Soft Sensing - Advanced Control and Supervision of Mineral Processing Plants - page 156

cess, such as their rate of change (or frequency contents) so that the relevant plant
dynamics may be captured by the model ( e.g. , through persistent excitation).
The classes of models may be broadly classified as [44, 45]:

• Phenomenological models or white box models: Models derived from first prin-

ciples for which all parameters are known. For example, in the case of a mechan-
ical system formed by the interconnection of a mass a spring and a damper with
known mass, spring constant and damper constant.

• Empirical models or black box models: Models where there is almost no knowl-

edge of the phenomenology or first principles governing the modeled system.
Such is the case of hydrocyclone models. This class of models is very important
in mineral processing, since phenomenological models are too complex. In many
cases they are combined with phenomenological properties to produce gray mod-
els.

• Gray models are models that have characteristics of both white and black box

models. Here the phenomenological part usually consists of dynamic balances
of masses, volumes, energy, etc. , while the empirical part is usually concerned
with internal processes that are difficult or impractical to model using first princi-
ples ( e.g. , particle breakage, classification in hydrocyclones and SAG mills, mill
power draw). Many models of mineral processing units belong to this class.

4.2.1 Optimality Criterion

The criterion to decide which model is the best is usually based on the one step
ahead prediction error [1, 44]

e ( t , κ )= y ( t )− y ( t | t − 1 , κ ), (4.1)

between the plant output y and its one step ahead prediction output given by the

model, where κ includes all the model parameters. The model fitness is measured
by the mean square prediction error

( t )

2 2

J = E {[ y ( t )− y ( t | t − 1 , κ )] } = E { e ( t , κ ) }, (4.2)

so the best model is the one for which (4.2) is minimum. For a given model structure
the parameter vector κgiving the best model is found by vector ˆ κwhich minimizes

2

κ ˆ κ

J ( )= E { e ( t , ) }. (4.3)

In the case of structure determination J also incorporates the effect of the number
of model parameters, i.e. , the dimension of κ, in relation to the length of the data
sets T (see (4.74)). Since the required probability functions are rarely available the
minimum of (4.3) must be estimated by minimizing the corresponding time average