Page 147 - Mathematical Techniques of Fractional Order Systems

Fractional Order Time-Varying-Delay Systems Chapter | 5 135

I and 0 denote the identity matrix and zero matrix of appropriate dimensions,
respectively. I n is the identity matrix of dimensions n 3 n; SymfXg is used to
T
denote X 1 X. The symbol “ ” corresponds to the convolution operator; and

“ ” denotes the corresponding part of a symmetric matrix.

5.2 PRELIMINARY RESULTS
This section will be started by providing a brief recall of some useful defini-
tions of the fractional order derivatives and Newton Leibniz formula.
The most known definitions of noninteger order derivatives, which are
used in the literature, are divided into two main classes. On one hand, the
Riemann Liouville derivative defined as (Podlubny, 1999; Das, 2008).

α
RL D fðtÞ 5 1 d n ð t fðτÞ dτ;
a t n α2n11 ðn 2 1Þ # α , n ð5:1Þ
Γðn 2 αÞ dt a ðt2τÞ
Or the Caputo derivative on the other (Das, 2008; Podlubny, 1999),
which arise as a result of a simple permutation of integral and derivative in
the Eq. (5.1). The noninteger order derivative of a causal time function intro-
duced by Caputo is then defined as
n
d fðτÞ
α
C D fðtÞ 5 1 ð t dt n dτ; ðn 2 1Þ , α , n ð5:2Þ
a t α2n11
Γðn 2 αÞ a ðt2τÞ
1
with nAℕ and αAR , where the Gamma function Γ: ð0;NÞ-R is
defined as
N
ð
ΓðαÞ 5 τ α21 expð2 τÞdτ; ð5:3Þ
0
The physical interpretation of the fractional derivatives and the solution
of fractional differential equations are given in Podlubny (2002) and Das
(2008). In Podlubny (1999), the author explain the differences between these
definitions, and when they are equivalent. In this work, the definition intro-
duced by Riemann Liouville, given in (5.1), is considered.
n
The diffusive representation of the fractional integral operator I given in
Trigeassou and Maamri (2009) which plays an important role to derive our
main result, is stated in the following definition.
Definition 1: Consider a linear system such as:

yðtÞ 5 hðtÞ uðtÞ ð5:4Þ
where is the convolution operator.