152
8 Filtering-Based Robust Fault Detection of Fuzzy Multi-model Jumping System
ii (r ) =
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
1 2 Q 13 P(r )A hi (r )
0
0
∗ 3 Q 23 −Y i (r )C hi (r ) 0
0
∗ ∗ 4
0
0
0
∗ ∗ ∗
Q 11
−Q 12 −Q 13
∗ ∗ ∗
∗
−Q 22 −Q 33
∗ ∗ ∗
∗
−Q 22 0
∗ ∗ ∗
∗
∗ −Q 33
∗ ∗ ∗
∗
∗
∗
∗ ∗ ∗
∗
∗
∗
∗ ∗ ∗
∗
∗
∗
∗ ∗ ∗
∗
∗
∗
P(r )B di (r ) P(r )B f i (r ) C
T
Fi (r ) + C
T
j (r )D
T
Fi (r )
−Y i (r )D di (r ) −Y i (r )D f i (r )
−C
T
Fi (r )
0
0
C
T
w
0
0
C
T
h j (r )D
T
Fi (r )
0
0
0
0
0
0
0
0
0
0
0
0
−y
2 I
0
0
∗
− γ
2 I
D
T
f j (r )D
T
Fi (r ) − D
T
w
∗
∗
− I
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
,
with
1 = P(r )A i (r ) + A
T
i (r )P(r ) +
N
j=1 π i j P(k) + Q 11 ,
2 = A
T
i (r )P(r ) − X
T
i (r ) − C
T
j (r )Y
T
i (r ) + Q 12 ,
3 = X i (r ) + X
T
i (r ) +
N
j=1 π i j P(k) + Q 22 ,
4 = P(r )A w + A
T
w P(r ) +
N
j=1 π i j P(k) + Q 33 .
Proof For T > 0, we present the following cost function for overall FDF system
(8.47):
J (T ) = E
T
0
r
T
e f (t)r e f (t) − γ
2
w
T
(t)w(t)dt
.
(8.50)
Then, we can rewrite the index J (T ) as:
J (T ) = E
T
0
r
T
e f (t)r e f (t)dt
− γ
2
T
0
w
T
(t)w(t)dt
= E
T
0
r
T
e f (t)r e f (t) − γ
2
w
T
(t)w(t) + +V
ˆ
x(T ), r T
dt
− E
V
ˆ
x(T ), r T
≤ E
T
0
ˆ
x
T
(t) ˆ
x
T
h w
T
(t)
i j (r ) + i j (r )
ˆ
x
T
(t) ˆ
x
T
h w
T
(t)
T ,
(8.51)