Covariance-Based Estimation for Clustered Sensor Networks Subject to Random Deception Attacks

sensors Article Covariance-Based Estimation for Clustered Sensor Networks Subject to Random Deception Attacks Raquel Caballero-Águila 1,*,† , Aurora Hermoso-Carazo 2,† and Josefa Linares-Pérez 2,† 1Dpto. de Estadística, Universidad de Jaén, Paraje Las Lagunillas, 23071 Jaén, Spain 2Dpto. de Estadística, Universidad de Granada, Avda. Fuentenueva, 18071 Granada, Spain *Correspondence: [email protected]; Tel.: +34-953-212-926 † These authors contributed equally to this work. Received: 30 May 2019; Accepted: 12 July 2019; Published: 14 July 2019   Abstract: In this paper, a cluster-based approach is used to address the distributed fusion estimation problem (filtering and fixed-point smoothing) for discrete-time stochastic signals in the presence of random deception attacks. At each sampling time, measured outputs of the signal are provided by a networked system, whose sensors are grouped into clusters. Each cluster is connected to a local processor which gathers the measured outputs of its sensors and, in turn, the local processors of all clusters are connected with a global fusion center. The proposed cluster-based fusion estimation structure involves two stages. First, every single sensor in a cluster transmits its observations to the corresponding local processor, where least-squares local estimators are designed by an innovation approach. During this transmission, deception attacks to the sensor measurements may be randomly launched by an adversary, with known probabilities of success that may be different at each sensor. In the second stage, the local estimators are sent to the fusion center, where they are combined to generate the proposed fusion estimators. The covariance-based design of the distributed fusion filtering and fixed-point smoothing algorithms does not require full knowledge of the signal evolution model, but only the first and second order moments of the processes involved in the observation model. Simulations are provided to illustrate the theoretical results and analyze the effect of the attack success probability on the estimation performance. Keywords: least-squares filtering; least-squares fixed-point smoothing; networked systems; cluster-based approach; stochastic deception attacks 1. Introduction Nowadays, communication networks are widely used to cope with signal estimation problems in engineering, economy, health or security, among others, since they generally provide more robust and precise estimators of the target signal than a single sensor. Conventional centralized and distributed fusion estimation architectures have been widely studied under a state-space approach (see e.g., [ 1 – 4 ] and references therein). In addition, assuming that the evolution model of the signal is not fully known and only covariance information is available, centralized and distributed fusion estimation algorithms have been proposed for sensor networks affected by different network-induced random uncertainties (see e.g., [ 5 – 7 ]). A comprehensive survey of recent developments in estimation and fusion for networked systems with randomly occurring phenomena can be found in [ 8 ]. The key theories and methodologies of distributed multisensor data fusion are comprehensively reviewed in [9] and a survey of distributed fusion estimation algorithms with applications in networked systems, including an interesting analysis of some network-induced uncertainties, is provided in [10]. These conventional centralized and distributed fusion architectures require connecting all sensor nodes to a central processor, which sometimes can involve a serious communication burden, especially Sensors 2019,19, 3112; doi:10.3390/s19143112 www.mdpi.com/journal/sensors Sensors 2019,19, 3112 2 of 16 when the number of sensors is large. Additionally, in the distributed fusion estimation architecture, equipping each single sensor with a local estimator may be sometimes unaffordable. To overcome these shortcomings, a usual practice in engineering consists of arranging the sensors in clusters and selecting one node per cluster (called cluster-head node) acting as a local processor. Each cluster head first collects measurements from the regular sensor nodes of its cluster to generate a local estimator and then all local estimators are collected in the central processor, where the distributed fusion estimator is generated. This structure, called hierarchical cluster-based estimation structure, provides an efficient way to analyze big data in a great variety of application fields [ 11 ]. Clustering is also widely used, for example, in underwater sensor networks established for military purposes, such as anti-submarine warfare, communications, positioning and guidance [ 12 ]. An essential issue when dealing with complex networked systems consisting of multiple clusters or individual subsystems is coordination control and the design of appropriate consensus protocols; relevant results concerning these problems in different kinds of networked systems are found in [ 13 – 16 ]. A state-of-the-art and comprehensive survey on clustering approaches can be found in [ 17 ] and a detailed outline of some modern energy-efficient clustering approaches to improve the lifetime of wireless sensor networks can be seen in [ 18 ]. Recently, a comprehensive review with comparisons and classifications of different optimized clustering approaches according to different metrics has been carried out in [19]. A remarkable disadvantage of using sensor networks for signal estimation is that their reliability can be compromised by possible cyber-attacks from adversaries. For this reason, cyber-security of networked systems is drawing considerable research attention. The most typical kinds of attacks are the denial-of-service attacks and the deception attacks [ 20 ]. While the first ones strike at data availability by obstructing the flow of information through the network, the second ones violate data integrity by injecting false information that modifies the real data packets. Such false information can include a wrong sensor measurement or control input, an incorrect time-stamp or a wrong identity of the sending device. The fusion estimation problem for stochastic signals from measured data coming from a sensor network subject to deception attacks is currently an important focus of research (see e.g., [ 20 – 26 ] and references therein). In [ 20 ], new insecurity conditions for the state estimation problem under false data injection attacks are proposed. In [ 21 ], the variance-constrained distributed filtering problem is studied for time-varying systems subject to multiplicative deception attacks with bounded attack noises. A distributed recursive filtering algorithm is designed in [ 22 ] for a class of discrete time-delayed systems subject to both uniform quantization and intermittent deception attacks. In [ 23 ], the security-guaranteed filtering problem is addressed for a class of nonlinear discrete time-delayed systems with both stochastic sensor saturations and deception attacks. The centralized security-guaranteed filtering problem for linear discrete time-invariant stochastic systems with multi-rate sensor nodes is addressed in [ 24 ], when deception attacks are launched during the transmission of information from the sensors to the centralized filter. In [ 25 ], the cluster-based covariance intersection fusion estimation problem under stochastic deception attacks is investigated. An integrated analysis of event-triggered fault detection and fault estimation is proposed in [ 26 ] for a class of discrete-time stochastic systems subject to unknown disturbances and stochastic deception attacks. Research motivation. In this paper, distributed fusion filtering and fixed-point smoothing algorithms are designed for clustered sensor networks subject to stochastic linear deception attacks. This study is motivated by the following challenges: (i) To find out fusion estimation structures for multi-sensor networked systems that reduce transmission burdens (with respect to both centralized and distributed fusion) and the cost of embedding a local processor in each sensor (with respect to distributed fusion). (ii) To design recursive and easily implementable estimation algorithms with high estimation accuracy, fusing unreliable multi-sensor data, corrupted by possible deception