Consensus Continuous-Discrete Gaussian Filtering Using Fully Symmetric Interpolatory Quadrature

Consensus-based estimators have been applied in the state estimation for cooperative multi-sensor systems, and most of current studies are for the continuous-time or discrete-time case. With regards to some engineering applications, such as ballistic target tracking, it is more suitable to adopt the continuous-discrete state-space model to formulate a dynamic system, which can capture the evolution characteristics of this state process more accurately. This paper presents a novel consensus continuous-discrete Gaussian filtering (CCDGF) estimator. On the basis of strong Taylor approximation for continuous state, the estimator utilizes the fully symmetric interpolatory quadrature (FSIQ) rule to numerically resolve the first two moments of propagated Gaussian density. Then, the average consensus protocol is leveraged to iterate the local innovations of Gaussian filtering framework at each sensor. The consensus estimates with odd-degree accuracy can be obtained through sufficient exchanges of neighborhood information. Finally, it is demonstrated by simulation examples that the CCDGF estimator can achieve performance close to its centralized counterpart, and has higher tracking accuracy with the increase of quadrature degree.


I. INTRODUCTION
Distributed sensor network is a type of multi-sensor system operating without a fusion center, where each sensor equips with the same processing capability and only communicates with locally connected neighbors. Due to the scalable architecture, enhanced fault tolerance, and fast parallel processing of distributed sensor networks, they are being applied in a wide range of areas [1]- [4]. In particular, dynamic state estimation in sensor network is an essential step for many practical applications related to military surveillance, environmental monitoring, and situational awareness. In addition, this problem may become more challenging in situations with limited resource and real-time operating requirements.
As one of the foundational research problems in information fusion, distributed state estimation for linear or nonlinear systems has received a lot of research efforts [5]- [8]. Therefore, numerous distributed estimation schemes have been developed based on the Kalman filter framework [9], such as sequential state-vector fusion (SSF) [10], [11], The associate editor coordinating the review of this manuscript and approving it for publication was Yan-Jun Liu.
Kalman consensus filter (KCF) [12], [14], gossip distributed Kalman filter (GDKF) [15], [16], and diffusion Kalman filter (DKF) [17], [18]. Among above types of distributed estimation algorithms, the KCF is an effective consensus-based distributed estimator where each sensor can iteratively compute the consensus estimates over the network without the requirement of specific communication topology. It is proven that the KCF can asymptotically converge to the global solution under the conditions of collective observability and network connectivity [14]. In the presence of naive sensor node, the performance of KCF will deteriorate drastically. To mitigate this issue, an information-weighted consensus filter (ICF) is developed by conducting average consensus on information weighting terms [19], [20]. In [21], an information-weighted Kalman consensus filter with improved performance is proposed, where the channel transmission noises between sensor nodes and unknown control input of target are both considered. For the state estimation of nonlinear dynamic systems, a consensus nonlinear filter is presented based on the extended Kalman filter (EKF) linearization paradigm [22], [23], in which consensus on measurement and consensus on information are performed VOLUME 8, 2020 This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ in parallel. In addition, a consensus-based estimator with fast finite-time convergence is proposed to reduce the communication costs [24], yet it requires the acyclic communication topology (e.g., tree graph). The consensus-based estimators reviewed above are designed for the continuous-time or discrete-time stochastic systems. However in practice, the state-space model may be the continuous-discrete type, where the underlying physics of target dynamic is a typically continuous-time process and the measurements are discretely sampled by digital devices [25]- [30]. To mathematically describing the evolution of hidden state, the state equations are generally modeled by a set of Itô-type stochastic differential equations (SDEs). In [31], the KCF and ICF are both extended to address the cooperative space object tracking problem by embedding the cubature rule, where a fourth-order Runge-Kutta scheme is employed to integrate the state dynamic. Using the same scheme for numerical integration, a universal KCF is presented for the asynchronous measurements case [32]. It may be more convenient to adopt a discrete-time model in the dynamic propagation. In [33], a composite weighted average consensus filter is proposed by merging two heterogeneous kinds of nonlinear filters, where an Euler approximation scheme is used for the state dynamic discretization. Strong Taylor approximation is a general technique to derive the discrete-time approximation of continuous state under strong convergence criterion [34]. The Euler approximation can be regarded as the simplest strong Taylor approximation scheme, which uses a 0.5-order Itô-Taylor expansion for the SDE discretization. To further reduce the discretization error, a higher-order strong Taylor approximation is a proper approach with a small amount of computational increase.
Under Gaussian assumption, the Bayesian solution for dynamic state estimation will reduce to the calculations of multi-dimensional integrals weighted by Gaussian density, which is commonly termed as the Gaussian filter. Compared with the first-order linearization of EKF, the point-based methods without derivatives can approximate the integral equations to a higher polynomial degree by selecting quadrature points according to certain rules. The Gauss-Hermite quadrature (GHQ) rule serially handles a tensor product of univariate integrals by using a grid of quadrature points [35]. The McNamee-Stenger rule directly chooses the quadrature points from the integration region [36], which is applicable to high-dimensional integrals. In [37], the fully symmetric interpolatory quadrature (FSIQ) rule is constructed to approximate the Gaussian-weighted integral of arbitrary dimension over infinite region. Based on the Lagrange interpolatory formula, it presents the weights in an explicit form, which is helpful for the theoretical analysis of the numerical stability and computational efficiency of the rule [38].
The goal of this paper is to develop a novel consensusbased filtering estimator for distributed state estimation in the continuous-discrete time domain. Based on the 1.5-order strong Taylor approximation, a discretized state dynamic under the strong convergence criterion is derived.
A continuous-discrete Gaussian filter (CDGF) is constructed to recursively propagate the mean and covariance of Gaussian posterior density, where the multi-dimensional integrals of nonlinear functions are efficiently approximated to odd degree polynomials by utilizing the FSIQ rule. Then, the average consensus protocol is leveraged to iteratively calculate the consensus values of the local innovations of the CDGF at each sensor. The resulting distributed estimator is referred to as the consensus continuous-discrete Gaussian filtering (CCDGF) estimator. Finally, cooperative ballistic target tracking scenario is used to test the performance of the CCDGF in terms of accuracy, convergence, and efficiency, which is a challenging problem with high-dimensional nonlinearity.
The remainder of the paper is structured as follows. Section II presents a formal formulation of the continuousdiscrete state estimation problem based on strong Taylor approximation, and then derives two quadrature rules of odd degree based on FSIQ. Section III develops the CCDGF estimator for distributed estimation. Section IV presents the application of the proposed estimator in cooperative ballistic target tracking. Section V provides the simulation studies to illustrate the performance of the proposed estimator. Conclusion and future work are drawn in the end.

