142
8 Filtering-Based Robust Fault Detection of Fuzzy Multi-model Jumping System
where the weighted matrix function W f (s) is assumed to be
˙
x f (t) = A w x f (t) + B w f (t),
r f (t) = C w x f (t) + D w f (t).
(8.10)
in which r f (s) is the reference residual, which shows sensitivity to faults. In this
design project, our primary purpose is that when the system model is fully known, the
reference model corresponds to an ideal solution to guarantee an optimal compromise
between the minimum false alarm rate and the maximum FD rate. Therefore, the
design intention is to obtain an optimal robust FDF that minimizes the difference
between the designed robust FDF and the reference model.
For convenience, we represent h i (μ(t)) as h i and let r e f (t) = r F (t) − r f (t) and
e(t) = x(t) − x F (t). We can obtain the overall robust FDF system as:
˙ ˆ
x(t) = ˆ
A i j (r ) ˆ
x(t) + ˆ
A hi j (r ) ˆ
x h + ˆ
B i j (r )w(t),
r e f (t) = ˆ
C i j (r ) ˆ
x(t) + ˆ
C hi j (r ) ˆ
x h + ˆ
D i j (r )w(t).
(8.11)
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 )
A i (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 ) + 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 )
B di (r )
B f i (r )
0 −B Fi (r )D d j (r ) −B Fi (r )D f j (r )
0
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
.
Therefore, the objective for robust FDF design is how to devise the appropriate
parameter matrices in (8.7) and to define an appropriate threshold and a residual
evaluation function to accomplish the following aims:
(A1) The overall robust FDF system is stochastically stable.
(A2) It shows the performance index J e as small as possible:
8 Filtering-Based Robust Fault Detection of Fuzzy Multi-model Jumping System
where the weighted matrix function W f (s) is assumed to be
˙
x f (t) = A w x f (t) + B w f (t),
r f (t) = C w x f (t) + D w f (t).
(8.10)
in which r f (s) is the reference residual, which shows sensitivity to faults. In this
design project, our primary purpose is that when the system model is fully known, the
reference model corresponds to an ideal solution to guarantee an optimal compromise
between the minimum false alarm rate and the maximum FD rate. Therefore, the
design intention is to obtain an optimal robust FDF that minimizes the difference
between the designed robust FDF and the reference model.
For convenience, we represent h i (μ(t)) as h i and let r e f (t) = r F (t) − r f (t) and
e(t) = x(t) − x F (t). We can obtain the overall robust FDF system as:
˙ ˆ
x(t) = ˆ
A i j (r ) ˆ
x(t) + ˆ
A hi j (r ) ˆ
x h + ˆ
B i j (r )w(t),
r e f (t) = ˆ
C i j (r ) ˆ
x(t) + ˆ
C hi j (r ) ˆ
x h + ˆ
D i j (r )w(t).
(8.11)
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 )
A i (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 ) + 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 )
B di (r )
B f i (r )
0 −B Fi (r )D d j (r ) −B Fi (r )D f j (r )
0
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
.
Therefore, the objective for robust FDF design is how to devise the appropriate
parameter matrices in (8.7) and to define an appropriate threshold and a residual
evaluation function to accomplish the following aims:
(A1) The overall robust FDF system is stochastically stable.
(A2) It shows the performance index J e as small as possible: