36
3 Finite-Time Robust Filtering for Multi-model Jumping System
Furthermore, the designed filter parameters are:
A Fi = P
−1
i X i , B Fi = P
−1
i Y i , C Fi = C Fi
(3.30)
Proof Recalling to inequality (3.12), we ˜
P i = diag [P i , P i ],
˜
Q = diag [Q, Q], and have
+ < 0
(3.31)
where
=
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎣
11i 12i 13i 0
15i
16i
∗ 22i 23i 0
25i
26i
∗
∗ −Q 0
0
0
∗
∗
∗ −Q
0
0
∗
∗
∗ ∗ −γ
2 e
−ηT I 0
∗
∗
∗ ∗
∗
−I
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎦
,
=
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎣
P i A i + A
T
1 P i A
T
1 P i P i A hi 0 0 0
P i A i
0
P i A hi 0 0 0
A
T
hi P i
A
T
hi P i
0
0 0 0
∗
∗
∗
0 0 0
∗
∗
∗
∗0 0
∗
∗
∗
∗∗0
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎦
,
11i = P i A i + A
T
i P i +
N
j=1 π i j P j − η P i + Q,
12i = A
T
i P i − A
T
Fi P i − C
T
i B
T
Fi P i ,
13i = P i A hi ,
15i = P i B i ,
16i = E
T
i − C
T
Fi ,
22i = P i A i + A
T
i P i +
N
j=1 π i j P j − η P i + Q,
23i = P i A hi − P i B Fi C hi ,
25i = P i B i − P i B Fi D i ,
26i = C Fi .
Recalling to Lemma 1.6, can be rewritten as:
= L 11 i (t)L 12 + L
T
12
T
i (t)L
T
11 < δ
−1
i L 11 L
T
11 + δ i L
T
12 L 12
(3.32)
where
L 11 (r ) = col [P i M 1i P i M 1i 0 0 0 0] ,
L 12 (r ) = [N i 0 N hi 0 0 0].
Then inequality (3.31) can be derived as