Robust locally optimal filters: Kalman and Bayesian estimation theory

Abstract

This paper presents a unified approach to local optimality, robustness, and Bayesian estimation theory concepts in deriving Kalman filtering equations in the case of non-Gaussian observation noise. In most of these derivations, an approximation criterion proposed by Masreliez is used: the prediction error density is approximately Gaussian. It is shown that locally optimal Kalman filters are equivalent to one-step MAP iterative procedures. The introduction of nonlinear iterative techniques adds a new dimension to the analysis of recursive non-Gaussian estimation. Moreover, to estimate the parameters of the Kalman model, a direct solution of MAP equations is obtained through an m-interval piecewise linear approximation (MIPLA) of the locally optimal nonlinearity (score function). Robustness is achieved by proper modification and approximation of the score function. Finally, the performance of the filters in question are tested under intensive Monte Carlo simulations. Satisfactory results are obtained even in the case of extremely impulsive observation noise (Cauchy contaminated Gaussian).