Environmental Engineering Reference
In-Depth Information
•
The simplest one, for software development, consists in selecting
C
I
,
i.e.
,as-
suming that all the states are measured. For states that are not actually measured,
corresponding variances of measurement errors are simply given very large val-
ues. The solution is thus directly obtained from (2.65) as
=
X
VM
T
MV M
T
)
−
1
M
=(
I
−
(
)
Y
.
(2.67)
This method is used in the BILMAT™ algorithm [22], and has never numerically
failed.
•
The system of conservation constraints can be partitioned using the redundancy
equations as follows (linear version of Equation 2.46):
RX
mr
=
0
,
(2.68)
Q
X
mr
Q
X
mnr
QX
o
=
+
.
The solution is then obtained in two steps. First, the reconciled values of the
measured redundant states are estimated by
X
mr
V
mr
R
T
RV
mr
R
T
)
−
1
R
=(
−
(
)
,
I
Y
mr
(2.69)
where
V
r
and
Y
r
are formatted to match with
X
mr
. Second, the following equation
is solved for
X
o
:
Q X
o
Q
X
mr
Q
Y
nr
=
+
.
(2.70)
Tuning of the measurement error matrix covariance.
Before using the steady-
state method, it is always necessary to test: (1) that the main contribution to the
variability of the process variables is related to the measurement and sampling er-
rors; and (2) that there is an underlying steady-state regime during the period of time
corresponding to the measurement and sampling process. In this case, the measure-
ment errors due to the process time variations, such as the integration error, should
be incorporated into the measurement error. In other words data obtained during
a transient regime of large magnitude in comparison with the measurement errors
should not be used for SSR. Methods to find the measurement variance matrix have
been discussed by [41],[73-78].
Four methods are proposed to estimate the
V
covariance matrix:
•
Estimation from the properties of the equipment involved in the measurement
process and the properties of the material flowing in the stream. For instance
the theory of particulate material sampling [56, 79, 80] can be applied as seen
in Section 2.4, in conjunction with a statistical analysis of the reliability of the
measuring devices for the measured specific material property.
•
Direct estimation of the variance of the measurement
Y
from a large set of data
for the same plant operated under the same operating conditions. In this case, the
estimate of
V
is obtained from the
Y
records by standard statistical estimation
techniques, and is noted
Var
. This estimate of
V
does not strictly consist of
measurement errors, since it also includes some process dynamic disturbances,
such as the integration error. However, this is the right approach, since the recon-
(
Y
)
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