Page 154 - Mathematical Techniques of Fractional Order Systems

142 Mathematical Techniques of Fractional Order Systems

Now, the time derivative of V 2 ðtÞ along the solution of (5.21a) is given as
follows

dV 2 ðtÞ 1 T 1 @τðtÞ T
5 x ðtÞxðtÞ2 12 x ðt2τðtÞÞxðt2τðtÞÞÞ
dt ð12@τ max Þ ð12@τ max Þ @t
ð5:32Þ
Under the Assumption (5.20), there exists an upper bound on the delay
first derivative. This allows us to obtain an upper bound of the Eq. (5.32)
dV 2 ðtÞ 1 T T
# x ðtÞxðtÞ 2 x ðt 2 τðtÞÞxðt 2 τðtÞÞ ð5:33Þ
dt ð1 2 @τ max Þ
Summing the two time derivatives of the Lyapunov functions V 1 ðtÞ and
V 2 ðtÞ yields to
N
ð
dVðtÞ T T
#2 2 ωμðωÞz ðω; tÞPzðω; tÞdω 1 2x ðtÞðPA 0 1 Λ 1 ÞxðtÞ
dt 0 ð5:34Þ
T T
1 2x ðtÞPA τ xðt 2 τðtÞÞ 2 x ðt 2 τðtÞÞΛ 2 xðt 2 τðtÞÞ
where Λ 1 5 1 I and Λ 2 5 1 I.
2ð1 2 @τ max Þ ð1 2 @τ max Þ
The inequality (5.34) is equivalent to
T
X ðtÞΩ 1 XðtÞ # 0 ð5:35Þ
where

2ðPA 0 1 Λ 1 Þ 2PA τ
Ω 1 5 ð5:36Þ
0 2Λ 2
and

XðtÞ 5 xðtÞ ð5:37Þ
xðt 2 τðtÞÞ
Notice that the matrix Ω 1 is not a symmetric matrix. That means inequal-
ity (5.35) can not be solved by using LMI solvers of Matlab, because it is not
a semidefinite programming (SDP) problem.
To overcome this situation, the matrix Ω 1 can be replaced by an equiva-
lent matrix Ω given by
Ω 1 1 Ω
T
Ω 5 1 ð5:38Þ
2
where

T PA τ
0
Ω 5 PA 0 1 A P 1 2Λ 1 ð5:39Þ
T
A P 2Λ 2
τ