Nonlinear Regression Huber–Kalman Filtering and Fixed-Interval Smoothing

Abstract

T HIS Note develops an improved Huber-based Kalman filtering approach using a nonlinear regression method for problems with non-Gaussian uncertainties and outliers. The robust technique relies on Huber’s generalized maximum likelihood approach to estimation [1]. Specifically, Huber’s method is a combinedminimum l1 and l2 norm estimation technique, which exhibits robustness with respect to deviations from the commonly assumed Gaussian error probability density functions, for which the least-squares or minimum l2 norm technique exhibits a severe degradation in estimation accuracy [2]. The Huber-based estimates are robust in the sense that they minimize the maximum asymptotic estimation variance when applied to contaminated Gaussian densities. The Huber technique was originally developed as a generalization of maximum likelihood estimation, applied first to estimating the center of a probability distribution in [1] and generalized to multiple linear regression in [2–4]. The Kalman filter is a recursive minimum l2 norm technique and therefore exhibits sensitivity to deviations in the true underlying error probability distributions. For this reason, the Huber technique has been further extended to dynamic estimation problems. Boncelet and Dickinson [5] first proposed to solve the robust filtering problem by means of the Huber technique. The solution was obtained by recasting the dynamic filtering problem into the form of a linear regression problem. This reformulation was then solved using Huber’s robust technique. Several variants of the Huber-based Kalman (and extendedKalman) filters have been applied to awide range of problems, including underwater vehicle navigation [6], speech signal processing [7], power system state estimation [8], radar tracking [9], spacecraft rendezvous [10], spacecraft attitude estimation [11], aircraft navigation [12], spacecraft relative navigation [13–15], satellite orbit determination [16], and atmospheric data assimilation [17]. In each instance, the Huber-based approaches perform favorably in the presence of non-Gaussian distributions and outliers. It is also shown in [10] that the additional computational cost of the Huber-based approach is small. The Huber-based filtering approach was extended to the class of sigma point Kalman filters (SPKFs) in [18–20] using the divided difference filter framework. The solution was obtained by reformulating the SPKF measurement update into an equivalent linear regression problem, which was then solved using Huber’s robust technique. The same approach was taken in the framework of the unscentedKalman filter in [21]. Other implementations of theHuberbased SPKF method have been investigated in [22–26]. In this Note, an improved Huber–Kalman filter approach is proposed based on a nonlinear regression model. In this approach, the measurement nonlinearity is maintained, and the Huber-based filtering problem is solved using a Gauss–Newton approach. This modification is motivated by [27], in which the iterative extended Kalman filter (IEKF) is derived from the standpoint of nonlinear regression theory. The purpose of this Note is to extend the IEKF nonlinear regression model to use Huber’s generalized maximum likelihood approach to provide robustness to non-Gaussian errors and outliers. In doing so, the filter performance is improved for problems with nonlinear measurement relations. The nonlinear regression Huber–Kalman approach is then extended to the fixed-interval smoothing problem, wherein the state estimates from a forward pass through the filter are smoothed back in time to produce a best estimate of the state trajectory given all available measurement data. The solution to the Huber-based fixedinterval smoothing problem is obtained by recasting the optimal smoother into the form of a nonlinear regression problem [28], which is then solved using the generalized maximum likelihood method. The solution turns out to be a generalization of the well-known Rauch–Tung–Striebel fixed-interval smoother [29] for nonlinear and non-Gaussian systems. Note that only the fixed-interval smoothing problem is considered here; fixed-lag and fixed-point smoothing are not addressed in this Note. Throughout this Note, estimates of the state x at time k, given measurement data through timem are denoted by xkjm. Following the usual nomenclature, determining estimates for k > m is known as prediction, determining estimates for k m is known as filtering, and determining estimates for k < m is known as smoothing. The remainder of this Note is organized as follows. Section II describes theHuber-basedKalman filteringmethodology. Section II.A provides a review of the linear-regression-based Kalman filtering method, wherein the measurement equation is linearized and then the filter update is recast as a regression problem that is solved using Huber’s robust technique. Section II.B describes a new result in this Note, which is the nonlinear regression Huber filtering approach. The nonlinear regression filter approach is then extended to the fixedinterval smoothing problem in Sec. III. Section IV shows numerical simulation results for a radar tracking problem, and Sec. V provides concluding remarks.