Particle filtering with applications in networked systems: a survey

Li, Wenshuo; Wang, Zidong; Yuan, Yuan; Guo, Lei

doi:10.1007/s40747-016-0028-2

Particle filtering with applications in networked systems: a survey

Survey and State of the Art
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Published: 27 October 2016

Volume 2, pages 293–315, (2016)
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Particle filtering with applications in networked systems: a survey

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Wenshuo Li¹,
Zidong Wang ORCID: orcid.org/0000-0002-9576-7401^2,3,
Yuan Yuan¹ &
…
Lei Guo¹

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Abstract

The particle filtering algorithm was introduced in the 1990s as a numerical solution to the Bayesian estimation problem for nonlinear and non-Gaussian systems and has been successfully applied in various fields including physics, economics, engineering, etc. As is widely recognized, the particle filter has broad application prospects in networked systems, but network-induced phenomena and limited computing resources have led to new challenges to the design and implementation of particle filtering algorithms. In this survey paper, we aim to review the particle filtering method and its applications in networked systems. We first provide an overview of the particle filtering methods as well as networked systems, and then investigate the recent progress in the design of particle filter for networked systems. Our main focus is on the state estimation problems in this survey, but other aspects of particle filtering approaches are also highlighted.

Particle Filter Optimization: A Brief Introduction

Particle filtering for nonlinear dynamic state systems with unknown noise statistics

Article 03 July 2014

Jaechan Lim

Nonlinear System Identification Using Particle Filters

Introduction

In recent years, many fruitful results have been published regarding Bayesian inference. Analytically tractable solutions to the Bayesian inference, however, exist only for a limited class of models. Enormous research efforts have, therefore, been put to develop approximate solutions for the Bayesian inference, among which the particle filtering methods, using the sampling-based approximation techniques, have gained increasing popularity due to their ability to provide an arbitrarily close approximation to the true probability density function (PDF). Particle filtering approaches have been found to be especially attractive for nonlinear/non-Gaussian filtering problems where the assumptions enabling the Kalman-type filters are violated.

On another research frontier, with the rapid development of wireless communication technology, networked systems have found applications in a wide range of areas such as process monitoring, formation control, tele-operations, etc. Despite their advantages such as structural flexibility and low cost, networked systems have given rise to new challenges to traditional state estimation approaches. The challenges come mainly from two aspects: (i) there may be certain degree of information loss due to the limited network resources and, (ii) the trade-off between accuracy and communication cost must be addressed, especially for large-scale networks. Furthermore, the constraints on communication cost are highlighted when particle filtering methods are applied to networked systems.

The purpose of this paper is to provide a thorough and timely review of existing results on particle filtering methods and their applications to networked systems. The rest of this paper is organized as follows. In Sect. 2, the basic idea of the particle filter is introduced, and some improvements on the existing particle filtering algorithms are discussed. State estimation for networked systems are studied in Sect. 3, where existing filtering algorithms addressing network-induced phenomena are discussed and distributed filtering methods are briefly reviewed. In Sect. 4, the applications of particle filtering methods to the state estimation of networked systems are investigated. Particle filtering under network-induced phenomena and particle filtering for networked systems are reviewed, respectively. Conclusions and future research topics are presented in Sect. 5.

Particle filter

Basic ideas

It is known that the Kalman-type filters are not suitable for state estimation for systems with non-Gaussian noises and/or strong nonlinearities since the Gaussian assumption on the state posterior is no longer valid. Particle filtering (PF), with the capability of approximating PDFs of any form, has received considerable attention among researchers and engineers since it was proposed in the 1990s. For some excellent research reviews on particle filtering method, the readers can refer to [1–3], to name just a few.

Next we will briefly introduce the basic principles of particle filtering. Consider the state space model given by

$$\begin{aligned} \left\{ {\begin{array}{l} {{x_{k+1}} = {f_k}\left( {{x_k},{v_k}} \right) }\\ {{z_k} = {h_k}\left( {{x_k},{n_k}} \right) } \end{array}} \right. \end{aligned}$$

(1)

where $\left\{ {{x_k},k \in N} \right\} $ is the state sequence which is of interest to us; $\left\{ {{z_k},k \in N} \right\} $ is the observed signal sequence; ${f_k}:{R^{{n_x}}} \times {R^{{n_v}}} \rightarrow {R^{{n_x}}}$ and ${h_k}:{R^{{n_x}}} \times {R^{{n_n}}} \rightarrow {R^{{n_z}}}$ are both nonlinear functions; ${v_k}$ and ${n_k}$ are the process noise and measurement noise, respectively, both of which are assumed to be an independent identical distributed (i.i.d.) process.

Our aim here is to estimate, in a recursive manner, the current state ${x_k}$ given the measurements up to time k, denoted by ${z_{1:k}} := \left\{ {{z_i},i = 1,2,...,k} \right\} $. The Bayesian method for state estimation seeks to construct the posterior PDF $p({x_k}|{z_{1:k}})$ in which the complete statistical information of ${x_k}$ is contained. Suppose that $p({x_{k-1}}|{z_{1:{k-1}}})$ is available at time k, then $p({x_k}|{z_{1:k}})$ can be obtained through the following two-stage calculation:

$$\begin{aligned}&p({x_k}|{z_{1:k - 1}}) = \int {p({x_k}|{x_{k - 1}})p({x_{k - 1}}|{z_{1:k - 1}})\text {d}{x_{k - 1}}}\end{aligned}$$

(2)

$$\begin{aligned}&p({x_k}|{z_{1:k}}) = \frac{{p({z_k}|{x_k})p({x_k}|{z_{1:k - 1}})}}{{\int {p({z_k}|{x_k})p({x_k}|{z_{1:k - 1}})\text {d}{x_k}} }} \end{aligned}$$

(3)

Therefore, as long as we know the initial PDF $p({x_0}|{z_0})$, we can, in theory, obtain $p({x_k}|{z_{1:k}})$ using formulas (2) and (3) recursively at each filtering period.

For linear systems with Gaussian additive noise, (2) and (3) can be solved analytically to obtain $p({x_k}|{z_{1:k}})$ and the state estimate ${\hat{x}_k}$ which maximizes $p({x_k}|{z_{1:k}})$ is exactly the solution of Kalman filter. For systems with a more general form, however, the analytical expression of $p({x_k}|{z_{1:k}})$ is usually unavailable due to the computationally intractable integrals in (2) and (3). When this is the case, one can apply particle filtering algorithm to obtain an approximation of $p({x_k}|{z_{1:k}})$. Particle filter is a sequential Monte Carlo method, the basic idea of which is to represent the state posterior $p({x_k}|{z_{1:k}})$ by a set of particles $\left\{ {x_k^i,i = 1,2,\ldots ,{N_s}} \right\} $ with associated weights $\left\{ {w_k^i,i = 1,2,\ldots ,{N_s}} \right\} $:

$$\begin{aligned} p({x_k}|{z_{1:k}}) \approx \sum \limits _{i = 1}^{{N_s}} {w_k^i\delta ({x_k} - x_k^i)} \end{aligned}$$

(4)

where ${N_s}$ is the number of particles, $\delta (\cdot )$ is the Dirac delta function, and the weights are normalized such that $\sum \nolimits _i {w_k^i} = 1$. In this way, the integrals in (2) and (3) are transformed into summations which are easier to calculate.

Now it is natural for one to ask that how could we sample from $p({x_k}|{z_{1:k}})$ which is unknown to us. Particle filter does this by means of importance sampling. Suppose that there is a probability density $q({x_k}|{z_{1:k}})$ which is known to us and from which we can draw samples easily. Typically we choose $q({x_k}|{z_{1:k}})$ such that the recursion

$$\begin{aligned} q({x_k}|{z_{1:k}}) = q({x_k}|x_{k - 1}^i,{z_k})q({x_{k - 1}}|{z_{1:k - 1}}) \end{aligned}$$

(5)

is satisfied. Then, the particle representation of $p({x_k}|{z_{1:k}})$ is given by

$$\begin{aligned} p({x_k}|{z_{1:k}}) \approx \sum \limits _{i = 1}^{{N_s}} {\tilde{w}_k^i\delta ({x_k} - \tilde{x}_k^i)} \end{aligned}$$

(6)

where $\tilde{x}_k^i \sim q({x_k}|{z_{1:k}}){} {} (i = 1,2,\ldots ,{N_s})$ is the set of particles; $\tilde{w}_k^i \propto \frac{{p({x_{k}^{i}}|{z_{1:k}})}}{{q({x_{k}^{i}}|{z_{1:k}})}}$ is the normalized weight of the ith particle. The density $q({x_k}|{z_{1:k}})$ is called importance density and its distribution is called proposal distribution.

In practical applications, one is normally more interested in sequential implementation of the Monte Carlo approximation in (6), that is, to represent $p({x_k}|{z_{1:k}})$ using $p({x_{k-1}}|{z_{1:{k-1}}})$. Suppose that at time k, we have the following discrete approximation of $p({x_{k-1}}|{z_{1:{k-1}}})$:

$$\begin{aligned} p({x_{k - 1}}|{z_{1:k - 1}}) \approx \sum \limits _{i = 1}^{{N_s}} {\tilde{w}_{k - 1}^i\delta ({x_{k - 1}} - \tilde{x}_{k - 1}^i)} \end{aligned}$$

(7)

Note that

$$\begin{aligned}&p({x_k}|{z_{1:k - 1}}) = \int {p({x_k},{x_{k - 1}}|{z_{1:k - 1}})\text {d}{x_{k - 1}}} \\&\qquad = \int {p({x_k}|{x_{k - 1}})p({x_{k - 1}}|{z_{1:k - 1}})\text {d}{x_{k - 1}}} \\&\qquad \approx \int {p({x_k}|{x_{k - 1}})\sum \limits _{i = 1}^{{N_s}} {\tilde{w}_{k - 1}^i\delta ({x_{k - 1}} - \tilde{x}_{k - 1}^i)} \text {d}{x_{k - 1}}} \\&\qquad = \sum \limits _{i = 1}^{{N_s}} {\left\{ {\int {\tilde{w}_{k - 1}^ip({x_k}|{x_{k - 1}})\delta ({x_{k - 1}} - \tilde{x}_{k - 1}^i)\text {d}{x_{k - 1}}} } \right\} } \\&\qquad = \sum \limits _{i = 1}^{{N_s}} {\tilde{w}_{k - 1}^ip({x_k}|\tilde{x}_{k - 1}^i)} \end{aligned}$$

Therefore, if we draw samples $\tilde{x}_k^i \sim p({x_k}|\tilde{x}_{k - 1}^i)$ for $i = 1,2,\ldots ,{N_s}$, the prediction density $p({x_k}|{z_{1:k - 1}})$ can be approximated by $\left\{ {\tilde{x}_k^i,\tilde{w}_{k - 1}^i} \right\} ,i = 1,2,\ldots ,{N_s}$. Furthermore, since

$$\begin{aligned} p({x_k}|{z_{1:k}})= & {} \frac{{p({z_k},{x_k}|{z_{1:k - 1}})}}{{p({z_k}|{z_{1:k - 1}})}} = \frac{{p({z_k}|{x_k})p({x_k}|{z_{1:k - 1}})}}{{p({z_k}|{z_{1:k - 1}})}}\\&\propto p({z_k}|{x_k})p({x_k}|{z_{1:k - 1}}) \end{aligned}$$

the importance weights of ${\tilde{x}_k^i}$ should be updated as follows:

$$\begin{aligned} \tilde{w}_k^i \propto \tilde{w}_{k - 1}^ip({z_k}|\tilde{x}_k^i) \end{aligned}$$

(8)

for ${i = 1,2,\ldots ,{N_s}}$.

The weight update formula in (8) can be extended directly to the more general case where the importance density is selected as $q({x_k}|{z_{1:k}})$ satisfying (5):

$$\begin{aligned} \tilde{w}_k^i \propto \tilde{w}_{k - 1}^i\frac{{p({z_k}|\tilde{x}_k^i)p(\tilde{x}_k^i|\tilde{x}_{k - 1}^i)}}{{q(\tilde{x}_k^i|\tilde{x}_{k - 1}^i,{z_k})}} \end{aligned}$$

(9)

The standard particle filtering algorithm introduced above is also termed as sequential importance sampling (SIS) algorithm. For clarity, we present its pseudo code in Algorithm 1.

Although [4] is widely recognized to be the work which lays the foundation for modern particle filtering, the history of the Monte Carlo method can trace back to the 1940s [5]. In [5], the Monte Carlo method is introduced by Metropolis as a branch of statistical mechanics where one’s major concern is the collective behaviour of a group of particles. A statistical study based on samples (particles) drawn from all possible events is suggested to avoid dealing with multiple integrals or multiplications of the probability matrices. Soon after that, the Markov Chain Monte Carlo (MCMC) method was proposed by Metropolis inspired by the search of thermodynamic equilibrium through simulation [6]. The major finding is that one does not have to know the exact dynamics of the system in the simulation; instead, he only needs to simulate a Markov chain which has the same equilibrium as the original system. This scheme for simulation is then referred to as Metropolis algorithm. The generalized version of Metropolis Algorithm, also known as Metropolis-Hasting MCMC (MH-MCMC) algorithm, has been proposed in 1970 [7]. A thorough introduction of the MCMC method could be found in [8] and [9]. In the following, we will briefly introduce MH-MCMC algorithm to show how the idea of MCMC algorithm is implemented.

To draw samples from $\pi (x)$, we construct a Markov chain with $\pi (x)$ as its invariant distribution, i.e.,

$$\begin{aligned} p({x_k}) \rightarrow \pi (x)\text {ask} \rightarrow \infty ,\quad \text {for}\, \text {any}\,p({x_0}) \end{aligned}$$

where $p({x_k})$ is the marginal probability of the Markov chain and $p({x_0})$ is its initial value. It is known that a Markov chain is characterized by the initial probability $p({x_0})$ and the transition probability $T({x_{k + 1}}|{x_k})$. In the MH-MCMC algorithm, $p({x_0})$ can be chosen arbitrarily and the state transition is realized by sampling from a proposal distribution $q(x'|{x_k})$ and accepting the new sample ${x'}$ with the following probability:

$$\begin{aligned} A(x',{x_k}) = \min \left( {1,\frac{{p(x')q({x_k}|x')}}{{p({x_k})q(x'|{x_k})}}} \right) . \end{aligned}$$

If the proposal is accepted, the Markov chain state will be updated by $x'$, i.e., ${x_{k + 1}} = x'$; otherwise it remains at ${x_k}$, i.e., ${x_{k + 1}} = {x_k}$.

The pseudo code of MH-MCMC algorithm is given in Algorithm 2.

The introduction of MCMC method here is out of three considerations. First, it shares the similar idea with particle filtering. Specifically, both of them represent the unknown target distribution by a set of randomly drawn samples to avoid the intractable integrals. Second, it has been combined with the standard particle filter, as shown later, to reduce the problem of sample depletion in particle filter. Third, the missing data problems, which are one major concern of this survey due to their universality in networked systems, have been tackled by researchers using certain forms of MCMC method, such as the Gibbs sampler [10] and other data augmentation methods.

Comparing the importance sampling with the MCMC method, we see that both can obtain samples from a PDF that is known to us up to a normalizing constant. Nevertheless, some key differences between them should be noted. In the importance sampling, no iterations are involved, but the samples obtained are associated with different weights, which implies a lower computational efficiency since particles with different weights take up the same amount of computational resources. By contrast, all the samples obtained are equally weighted in the MCMC method. However, the MCMC method requires iterative sampling from the proposal distribution to ensure that the invariant distribution is finally reached, which can be time prohibitive in some real-time applications.

Before moving on to discuss some related problems of particle filtering, we will at first give a brief review on the convergence results. In practice, one cannot apply the particle filtering method with confidence until satisfactory answers are given to the following questions:

Could the particle approximation of $p({x_k}|{z_{1:k}})$ converge asymptotically to the true $p({x_k}|{z_{1:k}})$ and in what sense?
Is there error accumulation over time?

In [11], the law of large numbers for particle filtering has been established. It is proved that the particle representation converges almost surely to the quantity of interest as the number of particles tends towards infinity. The law of large numbers, however, does not provide a measure of the approximation error which is usually of more interest to practitioners. Here it is natural for one to think of the central limit theorem which can offer the probability distribution of approximation error. Unfortunately, the classical central limit theorem, assuming that samples are drawn independently from the same distribution, does not apply to the analysis of particle filtering where there is interaction between particles.

A comprehensive survey of convergence results on particle filters could be found in [12] where almost sure convergence and mean square convergence of particle filtering are studied, respectively. Note that the mean square convergence results stated in [12] relies on some strict assumptions. For example, the convergence of mean square error has only been established for bounded functions, which has excluded $f(x) = x$, meaning that this convergence result does not apply to the classical mean square estimation. Furthermore, it has been shown in [12] that the error accumulation seems to be inevitable, unless certain mixing conditions on the dynamic model (thus the true optimal filter) are satisfied. This also explains why particle filtering is not suitable for (fixed) parameter estimation. To avoid error accumulation, one has to increase the number of particles with time, which may lead to a formidable computation load.

