Synthesis of fractional Laguerre basis for system approximation

Mohamed Aoun (1) , Rachid R. Malti (1) , François Levron (2) , Alain Oustaloup (1)

1 IMS - Laboratoire de l'intégration, du matériau au système
2 IMB - Institut de Mathématiques de Bordeaux

Fractional differentiation systems are characterized by the presence of non-exponential aperiodic multimodes. Although rational orthogonal bases can be used to model any $L_2[0, \infty[$ 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.

Mots clés: Orthonormal basis; fractional differentiation; Laguerre function; system approximation; identification

Domaines: Automatique / Robotique

Mohamed Aoun, Rachid R. Malti, François Levron, Alain Oustaloup. Synthesis of fractional Laguerre basis for system approximation. Automatica, 2007, 43 (9), pp.1640-1648. ⟨10.1016/j.automatica.2007.02.013⟩. ⟨hal-00180684⟩