Page 663 - Mathematical Techniques of Fractional Order Systems

634 Mathematical Techniques of Fractional Order Systems

21.3.1.2 Definition of Riemann Liouville
The derivative in the sense of Riemann Liouville is based on the first primi-
tive or integral of a function f(t), is defined as Eq. (21.5):

1 d n ð t fðνÞ
p
α D fðtÞ 5 n p2n11 dν ð21:5Þ
t
Γðn 2 pÞ dt α ðt2νÞ
where Γ(.) the Euler’s Gamma function is defined for all nA@ and pA[n-1,
n[ as:
ð 1N
Γ nðÞ 5 t n21 2t ð21:6Þ
e dt
0

21.3.1.3 Definition of Caputo
The definition of the fractional derivative of f(t) in the sense of Caputo incor-
porates the initial conditions of the function to be treated, as well as its inte-
ger derivatives, the derivative of a function f(t) in the Caputo sense is
defined by the following relation (21.7):
n
t
d fðνÞ
1
ð
c p dt n dν
α D fðtÞ 5 p2n11 ð21:7Þ
t
Γðp 2 nÞ α ðt2νÞ
where Γ(.) the Gamma function, n is the smallest integer larger than the frac-
tional order p and pA]0, 1].
21.3.2 The n-Dimensional Fractional Order Chaotic System
The n-dimensional fractional order chaotic system is defined as following
relation (21.8):
P
α D X t 5 fðX t ; X 0 ; ΘÞ ð21:8Þ
t
n
0
where X t 5 ðx 1t ; x 2t ; ...; x nt Þ Aℜ is the n-dimensional state vector of the
original system; X 0 is the initial state of the system, P 5 ðp 1 ; p 2 ; ...; p n Þ Aℜ n
0
is a set of fractional order of the original system verifying that for all i 5 1,
n
2,..., n 0 , p i # 1, f: ℜ -ℜ n is the linear or nonlinear function and
m
Θ 5 ðθ 1 ; θ 2 ; ...; θ m Þ Aℜ is the n-dimensional vector value of original sys-
0
tem parameters.
21.3.3 The General Numerical Solution of the Fractional
Differential Equation and Stability
A remarkable property that distinguishes a fractional order derivation from an
integer derivation is that a noninteger derivation assumes a global character in
contrast to an integer derivation. It turns out that the fractional order derivative