ENERGY OF FRACTIONAL ORDER TRANSFER FUNCTIONS Rachid MALTI * , Olivier COIS + , Mohammed AOUN + , François LEVRON ¤ and Alain OUSTALOUP + * Intelligence ds les Instrumentations et les Systèmes + Laboratoire d'Automatique et de Productique UMR 5131 IUT de Sénart-Fontainebleau – Université Paris 12 Université Bordeaux I – ENSEIRB Avenue Pierre Point, 77127 Lieusaint, France 351 cours de la Libération, 33405 Talence cedex, France Tel: +33 (0)1 64 13 51 83 Tel : +33 (0)556 842 418 Fax: +33 (0)1 64 13 4503 Fax : +33 (0)556 846 644 malti@univ-paris12.fr {cois, aoun, oustaloup}@lap.u-bordeaux.fr ¤ Institut de Mathématique de Bordeaux Université de Bordeaux I 351 cours de la Libération, 33405 Talence cedex, France levron@math.u-bordeaux.fr Abstract: The objective of the paper is to compute the impulse response energy of a fractional order transfer function having a single mode. The differentiation order n, defined in the sense of Riemann-Liouville, is allowed to be a strictly positive real number. A necessary and sufficient condition is established on n, in order for the impulse response to belong to the Lebesgue space L 2 [0, ∞[ of square integrable functions on [0, ∞[. Copyright © 2002 IFAC Keywords: fractional order differentiation, impulse response energy, transfer function, dynamical system, fractional calculus 1. INTRODUCTION Although as yet relatively unused in physics, the concept of differentiation to an arbitrary order (also called fractional differentiation) was defined in the 19th century by Riemann and Liouville. Their main concern was to extend differentiation by using not only integer but also non-integer (real or complex) orders. The n th order derivative of fractional order is defined as (Samko, 1993): () ( ) () ( ) () ( ) ( )         -       - Γ = ∫ - - t n m m t d x dt d n m t x 0 1 1 τ τ τ n D with t > 0, n > 0 and   1 + = n m .  n means integer part of n. Studies on real systems such as thermal or electrochemical (Battaglia et al., 2000, Cois et al., 2000), reveal inherent fractional differentiation behaviour. The use of classical models (based on integer order differentiation) is thus inappropriate in representing these fractional systems. Thus, a further class (called fractional) of mathematical models has been provided since 1983 by (Oustaloup) using the concept of fractional differentiation. These models are based on either a fractional differential equation or a fractional state-space representation, namely (Matignon, 1994, Oustaloup, 1995, Cois et al., 2001): ( ) () () () () () ()      + = + = t u t t y t u t t n D x C B x A x D . (1) where n is a real, integer or non integer, number. The modal decomposition of such a representation leads to express the system output as a linear combination of elements called eigenmodes governed by the following equation: () () () t u t x t x n = + λ D , where λ denotes a system eigenvalue. This latter equation can also be written (see (Oldham and Spanier, 1974) for example) in the Laplace domain as: () () () λ + = = Δ n n s s U s X s B 1 One of the inherent system characteristics in control engineering is its impulse response energy. For instance, if the energy is finite, it can be concluded that a system belongs to L 2 [0, ∞[, space of squared integrable functions. Our concern in this paper is to compute the impulse response energy of () s B n whatever the differentiation order n is. Up to our knowledge, the method presented herein is original since no work Copyright © 2002 IFAC 15th Triennial World Congress, Barcelona, Spain