Fractional order differentiation by integration: an application to fractional linear systems

Da-Yan Liu 1 Taous-Meriem Laleg-Kirati 1 Olivier Gibaru 2, 3
2 NON-A - Non-Asymptotic estimation for online systems
CRIStAL - Centre de Recherche en Informatique, Signal et Automatique de Lille (CRIStAL) - UMR 9189, Inria Lille - Nord Europe
Abstract : In this article, we propose a robust method to compute the output of a fractional linear system defined through a linear fractional differential equation (FDE) with time-varying coefficients, where the input can be noisy. We firstly introduce an estimator of the fractional derivative of an unknown signal, which is defined by an integral formula obtained by calculating the fractional derivative of a truncated Jacobi polynomial series expansion. We then approximate the FDE by applying to each fractional derivative this formal algebraic integral estimator. Consequently, the fractional derivatives of the solution are applied on the used Jacobi polynomials and then we need to identify the unknown coefficients of the truncated series expansion of the solution. Modulating functions method is used to estimate these coefficients by solving a linear system issued from the approximated FDE and some initial conditions. A numerical result is given to confirm the reliability of the proposed method.
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Da-Yan Liu, Taous-Meriem Laleg-Kirati, Olivier Gibaru. Fractional order differentiation by integration: an application to fractional linear systems. 6th IFAC Workshop on Fractional Differentiation and its Applications, Feb 2013, Grenoble, France. ⟨hal-00759819⟩

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