Robust particle filtering with enhanced outlier resilience and real-time disturbance compensation

Abstract

This paper is concerned with the state estimation problem for nonlinear/non-Gaussian systems suffered from both time-varying disturbances (TVD) and measurement outliers. Conventional particle filtering (PF) approach can be used to track the non-Gaussian probability density functions, but its sampling efficiency is degraded in the presence TVD. To address this problem, we propose a disturbance observer based PF (DOBPF) method where the knowledge on the dynamics of TVD are fully exploited and real-time disturbance compensation is achieved. Furthermore, to enhance the resilience of our method against outliers, we adopt the skew-$t$ distribution to characterize both skewness and heavy-tailedness of the measurement noise. On this basis, the variational Bayes approach is incorporated into the DOBPF under the marginalization PF framework to infer the noise statistics. Compared with conventional PF approaches, the proposed outlier-resilient DOBPF method exhibits improved resilience against measurement outliers and increased sampling efficiency in the presence of TVD. Simulation and experimental results confirm the effectiveness of the proposed algorithm.

Introduction

State estimation for nonlinear systems with non-Gaussian noises is quite challenging due primarily to the complicated form of posterior PDFs. When the posterior PDF is, or can be approximated as Gaussian, the Kalman-type filters can be adopted to obtain closed-form solutions to the state estimation problem. In such cases, the state PDF is parameterized by its first two moments, namely the mean and the covariance. In practice, however, the state PDFs are usually multi-modal, skewed or/and heavy-tailed. Under these circumstances, the Gaussian assumptions are no longer valid and the resulting non-Gaussian state estimation problem can only be solved by means of numerical approximation

Proposed in the early1990$\prime$s, the PF method has now become a popular numerical tool for nonlinear/non-Gaussian state estimation problems [16]. Since the PF does not rely on Gaussian approximation, it is suitable for a much broader class of practical systems than the Kalman-type filters. In the PF algorithm, the state PDF is predicted by generating a set of particles according to the prior knowledge of system dynamics. This predicted PDF is then corrected by adjusting the weight of each particle according to the current measurement [3], [6], [11], [12]. Hence, the performance of the PF method depends heavily on the accuracy of our knowledge about system dynamics. In fact, if there exists model uncertainties (such as external disturbances or unmodeled dynamics), the particles generated by the PF algorithm might fall into unimportant regions of the state space, resulting in poor approximation of the posterior PDF.

On the other hand, outliers are inevitable in today’s large-scale control systems, especially the so-called CPSs. For example, some adversarial attacker may launch electromagnetic attack towards the sensing system, causing the sensors to operate abnormally, or inject falsified data into the communication network [29], [40]. If not properly dealt with, sensor attacks can cause severe performance degradation to the state estimator. When sensor attacks are launched in an intermittent fashion, the abnormal observations produced by the compromised sensors can be treated as measurement outliers. Therefore, it is of great importance to improve the resilience of PF against outliers.

In this paper, we will deal with the state estimation problem for nonlinear/non-Gaussian systems suffered from both model uncertainties and measurement outliers. The results obtained in this paper could have a great potential of application as there are a wide range of practical scenarios where model uncertainties and outliers coexist. Typical examples include a) the integrated INS/skylight polarization AHRS where, on one hand, the drift of inertial sensors can be modeled as an additive TVD term in the state transition equation and, on the other hand, occlusion of polarization sensors will lead to outliered measurements [42]; b) the UWB based UAV positioning system where the unknown external wind can be regarded as the TVD and the large positive errors in the UWB data caused by obstacles can be viewed as outliers [24]; c) the SCADA system of smart grid where the unknown load variation can be viewed as TVD and the false data injection attack leads to outliers [45].

Model uncertainties can be regarded as an additive TVD term in the system equation. Instead of treating the TVD term as a norm-bounded variable, as done in the $H_{\infty }$ filtering schemes [1], [29], it is more desirable to make full use of the dynamic information about the TVD term. In fact, such information available or partially available to us in practice, which enables the development of DOBF approaches [5], [13], [17], [26]. In the DOBF scheme, the TVD is estimated in real time and its effect is counteracted by an additional disturbance compensation term that is adopted in the state prediction step of standard Kalman/$H_{\infty }$/particle filters. Up to now, DOBF has been applied to integrated navigation systems with enhanced anti-disturbance capability [5], [17].

On another research frontier, outlier-resilient filtering problems have been extensively studied and many elegant approaches have been proposed, including the $H_{\infty }$ filtering, the Huber based Kalman filtering, the minimum error entropy filtering, the maximum correntropy Kalman filtering, the student-$t$ distribution based Kalman filtering, etc. In the $H_{\infty }$ filtering schemes [15], [35], [44], bounded estimation error can be guaranteed in the worst case. This comes, however, at a cost of decreased nominal performance. In [7], [8], [14], [22], the Huber based Kalman filtering method is proposed where a robust cost function called Huber function is adopted. The Huber function exhibits $L_2$ norm property for small residuals and $L_1$ norm property for large ones, thus reducing the weights of outliered data. In order to better characterize the non-Gaussian PDF induced by outliers, the minimum error entropy filtering has been proposed to replace the traditional Kalman-type filters. By minimizing error entropy, the error distribution can be made as sharp as possible, thereby reducing the dispersion of estimation error in the presence of non-Gaussian noises [28], [33], [43]. More recently, the maximum correntropy criterion has been introduced and applied to both linear and nonlinear filtering problems [9], [23], [39]. Unlike the mean square error, the correntropy has taken higher order moments into account when describing the similarity between two random variables, hence is more effective in dealing with heavy-tailed noise induced by outliers. As revealed in [8], the maximum correntropy criterion can be seen as a special type of robust cost function that suppresses the impact of data with large residuals.

From a distribution-theoretic point of view, outlier resilience can be achieved via introducing heavy-tailed distribution to describe the measurement noise [2]. In [25], both the predicted and the posterior state PDFs are approximated with student-$t$ distributions which have heavier tails than the Gaussian. On this basis, the EM algorithm are embedded in the PF procedure to compute the maximum likelihood estimate of the parameters of student-$t$ distributions. Another thought is to modify the standard Kalman-type filters to account for the heavy-tailed noise distributions. In [2], the measurement noise is assumed to be a Gaussian random variable with the associated covariance matrix modeled as invert Wishart distributed, resulting in a $t$-distributed likelihood function. Then, robust filtering and smoothing schemes are designed by incorporating Kalman filter into the VB framework where the state and covariance matrix are inferred in an iterative fashion. This idea has been further pursued in [20], [46] and extended in [21], [41] to nonlinear systems. In the nonlinear settings, the PF and VB approaches are incorporated under the MPF framework [32], [36] to estimate the state vector and noise parameters respectively [41]. The nonlinear state estimation problem has also been studied in [19] where the process and measurement noises are both heavy-tailed. Unlike in [41], the nonlinear state evolution is now addressed via Gaussian approximation techniques rather than the PF. As a generalization of the $t$-distribution, the skew-$t$ distribution was considered in [30], [38] where the VB approach is adopted for parameter inference. The results were further refined in [31] where novel skew-$t$ filter and smoother are proposed to approximate the covariance matrix with improved accuracy.

The aforementioned works have dealt with either TVD or measurement outliers, but not both. In this paper, an outlier-resilient disturbance observer based PF (OR-DOBPF) algorithm will be proposed to cope with TVD and measurement outliers simultaneously. In our problem formulation, the TVD is regarded as the output of an exogenous dynamic system with state-dependent coefficient matrices and the stochasticity of the measurement noise is characterized by the skew-$t$ distributions. On this basis, a modified PF with TVD-compensated sampling stage is introduced to achieve robustness against TVD, and the VB techniques [34], [37] are employed to infer the statistics of measurement noises. Compared with the robust PF method proposed in [41], our approach is more robust to model uncertainties and has an increased sampling efficiency in the presence of TVD. Compared with the DOBPF introduced in [26], the OR-DOBPF approach can cope with the skew-$t$ distributed measurement noise, hence achieves enhanced outlier resilience. The contribution of this paper is summarized as follows: 1) we improve the robustness of PF algorithm against both uncertainty in the system dynamics and outliers in the measured data; and 2) we extend the existing outlier resilient PF approaches to the cases where the measurement noise is both heavy-tailed and asymmetric.

The rest of this paper is organized as follows. In Section 2, the non-Gaussian state estimation problem under both TVD and measurement outliers is formulated and some preliminaries about the PF are given. Our main algorithm, namely the OR-DOBPF, is presented in Section 3. Specifically, the disturbance observer based PF approach is introduced in Section 3.1 to deal with the TVD and the VB algorithm is adopted in Section 3.2 for noise parameter inference. Simulation and experimental results are presented in Section 4 to illustrate the effectiveness of the proposed method. Finally, some concluding remarks are made in Section 5.

Notation

${\mathbb{R}}^n$ denotes the $n$-dimensional Euclidean space. The superscripts “$-1$” and “$T$” denote inverse and transpose operations, respectively. “$\mathbf{0}$” and “$I$” stand for, respectively, zero and identity matrices with proper dimensions. $\mathbb{E}\left(x\right)$ ($\mathbb{E}\left(x\right|y)$) represent the mathematical expectation of $x$ (conditional on $y$). $x_{i:j}$ stands for the path of $x$ from time $i$ to time $j,$ i.e., $x_{i:j}:=(x_i,x_{i+1},\cdots ,x_j)$. $X[i,j]$ stands for the $(i,j)$th element of the matrix $X$. ${\mathbb{P}}_{\upsilon }(·)$ denotes the PDF of the random variable $\upsilon$. ${\delta }_x(·)$ denotes the Dirac delta mass located at $x$. $\mathcal{N}(m,\Sigma )$ stands for the Gaussian PDF with mean $m,$ and covariance $\Sigma$. ${\mathcal{N}}_{+}(m,\Sigma )$ denotes the truncated Gaussian PDF with positive support. $\mathcal{G}(\alpha ,\beta )$ is the gamma distribution with shape parameter $\alpha$ and rate parameter $\beta$. $\mathcal{ST}(\mu ,r,\delta ,\nu )$ is the skew-$t$ PDF with location parameter $\mu ,$ spread parameter $\sqrt{r},$ shape parameter $\delta$ and degrees of freedom $\nu$. Other notations will be given if necessary.