Page 591 - Mathematical Techniques of Fractional Order Systems

Control and Synchronization Chapter | 19 561

(Sharma and Kar, 2009a), robotics (Nakamura and Sekiguchi, 2001), biolog-
ical implants (Mormann et al., 2000), etc.

19.1.2 Introduction to Fractional Calculus
The story of fractional calculus begins with a question asked by L. Hospital,
some 300 years ago, to Leibniz regarding his publication for the nth deriva-
tive of a function, i.e., what would the result be if n 5 1/2. The answer given
by Leibniz is now well-known to the research fraternity, i.e., “an apparent
paradox from which 1-day useful consequences will be drawn.” Fractional
calculus is basically a generalization of integer order differentiation and
n-fold integration to arbitrary order. Due to the unavailability of solution tech-
niques, fractional calculus has not been explored much for almost 300 years,
although some pioneers contributed significantly during the 19th century
towards the development of fractional calculus (Samko et al., 1993).
Furthermore, a number of researchers have also helped in the advancement of
this area (e.g., Hardy and Beier, 1994; Oldham and Spanier, 1974; Srivastava
and Owa, 1989; Samko et al., 1993; Miller and Ross, 1993; Podlubny, 1998).
In view of contributions made by various researchers, in recent years, various
methods for approximation of the fractional derivative and integral came into
existence, which further enabled us to use it in wide areas such as bioengi-
neering (West, 2007), groundwater flow problem (Atangana and Bildik, 2013;
Atangana and Vermeulen, 2014), diffusion of heat (Angulo et al., 2000), sig-
¸
nal processing (Tseng, 2007), robotics (da Graca Marcos et al., 2008), quan-
tum theory (Laskin, 2002), electrical engineering (Carlson and Halijak, 1964;
Meghni et al., 2017a,b,c; Soliman et al., 2017; Tolba et al., 2017), etc.
Fractional calculus finds wide applicability in control systems too (Tenreiro
Machado et al., 2010; Tabari and Kamyad, 2013; Mishra and Chandra, 2014).
The application revolves around two pillars: one is of fractional order systems
and the other is fractional order control. It has been established that fractional
calculus gives a more realistic modeling of systems and the best performance
is achieved when a fractional order controller is employed for a fractional
order system (Podlubny, 1999). Also, it is confirmed now that the best frac-
tional order controller can be better than the best integer order controller.
Enough research has been done on fractional order PID controller and its tun-
ing that fractional order PID controllers (Podlubny, 1994; Petras, 2011)are
now being used in real-time applications as their performance is better than
the traditional ones.

19.1.3 Introduction to Fractional Order Chaotic Systems
Chaotic systems can also be modeled more accurately by noninteger order
differential equations and hence are called fractional order chaotic systems.
It is known that chaotic behavior is exhibited by the systems, for system