Centralized Fusion Approach to the Estimation Problem with Multi-Packet Processing under Uncertainty in Outputs and Transmissions

sensors Article Centralized Fusion Approach to the Estimation Problem with Multi-Packet Processing under Uncertainty in Outputs and Transmissions Raquel Caballero-Águila 1,*,† , Aurora Hermoso-Carazo 2,† and Josefa Linares-Pérez 2,† 1Departamento de Estadística, Universidad de Jaén, Paraje Las Lagunillas, 23071 Jaén, Spain 2Departamento de Estadística, Universidad de Granada, Avda. Fuentenueva, 18071 Granada, Spain; [email protected] (A.H.-C.); [email protected] (J.L.-P.) *Correspondence: [email protected]; Tel.: +34-95-321-2926 † These authors contributed equally to this work. Received: 8 July 2018 ; Accepted: 13 August 2018; Published: date   Abstract: This paper is concerned with the least-squares linear centralized estimation problem in multi-sensor network systems from measured outputs with uncertainties modeled by random parameter matrices. These measurements are transmitted to a central processor over different communication channels, and owing to the unreliability of the network, random one-step delays and packet dropouts are assumed to occur during the transmissions. In order to avoid network congestion, at each sampling time, each sensor’s data packet is transmitted just once, but due to the uncertainty of the transmissions, the processing center may receive either one packet, two packets, or nothing. Different white sequences of Bernoulli random variables are introduced to describe the observations used to update the estimators at each sampling time. To address the centralized estimation problem, augmented observation vectors are defined by accumulating the raw measurements from the different sensors, and when the current measurement of a sensor does not arrive on time, the corresponding component of the augmented measured output predictor is used as compensation in the estimator design. Through an innovation approach, centralized fusion estimators, including predictors, filters, and smoothers are obtained by recursive algorithms without requiring the signal evolution model. A numerical example is presented to show how uncertain systems with state-dependent multiplicative noise can be covered by the proposed model and how the estimation accuracy is influenced by both sensor uncertainties and transmission failures. Keywords: least-squares filtering; least-squares smoothing; networked systems; random parameter matrices; random delays; packet dropouts 1. Introduction 1.1. Background and Motivation With the active development of computer and communication technologies, the estimation problem in multi-sensor network stochastic systems has become an important research topic in the last few years. The significant advantages of multi-sensor systems in practical situations (low cost, remote operation, simple installation, and maintenance) are obvious, and have triggered wide use of these systems in many areas, such as target tracking, communications, the manufacturing industry, etc. Moreover, they usually provide more information than traditional communication systems with a single sensor alone. In spite of these advantages, a sensor network is not generally a reliable communication medium, and together with the communication capacity limitations (network Sensors 2018,18, 2697; doi:10.3390/s18082697 www.mdpi.com/journal/sensors Sensors 2018,18, 2697 2 of 19 bandwidths or service capabilities, among others), may yield different uncertainties during data transmission, such as missing measurements, random delays, and packet dropouts. The development of sensor networks motivates the necessity to desig fusion estimation algorithms which integrate the information from the different sensors and take these network-induced uncertainties into account to achieve a satisfactory performance. Depending on the way the information fusion is performed, there are two fundamental fusion techniques: the centralized fusion approach, in which the measurements from all sensors are sent to a central processor where the fusion is performed, and the distributed fusion