Analysis, Modeling and Stability of Fractional Order Differential Systems 1

2Frequency Approach to the Synthesis of the Fractional Integrator

2.1. Introduction

In Chapter 1, we defined fractional integration as the key tool for modeling and simulation of FDEs. Unfortunately, the definition of Riemann–Liouville integration

[2.1]

where

[2.2]

does not provide a suitable technique for the numerical simulation of fractional integration, which is not a classical integral, but in fact a convolution integral.

Therefore, in the first step, we relate this operation to a frequency approach based on the Laplace transform of h_n(t), thus:

[2.3]

In the second step (Chapter 6), we will relate fractional integration to a time approach, called the infinite state approach, moreover demonstrating that these two approaches are equivalent and complementary.

The synthesis of the fractional integrator thanks to a frequency methodology is in fact based on Oustaloup’s technique [OUS 00] for the synthesis of the fractional differentiator Dⁿ(s) = sⁿ.

Therefore, in this chapter, we present a frequency approximation of Iⁿ(s) which will be used to derive a modal formulation, basis of the numerical integration algorithm and of the fractional integrator state variables.

2.2. Frequency ...