Page 466 - Mathematical Techniques of Fractional Order Systems

452 Mathematical Techniques of Fractional Order Systems

Practical applications of synchronization can be found in communication
systems where it is used for encoding of information. It is also useful in
understanding of biological phenomena and robotics (Ojo and Ogunjo,
2012).
Different types of synchronization such as generalized synchronization,
lag synchronization, complete synchronization, phase synchronization, and
projective synchronization have been introduced (Vaidyanathan et al., 2015a,
b,c; Azar and Vaidyanathan, 2015b; Zhu and Azar, 2015; Vaidyanathan and
Azar, 2015a,b,c,d, 2016a,b,c,d,e,f; Boulkroune et al., 2016a,b). Furthermore,
different techniques for synchronization have been proposed. These include:
Open Plus Close loop (OPCL), active control, direct method, backstepping,
etc. Synchronization of chaotic systems began with identical systems. It has
evolved to include synchronization between different systems, increased and
reduced order systems, compound synchronization, combination combination
synchronization and many other forms (Vincent et al., 2015; Ojo et al.,
2013b, 2014a,b). Switching synchronization of chaotic systems is one in
which the different states of the response system are synchronized with the
desired state of the drive system (Uar et al., 2008; Zheng, 2016; Ajayi et al.,
2014). Vincent et al. (2015) opined that the synchronization of different states
of the slave system synchronizing with desired states of the master in the
master slave configuration will improve security of information transmission
through synchronization.
From the Gru ¨nwald Letnikov definition of fractional order systems, the
fractional order derivative of order α can be written as (Petras, 2011)

0 1
N α
α 1 X j
D fðtÞ 5 lim ð21Þ @ A fðt 2 jhÞ ð15:1Þ
j
t h-0 h α
j50
where the binomial coefficients can be written in terms of the Gamma func-
tion as
α Γðα 1 1Þ

5
j Γðj 1 1ÞΓðα 2 j 1 1Þ
The Riemann Liouville definition of fractional derivative is given as
2α 1 d n ð t fðτÞ
D fðtÞ 5 dτ ð15:2Þ
t n α11
Γðn 2 αÞ dt a ðt2τÞ
The Caputo fractional derivatives can be written as

α
D fðtÞ 5 1 ð t f ðnÞ ðτÞ dτ; n 2 1 , α , n ð15:3Þ
t α2n11
Γðn 2 αÞ a ðt2τÞ
If the order of fractional order are the same ðα 1 5 α 2 5 α 3 5 α n 5 αÞ, the
system is called commensurate; otherwise, it is called incommensurate