II. GAUSSIAN FILTERING FOR CONTINUOUS-DISCRETE STATE-SPACE SYSTEMS
A class of continuous-discrete nonlinear state-space systems, with random disturbances, can often be described by where x(t) is the d-dimensional state vector at time t; z k is the r-dimensional measurement vector at sampling instant t k = kT , T is the measurement sampling interval; f and h are the known nonlinear functions. W (t) is the standard Brownian motion, and its increment dW (t) is uncorrelated with x(t); σ is the square root matrix of the process noise gain; n k is independent white Gaussian measurement noise with covariance matrix E[n k n T l ] = R k δ kl . Bayesian inference can be used as the optimal solution framework for above continuous-discrete state estimation problem, which consists of the Fokker-Planck-Kolmogorov equation (FPKE) and Bayesian formula. The resulting posterior probability density completely reflects the statistical characteristics of the current state. Giving the prior density p(x k |z k ) conditioned on the past measurements z k = {z 1 , · · · , z k }, the Bayesian filter can be formulated as where p = p(x(t)|z k )(t ≤ t k+1 ) is the predictive density; p(x k+1 |z k+1 ) is the updated density; Q = σ 2 is the gain matrix; and tr(·) denotes the matrix trace operator. However, the propagation of the full probability density performed by (3) and (4) only provides a conceptual solution procedure to the continuous-discrete optimal estimation problem. Except for the linear process or Benes-type nonlinear process, a closed-form expression for the updated posterior cannot be determined in an analytical manner. Thus, we have to find an approximate solution of the FPKE.

