Robust Kalman filter for systems subject to parametric uncertainties

Abstract

State estimation plays a fundamental role in control systems that rely on the knowledge of the underlying system state, especially when it is not readily available. The Kalman filter is among the most popular techniques in this matter. Nonetheless, one of its well-known shortcomings is the assumption that an exact system model is available. The difficulty in fulfilling this premise justifies the demand for estimation strategies that limit the effect of model uncertainties. In this paper, we propose a robust Kalman filter for uncertain linear discrete-time systems. We assume a general setting in which all matrices of both the system and measurement models are subject to norm-bounded parametric uncertainties. By adopting a purely deterministic viewpoint and applying the penalty function method, we formulate a robust regularized least-squares estimation problem. We further analyze how the penalty parameter influences the estimation performance. The resulting filter is presented both in a symmetric matrix arrangement and as explicit algebraic expressions in a Kalman-like structure that is suitable for online applications. Under reasonable conditions, we show that the steady-state filter is stable and, for quadratically stable systems, guarantees a bounded error variance. With a numerical example, we evaluate the proposed filter and compare its performance with other robust filtering approaches.