7.1 Robust FDO Design for Fuzzy Multi-model Jumping System
125
2J 1 (T )
= E
T
0
2r
T
d (t)r d (t)dt
− 2λ
2 E
T
0
ω
T
(t)ω(t)dt
= E
T
0
2r
T
d (t)r d (t) − 2λ
2
ω
T
(t)ω(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 (t) ω
T
(t)
⎡
⎣
S
i=1
h i
S
j=1
h j
X i j (r ) + i j (r )
⎤
⎦
ˆ
x
T
(t) ˆ
x
T
h (t) ω
T
(t)
T
,
(7.30)
where
i j (r ) =
1
2
⎡
⎢
⎢
⎢
⎢
⎣
ˆ
C i j (r ) + ˆ
C ji (r )
T
ˆ
C hi j (r ) + ˆ
C h ji (r )
T
ˆ
D di (r ) + ˆ
D d j (r )
T
⎤
⎥
⎥
⎥
⎥
⎦
ˆ
C i j (r ) + ˆ
C ji (r )
ˆ
C hi j (r ) + ˆ
C h ji (r )
ˆ
D di (r ) + ˆ
D d j (r )
,
X i j (r ) =
⎡
⎢
⎣
i P(r )
ˆ
A hi j (r ) + ˆ
A h ji (r )
P(r )
ˆ
B di (r ) + ˆ
B d j (r )
−2Q
0
∗
∗
− 2λ
2 I
⎤
⎥
⎦.
As T → ∞, X i j (r ) + i j (r ) < 0, we can get E
∞
0 r d
T r d dt
λ
2 E
∞
0 ω
T
ωdt}. Letting P(r ) = diag {P 1 (r ), P 2 (r )} , Q =
Q 11 Q 12
∗ Q 22
, H i (r ) = P 2 (r )H i (r )
and recalling to Schur complements, we have:
S i j (r ) =
⎡
⎢
⎢
⎢
⎣
i P(r )
ˆ
A hi j (r ) + ˆ
A h ji (r )
P(r )
ˆ
B di (r ) + ˆ
B d j (r )
∗
− 2Q
0
∗
∗
− 2λ
2 I
∗
∗
∗
ˆ
C i j (r ) + ˆ
C ji (r )
T
ˆ
C hi j (r ) + ˆ
C h ji (r )
T
ˆ
D di (r ) + ˆ
D d j (r )
T
−2I
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎦
< 0.
(7.31)
It is clearly that i j (r ) > 0 and X i j (r ) + i j (r ) < 0 can lead to X i j (r ) < 0 and
inequality (7.12) when ω(t) = 0. Thus, the conclusion of the error dynamic multi-