The central limit theorem for particle filters has been established in [13] which, due to its minimal assumptions on the distribution of interest, applies to various forms of particle filtering algorithms. The asymptotic variance allows us to compare the relative efficiency of different algorithms and assess the stability of a given particle filter. More recently, the convergence result for a rather general class of unbounded functions has been obtained (see [14]). Shortly the result was extended to ${L_p}$-convergence in [15]. Notably, both the results in [14, 15] require that the unnormalized importance weights are point-wise bounded. This constraint has been relaxed in [16] where only boundedness of the second (for mean square convergence) or fourth (for ${L_4}$-convergence) order moment of importance weights is required.

Networked systems

Introduction of networked systems and network-induced phenomena

The development of modern science and technology has given birth to a class of large-scale systems where different components are distributed spatially but work in a collaborative fashion to accomplish certain tasks such as target tracking, environment perception, process monitoring, multi-agent formation control, etc. A key feature of such systems is that all the nodes are connected by a network through which local information is shared. In view of this, such systems are referred to as networked systems to distinguish from the traditional ones. Networked systems have many attractive characteristics such as lower cost, reduced energy consumption, configuration flexibility, enhanced reliability, etc. In target tracking scenarios, a major advantage of distributed sensor nodes is that there are always a portion of nodes close to the target, thus being able to provide measurements with a high signal-to-noise ratio even when low cost sensing devices are employed [51]. In this section, we will mainly focus on state estimation problems for networked systems, i.e., we will study how some specific problems arising in networked systems are treated and how different sensor nodes operate coordinately to provide an accurate estimation of the target state.

We can identify different strategies and architectures to implement state estimation algorithms for networked systems. The simplest idea is to send all the raw measurements obtained at different sensor nodes to a fusion center where these measurements are processed together. This strategy is called a centralized filtering scheme. When Kalman filtering algorithm is adopted in the fusion center to derive the final estimation, we call this scheme centralized Kalman filter (CKF). Theoretically, CKF can recover the performance of standard Kalman Filter, i.e., achieve optimal filtering performance in the mean-square sense for linear systems with Gaussian additive noise. This theoretical optimality, however, is based on the ideal assumptions that the fusion center has sufficiently large computation capability and there exists a perfect communication channel between the fusion center and each sensor node, i.e., there are no limitations in data capacity, signal fidelity, transmission rate, sampling rate, etc. Such conditions are rarely satisfied in most real-world applications. Even if they are satisfied, the filter system is still poor in robustness, i.e., it will crash in the event of fusion center failure. Since the 1970s, distributed estimation schemes, including distributed Kalman filter (DKF), have been developed to overcome the drawbacks of centralized filtering schemes. The fundamental idea of distributed estimation is to share the task of data processing among the whole network. In distributed estimation schemes, each node first processes its local measurements and then shares the processed data over the network to derive the global estimate. Distributed estimation has gained increasing concern since it is more robust to node failure, requires moderate communication and allows for parallel processing. Comprehensive review of the distributed estimation approaches could be found in [52, 53].

Unlike general state estimation methods, a distinguishing feature of state estimation for networked systems is that the limited capability of communication and local computation has to be taken into consideration in the algorithm design. In most cases, sensor nodes employed in the network are low cost devices (with limited power supply) connected by possibly unreliable channels; so it is unrealistic to expect a perfect communication performance from them. The communication problems, also termed as network-induced phenomena, will cause ambiguity and reduce the informativeness of the measurements. As a result, the estimation performance will be degraded to a certain degree. In the following, we will give a brief introduction about several network-induced phenomena that frequently occur in the networked systems.

Network-induced delay

Time delay is quite common in networked systems where there is limited access to the transmission medium. Time delay in networked systems may result from the following factors:

Nodal processing: refers to the time required to process local data and reach a routing decision; including data collecting and processing, bit error checking and output link determination;
Queuing: refers to the time waiting at the output link for transmission; usually depending on the congestion level of router;
Transmission delay: refers to the time required to push all the bits in packet on the communication medium in use; also known as store and forward delay; primarily due to the limited transmission rate of links;
Propagation delay: refers to the time required for the bit to reach the target node once it is pushed on to the communication medium; mainly due to the limited travel speed of light in a certain medium.

There are various delay models, including constant delay, random delay which is independent of previous transmission, and random delay with the probability distribution governed by a Markov chain. It is widely understood that time delay can cause performance degradation or even instability to the system. Existing research results have addressed the time delay in networked systems from two aspects: one is to determine an admissible upper bound of the time delay for which a prescribed performance can be guaranteed; the other is to design filtering algorithm for a given time delay to meet certain performance requirements.

Packet dropout

As another network-induced phenomenon, packet dropout occurs frequently in inter-node communication. Typically the network has its own mechanism to keep the packet dropout at a low level so that the system performance will not be affected. Once the loss rate approach five per cent or higher (the threshold depends on the applications, for real-time ones, it should be lower), the user will begin to notice the presence of communication problems. There are various reasons for packet dropout, including:

Congestion: in the Internet Standards, congestion and packet loss are treated as synonyms. The packet has to wait in a queue for its turn to be sent when it arrives at an intermediate node on its route. Once the length of the queue exceeds the maximum buffer capacity of the node, some data have to be discarded, which leads to packet loss.
Bit errors: during the process of data transmission, it is inevitable that some bits will be modified, leading to a mismatch between the value stored in the check bit and the actual checksum. Once the mismatch is detected by the receiving router, this packet will be considered as an erroneous one and hence discarded.
Limited processing capability: packet dropout will occur when certain local processors (router/switch) are unable to keep up with the speed of data traffic. This is a case of mismatch between communication bandwidth and processing capability.
Deliberate discard: some routers have packet discard policies that allow them to discard certain type of packets to make room for the ones with higher privilege.

Typically, a Bernoulli distribution is used to describe the randomness of packet loss, where the probability of loss is assumed to be fixed. This assumption can effectively simplify the filter design, and the results obtained have been extended to the cases where packet loss follows a more general distribution. Similarly to time delay, packet dropout can also have adverse effects on state estimation, the severity of which depend on the loss ratio. Existing studies on packet dropout in networked systems have been focused on the determination of maximum admissible packet loss rate and filter design in the presence of packet dropout. Note that with the maximum admissible loss rate, we can discard some redundant data artificially to save the network bandwidth without violating the performance requirement.

Quantization

Analog signals have infinitely variable amplitude and therefore have to be quantized before they are transmitted through the network. Quantization is involved in almost all digital signal processing. Examples of quantization processes include rounding and truncation. As a many-to-one mapping, quantization is inherently a lossy process. Considerable research efforts have been devoted to the selection of information that can be discarded without significant loss in performance. The module that realizes the quantization procedure is called quantizer. Existing types of quantizer include logarithmic quantizer and uniform quantizer.

Signal fading

Another common phenomenon in network communication is that the strength of received signals may vary over time, an effect also referred to as signal fading. It is closely related to multipath, a propagation phenomenon that results in signals reaching the receiving antenna by two or more paths. Multipath propagation can cause fading and phase shifting of the received signals. Movements of transmitter or receiver may also give rise to signal fading. Fading can occur in many forms. When all the frequency components transmitted are attenuated to the same degree, this type of fading is called flat fading. Otherwise we say there is a frequency-selective fading. Existing mathematical descriptions for the channel fading phenomena include analog erasure channel model, Rice fading channel model and Rayleigh channel model [54].

Existing results on state estimation for networked systems

In this section, we will first review some existing results on state estimation methods designed to address the network-induced phenomena mentioned above, and then proceed to investigate some distributed estimation approaches for networked systems.

Treatments of network-induced phenomena

Time delay Early results on filter design for time delay systems can be found in [55–57]. In [55], the standard Kalman–Bucy filter has been extended to include delayed measurements. The ordinary differential equations in Kalman filtering are replaced by partial differential equations together with boundary conditions which may not have an explicit solution. Note that this result is more closely related to the smoothing problem where the past state is estimated using current measurements. In [56], orthogonal projection has been employed to derive the optimal filter for discrete systems with multiple time delays. In [57], the same result as that in [56] has been obtained via maximum likelihood estimation for an augmented state. Despite the straightforwardness, this method is valid only when the random processes considered follow Gaussian distribution.

The ${H_\infty }$ filter for linear continuous systems with delayed measurements has been provided in [58]. Like ${H_\infty }$ filter design without delay, the filtering problem is transformed to seeking a bounded solution of a Riccati differential equation. In [59], a less conservative bounded real lemma (BRL) has been employed in the filter design for systems with known state delay to achieve a smaller overdesign. A robust ${H_\infty }$ filtering approach has been proposed in [60] for a type of uncertain discrete time-delay systems whose parameter matrices are assumed to belong to a convex bounded polytope. Similarly, the filtering method proposed in [61] also addresses parameter uncertainty and time delay, but in this case continuous-time systems are considered and the delay is assumed to be unknown (only the upper bound is available). A similar result has been obtained in [62] where an exponentially stable filter is designed for time-delay systems with norm-bounded parameter uncertainties. Quadratic matrix inequalities are adopted in the analysis and design of the filter. Note that robust filtering approaches in [60–62] have a common presumption that the original system is stable, which may to some extent limit the application scope of the results obtained. As an extension to [59], a robust ${H_\infty }$ filter has been proposed in [63] to deal with time-varying delay and polytopic type uncertainties using a more efficient BRL. The fact that the BRL is applied to the resulting error system can remove the requirement on a stable system matrix.

Robust filtering for nonlinear time-delay systems have been investigated in [64–66]. In [64], the nonlinearities are assumed to satisfy global Lipschitz conditions. A delay-dependent robust ${L_2}/{L_\infty }$ filter has been designed based on linear matrix inequalities (LMIs). In [65], full-order filter has been derived for a general class of nonlinear time-delay systems with guaranteed mean square boundedness of the error dynamics. A general class of nonlinear systems with randomly varying sensor delays have been considered in [66] where conditions for guaranteed ${H_\infty }$ performance are provided in terms of Hamilton–Jacobi–Isaacs (HJI) inequality.

Recently, extensive studies have been devoted to address the conservatism of ${H_\infty }$ filtering design due mainly to inequality scaling in the derivation. The conservatism of certain filtering method can be measured by the maximum admissible delay or the ${H_\infty }$ performance. In [67], the free-weighing matrix method has been adopted in the ${H_\infty }$ filter design for systems with time-varying interval delays to reduce conservatism of the existing results. In [68], a novel integral inequality has been used to establish the LMI conditions for the existence of ${H_\infty }$ filter without resorting to model transformation or bounding technique for cross terms both of which are sources of conservatism.

Recent results on Kalman filtering for time-delay systems can be found in [69–72]. The novel idea in [69] and [70] is to reorganize the measurements from different channels as a delay free system. This reorganized innovation is combined with the orthogonal projection formula to derive the optimal filter. It is shown that for systems with m delays, the obtained solution consists of m standard Kalman filters with the same dimension as the original systems. To reduce the computational complexity of reorganization innovation, the equivalent delay free system has been obtained in [71] by directly solving stochastic equations. The optimal filter and error propagation formula are then derived through Itô differentials of the state expectation conditioned on observation processes. In [72], a new sub-optimal filter has been proposed in the minimum variance framework. In this method, only instantaneous terms are used, thereby avoiding the computation of distributed terms. Also, the filter derived in [72] can be applied to any bounded delay function including non-continuous delays.

Packet dropout Filtering problems for networked systems with packet dropout have also received considerable research attention. Many elegant results have been obtained regarding this direction. Due to space limitation, we will only introduce a small portion of them. A more comprehensive review can be found in the excellent survey paper [73]. Intuitively, the missing probability will not affect the boundedness of the error covariance until it reaches a certain critical value. This value is identified in [74] based on novel matrix decomposition techniques. The optimal ${H_2}$ filtering for networked system with multiple packet dropouts has been considered in [75] where stochastic packed loss is assumed to follow a Bernoulli distribution. The stochastic ${H_2}$ norm is defined, which generalizes the norm of systems with both deterministic and stochastic inputs. The filter which minimizes such a norm is derived based on the solution of a set of LMIs. The phenomenon of multiple packet dropouts with a more general form of missing probability has been treated in [76] where random measurement loss is allowed to follow any discrete distribution taking values over the interval [0,1] with known occurrence probability. The extended Kalman filter is derived through minimizing the upper bound for the filtering error covariance. For multi-rate sensor fusion with missing measurements, an unknown input observer has been proposed in [99] to minimize the mean square error.

It is known that packet loss can cause instability to the filtering system. Stability analysis of Kalman filter for networked systems with random packet losses has been provided in, to name just a few, [77–81]. It is shown in [77] that there exists a critical value of observation arrival probability under which the expected error covariance is likely to grow unbounded. In [78], necessary and sufficient conditions have been obtained for stability analysis of networked systems with random packet losses characterized by a binary Markov chain. The Markovian packet loss has also been studied in [79] where the notion of stability in stopping times is introduced and its equivalence with the stability in sampling times is established, which simplifies the subsequent stability analysis. The necessary and sufficient conditions for mean square stability have been derived, respectively, for second order systems with different structures and high order systems with a certain structure. In [80], convergence of the error covariance has been analyzed for a rather general class of packet dropping models and alternative performance measures are introduced when the expectation of error covariance cannot be well defined. The asymptotic behavior of the random Riccati equations (RRE) which describe the evolution of estimation error covariance in Kalman filter has been studied in [81] where sufficient conditions for the existence and uniqueness of the invariant distribution for the RRE are derived.

Quantization State estimation for systems with quantized measurements has been studied in [82] where the quantizer and the estimator are designed jointly. A logarithmic quantizer is employed whose resulting quantization error can be regarded as a multiplicative noise. Choices of quantization density or number of quantization levels are discussed. Furthermore, a dynamic scaling parameter for the quantizer is introduced to ensure the convergence of estimation error in unstable systems. In [83], a quantized filtering scheme using decentralized Kalman filter has been proposed for linear discrete systems with multiple sensors. The innovation process of each local sensor, instead of local estimation, is quantized to avoid saturation of the quantizer. It is proved that stability of the filter can be achieved under sufficiently high bit rate even for unstable systems. The trade-off between quantization rate and state estimation error is analyzed. The problem of rate allocation among different sensors is also considered to enhance the asymptotic behavior of estimation error. The quantized gossip-based interactive Kalman filtering approach has been studied in [84] where it is proved that the error covariance sequence at a randomly selected sensor can converge weakly to a unique invariant measure even with information loss caused by quantization. In [85], a recursive filter is designed for power systems with quantized nonlinear measurements by minimizing the upper bound of the error covariance.

Signal fading Kalman filter for networked systems where local measurements are sent to the fusion center via fading wireless channels has been investigated in [86]. The expected error covariance of Kalman filter is proved to be bounded and converge to a steady state value. Exact recursive formulas are provided to calculate the upper bounds of error covariance which may serve as an alternative index to be optimized when the expression of error covariance is unavailable. Kalman filter with measurements transmitted over fading channels has been considered in [87] where it is assumed that the transmission power can be adjusted to alleviate the effects of fading channels. Sufficient conditions are obtained to ensure the boundedness of error covariance. These conditions are then used for power allocation to minimize the total power consumed by the network. In [88], the envelop-constrained ${H_\infty }$ filter has been presented for a class of time-varying discrete systems. The finite horizon case is considered and a novel envelope-constrained performance criterion is proposed to define transition performance of error dynamics. Borrowing idea from the set-membership filtering method, an ellipsoid description of estimation error has been utilized in [88] to transform the envelop constraints into a set of matrix inequalities solvable using standard software package.

Apparently, the network-induced phenomena mentioned above will coexist in a system. A large number of papers have been published on filter design for systems with multiple network-induced phenomena. For example, the simultaneous presence of time delay and packet loss has been addressed in [89–91]; both quantization and packet dropout have been taken into consideration in [92]; a filtering scheme has been recently proposed in [93] which is robust against both channel fadings and gain variations. For more related results, the readers are referred to [94–101].

Distributed estimation for networked systems

Another research direction of state estimation theory for networked systems is distributed estimation methods which aim to maintain an accurate estimation of certain network states at each sensor node using measurements from all sensor nodes in the network. Besides estimation accuracy, communication overhead and computational complexity also pose constraints on the filtering scheme to be designed due to the limited bandwidth and power supply of the network. Early treatments to the distributed estimation problem can be found in [102–104]. In [102], both measurements and local estimates are shared among neighboring nodes. It is shown that asymptotic agreement on the estimates can be achieved through infinitely frequent data exchange among sensors which form a communication ring. In [103], sufficient statistics are extracted from local measurements and transmitted to a fusion center where the centralized conditional distribution is exactly reconstructed. Sufficient conditions have been presented in [104] under which global sufficient statistics can be expressed as a function of the local ones.

