Robust Recursive Kalman–Filtering

We consider robust recursive filtering in the case of a linear, finite-dimensional and timediscrete state-space model with Euclidean state space. Insisting on recursivity for computational reasons, we come up with a new procedure, the rLS-filter, using a Huberized correction-step in the Kalman-filter recursions. Simulation results for ideal and contaminated data indicate that this procedure achieves robustness with respect to AO-contamination, still behaving well in the ideal model compared to the classically optimal procedure, the Kalman-filter. To attack the properties of this procedure theoretically, we consider the state-space model in innovation form. In this reduced setup, it is possible to derive optimal robust filters under SO-contamination—both in a “Lemma 5” approach—c. f. [3]—and in a minimax approach, the latter generalizing a result of [1]. As in the location case, both solutions coincide, and yield the rLS-filter, provided all inputs from the past are Gaussian. However, treated by the rLS-filter, normality of the past is actually lost. But, extending the SO-contamination neighborhood a little, the minimax and “Lemma 5”solution of the original SO-neighborhood remain valid, and we are able to show [numerically] that the process of filters/predictions generated by the rLS-filter stays in this extended [e]SO-neighborhood about some fictive Gaussian ideal process, which we base on the second moments of the classical Kalman-filter. We thus obtain the first robust optimality result for recursive procedures referring to distributional neighborhoods about the ideal state-space model.