130
7 Observer-Based Robust Fault Detection for Fuzzy Multi-model Jumping System
7.2.2 Design of FDO for Fuzzy Multi-model Jumping System
Theorem 7.3 For a given λ > 0, the error dynamic multi-model jumping system
(7.44) is stochastically stable and satisfies the condition (7.27), if there exist positive matrices P 1 (r ) = P
T
1 (r ), P 2 (r ) = P
T
2 (r ), Q 11 = Q
T
11 , Q 22 = Q
T
22 , matrix Q 12 ,
parameter matrix H i (r ) and scalars α i j (r ) satisfying the following LMIs:
i j (r ) =
⎡
⎢
⎢
⎢
⎢
⎣
1i j (r ) ) 2i j (r ) ) 3i j (r ) ) 4i j (r ) ) 5i j (r )
∗
6i j (r )
0
7i j (r )
0
∗
∗
8i j (r ) ) 9i j (r )
0
∗
∗
∗
−2I
0
∗
∗
∗
∗
−I
⎤
⎥
⎥
⎥
⎥
⎦
< 0,
(7.45)
where
1i j (r ) =
11i j (r ) + 11 ji (r )
Q 12
∗
12i j (r ) + 12 ji (r )
,
11i j (r ) = P 1 (r )A i (r ) + A
T
i (r )P 1 (r ) +
N
j=1 π i j P 1 (k) + Q 11 ,
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 −2Q 12
∗ −2Q 22
,
7i j (r ) =
0
C hi (r ) + C h j (r )
T
,