Editorial
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 integrodifferential 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 nonstandard 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 (ATSDNE) 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.
Anglais
