Abstract
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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Ichise, M., Nagayanagi, Y., and Kojima, T., ‘An analog simulation of non integer order transfer functions for analysis of electrode processes’, Journal of Electroanalytical Chemistry Interfacial Electrochemistry33, 1971, 253.
Darling, R. and Newman J., ‘On the short behavior of porous intercalation electrodes’, Journal of Electrochemical Society144(9), 1997, 3057–3063.
Battaglia, J. L., Cois, O., Puigsegur, L., and Oustaloup, A., ‘Solving an inverse heat conduction problem using a non-integer identified model’, International Journal of Heat and Mass Transfer44(14), 2001, 2671–2680.
Cois, O., ‘Systèmes linéaires non entiers et identification par modèle non entier: application en thermique’, {Ph.D. Thesis}, University of Bordeaux I, France, 2003.
Bode, H. W., Network Analysis and Feedback Amplifiers Design, Nostrand, New York, 1945.
Tustin, A., Allanson, J. T., Layton, J. M., and Jakeways, R. J., ‘The design of systems for automatic control of the position of massive object’, in Proceedings of Institution of Electrical Engineers, 1958, 105, Part C, Suppl. 1, pp. 1–57.
Oustaloup, A., Systèmes asservis linéaires d’ordre fractionnaire, Masson, Paris, 1983.
Al-Alaoui, M. A., ‘Novel IIR differentiator from the Simpson Integration rule’, IEEE Transactions on Circuits and Systems I. Fundamental Theory and Applications41(2), 1994, 186–187.
Vinagre, B. M., Podlubny, I., Hernandez, A., and Feliu, V., ‘Some approximations of fractional order operators used in control theory and applications’, Fractional Calculus & Applied Analysis3(3), 2000, pp. 231–248.
Petras, I., Podlubny, I., O’Leary, P., and Dorcak, L., ‘Analogue fractional-order controllers: Realization, tuning and implementation’, in Proceedings of the ICCC’2001, Krynica, Poland, 2001, pp. 9–14.
Chen, Y. Q. and Kevin, L. Moore, L., ‘Discretization schemes for fractional-order differentiators and integrators’, IEEE Transactions on Circuits and Systems-I: Fundamental Theory and Applications49(3), 2002, 363–367.
Chen, Y. Q., Vinagre, B., and Podlubny, I., ‘A new discretization method for fractional order differentiators via continued fraction expansion’, in ASME First Symposium on Fractional Derivatives and Their Applications, International Design Engineering Technical Conferences, Chicago, Illinois, 2003.
Podlubny, I., Fractional Differential Equations. Mathematics in Science and Engineering, Vol. III. Academic Press, San Diego, California, 1999.
Podlubny, I., Petras, I., Vinagre, B. M., O’Leary, P., and Dorcak, L., ‘Analogue realizations of fractional-order controllers’, Nonlinear Dynamics29(1–4), 2002, 281–296.
Petras, I., Podlubny, I., O’Leary, P., Dorcak, L., and Vinagre, B., Analogue Realization of Fractional Order Controllers, Technical University of Kosice, Kosice, Slovak Republic, 2002, p. 84.
Samko, S. G., Kilbas, A. A., and Marichev, O. I., Fractional Integrals and Derivatives: Theory and Applications. Gordon and Breach, Amsterdam, 1993.
Miller, K.S. and Ross, B., An Introduction to the Fractional Calculus and Fractional Differential Equations. Wiley, New York, 1993.
Oldham, K. B. and Spanier, J., The Fractional Calculus. Academic Press, New York, 1974.
Matignon D., ‘Représentations en variables d’état de modèles de guides d’ondes avec dérivation fractionnaire’, {Ph.D. Thesis}, Université de Paris-Sud, Orsay, France, 1994.
Oustaloup, A., La dérivation non Entière: Théorie, Synthèse et Applications, Hermès, Paris, 1995.
Matignon, D., ‘Stability properties for generalized fractional differential systems’, in ESAIM: Proceedings, Vol. 5, Systèmes Différentiels Fractionnaires – Modèles, Méthodes et Applications, Paris, 1998.
Tabak, D., ‘Digitalization of control systems’, Computer Aided Design32, 1971, 13–18.
Lin, J., ‘Modélisation et identification de systèmes d’ordre non entier’, {Thèse de Doctorat,} Université de Poitiers, France, 2001.
Oustaloup, A., Levron, F., Nanot, F., and Mathieu, B., ‘Frequency-band complex non integer differentiator: Characterization and synthesis’, IEEE Transaction on Circuits and Systems47(1), 2000.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Aoun, M., Malti, R., Levron, F. et al. Numerical Simulations of Fractional Systems: An Overview of Existing Methods and Improvements. Nonlinear Dyn 38, 117–131 (2004). https://doi.org/10.1007/s11071-004-3750-z
Received:
Accepted:
Issue Date:
DOI: https://doi.org/10.1007/s11071-004-3750-z