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Journal Européen des Systèmes Automatisés

Revues des Systèmes

 ARTICLE VOL 42/6-8 - 2008  - pp.625-626

Fractional order differentiation consists in the generalisation of classical integer differentiation to real or complex orders. From a mathematical point of view, several interpretations of fractional differentiation were proposed, but there is still a deep debate about it. However, all these interpretations demonstrate that fractional order differentiation cannot simply be connected to the slope at one point of the derivated function for instance. This lack of interpretation is in fact due to the definition of the fractional order operator. This is a non local operator based on an integral with a singular kernel. The same conclusion can be made for the fractional integrator operator, fractional differentiation operator definition being based on the fractional integrator operator definition. This situation explains why these operators are still not well defined and that several definitions still coexist. Since the first recorded reference work in 1695 up to the present day, many articles have been published on this subject, but much progress still to be done particularly on the relationship of these different definitions with the physical reality of a system (through taking into account the initial conditions for instance). A fractional order system is a system described by an integro-differential equation involving fractional order derivatives of its input(s) and/or output(s). From a physical point of view, linear fractional derivatives and integrals order systems are not quite conventional linear systems, and not quite conventional distributed parameter systems. They are in fact halfway between these two classes of systems, and are particularly suited for diffusion phenomena modelling. They also have been a modelling tool well suited to a wide class of phenomena with non-standard dynamic behaviour, and the applications of fractional order systems are now well accepted in the following disciplines: – Electrical engineering (modelling of motors, modelling of transformers, skin effect…); – Electronics, telecommunications (phase locking loops…); – Electromagnetism (modelling of complex dielectric materials…); – Electrochemistry (modelling of batteries and ultracapacitors…); – Thermal engineering (modelling and identification of thermal systems…); – Mechanics, mechatronics (vibration insulation…); – Rheology (behaviour identification of materials…); – Automatic control (robust control, system identification, observation and control of fractional systems…); – Robotics (modelling, path tracking, path planning…); – Signal processing (filtering, restoration, reconstruction, analysis of fractal noises…); – Image processing (fractal environment modelling, pattern recognition, edge detection…);– Biology, biophysics (electric conductance of biological systems, fractional models of neurons, muscle modelling…); – Physics (analysis and modelling of diffusion phenomenon…); – Economy (analysis of stock exchange signals…). In France, fractional differentiation and fractional order systems has been popularized mainly by the work of Professor Alain Oustaloup at Bordeaux 1 University. Today, numerous colleagues work in this field and their results are mostly presented during the “Action thématique Systèmes à Dérivation Non Entière (AT-SDNE)” meetings and are available on the website dedicated to these meetings. Also, important industrial groups have shown growing interest. Some of them now develop, in the context of universityindustry relations or for their own benefit, applications using fractional differentiation.




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