A Survey on Model-Based Distributed Control and Filtering for Industrial Cyber-Physical Systems <s> B. Non-Kalman-Based Distributed Filtering <s> This paper presents a distributed expectation-maximization (EM) algorithm over sensor networks. In the E-step of this algorithm, each sensor node independently calculates local sufficient statistics by using local observations. A consensus filter is used to diffuse local sufficient statistics to neighbors and estimate global sufficient statistics in each node. By using this consensus filter, each node can gradually diffuse its local information over the entire network and asymptotically the estimate of global sufficient statistics is obtained. In the M-step of this algorithm, each sensor node uses the estimated global sufficient statistics to update model parameters of the Gaussian mixtures, which can maximize the log-likelihood in the same way as in the standard EM algorithm. Because the consensus filter only requires that each node communicate with its neighbors, the distributed EM algorithm is scalable and robust. It is also shown that the distributed EM algorithm is a stochastic approximation to the standard EM algorithm. Thus, it converges to a local maximum of the log-likelihood. Several simulations of sensor networks are given to verify the proposed algorithm. <s> BIB001 </s> A Survey on Model-Based Distributed Control and Filtering for Industrial Cyber-Physical Systems <s> B. Non-Kalman-Based Distributed Filtering <s> This paper is concerned with a new distributed H"~-consensus filtering problem over a finite-horizon for sensor networks with multiple missing measurements. The so-called H"~-consensus performance requirement is defined to quantify bounded consensus regarding the filtering errors (agreements) over a finite-horizon. A set of random variables are utilized to model the probabilistic information missing phenomena occurring in the channels from the system to the sensors. A sufficient condition is first established in terms of a set of difference linear matrix inequalities (DLMIs) under which the expected H"~-consensus performance constraint is guaranteed. Given the measurements and estimates of the system state and its neighbors, the filter parameters are then explicitly parameterized by means of the solutions to a certain set of DLMIs that can be computed recursively. Subsequently, two kinds of robust distributed H"~-consensus filters are designed for the system with norm-bounded uncertainties and polytopic uncertainties. Finally, two numerical simulation examples are used to demonstrate the effectiveness of the proposed distributed filters design scheme. <s> BIB002 </s> A Survey on Model-Based Distributed Control and Filtering for Industrial Cyber-Physical Systems <s> B. Non-Kalman-Based Distributed Filtering <s> In this paper, the distributed H∞ filtering problem is addressed for a class of polynomial nonlinear stochastic systems in sensor networks. For a Lyapunov function candidate whose entries are polynomials, we calculate its first- and second-order derivatives in order to facilitate the use of Itô's differential rule. Then, a sufficient condition for the existence of a feasible solution to the addressed distributed H∞ filtering problem is derived in terms of parameter-dependent linear matrix inequalities (PDLMIs). For computational convenience, these PDLMIs are further converted into a set of sums of squares that can be solved effectively by using the semidefinite programming technique. Finally, a numerical simulation example is provided to demonstrate the effectiveness and applicability of the proposed design approach. <s> BIB003 </s> A Survey on Model-Based Distributed Control and Filtering for Industrial Cyber-Physical Systems <s> B. Non-Kalman-Based Distributed Filtering <s> In this paper, the problem of distributed H∞ filtering in sensor networks using a stochastic sampled-data approach is investigated. A set of general nonlinear equations described by sector-bounded nonlinearities is utilized to model the system and sensors in networks. Each sensor receives the information from both the system and its neighbors. The signal received by each sensor is sampled by a sampler separately with stochastic sampling periods before it is employed by the corresponding filter. By converting the sampling periods into bounded time-delays, the design problem of the stochastic sampled-data based distributed H∞ filters amounts to solving the H∞ filtering problem for a class of stochastic nonlinear systems with multiple bounded time-delays. Then, by constructing a new Lyapunov functional and employing both the Gronwall's inequality and the Jenson integral inequality, a sufficient condition is derived to guarantee the H∞ performance as well as the exponential mean-square stability of the resulting filtering error dynamics. Subsequently, the desired sampled-data