Environmental Engineering Reference
In-Depth Information
⎡
⎤
⎡
⎤
ϕ
T
(
t
−
T
+
1
)
y
(
t
−
T
+
1
)
⎣
.
⎦
,
⎣
.
⎦
,
X
=
Y
=
(4.63)
ϕ
T
(
t
)
y
(
t
)
so the estimator is given in terms of matrix operations instead of sums, which is
very convenient in computer programming. The covariance matrix of estimator θis
given by [44]
λ
0
t
∑
θ
ϕ
T
X
T
X
)
−
1
Cov
(
)=
ϕ
(
i
)
(
i
)
=
λ
0
(
,
(4.64)
i
=
t
−
T
+
1
where an unbiased estimator of λ
0
is
t
∑
1
θ
ˆ
λ
T
ϕ
T
=
1
(
y
(
i
)−
(
i
)
)
(4.65)
T
−
m
i
=
t
−
T
+
θ
and
m
is the dimension of ϕ. The diagonal of
Cov
(
)
are the variances of the pa-
rameter estimations.
4.2.2.2 Gray Models
The phenomenological part of gray models usually consists of dynamic balances
of masses, volumes, energy,
etc.
, as in the case of particle masses for different size
intervals in a mill. The empirical part is usually concerned with some internal pro-
cesses governing the relation between masses, volumes,
etc.
, for example, in a mill
model [50] the transfer of ore particles from one size range to another. Figure 4.5
gives a block diagram of this state model. Other examples of Gray models for soft
sensors are given in [11, 16, 17] and in [10, 19, 31]. In these cases nonlinear com-
binations of secondary variables having phenomenological meaning are used in the
definition of the basis functions ϕ
k
, but the models are LIP, hence NARX models.
An example is given in detail for the case of a +65# particle size soft sensor in
Section 4.3. One of the models determined there is
S
p
).
(4.66)
It can be seen that it is a NARX model, LIP but nonlinear in plant measurements:
mill power draw
J
BM
and solids concentration in the hydrocyclone feed flow
S
p
.
f
65
(
|
−
)=
(
−
)+
(
−
)
(
)+
(
−
)
(
−
)+
(
t
t
1
θ
0
f
65
t
1
θ
1
J
BM
t
1
t
θ
2
J
BM
t
3
S
p
t
3
θ
3
l
t
4.2.3 The Use of the Model as a Soft Sensor
Once the model has been determined it may be used as a soft sensor. Since the soft
sensor is replacing the actual sensor, no measurements are available for measure-
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