64
4 Finite-Frequency Robust Filtering for Multi-model Jumping System
By reconstruction, inequality (4.31) can be transformed into:
⊥
⊥T
< 0,
(4.32)
where
=
−I ˜
A ˜
B
, ,
⊥
=
˜
A
T I 0
˜
B
T I 0
, , =
0 I 0
,
=
⎡
⎣
I 0
0 I
0 0
⎤
⎦
−Q l P l
P l l
2 Q l
I 0 0
0 I 0
+
⎡
⎣
0 0
˜
C
T 0
˜
D
T I
⎤
⎦
I 0
0 −γ
2 I
0 ˜
C ˜
D
0 0 I
.
By Lemma 1.6, inequality (4.32) is equivalent to the following inequality:
F
T
T
+ F
T
+ < 0,
(4.33)
By recombination, it can be demonstrated that the equivalent requirement of inequality (4.33) is
⎡
⎣
−Q l
P l − F
T
0
∗ l
2 Q l + F ˜
A + ˜
A
T F
T F ˜
B
∗
∗
− γ
2 I
⎤
⎦ +
⎡
⎣
0
˜
C
˜
D
⎤
⎦
⎡
⎣
0
˜
C
˜
D
⎤
⎦
T
< 0
(4.34)
By Schur complement Lemma, inequality (4.34) can be transformed to
⎛
⎜
⎜
⎝
−Q l
P l − F
T
0
0
∗ l
2 Q l + F ˜
A + ˜
A
T F
T F ˜
B ˜
C
T
∗
∗
− γ
2 I ˜
D
T
∗
∗
∗ − I
⎞
⎟
⎟
⎠ < 0
(4.35)
Let
¯
A F = F 3 A F , ¯
B F = F 3 B F , ¯
C F = C F , ¯
D F = D F .
and choose the structures of Q l , P l , and F as,
P l =
P l1 P l2
∗ P l3
, Q l =
Q l1 Q l2
∗ Q l3
, F =
F 1 F 3
F 2 F 3
.