Page 157 - Mathematical Techniques of Fractional Order Systems

Fractional Order Time-Varying-Delay Systems Chapter | 5 145

By substituting in (5.46) the expression of the pseudo-state giving by
(5.41), the following expression

ð N
dV 1 ðtÞ T T
52 2 ωμðωÞz ðω; tÞPzðω; tÞdω 1 2x ðtÞPðA 0 xðtÞ 1 A τ xðt 2 τðtÞÞÞ
dt 0
ð5:47Þ
is obtained.
It is easy to remark that the following inequality
ð N
T
22 ωμðωÞz ðω; tÞPzðω; tÞd # 0 ð5:48Þ
0
holds, for any symmetric positive definite matrix P.
As a result, sufficient conditions are given such that the following
inequalities V 1 ðtÞ . 0 and dV 1 ðtÞ # 0 are fulfilled. These two inequalities are
dt
equivalent to
P . 0 ð5:49aÞ
T
2x ðtÞPðA 0 xðtÞ 1 A τ xðt 2 τðtÞÞÞ # 0 ð5:49bÞ
In addition, a second Lyapunov Krasovskii function V 2 ðtÞ is defined as
ð t
T
V 2 ðtÞ 5 x ðsÞQxðsÞds ð5:50Þ
t2τðtÞ
Thus, its time derivative along the solution of (5.21a) is given as follows

dV 2 ðtÞ T @τðtÞ T
5 x ðtÞQxðtÞ 2 1 2 x ðt 2 τðtÞÞQxðt 2 τðtÞÞÞ ð5:51Þ
dt @t
and is bounded by
dV 2 ðtÞ T T
# x ðtÞQxðtÞ 2 ð1 2 @τ m Þ x ðt 2 τðtÞÞQxðt 2 τðtÞÞ ð5:52Þ
dt |fflfflfflfflfflffl{zfflfflfflfflfflffl}
Λ 1
Now, a third Lyapunov functional candidates V 3 ðtÞ
ð 0 ð t
V 3 ðtÞ 5 _ xðsÞZ _ xðsÞds dθ ð5:53Þ
t1θ
2τ m
is defined, which has a time derivative given as

ð 0 ð t
dV 3 ðtÞ
5 _ xðsÞZ _ xðsÞds dθ ð5:54Þ
dt t1θ
2τ m
ð t
T
5 τ m _ x ðtÞZ _ xðtÞ 2 _ xðsÞZ _ xðsÞds ð5:55Þ
t2τ m