7.2 FDO Design for Fuzzy Multi-model Jumping System
131
8i j (r ) = −2λ
2 I,
9i j (r ) =
D di (r ) + D d j (r )
T .
Moreover, the parameter matrix of 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 (7.44) 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
.
(7.46)
Then, we can rewrite the index J 1 (T ) as:
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
,
(7.47)
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 f
T r f 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: