Distribution-Free Mode-Estimators for a Class of Discrete-Time Jump-Linear Systems

This paper is concerned with the development of recursive distribution-free mode-estimators for a class of discrete-time jump-linear systems. The cornerstone of the proposed filters consists of an algebraic manipulation of the dynamics equation of the continuous state. This equation turns out to be linear with respect to the mode vector, and, under the assumption of perfect state information, provides a linear observation equation for the mode. Appending this equation to the known linear dynamics equation of the mode yields a linear non-Gaussian state-space model. A first mode-estimator is then derived using standard Linear Filtering results. A second filter is developed as an application of a general discrete-time filter, which approximates the continuous-time optimal non-linear filter (the conditional mean estimator for continuous time) for small sample times. The second filter is preferred from a performance point of view. Model order reduction is applied in order to avoid singularity issues in the filters implementations. The second filter is envisioned as a useful tool in the analysis and design of dual controllers for this type of hybrid systems.