Long memory models: a first solution to the infinite energy storage ability of linear time-invariant fractional models

Abstract

In this paper, it is shown that linear time-invariant fractional models do not reflect the reality of physical systems in terms of energy storage ability. It is first shown that this property may result from poorly chosen asymptotic behaviors. Another reason is that a fractional model can be viewed as a doubly infinite model. Indeed, its real state is of infinite dimension as it is distributed. Moreover, this state is distributed on an infinite domain. It is precisely this last feature that induces the ability to store an infinite energy, even if the fractional behavior is limited to a frequency band. As a consequence, even if fractional models permit to capture accurately the input-output dynamical behavior of many physical systems, the obtained models do not reflect the internal behavior of the modelled system which implies hard theoretical problems. Such problems may be avoided by the use of other models that exhibit the same input-output behavior but that do not have an infinite energy storage ability. As a first attempt to solve this issue, a new class of models is thus introduced in the paper.