Environmental Engineering Reference
In-Depth Information
ciled values are expected to give an estimate of the underlying plant steady-state
(
i.e.
,average)behavior.
•
Indirect estimation from the constraint residuals (the node imbalances) in the
particular case where
C
I
(the identity matrix). The constraint residuals
r
are
calculated for each
Y
value of the data set and their variance estimated:
r
=
=
MY
=
MX
+
Me
,
(2.71)
MV M
T
Var
(
r
)=
.
Techniques have been proposed [77] for extracting
V
from this last equation. The
main advantage of this technique is that it takes account of the mass conservation
constraints.
•
Simultaneous estimation of
V
and of a rough model of the plant. For instance,
a mineral separation plant can be simply modeled by mineral separation coeffi-
cients at each separation node of the flowsheet. The plant model is then expressed
as
X
=
B
(
s
)
X
f
(2.72)
where
X
f
is the state vector of the feed streams and
s
the separation coefficient
vector. The variance of
Y
estimated from a measurement data set is then
T
(
)=
(
)
(
)
(
)
+
.
Var
Y
B
s
Var
X
f
B
s
V
(2.73)
From Equation 2.73, variance
V
can be extracted simultaneously to
s
and
Var
(
X
f
)
by a least-squares procedure [78].
2.7.2 The Stationary Case
The unconstrained stationary reconciliation problem is formulated as the following
particular case of (2.30):
⎧
⎨
X
T
V
−
1
ε
T
V
−
1
ε
=
arg min
X
[(
Y
−
CX
)
(
Y
−
CX
)+
ε
],
(2.74)
Y
=
CX
+
e
;
e
∼
N
(
0
,
V
),
⎩
ε
=
MX
; ε
∼
N
(
0
,
V
ε
).
The solution is obtained by canceling the derivatives of
J
:
dJ
dX
=
2
C
T
V
−
1
CX
2
C
T
V
−
1
Y
2
M
T
V
−
ε
MX
−
+
=
0
,
(2.75)
which leads to
X
C
T
V
−
1
C
M
T
V
−
ε
M
)
−
1
C
T
V
−
1
Y
=(
+
(2.76)
or alternatively, using a matrix inversion lemma,
X
Π
−
1
M
T
M
Π
−
1
M
T
)
−
1
M
Π
−
1
C
T
V
−
1
Y
=
[
I
−
(
V
ε
+
]
,
(2.77)
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