Fractional State Space Description: A Particular Case of the Volterra Equations

Fractional Behaviours Modelling with Volterra Equations: Application to a Lithium-Ion Cell and Comparison with a Fractional Model

This paper proposes to model fractional behaviors using Volterra equations. As fractional differentiation-based models that are commonly used to model such behaviors exhibit several drawbacks and are particular cases of Volterra equations (in the kernel definition), it appears legitimate in a modeling approach to work directly with Volterra equations. In this paper, a numerical method is thus developed to identify the kernel associated to a Volterra equation that describes the input–output behavior of a system. This method is used to model a lithium-ion cell using real data. The resulting model is compared to a fractional differentiation-based model with the same number of tunable parameters.

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

Using a small number of mathematical transformations, this article examines 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 in an infinite space domain. Fractional models can thus be viewed as doubly infinite dimensional models: infinite as distributed with a distribution in 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 behaviors without infinite memory are finally proposed.

Fractional-Order Derivatives Defined by Continuous Kernels: Are They Really Too Restrictive?

In the field of fractional calculus and applications, a current trend is to propose non-singular kernels for the definition of new fractional integration and differentiation operators. It was recently claimed that fractional-order derivatives defined by continuous (in the sense of non-singular) kernels are too restrictive. This note shows that this conclusion is wrong as it arises from considering the initial conditions incorrectly in (partial or not) fractional differential equations.