Analysis, Modeling and Stability of Fractional Order Differential Systems 1

6The Distributed Model of the Fractional Integrator

6.1. Introduction

In Part 1, and more specifically in Chapters 1 and 2, the Riemann–Liouville integration was defined, with a particular focus on the fractional order integrator. This operator was synthesized according to a frequency approach. Moreover, thanks to a partial fraction expansion, a modal representation was introduced. This result was not straightforward, considering the classical form of the integrator impulse response:

[6.1]

However, an important question is related to the limit of this modal model when the number of terms is increased to infinity.

The answer to this question is provided by the frequency distributed model of the fractional integrator, as a result of the inverse Laplace transform of . This model provides answers to fundamental questions, for example, on the relation between frequency and modal representations, the expression of the Riemann–Liouville integral at any initial instant t₀ and more globally on the internal distributed state z(ω, t) of the integrator.

Fundamentally, it will be demonstrated that the Riemann–Liouville integral of any function v(t) is a convolution between the impulse response h_n (t) and v(t) , i.e. this integral can be interpreted as the response of an infinite dimension ...