Environmental Engineering Reference
In-Depth Information
assuming that the independent variables have been selected to ensure
M
dep
invert-
ibility. Therefore X can be expressed as
X
ind
X
dep
X
ind
X
=
=
=
LX
ind
,
(2.57)
M
−
1
−
dep
X
ind
I
dep
T
M
−
1
with
L
.
The linear reconciliation criterion:
=
−
T
V
−
1
J
=(
CX
−
Y
)
(
CX
−
Y
)
(2.58)
can then be written as a quadratic expression of
X
ind
:
X
ind
L
T
C
T
V
−
1
CLX
ind
2
Y
T
V
−
1
CLX
ind
Y
T
V
−
1
Y
(
)=
−
+
.
J
X
ind
(2.59)
J
is minimized with respect to
X
ind
by writing that the derivatives of
J
with respect
to
X
ind
have zero values. One obtains a system of
n
X
−
q
equations system with
n
X
−
q
unknown states, which has the following solution:
X
ind
L
T
C
T
V
−
1
CL
)
−
1
L
T
C
T
V
−
1
Y
=(
.
(2.60)
Knowing
X
ind
, the dependent variable estimates can be calculated from Equation
2.56:
X
dep
)
−
1
M
ind
X
ind
= −(
M
dep
.
(2.61)
The Lagrange method can also be applied to solve the linear SSR problem. The
Lagrangian function is
T
V
−
1
λ
T
MX
L =(
CX
−
Y
)
(
CX
−
Y
)+
.
(2.62)
The
L
stationarity conditions are
d
dX
=
2
C
T
V
−
1
CX
2
C
T
V
−
1
Y
M
T
λ
−
+
=
,
0
(2.63)
d
d
λ
=
MX
=
0
.
(2.64)
The
X
solution of this
n
X
+
q
equation system with
n
X
+
q
unknown variables is
X
Π
−
1
M
T
M
Π
−
1
M
T
)
−
1
M
Π
−
1
C
T
V
−
1
Y
=
(
I
−
(
)
,
(2.65)
where
C
T
V
−
1
C
Π
=
.
(2.66)
There are other alternatives to the solution of the reconciliation problem. Two of
them should be mentioned, in the particular case where the measured variables are
states (
Z
=
X
m
):
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