based distributed H∞ filters are designed in terms of the solution to certain matrix inequalities that can be solved effectively by using available software. Finally, a numerical simulation example is exploited to demonstrate the effectiveness of the proposed sampled-data distributed H∞ filtering scheme. <s> BIB004 </s> A Survey on Model-Based Distributed Control and Filtering for Industrial Cyber-Physical Systems <s> B. Non-Kalman-Based Distributed Filtering <s> This paper deals with the distributed H"~ state estimation problem for a class of discrete time-varying nonlinear systems with both stochastic parameters and stochastic nonlinearities. The system measurements are collected through sensor networks with sensors distributed according to a given topology. The purpose of the addressed problem is to design a set of time-varying estimators such that the average estimation performance of the networked sensors is guaranteed over a given finite-horizon. Through available output measurements from not only the individual sensor but also its neighboring sensors, a necessary and sufficient condition is established to achieve the H"~ performance constraint, and then the estimator design scheme is proposed via a certain H"2-type criterion. The desired estimator parameters can be obtained by solving coupled backward recursive Riccati difference equations (RDEs). A numerical simulation example is provided to demonstrate the effectiveness and applicability of the proposed estimator design approach. <s> BIB005 </s> A Survey on Model-Based Distributed Control and Filtering for Industrial Cyber-Physical Systems <s> B. Non-Kalman-Based Distributed Filtering <s> Motivated by non-linear, non-Gaussian, distributed multi-sensor/agent navigation and tracking applications, we propose a multi-rate consensus/fusion based framework for distributed implementation of the particle filter (CF/DPF). The CF/DPF framework is based on running localized particle filters to estimate the overall state vector at each observation node. Separate fusion filters are designed to consistently assimilate the local filtering distributions into the global posterior by compensating for the common past information between neighboring nodes. The CF/DPF offers two distinct advantages over its counterparts. First, the CF/DPF framework is suitable for scenarios where network connectivity is intermittent and consensus can not be reached between two consecutive observations. Second, the CF/DPF is not limited to the Gaussian approximation for the global posterior density. A third contribution of the paper is the derivation of the exact expression for computing the posterior Cramer–Rao lower bound (PCRLB) for the distributed architecture based on a recursive procedure involving the local Fisher information matrices (FIMs) of the distributed estimators. The performance of the CF/DPF algorithm closely follows the centralized particle filter approaching the PCRLB at the signal to noise ratios that we tested. <s> BIB006 </s> A Survey on Model-Based Distributed Control and Filtering for Industrial Cyber-Physical Systems <s> B. Non-Kalman-Based Distributed Filtering <s> This paper addresses distributed state estimation over a sensor network wherein each node-equipped with processing, communication and sensing capabilities-repeatedly fuses local information with information from the neighbors. Estimation is cast in a Bayesian framework and an information-theoretic approach to data fusion is adopted by formulating a consensus problem on the Kullback-Leibler average of the local probability density functions (PDFs) to be fused. Exploiting such a consensus on local posterior PDFs, a novel distributed state estimator is derived. It is shown that, for a linear system, the proposed estimator guarantees stability, i.e. mean-square boundedness of the state estimation error in all network nodes, under the minimal requirements of network connectivity and system observability, and for any number of consensus steps. Finally, simulation experiments demonstrate the validity of the proposed approach. <s> BIB007 </s> A Survey on Model-Based Distributed Control and Filtering for Industrial Cyber-Physical Systems <s> B. Non-Kalman-Based Distributed Filtering <s> We develop a distributed particle filter for sequential estimation of a global state in a decentralized wireless sensor network. A global state estimate that takes into account the measurements of all sensors is computed in a distributed manner, using only local calculations at the individual sensors and local communication between neighboring sensors. The paper presents two main contributions. First, the likelihood consensus scheme for distributed calculation of the joint likelihood function (used by the local particle filters) is generalized to arbitrary local likelihood functions. This generalization overcomes the restriction to