98
6 Robust Fault Detection for Multi-model Jumping System
The FD filter design scheme can be summarized as the H ∞ filtering issue, that is,
(1) Given a constant γ > 0, the error dynamic multi-model jumping system (6.5)
subject to w(t) ∈ L 2 is stochastically stable and holds the H ∞ performance:
r e f (t) 2E < γw(t) 2 .
(6.7)
(2) Get the values of the filter parameters A Fi , B Fi , C Fi and D Fi .
For model-based FD, the rest of the crucial task for FD filter scheme is to evaluate
the generated residual. For detecting the faults, the generally selected method is to
find a proper threshold J th and decide the evaluation function f (r e f ). Based on the
fact that ω(t) is L 2 -norm bounded, then, the threshold J th is determined as:
J th =
sup
ω(t)∈L 2 , f (t)=0
E
t 0 +h
t 0
r
T
e f (t)r e f (t)dt
.
(6.8)
Besides, the evaluation function f (r ) is set as:
f (r ) =
t 0 +h
t 0
r
T
e f (t)r e f (t)dt,
(6.9)
where [t 0 , t 0 + h] is the finite-time window, h is the length and t 0 is the initial evaluation time. Accordingly, we provide a suitable logic for FD as:
f (r ) > J th → with faults,
f (r ) ≤ J th → no faults (faults free).
(6.10)
6.1.2 Design of Jumping FD Filter
Theorem 6.1 Given ξ > 0, the error dynamic multi-model jumping system (6.5) is
stochastically stable and holds the given H ∞ performance (6.6), if for all i, j ∈ ,
there exist mode-dependent symmetric matrices P i > 0, symmetric matrices Q 11 >
0, Q 22 > 0, Q 33 > 0, a set of matrices X i , Y i , C Fi , D Fi , Q 12 , Q 13 , Q 23 , and modedependent scalars λ i > 0, holding the following LMI:
6 Robust Fault Detection for Multi-model Jumping System
The FD filter design scheme can be summarized as the H ∞ filtering issue, that is,
(1) Given a constant γ > 0, the error dynamic multi-model jumping system (6.5)
subject to w(t) ∈ L 2 is stochastically stable and holds the H ∞ performance:
r e f (t) 2E < γw(t) 2 .
(6.7)
(2) Get the values of the filter parameters A Fi , B Fi , C Fi and D Fi .
For model-based FD, the rest of the crucial task for FD filter scheme is to evaluate
the generated residual. For detecting the faults, the generally selected method is to
find a proper threshold J th and decide the evaluation function f (r e f ). Based on the
fact that ω(t) is L 2 -norm bounded, then, the threshold J th is determined as:
J th =
sup
ω(t)∈L 2 , f (t)=0
E
t 0 +h
t 0
r
T
e f (t)r e f (t)dt
.
(6.8)
Besides, the evaluation function f (r ) is set as:
f (r ) =
t 0 +h
t 0
r
T
e f (t)r e f (t)dt,
(6.9)
where [t 0 , t 0 + h] is the finite-time window, h is the length and t 0 is the initial evaluation time. Accordingly, we provide a suitable logic for FD as:
f (r ) > J th → with faults,
f (r ) ≤ J th → no faults (faults free).
(6.10)
6.1.2 Design of Jumping FD Filter
Theorem 6.1 Given ξ > 0, the error dynamic multi-model jumping system (6.5) is
stochastically stable and holds the given H ∞ performance (6.6), if for all i, j ∈ ,
there exist mode-dependent symmetric matrices P i > 0, symmetric matrices Q 11 >
0, Q 22 > 0, Q 33 > 0, a set of matrices X i , Y i , C Fi , D Fi , Q 12 , Q 13 , Q 23 , and modedependent scalars λ i > 0, holding the following LMI: