Page 148 - Mathematical Techniques of Fractional Order Systems

136 Mathematical Techniques of Fractional Order Systems

Its impulse response hðtÞ verify the pseudo-Laplace transform definition
is given by (Matignon, 1994):

ð N
hðtÞ 5 μðωÞe 2ωt dω ð5:5Þ
0
where μðωÞ is called the diffusive representation (Montseny, 1998) (or
frequency weighting function) of the impulse response hðtÞ, which has the
following form

sinðαπÞ 2α
μðωÞ 5 ω ð5:6Þ
π
&

Lemma 1: (Trigeassou et al., 2011) Consider a nonlinear fractional deriva-
tive system:
RL α
D xðtÞ 5 fðxðtÞÞ ð5:7Þ
t 0 t
due to the continuous frequency distributed model of the fractional integra-
tor, the nonlinear fractional system (5.7) can be expressed as:
@zðω; tÞ
52 ωzðω; tÞ 1 fðxðtÞÞ ð5:8aÞ
@t
N
ð
xðtÞ 5 μðωÞzðω; tÞdω ð5:8bÞ
0
where μðωÞ is given by (5.6)

Remark 1: In the sequel, the initial time t 0 is equal to zero. Then, the nota-
α
α
tion RL D can be replaced by D without loss of generality.
a t
In Komornik (2016), it is shown that the following Newton Leibniz
formula
ð t
_ xðsÞds 5 xðtÞ 2 xðt 2 τðtÞÞ ð5:9Þ
t2τðtÞ
is one of the most important formula used in the theorems proofs of classical
analysis, and its validity it was extended to the Lebesgue integrable func-
tions. This formula will be used in this work later to derive a delay-
dependent condition stability.
In addition, the following relationship between the terms of the equality
(5.9)

ð t
T T

2 x ðtÞY 1 x ðt 2 τðtÞÞT 3 xðtÞ 2 xðt 2 τðtÞ 2 _ xðsÞds 5 0 ð5:10Þ
t2τðtÞ
|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}
0
is satisfied, for any matrices Y and T with appropriate dimension.