104
6 Robust Fault Detection for Multi-model Jumping System
i = ˜
A
T
i P i + P i ˜
A i +
N
j=1
π i j P j + P i < 0.
(6.27)
To reduce the impact of the external perturbation for the residual signals, the FD
problem is regulated when f (t) = 0 to solve the observer parameter H i which can
make the error dynamic multi-model jumping system (6.26) be stochastically stable
and
E
t
0
r
T
ω (t)r ω (t)dt
≤ γ
2
1 E
t
0
ω
T
(t)ω(t)dt
| f (t)=0 .
(6.28)
holds for all non-zero ω(t) ∈ L 2 [0, ∞).
Inequality (6.28) provides the plan of minimizing the influences of disturbances
on the residual and the following objective is to render the difference between the
residual and the disturbances as small as possible.
Likewise, we provide another performance index when devising the observer
parameter H i such that the error dynamic multi-model jumping system (6.26) is
stochastically stable and
E
t
0
r
T
f (t)r f (t)dt
≥ γ
2
2 E
t
0
f
T
(t) f (t)dt
| ω(t)=0 .
(6.29)
holds for all non-zero ω(t) ∈ L 2 [0, ∞).
Inequality (6.29) provides the plan of improving the effects of faults on the residual
and then the difference between the residual and the faults is built as large as possible.
Then, we take a generally utilized strategy to introduce a suitable threshold J th
and decide the evaluation function f (r ).
Suppose ω(t) 2 ≥ ω, and define the following threshold:
J th =
sup
ω(t)∈L 2 , f (t)=0
t
0
r
T
eo (t)r eo (t)dt.
(6.30)
Thus, the following logic is suitable for the FD scheme:
⎧
⎨
⎩
f (r eo ) = E
t
0 r
T
eo (t)r eo (t)dt
> J th → with fault,
f (r eo ) = E
t
0 r
T
eo (t)r eo (t)dt
≤ J th → fault-free.
(6.31)
where (0, t] is the finite-time window.
Consider the fact that the evaluation of residual signal during the entire time scope
is unpractical. Thus, the valuation time window t is considered to be limited.
Inequality (6.28) describes the criterion of the worst situation for the impact
of disturbances on the residuals, and inequality (6.29) represents the criterion of the
worst situation for the sensitivity of residuals to faults. They give a direct quantitative
standard for the robustness and sensitivity of FD. Ordinarily, to get a suitable FD
6 Robust Fault Detection for Multi-model Jumping System
i = ˜
A
T
i P i + P i ˜
A i +
N
j=1
π i j P j + P i < 0.
(6.27)
To reduce the impact of the external perturbation for the residual signals, the FD
problem is regulated when f (t) = 0 to solve the observer parameter H i which can
make the error dynamic multi-model jumping system (6.26) be stochastically stable
and
E
t
0
r
T
ω (t)r ω (t)dt
≤ γ
2
1 E
t
0
ω
T
(t)ω(t)dt
| f (t)=0 .
(6.28)
holds for all non-zero ω(t) ∈ L 2 [0, ∞).
Inequality (6.28) provides the plan of minimizing the influences of disturbances
on the residual and the following objective is to render the difference between the
residual and the disturbances as small as possible.
Likewise, we provide another performance index when devising the observer
parameter H i such that the error dynamic multi-model jumping system (6.26) is
stochastically stable and
E
t
0
r
T
f (t)r f (t)dt
≥ γ
2
2 E
t
0
f
T
(t) f (t)dt
| ω(t)=0 .
(6.29)
holds for all non-zero ω(t) ∈ L 2 [0, ∞).
Inequality (6.29) provides the plan of improving the effects of faults on the residual
and then the difference between the residual and the faults is built as large as possible.
Then, we take a generally utilized strategy to introduce a suitable threshold J th
and decide the evaluation function f (r ).
Suppose ω(t) 2 ≥ ω, and define the following threshold:
J th =
sup
ω(t)∈L 2 , f (t)=0
t
0
r
T
eo (t)r eo (t)dt.
(6.30)
Thus, the following logic is suitable for the FD scheme:
⎧
⎨
⎩
f (r eo ) = E
t
0 r
T
eo (t)r eo (t)dt
> J th → with fault,
f (r eo ) = E
t
0 r
T
eo (t)r eo (t)dt
≤ J th → fault-free.
(6.31)
where (0, t] is the finite-time window.
Consider the fact that the evaluation of residual signal during the entire time scope
is unpractical. Thus, the valuation time window t is considered to be limited.
Inequality (6.28) describes the criterion of the worst situation for the impact
of disturbances on the residuals, and inequality (6.29) represents the criterion of the
worst situation for the sensitivity of residuals to faults. They give a direct quantitative
standard for the robustness and sensitivity of FD. Ordinarily, to get a suitable FD