6.1 Robust FD Filter Design for Multi-model Jumping System
99
˜
i =
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎣
˜
11 12 13 14 15 16
∗ −ξ
2 I 0 24 0
0
∗
∗ 33 34 35 0
∗
∗
∗ −I 0
0
∗
∗
∗ 0 −λ i I 0
∗
∗
∗ ∗
∗ 66
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎦
< 0,
(6.11)
where
˜
11 =
⎡
⎣
˜
11 12 Q 13
∗ 22 Q 23
∗ ∗ 33
⎤
⎦ ,
˜
11 = P i A i + A
T
i P i + Q 11 + (π ii − β)P i ,
12 = A
T
i P i − X
T
i − C
T
i Y
T
i + Q 12 ,
22 = X
T
i + X i +
N
j=1 π i j P j + Q 22 − β P i ,
33 = P i A w + A
T
w P i +
N
j=1 π i j P j + Q 33 − β P i ,
12 =
⎡
⎣
P i B i P i B di
P i B f i
P i B i −Y
T
i D di P i B f i − Y
T
i D f i
0
0
P i B w
⎤
⎦ ,
13 =
⎡
⎣
P i A hi 0 0
−Y i C hi 0 0
0
0 0
⎤
⎦ ,
14 =
C Fi + D Fi C i −C Fi −C w
T ,
15 =
⎡
⎣
λ i P 1 M N i
0
0
0
0
⎤
⎦ ,
16 =
√ π i1 ˜
P i , · · · ,
√
π ii−1 ˜
P i ,
√ π ii+1 ˜
P i , · · · ,
√
π i N ˜
P i
,
24 =
0 D Fi D di D Fi D f i − D w
T ,
33 =
⎡
⎣
−Q 11 −Q 12 −Q 13
∗ −Q 22 −Q 23
∗
∗ −Q 33
⎤
⎦ ,
34 =
D Fi C hi 0 0
T ,
35 =
⎡
⎣
0 N hi
0 0
0 0
⎤
⎦ ,
66 = −diag
˜
P i , · · · , ˜
P i , ˜
P i , · · · , ˜
P i
,
˜
P i = diag{P i , P i , P i }.
In addition, the FD filter parameters are given as:
A Fi = P
−1
i X i , B Fi = P
−1
i Y i , C Fi = C Fi , D Fi = D Fi .
Proof Select the following stochastic Lyapunov–Krasovskii functional as: