2011 |
Aoun, M.; Amairi, M.; Najar, S.; Abdelkrim, M. N. 2011, (Cited by: 2). Résumé | Liens | BibTeX | Étiquettes: ACFDR, Approximation theory, CFDR, Diethelm’s approximation, Differentiation (calculus), Fractional derivatives, Fractional systems, Pole-zero distribution, simulation @conference{Aoun2011d,This paper proposes a new method for the simulation of the fractional systems. It deals with the approximation methods of the fractional derivative. It compare the approximation based on Grnwald with Caputo approximation. The efficiency of each approximation methods in termes of execution time and quadratic error is evaluated for different differential orders and stepsize. The best approximation method is used to develop an original simulation method to demonstrate its effectiveness. © 2011 IEEE. |
2007 |
Aoun, M.; Malti, R.; Levron, F.; Oustaloup, A. Synthesis of fractional Laguerre basis for system approximation Article de journal Dans: Automatica, vol. 43, no. 9, p. 1640 – 1648, 2007, (Cited by: 115; All Open Access, Green Open Access). Résumé | Liens | BibTeX | Étiquettes: Approximation theory, Asymptotic stability, Fractional differentiation, Identification (control systems), Integer programming, Laguerre function, Mathematical models, Orthonormal basis @article{Aoun20071640b,Fractional differentiation systems are characterized by the presence of non-exponential aperiodic multimodes. Although rational orthogonal bases can be used to model any L2 [0, ∞ [ system, they fail to quickly capture the aperiodic multimode behavior with a limited number of terms. Hence, fractional orthogonal bases are expected to better approximate fractional models with fewer parameters. Intuitive reasoning could lead to simply extending the differentiation order of existing bases from integer to any positive real number. However, classical Laguerre, and by extension Kautz and generalized orthogonal basis functions, are divergent as soon as their differentiation order is non-integer. In this paper, the first fractional orthogonal basis is synthesized, extrapolating the definition of Laguerre functions to any fractional order derivative. Completeness of the new basis is demonstrated. Hence, a new class of fixed denominator models is provided for fractional system approximation and identification. © 2007 Elsevier Ltd. All rights reserved. |
Aoun, M.; Malti, R.; Levron, F.; Oustaloup, A. Synthesis of fractional Laguerre basis for system approximation Article de journal Dans: Automatica, vol. 43, no. 9, p. 1640 – 1648, 2007, (Cited by: 115; All Open Access, Green Open Access). Résumé | Liens | BibTeX | Étiquettes: Approximation theory, Asymptotic stability, Fractional differentiation, Identification (control systems), Integer programming, Laguerre function, Mathematical models, Orthonormal basis @article{Aoun20071640,Fractional differentiation systems are characterized by the presence of non-exponential aperiodic multimodes. Although rational orthogonal bases can be used to model any L2 [0, ∞ [ system, they fail to quickly capture the aperiodic multimode behavior with a limited number of terms. Hence, fractional orthogonal bases are expected to better approximate fractional models with fewer parameters. Intuitive reasoning could lead to simply extending the differentiation order of existing bases from integer to any positive real number. However, classical Laguerre, and by extension Kautz and generalized orthogonal basis functions, are divergent as soon as their differentiation order is non-integer. In this paper, the first fractional orthogonal basis is synthesized, extrapolating the definition of Laguerre functions to any fractional order derivative. Completeness of the new basis is demonstrated. Hence, a new class of fixed denominator models is provided for fractional system approximation and identification. © 2007 Elsevier Ltd. All rights reserved. |
2004 |
Aoun, Mohamed; Malti, Rachid; Levron, François; Oustaloup, Alain Numerical simulations of fractional systems: An overview of existing methods and improvements Article de journal Dans: Nonlinear Dynamics, vol. 38, no. 1-4, p. 117 – 131, 2004, (Cited by: 117). Résumé | Liens | BibTeX | Étiquettes: Approximation theory, Computer simulation, Continuous time model, Differential equations, Discrete time model, Fractional calculus, Fractional model, Functions, Identification (control systems), Laplace transforms, Mathematical models, Time domain analysis @article{Aoun2004117b,An overview of the main simulation methods of fractional systems is presented. Based on Oustaloup’s recursive poles and zeros approximation of a fractional integrator in a frequency band some improvements are proposed. They take into account boundary effects around outer frequency limits and simplify the synthesis of a rational approximation by eliminating arbitrarily chosen parameters. © 2004 Kluwer Academic Publishers. |
