Page 156 - Mathematical Techniques of Fractional Order Systems

144 Mathematical Techniques of Fractional Order Systems

hold for a given scalars τ m and @τ m satisfy the Assumption (5.20), where
denotes the corresponding part of a symmetric matrix.
&

Proof 2: From Lemma 1, Eq. (5.21) can be expressed as
@zðω; tÞ
52 ωzðω; tÞ 1 A 0 xðtÞ 1 A τ xðt 2 τðtÞÞ ð5:41aÞ
@t
ð N
xðtÞ 5 μðωÞzðω; tÞdω ð5:41bÞ
0
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 ð5:42Þ
is considered, where P 5 P AR n 3 n is a symmetric positive definite matrix.
T
Then, the derivative 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 ð5:43Þ
@zðω; tÞ

and

@v 1 ðω; tÞ @v 1 ðω; tÞ @zðω; tÞ
5
@t @zðω; tÞ @t ð5:44Þ
T
5 2z ðω; tÞPð2 ωzðω; tÞ 1 A 0 xðtÞ 1 A τ xðt 2 τðtÞÞÞ
Furthermore, a Lyapunov function candidate V 1 ðtÞ is defined by summing
all the monochromatic v 1 ðω; tÞ with the weighting function μðωÞ, i.e., V 1 ðtÞ is
given by
ð N
V 1 ðtÞ 5 μðωÞv 1 ðω; tÞdω ð5:45Þ
0
The Lyapunov function (5.45) dynamics along the solution trajectories of
(5.41) is given as

ð N
dV 1 ðtÞ @v 1 ðω;tÞ
5 μðωÞ dω
dt 0 @t
N
ð
T
5 μðωÞ2z ðω;tÞPð2ωzðω;tÞ1A 0 xðtÞ1A τ xðt2τðtÞÞÞ dω
N ð5:46Þ
0 ð
T
522 ωμðωÞz ðω;tÞPzðω;tÞ dω
0
ð N
T
12 μðωÞz ðω;tÞP dω ðA 0 xðtÞ1A τ xðt2τðtÞÞÞ
0