For distributed estimation, consensus is an important concept and a lot of research efforts have been devoted to it. Generally, consensus refers to agreement among all the members of a group. In the specific case of state estimation for networked systems, we say the network reaches consensus if all the nodes hold an identical estimate of a certain quantity of interest. As a fully distributed framework, consensus based distributed estimation allows for cooperation over the network without the participation of a fusion center, thereby avoiding over reliance on a certain node. Various forms of consensus algorithms have been developed to establish the rule following which inter-node communications are implemented to reach an agreement among nodes. Average consensus and gossip consensus are the two major consensus strategies that have been investigated extensively in recent years. Readers are referred to [105] to obtain a full understanding of consensus algorithm for networked systems.

A fully distributed Kalman filter has been proposed in [106] for sparsely connected, large-scale systems. The global dynamic model is decomposed into low-dimensional subsystems for which local filters are designed. These subsystems overlap and the common states shared by several nodes are estimated using fusion algorithm. The centralized error covariance is derived through a distributed computation algorithm for matrix inversion, called distributed iterate-collapse inversion algorithm, which assimilates local error covariances with computation complexity independent of the system dimension. In the proposed method, each sensor node only needs to deal with a portion of the system state, which has significantly reduced the communication and computation requirements.

In [107], the authors consider a two-stage distributed Kalman filter which consists of a measurement update step and a fusion step using consensus algorithm. The interaction between the filter gain, the consensus matrix and the number of communications is analyzed in depth. It is proved that the common practice of minimizing the spectral radius of consensus matrix for fastest convergence is not necessarily the optimal strategy when only a small number of communications are available between two consecutive samples. It is also shown that the joint optimization of filter gain and consensus matrix is non convex and can be analytically characterized only in some special cases.

A similar two-stage distributed filter has been investigated in [108]. Sufficient conditions have been obtained to judge the distributed detectability of the networked system, i.e., the existence of filter gains which ensure an asymptotically stable error dynamic given a specific choice of consensus weights. A sub-optimal filtering scheme is then developed through minimizing an upper bound on a quadratic cost, and convergence analysis has been carried out for the time-invariant case.

One significant feature of the two-stage filtering schemes discussed above is that the consensus communication occurs at a much shorter time scale than the operation of local filters, i.e., it is assumed that there is sufficient time for the network to achieve consensus through intensive inter-node communications before the arrival of the next observation. This assumption, however, does not apply to the cases where there is a fast target dynamic and/or a high sampling rate. Besides, the high rate of consensus communication has blurred the line between distributed estimation and centralized estimation, as argued in [110]. To address this problem, Kar et al. proposes the gossip interactive Kalman filtering (GIKF) where consensus communication and observations take place at the same time scale. As a communication protocol inspired by social network phenomena, the gossip protocol has found broad applications, especially in networks with large scale or inconvenient structures. Readers can refer to [109] for a detailed overview of recent studies on gossip algorithms. In [110], the convergence of the GIKF scheme has been analyzed and the error covariance is shown to evolve according to a switched system of random Riccati operators where the switching is governed by a Markov chain determined by the network topology. Stochastic boundedness of estimation error and weak consensus of the error covariance have been established under weak assumptions about detectability and connectivity of the networked system. The method in [110] requires transmission of the error covariance, which may be burdensome for high-dimensional systems. An alternative approach has been given in [111] based on dynamic consensus on pseudo-observations (DCPO). The network tracking capacity (NTC) is used to characterize the influence of network topology and observation models on the stabilizability of DCPO error process. An explicit expression of the NTC is derived and asymptotic stability of DCPO error dynamics is established. The averaged pseudo-observations obtained are then used to construct local filters whose gains are designed to minimize the mean square error (MSE). It is shown that the method in [111] can achieve lower MSE while maintaining the major advantage of GIKF, i.e., the inter-node communication occurs no more frequently than sensor sampling.

Consensus-based distributed least square (LS) estimation problems have been studied in [112, 113]. The authors of [112] consider the total least square (TLS) estimation for overdetermined systems where both the input data matrix and the data vector are assumed to be noisy. A semi-definite relaxation technique is used to transform the nonconvex TLS problem into an equivalent convex semidefinite program (SDP). At this point, the dual based subgradient algorithm (DBSA) can be used to solve the distributed TLS problem without reliance on the computationally expensive SDP optimization procedure. In [113], underdetermined least square estimation problem has been considered. The requirement for consensus is expressed as constraints where an auxiliary variable is introduced to facilitate parallel processing, and the resulting constrained optimization problem is solved under the Augmented Lagrangian framework.

Distributed estimation schemes based on consensus algorithms aim to reach an agreement among all the nodes in the network. However, the ultimate purpose of state estimation problems is to achieve at each node an estimate that minimizes a predefined cost function, which does not necessarily require that all nodes provide the same result. Moreover, it is shown in [115] that the consensus network can become unstable even if all the local filters are stable, i.e., cooperation by means of consensus algorithms may lead to disastrous consequences. Motivated by such observations, the estimation schemes based on diffusion strategies have been proposed. As another class of fully distributed estimation methods, diffusion algorithms make several key improvements upon the consensus ones, among which the fundamental one is that agreement is no longer the goal. In the diffusion Kalman filtering proposed in [114], each node first adapt its local estimate using measurements from neighboring sensors, obtaining an intermediate estimate, which is refer to the incremental update step. Then a diffusion update step is performed by combining (through calculating a weighted average) the intermediate estimates received from neighboring nodes. Diffusion filter belongs to the single time-scale estimation scheme, i.e., the communication requirement of distributed filter is comparable to that of the gossip filter, but diffusion networks can achieve faster convergence rate and lower MSE than consensus networks. In addition, it is proved in [115] that the stability of local filters is sufficient to guarantee global stability of network under the diffusion framework, regardless of the choice of combination weights.

Particle filter in networked systems

In the previous sections, we have investigated the particle filtering methods and state estimation for networked systems, respectively. In the cases where nonlinear systems or non-Gaussian PDFs are addressed, particle filtering becomes a more suitable choice. The applications of PF to networked systems, however, have given rise to some new challenges. This is mainly because in particle filtering methods, a large amount of particles are required to represent the posterior PDF, which implies a huge communication burden if local information is exchanged among nodes. One major concern of particle filtering for networked systems is, therefore, how to achieve an affordable communication cost while maintaining an acceptable accuracy. On the other hand, the network-induced phenomena should also be addressed when PF is applied to networked systems. In this section, we will discuss the applications of PF algorithms to the filtering problems of networked systems, highlighting how the network-induced phenomena are treated by particle filters and how, in networked systems, local particle filters work coordinately to accomplish the state estimation task with an affordable communication cost.

Particle filter with network-induced phenomena

Particle filtering for systems with missing data

Generally speaking, there are two categories of methods for missing data problems: guaranteed cost ones and data imputation based ones. The methods introduced in Sect. 2 obviously belong to the former category. Next we will introduce some data imputation based approaches which are more commonly used in Monte Carlo methods. The basic idea of data imputation is to generate some data which obey the same distribution as the missing ones. These artificially generated data will then be regarded as real measurements in the subsequent processing. It is typically difficult to calculate the distribution of missing data conditioned on the observed ones. This being the case, one has to resort to sampling-based approaches such as the MCMC method introduced in Sect. 2 of this paper.

As a special form of the MCMC approach, Gibbs Sampler has attracted extensive attention among researchers since its introduction in [116]. Essentially, Gibbs sampler is an iterative algorithm to draw samples from an unknown distribution without direct calculation of the density. The pseudo code of Gibbs Sampler is given in Algorithm 3 where the case of three random variables, x, y and z, is considered and the marginal distribution p(x) is of interest to us. The readers can also refer to [117] for an explanation of the underlying principle of Gibbs sampler.

Kong et al. extended the Gibbs sampler to sequential imputation which does not require iterations, thus reducing the computational burden [18]. The sequential imputation uses samples and associated weights to approximate the unknown distribution in the presence of missing data, and thus can be seen as a combination of Gibbs sampler and sequential importance sampling. As new data arrive, a new sample is drawn and the augmented data set is updated to include this sample. The corresponding weight is then determined according to the quality of the augmented data set. Some related problems, such as effective sample size, the order of imputation, the behavior of weights, and sensitivity to the choice of prior distribution, are also analyzed in detail. An interesting finding, which is also illustrated by simulation results, is that the order of imputation can have a huge impact on the approximation accuracy. To be brief, a “good” data set cannot play its due role if it is processed after the “bad” ones because the trail distribution has already been corrupted by early imputations.

