Multiple Outliers Suppression Derivative-Free Filter Based on Unscented Transformation

Abstract

D UE to its familiarity and computational feasibility, the Kalman filter (KF) has found numerous applications in automatic control, navigation, and communications since its introduction [1–3]. Many nonlinear Kalman type filters (KTFs) have been introduced to extend the KF to nonlinear dynamic and measurement models by forming a Gaussian approximation to the posterior state distribution. Among these nonlinear KTFs, the unscented Kalman filter (UKF) is most celebrated due to its easy implementation, appropriate performance and computational feasibility [4]. However, it is well documented that when the state or measurement noises are contaminated by outliers, theKTF’s performance can severely degrade, because they rely on weighted least-squares (WLS) criteria which is susceptible to outliers [5]. To handle these outliers, several algorithms based on the concept of robust statistics have been proposed. By minimizing the worst-case estimation error averaged over all samples, theH1 basedKF can be used to treat process noises, measurement noises, and model uncertainties [6]. However, it breaks down in the presence of randomly occurring outliers since the design matrices of the H1 filter cannot accommodate well the outliers induced by the thick tails of a noise distribution [7]. Other approaches are robust to either state or measurement outliers and they can not cope with both types of outliers jointly [8–10]. Therefore, they may yield unreliable results when state andmeasurement outliers occur simultaneously. Converting the classical recursive approach into a batch-mode regression form and solving it iteratively, Gandhi proposed a generalized maximum likelihood (GM) type KF which is robust to both the state and measurement outliers [7]. Through auxiliary unknown variables that are jointly estimated along with the state based on the least-squares criterion regularized with the l1 norm, both the state and measurement outliers can also be handled in [11]. However, these methods of [7,11] are both limited to the linear case. Although Gandhi has extended his method to the nonlinear case making use of the extended Kalman filter (EKF) [12], the crude approximation and the cumbersome derivation and evaluation of Jacobian matrices in the EKF may degrade its performance. References [13–16] have combined the derivative-free filters with the M estimator mainly to handle the measurement outliers. Although the structure proposed in [13–16] can be used to handle the state outliers with many iterations, unfortunately, only one iteration is suggested in [13–16], which is not enough to suppress the state outliers. Moreover, in [13–16] the nonlinear measurement functions are also statistically linearized and haven’t been used directly. To the best knowledge of the authors, robust derivative-free filters without linear or statistically linear approximation that addresses both the state and measurement outliers haven’t been studied before. This Note will focus on the robust state estimation problem with outliers using the UKF framework. Based on the GM perspective on the KF, the quadratic cost function in the KF framework is modified by the robust cost function [9,13], to robustify the KF. In this respect, the robust cost function is virtually used to reformulate the predicted state covariance and the measurement noise covariance. Then the reformulated covariance is propagated through the UKF. To handle the state outliers, the measurement update of the UKF should be iterated, which is accomplished by a modification of the iterated UKF [17]. The rest of this Note is organized as follows. Section II presents a GM perspective on the KF and points out how the robustM estimate methodology can be embedded into the UKF framework without linear or statistically linear approximation. Section III is devoted to derive the robust derivative-free filter called the outlier robust unscented Kalman filter (ORUKF) algorithm. Some discussions of the proposed algorithm and comparisonswith existing algorithms are the subject of Sec. IV. Simulation results and comparisons are presented in Sec. V. Finally, conclusions are drawn in Sec. VI.