approach, in which the measurements from each sensor are processed independently to obtain local estimators before being sent to the fusion center. The survey papers [ 1 – 3 ] can be examined for a wide view of these and other multi-sensor data fusion techniques. As already indicated, centralized fusion architecture is based on a fusion centre that is able to receive, fuse, and process the data coming from every sensor; hence, centralized fusion estimation algorithms provide optimal signal estimators based on the measured outputs from all sensors and, consequently, when all of the sensors work correctly and the connections are perfect, they have the optimal estimation accuracy. In light of these concerns, it is not surprising that the study of the centralized and distributed fusion estimation problems in multi-sensor systems with network-induced uncertainties (in both the sensor measured outputs and the data transmission) has become an active research area in recent years. The estimation problem in systems with uncertainties in the sensor outputs (such as missing measurements, stochastic sensor gain degradation and fading measurements) is addressed in refs. [ 4 – 6 ], among others. In refs. [ 7 – 10 ], systems with failures during transmission (such as uncertain observations, random delays, and packet dropouts) are considered. Also, recent advances in the estimation, filtering, and fusion of networked systems with network-induced phenomena can be reviewed in refs. [ 11 , 12 ], where detailed overviews on this field are presented. Since our aim in this paper is the design of centralized fusion estimators in multi-sensor network systems with measurements perturbed by random parameter matrices subject to random transmission failures (delays and packet dropouts), and multi-packet processing is considered, we discuss the research status of the estimation problem in networked systems with some of these characteristics. 1.2. Multi-Sensor Measured Outputs with Random Parameter Matrices It is well known that in sensor-network environments, the measured outputs can be subject not only to additive noises, but also to multiplicative noise uncertainties due to several reasons, such as the presence of an intermittent sensor or hardware failure, natural or human-made interference, etc. For example, measurement equations that model the above-mentioned situations involving degradation of the sensor gain, or missing or fading measurements must include multiplicative noises described by random variables with values of [ 0,1 ] . So, random measurement parameter matrices provide a unified framework to address different simultaneous network-induced phenomena, and networked systems with random parameter matrices are used in different areas of science (see, e.g., refs. [ 13 , 14 ]). Also, systems with random sensor delays and/or multiple packet dropouts are transformed into equivalent observation models with random measurement matrices (see, e.g., ref. [ 15 ]). Hence, the estimation problem for systems with random parameter matrices has experienced increasing interest due to its diverse applications, and many estimation algorithms for such systems have been proposed over the last few years (see, e.g., refs. [16–24], and references therein). 1.3. Transmission Random Delays and Losses: Observation Predictor Compensation Random delays and packet dropouts in the measurement transmissions are usually unavoidable and clearly deteriorate the performance of the estimators. For this reason, much effort has been made towards the study of the estimation problem to incorporate the effects of these transmission uncertainties, and several modifications of the standard estimation algorithms have been proposed (see, e.g., refs. [ 25 – 27 ], and references therein). In the estimation problem from measurements subject to Sensors 2018,18, 2697 3 of 19 transmission losses, when a packet drops out, the processor does not recieve a valid measurement and the most common compensation procedure is the hold-input mechanism which consists of processing the last measurement that was successfully transmitted. Unlike the approach to deal with losses, in ref. [ 28 ] the estimator of the lost measurement based on the information received previously is proposed as the compensator; this methodology significantly improves the estimations, since in cases of loss, all the previously received measurements are considered, instead of using only the last one. In view of this clear improvement of the estimators, the compensation strategy developed in ref. [ 28 ] has been adopted in some other recent investigations (see, e.g., refs. [29,30], and references therein). 