Reduced-order filtering of jump Markov systems with noise-free measurements

In continuous-time Kalman filtering for jump Markov systems, it is required that the measurement noise covariance be nonsingular. In this work, the case of noise-free measurements is considered and it is proposed that a reduced-order filter be used to overcome this singularity problem. This filter is optimal in the minimum variance sense and is of dimension (n−p) where n and p are the state and measurement vector dimensions, respectively. After the optimal filter equations are derived for the finite-time case, we focus on the infinite-time case and characterize the set of all assignable estimation error covariances and parametrize the set of all estimator gains. The conditions for the existence of the optimal steady-state filter are obtained in terms of the system theoretic properties of the original signal model.