Analysis, Modeling and Stability of Fractional Order Differential Systems 2

5Fractional Model-based Control of the Diffusive RC Line

5.1. Introduction

In this chapter, we intend to demonstrate that the state control of a fractional differential system (FDS) is not only a theoretical problem; on the contrary, it can be applied to control the distributed state of a diffusive system. For this purpose, this study is based on two main principles. It was demonstrated in Chapter 4 of Volume 1 that a diffusive interface can be approximated by a fractional model. Moreover, as related in Chapter 6 of Volume 1, an infinite length RC line behaves like a fractional integrator, and there is a connection between their distributed internal variables v(x,t) and z(ω,t). Hence, we intend to demonstrate that Ẕ(ω,t) state control of the fractional model of the RC line is equivalent to the control of the distributed RC line variable v(x,t).

Modeling of the diffusive interface was performed in Chapter 4 of Volume 1 using a frequency identification technique. This methodology is reserved to a theoretical analysis, making possible to justify and discriminate different models in a physical context. In practical situations, we mainly use fractional identification techniques based on time measurements [BEN 08b, MAL 08, GAB 11a, GAB 11b].

For the identification of a fractional model of the RC line, input and output data {u(t),y(t)} are generated by a numerical simulation. Consequently, these measurements are not disturbed by noise; therefore, we propose to use an elementary algorithm ...