Environmental Engineering Reference
In-Depth Information
z
−
1
z
−
1
z
−
1
H
(
)
y
(
t
|
t
−
1
)=
G
(
)
u
(
t
)+(
H
(
)−
1
)
y
(
t
|
t
−
1
),
(4.70)
z
−
1
y
(
t
|
t
−
1
)=
G
(
)
u
(
t
).
(4.71)
For example in the ARMAX case, by (4.19)
z
−
1
B
(
)
z
−
1
z
−
1
y
(
t
|
t
−
1
)=
u
(
t
)
A
(
)
y
(
t
|
t
−
1
)=
B
(
)
u
(
t
).
(4.72)
A
(
z
−
1
)
In the ARX case, by (4.21)
z
−
1
z
−
1
A
(
)
y
(
t
|
t
−
1
)=
B
(
)
u
(
t
),
(4.73)
so the prediction has the same form as (4.72) for the ARMAX case. However, in
order to obtain a correct estimate of parameters in
A
and
B
, parameter estimation
corresponding to the ARMAX case must be used.
4.2.4 Development of Soft Sensor Models
The process of finding the best model for a soft sensor may be separated into two
stages:
•
structure determination;
•
parameter estimation.
Prior to using the data in model determination it must be analyzed and subjected
to certain processing, as indicated in Section 4.3 and by Fortuna
et al.
[1] and Ljung
[44] (see also [31]).
4.2.4.1 Model Structure
The determination of model structure is concerned with finding a form for the soft
sensor model. In particular in the case of ARX and NARX class of models this in-
volves finding the most significant basis functions ϕ
k
correlated with the primary
measurement which will be used in the model. In the neural network case, the struc-
ture comprises the type of network, the network topology in general, the most sig-
nificant measurements to be used as inputs, the number of layers, the selection of the
activation functions, and the number of nodes in each layer. Composite components
may also be considered as suitable inputs for a neural network. A schematic dia-
gram of the determination of model structure is shown in Figure 4.6. The selection
of plant measurements is determined by the measurements required by the selected
components or basis functions.
A model having a given structure may be valid for a large operating region of
a process, but its parameters may need to be updated as the operating point un-
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