Kalman Filtering for Two Broad Classes of Caputo Fractional-Order Differential Equations

Abstract This work aims to develop novel strategies to carry out Kalman filtering for two broad classes of linear constant-coefficient fractional differential equations (FDEs) involving Caputo fractional derivatives. In the first part of this work, the closed-form solutions for such FDEs are obtained in the Laplace domain, followed by their time-domain representation involving the fundamental matrix Φ(t) and the convolution integral of the Φ(t)-dependent function Ψ(t) with the known forcing f(t). Moreover, the components of Ψ(t) are determined to depend on those of Φ(t) through problem-specific fractional-order integral equations. The specific structure of the FDEs being studied presents a notable challenge in deriving Laplace domain solutions. Nevertheless, the subsequent time-domain representation of these solutions makes them amenable to the well-known Kalman filtering. In the second part, we achieved a systematic time-domain discretization of the solutions for these FDEs and applied the discrete Kalman filter to the resulting set of discretized equations. The efficacy of this work is evident from the successful application of the Kalman filter to a variety of FDEs. Consistently excellent results have been obtained in all the cases.