Bayesian state estimation on finite horizons: The case of linear state-space model

The finite impulse response (FIR) filter and infinite impulse response filter including the Kalman filter (KF) are generally considered as two different types of state estimation methods. In this paper, the sequential Bayesian philosophy is extended to a filter design using a fixed amount of most recent measurements, with the aim of exploiting the FIR structure and unifying some basic FIR filters with the KF. Specifically, the conditional mean and covariance of the posterior probability density functions are first derived to show the FIR counterpart of the KF. To remove the dependence on initial states, the corresponding likelihood is further maximized and realized iteratively. It shows that the maximum likelihood modification unifies the existing unbiased FIR filters by tuning a weighting matrix. Moreover, it converges to the Kalman estimate with the increase of horizon length, and can thus be considered as a link between the FIR filtering and the KF. Several important properties including stability and robustness against errors of noise statistics are illustrated. Finally, a moving target tracking example and an experiment with a three degrees-of-freedom helicopter system are introduced to demonstrate effectiveness.