Kalman-like Observer for Hybrid Systems with Linear Maps and Known Jump Times

We propose a hybrid Kalman-like observer for general hybrid systems with linear (time-varying) dynamics and output maps, where the solutions' jump times are exactly known. After defining a hybrid observability Gramian and the corresponding hybrid uniform complete observability, we show that the estimate provided by this observer converges asymptotically to the system solution if this observability holds together with some boundedness and invertibility conditions along the considered system solution. Then, under additional uniformity and strictness of the forgetting factors, we show exponential stability of the estimation error with an arbitrarily fast rate. The robust stability of this error against input disturbances and measurement noise is also studied. The results are illustrated on several benchmark examples, including switched systems, hybrid systems with discontinuous solutions, and continuous- time systems with multi-rate sporadic outputs.