Page 153 - Mathematical Techniques of Fractional Order Systems

Fractional Order Time-Varying-Delay Systems Chapter | 5 141

@zðω; tÞ
52 ωzðω; tÞ 1 A 0 xðtÞ 1 A τ xðt 2 τðtÞÞ ð5:26aÞ
@t
ð N
xðtÞ 5 μðωÞzðω; tÞdω ð5:26bÞ
0
Then, the derivatives of the Lyapunov function v 1 ðtÞ with respect to zðω; tÞ
and t are given by

@v 1 ðω; tÞ T
5 2z ðω; tÞP ð5:27Þ
@zðω; tÞ
and

@v 1 ðω; tÞ @v 1 ðω; tÞ @zðω; tÞ
5
@t @zðω; tÞ @t ð5:28Þ
T
5 2z ðω; tÞPð2 ωzðω; tÞ 1 A 0 xðtÞ 1 A τ xðt 2 τðtÞÞÞ
The dynamics of the Lyapunov function (5.24a) along the solution trajec-
tories of (5.26) is given as

ð N
dV 1 ðtÞ @v 1 ðω; tÞ
5 μðωÞ dω
dt 0 @t
ð N
T
5 μðωÞ2z ðω; tÞPð2 ωzðω; tÞ 1 A 0 xðtÞ 1 A τ xðt 2 τðtÞÞÞ dω
N ð N
0 ð
T
T
52 2 ωμðωÞz ðω; tÞPzðω; tÞ dω 1 2 μðωÞz ðω; tÞPdω ðA 0 xðtÞ
0 0
1 A τ xðt 2 τðtÞÞÞ
ð5:29Þ
By substituting the pseudo-state giving by diffusive representation (5.26)
in the Eq. (5.29) leads to
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
ð5:30Þ
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 ð5:31aÞ

T
2x ðtÞPðA 0 xðtÞ 1 A τ xðt 2 τðtÞÞÞ , 0 ð5:31bÞ