# Fractional Order Differentiation by Integration and Error Analysis in Noisy Environment

Handle URI:
http://hdl.handle.net/10754/622551
Title:
Fractional Order Differentiation by Integration and Error Analysis in Noisy Environment
Authors:
Liu, Dayan; Gibaru, Olivier; Perruquetti, Wilfrid; Laleg-Kirati, Taous-Meriem ( 0000-0001-5944-0121 )
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 propose to extend this method from the integer order to the fractional order to estimate the fractional order derivatives of noisy signals. Firstly, two 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 for both continuous-time and discrete-time models in on-line or off-line applications. Secondly, some error bounds are provided for the corresponding estimation errors. These bounds allow to study the design parameters' influence. The noise error contribution due to a large class of stochastic processes is studied in discrete case. The latter shows that the differentiator based on the Caputo fractional order derivative can cope with a class of noises, whose mean value and variance functions are polynomial time-varying. Thanks to the design parameters analysis, the proposed fractional order differentiators are significantly improved by admitting a time-delay. Thirdly, in order to reduce the calculation time for on-line applications, a recursive algorithm is proposed. Finally, the proposed differentiator based on the Riemann-Liouville fractional order derivative is used to estimate the state of a fractional order system and numerical simulations illustrate the accuracy and the robustness with respect to corrupting noises.
KAUST Department:
Computer, Electrical and Mathematical Sciences and Engineering (CEMSE) Division
Citation:
Liu D-Y, Gibaru O, Perruquetti W, Laleg-Kirati T-M (2015) Fractional Order Differentiation by Integration and Error Analysis in Noisy Environment. IEEE Transactions on Automatic Control 60: 2945–2960. Available: http://dx.doi.org/10.1109/TAC.2015.2417852.
Publisher:
Institute of Electrical and Electronics Engineers (IEEE)
Journal:
IEEE Transactions on Automatic Control
Issue Date:
31-Mar-2015
DOI:
10.1109/TAC.2015.2417852
Type:
Article
ISSN:
0018-9286; 1558-2523
http://ieeexplore.ieee.org/document/7072464
Appears in Collections:
Articles; Computer, Electrical and Mathematical Sciences and Engineering (CEMSE) Division

DC FieldValue Language
dc.contributor.authorLiu, Dayanen
dc.contributor.authorGibaru, Olivieren
dc.contributor.authorPerruquetti, Wilfriden
dc.contributor.authorLaleg-Kirati, Taous-Meriemen
dc.date.accessioned2017-01-02T09:55:29Z-
dc.date.available2017-01-02T09:55:29Z-
dc.date.issued2015-03-31en
dc.identifier.citationLiu D-Y, Gibaru O, Perruquetti W, Laleg-Kirati T-M (2015) Fractional Order Differentiation by Integration and Error Analysis in Noisy Environment. IEEE Transactions on Automatic Control 60: 2945–2960. Available: http://dx.doi.org/10.1109/TAC.2015.2417852.en
dc.identifier.issn0018-9286en
dc.identifier.issn1558-2523en
dc.identifier.doi10.1109/TAC.2015.2417852en
dc.identifier.urihttp://hdl.handle.net/10754/622551-
dc.description.abstractThe integer order differentiation by integration method based on the Jacobi orthogonal polynomials for noisy signals was originally introduced by Mboup, Join and Fliess. We propose to extend this method from the integer order to the fractional order to estimate the fractional order derivatives of noisy signals. Firstly, two 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 for both continuous-time and discrete-time models in on-line or off-line applications. Secondly, some error bounds are provided for the corresponding estimation errors. These bounds allow to study the design parameters' influence. The noise error contribution due to a large class of stochastic processes is studied in discrete case. The latter shows that the differentiator based on the Caputo fractional order derivative can cope with a class of noises, whose mean value and variance functions are polynomial time-varying. Thanks to the design parameters analysis, the proposed fractional order differentiators are significantly improved by admitting a time-delay. Thirdly, in order to reduce the calculation time for on-line applications, a recursive algorithm is proposed. Finally, the proposed differentiator based on the Riemann-Liouville fractional order derivative is used to estimate the state of a fractional order system and numerical simulations illustrate the accuracy and the robustness with respect to corrupting noises.en
dc.publisherInstitute of Electrical and Electronics Engineers (IEEE)en
dc.relation.urlhttp://ieeexplore.ieee.org/document/7072464en
dc.subject, Jacobi orthogonal polynomial filteren
dc.subjectDigital fractional order differentiatoren
dc.subjectError analysisen
dc.subjectError analysisen
dc.subjectRecursive algorithmen
dc.subjectTime-delayen
dc.titleFractional Order Differentiation by Integration and Error Analysis in Noisy Environmenten
dc.typeArticleen
dc.contributor.departmentComputer, Electrical and Mathematical Sciences and Engineering (CEMSE) Divisionen
dc.identifier.journalIEEE Transactions on Automatic Controlen
dc.contributor.institutionINSA Centre Val de Loire, Université d'Orléans, Bourges Cedex, 18022, Franceen
dc.contributor.institutionLSIS (CNRS, UMR 7296), Arts et Métiers Paris Tech, Lille Cedex, 59046, Franceen
dc.contributor.institutionÉquipe Projet Non-A, INRIA Lille- Nord Europe, Parc Scientifique de la Haute Borne, Villeneuve d'Ascq, 59650, Franceen
dc.contributor.institutionCRIStAL (CNRS, UMR 9189), École Centrale de Lille, Villeneuve d'Ascq, 59650, Franceen
kaust.authorLiu, Dayanen
kaust.authorLaleg-Kirati, Taous-Meriemen