Soft Sensing - Advanced Control and Supervision of Mineral Processing Plants - page 175

When the sensor becomes unavailable at t f there are three choices for the soft
sensor parameters. On the one hand the values that had been estimated up to time t f
may be used (frozen parameters). But a better choice might be to use the average of
the estimated parameters or, better still, a prediction of the future parameter values
using a parameter evolution sub-model, whenever appropriate conditions are met
and the sub-model parameters may be estimated while the soft sensor is working.
The sub-model is then used to give an optimal prediction of the future value of the
soft sensor model parameters for t t f . This optimal prediction for a given param-

eter is its conditional expectation given its value prior to the time of failure t f [19].
As an example a first order sub-model for each parameter of a soft sensor for parti-
cle size measurement in an industrial grinding plant has been used. The prediction
evolves exponentially from the initial value at t f to its unconditional expected value
for t

>

t f . As should be expected, the frozen parameter choice gives better pre-
diction for times near the fault, while the unconditional expected value parameters
give a better prediction for t

>>

>> t f [19].

4.2.5.1 Region of Validity

Once a sensor fails, the parameter updating must cease, since it is no longer possible
to determine the error between the soft sensor output and the actual measurement.
The soft sensor signal then begins to perform its job, providing a virtual measure-
ment of the missing one, using the secondary measurements or composite com-
ponents as bases. But plant characteristics may change during the period after the
sensor failure, e.g. , because the operation point shifts due to disturbances or control
actions. If the model involves a simplification of reality, as is usually the case, it may
be expected that, except in very simple instances, the model will likely not be able to
represent the part of the actual plant to be modeled for all possible operating points
or operating trajectories. This is clearly the situation when for the sake of having a
simple model, a class of linear models is selected containing the candidates that rep-
resent a nonlinear plant. As the operating point changes, the optimal parameter set
should change, so a frequent updating of the model parameters may be required. But
since the primary sensor signal is no longer available this updating is no longer pos-
sible, and the soft sensor error becomes unacceptable since its region of validity has
been surpassed. One way to palliate this performance degradation is to increase the
region of validity of the soft-sensor model by incorporating more structure into it,
e.g. , using NARX or NARMAX models having physically significant components
( i.e. , gray models),[10, 11, 16, 17, 31] or sufficiently complex neural networks. In
any case, on-line adaptation requires a compromise to be made between simple soft
sensor models requiring frequent but fast adaptation ( e.g. ARX models) and com-
plex models requiring a less frequent but slower adaptation process, e.g. ,neural
networks of increasing complexity [12]. In between are those models that although
nonlinear are LIP ( e.g. , NARX models), and hence may undergo faster adaptation
even though they may be nonlinear in the secondary measurements. Another way is
to use clustering, as explained above.