A. STRONG TAYLOR APPROXIMATION FOR CONTINUOUS-TIME STATE EQUATION
The state equation (1) can be discretized over the time interval (t, t + δ) by using the 1.5-order Itô-Taylor expansion Equation (5) consists of two components: a noise-free state function f d (x(t), t), and a pair of correlated Gaussian random variables (w, y), which can be written as where the random variables u 1 and u 2 obey the standard Gaussian distribution, thus w ∼ N (0, δI), y ∼ N (0, (δ 3 3)I), and E[wy T ] = (δ 2 2)I, where 0 denotes the zero vector, I denotes the identity matrix. In addition, two differential operators L 0 and L j in (5) can be defined as Note that the term Lf represents a d-dimensional square matrix with the (i, j)-th element being L j f i . Using the discretized form of the state equation and assuming that the posterior density p(x k |z k ) is approximated by the Gaussian distribution, we formulate the first two moments of the predictive density with δ = T aŝ Remark 1: In the derivations for the mean and covariance of p(x k+1 |z k ), we use the assumptions that the Gaussian noises w and y are zero-mean and uncorrelated with the system state, and substitute Lf (x k , t k ) with Lf (x k , t k ). Under the state process being not severely nonlinear, e.g. stiff process [29], (9) and (10) can be used to compute the predicted system state and error covariance accurately.

B. FULLY SYMMETRIC INTERPOLATORY QUADRATURE
The multi-dimensional integral weighted by the Gaussian distribution N (x;x, P) can be formulated as (11) where g is a nonlinear vector function, andg(x) = g(Sx +x), P = SS T .
By using the FSIQ rule, (11) can be approximated as where where |p| = d i=1 p i . In (11), the integral region R d and the weight function w(x) = N (x; 0, I) are fully symmetric. This implies that if u ∈ R d and v is generated by permutations and sign changes of the coordinates of u, then w(u) = w(v). Thus, the fully symmetric sumg{λ p } and weights w (m,d) p are given bỹ where p denotes the set of all permutations of p; (s 1 λ q 1 , s 2 λ q 2 , · · · , s d λ q d ) is the symmetric point generated from the FSIQ rule, where s i = ±1 for λ q i = 0; K is the number of nonzero elements of p. For i > 0, a i is given by with a 0 = 1. Remark 2: Theoretically, the FSIQ rule achieves the algebraic accuracy of odd degree. Because the rule (12) is exact for all monomials d i=1 x α i i of the total degree d i=1 α i ≤ 2m + 1 with α i being the nonnegative integer, that is, the equality in (12) is set up. Meanwhile, there is one monomial of degree 2m + 2 at least that makes (12) inexact.
Remark 3: In Table 1, we summarize the parameter values of the FSIQ rule that are consistent with several typical Gaussian filters [39]- [43]. In the UKF with 3rd-degree accuracy, the tunable parameter is recommended to be λ 1 = √ 3 (i.e. κ = 3 − d) for Gaussian distributions. The total number of function evaluations is V (m,d) = p∈P (m,d) N d p for the FSIQ rule without zero weights, where N d p = 2 K d!/((d − K )!i 1 !i 2 ! · · · i k !) and (i 1 , i 2 , · · · , i k ) are the corresponding multiplicities of k distinct nonzero elements in p. If we use the total function evaluations as the metric of the computational complexity, then the computational complexity of the CDGF increases polynomially with the dimension d for a given m.

III. DISTRIBUTED FILTERING FOR CONTINUOUS-DISCRETE STATE-SPACE SYSTEMS
As the benchmark, the centralized extended information filtering (EIF) estimator for continuous-discrete state-space system is introduced in this section. Then, a consensus-based filtering estimator in a continuous-discrete setting is derived by using the consensus of the local innovations of the continuous-discrete Gaussian filter.

A. CENTRALIZED EXTENDED INFORMATION FILTERING
Information filter can present the global information contribution over the whole sensor network in the sum form of local contributions, so it is more computationally appropriate for multi-sensor fusion. For the continuous-discrete statespace system, the centralized EIF paradigm is formulated as followŝ where k+1,j and I k+1,j are the local information contributions,ŷ k+1|k andŷ k+1 are the information states, Y k+1|k and Y k+1 are the information matrices, N s is the total sensor numbers. Note thatx k+1|k and P k+1|k are predicted by the Euler approximation, where F k is the Jacobian matrix of the state function f . In addition,ẑ k+1|k,j is predicted by the measurement function h j of node j, H k+1,j is the linearized measurement matrix (i.e. Jacobian matrix of h j that evaluated atx k+1|k ).