attacks, under general assumptions on the target signal that do not require the full knowledge of the system state-space model. Sensors 2019,19, 3112 3 of 16 Paper contributions. In light of our motivations, the main contributions of this paper are highlighted as follows: (i) The fusion estimation problem in multi-sensor networked systems when the sensors are grouped into clusters is investigated assuming, on the one hand, that the measurements are subject to stochastic deception attacks and, on the other, that the signal evolution model is not necessarily known, but only information about its first and second order statistical properties (covariance information) is available. This covariance-based estimation approach is more general than the conventional one based on the full knowledge of the state-space model of the system since, in that case, the mean and autocovariance function of the signal process can be calculated. Hence, this estimation approach provides a comprehensive framework to deal with a great variety of stochastic signals. (ii) A two-stage fusion estimation algorithm is designed for both filtering and fixed-point smoothing problems. In the first stage, each local processor collects measurements (subject to random deception attacks) from its cluster to generate local least-squares linear estimators; in this stage, the innovation approach is used to design the local estimation algorithms, which are recursive and computationally simple. In the second stage, in order to improve the estimation performance, all these local estimators are transmitted to the fusion center, where fusion estimators are obtained by matrix-weighted linear combinations of the local estimators; the weight matrices are computed by minimizing the mean squared estimation error, for which the cross-correlation matrices between any two local estimators need to be previously calculated. In contrast with previous papers concerning the fusion estimation problem under a covariance-based approach, this is the first time that a cluster structure of the sensor network is used and stochastic deception attacks are considered. In comparison with the existing literature about the estimation problem in clustered sensor networks, either subject to random deception attacks or not, the main difference is the kind of information required for the derivation of the algorithms: full knowledge of the state-space model in the existing literature and only covariance information (which covers the conventional formulation based on state-space model and also more general situations) in the current paper. Paper structure. The measurement model framework is presented in Section 2, where the equations of the clustering sensor measurements and the stochastic deception attacks, together with the hypotheses under which the distributed estimation problem will be addressed, are specified. In Sections 3and 4the first and second stages of the proposed cluster-based fusion estimation structure are developed. The local least-squares linear filtering and fixed-point smoothing estimators, as well as formulas for the error covariance matrices, are obtained by recursive algorithms in Section 3. The proposed distributed fusion estimators are proposed in Section 4by a matrix-weighted linear combination of the local estimators using the mean squared error as optimality criterion; also, to measure the estimation accuracy, recursive formulas for the error covariance matrices are derived. The performance of the proposed estimation algorithms is analyzed in a numerical simulation study, carried out in Section 5. Finally, some concluding remarks are provided in Section 6. Notation. Unless otherwise stated, the notation used in the paper is fairly standard. Rn denotes the n -dimensional Euclidean space and Rm×n the set of all m×n real matrices. When the dimensions of vectors or matrices are not specified, they are assumed to be compatible with algebraic operations. 