Analysis, Modeling and Stability of Fractional Order Differential Systems 1

5Modeling of Physical Systems with Fractional Models: an Illustrative Example

5.1. Introduction

In Chapter 4, we demonstrated that the diffusive heat transfer interface, governed by a partial differentiation equation, can be modeled by a fractional transfer function, independently of the physics of the interface. More generally, this means that the dynamical behavior of a physical system can be approximated by a differential mathematical model (linear or nonlinear) with integer order derivatives or fractional order ones. The heat transfer example has justified the interest of fractional modeling for systems governed by a diffusive equation, i.e. with dynamics characterized by long memory behavior. We also demonstrated that the structure cannot be chosen arbitrarily; on the contrary, these models must satisfy structural constraints, in particular for identification purposes.

More generally, it is necessary to remember some basic principles for modeling and identification related to the system theory. In particular, it is important to differentiate between modeling concepts such as the knowledge model, the behavior model or the black box model and the intermediate category called the gray box model [TUL 93, WAL 97, TRI 01].

In order to specify these essential notions, we consider an illustrative example, the induction motor [ALG 95]. During the last decades, it has gradually replaced the DC motor in variable speed applications, mainly for rail traction [BOS 86, HUS 09]. The ...