42
3 Finite-Time Robust Filtering for Multi-model Jumping System
ϒ i =
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎣
ϒ 11i ϒ 12i ϒ 13i 0 ϒ 15i ϒ 16i
∗ ϒ 22i ϒ 23i ϒ 24i ϒ 25i ϒ 26i
∗
∗ −Q 0
0
0
∗
∗
∗ −Q 0
0
∗
∗
∗
∗ −I 0
∗
∗
∗
∗
0 −δ I
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎦
< 0
(3.57)
⎡
⎣
−P i 0 E
T
i − C
T
Fi
∗ −P i
C
T
Fi
∗ ∗
−γ
2 I
⎤
⎦ < 0
(3.58)
R i < P i < σ 1 R i
(3.59)
0 < Q < σ 2 R i
(3.60)
−e
−αT c 2 + c 1 τ σ 2 +
α
1 − e
−αT
√
c 1
√ c 1
−σ
< 0
(3.61)
where
ϒ 11i = P i A i + A
T
i P i +
N
j=1 π i j P j + Q − αP i + δ i N
T
i N i ,
ϒ 12i = A
T
i P i − C
T
i Y i − X
T
i ,
ϒ 13i = P i A hi + δ i N
T
i N hi ,
ϒ 15i = P i B i ,
ϒ 16i = P i M i ,
ϒ 22i = X i + X
T
i +
N
j=1 π i j P j + Q − αP i ,
ϒ 23i = −Y i C hi ,
ϒ 24i = P i A hi ,
ϒ 25i = P i B i − Y i D i ,
ϒ 26i = P i M i .
Furthermore, the designed filter parameters can be derived as
A Fi = P
−1
i X i , B Fi = P
−1
i Y i , C Fi = C Fi
(3.62)
Proof Recalling to Theorem 3.3, we set ˜
P i = diag [P i P i ] and ˜
Q = diag [Q Q].
Inequality (3.42) can be written as
+ < 0
(3.63)
where