Adaptable Recursive Update Filter

Abstract

T HE Kalman filter [1,2] is an optimal linear estimator for linear dynamic systems subject to linear measurements. The extended Kalman filter [3] (EKF) is a nonlinear approximation of the Kalman filter, which assumes small estimation errors and approximates them to first order to calculate their covariance matrix. The EKF is a linear estimator that relies on the additional assumption that the first-order approximation is valid. Nonlinear filters with a polynomial update up to arbitrary order are known [4]; knowledge of moments of the conditional estimation error distribution higher than the second are needed for these updates. Techniques exist to overcome some of the limitations of the EKF linearization assumption. The Gaussian second-order filter [5] takes into account second-order terms assuming the prior error distribution is Gaussian. The iterated extended Kalman filter (IEKF) [3] recursively improves the center of theTaylor series expansion for a better linearization. The unscented Kalman filter [6] is able to retain higher-order terms of the Taylor series expansion. Underweighting [7] is an ad hoc technique to compensate for the second-order effects without actually computing them. The recursive update filter (RUF) [8] applies the update gradually and relinearizes at each recursion, hence avoiding linearization problems and providing a consistent covariance. The number of recursions of the RUF is a user-defined parameter that needs to be selected by addressing two conflicting design objectives. On the one hand, the number of iterations should be chosen high to improve performance; themore recursions, themore often the algorithm relinearizes and the better the nonlinearity of the measurement is followed. On the other hand, whenever computational time is of concern, it is desirable to reduce the number of recursions. In general, the higher the degree of nonlinearity of themeasurement, the more recursions are needed. A good indicator of the performance of the algorithm is the postupdate residual (actual measurement minus estimated measurement computed with the updated state). The residual should be consistent with its predicted covariance. Discrepancies between the two indicate the nonlinear effects are of concern and more iterations are needed. The estimate x provided by the IEKF is equivalent to using a Gauss–Newton method [9] to minimize the following nonlinear least-squares problem: