Environmental Engineering Reference
In-Depth Information
Let's now define the possible terms of a reconciliation criterion. They are written
below at time
t
only, but can be extended in a finite size window, or a window
beginning at time zero, thus usually leading to optimal observation. First, there are
terms representing measurement errors,
i.e.
squared residuals according to
T
V
−
1
J
m
(
t
)=[
Y
(
t
)−
g
[
X
(
t
)]]
[
Y
(
t
)−
g
[
X
(
t
)]].
(2.126)
Squared node imbalances can be put into the criterion when appropriate. They are
T
M
T
V
−
1
ε
J
ε
(
t
)=
f
(
t
)
(
0
)
Mf
(
t
).
(2.127)
0) is to
be taken into account, the vector
f
is replaced by the stacked values of
f
at previous
times, and the weighting matrix accordingly augmented with the autocovariance
terms.
The model uncertainties δ and the driving white noise signals, either ξ, ξ
or ω,
which are functions of the nature of the state variables to be estimated, should also
be part of the reconciliation criteria and lead to the following quadratic terms:
When time correlation between the node imbalances (
V
ε
(
τ
) =
0forτ
>
T
V
−
1
(
)=
(
)
(
),
J
ς
t
ς
t
ς
t
(2.128)
ς
where ςis either δ, ξ, ξ
, ω, depending upon the model used. Moreover, if a smooth-
ing horizon is considered in the criterion formulation, past terms can also be used.
Table 2.6 summarizes some of the possibilities of coupling state equations, obser-
vation equations and quadratic reconciliation criteria. It gives the models and mea-
surements that are used as well as the quadratic criteria. The methods are optimal or
sub-optimal depending on the consistency between the modeling assumptions and
the reconciliation criterion used. For instance, when the time correlation is quanti-
fied in the model while the criterion contains only present information (thus freez-
ing past estimates) the method is sub-optimal. On the contrary, when the smoothing
horizon is extended to all past information, the methods are optimal.
The criterion minimization may be a LQ optimization problem, but most fre-
quently it is a non-linear optimization problem that requires using non-linear pro-
gramming methods (see for instance [38]) or sequentially linearizing the observa-
tion equations. In the case where the estimation horizon starts at time zero and a full
model is used, one could apply either the Kalman filter in the LQ case [99], or the
extended Kalman filter (EKF) in the non-linear case. When a sub-model is used the
generalized version of the Kalman filter (GKF) can be used [37, 96, 100].
It is also possible to use sub-models for the steady-state part of the model (thus
ignoring the process gains) and a full model for its dynamic part [93]. This allows
some kind of synchronization of the data, while avoiding constructing a complete
model of the process. This technique has been used to deal with processes exhibiting
large pure delays, as those occurring in pipe-lines [101].
There are also methods that keep the bilinear structure of the mass conserva-
tion equations (products of flowrates and concentrations), instead of using species
flowrates as above, thus leading to linear observation equations. Estimation algo-
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