8.2 FDF Design for Fuzzy Multi-model Jumping System
153
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
⎤
⎦ .
(8.52)
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 )
⎫
⎬
⎭
< 0,
(8.53)
where
i j (r ) =
1 (r ) ) 2 (r )
∗ 3 (r )
,
1 (r ) =
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎣
11 (r ) ) 12 (r ) Q 13
P(r )A hi (r )
0
0
∗ 22 (r ) Q 23 −P(r )B Fi (r )C hi (r ) 0
0
∗
∗
A 33 (r )
0
0
0
∗
∗
∗
−Q 11
−Q 12 −Q 13
∗
∗
∗
∗
−Q 22 −Q 23
∗
∗
∗
∗
∗ −Q 33
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎦
,
A 2 (r ) =
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎣
P(r )B i (r )
A 0108 (r )
A 0109 (r )
A 0110 (r )
0
−P(r )B Fi (r )D di (r ) −P(r )B Fi (r )D f i (r ) ) 0210 (r )
0
0
0
A 0310 (r )
0
0
0
0410 (r )
0
0
0
0
0
0
0
0
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎦
,
A 3 (r ) =
⎡
⎢
⎢
⎣
−γ
2 I 0
0
0
∗ −γ
2 I 0 A 0810 (r )
∗
∗ −γ
2 I 0910 (r )
∗
∗
∗
−I
⎤
⎥
⎥
⎦ ,
11 (r ) = P(r )A i (r ) + A
T
i (r )P(r ) +
N
j=1 π i j P(k) + Q 11 ,
12 (r ) =
A i (r ) − A Fi (r ) − B Fi (r )C j (r )
T P(r ) + Q 22 ,