Page 162 - Mathematical Techniques of Fractional Order Systems

150 Mathematical Techniques of Fractional Order Systems

2 3
Γ 11 Γ 12 Y
Ω 5 Γ 22 T 5 $ 0 ð5:64bÞ
4
Z
where

~ ~ T T T ~ ~ T ~ ~
ϒ 11 5 A 0 P 1 PA 1 B 0 Y 0 1 Y B 1 Y 1 Y 1 τ m Γ 11 1 Q
0 0 0
~ ~ T ~
ϒ 12 5 A τ 2 PY 1 T 1 τ m Γ 12
~ T
ϒ 13 5 τ m PA Z
0
T
ϒ 22 52 Λ 1 Q 2 T 2 T 1 τ m Γ 22
T
ϒ 23 5 τ m A Z
τ
ϒ 33 52 τ m Z
and
~
~
~
~
~
~
~
~
~
~
Y 5 PY; Q 5 PQ; T 5 T P; Γ 11 5 PΓ 11 ; Γ 12 5 PΓ 12 :
hold for a given scalars τ m and @τ m satisfying the Assumption (5.20),
where denotes the corresponding part of a symmetric matrix.
~
The pseudo-state feedback controller gain matrix is given as K 0 5 Y 0 P.
Proof 4: The proof of this Theorem is similar to the one given for the
Theorem 2, the explanation here is omitted for brevity. After replacing the
matrix A 0 by A 0 1 B u K 0 and following the same procedure as the proof of
Theorem 3, i.e., pre- and postmultiply the obtained inequality by the follow-
21 21
2 3 2 3 T
P 0 0 P 0 0
ing matrices 4 0 I 0 5 and 4 0 I 0 5 , respectively, and making
0 0 I 0 0 I
~ 21
some change of variables, where P 5 P yields the inequality (5.64). It
completes the proof.
We can remark that the feedback stabilization problem based on the
time-delay-dependent stability condition given by the inequality (5.64) is a
nonconvex problem. This is due to the presence of decision variable Zalone
~
T
and in the product PA Z, such that the matrix A 0 is not necessarily nonsin-
0
gular, which leads to a bilinear matrix inequality (BMI) structure. Then, the
inequality (5.64) can not be solved for the given decision variables in the
same time. Thanks to the Schur lemma, all diagonal components of a matrix
must satisfy the inequality. Therefore, a two-step procedure is proposed to
resolve this problem, as in Boukal et al. (2016a). Firstly, solving the first
~
~
component of the inequality (5.64) allows us to compute P and Y 0 . Then,
~
~
after obtaining P and Y 0 , replacing them into the inequality (5.4) by their
value leads to a feasible LMI.