A superposition principle for the Kalman filter

In this paper, decomposition formulas for the discrete-time Kalman filter are presented. Both the state estimate and the error covariance matrix are expressed as the sum of two terms, the first being the estimate corresponding to zero initial conditions, and the second being an explicit function of the initial values x^0 and P0P0. The representation is updated in time by well-behaved finite complexity matrix recursions, and allows for a direct evaluation of the estimates for variable initial conditions. Applications to stochastic hybrid filtering are discussed.