A maximum-likelihood Kalman filter for switching discrete-time linear systems

State estimation is addressed for a class of discrete-time systems that may switch among different modes taken from a finite set. The system and measurement equations of each mode are assumed to be linear and perfectly known, but the current mode of the system is unknown. Moreover, additive, independent, normally distributed noises are assumed to affect the dynamics and the measurements. First, relying on a well-established notion of mode observability developed “ad hoc” for switching systems, an approach to system mode estimation based on a maximum-likelihood criterion is proposed. Second, such a mode estimator is embedded in a Kalman filtering framework to estimate the continuous state. Under the unique assumption of mode observability, stability properties in terms of boundedness of the mean square estimation error are proved for the resulting filter. Simulation results showing the effectiveness of the proposed filter are reported.