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264 Mathematical Techniques of Fractional Order Systems
Crisp input Fuzzy Crisp output
Fuzzification inference Defuzzification
system
Knowledge
base
FIGURE 9.7 Fuzzy logic block diagram.
0.9
0.8
0.7
0.6
Output 0.5
0.4
0.3
0.2
0.1
–1 –0.8 –0.6 –0.4 –0.2 0 0.2 0.4 0.6 0.8 1
Sliding surface (S)
FIGURE 9.8 Input output surface plot.
9.5.3 Fractional Order Operator Implementation
The combination of fractional order mathematical operators with SMC
increases the DOF and as a result accurate solutions are attained which have
been used in the field of control in various applications (Sharma et al., 2014,
Kumar et al., 2015, 2016; Mishra et al., 2015). A fractional order differentia-
α
tor and integrator of a function gtðÞ is represented as, D gtðÞ and D 2β gtðÞ,
respectively, where 0 , α , 1 and 0 , β , 1. In the present work,
Gru ¨nwald Letnikov (GL) method (9.63) has been used to implement the
fractional order operators with a memory size of 100.
γ
γ 1 ½ðt2aÞ=h
X
a D gðtÞ 5 lim h-0 γ ð2 1Þ gðt 2 ihÞ ð9:63Þ
t
h i
i50
where D represents the derivative/integrator operator and γ (α, β) is the
order of fractional operator. In the present work, α is chosen as the order of
differentiator, whose value is found by GA.