1.4. Multi-Packet Processing Another concern at the forefront of research in networked systems subject to random delays and packet dropouts is the number of packets that are processed to update the estimator at each moment, and different observation models have been proposed to deal with this issue. For example, to avoid losses as much as possible, in ref. [ 16 ] it is assumed that each packet is transmitted several times. In contrast, to avoid the network congestion that may be caused by multiple transmissions, ref. [ 31 ] the packets are sent just once. These papers also assume that each packet is either received on time, delayed for, at most, one sampling time, or lost, and only one packet or no packets are processed to update the estimator at each moment. However, in refs. [ 32 – 34 ] two packets were able to arrive at each sampling time, in which case, both were used to update the estimators, thus improving their performance. In these papers, different packet dropout compensation procedures have been employed. The last available measurement was used as compensation in refs. [ 32 , 34 ], while the observation predictor was considered in ref. [34]. 1.5. Addressed Problem and Paper Contributions Based on the considerations made above, we were motivated to address the study of the centralized fusion estimation problem for multi-sensor networked systems perturbed by random parameter matrices. This problem is discussed under the following assumptions: (a) Each sensor transmits their measured outputs to a central processor over different communication channels and random delays, and packet dropouts are assumed to occur during the transmission; (b) in order to avoid the network congestion, at each time instant, the different sensors send their packets only once, but due to the transmission random failures, the processing center can receive more than one packet; specifically, either one packet, two packets, or nothing; and (c) the measurement output predictor is used as a loss compensation strategy. The main contributions and advantages of the current work are summarized as follows: (1) A unified framework to deal with different network-induced phenomena in the measured outputs, such as missing measurements or sensor gain degradation, is provided by the use of random measurement matrices. (2) Besides the uncertainties in the measured outputs, random one-step delays and packet dropouts are assumed to occur during the transmission at different rates at each sensor. Unlike previous authors’ papers concerning random measurement matrices and random transmission delays and losses where only one packet is processed to update the estimator at each moment, in this paper, the estimation algorithms are obtained under the assumption that either one packet, two packets, or nothing may arrive at each sampling time. (3) Concerning the compensation strategy, the use of the measurement predictor as the loss compensator combined with the simultaneous processing of delayed packets provides better estimators in comparison to other approaches where the last measurement successfully received is used to compensate the data packets and only one packet is processed to update the estimator at each moment. (4) The centralized fusion estimation problem is addressed using covariance information, without requiring full knowledge of the state-space model generating the signal process, thus providing a general approach to deal with different kinds of signal processes. (5) The innovation approach is used to obtain recursive prediction, filtering, and fixed-point smoothing algorithms which are recursive and computationally simple, and thus aresuitable for Sensors 2018,18, 2697 4 of 19 online implementation. In contrast to the approaches where the state augmentation technique is used, the proposed algorithms are deduced without making use of augmented systems; therefore, they have lower computational costs than those based on the augmentation method. 