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Fractional order differentiation by integration and error analysis in noisy environment: Part 2 discrete case

Da-Yan Liu 1 Olivier Gibaru 2, 3 Wilfrid Perruquetti 4, 2 Taous-Meriem Laleg-Kirati 1
2 NON-A - Non-Asymptotic estimation for online systems
Inria Lille - Nord Europe, CRIStAL - Centre de Recherche en Informatique, Signal et Automatique de Lille (CRIStAL) - UMR 9189
4 SyNeR - Systèmes Non Linéaires et à Retards
CRIStAL - Centre de Recherche en Informatique, Signal et Automatique de Lille (CRIStAL) - UMR 9189
Abstract : In the first part of this work, the differentiation by integration method has been generalized from the integer order to the fractional order so as to estimate the fractional order derivatives of noisy signals. The estimation errors for the proposed fractional order Jacobi differentiators have been studied in continuous case. In this paper, the focus is on the study of these differentiators in discrete case. Firstly, the noise error contribution due to a large class of stochastic processes is studied in discrete case. In particular, it is shown that the differentiator based on the Caputo fractional order derivative can cope with a class of noises, the mean value and variance functions of which are time-variant. Secondly, by using the obtained noise error bound and the error bound for the bias term error obtained in the first part, we analyze the design parameters' influence on the obtained fractional order differentiators. Thirdly, according to the knowledge of the design parameters' influence, the fractional order Jacobi differentiators are significantly improved by admitting a time-delay. In order to reduce the calculation time for on-line applications, a recursive algorithm is proposed. Finally, numerical simulations show their accuracy and robustness with respect to corrupting noises.
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https://hal.inria.fr/hal-00779182
Contributor : Dayan Liu <>
Submitted on : Monday, January 21, 2013 - 5:40:42 PM
Last modification on : Saturday, October 3, 2020 - 3:19:34 AM
Long-term archiving on: : Monday, April 22, 2013 - 3:57:11 AM

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Da-Yan Liu, Olivier Gibaru, Wilfrid Perruquetti, Taous-Meriem Laleg-Kirati. Fractional order differentiation by integration and error analysis in noisy environment: Part 2 discrete case. 2013. ⟨hal-00779182⟩

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