150
8 Filtering-Based Robust Fault Detection of Fuzzy Multi-model Jumping System
⎧
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
˙
x(t) =
S
i=1
h i (μ(t))[A i (r )x(t) + B i (r )u(t) + A hi (r )x h
+ B di (r )ω(t) + B f i (r ) f (t)],
y(t) =
S
i=1
h i (μ(t))
C i (r )x(t) + C hi (r )x h + D di (r )ω(t) + D f i (r ) f (t)
,
x(t) =η(t), r = r 0 , t ∈ [−τ 0], i = 1, 2, . . . , S.
(8.42)
where μ(t) = [μ 1 (t) μ 2 (t) · · · μ S (t)] . And for ∀i = 1, 2, . . . , S, we have:
h i (μ(t)) = u i (μ(t))/
S
i=1 u i (μ(t)),
u i (μ(t)) =
g
l=1 F
i
l (μ l (t)) .
(8.43)
in which F
i
l (μ l (t)) is the grade of membership of μ l (t) in the fuzzy set F
i
l with
u i (μ(t)) ≥ 0 and
S
i=1 u i (μ(t)) > 0. It follows that
S
i=1 h i (μ(t)) = 1,
0 ≤ h i (μ(t)) ≤ 1, i = 1, 2, . . . , S.
(8.44)
Moreover, we construct the following fuzzy jump FDF system:
Filter Rule i:
IF μ 1 (t) is F
i
1 , μ 2 (t) is F
i
2 , and . . . , μ g (t) is F
i
g , THEN
⎧
⎨
⎩
˙
x F (t) = A Fi (r )x F (t) + B Fi (r )y(t)
r F (t) = C Fi (r )x F (t) + D Fi (r )y(t)
x F (0) = 0
(8.45)
where x F (t) ∈ R
n is the filter state and r F (t) ∈ R
m is the filter output. A Fi (r ),
B Fi (r ), C Fi (r ) and D Fi (r ) are the filter gain matrices to be determined. Then, the
overall global FDF dynamics are expressed as:
⎧
⎨
⎩
˙
x F (t) =
S
i=1 h i (μ(t)) [A Fi (r )x F (t) + B Fi (r )y(t)] ,
r F (t) =
S
i=1 h i (μ(t)) [C Fi (r )x F (t) + D Fi (r )y(t)] ,
x F (0) = 0.
(8.46)
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 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).
(8.47)