Fractional order differentiation by integration and error analysis in noisy environment: Part 1 continuous 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 : The integer order differentiation by integration method based on the Jacobi orthogonal polynomials for noisy signals was originally introduced by Mboup, Join and Fliess. We are going to generalize this method from the integer order to the fractional order so as to estimate the fractional order derivatives of noisy signals. For sake of clarity, this work has been divided into two parts. The first part presented in this paper focuses on the continuous case while the second part that has been presented in another paper deals with the discrete case with on-line applications. In this paper, two kinds of fractional order differentiators are deduced from the Jacobi orthogonal polynomial filter, using the Riemann-Liouville and the Caputo fractional order derivative definitions respectively. Exact and simple formulae for these differentiators are given by integral expressions. Hence, they can be used both for continuous-time and discrete-time models in on-line or off line applications. Then, some error bounds are provided for the corresponding estimation errors in continuous case. These bounds will be used to study the design parameters' influence on the obtained fractional order differentiators in the second part. Finally, numerical simulations are given to show the accuracy and the robustness with respect to corrupting noises of the proposed differentiators in off-line applications. The properties of our differentiators in on-line applications will be shown in the second part.
Type de document :
Pré-publication, Document de travail
2013
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  • HAL Id : hal-00779176, version 1

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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 1 continuous case. 2013. 〈hal-00779176〉

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