Kalman filtering for discrete‐time linear fractional‐order singular systems

Abstract

This study considers the optimal linear estimation problem for the discrete-time stochastic fractional-order system in its more general formulation. The system is allowed to be in singular form, rectangular, with dynamical and measurement noises correlated. First, some new conditions for the solvability, regularity and causality to discrete-time linear stochastic fractional-order singular (FOS) systems are given, and then, a new Kalman filter (KF) fractional singular KF (FSKF) is designed for such systems. This general form of filter is derived using deterministic arguments in a completely self-contained way besides the stochastic reasoning and covers the nominal singular and fractional KFs. Instead of the standard stochastic formulation, the filter recursions are obtained as a solution of a convenient organisation of the deterministic data-fitting estimate of an entire state trajectory given the measurements for both time-invariant and time-varying cases. To present the efficiency of the proposed algorithm, results of numerical simulations are presented.