Environmental Engineering Reference
In-Depth Information
is a vector of the selected inputs to the model, which for
k
>
1 may contain com-
mands or manipulated variables
u
(
t
−
k
)
, measured disturbances
v
(
t
−
k
)
, delayed
primary variable outputs
y
(
t
−
k
)
, other plant outputs η
(
t
−
k
)
, delayed predictions
y
[1,45].Theselectionof
basis functions ϕ
k
in the model structure determination phase is covered in Section
4.2.4.
For the case of the soft sensor models covered in this Chapter, in (4.8) vector κ
contains all the model parameters to be determined,
i.e.
, θ and μ:
(
t
|
t
−
1
−
k
,
ˆκ
)
, and prediction errors (residuals)
e
(
t
−
k
,
κ
)
θ
T
μ
T
T
T
T
κ
=[
]
,
θ
=[
θ
1
θ
2
...
θ
m
]
,
μ
=[
μ
1
μ
2
...
μ
m
]
.
(4.10)
appear, (4.8)
is just the estimated output obtained and (4.1) becomes the modeling error.
Specific black box models turn out to be special cases of this general form, in-
cluding neural network models. Let
In static models and in dynamic models where present values ξ
j
(
t
)
T
ϕ
(
t
,
μ
)=[
ϕ
1
(
ξ
(
t
),
μ
1
)
ϕ
2
(
ξ
(
t
),
μ
2
) ...
ϕ
m
(
ξ
(
t
),
μ
m
)]
.
(4.11)
Equation 4.8 may be written, using (4.10) and (4.11),
ϕ
T
(
|
−
,
)=
(
(
),
)
y
t
t
1
κ
ξ
t
μ
θ
(4.12)
For example, if the ϕ
k
are radial basis functions or wavelets, parameters in μare
related to scaling and delays or shifts [45]. The following abbreviated notation shall
be used:
ϕ
k
(
ξ
(
t
),
μ
k
)=
ϕ
k
(
t
,
μ
).
(4.13)
Figure 4.4 has a schematic representation of (4.8).
If in (4.11) functions ϕ
k
do not contain model parameters, or if their parameters
are fixed and known, let this fact be reflected in the notation
ϕ
k
(
ξ
(
t
),
μ
k
)=
ϕ
k
(
ξ
(
t
)) =
ϕ
k
(
t
), {
k
=
1
, ...,
m
},
(4.14)
ϕ
T
y
(
t
|
t
−
1
,
θ
)=
(
ξ
(
t
))
θ
,
(4.15)
and in this case κ
t
is not a model
input ξ
r
because it is a function of parameter vector θ, implying that some of the ϕ
k
would also be functions of θ. In addition, let the set (4.14) be a linearly independent
set. Then the set of functions (4.14) constitute a basis
B
m
for
y
=
θ. It must be noted that here
y
(
i
|
i
−
1
,
θ
)
,for
i
<
as seen in
Section 4.3. Linear combinations of the basis functions given by (4.12) span a linear
space to which the LIP model output
y
(
t
|
t
−
1
,
θ
)
(
t
|
t
−
1
,
θ
)
belongs, since
ϕ
T
(
|
−
,
)=
(
)+
(
)+...+
(
)=
(
(
))
.
y
t
t
1
θ
θ
1
ϕ
1
t
θ
2
ϕ
2
t
θ
m
ϕ
m
t
ξ
t
θ
(4.16)
Basis functions ϕ
k
provide a means for introducing nonlinearities in a model
which is LIP, since they may be nonlinear combinations of the ξ
j
,
e.g.
, of direct
plant measurements contained in ξ. Basis functions ϕ
k
may have physical signifi-
Search WWH ::
Custom Search