Page 8 - Mathematical Techniques of Fractional Order Systems

Preface

Fractional calculus, as generalization of integer order integration and differ-
entiation to its noninteger (fractional) order counterpart, has proved to be a
valuable tool in the modeling of many physical phenomena and engineering
problems. Fractional derivatives provide an excellent instrument for the
description of memory and hereditary properties of various materials and
processes.
The main reason for using integer order models was the absence of solu-
tion methods for fractional differential equations. The advantages or the real
objects of the fractional order systems are that we have more degrees of free-
dom in the model and that a “memory” is included in the model. One of the
very important areas of application of fractional calculus is the chaos theory.
Chaos is a very interesting nonlinear phenomenon which has been inten-
sively studied. It is found to be useful or has great application potential in
many fields such as secure communication, data encryption, financial sys-
tems, and biomedical engineering. The research efforts have been devoted to
chaos control and chaos synchronization problems in nonlinear science
because of its extensive applications.
Recently, studying fractional order systems has become an active
research area. The chaotic dynamics of fractional order systems began to
attract much attention in recent years. It has been shown that the fractional
order systems can also behave chaotically, such as the fractional order
Chua’s system, the fractional order Lorenz system, the fractional order Chen
system, the fractional order Ro ¨ssler system, the fractional order modified
Duffing system, the fractional order Newton Leipnik system, the fractional
order Lotka Volterra system, and the fractional order Liu system.
Moreover, recent studies show that chaotic fractional order systems can also
be synchronized. Many scientists who are interested in this field have found
it to be a challenging research problem to achieve the synchronization of
fractional order chaotic systems, and this research area has potential applica-
tions in secure communication and cryptography.
A wide variety of mathematical methods and techniques have been used
in this book to study the dynamics, control, design, and synchronization of
the fractional order control systems such as sliding mode controller, active
and adaptive control methods, fuzzy logic control, nonlinear control

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