1n denotes the n -dimensional column vector with all ones, while the identity and zero matrices are denoted by I and 0, respectively. For any function Γh,k , depending on h and k , and Ξ(rs) , depending on r and s , we will write Γk=Γk,k and Ξ(r)=Ξ(rr) , respectively, for simplicity. The Kronecker product of the matrices A∈Rm×n and B∈Rm0×n0 , which is a matrix in Rmm0×nn0 , is denoted by A⊗B . The Hadamard product of the matrices C , D∈Rm×n , which is also a matrix in Rm×n , is denoted by C◦D and it is defined by (C◦D)ij =CijDij . δk,s is the Kronecker delta function, i.e., δk,s= 1 when k=s and zero otherwise. The autocovariance function of a second-order process {αk}k≥1 is defined as E(αk−E[αk])(αs−E[αs])T,k,s≥1, where E[·]stands for the mathematical expectation operator. Sensors 2019,19, 3112 4 of 16 2. Estimation Problem Formulation and Measurement Model In this paper, we consider the distributed fusion estimation problem of a discrete-time stochastic signal from a set of measurements provided by multiple sensors. A cluster-based estimation structure will be adopted: the sensors are grouped into different clusters, each of them connected with a local processor which, in turn, is connected with a global fusion center (FC). The distributed fusion estimation process operates in two stages. In the first one, the measured outputs provided by the sensors of each cluster are sent to the corresponding local processor, where local least-squares (LS) linear estimators are obtained; it is assumed that, during the transmission to the local processors, the clustering sensor measurements are subject to stochastic linear deception attacks. In the second stage, the local estimators received from all the local processors are gathered and fused in the FC, where the proposed distributed signal estimators are generated by a matrix-weighted linear combination of the local estimators using the mean squared error as optimality criterion. The aforementioned estimation problem will be addressed without requiring full knowledge of the evolution model generating the signal process. Instead, it will be assumed that the signal mean function is zero and its covariance function is factorizable (covariance-based estimation approach). More precisely, the following assumption is required ([7]): (A1) The nx -dimensional signal {xk}k≥1 is a zero-mean second-order process and its autocovariance function is expressed in a separable form; namely, ExkxT h=AkBT h , h≤k ,where Ak , Bh∈Rnx×n are known matrices. The signal process of the most common signal evolution models (e.g., the signal of linear systems, or that of uncertain systems with a sum of multiple multiplicative noise terms) meets this assumption (A1) and, hence, the covariance-based estimation approach provides a comprehensive framework to cope with different signal evolution models, thus overcoming the necessity of deriving specific algorithms for each situation. 2.1. Clustering Sensor Measurements and Stochastic Deception Attacks Consider a sensor network and assume that the sensor nodes are grouped into L clusters to measure the stochastic signal of interest. Specifically, assume that each cluster r= 1, . . . , L is made up of mrsensors that provide measurements of the signal according to the following model: z(r(i)) k=C(r(i)) kxk+v(r(i)) k,k≥1; i=1, . . . , mr,r=1, . . . , L, (1) where C(r(i)) k are known matrices and z(r(i)) k∈Rnz is the signal measured output from the i -th sensor of the r -th cluster at time k , which is transmitted to the r -th local processor to obtain the local LS linear signal estimators. The following assumption is required on the measurement noises, nv(r(i)) kok≥1,i=1, . . . , mr,r=1, . . . , L: (A2) The measurement noises of different clusters are independent and, for each r= 1, . . . , L , the noises nv(r(i)) kok≥1 , i= 1, . . . , mr , are zero-mean second-order white processes with known covariance matrices Ehv(r(i)) kv(r(j))T hi=R(r(ij)) kδk,h,k,h≥1; i,j=1, . . . , mr. For each cluster r= 1, . . . , L , the transmissions of the measured outputs z(r(i)) k , i= 1, . . . , mr , to the r -th local processor are affected by random linear deception attacks and the deceptive signal injected by the attackers, ξ(r(i)) k, is described by: ξ(r(i)) k=−z(r(i)) k+w(r(i)) k,k≥1; i=1, . . . , mr,r=1, . . . , L. (2) Sensors 2019,19, 3112 5 of 16 This signal involves two parts: the first one neutralizes the true information and the second one is the blurred information (noise) added