Modélisation et identification des systèmes d’ordre fractionnaire

Abstract

"It has been observed that many physical systems are well characterized by linear fractional order models. Hence, their identification is attracting more and more interest of the scientific community. However, they pose a more difficult identification problem than the integer order systems because it requires not only the estimation of the model coefficients but also the determination of the fractional orders with the tedious calculation of fractional order derivatives. This thesis focuses on the identification in the time domain of the dynamic fractional order systems described by linear fractional order differential equations. The proposed identification method is based on the recursive least squares algorithm applied to an ARX structure derived from the linear fractional order differential equation using a numerical fractional differentiator of adjustable order. In the first place, this identification method has been used to estimate the parameters with a prior knowledge of the fractional differentiation orders of the fractional order linear differential equation representing the linear fractional order system under investigation. Then, it has been used to estimate the parameters of a linear fractional system of commensurate order without a prior knowledge of the commensurate fractional order which is obtained among several values as the one when the square error between the measured data and the estimated model is the smallest one. Finally, an extension of the proposed identification method has been done to estimate the parameters and the order at the same time of the fundamental linear fractional order system. Illustrative examples are also presented to validate the usefulness of the proposed identification methods."