1.6. Paper Structure and Notation The remaining sections of the paper are organized as follows. Section 2presents the assumptions for the signal process, the mathematical models of the multi-sensor measured outputs with random parameter matrices, and the measurements received by the central processor with random delays and packet losses. Section 3provides the main results of the research, namely, the covariance-based centralized least-squares linear prediction and filtering algorithm (Theorem 1) and fixed-point smoothing algorithm (Theorem 2). A numerical example is presented in Section 4to show the performance of the proposed centralized estimators, and some concluding remarks are drawn in Section 5. The proofs of Theorems 1 and 2 are presented in the Appendix Aand Appendix B, respectively. The notations used throughout the paper are standard. Rn and Rm×n denote the n -dimensional Euclidean space and the set of all m×n real matrices, respectively. AT and A−1 denote the transpose and inverse of a matrix ( A ), respectively. In and 0 n denote the n×n identity matrix and zero matrix, respectively. 1n denotes the all-ones vector. Finally, ⊗ and ◦ are the Kronecker and Hadamard products, respectively, and δk,sis the Kronecker delta function. 2. Observation Model and Preliminaries The aim of this section is to design a mathematical model to allow the observations to be processed in the least-squares (LS) linear estimation problem of discrete-time signal processes from multi-sensor noisy measurements transmitted through imperfect communication channels where random one-step delay and packet dropouts may arise in the transmission process. More specifically, in order to avoid the network congestion, at every sampling time, it is assumed that the measured outputs from each sensor, which are perturbed by random parameter matrices, are transmitted just once to a central processor, and due to random delays and losses, the processing center (PC) may receive, from each sensor, either one packet, two packets, or nothing at each time instant. In this context, our goal is to find recursive algorithms for the LS linear prediction, filtering, and fixed-point smoothing problems using the centralized fusion method. We assume that only information about the mean and covariance functions of the signal process is available, and this information is specified in the following hypothesis: (H1) The nx -dimensional signal process, {xk}k≥1 , has a zero-mean, and its autocovariance function is expressed in a separable form, E[xkxT s] = AkBT s , s≤k , where Ak , Bs∈Rnx×M are known matrices. 2.1. Multi-Sensor Measured Outputs with Random Parameter Matrices We assume that there are m sensors which provide measured outputs of the signal process that are affected by random parameter matrices according to the following model: z(i) k=H(i) kxk+v(i) k,k≥1, i=1, . . . , m, (1) where z(i) k∈Rnz is the signal measurement in the i -th sensor at time k , H(i) k are random parameter matrices, and v(i) k are the measurement noises. We assume the following hypotheses for these proceses: (H2) {H(i) k}k≥1 , for i= 1, . . . , m , are independent sequences of independent random parameter matrices. For p= 1, . . . , nz and q= 1, . . . , nx , we denote h(i) pq (k) as the (p , q) -th entry of H(i) k , which has known first and second order moments, and H(i) k=E[H(i) k]. Sensors 2018,18, 2697 5 of 19 (H3) The measurement noises {v(i) k}k≥1 , i= 1, . . . , m , are zero-mean second-order white processes with E[v(i) kv(j)T s] = R(ij) kδk,s. 2.2. Observation Model. Properties To address the estimation problem with the centralized fusion method, the observations coming from the different sensors are gathered and jointly processed at each sampling time to yield the optimal signal estimator. So, the problem is addressed by considering, at each time ( k≥ 1), the vector constituted by the measurements received from all sensors and for this purpose, Equations (1) were combined to yield the following stacked measured output equation: zk=Hkxk+vk,k≥1, (2) where zk=z(1)T k, . . . , z(m)T kT ,Hk=H(1)T k, . . . , H(m)T kT ,vk=v(1)T k, . . . , v(m)T kT. As already indicated, random one-step delays and packet dropouts occur during the transmissions to the PC. To model these failures, we introduced the following sequences of random variables: • {γ(i) k}k≥1 , i= 1, . . . , m , are sequences of Bernoulli random variables. Each i= 1, . . . , m , γ(i) k= 0 means that the output at the current sampling time, z(i) k , arrives on time to be processed for the estimation, while γ(i) k=1 means that this output is either delayed or dropped out; and • {ψ(i) k}k≥2 , i= 1, . . . , m , are sequences of Bernoulli random variables. For each i= 1, . . . , m , ψ(i) k= 1 means that z(i) k−1 is processed at sampling time k (because it was one-step delayed) and ψ(i) k= 0 means that z(i) k−1 is not processed at sampling time k (because it was either received at time k− 1 or dropped out). Since γ(i) k−1= 0 implies ψ(i) k= 0, it is clear that the value of ψ(i) k is conditioned by that of γ(i) k−1. For the previous sequences of Bernoulli variables, we assume the following hypothesis: (H4) nγ(i) k,ψ(i) k+1Tok≥1 , i= 1, . . . , m , are independent sequences of independent random vectors, such that • {γ(i) k}k≥1 , i= 1, . . . , m , are sequences of Bernoulli random variables with known probabilities, Pγ(i) k=1=γ(i) k,k≥1; and • {ψ(i) k}k≥2 , i= 1, . . . , m , are sequences of Bernoulli random variables such that the conditional probabilities (Pψ(i) k=1/γ(i) k−1=1) are known. Thus, ψ(i) k≡Pψ(i) k=1=Pψ(i) k=1/γ(i) k−1=1γ(i) k−1,k≥2. Moreover, the mutual independence hypothesis of the involved processes is also necessary: (H5) For i= 1, . . . , m , the signal process {xk}k≥1 , the random matrices {H(i) k}k≥1 , and the noises {v(i) k}k≥1and nγ(i) k,ψ(i) k+1Tok≥1are mutually independent. Remark 1. From hypothesis (H4), for i,j=1, . . . , m, the following correlations are clear: E[γ(i) kγ(j) k] = (γ(i) k,i=j, γ(i) kγ(j) k,i6=j.E[ψ(i) kψ(j) k] = (ψ(i) k,i=j, ψ(i) kψ(j) s,i6=j. E[ψ(i) k+1(1−γ(j) k)] = (0, i=j, ψ(i) k+1(1−γ(j) k),i6=j. (3) Sensors 2018,18, 2697 6 of 19 In order to write jointly the sensor measurements to be processed at each sampling time, we efined the matrices Γk≡Diagγ(1) k , . . . , γ(m) k⊗Inz . and Ψk≡Diagψ(1) k , . . . , ψ(m) k⊗Inz , k≥ 1. From the definition of variables γ(i) k , i= 1, . . . , m ,, it is clear that the non-zero components of vector (Imnz−Γk)zk are those of zk that arrive on time at the PC and, consequently, those processed at time k . The other components of zk are delayed or lost, and as compensation, the corresponding components of the predictor bzk/k−1 , specified in Γkbzk/k−1 , are processed. Similarly, the non-zero components of Ψkzk−1 are those of zk−1 that are affected by one-step delay, and consequently, they are also processed at time k. Hence, the processed observations at each time are expressed by the following model: yk= (Imnz−Γk)zk+Γkbzk/k−1 Ψkzk−1!,k≥2; y1= (Imnz−Γ1)z1 0!, (4) or equivalently, yk=C0(Imnz−Γk)zk+C0Γkbzk/k−1+C1Ψkzk−1,k≥2; y1=C0(Imnz−Γ1)z1,(5) where C0= ( Imnz, 0mnz)Tand C1= ( 0mnz,Imnz)T. Remark 2. For a better understanding of Model (4) for the measurements processed after the possible transmission one-step delays and losses, a single sensor is considered in the following comments. On the one hand, note that γk= 0means that the output at the current sampling time ( zk ) arrives on time to be processed. Then, if ψk= 1, the measurement processed at time k is yk=zT kzT k−1T , while if ψk= 0, then yk=zT k0T . On the other hand, if γk= 1, the output zk is either delayed or dropped out, and its predictor bzk/k−1 is processed at time k . Then, if ψk= 1, the measurement processed at time k is yk=bzT k/k−1zT k−1T , while if ψk= 0, then yk=bzT k/k−10T . Table 1displays ten iterations of a specific simulation of packet transmission. From Table 1, it can be observed that z1 , z3 , z6 , z7 , and z9 arrive on time to be processed; z2 , z4 and z8 are one-step delayed; and z5 and z10 are lost. So, Model (4) describes possible one-step random transmission delays and packet dropouts in networked systems, where one or two packets can be processed for the estimation. Finally, note that the predictors bzk/k−1 , k= 2,4,5, 8,10 are used to compensate for the measurements that do not arrive on time. Table 1. Measurements processed to update the estimators. Time k1 2 3 4 5 6 7 8 9 10 γk0 1 0 1 1 0 0 1 0 1 ψk0 1 0 1 0 0 0 1 0 ykz1 0 bz1/0 0 z3 z2 bz4/3 0 bz5/4 z4 z6 0 z7 0 bz8/7 0 z9 z8 bz10/9 0 The problem is then formulated as that of obtaining the LS linear estimator of the signal, xk based on the observations {y1, . . . , yL} given in (5). Next, some statistical properties of the processes involved in observation models (2) and (5), which are necessary to address the LS linear estimation problem, are specified: (P1) {Hk}k≥1 is a sequence of independent random matrices with known means: Hk≡E[Hk] = H(1)T k, . . . , H(m)T kT,k≥1. (P2) The sequence {vk}k≥1 is a zero-mean second-order process with E[vkvT s] = Rkδk,s , where Rk=R(ij) ki,j=1,...,m. Sensors 2018,18, 2697 7 of 19 (P3) The random matrices Γk,Ψk+1k≥1are independent, and their means are given by Γk≡E[Γk] = Diagγ(1) k, . . . , γ(m) k⊗Inz,Ψk≡E[Ψk] = Diagψ(1) k, . . . , ψ(m) k⊗Inz. (P4) The signal process, {xk}k≥1 and the processes {Hk}k≥1{vk}k≥1 and Γk,Ψk+1k≥1 are mutually independent. (P5) {zk}k≥1 is a zero-mean process with covariance matrices Σz k,s≡E[zkzT s] , for s≤k which, from (P4), are given by Σz k,s=EHkAkBT sHT s+Rkδk,s,s≤k, with E[HkAkBT sHT s] = HkAkBT sHT s, for s<k, and E[HkAkBT kHT k] = E[H(i) kAkBT kH(j)T k]i,j=1,...,m, where the (p,q)-th entries of the matrices E[H(i) kAkBT kH(j)T k]are given by E[H(i) kAkBT kH(j)T k]pq= nx ∑ a=1 nx ∑ b=1 E[h(i) pa (k)h(j) qb (k)]AkBT kab ,p,q=1, . . . , nz. Remark 3. By denoting γk=γ(1) k , . . . , γ(m) kT⊗1nz and ψk=ψ(1) k , . . . , ψ(m) kT⊗1nz , it is clear that K1−γ k≡E(1mnz−γk)(1mnz−γk)T and Kψ k≡EψkψT k are known matrices whose entries are given in (3). Now, by defining ξk=C0(Imnz−Γk)zk+C1Ψkzk−1,k≥2; ξ1=C0(Imnz−Γ1)z1,(6) and taking the Hadamard product properties into account, it is easy to check that the covariance matrices (Σξ k≡EξkξT k) are given by Σξ k=C0(K1−γ k◦Σz k)CT 0+C1(Kψ k◦Σz k−1)CT 1 +C0(Imnz−Γk)Σz k,k−1ΨkCT 1+C1ΨkΣzT k,k−1(Imnz−Γk)CT 0,k≥2; Σξ 1=C0(K1−γ 1◦Σ1 k)CT 0. (7) 3. Centralized Fusion Estimators This section is concerned with the problem of obtaining recursive algorithms for the LS linear centralized fusion prediction, filtering, and fixed-point smoothing estimators. For this purpose, we used an innovation approach. Also the estimation error covariance matrices, which are used to measure the accuracy of the proposed estimators when the LS optimality criterion is used, were derived. The centralized fusion structure for the considered networked systems with random uncertainties in the measured outputs and transmission is illustrated in Figure 1. Sensors 2018,18, 2697 8 of 19 Sensor 1 Transmission random delays and losses Signal Sensor 2 Sensor m . . . Processing center Random failures Filtering estimator Sensor gain degradation Missing measurements Multiplicative noises ... Figure 1. Centralized fusion filtering estimation with random uncertainties in measured outputs and transmission. 3.1. Innovation Technique As indicated above, our aim was to obtain the optimal LS linear estimators, b xk/L , of the signal xk based on the measurements {y1, . . . , yL} , given in (5) , by recursive algorithms. Since the estimator b xk/Lis the orthogonal projection of the signal xkonto the linear space spanned by the nonorthogonal vectors {y1, . . . , yL} , we used an innovation approach in which the observation process {yk}k≥1 was transformed into an equivalent one (innovation process) of orthogonal vectors {µk}k≥1 ; the equivalence means that each set {µ1, . . . , µL}spans the same linear subspace as {y1, . . . , yL}. The innovation at time k is defined as µk=yk−b yk/k−1 , where b y1/0 =E[y1] = 0 and, for k≥ 2, b yk/k−1 , the one-stage linear predictor of yk is the projection of yk onto the linear space generated by {µ1, . . . , µk−1} . Due to the orthogonality property of the innovations and since the innovation process is uniquely determined by the observations, by replacing the observation process by the innovation one, the following general expression for the LS linear estimators of any vector wk based on the observations {y1, . . . , yL}was obtained b wk/L= L ∑ h=1 EwkµT hEµhµT h−1µh. (8) This expression is derived from the uncorrelation property of the estimation error with all of the innovations, which is guaranteed by the Orthogonal Projection Lemma (OPL). As shown by (8), the first step to obtain the signal estimators is to find an explicit formula for the innovation or, equivalently, for the one-stage linear predictor of the observation. One-Stage Observation Predictor To obtain b yk/k−1 , the projection of yk onto the linear space generated by {µ1, . . . , µk−1} , we used (5) and we note that Ψk and Hk−1 are correlated with the innovation µk−1 . So, to simplify the derivation of b yk/k−1, the observations (5) were rewritten as follows: yk=C0(Imnz−Γk)zk+C1ΨkHk−1xk−1+C0Γkbzk/k−1+Vk−1,k≥2, Vk=C1Ψk+1zk−C1Ψk+1Hkxk,k≥1. (9) Sensors 2018,18, 2697 9 of 19 Taking into account that Ψk+1 and Hk are independent of µh , for h≤k− 1, it is easy to see that EVkµT h= 0 for h≤k− 1. So, from the general expression (8), we obtained b Vk/k=VkΠ−1 kµk , k≥ 1, where Vk≡EVkµT k=EVkyT k. Hence, according to the projection theory, b yk/k−1satisfies b yk/k−1=C0Hkb xk/k−1+C1ΨkHk−1b xk−1/k−1+Vk−1Π−1 k−1µk−1,k≥2. (10) This expression for the one-stage observation predictor along with (8) for the LS linear estimators are the starting points to get the recursive prediction, filtering, and fixed-point smoothing algorithms. 3.2. Centralized Fusion Prediction, Filtering, and Smoothing Algorithms The following theorem presents a recursive algorithm for the LS linear centralized fusion prediction and filtering estimators b xk/L , L≤k , of the signal xk based on the observations {y1, . . . , yL} given in (5) or equivalently, in (9). Theorem 1. Under hypotheses (H1)–(H5), the LS linear centralized predictors and filter b xk/L , L≤k and the corresponding error covariance matrices b Σk/L≡E[(xk−b xk/L)(xk−b xT k/L)] are obtained by b xk/L=AkeL,b Σk/L=Ak(Bk−AkΣe L)T,L≤k, (11) where the vectors eLand the matrices Σe L≡EeLeT Lare recursively obtained from eL=eL−1+ELΠ−1 LµL,L≥1; e0=0, (12) Σe L=Σe L−1+ELΠ−1 LET L,L≥1; Σe 0=0, (13) and the matrices EL≡EeLµT Lsatisfy EL=HT BL−Σe L−1HT AL− EL−1Π−1 L−1VT L−1,L≥2; E1=HT B1, (14) where HDL, for D =A,B,is defined by HDL=C0Imnz−ΓLHLDL+C1ΨLHL−1DL−1,L≥2; HD1=C0(Imnz−Γ1)H1D1.(15) The innovations µL=yL−b yL/L−1are given by µL=yL−HAL+C0ΓLHLALeL−1− VL−1Π−1 L−1µL−1,L≥2; µ1=y1,(16) and the coefficients VL=EVLµT L,are obtained by VL=C1Kψ(1−γ) L+1,L◦(Σz L−HLALΣe L−1AT LHT L) −ΨL+1HLALBL−ALΣe L−1THT L(I−ΓL)CT 0,L≥1, (17) where Kψ(1−γ) L+1,L≡EψL+1(1mnz−γL)T, whose entries are given in (3). The innovation covariance matrices ΠL≡EµLµT Lsatisfy ΠL=Σξ L−C0Kγ L◦(HLALΣe L−1AT LHT L)CT 0+OL,L−1AT LHT LΓLCT 0 −HALOT L,L−1− VL−1Π−1 L−1YT L,L−1,L≥2; Π1=Σξ 1,(18) Sensors 2018,18, 2697 16 of 19 Author Contributions: All authors have contributed equally to this work. R.C.-Á., A.H.-C. and J.L.-P. provided original ideas for the proposed model and they all collaborated in the derivation of the estimation algorithms; they participated equally in the design and analysis of the simulation results; and the paper was also written and reviewed cooperatively. Funding: This research is supported by Ministerio de Economía, Industria y Competitividad, Agencia Estatal de Investigación and Fondo Europeo de Desarrollo Regional FEDER (grant no. MTM2017-84199-P). Conflicts of Interest: The authors declare no conflict of interest. Appendix A Proof of Theorem 1. Based on the general expression (8), to obtain the LS linear estimators b xk/L , L≤k , it is necessary to calculate the coefficients Xk,h≡ExkµT h=ExkyT h−Exkb yT h/h−1,h≤k. Using (5) for yh , the independence hypotheses and the factorization of the signal covariance (H1) lead