8.2 FDF Design for Fuzzy Multi-model Jumping System
151
where
ˆ
x(t) =
x
T
(t) e
T
(t) x
T
f (t)
T , w(t) =
u
T
(t) ω
T
(t) f
T
(t)
T ,
ˆ
A i j (r ) =
S
i=1 h i
⎧
⎨
⎩
S
j=1 h j
⎡
⎣
A i (r )
A i (r ) − A Fi (r ) − B Fi (r )C j (r )
0
0
0
A Fi (r ) 0
0
A w
⎤
⎦
⎫
⎬
⎭
,
ˆ
A hi j (r ) =
S
i=1 h i
⎧
⎨
⎩
S
j=1 h j
⎡
⎣
A hi (r )
0 0
−B Fi (r )C h j (r ) 0 0
0
00
⎤
⎦
⎫
⎬
⎭
,
ˆ
B i j (r ) =
S
i=1 h i
⎧
⎨
⎩
S
j=1 h j
⎡
⎣
B i (r )
0
0
B di (r )
B f i (r )
−B Fi (r )D d j (r ) −B Fi (r )D f j (r )
0
B w
⎤
⎦
⎫
⎬
⎭
,
ˆ
C i j (r ) =
S
i=1 h i
S
j=1 h j
C Fi (r ) + D Fi (r ) C j (r ) − C Fi (r ) C w
,
ˆ
C hi j (r ) =
S
i=1 h i
S
j=1 h j
D Fi (r )C h j (r ) 0 0
,
ˆ
D i j (r ) =
S
i=1 h i
S
j=1 h j
0 D Fi (r )D d j (r ) D Fi (r ) D f j (r ) − D w
.
8.2.2 Design of Fuzzy Jump FDF
Theorem 8.2 For a given positive scalar γ, the overall FDF system (8.47) is stochastically stable, if there exist positive definite symmetric matrices P(r ), Q 11 , Q 22 , Q 33 ,
a series of matrices X i (r ), Y i (r ), C Fi (r ), D Fi (r ), Q 12 , Q 13 , Q 23 , and positive numbers λ i (r ) > 0, meeting the following coupled inequalities:
ii (r ) < 0, i = 1, 2, . . . , S,
(8.48)
i j (r ) + ji (r ) < 0, i < j, i = 1, 2, . . . , S.
(8.49)
where