Expectation Maximization (EM) algorithm is another powerful tool to address missing data problems. An excellent introduction of it can be found in [118]. The EM algorithm consists of an expectation (E) step and a maximization (M) step. These two steps are executed alternately: in the E step, expected values of the missing data are computed, which will then be used in the M step; while in the M step, the maximum likelihood estimate of unknown parameters are calculated, to be used in the next E step. It is guaranteed that the likelihood function will not decrease after every such iteration. This implies that only local maximum can be reached, which may be seen as a deficiency of the EM algorithm. The basic EM procedure for estimating unknown parameter $\theta $ in the presence of missing data $z_{k}$ is illustrated in Fig. 1 below.

In [119], three EM algorithms have been adopted to treat the maximum a posteriori (MAP) state estimation for JMLS. Both the Markov chain and continuous states are unknown and to be determined. The first algorithm addresses MAP estimation of the Markov chain. The unknown continuous states are regarded as missing data and estimated with a fixed-interval Kalman smoother in the E step. Assuming the continuous states to be known, the MAP estimation of Markov chain is then obtained through dynamic programming in the M step. The second algorithm aims to estimate the continuous states with the unknown Markov chain viewed as missing data. A forward and backward recursion is applied in the E step to calculate the probability of Markov chain and a Kalman smoother is used in the M step to compute MAP estimation of the continuous states. The last algorithm deals with joint estimation of Markov chain and continuous states, which is realized through an alternate execution of fixed-interval Kalman smoother and dynamic programming. To overcome local convergence of the EM method, stochastic sampling based algorithms have been proposed in [120] for state estimation of JMLS. Three data augmentation (DA) algorithms (DA, stochastic annealing DA, and Metropolis-Hasting DA) are employed and an acceptable computational cost is achieved. As a special form of MCMC method, the DA algorithm can ensure convergence to the globally optimal solution. In [121], the EM algorithm is applied to estimate missing data in the Moderate Resolution Imaging Spectroradiometer (MODIS) time series for forest growth prediction.

Multiple imputation particle filter (MIPF) has been introduced in [122] where particle approximation is utilized to perform multiple imputation. This method resembles both the Gibbs sampler and EM algorithm. First, multiple imputations are drawn from a proposal distribution where the true states are replaced with particle representation which is calculated regardless of the missing observations. Next, for each imputation, a particle filter is performed to obtain an approximation of the state posterior. We then combine the approximations derived from different particle filters to give the final particle representation of the target density. Almost sure convergence of the MIPF method is established in a later work [123].

Particle filtering for time-delay systems

As mentioned in Sect. 2, transmission delay occurs frequently in a networked environment, which gives rise to the so-called out-of-sequence measurements (OOSMs). A great deal of research has been done to address this phenomenon, but mostly within the framework of Kalman filtering. The particle filtering algorithm in the presence of OOSMs has been studied in [124], where the basic idea is to rerun the particle filter to incorporate the delayed measurements. A major drawback of this method is that we have to store the particles and corresponding weights at each sampling period, which poses severe challenges to the storage capability of the processor, especially when the required number of particles is large. The proposed method in [124] also suffers from the problem particle degeneracy, but this can be mitigated via an MCMC step [125]. In [126, 127], a backward information filter has been adopted to retrodict particles corresponding to the delayed measurements. These particles are then used to recalculate the current weights. The implementation of backward information filter, however, involves intensive computation, which may be formidable in some practical applications.

The above-mentioned methods suffer from either excessive memory requirement or huge computational burden. A storage efficient particle filter has been proposed in [128] where only mean and covariance of the particle set need to be stored. The memory requirement of this method is dramatically decreased compared with that of [124], but the problem of particle degeneracy remains, especially when the OOSMs contain such a great amount of information that the original support is not enough to describe the filtering distribution. A selective procedure has been proposed in [129] to distinguish the OOSMs according to their utility. At every sampling period, a threshold for measurement selection is firstly calculated through a constrained optimization problem. The measurements whose utility exceeds the threshold are identified as informative and processed in the subsequent filtering step, while those with utility below the threshold are simply discarded. To reduce computational cost associated with solving the optimization problem, Gaussian approximation and linearization are employed for a rapid prediction of the mean square error (MSE) reduction brought by each delayed measurement. In addition, another threshold test is conducted to detect particle degeneracy. Once the effective sample size is found to drop dramatically, which implies that current support can no longer give an accurate description of the filtering distribution, the OOSMs are reprocessed through another filtering procedure which allows for simultaneous adjustment of the weights and locations of particles, thus avoiding particle degeneracy.

The exact Bayesian solution to the filtering problem for OOSMs has been derived in [130]. Different from the storage efficient particle filter [128] whose performance degrades when the target states do not follow a unimodal distribution, the exact Bayesian algorithm uses all the past particles and weights to achieve optimal performance. The cost of optimality, however, is a huge computational overhead, which limits the application scope of the exact Bayesian algorithm to low-dimensional problems or offline processing. For nonlinear model containing a linear substructure, two Rao-Blackwellized particle filtering algorithms have been presented in [131] to yield efficient execution and high accuracy, respectively.

Particle filtering problem for target tracking in the presence of signal propagation delay has been considered in [132]. Due to the interaction between target dynamics and propagation delay model, neither kinematic state of the target nor the propagation delay can be determined independently, which substantially complicates the problem. To tackle this difficulty, an augmented state vector is defined which includes both the time delay and the target state with a stochastic time stamp. The key of this method is to solve the unknown delay from an implicit equation. It is shown that iterative techniques can be used to obtain an approximate solution to the implicit equation as long as a fairly weak convergence condition is satisfied. The bootstrap particle filter is employed where iterations are incorporated in the time-update step to predict the time delay of current measurements. The resultant particles have different time stamps with each other, therefore a time synchronization is performed before the final estimate is derived.

Particle filtering algorithm with one-step delayed measurements has been studied in [133]. The standard particle filtering algorithm is modified to take the probable delay into account. When the latency probability is unknown to the designer, a maximum likelihood algorithm is proposed to identify it. The result has been extended to handle multiple-step randomly delayed measurements in [134], but only the case of known latency probability is considered.

Particle filtering for systems with signal quantization

For estimation problems with quantized measurements, the Cramer-Rao lower bound (CRLB) has been derived in [39], giving an indication on the information loss caused by signal quantization. Both Kalman filtering and particle filtering algorithms are developed to handle measurement quantization, and the superiority of particle filtering over Kalman filtering is demonstrated through numerical experiments. Measurement quantization will induce big error to the filter system when the values of observed data are large. The filtering problem with innovation quantization has been studied in [135]. A counterexample is constructed to show that the Kalman filtering may perform below expectation or even diverge in the presence of quantized innovation. The particle filter, instead, seems capable of approximating the optimal filter in the same situation.

It is revealed in [136] that the state conditional on quantized observations can be decomposed into the sum of two independent random variables, one of whom follows Gaussian distribution while the other is a truncated Gaussian random vector. The authors of [136] point out that we only need to propagate the truncated Gaussian variable, rather than the sum, since the truncated Gaussian variable has a probability density whose covariance is much smaller than that of the conditional state density. Taking advantage of the Gaussian properties, the authors design a Kalman-like particle filter (KLPF) where a group of Kalman filters are processing in parallel to obtain minimum mean square estimate of the state conditioned on perfect observations. One major advantage of KLPF is that the required number of particles is dramatically reduced compared with directly using particle filter as in [135].

Particle filter for cooperative estimation in networked systems

As state estimation problem for networked systems has gained increased research attention, a great variety of particle filtering schemes for networked systems have been published in the literature [137]. We can, similarly to that of Sect. 2, define centralized particle filtering (CPF) and distributed particle filtering (DPF) according to whether local processing is performed at each agent. It should be noted that the CPF method is nothing different from the general form of particle filtering introduced in Sect. 1, and has the similar shortcomings as the CKF discussed in the previous section. Therefore, we will present a brief review of the existing CPF methods in the next paragraph and reserve most of our attention for the discussion of DPF methods. For clarity, we first present a taxonomy of different particle filtering approaches for networked systems in Fig. 2.

Centralized particle filtering

Theoretically, the CPF can give the optimal state estimate if we ignore the error caused by particle representation of the continuous posterior distribution. The optimality, however, comes with a high cost, both in communication burden and computation complexity. The requirement on communication could be formidable in sensor networks where each node has limited power supply and thus limited communication capability. In [138, 139], a CPF method based on state partition and parallel EKF has been proposed for target tracking using collaborative acoustic sensors. The major computation task is efficiently done in the fusion center, thus freeing sensor nodes from local data processing. To obtain a proposal distribution which is closer to the state posterior, a bank of EKFs are used to process data from all activated sensors concurrently, and a weight sum of these EKF estimations is calculated and taken as the proposal distribution. An efficient way to store particles has been introduced in [140] where a compression step is taken before storing particle states. Simulation results suggest that this scheme can significantly reduce memory requirement with minimal performance loss.

