Analysis, Modeling and Stability of Fractional Order Differential Systems 1

8Fractional Differentiation

8.1. Introduction

Fractional calculus is generally referred to the fractional derivatives of Riemann–Liouville and Caputo [DIE 10]. Paradoxically, from the beginning of this book, we have addressed fractional calculus only with the definition of the Riemann–Liouville integral and the fractional integrator Fractional differentiation is mentioned only through its Laplace transform sⁿ and with the Grünwald–Letnikov derivative. This surprising approach is motivated by the definition of fractional differentiation which requires the integer order differentiation and the fractional order integration [TRI 13d]. Moreover, the use of Riemann–Liouville or Caputo derivatives requires us to specify their initial conditions [POD 99]. However, as we will demonstrate it, these initial conditions require us to introduce the initial state of the associated Riemann–Liouville integral. Consequently, usual initial conditions of these derivatives have to be revisited [HAR 09a, LOR 11], as highlighted by many research works [SAB 10a, TRI 10b, TRI 13a, TRI 15].

Let us finally recall that the usual approach to the analysis of fractional systems dynamics is based on the properties of these derivatives, particularly on their initial conditions [MAT 96, BET 06, MON 10]. As demonstrated in Chapter 7, FDS transients can be expressed with the initial conditions of internal integrators. ...