https://orcid.org/0000-0002-4052-3493
This work develops a novel method to carry out the discrete Kalman filter algorithm for two broad types of linear constant-coefficient Caputo fractional differential equations (FDEs), namely multi-term FDEs and multi-order system of FDEs. Firstly, the non-trivial closed-form Laplace-domain solutions are obtained for the FDEs under consideration. Taking the inverse Laplace transform then yields the time-domain representation of solutions involving the fundamental matrix Φ(t) and convolution integral of the Φ(t)-dependent function Ψ(t) with the known forcing f(t). The matrix Φ(t) and thereafter the function Ψ(t) are computed numerically. Subsequently, the time-domain discretization of these solutions is achieved through a systematic procedure, which makes them readily amenable to the discrete Kalman filter algorithm. The non-local nature of fractional-order operators makes such discretization challenging. It is worth mentioning that the time-domain representation of solutions plays the central role in obtaining the discretization. Finally, the robustness of the proposed strategy is tested using a variety of examples. Despite huge process and measurement noise, the results indicate a very good match between the true and the Kalman filter estimated states. A salient feature of the present work is that the classical discrete Kalman filter algorithm is implemented without any modifications, which guarantees optimal performance.
This work develops a novel method to carry out the discrete Kalman filter algorithm for two broad types of linear constant-coefficient Caputo fractional differential equations (FDEs), namely multi-term FDEs and multi-order system of FDEs. Firstly, the non-trivial closed-form Laplace-domain solutions are obtained for the FDEs under consideration. Taking the inverse Laplace transform then yields the time-domain representation of solutions involving the fundamental matrix Φ(t) and convolution integral of the Φ(t)-dependent function Ψ(t) with the known forcing f(t). The matrix Φ(t) and thereafter the function Ψ(t) are computed numerically. Subsequently, the time-domain discretization of these solutions is achieved through a systematic procedure, which makes them readily amenable to the discrete Kalman filter algorithm. The non-local nature of fractional-order operators makes such discretization challenging. It is worth mentioning that the time-domain representation of solutions plays the central role in obtaining the discretization. Finally, the robustness of the proposed strategy is tested using a variety of examples. Despite huge process and measurement noise, the results indicate a very good match between the true and the Kalman filter estimated states. A salient feature of the present work is that the classical discrete Kalman filter algorithm is implemented without any modifications, which guarantees optimal performance.
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 and the convolution integral of the -dependent function with the known forcing . Moreover, the components of are determined to depend on those of 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.