exponential-family likelihood functions that limited the applicability of the original likelihood consensus (Hlinka , “Likelihood consensus and its application to distributed particle filtering,” IEEE Trans. Signal Process., vol. 60, pp. 4334–4349, Aug. 2012). The second contribution is a consensus-based distributed method for adapting the proposal densities used by the local particle filters. This adaptation takes into account the measurements of all sensors, and it can yield a significant performance improvement or, alternatively, a significant reduction of the number of particles required for a given level of accuracy. The performance of the proposed distributed particle filter is demonstrated for a target tracking problem. <s> BIB008 </s> A Survey on Model-Based Distributed Control and Filtering for Industrial Cyber-Physical Systems <s> B. Non-Kalman-Based Distributed Filtering <s> The paper considers a distributed robust estimation problem over a network with Markovian randomly varying topology. The objective is to deal with network variations locally, by switching observer gains at affected nodes only. We propose sufficient conditions which guarantee a suboptimal $H_\infty$ level of relative disagreement of estimates in such observer networks. When the status of the network is known globally, these sufficient conditions enable the network gains to be computed by solving certain LMIs. When the nodes are to rely on a locally available information about the network topology, additional rank constraints are used to condition the gains, given this information. The results are complemented by necessary conditions which relate properties of the interconnection graph Laplacian to the mean-square detectability of the plant through measurement and interconnection channels. <s> BIB009 </s> A Survey on Model-Based Distributed Control and Filtering for Industrial Cyber-Physical Systems <s> B. Non-Kalman-Based Distributed Filtering <s> This paper is concerned with event-triggered generalized dissipativity filtering for a neural network (NN) with a time-varying delay. The signal transmission from the NN to its filter is completed through a communication channel. It is assumed that the network measurement of the NN is sampled periodically. An event-triggered communication scheme is introduced to design a suitable filter such that precious communication resources can be saved significantly while certain filtering performance can be ensured. On the one hand, the event-triggered communication scheme is devised to select only those sampled signals violating a certain threshold to be transmitted, which directly leads to saving of precious communication resources. On the other hand, the filtering error system is modeled as a time-delay system closely dependent on the parameters of the event-triggered scheme. Based on this model, a suitable filter is designed such that certain filtering performance can be ensured, provided that a set of linear matrix inequalities are satisfied. Furthermore, since a generalized dissipativity performance index is introduced, several kinds of event-triggered filtering issues, such as $H_\infty $ filtering, passive filtering, mixed $H_\infty $ and passive filtering, $(Q,S,R)$ -dissipative filtering, and $\mathcal L_{2}$ – $\mathcal L_{\infty }$ filtering, are solved in a unified framework. Finally, two examples are given to illustrate the effectiveness of the proposed method. <s> BIB010 </s> A Survey on Model-Based Distributed Control and Filtering for Industrial Cyber-Physical Systems <s> B. Non-Kalman-Based Distributed Filtering <s> We introduce in this paper a novel, fully distributed diffusion Bernoulli filter based on random information dissemination over a partially connected sensor network. The proposed algorithm allows the network nodes to cooperatively perform joint multiframe detection and tracking of an emitter that randomly appears in and disappears from a surveillance region. In addition, we also introduce an alternative flooding-based algorithm that reproduces exactly, in a fully distributed fashion, the optimal centralized Bernoulli filter based on all network node measurements. We derive two alternative low bandwidth implementations of the proposed filters, using respectively a marginalized sequential Monte Carlo (SMC) method and a hybrid Gaussian mixture model (GMM)/SMM scheme. The algorithms were evaluated in a simulated scenario where network nodes directly assimilate raw received signal strength measurements, subject only to possible measurement censoring. <s> BIB011 </s> A Survey on Model-Based Distributed Control and Filtering for Industrial Cyber-Physical Systems <s> B. Non-Kalman-Based Distributed Filtering <s> In this paper, the event-based finite-horizon H∞ state estimation problem is investigated for a class of discrete time-varying stochastic dynamical networks with state- and disturbance-dependent noises [also called (x,v) -dependent noises]. An event-triggered scheme is proposed to decrease the frequency of the data transmission between the sensors and the estimator, where the signal is transmitted only when certain conditions are satisfied. The purpose of the problem addressed is to design a time-varying state estimator in order to estimate the network states through available output measurements. By employing the completing-the-square technique and the stochastic analysis approach, sufficient conditions are established to ensure that the error dynamics of the state estimation satisfies a prescribed H∞ performance constraint over a finite horizon. The desired estimator parameters can be designed via solving coupled backward recursive Riccati difference equations. Finally, a numerical example is exploited to demonstrate the effectiveness of the developed state estimation scheme. <s> BIB012 </s> A Survey on Model-Based Distributed Control and Filtering for Industrial Cyber-Physical Systems <s> B. Non-Kalman-Based Distributed Filtering <s> This paper proposes an event-based networked set-membership filtering method to detect islanding fault for distributed grid-connected solar photovoltaic generation systems. The method enables each set-membership filter to offer an ellipsoidal estimation set, which is used to judge whether or not the islanding fault happens. When islanding fault happens, the intersection of the ellipsoids is empty, and when islanding fault is free, the intersection of the ellipsoids is nonempty. In the filtering scheme, a novel event-triggered mechanism is proposed to reduce the transmission frequency for saving the communication resources. The condition of the existence of the set-membership algorithm is derived by a time-varying convex optimization approach. A simulation experiment and a comparative experiment are provided using Sim-Power-Systems implementation based on a 2-kW single-phase grid-connected power generation system to illustrate the effectiveness of the proposed method for the detection of the islanding fault and the reduction of the resource consumption, respectively. <s> BIB013 </s> A Survey on Model-Based Distributed Control and Filtering for Industrial Cyber-Physical Systems <s> B. Non-Kalman-Based Distributed Filtering <s> Diffusion networks where nodes collaboratively estimate the parameters of stochastic models from shared observations and other estimates have become an established research topic. In this paper the problem of sequential estimation where information in the network diffuses with time is formulated abstractly and independently from any particular model. The objective is to reach a generic solution that is applicable to a wide class of popular models and based on the exponential family of distributions. The adopted Bayesian and information-theoretic paradigms provide probabilistically consistent means for incorporation of shared observations in the implemented estimation of the unknowns by the nodes as well as for effective combination of the “knowledge” of the nodes over the network. It is shown and illustrated on four examples that under certain conditions, the resulting algorithms are analytically tractable, either directly or after simple approximations. The examples include linear regression, Kalman filtering, logistic regression, and inference of an inhomogeneous Poisson process. The first two examples have their more or less direct counterparts in the state-of-the-art diffusion literature whereas the latter two are new. <s> BIB014 </s> A Survey on Model-Based Distributed Control and Filtering for Industrial Cyber-Physical Systems <s> B. Non-Kalman-Based Distributed Filtering <s> This brief is concerned with the problem of neural state estimation for static neural networks with time-varying delays. Notice that a Luenberger estimator can produce an estimation error irrespective of the neuron state trajectory. This brief provides a method for designing such an estimator for static neural networks with time-varying delays. First, in-depth analysis on a well-used reciprocally convex approach is made, leading to an improved reciprocally convex inequality. Second, the improved reciprocally convex inequality and some integral inequalities are employed to provide a tight upper bound on the time-derivative of some Lyapunov–Krasovskii functional. As a result, a novel bounded real lemma (BRL) for the resultant error system is derived. Third, the BRL is applied to present a method for designing suitable Luenberger estimators in terms of solutions of linear matrix inequalities with two tuning parameters. Finally, it is shown through a numerical example that the proposed method can derive less conservative results than some existing ones. <s> BIB015 </s> A Survey on Model-Based Distributed Control and Filtering for Industrial Cyber-Physical