Page 152 - Mathematical Techniques of Fractional Order Systems

140 Mathematical Techniques of Fractional Order Systems

xðtÞ 5 ψðtÞ; tA½ 2 τ m ; 0Š; 0 , α , 1 ð5:21bÞ
In the following theorem, the stability analysis of the system (5.21), by
using the indirect Lyapunov approach, leads to a sufficient time-delay-
independent stability condition given in a LMI formulation, which can be
solved easily.

Theorem 1: The unforced fractional order time-delay system (5.21) is
asymptotically stable if there exists a positive definite matrix P which satis-
fies the following LMI

T PA τ
Ω 5 PA 0 1 A P 1 2Λ 1 2Λ 2 , 0 ð5:22Þ
0

where
1 1
Λ 1 5 I; Λ 2 5 I:
2ð1 2 @τ max Þ ð1 2 @τ max Þ

and denotes the corresponding part of a symmetric matrix.
&

Proof 1: Firstly, The Lyapunov Krasovskii functional candidate as

VðtÞ 5 V 1 ðtÞ 1 V 2 ðtÞ ð5:23Þ

is chosen, where the first term V 1 ðtÞ of the Lyapunov function candidate VðtÞ
is defined as
ð N
V 1 ðtÞ 5 μðωÞv 1 ðω; tÞdω ð5:24aÞ
0
by summing all the monochromatic v 1 ðω; tÞ with the weighting function μðωÞ,
and the second term V 2 ðtÞ is defined as
t
ð
1
T
V 2 ðtÞ 5 x ðsÞxðsÞds ð5:24bÞ
ð1 2 @τ max Þ
t2τðtÞ
Now, to analyze the stability of system (5.21), the following monochro-
matic Lyapunov function candidate for the elementary frequency ω
T
v 1 ðω; tÞ 5 z ðω; tÞPzðω; tÞ ð5:25Þ
is considered, where PAR n 3 n is a symmetric positive definite matrix.
Following the idea of Trigeassou et al. (2011), by introducing the diffu-
n
sive representation of the fractional integral operator I given in Trigeassou
and Maamri (2009), the system (5.21) is rewritten in a diffusive form as
given by Lemma 1