Analysis, Modeling and Stability of Fractional Order Differential Systems 1

7Modeling of FDEs and FDSs

7.1. Introduction

Chapters 1–3 were dedicated to the numerical simulation of FDEs using fractional integrators. As in the integer order case, the integration operator is the key tool for the modeling of FDSs. In Chapter 6, a theoretical framework was provided to the analysis of this operator. Moreover, it was proved that the Riemann–Liouville integration corresponds basically to a convolution process, obeying the well-known principles of the system theory [KAI 80].

This convolution interpretation provides a natural solution to the initialization of FDSs, recognized as a major problem of fractional calculus for a long time [LOR 01, FUK 04, TEN 14]. Numerical simulations highlight the role of the distributed state variable z(ω,t) in the interpretation of system transients and reciprocally the inability of pseudo-state variables x(t) to predict them [SAB 10a, TRI 09b, TAR 16a]. This type of modeling, based on fractional integrators, can be qualified as the closed-loop or internal representation [TRI 12b, TRI 12c].

Nevertheless, another representation of linear commensurate order fractional systems exists, which is not based on fractional integrators. The inverse Laplace transform of the elementary transfer function provides another distributed model of this system, known as the diffusive representation [MON 98, HEL 98, MON 05a, MON 05b], which can be ...