Kalman filter with both adaptivity and robustness

Adaptive and robust methods are two opposite strategies to be adopted in the Kalman filter when the difference between the predictive observation and the actual observation, i.e. the innovation vector is abnormally large. The actual observation is more weighted in the former one, and is less weighted in the later one. This article addresses the subject of making a choice between the adaptive and robust methods when abnormal innovation occurs. An adaptive method with fading memory and a robust method with enhancing memory is proposed in the Kalman filter based on the chi-square distribution of the square of the Mahalanobis distance of the innovation. A heuristic method of recursively choosing among the adaptive, the robust, and the standard Kalman filter approaches in the occurrence of abnormal innovations is proposed through incorporating the observations at the next instance. The proposed method is both adaptive and robust, i.e. having the ability of strongly tracking the variation of the state and being insensitive to gross errors in observation. Numerical simulations of a simple illustrating example validate the efficacy of the proposed method.