4.2 Multiple Frequency Robust Filtering for Multi-model Jumping System
63
0, Q l =
Q l1 Q l2
∗ Q l3
> 0, Q h =
Q h1 Q h2
∗ Q h3
> 0 and matrices F 1 , F 2 , F 3 , ¯
A F , ¯
B F ,
¯
C F , ¯
D F with approximate dimensions such that
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎣
−Q l1 −Q l2 P l1 − F
T
1 P l2 − F
T
2
0
0
∗ −Q l3 P
T
l2 − F
T
3 P l3 − F
T
3
0
0
∗
∗
M l33
M l34 F 1 B d + ¯
B F D d
C
T
2
∗
∗
∗
M l44 F 2 B d + ¯
B F D d − ¯
C
T
F
∗
∗
∗
∗
−γ
2 I
D d − ¯
D
T
F
∗
∗
∗
∗
∗
−I
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎦
< 0,
(4.28)
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎣
Q h1 Q h2 P h1 − F 1 P h2 − F 3
0
0
∗ Q h3 P
T
h2 − F 2 P h3 − F 3
0
0
∗ ∗
M h33
M h34 F 1 B d + ¯
B F D d
C
T
2
∗ ∗
∗
M h44 F 2 B d + ¯
B F D d − ¯
C
T
F
∗ ∗
∗
∗
−ρ
2 I
D d − ¯
D
T
F
∗ ∗
∗
∗
∗−I
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎦
< 0,
(4.29)
A
T F
T
1 + F 1 A + C
T
1
¯
B F
T + ¯
B F C 1 A
T F
T
2 + C
T
1
¯
B F
T + ¯
A F
∗
¯
A F
T + ¯
A F
< 0,
(4.30)
where
M l33 = l
2 Q l1 + F 1 A + ¯
B F C 1 +
F 1 A + ¯
B F C 1
T ,
M l34 = l
2 Q l2 + ¯
A F +
F 2 A + ¯
B F C 1
T ,
M l44 = l
2 Q l3 + ¯
A F + ¯
A F
T ,
M h33 = − h
2 Q h1 + F 1 A + ¯
B F C 1 +
F 1 A + ¯
B F C 1
T ,
M h34 = − h
2 Q h2 + ¯
A F +
F 2 A + ¯
B F C 1
T ,
M h44 = − h
2 Q h3 + ¯
A F + ¯
A F
T .
Then, the filtering error dynamic system (4.24) is said to be stable and satisfies the
predefined multiple frequency performances. Furthermore, the corresponded multiple frequency filter parameters could be obtained by
A F = F
−1
3
¯
A F , B F = F
−1
3
¯
B F , C F = ¯
C F , D F = ¯
D F .
Proof From GKYP Lemma, the low-frequency constraint (4.25) is equivalent to
˜
A ˜
B
I 0
T −Q l P l
P l l
2 Q l
˜
A ˜
B
I 0
+
˜
C ˜
D
0 I
T
I 0
0 −γ
2 I
˜
C ˜
D
0 I
< 0.
(4.31)