Page 592 - Mathematical Techniques of Fractional Order Systems

562 Mathematical Techniques of Fractional Order Systems

order greater than three, but recently, researchers have established that some
fractional order differential systems exhibit chaotic behavior for total order
less than three, such as fractional order Chen system (Wang et al., 2006;
Li and Peng, 2004), fractional order Lorenz system (Grigorenko and
Grigorenko, 2003; Yu et al., 2009). Various other fractional order chaotic
systems have also been studied in recent years such as, Chua’s circuit
(Hartley et al., 1995), Ro ¨ssler’s system (Li and Chen, 2004), Lu system
(Lu, 2006; Ouannas et al., 2017a,b,c,d,e,i), etc. Some fractional order
hyperchaotic systems have also been analyzed and presented in literature
(Yang and Liu, 2013; Liu and Lu, 2010), such as hyperchaotic Ro ¨ssler’s sys-
tem (Zhang et al., 2005), hyperchaotic Henon map (Shukla and Sharma,
2017e), Lorenz Stenflo system (Wang et al., 2014), etc. Various techniques
mentioned above have also been utilized for control of various classes of
chaotic systems such as sliding mode control (Chen et al., 2012; Aghababa,
2012), active control (Shukla and Sharma, 2015; Azar et al., 2017a) etc.
Synchronization of integer order systems has been studied extensively and
nonlinear control methodologies used for such systems are also extended to
synchronize fractional order systems, (Pham et al., 2017; Asheghan et al.,
2011; Mohadeszadeh and Delavari, 2017; Wang and Zhang, 2006; Agrawal
et al., 2012; Borah et al., 2016).
Backstepping control developed by Krstic et al. (1995) is one of the most
popular nonlinear techniques of controller design and has been used inten-
sively for the control and synchronization of integer order nonlinear systems.
It employs a systematic approach based on Lyapunov theory where the uncer-
tainty can be linearly parameterized and guarantees global stability of systems.
The state variables are treated as independent controllers for subsystems and
subsequently each step leads to an updated control law for the following step.
The controller algorithm for each step is chosen such that the corresponding
Lyapunov function is satisfied which further guarantees the stability of each
subsystem. A similar approach can be used for controller design for fractional
order nonlinear systems provided there exists some methodology to exploit
the existing Lyapunov stability analysis. Extension of Lyapunov theory to
fractional order nonlinear systems along with development of Mittag Leffler
stability concept is presented in Li et al. (2010). Use of these stability con-
cepts for design of nonlinear feedback controllers for fractional order nonlin-
ear systems is the topic of interest and it has been exploited in various
research works (Wei et al., 2016; Zhao et al., 2016; Ding et al., 2015; Shukla
and Sharma, 2017a,b,c,d,e; Wang et al., 2006; Shukla and Sharma, 2018).
Active backstepping approach (Zhang et al., 2005) is a combination of
active control and backstepping approach. Unlike backstepping strategy, it
involves design of multiple controllers but the step-wise procedure is derived
from backstepping approach design methods to control chaos. For a wider
class of strict-feedback system, backstepping design guarantees global
asymptotic stabilities, tracking, and transient performance. Active control