80
5 Higher Order Moment Robust Filtering …
⎡
⎢
⎢
⎣
P 1 A tp + A
T
tp P 1 M 12 P 1 B dt p C
T
2tp − C
T
1tp D
T
F
∗
M 22 B F D dt p
−C
T
F
∗
∗ − γ
2 I
−D
T
dt p D
T
F
∗
∗
∗
−I
⎤
⎥
⎥
⎦ < 0,
(5.29)
where
M 12 = P 1 A tp + A
T
tp P 2
T
+ C
T
1tp B
T
F + A
T
F ,
M 22 = P 2 A tp + A
T
tp P 2
T
+ C
T
1tp B
T
F + B F C 1tp ,
then a filter in the form of (5.16) must satisfy the stability of the high-order moment
filtering error dynamic system (5.17) with a required finite frequency performance
γ. However,inequality (5.20) can not be solved directly by LMI toolbox due to the
unknown filter parameters inside. We need to change the non-LMI (5.20) into a LMI
form by variable alternation. Choose
P =
P 11 P 12
P
T
12 P 22
,
P
−1
=
S 11 S 12
S
T
12 S 22
.
Due to P P
−1
= I , we get
I − P 11 S 11 = P 12 S
T
12 .
Define
J =
S 11 I
S
T
12 0
, ˜
J =
I P 11
0 P
T
12
,
then P J = ˜
J . Pre- and post-multiply inequality (5.20) by diag{J
T
, I, I } its inverse,
and diag{S
−1
11 , I, I } gives
⎡
⎢
⎢
⎣
S
−1
11 A tp + A
T
tp S
−1
11 S
−1
11 A tp + A
T
tp P 11 + C
T
1tp Z
T
+ S
−1
11 Z
T
∗
A
T
tp P 11 + P 11 A tp + C
T
1tp Z
T
+ ZC 1tp
∗
∗
∗
∗
S
−1
11 B dt p
C
T
2tp − C
T
1tp D
T
f − S
−1
11
˜
Z
T
P 11 B dt p + Z D dt p
C
T
2tp − C
T
1tp D
T
f
−I
−D
T
dt p D
T
f
∗
− γ
2 I
⎤
⎥
⎥
⎦ < 0,
(5.30)
where
Z = P 12 B f , ˜
Z = C f S
T
12 , ˆ
Z = P 12 A f S
T
12 .