by the attackers. These noises, nw(r(i)) kok≥1 , i= 1, . . . , mr , r= 1, . . . , L , are assumed to satisfy the following requirement: (A3) The attack noises of different clusters are independent and, for each r= 1, . . . , L , the noises nw(r(i)) kok≥1 , i=1, . . . , mr, are zero-mean second-order white processes with known covariance matrices Ehw(r(i)) kw(r(j))T hi=S(r(ij)) kδk,h,k,h≥1; i,j=1, . . . , mr. 2.2. Measurements Received by the Local Processors Usually, in practice, the attacks may randomly succeed or not. So, taking into account this random nature of the attacks, for every r= 1, . . . , L , the measurements received by the r -th local processor are modelled by introducing different sequences of Bernoulli random variables, nλ(r(i)) kok≥1 , i= 1, . . . , mr . For each r= 1, . . . , L , and i= 1, . . . , mr , the value λ(r(i)) k= 1 models a successful attack into the i -th communication channel from the r -th cluster, meaning that only noise w(r(i)) k arrives to the r -th local processor; conversely, the value λ(r(i)) k= 0 models a failed attack, which means that the real measured output z(r(i)) k is received by the r -th local processor. Taking these considerations into account, the following model for y(r(i)) k, the measurements received by the r-th local processor, is considered: y(r(i)) k=z(r(i)) k+λ(r(i)) kξ(r(i)) k,k≥1; i=1, . . . , mr,r=1, . . . , L, (3) or, equivalently, by substituting (1) and (2) into (3), we have: y(r(i)) k=1−λ(r(i)) kC(r(i)) kxk+v(r(i)) k+λ(r(i)) kw(r(i)) k,k≥1; i=1, . . . , mr,r=1, . . . , L. (4) The following assumption is imposed on the Bernoulli random variables describing the success or failure of attacks: (A4) nλ(r(i)) kok≥1 , r= 1, . . . , L , i= 1, . . . , mr , are independent sequences of independent Bernoulli random variables with known probabilities P λ(r(i)) k=1=λ(r(i)) k. From this assumption, if we denote λ(r) k=λ(r(1)) k, . . . , λ(r(mr)) kT⊗1nz , for r= 1, . . . , L , then the correlation matrices Kλ(r) k≡Ehλ(r) kλ(r)T ki and K1−λ(r) k≡Eh1mrnz−λ(r) k1mrnz−λ(r) kTi are known and their entries are easily calculated taking into account that Ehλ(r(i)) kλ(r(j)) ki=     λ(r(i)) k,i=j, λ(r(i)) kλ(r(j)) k,i6=j. (5) Finally, the following independence hypothesis is also assumed: (A5) For r= 1, . . . , L and i= 1, . . . , mr , the signal process {xk}k≥1 and the processes nv(r(i)) kok≥1 , nw(r(i)) kok≥1and nλ(r(i)) kok≥1are mutually independent. 3. First Stage: Local LS Linear Estimators To start with, as indicated previously, our aim is to calculate in every local processor, r=1, . . . , L, LS linear estimators of the signal based on the measurements received from all the sensors of the Sensors 2019,19, 3112 6 of 16 r -th cluster. Therefore, to estimate the signal xk in the local processor r at time k+N , we consider all the measurements received from all the sensors i= 1, . . . , mr , of the r -th cluster, up to time k+N ; that is, the measurement set ny(r(i)) h,h≤k+N,i=1, . . . , mro . So defining the vectors y(r) h=y(r(1))T h, . . . , y(r(mr))T hT, made up of all the measurements received by the r -th processor at each sampling time h , the problem at hand is formulated as that of determining local LS linear estimators of the signal xkbased on the vectors ny(r) h,h≤k+No, for r=1, . . . , L. 3.1. Stacked Model for the Measurements Received by the Local Processors Taking into account Equation (4) for the measurements received by the r -th local processor, the following model for the above-defined vectors y(r) kis clearly deduced: y(r) k=I−Λ(r) kC(r) kxk+v(r) k+Λ(r) kw(r) k,k≥1; r=1, . . . , L, (6) where C(r) k=    C(r(1)) k. . . C(r(mr)) k    ,v(r) k=    v(r(1)) k. . . v(r(mr)) k    ,w(r) k=    w(r(1)) k. . . w(r(mr)) k    ,Λ(r) k=    λ(r(1)) k· · · 0 . . ..... . . 