Malti, Rachid; Aoun, Mohamed; Oustaloup, Alain Synthesis of fractional Kautz-like basis with two periodically repeating complex conjugate modes Conférence 2004, (Cited by: 19). Résumé | Liens | BibTeX | Étiquettes: Approximation theory, Differential equations, Dynamical systems, Extrapolation, Fractional calculus, Fractional derivatives, Generalized orthogonal basis (GOB), Identification (control systems), Laplace transforms, Linear control systems, Mathematical models, Model reduction, Orthogonal functions, Polynomials, Set theory, System stability, Transfer functions @conference{Malti2004835b,Fractional Kautz-like functions are synthesized extrapolating the definition of classical Kautz functions to fractional derivatives. The synthesized bases has two periodically repeating complex conjugate modes. The new basis extends the definition of the fractional Laguerre basis, by allowing the modes to be complex. An example is then provided in system identification context. |
Aoun, Mohamed; Malti, Rachid; Levron, François; Oustaloup, Alain Numerical simulations of fractional systems: An overview of existing methods and improvements Article de journal Dans: Nonlinear Dynamics, vol. 38, no. 1-4, p. 117 – 131, 2004, (Cited by: 117). Résumé | Liens | BibTeX | Étiquettes: Approximation theory, Computer simulation, Continuous time model, Differential equations, Discrete time model, Fractional calculus, Fractional model, Functions, Identification (control systems), Laplace transforms, Mathematical models, Time domain analysis @article{Aoun2004117,An overview of the main simulation methods of fractional systems is presented. Based on Oustaloup’s recursive poles and zeros approximation of a fractional integrator in a frequency band some improvements are proposed. They take into account boundary effects around outer frequency limits and simplify the synthesis of a rational approximation by eliminating arbitrarily chosen parameters. © 2004 Kluwer Academic Publishers. |
Malti, Rachid; Aoun, Mohamed; Oustaloup, Alain Synthesis of fractional Kautz-like basis with two periodically repeating complex conjugate modes Conférence 2004, (Cited by: 19). Résumé | Liens | BibTeX | Étiquettes: Approximation theory, Differential equations, Dynamical systems, Extrapolation, Fractional calculus, Fractional derivatives, Generalized orthogonal basis (GOB), Identification (control systems), Laplace transforms, Linear control systems, Mathematical models, Model reduction, Orthogonal functions, Polynomials, Set theory, System stability, Transfer functions @conference{Malti2004835,Fractional Kautz-like functions are synthesized extrapolating the definition of classical Kautz functions to fractional derivatives. The synthesized bases has two periodically repeating complex conjugate modes. The new basis extends the definition of the fractional Laguerre basis, by allowing the modes to be complex. An example is then provided in system identification context. |
Publications
2011 |
2011, (Cited by: 2). |
2007 |
Synthesis of fractional Laguerre basis for system approximation Article de journal Dans: Automatica, vol. 43, no. 9, p. 1640 – 1648, 2007, (Cited by: 115; All Open Access, Green Open Access). |
Synthesis of fractional Laguerre basis for system approximation Article de journal Dans: Automatica, vol. 43, no. 9, p. 1640 – 1648, 2007, (Cited by: 115; All Open Access, Green Open Access). |
2004 |
Numerical simulations of fractional systems: An overview of existing methods and improvements Article de journal Dans: Nonlinear Dynamics, vol. 38, no. 1-4, p. 117 – 131, 2004, (Cited by: 117). |
Synthesis of fractional Kautz-like basis with two periodically repeating complex conjugate modes Conférence 2004, (Cited by: 19). |
Numerical simulations of fractional systems: An overview of existing methods and improvements Article de journal Dans: Nonlinear Dynamics, vol. 38, no. 1-4, p. 117 – 131, 2004, (Cited by: 117). |
Synthesis of fractional Kautz-like basis with two periodically repeating complex conjugate modes Conférence 2004, (Cited by: 19). |