Optimal State Estimation For Lti Systems With Imperfect Observations

State estimation under imperfect observations is a highly realistic and practically significant problem. These imperfections often manifest themselves in various forms, such as partial/total packet losses, and multiplicative degradation in sensor performance, making estimator design for such systems challenging. A method to find the minimum mean squared error estimate for such systems is presented in this paper, based on an equivalent Markovian Jump Linear System (M-JLS) modeling of the system under consideration. The estimation problem for the M-JLS is then solved using a projection based approach, and the corresponding modified Riccati equation is derived. An illustrative example is then presented for sensor malfunctions in a vehicle-to-vehicle communication channel for autonomous cars operating in a single lane.