Environmental Engineering Reference
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variables is
V
Y
=
V
−
V
in
,
(2.108)
where
V
in
is the variance of the innovations
in
, which are residuals of the observer,
i.e.
, the differences between the measured and reconciled values:
Y
C X
in
=
Y
−
=
Y
−
.
(2.109)
Since
V
in
has, by definition, positive diagonal terms, one obtains
V
Y
(
i
,
i
) ≤
V
(
i
,
i
)
for
i
=
1to
m
,
(2.110)
an inequality showing the variance reduction due to data reconciliation. A global
variance reduction factor ρ
v
can be defined and calculated for the case where a sub-
set of states is measured (
Z
=
X
m
):
m
i
=
1
V
(
i
,
i
)−
V
Y
(
i
,
i
)
1
m
1
m
[
ρ
v
=
=
q
+
m
−
n
X
].
(2.111)
V
(
i
,
i
)
This equation shows that the reduction factor is directly related to the degree of
redundancy ρ
r
of the information on the system (see Equation 2.34). The expression
(2.111) is valid also for bilinear SSR problems, providing that the mass conservation
constraints are linearized around the reconciled states [44].
In addition to this difference between the variances before and after reconcili-
ation, there is a strong difference that appears in the covariance terms. It is here
assumed that the measurement errors are not correlated. However, the estimation er-
rors are now correlated because of the correlation induced by the conservation equa-
tions. Thus, matrices
V
X
and
V
Y
exhibit covariance terms that correspond to the state
consistency induced by the reconciliation procedure. This last property has a strong
impact on the reliability of the process performance indicators or process model
parameters subsequently calculated using reconciled states instead of raw measure-
ments [89, 90]. When the stationary reconciliation method is used rather than the
steady-state method, the covariance terms are usually smaller since the conserva-
tion equations are given more flexibility by relaxing these constraints through the
tuning of
V
ε
.
Example 1.
The impact of the variance reduction and the role of the covariance terms
have been illustrated for the calculation of separation plant recoveries in [89]. Fig-
ure 2.11 shows a separation node whose separation efficiency of component
i
can
be calculated by one of the two possible formulae:
F
2
x
2
i
F
1
x
1
i
,
R
i
=
(2.112)
x
3
i
−
x
1
i
x
2
i
x
1
i
.
R
i
=
(2.113)
x
3
i
−
x
2
i
This last formula is known as the two-product formula. When using raw data,
one obtains quite different results depending on the expression used to calculate the
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