B. CONSENSUS CONTINUOUS-DISCRETE GAUSSIAN FILTERING
The network topology for distributed sensor fusion can be modeled as a dynamic undirected graph H(t) = (V(t), E(t)), where V(t) = {1, 2, · · · , N s (t)} and E(t) = {(j, i) : j, i ∈ V(t)} denote the node set and edge set of the sensor network respectively. If the edge (j, i) ∈ E(t), then the node i can be called the neighbor of j in graph H(t). The neighbor set of node j is denoted by j (t) = {i ∈ V(t) : (j, i) ∈ E(t)}, and the connectivity is the number of its neighbors, which is denoted by d j (t) = j (t) .
In the following, we use the 3rd-degree FSIQ rule I 3 as an example to derive the CCDGF. The prediction stage for the CCDGF is identical to that of the CDGF, and performed at each sensor node separately. Combining (9), (10), and (20), the predicted state and its covariance can be formulated aŝ where χ i k,j is the function sampling point obtained by the linear transformation χ i k,j = S k,j ξ i +x k,j , S k,j is the matrix square root of P k,j ; the symmetric points ξ i and weights W i are generated by the rule I 3 In Gaussian filtering framework, the linear measurement matrix in the update stage of the KCF can be substituted with the quadrature approximation. According to the statistical linear error propagation, the following explicit formula of the measurement matrix H k+1,j can be obtained where the cross-covariance matrix P xz k+1|k,j is approximated by the rule I 3 aŝ where χ i k+1|k,j = S k+1|k,j ξ i +x k+1|k,j , S k+1|k,j is the matrix square root of P k+1|k,j .
If H is fully connected, each node can update its own state information as the centralized estimator. Substituting (39) into (31) and (32), the updated information state and information matrix can be written aŝ In (42) and (43), the second term is the total innovation from the measurements available in the network.
For a partially connected network, average consensus allows all sensor nodes to compute the average value of initial states {α i : j = 1, 2, · · · , N s } in a distributed manner. The true averages of the innovations in the overall network are defined as follows To acquire these two averages, we can use the average consensus protocol for the local innovations from the node and its immediate neighbors. Let α j (τ ) denote the state value of the iteration τ , then each node can update its state value by where A is a single iteration of average consensus; ε jl is the time-invariant weight of node j to node l, and ε jl = 0 for l / ∈ j . Note that weight matrix = [ε jl ] N s ×N s is jointly defined by the size N s and connectivity {d j } of the network, and its second largest eigenvalue determines the convergence rate for the iteration of A. Moreover, two design schemes of the weight matrix are presented in [44], which are called the Maximum-degree weights and Metropolis weights respectively.
The input states of A are initialized to be m k+1,j and M k+1,j first in the update stage of the CCDGF. Through a sufficient amount of consensus iterations, the output statesm k+1,j and M k+1,j can converge to their true averages m k+1 and M k+1 . Hence, the information state and information matrix of each node can be updated bŷ Moreover,x k+1,j and P k+1,j can be recovered by (35).

IV. APPLICATION TO COOPERATIVE BALLISTIC TARGET TRACKING
In this section, we use the CCDGF for cooperative ballistic target tracking. To this end, it is necessary to present the continuous-discrete state-space model of the ballistic target tracking problem. 1) State model. Generally, a ballistic missile switchs off its thrusters at a given instant, and then enters the ballistic flight mode characterized by a Keplerian orbit. Hence, the state equation describing the ballistic motion can be modeled aṡ where x = [x, y, z,ẋ,ẏ,ż] T is the 6-dimensional state vector; W = [W x , W y , W z , Wẋ, Wẏ, Wż] T is the noise term with the square root of noise gain σ = diag([0, 0, 0, σ 1 , σ 1 , σ 1 ]); a G = − µ r 3 r + a J 2 is the gravitational acceleration term. Here a J 2 is the instantaneous perturbation due to the second-order zonal coefficient J 2 of the Earth's gravity potential, which is defined as where µ is the gravitational coefficient, R e is the equatorial radius, r = [x, y, z] T is the position component of target in the Earth-centered Earth-fixed (ECEF) coordinate system, with r = x 2 + y 2 + z 2 . Since the ECEF system is a noninertial reference frame, (49) contains the Coriolis and centrifugal acceleration terms induced by the the angular velocity ω of the Earth's rotation.
By using the 1.5-order Itô-Taylor expansion for (49), the discretized form of the state equation can be obtained as follows where all components in L 0 f (x) are given by 2) Measurement model. The motion of a ballistic target with respect to a radar station is often be described by the topocentric Earth-North-Up (ENU) coordinate system. Three reference axes of the ENU system are parallel to the east, north and zenith direction, respectively. Let r e = [x e , y e , z e ] T denote the target position in the ENU system. The measurement equation can be modeled as where [ρ, θ, ϕ] T is the range-azimuth-elevation coordinate of the target, n = [n ρ , n θ , n ϕ ] T is the measurement noise with covariance R = diag([σ 2 ρ , σ 2 θ , σ 2 ϕ ]). r e can be obtained from the coordinate transformation r e = T E F ( )(r − r R ), where r R = [x R , y R , z R ] T is the radar position in the ECEF system, and T E F ( ) is the rotation matrix from ECEF to ENU system where = [L, B, H ] T is the geodetic coordinate of the radar. Finally, a single filtering loop for cooperative ballistic target tracking using the CCDGF estimator is summarized in Table 2. It should be noted that a multi-step prediction over the time interval (t k , t k+1 ) is employed to handle the significant nonlinearity of the state model, where the total prediction steps T p > 1 with the interval δ = T T p . Moreover, if the number of consensus iterations T c is not adequate, the updated outputs may not truly reach consensus.

