8.1 Robust FDF Design for Fuzzy Multi-model Jumping System
147
Then, we can rewrite the index J (T ) as:
J (T ) = E
T
0
r
T
e f (t)r e f (t)dt
− γ
2
T
0
w
T
(t)w(t)dt
= E
T
0
r
T
e f (t)r e f (t) − γ
2
w
T
(t)w(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 w
T
(t)
i j (r ) + i j (r )
ˆ
x
T
(t) ˆ
x
T
h w
T
(t)
T ,
(8.35)
where
i j (r ) =
⎡
⎣
i j (r ) ˆ
P(r ) ˆ
A hi j (r ) ˆ
P(r ) ˆ
B i j (r )
∗
− ˆ
Q
0
∗
∗
− γ
2 I
⎤
⎦ ,
i j (r ) =
⎡
⎢
⎣
ˆ
C
T
i j (r )
ˆ
C
T
hi j (r )
ˆ
D
T
i j (r )
⎤
⎥
⎦
ˆ
C i j (r ) ˆ
C hi j (r ) ˆ
D i j (r )
.
As T → ∞, , i j (r ) + i j (r ) < 0 gives J (∞) < −V (∞) < 0, we can obtain
E
∞
0 r
T
e f (t)r e f (t)dt
≤ γ
2 E
∞
0 w
T
(t)w(t)dt
. To drive the next proof, we let
ˆ
P(r ) = diag{P(r ), P(r ), P(r )}, ˆ
Q =
⎡
⎣
Q 11 Q 12 Q 13
∗ Q 22 Q 23
∗ ∗ Q 33
⎤
⎦ ,
(8.36)
where P(r ) = P
T
(r ) are mode-dependent positive matrices, Q ii ∈ R
n×n
, i = 1, 2, 3
are a set of positive matrices and Q 12 , Q 13 , Q 23 are nonsingular. Then i j (r ) +
i j (r ) < 0 equals to the following condition:
S
i=1
h i
⎧
⎨
⎩
S
j=1
h j
i j (r ) + i j (r )
⎫
⎬
⎭
< 0,
(8.37)
where
i j (r ) =
1 (r ) ) 2 (r )
∗ 3 (r )
, , i j (r ) =
1 (r ) ) 2 (r )
∗
0
,