Environmental Engineering Reference
In-Depth Information
KnowingM,C,Y,V,V
ε
,L
F
, and K
F
, decide between the two following
hypotheses:
=
=
,
H
0:
F
X
i.e., there are no faults in the process
;
H
1:
either F
X
or F
Y
or both are
F
Y
0
0
, i.e., there is at least
one fault in the process.
=
Most of the time, detection of gross errors is performed after the reconciliation
procedure by testing innovations,
i.e.
, corrections brought to the measured process
variable values. When the statistical tests are sequentially applied to single inno-
vations, these methods are incorrect since they ignore the fundamental correlation
brought to the measured variable estimates by the reconciliation procedure. A gross
error in any process variable usually contaminates the whole set of state estimates,
and therefore might lead to wrong diagnosis. An alternative to the innovation resid-
uals obtained through data reconciliation is to directly use the residuals of the re-
dundancy equations. Various fault detection tests applied to the residuals of the re-
dundancy equations (see Figure 2.5 and Equation 2.38) are available [7, 8]. Only the
parity space approach is presented here. Each residual (element of the parity vector,
[114]) can be tested against a normal distribution, or they can be tested together. The
latter approach, called the global detection test, is applied to the following quadratic
term:
r
T
V
−
1
J
r
=
r
,
(2.131)
r
where
r
represents the residuals of the redundancy equations (Figure 2.5 and Equa-
tion 2.38), hence a normal centred vector in the absence of faults since it depends
only on
e
and ε.
V
r
, the variance matrix of the residuals, is directly calculable, in
the linear case, since
r
consists of linear functions of
e
and ε (Equation 2.45). This
quadratic term is a fault signature in the most usual case where the set of poten-
tial faults is not specified (no
L
F
and
K
F
matrices structuring the fault distribution).
Since the observer uncertainties have been assumed to be normal,
J
r
follows a χ
2
statistical distribution with
m
n
X
degrees of freedom and a non-central pa-
rameter only depending on
F
X
and
F
Y
, which are zero when there is no fault in the
process.
J
r
is tested against a given level of false alarms.
When
H
1 is the conclusion of the fault detection step, the fault isolation step
becomes [118]
Deciding between the two following hypotheses:
+
q
−
H
0:
F
=
0
,F
=
0
,
a
b
H
1:
F
=
0
,F
=
0
,
a
b
F
X
F
Y
where
F
a
,
F
b
are two exhaustive subsets of the
F
set of faults
(
)
.
By repetitive application of this diagnosis test to various fault subsets, one can
isolate the most probable active faults. The statistical isolation tests are performed
on residuals derived from the redundant equations (Equation 2.45). The generalized
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