Systems <s> B. Non-Kalman-Based Distributed Filtering <s> The discrete-time Distributed Bayesian Filtering (DBF) algorithm is presented for the problem of tracking a target dynamic model using a time-varying network of heterogeneous sensing agents. In the DBF algorithm, the sensing agents combine their normalized likelihood functions in a distributed manner using the logarithmic opinion pool and the dynamic average consensus algorithm. We show that each agent’s estimated likelihood function globally exponentially converges to an error ball centered on the joint likelihood function of the centralized multi-sensor Bayesian filtering algorithm. We rigorously characterize the convergence, stability, and robustness properties of the DBF algorithm. Moreover, we provide an explicit bound on the time step size of the DBF algorithm that depends on the time-scale of the target dynamics, the desired convergence error bound, and the modeling and communication error bounds. Furthermore, the DBF algorithm for linear-Gaussian models is cast into a modified form of the Kalman information filter. The performance and robust properties of the DBF algorithm are validated using numerical simulations. <s> BIB016 </s> A Survey on Model-Based Distributed Control and Filtering for Industrial Cyber-Physical Systems <s> B. Non-Kalman-Based Distributed Filtering <s> This paper is concerned with the distributed set-membership estimation for a discrete-time linear time-varying system over a resource-constrained wireless sensor network under the influence of unknown-but-bounded (UBB) process and measurement noise. Sensors collaborate among themselves by exchanging local measurements with only neighboring sensors in their sensing ranges. First, a new dynamic event-triggered transmission scheme (ETS) is developed to schedule the transmission of each sensor’s local measurement. In contrast with the majority of existing static ETSs, the newly proposed dynamic ETS can result in larger average interevent times and thus less totally released data packets. Second, a criterion for designing desired event-triggered set-membership estimators is derived such that the system’s true state always resides in each sensor’s bounding ellipsoidal estimation set regardless of the simultaneous presence of UBB process and measurement noise. Third, a recursive convex optimization algorithm is presented to determine optimal ellipsoids as well as the estimator gain parameters and the event triggering weighting matrix parameter. Furthermore, the proposed dynamic ETS is applied to address the distributed set-membership estimation problem for a discrete-time linear time-varying system with a nonlinearity satisfying a sector constraint. Finally, an illustrative example is given to show the effectiveness and advantage of the developed approach. <s> BIB017 </s> A Survey on Model-Based Distributed Control and Filtering for Industrial Cyber-Physical Systems <s> B. Non-Kalman-Based Distributed Filtering <s> This paper is concerned with distributed event-triggered $\boldsymbol {H_{\infty }}$ consensus filtering for a discrete-time linear system over a sensor network. Different from some existing event-triggered communication schemes (ETCSs), a new distributed ETCS is first developed to reduce the communication frequency of neighboring sensors, where the threshold parameter in an event triggering condition is time-varying with attainable upper and lower bounds. Then a threshold-parameter-dependent approach is proposed to derive criteria for designing the desired $\boldsymbol {H_{\infty }}$ consensus filters and the ETCS such that the resultant filtering error system is asymptotically stable with the prescribed $\boldsymbol {H_{\infty }}$ performance while maintaining satisfactory resource efficiency. Furthermore, a polytope-like transformation with regard to time-varying threshold parameters is performed and a recursive algorithm is presented to determine the threshold-parameter-dependent filter matrix sequences and event triggering weighting matrix sequence. Two illustrative examples are employed to show the effectiveness of the developed approach. <s> BIB018 </s> A Survey on Model-Based Distributed Control and Filtering for Industrial Cyber-Physical Systems <s> B. Non-Kalman-Based Distributed Filtering <s> The traditional consensus-based filters are widely used in distributed sensor networks. However, they suffer from divergence when outliers occur. This paper proposes a robust consensus nonlinear information filter for distributed state estimation with measurement outliers. Unlike the Gaussian assumption in traditional consensus filers, the measurement of each sensor node is modeled here as a multivariate Student- ${t}$ process with unknown parameters of the sufficient statistic. The variational Bayesian method is employed to jointly estimate the state and the parameters. As