Analysis of fractional models physical consistency

Abstract

Recently, for fractional order model, a distinction has emerged between real state and pseudo state. Pseudo state is a vector of finite dimension but does not have the property of a state (it does not allow to determine future behavior of the system for instance). The real state is of infinite dimension as it is distributed, but is distributed on an infinite domain. A fractional model can thus be viewed as a doubly infinite model (distributed model on an infinite domain). It is shown in the paper, that this last feature induces the real state ability to store an infinite amount of energy using an electrical interpretation of fractional models. Thus, fractional models do not reflect the reality of macroscopic physical systems in terms of energy storage ability. As a consequence, even if fractional models permit to capture accurately the input-output dynamical behavior of many physical systems, such a property highlights a physical inconsistence of fractional models. They do not reflect the internal behavior of the modelled system. This analysis is made for explicit and implicit differentiation based fractional order models.