V. SIMULATION STUDIES
The performances of the CDGF and CCDGF are verified by the ballistic target tracking problem in this section. In the simulation scenario, initial position and velocity components of target in the ECEF system are r 0 = [−3325.18km, 3562.44km, 4575.03km] T , v 0 = [3.2698km/s, 4.3100km/s, 2.5473km/s] T , respectively. The initial true state is given by To generate the motion trajectory of the ballistic target, the fourthorder Runge-Kutta method with the step size of 2 −6 s is used as the integration tool. The initial simulation time is set to be 0s, then the target will re-enter the atmosphere in about 1128s. Four ground stations are used to simultaneously observe the moving target, and their geodetic coordinates  We first consider an example of using a single radar located at [110 • , 38 • , 0km] T to track a ballistic target. The initial state estimatex 0 is generated from a normal distribution randomly. The mean of this distribution is the initial true state x 0 , and the covariance matrix is assumed to be P 0 = diag([(50km) 2 , (50km) 2 , (50km) 2 , (0.1km/s) 2 , (0.1km/s) 2 , (0.1km/s) 2 ]). Moreover, the parameters of the noise terms in the state-space model are set as follows: σ 1 = 10 −8 km/s, σ ρ = 0.025km, σ θ = 0.015 • and σ ϕ = 0.015 • .
As the size of measurement interval and the number of prediction steps affect the estimator performance, we first choose a small sampling interval T = 1s and set the total prediction steps as T p = 2 3 , 2 4 in the experiment. The CDGFs using the suggested parameters in Table 1 are compared with the continuous-discrete extended Kalman filter (CDEKF), where the latter uses the Euler approximation for discretization. Fig. 3 depicts the root mean square errors (RMSEs) in position and velocity estimate by 50 Monte Carlo runs. It is seen that the CDGFs can stably converge to a more accurate state estimate compared with the CDEKF. Besides, the increase of T p has little improvement on the performances of the CDGFs, but helps to reduce the divergence rate of the CDEKF. Note that all the CDGFs exhibit close estimation accuracy, which indicates that the diversities of the parameters λ j and the polynomial degree 2m + 1 have no noticeable effect on the performances of the CDGFs for this experiment.
In Fig. 4, we increase the sampling interval, and present the averaged root mean square errors (ARMSEs) of the CDGFs with respect to T p . Here, the ARMSE is defined as the average of steady-state RMSE over the time interval of 450~1128s. Because the same-degree CDGFs are indistinguishable in accuracy, Fig. 4 depicts the errors of the 3rd CDGF and 5th CDGF with only one group of the parameters. The results show that the 5th CDGF has higher accuracy than the 3rd CDGF. Moreover, as T increases, the performance of the CDGF is obviously improved by multi-step prediction, especially for T p = 2. Compared with the CDEKF, the CDGF can capture the evolution characteristics of continuous state and the uncertainty of process more accurately.

