On Set-Valued Kalman Filtering and Its Application to Event-Based State Estimation

Motivated by challenges in state estimation with event-based measurement updates, the properties of the exact and approximate set-valued Kalman filters with multiple sensor measurements for linear time-invariant systems are investigated in this work. First, we show that the exact and the proposed approximate set-valued filters are independent of the fusion sequence at each time instant. Second, the boundedness of the size of the set of estimation means is proved for the exact set-valued filter. For the approximate set-valued filter, if the closed-loop matrix is contractive, then the set of estimation means has a bounded size asymptotically; otherwise a nonsingular linear transform is constructed such that the size of the set of estimation means for the transformed states is asymptotically bounded. Third, the effect of set-valued measurements on the size of the set of estimation means is analyzed and conditions for performance improvement in terms of smaller size of the set of estimation means are proposed. Finally, the results are applied to event-based estimation, which allow the event-triggering conditions to be designed by considering requirements on performance and communication rates. The efficiency of the proposed results are illustrated by simulation examples and comparison with the approximate event-based MMSE estimator and the Kalman filter with intermittent observations.