Page 627 - Mathematical Techniques of Fractional Order Systems

598 Mathematical Techniques of Fractional Order Systems

documented in many control theory or in the application literature (Azar
et al., 2017a; Meghni et al, 2017a,b,c; Soliman et al., 2017; Tolba et al.,
2017; Ghoudelbourk et al., 2016; Ouannas et al., 2016b, 2017a,b,c,d,e,I;
Ladaci and Charef, 2006). Fractional calculus deals with derivatives and
integrations of arbitrary order (Lin et al., 2011; Podlubny, 1999) and has
found many applications in many fields of physics, applied mathematics, and
engineering. Moreover, many real-world physical systems are well character-
ized by fractional order differential equations, i.e., equations involving both
integer and noninteger order derivatives. It is observed that the description of
some systems is more accurate when the fractional derivative is used. For
instance, electrochemical processes and flexible structures are modeled by
fractional order models (Lin et al., 2011; Lin and Kuo, 2011; Ladaci et al.,
2008). Nowadays, many fractional order differential systems behave chaoti-
cally, such as the fractional order Chua system, the fractional order Duffing
system (Arena et al., 1997), the fractional order Lu system, the fractional
order Chen system (Petra ´ˇ s, 2006), the fractional order cellular neural network
(Arena and Caponetto, 1998; Petra ´ˇ s, 2006; Lamamra et al., 2017).
The synchronization problem of fractional order chaotic systems was
first investigated by Deng and Li who carried out synchronization in the
case of the fractional Lu ¨ system. Afterwards, they studied chaos synchroni-
zation of the Chen system with a fractional order in a different manner
(Hartley et al., 1995; Hilfer, 2001; Pham et al., 2017; Ouannas et al.,
2016b, 2017a,b,c,d,e,i).
In this chapter we are interested by the problem of uncertain fractional
order chaotic systems synchronization by means of robust adaptive fuzzy
control. Variable structure control is a very suitable method for handling
such nonlinear systems because of low sensitivity to disturbances and plant
parameter variations and its order reduction property, which relaxes the bur-
den of the necessity of exact modeling (Lin et al., 2004; Utkin,1977; Hung
et al., 1993).
Based on the universal approximation theorem (Wang and Mendel, 1992;
Wang, 1992; Wang, 1994; Lin et al., 2012) (fuzzy logic controllers are gen-
eral enough to perform any nonlinear control actions), there is rapidly grow-
ing interest in systematic design methodologies for a class of nonlinear
systems using fuzzy adaptive control schemes. An adaptive fuzzy system is a
fuzzy logic system equipped with a training algorithm in which an adaptive
controller is synthesized from a collection of fuzzy IF THEN rules and the
parameters of the membership functions characterizing the linguistic terms in
the IF THEN rules change according to some adaptive law for the purpose
of controlling a plant to track a reference trajectory.
By incorporating the H N tracking design technique (Chen et al., 1996,
Bensafia et al., 2017) and based on the fractional Lyapunov stability theorem
(Aguila-Camacho et al., 2014; Duarte-Mermoud et al., 2015; Sastry and
Bodson, 1989; Khettab et al., 2017a), an efficient adaptive control algorithm