Page 174 - Mathematical Techniques of Fractional Order Systems
P. 174
162 Mathematical Techniques of Fractional Order Systems
6.2 PRELIMINARIES
In this section, some Definitions, Lemmas, and Theorems are presented,
which will be used in the analyses performed throughout the chapter.
6.2.1 Fractional Calculus
Since the first known mention of derivatives of noninteger order, made by
Leibinz in a note to L’Hopital in 1695, the theory of fractional calculus
was developed mainly as a pure theoretical field of mathematics, useful
only for mathematicians. However, during the last decades of the 20th cen-
tury, some applications of fractional operators in science appeared, and the
interest of using these operators in more applied fields started to grow sig-
nificantly. Nowadays, it has been recognized that fractional calculus plays
an important role in science, with an increased use of fractional operators
in engineering.
In the time domain, the fractional order derivative (FOD) and the frac-
tional order integral (FOI) operators are defined by a convolution operation.
The Riemann Liouville FOI is one of the main concepts of fractional calcu-
lus, and is presented in Definition 1.
Definition 1: Riemann Liouville FOI (Kilbas et al., 2006). The
Riemann Liouville FOI of order α . 0 of a function f tðÞAℝ is defined as
α
I fðtÞ 5 1 ð t fðτÞ dτ; t . t 0 ; ð6:1Þ
t 0 12α
ΓðαÞ ðt2τÞ
t 0
where ΓαðÞ corresponds to the Gamma Function (Kilbas et al., 2006).
Regarding the FOD of order α . 0 of a function fðtÞAℝ, there exist sev-
eral definitions. The results presented in this work use the Caputo definition
given in Definition 2, which has been extensively used in the literature for
systems modeling and control.
Definition 2: Caputo FOD (Diethelm, 2010). Let α $ 0 and n 5 α ½, i.e.,
the integer part of α. According to (Diethelm, 2010,Def. 3.1), the
Caputo fractional derivative of order α . 0 of afunctionfðtÞAℝ is
defined as
1 ð t f ðnÞ ðτÞ
a
C D fðtÞ 5 dτ
t 0 α2n11 ð6:2Þ
Γðn 2 αÞ ðt2τÞ
t 0
whenever f ðnÞ AL 1 t 0 ; t.
½
