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8 Filtering-Based Robust Fault Detection of Fuzzy Multi-model Jumping System
8.1 Robust FDF Design for Fuzzy Multi-model Jumping
System
8.1.1 System Description
We construct the following fuzzy multi-model jumping system 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 , and . . . , μ g (t) is F
i
g , THEN
⎧
⎪ ⎪ ⎪ ⎨
⎪ ⎪ ⎪ ⎩
˙
x(t) = [A i (r t ) + A i (t, r t )] x(t) + [A hi (r t ) + A hi (t, r t )] x(t − τ )
+ B i (r t ) u(t) + 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,
(8.1)
where x(t) ∈ R
n is the state, x(t − τ ) ∈ R
n is the time-delayed state, y(t) ∈ R
l
is the measured output, u(t) ∈ R
r is the controlled input, ω(t) ∈ L
m
2 [0, +∞) is
the unknown disturbance, f (t) ∈ L
p
2 [0, +∞) is the fault to be detected. τ > 0 is
the delay scalar, η(t) is a continuous vector-valued initial function assumed to be
continuously differentiable on [−τ 0] and r 0 is the initial mode. μ 1 (t), μ 2 (t), . . .,
μ g (t) are the premise variables. F
i
l , i = 1, 2, . . . , S, l = 1, 2, . . . , g are the fuzzy
sets, S is the number of subsystem i. A i (r t ), A hi (r t ), B i (r t ), B di (r t ), B f i (r t ),
C i (r t ), C hi (r t ), D di (r t ), D f i (r t ) are known matrices with appropriate dimensions.
For presentation convenience, we denote x(t − τ ), A i (r t ), A i (t, r t ), A hi (r t ),
A hi (t, r t ), B i (r t ), B di (r t ), B f i (r t ), C i (r t ), C hi (r t ), D di (r t ), D f i (r t ) as x h , A i (r ),
A i (r ), A hi (r ), A hi (r ), B i (r ), B di (r ), B f i (r ), C i (r ), C hi (r ), D di (r ), D f i (r )
respectively.
The time-varying uncertainties A i (t, r t ) , ,A hi (t, r t ) in (8.1) satisfy
[A i (r ) )A hi (r )] = M i (r )) i (r, t) [N 1i (r ) N hi (r )] .
(8.2)
where M i (r ), N 1i (r ), N hi (r ) are known matrices and i (r, t) is the time-varying
function satisfying
T
i (t, r )) i (t, r ) ≤ I .
Definition 8.1 The nonlinear time-delay multi-model jumping system (8.1) (when
ω(t), f (t) ≡ 0 ) is said to be stochastically stable, if the following condition satisfies:
lim
T →∞
E
T
0
x (t, η(t), r 0 )
2 dt | r 0 , x(t) = η(t), t ∈ [−τ 0]
< ∞.
(8.3)
Applying the T-S fuzzy model, the fuzzy multi-model jumping system (8.1) is
expressed as:
8 Filtering-Based Robust Fault Detection of Fuzzy Multi-model Jumping System
8.1 Robust FDF Design for Fuzzy Multi-model Jumping
System
8.1.1 System Description
We construct the following fuzzy multi-model jumping system 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 , and . . . , μ g (t) is F
i
g , THEN
⎧
⎪ ⎪ ⎪ ⎨
⎪ ⎪ ⎪ ⎩
˙
x(t) = [A i (r t ) + A i (t, r t )] x(t) + [A hi (r t ) + A hi (t, r t )] x(t − τ )
+ B i (r t ) u(t) + 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,
(8.1)
where x(t) ∈ R
n is the state, x(t − τ ) ∈ R
n is the time-delayed state, y(t) ∈ R
l
is the measured output, u(t) ∈ R
r is the controlled input, ω(t) ∈ L
m
2 [0, +∞) is
the unknown disturbance, f (t) ∈ L
p
2 [0, +∞) is the fault to be detected. τ > 0 is
the delay scalar, η(t) is a continuous vector-valued initial function assumed to be
continuously differentiable on [−τ 0] and r 0 is the initial mode. μ 1 (t), μ 2 (t), . . .,
μ g (t) are the premise variables. F
i
l , i = 1, 2, . . . , S, l = 1, 2, . . . , g are the fuzzy
sets, S is the number of subsystem i. A i (r t ), A hi (r t ), B i (r t ), B di (r t ), B f i (r t ),
C i (r t ), C hi (r t ), D di (r t ), D f i (r t ) are known matrices with appropriate dimensions.
For presentation convenience, we denote x(t − τ ), A i (r t ), A i (t, r t ), A hi (r t ),
A hi (t, r t ), B i (r t ), B di (r t ), B f i (r t ), C i (r t ), C hi (r t ), D di (r t ), D f i (r t ) as x h , A i (r ),
A i (r ), A hi (r ), A hi (r ), B i (r ), B di (r ), B f i (r ), C i (r ), C hi (r ), D di (r ), D f i (r )
respectively.
The time-varying uncertainties A i (t, r t ) , ,A hi (t, r t ) in (8.1) satisfy
[A i (r ) )A hi (r )] = M i (r )) i (r, t) [N 1i (r ) N hi (r )] .
(8.2)
where M i (r ), N 1i (r ), N hi (r ) are known matrices and i (r, t) is the time-varying
function satisfying
T
i (t, r )) i (t, r ) ≤ I .
Definition 8.1 The nonlinear time-delay multi-model jumping system (8.1) (when
ω(t), f (t) ≡ 0 ) is said to be stochastically stable, if the following condition satisfies:
lim
T →∞
E
T
0
x (t, η(t), r 0 )
2 dt | r 0 , x(t) = η(t), t ∈ [−τ 0]
< ∞.
(8.3)
Applying the T-S fuzzy model, the fuzzy multi-model jumping system (8.1) is
expressed as: