146
8 Filtering-Based Robust Fault Detection of Fuzzy Multi-model Jumping System
where i j (r ) = [A mn (r )] 11×11 , with
A 0101 (r )=P(r )A i (r ) + A T
i (r )P(r ) +
N
j=1 π i j P(k) + Q 11 + λ i (r )N T
1i (r )N 1i (r ),
A 0102 (r ) = A
T
i (r )P(r ) − X
T
i (r ) − C
T
j (r )Y
T
i (r ) + Q 12 ,
A 0103 (r ) = Q 13 ,
A 0104 (r ) = P(r )A hi (r ) + λ i (r )N
T
1i (r )N hi (r ),
A 0107 (r ) = P(r )B i (r ) + λ i (r )N
T
1i (r )N 2i (r ),
A 0108 (r ) = P(r )B di (r ),
A 0109 (r ) = P(r )B f i (r ),
A 0110 (r ) = C
T
Fi (r ) + C
T
j (r )D
T
Fi (r ),
A 0111 (r ) = P(r )M i (r ),
A 0202 (r ) = X i (r ) + X
T
i (r ) +
N
j=1 π i j P(k) + Q 22 ,
A 0203 (r ) = Q 23 ,
A 0204 (r ) = −Y i (r )C hi (r ),
A 0208 (r ) = −Y i (r )D di (r ),
A 0209 (r ) = −Y i (r )D f i (r ),
A 0210 (r ) = −C
T
Fi (r ),
A 0303 (r ) = P(r )A w + A
T
w P(r ) +
N
j=1 π i j P(k) + Q 33 ,
A 0310 (r ) = C
T
w ,
A 0404 (r ) = Q 11 + λ i (r )N
T
hi (r )N hi (r ),
A 0405 (r ) = −Q 12 ,
A 0406 (r ) = −Q 13 ,
A 0407 (r ) = λ i (r )N
T
hi (r )N 2i (r ),
A 0410 (r ) = C
T
h j (r )D
T
Fi (r ),
A 0505 (r ) = −Q 22 ,
A 0506 (r ) = −Q 33 ,
A 0606 (r ) = −Q 33 ,
A 0707 (r ) = −γ
2 I + λ i (r )N
T
2i (r )N 2i (r ),
A 0808 (r ) = −y
2 I,
A 0810 (r ) = D
T
d j (r )D
T
Fi (r )
A 0009 (r ) = −γ
2 I,
A 0910 (r ) = D
T
f j (r )D
T
Fi (r ) − D
T
w ,
A 1010 (r ) = −1,
A 1111 (r ) = −λ i (r )I,
A mn (r ) = 0.
Furthermore, the parameter matrices of the robust FDF system are determined as:
A Fi (r ) = P −1 (r )X i (r ), B Fi (r ) = P −1 (r )Y i (r ), C Fi (r ) = C Fi (r ), D Fi (r ) = D Fi (r ).
Proof For T > 0, we present the following cost function for the overall FDF system
(8.11):
J (T ) = E
T
0
r
T
e f (t)r e f (t) − γ
2
w
T
(t)w(t)dt
.
(8.34)