Page 159 - Mathematical Techniques of Fractional Order Systems

Fractional Order Time-Varying-Delay Systems Chapter | 5 147

The inequality ϒ 1 , 0 is equivalent by using the Schur complement for-
mula to
~ ~ ~
2 3
Φ 11 Φ 12 Φ 13
~ ~ ~ ~
ϒ 1 5 Φ 21 Φ 22 Φ 23 5 , 0 ð5:59Þ
4
~ ~ ~
Φ 31 Φ 32 Φ 33
where
~
Φ 11 5 2PA 0 1 2Y 1 τ m Γ 11 1 Q
~
Φ 12 5 2PA τ 2 2Y 1 τ m Γ 12
~ T
Φ 13 5 τ m A Z
0
~ T
Φ 21 51 τ m Γ 1 2T
12
~
Φ 22 52 Λ 1 Q 2 2T 1 τ m Γ 22
~ T
Φ 23 5 τ m A Z
τ
~
Φ 31 5 τ m ZA 0
~
Φ 32 5 τ m ZA τ
~
Φ 33 52 τ m Z
~
Notice that at this stage the matrices ϒ 1 and Ω 1 are not symmetric ones.
~
That means, inequalities ϒ 1 , 0 and Ω 1 $ 0 can not be solved by using LMI
solvers of Matlab, because it is not an SDP problem.
~
To overcome this situation, matrices ϒ 1 and Ω 1 can be replaced by
equivalent ones as
~ ~ T
ðϒ 1 1ϒ Þ
1
ϒ5
2
2 T T T T T 3
PA 0 1A P 1Y1Y 1τ m Γ 11 1Q PA τ 2Y1T 1τ m Γ 12 τ m A Z
0 0
T
T
5 4 2Λ 1 Q2T 2T 1τ m Γ 22 τ m A Z ,0
τ
5
2τ m Z
ð5:60aÞ
T
ðΩ 1 1Ω Þ
1
Ω 5
2
2 3 ð5:60bÞ
Γ 11 Γ 12 Y
5 Γ 22 T 5 $0
4
Z
Then, under the assumption (5.20), the system (5.21) is asymptotically
stable if the LMI conditions given by Eq. (5.60) are satisfied. This completes
the proof.
Remark 2: By requiring that the matrices T ; Y and Γ are equal to zero,
and the spectral radius of the matrix Z is chosen small enough, the inequal-
ities (5.60) given in Theorem 2 are reduced to the delay-independent stabil-
ity condition as given in Theorem 1 in Boukal et al. (2016b).