Page 594 - Mathematical Techniques of Fractional Order Systems

564 Mathematical Techniques of Fractional Order Systems

q is the order which can be any real number. For the purpose of handling
fractional calculus, the integro-differential operator is defined as:

d
8 q
> q . 0
>
dt
< q
q
D 5 ð19:1Þ
a t
1 q 5 0
>
>
Ð t
: 2q
ð dτÞ q , 0
a
The available literature mainly highlights three prominent definitions
which are used for general fractional order derivative or integral and these
are Grunwald Letnikov (GL), Riemann Liouville (RL), and the Caputo
definition. These popular definitions are elaborated as follows:
A. Grunwald Letnikov Definition
N
1 X j q
GL q
a D ftðÞ 5 lim q ð 21Þ fðt 2 jhÞ ð19:2Þ
t
h-0 h j
j50
B. Riemann Liouville Definition
For integral:
1 ð t
q
ℐ fðtÞ9 ð t2τÞ q21 f τðÞdτ ð19:3Þ
ΓðqÞ 0
For Derivative:
1 d m ð t fðτÞ
q
RL D ftðÞ 5 dτ
a t m q2m11
Γðm 2 qÞ dt a t2τð Þ
1
where, Γ mðÞ 5 m 2 1Þ!, t . 0; qAR m 2 1 , q , m
ð
C. Caputo Definition (Derivative)
t m
1 f ðτÞ
ð
C D ftðÞ 5 f τðÞdτ
q
a t q2m11 ð19:4Þ
Γðm 2 qÞ 0 t2τð Þ
In the case of fractional order control, the above three definitions are used
commonly. GL definition gives slightly inaccurate results during the initial
phase of the simulation. RL definition is basically for fractional order integra-
tions and it cannot be directly used for fractional order differentiation. The most
suitable definition out of the abovementioned definitions is the Caputo defini-
tion as in this case the initial conditions for fractional order differential equa-
tions (FODE) are similar to those for integer order differential equations. Under
the homogenous initial conditions, the RL and the Caputo derivatives are equiv-
alent. The Caputo definition is related to RL definition as,
m21 k2q
X ð t2τÞ
q
C
RL D ftðÞ 5 D ftðÞ 1 k
q
a t a t f ðaÞ
Γðk 2 q 1 1Þ
k50