On the Effects of the Initial Condition in State Estimation for Discrete-Time Linear Systems

We consider the one-step prediction problem for discrete-time linear systems in correlated Gaussian white plant and observation noises, and non-Gaussian initial conditions. Explicit representations are obtained for the MMSE and LLSE (or Kalman) estimates square of their difference. These formulae are obtained with the help of the Girsanov transformation for Gaussian white noise sequences, and explicitly display the effects of the distribution of the initial condition. With the help of these formulae, we investigate the large-time asymptotics of et , the expected squared difference between the MMSE and LLSE estimates at time t. We characterize the limit of the error sequence {et, t = 1,2,...} and obtain some related rates of convergence. A complete large-time analysis is provided for the scalar case