to E[xkyT h] = AkHT Bh+E[xkb xT h/h−1]HT hΓhCT 0 , 2 ≤h≤k , and E[xkyT 1] = AkHT B1 , with HBh given in (15). Now, using expression (10) for b yh/h−1 , together with (8) for b xh/h−1 and b xh−1/h−1 , the coefficients Xk,h, 1 ≤h≤k, are expressed as follows: Xk,h=AkHT Bh− h−1 ∑ j=1 Xk,jΠ−1 jXT h,jHT h(Imnz−Γh)CT 0+XT h−1,jHT h−1ΨhCT 1− Xk,h−1Π−1 h−1VT h−1, 2 ≤h≤k; Xk,1 =AkHT B1, which guarantees that Xk,h=AkEh, 1 ≤h≤k, with Ehgiven by Eh=HT Bh− h−1 ∑ j=1 EjΠ−1 jEjHT Ah− Eh−1Π−1 h−1VT h−1,h≥2; E1=HT B1. Then, by defining eL≡ L ∑ h=1 EhΠ−1 hµh and Σe L≡EeLeT L= L ∑ h=1 EhΠ−1 hEh , for L≥ 1, and taking into account that E[eLµT h] = Eh, for h≤L, it is easy to obtain expressions (11)–(16). Next, the expression (17) for VL=EVLµT L=EVLyT L is derived. Using (9) for VL , we write VL=C1E[ΨL+1zLyT L]−ΨL+1HLE[xLyT L], and we calculate each of these expectations: • From (5), we write yL=C0(Imnz−ΓL) (zL−bzL/L−1)+C1ΨLzL−1+C0bzL/L−1 , and from the independence properties, it is clear that E[ΨL+1zLyT L] = E[Ψk+1zL(zL−bzL/L−1)T(Imnz−ΓL)]CT 0 +ΨL+1E[zLzT L−1]ΨLCT 1+ΨL+1E[zLbzT L/L−1]CT 0. Now, from the Hadamard product properties, we obtain E[ΨL+1zL(zL−bzL/L−1)T(Imnz−ΓL)] = (Kψ(1−γ) L+1,L◦(Σz L−E[zLbzL/L−1]T) ; from property (P5), E[zLzT L−1] = HLALBT L−1HT L−1 , and using the OPL and (2), E[zLbzT L/L−1] = E[bzL/L−1bzT L/L−1] = HLE[b xL/L−1b xT L/L−1]HT L . Then, using (11) and the definition of Σe L, the following expression is obtained: E[ΨL+1zLyT L]=(Kψ(1−γ) L+1,L◦(Σz L−HLALΣe L−1AT LHT L))CT 0 +ΨL+1HLALBT L−1HT L−1ΨLCT 1+Σe L−1AT LHT LCT 0. Sensors 2018,18, 2697 17 of 19 •Using (2), (5) again and the OPL, together with Hypothesis (H1) and (15) for HBL, we have ΨL+1HLE[xLyT L] = ΨL+1HLALHT BL+Σe L−1AT LHT LΓLCT 0. From the above items and using (15), BT L−1HT L−1ΨLCT 1− HT BL=−BT LHT L(Imnz−ΓL)CT 0 , expression (17) is deduced with no difficulty. To obtain expression (18) for ΠL=E[µLµT L] , we apply the OPL to write ΠL=EyLyT L− Eb yL/L−1b yT L/L−1. On the one hand, using the OPL again, we express Eb yL/L−1b yT L/L−1=Eb yL/L−1yT L which, takes (16) into account for b yL/L−1, and the definitions of OL,L−1and YL,L−1, clearly satisfy Eb yL/L−1yT L=HAL+C0ΓLHLALOT L,L−1+VL−1Π−1 L−1YT L,L−1,L≥2. On the other hand, to obtain EyLyT L , we use (9) and (6) to write yL=ξL+C0ΓLbzL/L−1 , and since bzL/L−1=HLALeL−1 , the following expression is obtained from the definition of Σe L after some manipulations: EyLyT L=Σξ L−C0Kγ L◦(HLALΣe L−1AT LHT L)CT 0+C0ΓLHLALOT L,L−1+OL,L−1AT LHT LΓLCT 0,L≥2. From the above expectations, again, after some manipulations, expression (18) for ΠL is obtained. To complete the proof, expression (19) for OL,L−1=E[yLeT L−1] and YL,L−1=E[yLµT L−1] is derived. Using the OPL, we have E[yLeT L−1] = E[b yL/L−1eT L−1] and E[yLµT L−1] = E[b yL/L−1µT L−1] , and from (16) for b yL/L−1, expression (19) is straightforward. Then, the proof of Theorem 1 is complete. Appendix B Proof of Theorem 2. Using (8), the signal estimators are written as b xk/k+h= k+h ∑ l=1 Xk,lΠ−1 lµl , h≥ 1, from which it is immediately deduced that the smoothers are recursively obtained by (20) from the filter b xk/k. Taking into account that Xk,k+h=ExkyT k+h−Exkb yT k+h/k+h−1 , h≥ 1, the recursive relation (21) is derived by just calculating each of these expectations as follows: •Hypothesis (H1) together with (15), leads to ExkyT k+h=BkHT Ak+h+ExkeT k+h−1AT k+hHT k+hΓk+hCT 0,h≥1. •From (16) for b yk+h/k+h−1, it is clear that Exkb yT k+h/k+h−1=ExkeT k+h−1HAk+h+C0Γk+hHk+hAk+hT+Xk,k+h−1Π−1 k+h−1VT k+h−1,h≥1. From the above items, (21) is proven simply by denoting Ek,k+h=ExkeT k+h , whose recursive expression (22) is also obvious by using (12) for ek+h. Finally, using (20) for the smoothers b xk/k+h , the recursive formula for the fixed-point smoothing error covariance matrices b Σk/k+his immediately deduced. References 1. Castanedo, F. A review of data fusion techniques. Sci. World J. 2013,2013, 704504. [CrossRef] [PubMed] 2. Khaleghi, B.; Khamis, A.; Karray, F.O.; Razavi, S.N. Multisensor data fusion: A review of the state-of-the-art. Inform. Fusion 2013,14, 28–44. 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