Fractional Order Models Are Doubly Infinite Dimensional Models and Thus of Infinite Memory: Consequences on Initialization and Some Solutions

Abstract

Using a few mathematical transformations, this article sheds light on the nature of fractional models (described by fractional differential equations or pseudo state space descriptions). Computation of the impulse response of a fractional model using the Cauchy method shows that they exhibit infinitely small and high time constants. This impulse response can be rewritten as a diffusive representation whose Fourier transform permits a representation of a fractional model by a diffusion equation on an infinite space domain. Fractional models can thus be viewed as doubly infinite dimensional models (distributed and therefore infinite, on an infinite domain). This infinite domain or the infinitely large time constants of the impulse response reveal a property intrinsic to fractional models: their infinite memory. Solutions to generate fractional behaviours without infinite memory are finally proposed.KeywordsFractional modelsInfinite memoryInitial conditions