Environmental Engineering Reference
In-Depth Information
T
i
=
1
[
y
(
t
)−
y
(
t
|
t
−
1
,
ˆ
κ
T
i
=
1
e
(
t
,
κ
1
T
1
T
2
2
V
=
)]
=
)
.
(4.4)
(
)
(
|
−
,
)
The fact that
y
are in general stochastic processes implies that
V
is a random variable having an expected value and a variance depending on
T
.For
the estimate
V
to be useful it should be unbiased,
i.e.
,
J
t
and
y
t
t
1
κ
=
{
}
, and its variance
should tend to zero as
T
tends to infinity. This situation propagates to the estimation
of the model parameters (see Section 4.3.2).
Applying the necessary condition for the minimization of (4.3)
E
V
∂
∂κ
E
∂
∂
ˆ
κ
e
∂
∂κ
e
2
2
ˆ
κ
ˆ
κ
ˆ
κ
κ
{
e
(
t
,
)
} =
E
{
(
t
,
)
} =
0
,
E
{
e
(
t
,
)
(
t
,
)} =
0
.
(4.5)
From (4.1),
ˆ
κ
∂
∂ˆκ
e
∂
∂κ
[
∂
y
(
t
|
t
−
1
,
)
(
t
,
κ
)=
y
(
t
)−
y
(
t
|
t
−
1
,
ˆκ
)] = −
(4.6)
∂κ
does not depend on the model parameters κ.
Then from (4.1),(4.5) and (4.6) the necessary condition for determining the optimal
κis
since, obviously, the plant output
y
(
t
)
∂
y
(
t
|
t
−
1
,
ˆκ
)
E
{[
y
(
t
)−
y
(
t
|
t
−
1
,
ˆκ
)]
} =
0
.
(4.7)
∂κ
Figure 4.2
Concentrate grade sensor measurement (dotted line) and soft sensor output (solid line),
where differences may be explained by the effect of important unmeasured disturbances
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