A closed-form solution to the discrete-time Kalman filter and its applications

This paper presents a closed-form solution to the discrete-time Kalman filter and its applications. We first represent the Kalman filter in terms of model parameters without using the Riccati equation and requiring any artificial conditions such as invertibility of a system matrix and no system noises. Replacing the initial time with the fixed-lag time to achieve the finite memory with respect to inputs and outputs, and choosing the proper initial covariances on the recent finite horizon, we easily obtain the minimum variance finite memory filter and then propose its iterative computation algorithm. As another application, a closed-form solution to the difference Riccati equation on the finite horizon is utilized to obtain a stabilizing gain matrix of a Luenberger-type filter as in Ackermann’s formula.