0· · · λ(r(mr)) k    ⊗I. The following statistical properties of the processes involved in model (6), which will be used to address the LS linear estimation problem, are easily inferred from the model assumptions (A1)–(A5) stated in Section 2: (P1) nv(r) kok≥1,r=1, . . . , L, are independent zero-mean noise processes with Ehv(r) kv(s) hi=R(r) kδk,hδr,s, where R(r) k=R(r(ij)) ki,j=1,...,mr . (P2) nw(r) kok≥1,r=1, . . . , L, are independent zero-mean noise processes with Ehw(r) kw(s) hi=S(r) kδk,hδr,s, where S(r) k=S(r(ij)) ki,j=1,...,mr . (P3) nΛ(r) kok≥1 , r= 1, . . . , L , are independent sequences of independent random matrices with known means, Λ(r) k=Diag λ(r(1)) k, . . . , λ(r(mr)) k⊗I . Moreover, the Hadamard product properties guarantee that, for any random matrix G independent of Λ(r) k , EΛ(r) kGΛ(r) k=Kλ(r) k◦E[G] , where Kλ(r) kis calculated from (5). (P4) For r= 1, . . . , L , the signal process, {xk}k≥1 , and the processes nv(r) kok≥1 , nw(r) kok≥1 and nΛ(r) kok≥1are mutually independent. (P5) ny(r) kok≥1 , r= 1, . . . , L , are zero-mean processes with covariance matrices Σy(rs) k≡E[y(r) ky(s)T k] given by Σy(rs) k=       K1−λ(r) k◦C(r) kAkBT kC(r)T k+R(r) k+Kλ(r) k◦S(r) k,r=s, I−Λ(r) kC(r) kAkBT kC(s)T kI−Λ(s) k,r6=s, (7) where K1−λ(r) kand Kλ(r) kare calculated from (5). Sensors 2019,19, 3112 7 of 16 3.2. Recursive Local LS Linear Filtering and Fixed-Point Smoothing Algorithms This subsection is devoted to the design of recursive algorithms, at each local processor r=1, . . . , L, for the LS linear filtering and fixed-point smoothing estimators based on the measurements received from all the sensors of the r -th cluster. In other words, for the r -th cluster, r= 1, . . . , L , the aim in this subsection is the design of algorithms to obtain the local LS linear estimators, b x(r) k/k+N , N≥ 0, of the signal xk based on the vectors y(r) h , h≤k+N , given by (6); specifically, a recursive algorithm for the local LS filter b x(r) k/k , k≥ 1, and a recursive algorithm for the local LS smoother b x(r) k/k+N , for fixed k≥1 and N=1,2, . . ., will be derived. For this purpose, we will use the measurement innovations rather than the raw measurements (innovation approach), where the innovation at time kis defined as µ(r) k≡y(r) k−b y(r) k/k−1, being b y(r) k/k−1 the one-stage observation predictor (LS linear estimator of y(r) k based on y(r) h , h≤k− 1). Taking into account the properties of the innovation process, the following expression for the LS linear estimators is derived: b x(r) k/H= H ∑ h=1 Ehxkµ(r)T hiEhµ(r) hµ(r)T hi−1µ(r) h. (8) Taking orthogonal projections in (6) and using properties (P2)–(P4) , we get that the one-stage observation predictor is b y(r) h/h−1=I−Λ(r) hC(r) hb x(r) h/h−1 and, consequently, the innovation is given by µ(r) h=y(r) h−I−Λ(r) hC(r) hb x(r) h/h−1. (9) Furthermore, from property (P5) , it is clear that the innovation covariance matrix, Π(r) h≡Ehµ(r) hµ(r)T hi=Ehy(r) hy(r)T hi−Ehb y(r) h/h−1b y(r)T h/h−1i, satisfies Π(r) h=Σy(r) h−I−Λ(r) hC(r) hEhb x(r) h/h−1b x(r)T h/h−1iC(r)T hI−Λ(r) h. Using now (6) for y(r) h and (8) for b x(r) h/h−1 , we obtain that the coefficients X(r) k,h=Ehxkµ(r)T hi satisfy X(r) k,h=AkO(r) h,h≤k, where O(r) his a matrix function such that O(r) h= BT h− h−1 ∑ l=1 O(r) lΠ(r)−1 lO(r)T lAT h!C(r)T hI−Λ(r) h. (10) Then, by defining o(r) k= (1−δk,0) k ∑ h=1 O(r) hΠ(r)−1 hµ(r) h,k≥1, Σo(r) k= (1−δk,0) k ∑ h=1 O(r) hΠ(r)−1 hO(r)T h,k≥1, (11) we obtain that the estimators b x(r) k/L , L≤k , are given by b x(r) k/L=Ako(r) L , and, by substitution in (9), we obtain that µ(r) h=y(r) h−I−Λ(r) hC(r) hAho(r) h−1. Finally, since (11) guarantees that h−1 ∑ l=1 O(r) lΠ(r)−1 lO(r)T l=Σo(r) h−1 , by substitution in (10), it is deduced that O(r) h=BT h−Σo(r) h−1AT hC(r)T hI−Λ(r) h . Bearing in mind the above results, the following filtering algorithm is derived. Sensors 2019,19, 3112 8 of 16 Recursive Local LS Linear Filtering Algorithm. For the r -th cluster, r= 1, . . . , L , the local filtering estimators, b x(r) k/k , and the error covariance matrices, b Σ(r) k/k≡Ehxk−b x(r) k/kxk−b x(r) k/kTi , are recursively obtained by b x(r) k/k=Ako(r) k,k≥1, b Σ(r) k/k=AkBk−AkΣo(r) kT,k≥1. (12) The vectors o(r) kand the matrices Σo(r) k≡Eho(r) ko(r)T ki, defined in (11), are recursively calculated from o(r) k=o(r) k−1+O(r) kΠ(r)−1 kµ(r) k,k≥1; o(r) 0=0, (13) Σo(r) k=Σo(r) k−1+O(r) kΠ(r)−1 kO(r)T k,k≥1; Σo(r) 0=0. The matrices O(r) k≡Eho(r) kµ(r)T ki, given in (10), satisfy O(r) k=Bk−AkΣo(r) k−1TC(r)T kI−Λ(r) k,k≥1. The innovations, µ(r) k=y(r) k−b y(r) k/k−1, are given by µ(r) k=y(r) k−I−Λ(r) kC(r) kAko(r) k−1,k≥1, (14) and the innovation covariance