M-estimator-based robust Kalman filter for systems with process modeling errors and rank deficient measurement models

Abstract

The aim of this paper was to develop a robust version of the Kalman filter (KF) to address process modeling errors in linear system with rank deficient measurement models. An important infinitesimal robustness metric, called the influence function, which is widely used in location, scale, and regression models, is introduced into KF. After recasting KF’s update as an artificial linear regression, the influence function of KF is derived in detail. The lack of robustness of KF is clearly demonstrated by the nonboundedness of its influence function. Huber estimator, the most important member of the robust M-estimator family, is introduced into the recast linear regression. The iteration and initialization to iteratively solve the M-estimator are emphasized to address the specific case with process modeling errors and rank deficient measurement models. The iterative reweighted least-squares method together with a covariance construction method is preferred to the standard and simplified Newton’s method and the asymptotic covariance. A new initial value constructed through correcting the a priori estimate to accord with the actual measurement exactly is proposed using the Moore–Penrose pseudoinverse method. A simple but illustrative example is simulated to check the feasibility of the proposed method.