8.1 Robust FDF Design for Fuzzy Multi-model Jumping System
141
⎧
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
˙
x(t) =
S
i=1
h i (μ(t)) [(A i (r ) + A i (r )) x(t) + B i (r )u(t)
+ (A hi (r ) + 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.
(8.4)
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)) .
(8.5)
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.
(8.6)
Furthermore, we reconstruct the fuzzy robust FDF systems as:
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
(8.7)
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 FDF gain matrices to be devised. Then, the global
robust FDF dynamics is 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.
(8.8)
Definition 8.2 Referring to [119], in order to facilitate the detection of the faults,
we select a appropriate weighted matrix function W f (s) to confirm the faults and
improve dynamics performance. Thus, we can give the following reference residual
model:
r f (s) = W f (s) f (s).
(8.9)