Brief paper: Synthesis of fractional Laguerre basis for system approximation

Authors:
M. Aoun;R. Malti;F. Levron;A. Oustaloup
Affiliations:
IMS, UMR 5218 CNRS, Département LAPS, Université Bordeaux 1, 351 cours de la Libération, F 33405 Talence cedex, France and MACS, ícole Nationale d'Ingénieurs de Gabès ...;IMS, UMR 5218 CNRS, Département LAPS, Université Bordeaux 1, 351 cours de la Libération, F 33405 Talence cedex, France;IMB, UMR 5251 CNRS, Université Bordeaux 1, 351 cours de la Libération, F 33405 Talence cedex, France;IMS, UMR 5218 CNRS, Département LAPS, Université Bordeaux 1, 351 cours de la Libération, F 33405 Talence cedex, France
Venue:
Automatica (Journal of IFAC)
Year:
2007

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Abstract

Fractional differentiation systems are characterized by thepresence of non-exponential aperiodic multimodes. Although rationalorthogonal bases can be used to model any L2[0,∞[system, they fail to quickly capture the aperiodic multimodebehavior with a limited number of terms. Hence, fractionalorthogonal bases are expected to better approximate fractionalmodels with fewer parameters. Intuitive reasoning could lead tosimply extending the differentiation order of existing bases frominteger 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 isnon-integer. In this paper, the first fractional orthogonal basisis synthesized, extrapolating the definition of Laguerre functionsto any fractional order derivative. Completeness of the new basisis demonstrated. Hence, a new class of fixed denominator models isprovided for fractional system approximation and identification.