128
7 Observer-Based Robust Fault Detection for Fuzzy Multi-model Jumping System
J e = min
H i (r )
λ
γ
,
s.t. LMIs (7.28) and (7.34).
(7.37)
7.2 FDO Design for Fuzzy Multi-model Jumping System
7.2.1 System Description
Without considering the uncertainties, we conclude the following multi-model jumping system (7.38) defined on the probability space ((, F, P):
Plant Rule i:
If μ 1 (t) is F
i
1 , μ 2 (t) is F
i
2 , . . . , μ g (t) is F
i
g , Then
⎧
⎪ ⎨
⎪ ⎩
˙
x(t) =A i (r t ) x(t) + A hi (r t ) x(t − τ ) + B di (r t ) ω(t) + B f i (r t ) f (t),
y(t) =C i (r t ) x(t) + C hi (r t ) x(t − τ ) + D di (r t ) ω(t) + D f i (r t ) f (t),
x(t) =η(t), r (t) = r 0 , t ∈ [−τ 0], i = 1, 2, . . . , S.
(7.38)
By using T-S fuzzy model, the fuzzy multi-model jumping system in (7.38) is
described by:
⎧
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
˙
x(t) =
S
i=1
h i (μ(t))
A i (r t ) x(t) + A hi (r t ) x h + B di (r t ) ω(t) + B f i (r t ) f (t)
,
y(t) =
S
i=1
h i (μ(t)) [C i (r t ) x(t) + C hi (r t )x h +D di (r t ) ω(t) + D f i (r t ) f (t)
,
x(t) =η(t), r t = r 0 , t ∈ [−τ 0], i = 1, 2, . . . , S,
(7.39)
where μ(t) = [μ 1 (t) μ 2 (t) · · · μ S (t)]. In addition, for ∀i = 1, 2, . . . , S,
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)) ,
(7.40)
in which F
i
j
μ j (t)
is the grade of membership of μ j (t) in the fuzzy set F
i
j . Here,
if we assume that u i (μ(t)) 0 and
S
i=1 u i (μ(t)) > 0, then the following relations
hold:
S
i=1 h i (μ(t)) = 1
0 ≤ h i (μ(t)) ≤ 1, i = 1, 2, . . . , S
(7.41)
Next, we constructed the fuzzy FDO systems as:
Filter Rule i: