July 14, 2009 11:14 WSPC/INSTRUCTION FILE paper MULTIPLE-FRACTIONAL ORDER DERIVATIVE Z. SANTOUH and A. CHAREF § This paper is concerned with the approximation of multiple-fractional power zero sys- tems in the frequency domain. This type of systems is often encountered in the modeling and analysis of complex dynamical ones such as dielectric, viscoelastic, magnetic mate- rials and interfacial polarization processes. What distinguishes this method with respect to previous ones is that it allows the approximation of whatever complex systems en- tailing multiple fractal power zeros of any fractional power. Illustrative examples are presented to confirm the validity and the accuracy of the approximation method. Keywords : Approximation, fractional power pole, singularity method, fractional power zero, multiple-fractional power zero, irrational transfer function. Received (Day Month Year) Revised (Day Month Year) Accepted (Day Month Year) 1. Introduction In the last two decades, huge research efforts have been invested in the investiga- tion and development of fractional order systems theory by mathematicians and scientist. A great effort was also made to put into practice some already established results, but only recently that one can find significant progress in theoretical works which serve as a foundation for an increasing number of applications in various fields 1,6 . Intensive research work is still underway in many disciplines towards the appli- cation of fractional order systems concepts known for their remarkable dynamical performances. These systems have irrational transfer functions which are difficult to analyze and study in the time domain hence the need to approximate them with rational functions using a technique based on the singularity function method 9,10 . More recently, there is a new trend to investigate the control and dynamics of fractional order dynamical systems 15,? . In this paper a new way to approximating the multiple fractional power zero system (MFPZ) is presented. This method is of very important use in the identification an approximation of real physical systems revealing fractional differentiation behaviors 14 . This paper is organized as follows. First, we describe the singularity function approximation method as applied to the single fractal system, then we describe a way to generalize this method to the multi- ple fractional order system of any fractional power order. Next, numerical examples demonstrating the validity of the approximation in the frequency domain are given. Finally, we end up with some concluding remarks and conclusions. 1