124
7 Observer-Based Robust Fault Detection for Fuzzy Multi-model Jumping System
12i j (r ) =P 2 (r )A i (r ) + A
T
i (r )P 2 (r ) − H i (r )C j (r ) − C
T
j (r )H
T
i (r )
+
N
j=1
π i j P 2 (k) + Q 22 ,
2i j (r ) =
21i j (r )
0
0
22i j (r )
,
21i j (r ) = P 1 (r )
A hi (r ) + A h j (r )
,
22i j (r ) = P 2 (r )
A hi (r ) + A h j (r )
− H i (r )C h j (r ) − H j (r )C hi (r ),
3i j (r ) =
P 1 (r )
B di (r ) + B d j (r )
P 2 (r )
B di (r ) + B d j (r )
− H i (r )D d j (r ) − H j (r )D di (r )
,
4i j (r ) =
0
C i (r ) + C j (r )
T
,
5i j (r ) =
P 1 (r )
M i (r ) + M j (r )
P 2 (r )
M i (r ) + M j (r )
,
6i j (r ) =
−2Q 11 + α i j (r )N
T
hi (r )N hi (r ) + α ji (r )N
T
h j (r )N h j (r ) −2Q 12
∗
− 2Q 22
,
7i j (r ) =
0
C hi (r ) + C h j (r )
T
,
8i j (r ) = −2λ
2 I,
9i j (r ) =
D di (r ) + D d j (r )
T ,
Then, the parameter matrix of robust FDO can be obtained as H i (r ) = P
−1
2 (r )
H i (r ).
Proof For T > 0, the cost function for the error dynamic multi-model jumping
system (7.26) can be selected as:
J 1 (T ) = E
T
0
r
T
d (t)r d (t) − λ
2
ω
T
(t)ω(t)dt
.
(7.29)
Moreover, we can rewrite the index J 1 (T ) as: