Environmental Engineering Reference
In-Depth Information
linear combinations of them, and represent component flowrates. From the state
estimates, one can subsequently correct the values of the measured variables and
estimate the unmeasured process variables.
The optimization method that can be used to solve the unconstrained problem
defined by (2.30) is the usual approach where the reconciled states are the
X
values
that cancel the derivatives of
J
with respect to
X
. The resolution methods are slightly
different for the degenerate cases when either εor
e
is zero.
When ε
=
0, the optimization problem
min
X
J
(
X
)
subject to
f
(
X
)=
0
(2.47)
can be processed by one of two main approaches. The substitution method consists
in exploiting the equality constraints to decrease the number of variables the crite-
rion has to be minimized with respect to. The Lagrange method, on the contrary,
increases the number of search variables to optimize the criterion by incorporating
the equality constraints into it. New variables λ, called the Lagrange multipliers, are
associated to these constraints.
•
The substitution method consists of the following steps:
1. Select a set of independent variables among
X
. Such a set is composed of the
smallest number of variables
X
ind
that, if they were measured, would allow the
remaining variables from the system
f
(
X
)=
0 (the dependent ones
X
dep
)to
q
independent variables
X
ind
are
selected in such a way that the
q
variables
X
dep
are at minimal observability.
2. Express
X
dep
as a function of
X
ind
by solving
take unique values. In other words, the
n
X
−
(
,
)=
f
X
ind
X
dep
0
(2.48)
with respect to
X
dep
:
X
dep
=
h
(
X
ind
).
(2.49)
3. Replace
X
in the criterion
J
by its expression as a function of
X
ind
. The initial
constrained minimization problem with respect to
X
is thus transformed into
an unconstrained minimization problem with respect to
X
ind
:
min
X
ind
J
(
X
ind
).
(2.50)
4. Find the
X
ind
value and calculate
X
dep
from (2.49).
•
The Lagrange method consists in integrating the constraint
f
(
X
)=
0 into the
J
criterion to form a new function, the Lagrangian
, which has to be optimum
with respect to the variables
X
and λ , the latter being the Lagrange multipliers:
L
∑
i
λ
i
f
i
(
X
),
L =
(
)+
J
X
(2.51)
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