Optimal FIR Filtering of State-Space Models in non-Gaussian Environment with Uncertainties

In industrial applications, optimal estimators of system state face a necessity to work in non-Gaussian environment in the presence of uncertainties. This lecture introduces readers to the recently developed p-shift iterative Kalman-like finite impulse response (FIR) unbiased estimation (UE) algorithm intended for filtering (p = 0), prediction (p > 0), and smoothing (p > 0) under such conditions of linear discrete time-varying state-space models. The algorithm was designed with no requirements for noise and initial conditions and thus has strong engineering features. A solution is first found in a batch form and then represented in the computationally efficient iterative Kalman-like one with the following advantages peculiar to FIR structures: guarantied bounded input/bounded output (BIBO) stability, better robustness against temporary model uncertainties and round-off errors, and low sensitivity to noise and initial conditions. It is shown than the estimator proposed overperforms the Kalman one when the noise covariances and initial conditions are not known exactly, if noise is not white sequence, and when both the system and measurement noise components need to be filtered out. Otherwise, the estimators produce similar errors. Extensive investigations of FIR UE have been carried out for the standard Kalman filter regarding different models. Examples of applications have been taken from signal and image processing, clock synchronization, and control. All the way, we lay stress on the trade-off with the Kalman filter in the Gaussian and non-Gaussian environments allowing for temporary model and measurement uncertainties, as well as outliers.