Environmental Engineering Reference
In-Depth Information
J
p
y
u
Class of
Signals
U
Criterion
PLANT
y d
v
η
Delays
Class of
Models
y
M
Candidate
Model
Class
Delays
Model M
Figure 4.3 A general approach to modeling. Vector u contains the manipulated variables, v is the
vector of measured disturbances, p is the vector of unmeasured disturbances, y is the plant output
to be modeled, ηare other measured plant outputs and y is the model output
4.2.2 A General Class of Black Box Models
Black box and gray box models are especially important for soft sensors in mineral
processing plants because the complexities of the involved processes entail very in-
tricate phenomenological models. As a consequence it is extremely difficult to iden-
tify these models accurately and reliably with plant data. Simpler back box models,
on the other hand may be much more easily identified, but they usually require up-
dating as the operation point shifts. However, gray models combining phenomenol-
ogy with empirical or black box models afford a possible and useful compromise
between simplicity and the need for too frequent updating. A unified approach to
black box modeling is given by Sjoberg et al. [45] and Bakshi et al. [46] which cov-
ers all classes of black box models: FIR, ARX, ARMAX, OE, NFIR, NARMAX,
NOE, nonlinear LIP models such as NARX models, wavelet models, radial basis
function models, neural network models, Takagi and Sugeno, PCA and PLS based
models, and other model structures. As an example in [31] ARX, NARX, PLS, Tak-
agi and Sugeno, and wavelet models are used in the design of soft sensors for the
concentrate grade of a rougher flotation bank of an industrial copper concentrator.
What follows is based on the general class of black box models proposed by
Sj oberg et al. [45]. Let this general model be
m
k = 1 θ k ϕ k (
y
(
t
|
t
1
,
κ
)=
ξ
(
t
),
μ k
),
(4.8)
where μ k is a row vector of parameters for function ϕ k and
T
ξ
(
t
)=[
ξ 1
(
t
)
ξ 2
(
t
) ...
ξ q
(
t
)]
(4.9)
 
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