In a sensor network where the communication capability is limited, data are always quantized at each sensor node before transmission. This, plus the imperfect nature of the communication channels, should be taken into account by the fusion center to gain better performance. A channel-aware particle filtering scheme is put forward in [141] to address the quantized measurements and fading channels simultaneously. The likelihood function, in which both data quantization and channel imperfection are considered, is calculated in three different scenarios. The posterior CRLBs for the proposed method are also derived. When there is a constraint on the total number of bits that can be transmitted, bit allocation becomes necessary. In [142], a dynamic bit allocation algorithm based on approximate dynamic programming has been presented. It is shown that the proposed algorithm can save much of the computational cost while achieving comparable accuracy with other existing methods. The amount of transmitted data can be significantly reduced if sensor nodes are able to distinguish informative measurements from uninformative ones. The data censoring performed at each sensor node does exactly this. The particle filtering with censored data has been studied in [143] where it is pointed out that even though uninformative measurements are not to be transmitted to the fusion center, the fact that they are uninformative also delivers some useful information for data processing. This information is exploited in the proposed filtering method through a full likelihood function, which contributes to an enhanced performance. Strictly speaking, the method in [143] does not belong to CPF since KF update is run at each node to obtain the variance of local innovation based on which the censoring threshold is identified. However, we introduce it here because, like other CPF methods, it requires transmission of raw measurements to eliminate the dependence between local data.

Distributed particle filtering

In the remainder of this section, we will focus on DPF for state estimation in agent networks. Hlinka et al. presented a detailed classification of the existing DPF methods (see [144]). A fundamental distinction between different DPF methods is whether a fusion center (FC) is present. In the FC-based DPF scheme, each agent processes its own measurements with a local PF and reports the obtained posterior to the FC according to a predefined communication protocol [145, 146]. This scheme is suitable for those applications where global knowledge is required only in the FC [147]. However, it suffers from two major drawbacks: (1) the filtering performance relies highly on the FC, which implies a poor robustness against the FC faults; (2) the communication path is highly dependent on the network topology, once the topology changes, which is very common in mobile networks, the total route table has to be re-established; (3) excessive communication burden is imposed on the nodes which are closer to the FC. To reduce long-distance communication, a two-tiered network structure has been proposed in [148] where some selected nodes, referred to as cluster heads (CHs), are responsible for processing raw measurements of the nearby sensors and sending the obtained local estimate to the FC for a further fusion. In this way, only the CHs are required to be capable of directly communicating with the FC.

DPF schemes without an FC is also referred to as decentralized particle filtering. We can further classify various decentralized particle filtering methods according to whether all the agents run the particle filter simultaneously or only a portion of them are in charge of data processing. We refer to the schemes where a portion of activated agents take the charge of global estimation as leader agent (LA)-based DPF (see, for example, [149–151]), and term those with all agents performing particle filtering algorithm consensus-based DPF (see, for example, [156, 158, 164]). In the LA-based schemes, a sequence of adjacent nodes form a LA path along which the local estimation is accumulated. Typically, this LA path is constructed dynamically and adaptively, i.e., current LA is in charge of selecting the next LA among its neighbors based on the assessment of their informativeness. The selection scheme can affect the estimation accuracy and energy usage to a large degree [152]. Compared with the LA-based schemes, consensus-based DPFs can achieve enhanced scalability and robustness against changes in network topology or node failures. The price for these advantages is a heavier demand on communication and the likely delay due to consensus iterations. In view of the fact that a detailed introduction of both LA-based and consensus-based schemes has already been provided in [144], we will, in the following, investigate from a different perspective, i.e., we will focus on how different decentralized particle filtering methods make a reasonable trade-off between multiple performance index including accuracy, communication burden and computation complexity.

Ideally, we hope the performance of decentralized particle filtering methods can reach that of the centralized ones which is theoretically optimal. Decentralized particle filtering for blind equalization has been studied in [153] where each node evaluates the likelihood function of its local observations and then broadcasts it to the entire network. The local PF performed at each node thereby has access to the global likelihood function, and is guaranteed to converge to the optimal filter asymptotically. Synchronization is required to ensure that an identical set of particles are generated at different nodes. It is also shown that the filtering performance could be enhanced via the optimal importance function, which, however, asks for an extra amount of broadcast. A modified method has been proposed in [154] to reduce inter-node communication by employing parametric approximations of the remote likelihood function. This method achieves significant communication reduction with only moderate performance degradation. The communication requirement could be further cut down through a protocol where inter-node connection for message exchange is established randomly and each node transmits its local data to only one remote sensor at each time step [155].

The methods mentioned in the previous paragraph rely on broadcasting and are thus suitable only for fully connected networks. Two consensus-based methods have been provided in [155] where inter-node communications are limited within adjacent nodes. Both methods involve evaluating the global likelihood function. The first one uses average consensus to calculate the global likelihood at each node and a quantization step to eliminate the discrepancy between particle sets at different nodes. To avoid performance degradation caused by quantization, the second method adopts a modified minimum consensus algorithm, which borrows the idea of flooding scheme, to obtain an ordered list of local likelihood functions shared by all the sensor nodes. The merit of this method lies in that it, unlike the first one, does not require an infinite number of consensus iterations for a guaranteed performance.

The consensus-based schemes reduce inter-node communication at the cost of heavier local computation. This is desirable in most applications since communication between nodes typically consumes more energy than local computation. Similar ideas have been discussed in [156] where a likelihood consensus (LC)-based DPF method is developed. The basic idea of LC-based DPF is that the underlying sufficient statistics, rather than raw measurements, should be exploited in the DPF to avoid redundant communication. A key step in the proposed algorithm is to obtain a parametric representation of the local likelihood functions. Specifically, local likelihood functions are approximated by the sum of a series of basis functions multiplied by respective coefficients. In this way, the useful information contained in local measurements is compressed into certain coefficients. Since the basis functions are known to all the nodes, the knowledge of corresponding coefficients is sufficient to reconstruct the likelihood functions. With this approximation, a consensus procedure is carried out to calculate the sum of the exponential term of global likelihood function which can be expressed as the product of all the local likelihood functions. The pseudo code of LC-based DPF is presented in Algorithm 4. An attractive feature of the LC-based DPF method is that its communication cost does not depend on measurement dimensions, which makes it particularly suitable in applications which involve high-dimensional measurements. Also note that synchronization is no longer a requirement in the LC-based method since the likelihood functions exchanged between nodes are in a parametric form rather than represented by discrete values.

One shortcoming of the method proposed in [156] is that current measurements have not been incorporated in the proposal density. This problem has been treated in [157] via proposal adaptation which is implemented in a distributed manner. Gaussian distribution is used to approximate both local and global posteriors, and EKF/UKF is employed to incorporate local measurements in the local posterior. Fusing local information to obtain a global proposal density now amounts to running consensus algorithms to calculate two sums, for global mean and global covariance, respectively, over all sensor nodes. The benefit gained from the adapted proposal density is twofold: on the one hand, particles are used with higher efficiency since they are located in a region of higher likelihood; on the other hand, the least square approximation in the likelihood consensus will also have an improved accuracy since the local likelihood functions are approximated in a smaller region.

Another efficient way to implement proposal adaptation has been proposed in [158] where a set-membership constrained particle filtering scheme is developed. It is argued that the importance sampling step can be implemented in a distributed fashion if particles are drawn from the prior distribution. The prior distribution, however, is a non-adapted one which has not taken current measurements into account. This implies that we have to use a large number of particles to ensure that the posterior density is well characterized by these particles because it is possible that the likelihood function is a very peaked one compared with the prior PDF. A large number of particles will in turn reduce the efficiency of DPF algorithm because the computational complexity is positively correlated with the number of particles in a consensus-based scheme. To overcome this dilemma, we seek to develop a proposal adaptation scheme with affordable distributed implementation. The set-membership based adaptation proposed in [158] does exactly this. In this method, each sensor first determines a local set which approximates the local posterior density. Local sets at different nodes are then combined using consensus algorithms to construct the global set which will serve as the global proposal density in the subsequent importance sampling. The consensus algorithms are guaranteed to converge in finite iterations, which further reduces the overall communication cost. The pseudo code of set-membership constrained DPF approach is presented in Algorithm 5.

