Optimal estimation for continous system with jump process

The minimum variance estimator algorithm is derived for a class of linear continuous systems modulated by a multivalued jump Markov process. The approach adopted in this paper is as follows. First, we express the jump Markov process in terms of its initial value, the jump times and the values taken by the jump process after the jump, and then we apply the Bayes´ rule and the general likelihood-ratio formula to obtain the a posteriori probability distribution of the jump process. The minimum variance estimate is given in terms of the a posteriori probability distribution of the jump process and the Kalman-filter estimates corresponding to the admissible values of the jump process. Simulation studies are also carried out to illustrate the behavior of the optimal estimator presented here.