Robust Kalman Filter Based on a Generalized Maximum-Likelihood-Type Estimator

Abstract

A new robust Kalman filter is proposed that detects and bounds the influence of outliers in a discrete linear system, including those generated by thick-tailed noise distributions such as impulsive noise. Besides outliers induced in the process and observation noises, we consider in this paper a new type called structural outliers. For a filter to be able to counter the effect of these outliers, observation redundancy in the system is necessary. We have therefore developed a robust filter in a batch-mode regression form to process the observations and predictions together, making it very effective in suppressing multiple outliers. A key step in this filter is a new prewhitening method that incorporates a robust multivariate estimator of location and covariance. The other main step is the use of a generalized maximum likelihood-type (GM) estimator based on Schweppe's proposal and the Huber function, which has a high statistical efficiency at the Gaussian distribution and a positive breakdown point in regression. The latter is defined as the largest fraction of contamination for which the estimator yields a finite maximum bias under contamination. This GM-estimator enables our filter to bound the influence of residual and position, where the former measures the effects of observation and innovation outliers and the latter assesses that of structural outliers. The estimator is solved via the iteratively reweighted least squares (IRLS) algorithm, in which the residuals are standardized utilizing robust weights and scale estimates. Finally, the state estimation error covariance matrix of the proposed GM-Kalman filter is derived from its influence function. Simulation results revealed that our filter compares favorably with the H¿-filter in the presence of outliers.