Estimation of maneuvering target in the presence of non-Gaussian noise: A coordinated turn case study

Highlights

• The novel maximum-correntropy-criterion-based extended Kalman filters are devised for treating continuous-time nonlinear stochastic models with non-Gaussian noise.

• The radar tracking scenarios, where an aircraft executes a coordinated turn, are set up with impulsive and mixed-Gaussian noises.

• The maximum-correntropy-criterion-based extended Kalman filters are examined numerically and compared to the continuous-discrete extended, cubature and unscented Kalman filters.

• The contemporary cubature- and unscented-type Kalman filters outperform all their competitors in the accuracy of state estimation in the non-Gaussian target tracking case studies.

Abstract

This paper explores performance of various methods for state estimation of radar tracking models. A coordinated turn case study of maneuvering target in the presence of non-Gaussian noise is of particular interest. We aim at evaluating the estimation potential of recently presented filters grounded in the Maximum Correntropy Criterion (MCC). Various investigations confirm the outstanding performance of such filters for treating stochastic systems disturbed with impulsive (shot) and mixed-Gaussian noises. However, those filters are intended for linear models, and the success of the MCC-based state estimation in a nonlinear continuous-time stochastic environment, which often underlies radar tracking modeling, is still debatable. First, we extend the MCC-based filters, which are designed presently for linear discrete-time stochastic models, to nonlinear continuous-discrete systems. We devise the conventional (non-square-root) filtering and its square-root version as well. Second, we fulfil a comprehensive examination of these new methods in severe conditions of tackling a seven-dimensional radar tracking problem, where an aircraft executes a coordinated turn, in the presence of both impulsive (shot) and mixed-Gaussian noises. In addition, the novel MCC-based filters are compared to various contemporary extended, cubature and unscented Kalman-like state estimators.

Introduction

The conventional option in estimating hidden states of many stochastic models is the celebrated Kalman Filter (KF) designed in [1]. The latter method has been used successfully in fault detection, target tracking and many other realms of applied science and engineering for decades and resulted in a great variety of fast, numerically stable and other algorithms presented in [2], [3], [4], [5], [6] and so on. The cited literature proves that the KF will give the optimal minimum-variance solution of a linear discrete-time state estimation task if its process and measurement noises are white, independent and Gaussian distributed.

Despite extensive utilizations of linear discrete-time stochastic models, with the associated KF, in many practical studies, the contemporary trend in mathematical modeling and simulation focuses on using advanced nonlinear continuous-discrete stochastic systems of the form

$$\begin{array}{ccc}\hfill dX\left(t\right)& =& F\left(X\right(t\left)\right)dt+G\left(t\right)dW\left(t\right),t>0,\hfill \end{array}$$

$$\begin{array}{ccc}\hfill Z_k& =& h\left(X_k\right)+V_k,k\ge 1.\hfill \end{array}$$

The continuous-time process model (1) is a standard It$\widehat{\mathrm{o}}$-type Stochastic Differential Equation (SDE), which simulates the model’s dynamic behavior, where $X\left(t\right)\in {\mathbb{R}}^n$ is the n-dimensional vector of system’s state at time t, $F:{\mathbb{R}}^n\to {\mathbb{R}}^n$ is a sufficiently smooth drift function, G(t) is a time-variant matrix of size n × q and {W(t), t > 0} is a multidimensional white stochastic process with zero mean and time-variant covariance matrix Q(t) > 0 of size q × q. The initial state X(0) of SDE (1) can also be a random variable with mean ${\overline{X}}_0$ and covariance Π₀. Besides, the measurement model (2) is of a discrete-time fashion with k standing for a discrete time index (i.e. X_k implies X(t_k)), $Z_k\in {\mathbb{R}}^m$ denoting the measurement information available at time t_k, $h:{\mathbb{R}}^n\to {\mathbb{R}}^m$ referring to a differentiable function and with the measurement noise V_k being a zero-mean white-noise sequence with covariance matrices R_k > 0. In what follows, the measurement samples are supposed to arrive equidistantly and with a sampling rate $\delta =t_k-t_{k-1}$. This time interval δ is referred to as the sampling period (or waiting time) in filtering theory. We point out that the sampling may be irregular in practice, and all our variable-stepsize Kalman filters elaborated in Sections 3 and 4 are applicable to such sort of state estimation tasks as these are. At the same time, the equidistant fashion of the sampling accepted here is obligatory in the state-of-the-art fixed-stepsize cubature and unscented Kalman filters presented in [7], [8], which are also under examination and discussed in Section 4 of our paper. In addition, all realizations of the noises W(t), V_k and the initial state X(0) are taken from mutually independent distributions. Further arguments for using stochastic systems of the form (1), (2) in practical state estimation studies are outlined in [9].