B. CASE 2: MULTI-SENSOR COOPERATIVE TRACKING OF BALLISTIC TARGET
A class of ring graph topology is adopted to restrict the information flow across the network, where the j-th node communicates with the (j + l)-th and the (j − l)-th node, and l = 1, 2, · · · , n b . The number of neighbors for each node is assumed to be N b = 2n b = 2, then the network topology in Fig. 1   In addition, the parameters of dynamic system and initial conditions of the filter are identical to case 1.
To evaluate the performance of the CCDGF in cooperative ballistic target tracking, the centralized information filters are used as the benchmark algorithms, including the EIF and the information filtering form of the CDGF (CDGIF). The RMSEs of the centralized information filters for the sampling interval T = 1s are presented in Fig. 5. The results indicate that the CDGIFs are considerably better than the EIF on tracking performance, and the 3rd CDGIF show indistinguishable difference with the 5th CDGIF. Compared with the CDEKF and CDGF in Fig. 3, it can be observed that multi-sensor fusion algorithms perform better than the filtering algorithms using a single sensor in terms of tracking accuracy and stability.
In fact, it is difficult to exactly know how many iterations are needed for the CCDGF to truly achieve consensus. In Fig. 6, the CCDGF with different number of consensus iterations is compared with the CDGIF. In the experiment, the Metropolis weight matrix is adopted and the prediction steps is set as T p = 2. Note that the CCDGF with 3rd-degree accuracy is used to represent all CCDGFs due to close accuracy between the diverse-degree FSIQ rules. It is shown that the CCDGF asymptotically approaches the CDGIF in performance with the increase of T c , and needs 15 iterations at least to reach consensus closely. Additionally, we add one sensor between two adjacent sensors in space to analyze the effect on performance as the   It is obvious that the convergence rate of the CCDGF increases with the increase of N b for fixed T c . Because each node can obtain more information from the network, the CCDGF with a larger number of neighbors will take less iterations to reach the consensus estimate.
To further investigate the effect of the polynomial degree of the FSIQ rule on the estimator performance, we increase the initial estimate error and measurement sampling interval simultaneously. The initial covariance is set as P 0 = diag([(200km) 2 , (200km) 2 , (200km) 2 , (1.5km/s) 2 , (1.5km/s) 2 , (1.5km/s) 2 ]), and the sampling interval is fixed at T = 16s. Fig. 8 depicts the ARMSEs of the CCDGFs of diverse degree for T c = 15, where the ARMSE is defined as the average of RMSE over the node set V. It is shown that the 5th CCDGFs have no noticeable difference from each other, and exhibit better performance than the 3rd CCDGFs. The 3rd CCDGFs are more sensitive to the parameters λ j , because the approximation accuracy of the 3rd-degree FSIQ rules are not enough. However, the 5th-degree FSIQ rules with different parameters ensure 5th-degree accuracy, thus the integral calculations can be approximated more accurately. Moreover, in order to optimize the performance of the 3rd CCDGF, the parameter λ 1 should be set to √ 3, which is consistent with the suggested scaling parameter κ in the 3rd UKF [39].
Finally, Table 3 shows the computational time of above cooperative tracking algorithms relative to that of the EIF, where T p = 2 and T c = 15. It is shown that computational time is proportional to the number of points with  nonzero weight. The computation cost of distributed estimator is about 3 times than that of centralized estimator, because the computer simulations are not executed in the parallel manner. According to the trade between tracking accuracy, computational efficiency, and robustness, the 3rd CCDGF with λ 1 = √ 6 is recommended for practical applications.

VI. CONCLUSIONS
For distributed estimation in the continuous-discrete time domain (e.g. cooperative ballistic target tracking), a consensus continuous-discrete Gaussian filtering estimator is presented. Based on the 1.5-order strong Taylor approximation for continuous state, the CCDGF adopts the FSIQ rule to calculate the mean and covariance of propagated Gaussian density, and employs the average consensus on local innovation to derive the optimal estimates of local sensors.
Simulation results indicate that the CCDGF can ensure the globally asymptotic convergence of local estimates at each sensor, and capture the evolution characteristics of state process more accurately than the traditional centralized EIF. Furthermore, the proposed estimator can achieve improved tracking accuracy with the utilization of higher-degree FSIQ rule. It is worth mentioning that the correlation (quantified by the cross-covariance) between the prior states of different nodes is not taken into account, which may lead to the degradation of the network performance. In our future work, the consensus on the prior information will be incorporated into the fusion framework. Moreover, some practical issues such as transmission delays, target maneuver, and multiple targets in limited field-of-view, that cause difficulty to distributed fusion estimation over networks, will be also investigated.