Page 358 - Mathematical Techniques of Fractional Order Systems

Dual Combination Synchronization Scheme Chapter | 12 349

another area of research. Therefore, chaos and chaos control in fractional
order systems have become a hot topic in research. Since there was lack of
appropriate mathematical tools, fractional order dynamic systems had not
been studied much in the design and practice of control systems during last
few decades. In recent years, after emergence of effective methods fractional
order systems have become more and more attractive for the systems control
α
community. The TID controller, the fractional PD controller, the fractional
α λ
PI controller, the fractional PI Dμ controller, and the fractional lead-lag
compensator are a few examples of well-known fractional order controllers.
In some recent works, it has come into the picture that the fractional order
controllers have better disturbance rejection ratios and less sensitivity to
plant parameter variations compared to the traditional controllers. At the
present time scientists and engineers are using fractional controllers in many
practical applications, viz., control of main irrigation canals, lateral and lon-
gitudinal control of autonomous vehicles, control of robotic time-delay sys-
tems, control of hexapod robots, reducing engine vibrations in automobiles,
control of electromechanical systems, and flexible spacecraft attitude control.
The most important achievement in the research of chaos is that chaotic
systems can be made to synchronize with each other. Synchronization is a
phenomenon that is usually treated as a regime in which two or more coupled
or periodic or even chaotic systems exhibit correlated, and sometimes even
identical, oscillations. The idea of synchronizing chaotic systems was first
introduced by Pecora and Carroll (1990); they showed that it is possible to
synchronize chaotic systems through a simple coupling. Chaos synchroniza-
tion has attracted extensive attentions for its potential applications in physical
systems (Lakshmanan and Murali, 1996), chemical systems (Hanetal.,1995),
neural networks (Wu et al., 2012; Moskalenko et al., 2010), neuron systems
(Shuai and Wong, 1998), secure communications (Chai et al., 2012; Chen
et al., 2014), ecological system (Blasius et al., 1999), and so on. In recent
years, many effective methods have been presented for synchronizing identi-
cal and nonidentical chaotic systems. Time-delay feedback approach, active
control, adaptive control (Park and Kwon, 2005; Srivastava et al., 2013a,b;
Huang et al., 2009), and synchronization via nonlinear feedback control tech-
niques (Grassi et al., 2017; Ouannas et al., 2016b,c; Huang et al., 2004; Chen
and Lu, 2002; Singh et al., 2017; Ouannas et al., 2017a,b,c,d,e,h; Boulkroune
et al, 2016b; Vaidyanathan et al, 2015a,b,c; Wang et al., 2017; Azar and Zhu,
2015; Azar and Vaidyanathan, 2015a,b,c, 2016; Zhu and Azar, 2015;
Vaidyanathan and Azar, 2015a,b,c,d, 2016a,b,c,d,e,f,g, Vaidyanathan et al.,
2017a,b,c; Azar et al., 2017a,b,c) have been effectively applied to application
of chaos synchronization problem. The conception of synchronization can be
extended to complete synchronization, antisynchronization, Projective syn-
chronization, Function projective synchronization, and hybrid synchronization
(Yu and Liu, 2003; Liu, 2006; Si et al., 2012; Yadav et al., 2017a; Zhou and
Zhu, 2011; Azar et al., 2017a; Ouannas et al., 2017c,d,e), etc.