for 𝒳_{𝑘,𝑘−1}=𝐸[𝑥_𝑘𝜇^𝑇_𝑘−1] is immediately clear.

Expression (20) for the prediction error covariance matrix, 𝑃_{𝑘/𝑘−1} is easily obtained by using (1) and (16); and, from (1) and (15), expression (21) for 𝒥_𝑘=𝐸[𝑥_𝑘/𝑘𝑤^𝑇_𝑘]=𝐸[𝑥_𝑘𝑤^𝑇_𝑘]−𝐸[𝑥_𝑘/𝑘𝑤^𝑇_𝑘] is also obvious. Expression (22) for the filtering error covariance matrix, 𝑃_𝑘/𝑘, is immediately derived by using (15).

Finally, we prove expression (23) for the innovation covariance matrix Π_𝑘,𝑘=𝐸[𝑦_𝑘𝑦^𝑇_𝑘]−𝐸[𝑦_{𝑘/𝑘−1}𝑦^𝑇_{𝑘/𝑘−1}]. From (6) and using (9), we have that 𝐸𝑦_𝑘𝑦^𝑇_𝑘=𝐸𝜃_𝑘𝜃^𝑇_𝑘∘𝐻_𝑘𝐷_𝑘𝐻^𝑇_𝑘+𝑅_𝑘,𝑘+Θ_𝑘𝐻_𝑘𝐸_𝑘+𝐸^𝑇_𝑘𝐻^𝑇_𝑘Θ_𝑘. (A.6) Using now (A.3) for 𝑦_{𝑘/𝑘−1} and property (9), and taking into account that, from the OPL, 𝐸[𝑥_{𝑘/𝑘−1}𝜇^𝑇_𝑘−1]=𝐸[𝑥_𝑘𝜇^𝑇_𝑘−1]=𝒳_{𝑘,𝑘−1}, the following identity holds: 𝐸𝑦_{𝑘/𝑘−1}𝑦^𝑇_{𝑘/𝑘−1}=𝜃_𝑘𝜃^𝑇_𝑘∘𝐻_𝑘𝐷_𝑘−𝑃_{𝑘/𝑘−1}𝐻^𝑇_𝑘+𝒱_{𝑘,𝑘−1}Π⁻¹_{𝑘−1,𝑘−1}𝒱^𝑇_{𝑘,𝑘−1}+Θ_𝑘𝐻_𝑘𝒳_{𝑘,𝑘−1}Π⁻¹_{𝑘−1,𝑘−1}𝒱^𝑇_{𝑘,𝑘−1}+𝒱_{𝑘,𝑘−1}Π⁻¹_{𝑘−1,𝑘−1}𝒳^𝑇_{𝑘,𝑘−1}𝐻^𝑇_𝑘Θ_𝑘. (A.7)

From the above expectations and after some manipulations, expression (23) for the innovation covariance matrix Π_𝑘,𝑘 is obtained.

Acknowledgments

This research is supported by Ministerio de Ciencia e Innovación (Programa FPU and Grant no. MTM2011-24718) and Junta de Andalucía (Grant no. P07-FQM-02701).