In [159], a Gaussian mixture model (GMM) has been employed to represent the posterior PDF to circumvent the transmission of a huge number of particles. In this method, each node obtains the global statistics through an average consensus which diffuses local statistics over the network. Based on the global statistics, an EM algorithm is performed to estimate the global GMM (see also [160]). The transmission of GMM representations, however, can be inefficient for high-dimensional systems since the amount of transmitted data grows with the volume of covariance matrix which is proportional to the square of state dimension. To achieve better scalability, a Markov chain distributed particle filter (MCDPF) has been proposed in [161, 162] based on random walks on the graph. In the MCDPF method, particles traverse the network along a random path and update the associated weights according to the local measurements at each node they pass by. The communication overhead of MCDPF approach depends linearly on the state dimension, which is particularly suitable for high-dimensional systems. Note that although data exchange is limited within neighboring nodes, the MCDPF does not belong to the consensus-based approaches since no consensus iterations are required. However, convergence to the optimal filter can only be established when the number of particles and the length of the random path both goes to infinity. It is also pointed out in [162] that, for low dimensional systems, the MCDPF algorithm is inefficient and GMM representation may be a better choice. Therefore, one needs to select the most suitable scheme on a case-by-case basis.

Gaussian mixture model has also been employed in [163] to develop a soft-data-constrained DPF. In this method, the global GMM is calculated from local ones using the consensus propagation algorithm. Instead of representing the global posterior from which the new sets of particles are drawn, the global GMM is used to pose soft-data-constraints according to which the local particles are reweighed. The resultant local particles at each node represent a local posterior closer to the global one, which implies an enhanced robust against noise and failures.

Up to now, the consensus-based DPF methods mentioned have a basic requirement that consensus be achieved before the arrival of next measurement. In a network with intermittent communication connectivity, however, convergence of the consensus algorithm between every two consecutive measurements cannot be guaranteed, which may lead to severe performance degradation. A consensus/fusion based DPF method has been presented in [164] to address this problem. In this method, an extra filter, referred to as the fusion filter, is employed at each node in addition to the local particle filter to diffuse local estimate and reach consensus across the entire network. Note that the fusion filter is allowed to run at a different rate from the local one, thereby removing the requirement on convergence between successive measurements. Another function of the fusion filter is to compensate for the common information contained in different local estimates which is a general problem for DPF schemes where local estimates rather than raw measurements are diffused [165].

A constrained sufficient statistic-(CSS) based DPF method has been provided in [166]. Similarly to the LC-based approach [156], this method also seeks to fuse local sufficient statistics (LSS) to the global one. However, no approximation of the global sufficient statistics (GSS) is involved, which implies an enhanced accuracy. Communication overhead per iteration is no longer dependent on the state dimensions (as is the case with [159, 167]) or the number of particles (as is the case with [158]). It is, instead, proportional to the number of GSS parameters which is much lower compared with either scenario mentioned above. To adapt the proposed method to error prone networks with intermittent connectivity, the authors of [166] further combine the CSS based DPF with distributed unscented particle filtering (DUPF) to achieve a guaranteed performance with fewer number of iterations per consensus run.

Conclusion and outlook

In this survey, we have reviewed existing results on particle filter and its applications in networked systems. As a simulation-based method, particle filter has particular advantages in complex systems where nonlinearities and non-Gaussian noises are ubiquitous. It can be seen that the application of particle filter is still limited by the hardware resources, therefore, existing results on particle filter design have mainly focused on the trade-off between estimation accuracy and computational complexity. It is believed, however, that with the development of hardware technology and improvement of computational power, particle filter will find more extensive applications in various fields.

At last, we point out the following research directions in the area of particle filter which are worthy of further studies:

How to incorporate prior knowledge into the design of particle filter: the efficiency of particle filter is highly dependent on the number of particles employed to represent the posterior PDF that is of interest. Without any prior knowledge, one can only use a large number of particles for an exhaustive exploration of the state space, which will result in an excessive computational burden especially in real-time applications. In the standard SIR algorithm, prior knowledge can be incorporated in either the sampling step or the importance step. In the sampling step, knowledge about the model information or the high likelihood region can be reflected in the construction of proposal distribution. Some schemes for proposal distribution adaptation, such as APF, EPF and UPF, have already been proposed in the existing literatures. This idea can be further extended to tackling other types of prior knowledge such as state constraints and time delay. In the importance step, one can address, say, signal fading or quantization, by evaluating a full likelihood function in which the corresponding occurrence probability has been incorporated.
How to deal with model uncertainty and norm-bounded noise: most particle filtering methods proposed in the literature have relied on perfect knowledge about the model information and noise statistics. This is largely due to the fact that one is unable to simulate a random signal without its statistic information. In practical applications, however, one has to deal with model uncertainty and random noise without accurate statistics. This is especially true for networked systems where the accurate occurrence probability of network-induced phenomena is generally unavailable and only an upper bound is known. In the existing literatures, the CRPF method has been proposed as an attempt to incorporate the prescribed cost function into particle filter design. This approach can be further developed so that the existing results and computational tools in the field of guaranteed cost filtering can be applied in particle filter design.
How to achieve further variance reduction: for particle filtering, variance reduction can be achieved in several ways. First, the linear substructure of the dynamic model should be fully exploited. Linear filtering methods can be combined with particle filter to derive an analytical solution to the estimation of conditionally linear states. This is justified by the Rao-Blackwell theorem which reveals that any redundant random variable present in the estimator will cause extra variance. Second, the resampling step should be performed with more flexibility. On one hand, in view of the extra variance that resampling has inevitably introduced, further studies should be focused on how to circumvent resampling while maintaining an acceptable ESS; On the other hand, novel resampling schemes should be developed to address the trade-off between particle diversity and variance reduction.
How to apply particle filtering approach in controller design: the controller design for nonlinear non-Gaussian systems is a quite challenging problem. Recently, it has been suggested that the controller design for nonlinear non-Gaussian systems should be based on the posterior PDF of system states, i.e., the aim of control is to shape the PDF which is represented by the particles and the corresponding weights. Therefore, it is of interest to investigate the interaction between control input and importance weights, and how the error of particle representation affects the performance of control system.

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School of Automation Science and Electrical Engineering, Beihang University, Beijing, 100191, China
Wenshuo Li, Yuan Yuan & Lei Guo
College of Electrical Engineering and Automation, Shandong University of Science and Technology, Qingdao, 266590, China
Zidong Wang
Department of Computer Science, Brunel University London, UB8 3PH, Uxbridge, Middlesex, UK
Zidong Wang

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Wenshuo Li
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Zidong Wang
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Yuan Yuan
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Lei Guo
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Correspondence to Zidong Wang.

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The work of W. Li, Y. Yuan and L. Guo were supported in part by Chinese 973 project under Grant 2012CB720003, the Foundation for Innovative Research Groups of the National Natural Science Foundation of China under Grant 61421063, the National Natural Science Foundation of China under Grant 61320106010, Special Program for International Science and Technology Cooperation from Ministry of Science and Technology of China under Grant 2013DFE13040. The work of Z. Wang was supported in part by the Royal Society of the UK, the Research Fund for the Taishan Scholar Project of Shandong Province of China, the National Natural Science Foundation of China under Grant 61273156, and the Alexander von Humboldt Foundation of Germany.

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Li, W., Wang, Z., Yuan, Y. et al. Particle filtering with applications in networked systems: a survey. Complex Intell. Syst. 2, 293–315 (2016). https://doi.org/10.1007/s40747-016-0028-2

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Received: 20 September 2016
Accepted: 20 October 2016
Published: 27 October 2016
Issue Date: December 2016
DOI: https://doi.org/10.1007/s40747-016-0028-2

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Particle filtering with applications in networked systems: a survey

Abstract

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Particle Filter Optimization: A Brief Introduction

Particle filtering for nonlinear dynamic state systems with unknown noise statistics

Nonlinear System Identification Using Particle Filters

Introduction

Particle filter

Basic ideas

Related problems and improvements

Degeneracy problem

Variance reduction

Robust particle filter

Efficient implementation of particle filtering algorithms

Networked systems

Introduction of networked systems and network-induced phenomena

Network-induced delay

Packet dropout

Quantization

Signal fading

Existing results on state estimation for networked systems

Treatments of network-induced phenomena

Distributed estimation for networked systems

Particle filter in networked systems

Particle filter with network-induced phenomena

Particle filtering for systems with missing data

Particle filtering for time-delay systems

Particle filtering for systems with signal quantization

Particle filter for cooperative estimation in networked systems

Centralized particle filtering

Distributed particle filtering

Conclusion and outlook

References

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