Page 295 - Mathematical Techniques of Fractional Order Systems

284 Mathematical Techniques of Fractional Order Systems

Vaidyanathan, 2016; Pham et al., 2017a; Moysis and Azar, 2017; Lamamra
et al., 2017). Fractional order complex chaotic and hyperchaotic systems
have been recently studied nowadays due to the vast number of physical sys-
tems in which this phenomenon is found; systems such as mechanical, elec-
trical, chemical, biological, renewable energy, and control systems (Zhang
et al., 2015; Azar et al., 2018a,b, 2017; Ouannas et al., 2016a,b, 2017d,e,f,g,
h,i,j; Pham et al., 2017a,b; Tolba et al., 2017a,b; Meghni et al., 2017;
Boulkroune et al., 2016a; Ghoudelbourk et al., 2016). For this reason, it’s
important to design efficient control and synchronization strategies to stabi-
lize these kinds of system to reach the equilibrium point or to synchronize a
response system following a reference from a drive system that can be iden-
tical or nonidentical.
The stabilization problem for integer order or fractional order complex cha-
otic system has been recently studied. For example, in El-Sayed et al. (2016),
the circuit realization of a fractional order hyperchaotic system is done by
designing a suitable control law and studying the effects of the fractional order
derivatives. In Zhang et al. (2008) an interesting example in which an analysis
of the chaotic and hyperchaotic behavior of a nonautonomous rotational
machine is shown where a study of the eigenvalues and the Lyapunov expo-
nents of this mechanical system is done. Something similar is found in Zhang
et al. (2015), where a stability analysis of a fractional order nonlinear system
is provided considering an order between 0 and 2. Another important work is
evinced in Li (2016), where an adaptive tracking control for a fractional order
chaotic system is shown considering and not considering the system uncertain-
ties. An interesting example of a fractional order chaotic system is shown in
Huang et al. (2014) where a fuzzy state feedback controller is designed, and
one important remark of this work is that the state feedback gains are found
by linear matrix inequalities (LMIs). A robust controller for a four-wing frac-
tional order hyperchaotic system is shown in Li et al. (2013b) where by using
a Lyapunov fractional order stability theorem a state feedback law is found. In
Li and Li (2015), an interesting example in which an adaptive integer and
fractional order controller is implemented for the stabilization of a fractional
order chaotic systems where the Barbalat’s Lyapunov-like stability theorem is
implemented to derive the proposed controller strategy. Finally in Soukkou
et al. (2016), a fractional order hyperchaotic system is stabilized by a general-
ized prediction based control where the control law is obtained by the
Lyapunov and fractional order system stability theorems.
The synchronization of fractional order chaotic and hyperchaotic systems
has been reported in the literature. For example, in Abedini et al. (2014),an
identical system synchronization using a fractional adaptation law for a 4D
Lu hyperchaotic system is shown where the fractional adaptation law is
implemented to reduce the convergence of the system parameters. In Su
et al. (2016), an interesting example related to the efficient numerical simula-
tion of fractional chaotic systems along with its synchronization is shown