This paper provides an optimal approximation of the fundamental linear fractional order transfer function using a distribution of the relaxation time function. Simple methods, useful in systems and control theories, which can be used to approximate the irrational transfer function of a class of fractional systems for a given frequency band by a rational function are presented. The optimal parameters of the approximated model are obtained by minimizing simultaneously the gain and the phase error between the irrational transfer function and its rational approximation. A simple analog circuit which can serve as a fundamental analog fractional system is obtained. Illustrative examples are presented to show the quality and usefulness of the approximation method.

This paper proposes a new wavelet family based on fractional calculus. The Haar wavelet is extended to fractional order by a generalization of the associated conventional low-pass filter using the fractional delay operator Z-D. The high-pass fractional filter is then designed by a simple modulation of the low-pass filter. In contrast, the scaling and wavelet functions are constructed using the cascade Daubechies algorithm. The regularity and orthogonality of the obtained wavelet basis are ensured by a good choice of the fractional filter coefficients. An application example is presented to illustrate the effectiveness of the proposed method. Thanks to the flexibility of the fractional filters, the proposed approach provides better performance in term of image denoising.

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In previous decades, it has been observed that many physical systems are well characterised by fractional order models. Hence, their identification is attracting more and more interest of the scientific community. However, they pose a more difficult identification problem, which requires not only the estimation of model coefficients but also the determination of fractional orders with the tedious calculation of fractional order derivatives. This study focuses on an identification scheme, in the time domain, of dynamic 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 adjustable fractional order differentiators. The basic ideas and the derived formulations of the identification scheme are presented. Illustrative examples are presented to validate the proposed linear fractional order system identification approach.