the state and parameters are coupled, the updated equation can be solved by fixed point iteration. The centralized outliers robust information filter is first derived for multiple sensors. It is then extended to a distributed version to fuse information from multiple interconnected local estimators. The integral of the consensus-based nonlinear filter is approximated by Gaussian approximation under the framework of the information filter. The consensuses are based on both likelihoods and prior probability distributions. The consensus and convergence of the proposed method are also analyzed. Simulation results show that the proposed approach is effective in dealing with outliers. <s> BIB019 </s> A Survey on Model-Based Distributed Control and Filtering for Industrial Cyber-Physical Systems <s> B. Non-Kalman-Based Distributed Filtering <s> For multisensor linear time-varying system with non-Gaussian measurement noise, how to design distributed robust estimator to increase the accuracy and robustness to outliers at a relatively low computation and communication cost is a fundamental task. This paper proposes a fast distributed variational Bayesian (VB) filtering algorithm to recursively estimate the state and noise distribution over three conventional sensor networks: 1) incremental-based; 2) diffusion-based; and 3) consensus-based. To be specific, the non-Gaussian measurement noise of each sensor is modeled as Student- ${t}$ distribution, and the system state and the parameters of the distribution are estimated via VB approach in each iteration step. An interaction scheme is then added to obtain the global optimal parameter by fusing the local optimal parameters over incremental, diffusion, and consensus communication topology. An efficient sensor selection criterion under these topologies based on the Cramer–Rao lower bound is proposed to reduce the communication and computation burden. Compared with the existing centralized VB filtering algorithms, the proposed algorithm in this paper can extensively increase the robustness to node or link failure at a lower computation cost with acceptable estimation performance and communication load. The theoretic results and simulation results are given to show the efficiency of our proposed algorithm. <s> BIB020
In parallel with the evolution of Kalman-based filtering algorithms, the finite-horizon distributed filtering of various timevarying systems receives an ever-increasing interest. Different from the consensus or diffusion strategies adopted in Kalmanbased filtering algorithms, the improvement of filtering performance mainly relies on the distributed fusion of neighbors's information, which is realized via designed filter structures where the filter gains act as the role of fusion weights. Denote the innovation and the consensus errors as , and L ij,k for different scenarios. Taking the available information into account, some typically distributed filter structures are summarized in Table II . Obviously, Structure I is a typical Luenberger observer, and Structures II and IV can be regarded as its variant via replacing neighboring innovations by consensus errors. Furthermore, Structure III with most design parameters is usually employed to deal with the filtering issue of complex plants with unknown time delays and uncertainties as well as weak nonlinearities. From a technical point of view, there are three representative approaches to design desired gains, namely the backward recursive Riccati difference equation (BRDE), the recursive linear matrix inequality (LMI), and the set-membership approach. With the help of the Moore-Penrose pseudoinverse, the BRDE approach is first developed in BIB005 by blending the finite-horizon H ∞ performance and the nominal H 2 cost, and further adopted to deal with some fashionable topics with various communication protocols (see BIB012 and the references therein). Furthermore, the recursive LMI tool is employed to handle the distributed filtering issue in BIB002 , under which the conception of finite-horizon H ∞ -consensus is creatively defined to quantify bounded consensus regarding to filtering errors. In addition to these two approaches, the set-membership approach serves as another effective tool for handling filtering issues of time-varying systems. Different from popular point-estimation ones, such an approach provides a reliable interval involving true values and therefore satisfies various hard constraints from practical engineering. Here, we refer the readers to BIB017 - BIB013 for some most cutting-edge researches on this topic. It is worth pointing out that, in almost all literature considering randomly occurring network-induced phenomena, the augmentation of filtering errors is usually inevitable, which leads to the obtained design condition dependent on the global topology information. This typical characteristic extremely