matrices, Π(r) k≡Ehµ(r) kµ(r)T ki, satisfy Π(r) k=Σy(r) k−I−Λ(r) kC(r) kAkΣo(r) k−1AT kC(r)T kI−Λ(r) k,k≥1, where the matrices Σy(r) kare given in (7). The general expression (8) for the LS linear estimators is also the starting point to derive the following covariance-based recursive fixed-point smoothing algorithm. The derivation is omitted, since this algorithm can easily be deduced by an analogous reasoning to that used in Theorem 2 of [ 6 ]. Recursive Local LS Linear Fixed-point Smoothing Algorithm. For the r -th cluster, r= 1, . . . , L , starting from the filter, b x(r) k/k, the local LS linear fixed-point smoothers, b x(r) k/k+N, N ≥1, are calculated as b x(r) k/k+N=b x(r) k/k+N−1+X(r) k,k+NΠ(r)−1 k+Nµ(r) k+N,N≥1, k≥1, (15) where the matrices X(r) k,k+N≡Exkµ(r)T k+Nare recursively calculated by X(r) k,k+N=Bk−M(r) k,k+N−1AT k+NC(r)T k+NI−Λ(r) k+N,N≥1, k≥1, with initial condition X(r) k,k=AkO(r) k , k≥ 1. The matrices M(r) k,k+N≡E[xko(r)T k+N] , of the expression of X(r) k,k+N , obey the following recursive formula M(r) k,k+N=M(r) k,k+N−1+X(r) k,k+NΠ(r)−1 k+NO(r)T k+N,N≥1, k≥1; M(r) k,k=AkKo(r) k,k≥1. Starting from the error covariance matrix of the filter b Σ(r) k/k , the fixed-point smoothing error covariance matrices, b Σ(r) k/k+N≡Ehxk−b x(r) k/k+Nxk−b x(r) k/k+NTi=EhxkxT ki−Ehb x(r) k/k+Nb x(r)T k/k+Ni, Sensors 2019,19, 3112 9 of 16 are recursively obtained by b Σ(r) k/k+N=b Σ(r) k/k+N−1− X (r) k,k+NΠ(r)−1 k+NX(r)T k,k+N,N≥1, k≥1. 4. Second Stage: Distributed Signal Estimators As we have already mentioned, once the local LS linear estimators, b x(r) k/k+N , r= 1, . . . , L , have been obtained, they are sent to the FC and our goal is to fuse these local estimators to obtain distributed estimators, b xk/k+N , N≥ 0, as matrix-weighted linear combinations that minimize the mean squared estimation error. Defining the stacked vectors by comprising all the local estimators, b Xk/k+N=bx(1)T k/k+N,. ..,bx(L)T k/k+NT , and applying the LS criterion, the proposed distributed fusion estimators are given by: b xk/k+N=Ehxkb XT k/k+NiEhb Xk/k+Nb XT k/k+Ni−1b Xk/k+N,N≥0, k≥1, (16) where Ehb Xk/k+Nb XT k/k+Ni=Ehb x(r) k/k+Nb x(s)T k/k+Nir,s=1,...,L and, from the Orthogonal Projection Lemma (OPL), Ehxkb XT k/k+Ni=Ehb x(1) k/k+Nb x(1)T k/k+Ni, . . . , Ehb x(L) k/k+Nb x(L)T k/k+Ni . Hence, the derivation of the distributed estimators in (16) requires to obtain the cross-covariance matrices between the local ones Σb x(rs) k/k+N≡Ehb x(r) k/k+Nb x(s)T k/k+Ni , r , s= 1, . . . , L , k≥ 1, N≥ 0. Since the initial condition of the recursive Formula (15) for the local smoothing estimators is the local filter, the cross-covariance matrices, Σb x(rs) k/k+N , N≥ 1, between the local smoothers will be recursively obtained by starting from the cross-covariance matrices, Σb x(rs) k/k, between the local filters. 4.1. Cross-Covariance Matrices between Local Filtering Estimators Σb x(rs) k/k=Ehb x(r) k/kb x(s)T k/ki From expression (12) for the filter b x(r) k/k , denoting Σo(rs) k≡Eho(r) ko(s)T ki , it is clear that the cross-covariance matrices between any two local filtering estimators b x(r) k/kand b x(s) k/ksatisfy: Σb x(rs) k/k=AkΣo(rs) kAT k,k≥1; r,s=1, . . . , L. (17) • Using (13) for o(r) k and denoting O(rs) h,k≡Eho(r) hµ(s)T ki , for h=k− 1, k , we get that Σo(rs) k is recursively obtained by: Σo(rs) k=Σo(rs) k−1+O(rs) k−1,kΠ(s)−1 kO(s)T k+O(r) kΠ(r)−1 kO(sr)T k,k≥1, Σo(rs) 0=0; r,s=1, . . . , L. • Using again (13) and denoting Π(rs) k≡Ehµ(r) kµ(s)T ki , the following expression for O(rs) k=Eho(r) kµ(s)T kiis immediately obtained: O(rs) k=O(rs) k−1,k+O(r) kΠ(r)−1 kΠ(rs) k,k≥1; r,s=1, . . . , L. • Next, we derive an expression for O(rs) k−1,k=Eho(r) k−1µ(s)T ki . Using (14) for µ(s) k , with (6) for y(s) k , and taking into account that, from the OPL, Eho(r) k−1xki=Eho(r) k−1b x(r)T k/k−1i, we obtain: O(rs) k−1,k=Σo(r) k−1−Σo(rs) k−1AT kC(s)T kI−Λ(s) k,k≥1; r,s=1, . . . , L Sensors 2019,19, 3112 16 of 16 18. Akila, I.S.; Manisekaran, S.V.; Venkatesan, S.V. Modern Clustering Techniques in Wireless Sensor Networks. In Wireless Sensor Networks—Insights and Innovations; Sallis, P.J.; Ed.; InTech Open: London, UK, 2017; pp. 141–156. 19. 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