Evidently, the original Kalman’s theory developed for the optimal minimum-variance state estimation in linear discrete-time Gaussian models may not be applied to the more complicated stochastic state-space system (1), (2) because of its nonlinear and continuous-time fashion. Therefore, it must be amended in some way for covering the latter problems. The most common method of coping with them consists of linearization and discretization of given stochastic systems (1), (2) with further application of the standard KF technique, whose various versions are presented in [2], [3], [4], [5], [6] in detail. The outlined state estimation approach is referred to as the Extended Kalman Filter (EKF) in [2], [3], [5], [6], [10]. It is the simplest, suboptimal but successful state estimator, which has been in use in applied science and engineering for decades. At the same time, the EKF is often shown to be ineffective in dealing with non-Gaussian stochastic systems, i.e. when the process and/or measurement models are corrupted by heavy-tailed distributed random variables. Such state estimation tasks may also include impulsive (shot) or mixed-Gaussian noises.

In practice, the growing attention is paid to treating stochastic models disturbed with non-Gaussian noises. Various approaches have succeeded in state estimation of non-Gaussian systems. First, these include special filters developed for stochastic systems corrupted by heavy-tailed distributed (or t-distributed) random variables, as those in [11], [12], [13], [14]. Second, another approach to handling non-Gaussian noise in state estimation tasks is to approximate it with a finite sum of Gaussian noises. Then, the Gaussian sum filter based on a bank of KF methods, which stem from this multi-model Gaussian environment, is to be applied. The latter is addressed, for instance, in [15], [16], [17], [18], [19]. Third, promising applications of Monte Carlo sampling methods are also reported in estimating stochastic systems with non-Gaussian noises. The Monte Carlo sampling enables a proper approximation of general probability distributions when the set of random samples is sufficiently large. This idea underlies known Particle Filters and Ensemble Kalman methods, which have proven their high performance in non-Gaussian state estimation tasks [19], [20], [21], [22], [23], [24]. However, randomly selected samples utilized in Monte Carlo-type filters possess some shortcomings, which include their high computational burden, degeneracy problem and sample impoverishment [24]. So, the latter sort of methods was extended with the unscented and cubature Kalman filters, where the probability distribution is approximated with a deterministically selected limited number of sampling nodes, which reduce considerably the computational effort of such state estimations, as explained in [7], [24], [25], [26], [27], [28]. Fourth, an interesting approach to state estimation in non-Gaussian stochastic systems can also be grounded in the novel concept introduced in information theoretic learning and called correntropy in [29], [30], [31]. The correntropy is a similarity measure of two random variables, which possesses many useful properties discussed in the cited literature in detail. Implementing the Maximum Correntropy Criterion (MCC) in the cost function (known also as the performance index) of the filtering solution results in various MCC-based state estimators, which can be considered as methods of choice in signal processing and machine learning due to their robustness against large outliers (or impulsive noises) and, additionally, because of their computational effort, which is comparable to that of the classical KF. All this is elaborated extensively in [32], [33], [34], [35], [36], [37], [38], [39], [40], [41], [42], [43]. In particular, Izanloo et al. [41] show that their MCC-based filter outperforms many other counterparts including such an effective technique as the unscented Kalman filter in a target tracking scenario. However, the conclusion drawn on the filters’ performance in the cited paper relates to linear discrete-time filtering and its validity for nonlinear continuous-time methods is still debatable because nonlinear stochastic systems are more challenging for decent state estimations, as evidenced in [44].

In what follows, we explore this issue for the continuous-discrete MCC-based methods in detail. First, starting with a brief introduction to the MCC-based filtering in Section 2, we devise conventional (non-square-root) and square-root versions of such methods in Section 3. In Section 6, our continuous-discrete MCC-based filters are examined in severe conditions of tackling a seven-dimensional radar tracking case study, where an aircraft executes a coordinated turn. The latter scenario is used commonly for assessment of state estimation potential of various target tracking methods [7], [8], [45], [46], [47], [48], [49], [50] and elaborated in Section 5 at large. Following Izanloo et al. [41], both impulsive (shot) and mixed-Gaussian noises are considered there. In addition, the novel MCC-based filters are compared to such effective state estimators as the conventional and square-root Accurate Continuous-Discrete Extended Kalman Filters, the conventional and square-root Accurate Continuous-Discrete Cubature and Unscented Kalman Filters, including their mixed-type versions, designed recently in [8], [46], [47], [48], [49], [50], [51]. All these methods are outlined briefly in Section 4. Finally, our concluding remarks and future plans are summarized in Section 7.