limits the application in largescale sensor networks. In other words, in order to execute the filtering task online, the network scale is commonly required to be small and the topology information is usually open. Additionally, their calculation complexity is dependent on the number of dimensions of linear matrix inequalities and the number of decision variables. As such, Algorithm III owns the highest calculation complexity. Furthermore, Algorithms II-IV all need to transmit 2(Θ i + 1)n data and thus have the same communication burden. The concrete performance is summarized in Table III . When the research concern is extended into infinite horizon, various distributed filtering schemes based on stability or H ∞ performance analysis were developed in recent years (see BIB003 , BIB009 , and the references therein). For instance, the distributed H ∞ filtering of polynomial nonlinear stochastic systems is investigated in BIB003 by constructing a polynomial Lyapunov function candidate, and the filter gains are designed in terms of the solution of parameter-dependent LMIs. For sensor networks with a randomly varying topology described by Markovian, some sufficient conditions are derived in BIB009 to satisfy the predetermined H ∞ level evaluating relative disagreement of estimated states both in the mean-square sense and with probability 1. Furthermore, some interesting results can be found in BIB018 for the case with event-triggered communication, in BIB004 for the case with sampled-data, and in BIB015 for the case subject to time delays. Compared with the finite-horizon case, the filter parameters can be obtained offline and therefore their scalability can be improved to some extent. Note that, due to the unavoidable impact from various network-induced phenomena, the obtained design condition applying to networked systems also relies on the augmented filtering error dynamics and their disadvantages in performance analysis are similar to that in finite-horizon cases. Additionally, the conservatism from time delays can be reduced possibly via various tools of time-delayed analysis adopted in BIB010 . Generally speaking, the application of these tools may sacrifice the computing performance partly. Finally, let us briefly review the state-of-the-art of distributed filtering in the frameworks of Bayesian filtering and particle filtering (PF). Under the probability perspective, it is interesting to utilize the well-known Bayesian interpretation to disclose: what kind of posterior probability distribution of unknown states is reasonable in the formalization of distributed Kalman filtering; and how it is compared to the optimal posterior distribution of states conditioned on all present and past network data  . Usually, the locally available measurements Y i,k = {y j,k , j ∈ N i ∪ {i}} together with mutual independence assumptions are first used to update the desired posterior PDF p i (x k |Y i,k ) , and then the obtained PDFs are assimilated into α j BIB014 via optimizing average Kullback-Leibler divergence BIB007 or Logarithmic opinion BIB016 where α j = 1. Recently, various alternative schemes are developed to overcome the deficiency of the abovementioned fusion approach in communication and computation. For instance, in the framework of consensus, alternating direction methods of multipliers combined with variational Bayesian algorithms are proposed to carry out the fusion in  and BIB019 for systems with non-Gaussian noises or Gaussian noises with unknown covariance. Additionally, discussing the Cramér-Rao lower bound can realize the efficient sensor selection BIB020 , and adopting the structure of information filter can obtain a compact form of filtering algorithm BIB019 . It should be pointed out that the calculation of PDFs results in a great challenge of the implementation of distributed Bayesian filters. As an alternative scheme, their approximation can be performed via Monte Carlo simulation and importance sampling, which is the main conception of PF. In the distributed fashion, each sensor runs a localized PF, in existing many literatures, and then disseminates and fuses local information by resorting to various distributed schemes, such as consensus and gossip as well as diffusion BIB006 . Recently, the consensus-based algorithms are receiving considerable research attention due to no requirement of both routing protocols and global knowledge. According to the types of quantities, the developed algorithms can be roughly divided into three categories  : algorithms focusing on calculation of particle weights BIB006 , algorithms focusing on calculation of posterior parameters BIB001 , and those focusing on calculation of likelihood parameters BIB008 . Note that distributed PFs usually outperform standard Kalman filtering when the underlying plant and/or measurements models are nonlinear and/or non-Gaussian BIB011 .