Environmental Engineering Reference
In-Depth Information
Mass transport
sub.model
Grinding sub-model
dW
M
*
ψ
dw
(
p
-
1
P
E
=
-
R
=
K
Rw
dt
dt
w
Water feed flow
Grinding
Chamber
F
A
Ore feed flow
F
P
Grate
+
w
W
A
c
f
f
p*
p
K
E
Size
distribution
z*=C p*
Classification sub-
model
z
i
= C
i
(c
f
) p
i
Power draw M
p
sub-model
W
W
W
2
5
+
+
a
b
M
=
k
D
L
(
-
A
J
)
f
(
φ
)
p
V
mol
Figure 4.5
Gray state model for a SAG mill in which dynamic mass balances are combined with
empirical (black box) relations [50]
ments involving the delayed versions
y
of the primary variable prior to the
present time
t
. Instead, the previous estimated measurements are used. For example
let, the one step ahead prediction be the one found in Example 1. Then replacing
(
t
−
k
)
y
(
t
−
k
)=
y
(
t
−
k
|
t
−
k
−
1
)
(4.67)
the soft sensor output is given by
y
(
t
|
t
−
1
)=−
c
1
y
(
t
−
1
|
t
−
2
)−
c
2
y
(
t
−
2
|
t
−
3
)+
b
1
u
(
t
−
1
)
−(
a
1
−
c
1
)
y
(
t
−
1
|
t
−
2
)−(
a
2
−
c
2
)
y
(
t
−
2
|
t
−
3
)−
a
3
y
(
t
−
3
|
t
−
).
(4.68)
4
Collecting terms,
y
(
t
|
t
−
1
)=−
a
1
y
(
t
−
1
|
t
−
2
)−
a
2
y
(
t
−
2
|
t
−
3
)−
a
3
y
(
t
−
3
|
t
−
4
)+
b
1
u
(
t
−
).
(4.69)
1
This is a difference equation for the soft sensor output
y
(
t
)
driven by input
u
.The
solution depends on initial conditions,
e.g.
,
y
. These initial conditions could
be arbitrary, but a better choice could be some reasonable value derived from knowl-
edge of the plant, ideally their expected values. In any case, if (4.69) is asymptoti-
cally stable, the effect of the initial conditions vanish after a transient. This requires
that the roots of
z
3
(
1
),
y
(
0
)
a
1
z
2
+
+
a
2
z
+
a
3
=
0 must be inside the unit circle in the complex
plane [44].
Considering now the general case (4.18) and using (4.67),
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