4.1 Robust Finite Frequency Filter Design for Multi-model Jumping System
57
κ 2 =
I 0
0 − γ
2 I
0 ˜
C ˜
D
0 0 I
.
Define
=
−I ˜
A ˜
B
, , =
0 I 0
,
=
⎡
⎣
I 0
0 I
0 0
⎤
⎦ κ 3
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
,
where
κ 3 =
−Q
P + j c Q
P − j c Q − 1 2 Q
.
Define
=
−I ˜
A ˜
B
,
=
0 I 0
.
Then according to Lemma 4.1, the obtained inequality is equivalent to the following expression:
F
T
T
+ F
T
+ < 0,
(4.12)
By recombination, it can be demonstrated that the equivalent requirement of inequality (4.12) is
⎡
⎣
−Q
P+ j c Q − F
T
0
∗ − 1 2 Q + F ˜
A + ˜
A
T F
T F ˜
B
∗
∗
− γ
2 I
⎤
⎦ +
⎡
⎣
0
˜
C
˜
D
⎤
⎦
⎡
⎣
0
˜
C
˜
D
⎤
⎦
T
< 0.
(4.13)
Based on Lemma 1.3, we can get inequality (4.13) from (4.9). It implies that the
desired disturbance attenuation index G ˜
e ˜
w ( jω ω ω)
1 <ω ω ω< 2
∞
< γ is satisfied by Theorem 4.1 in the middle-frequency bands. This completes the proof.
B. Finite Frequency Filter Design
Theorem 4.2 For given scalars 2 > 1 and γ, if there exist symmetric matrices
P =
P 1 P 2
∗ P 3
> 0, Q =
Q 1 Q 2
∗ Q 3
> 0 and matrices F 1 , F 2 , F 3 , ¯
A F , ¯
B F , ¯
C F with
approximate dimension such that