Soft Sensing - Advanced Control and Supervision of Mineral Processing Plants - page 178

measurements of plant variables from sensors which are close to the real sensor to
be replaced. In this way, only a part of the plant might be modeled, so that a con-
trol loop using the soft sensor contains a modeled and an unmodeled part of the
plant. Coupling between the modeled and unmodeled part of the plant may lead to
off-specification control performance - even instability - unless the fact that a soft
sensor is eventually going to replace the actual sensor is considered in the design of
the control loop [3, 35, 36].
4.3 Geometrical View of Modeling for Linear in the Parameter
and Nonlinear Models
The purpose of this section is to provide geometrical insight for modeling and iden-
tification problems. A general approach appears to be possible for linear and nonlin-
ear models which are LIP or that may be transformed into LIP models. In such case,
this approach covers a broad class of variables involved in modeling, e.g. , random
variables, stochastic processes (random functions or random sequences [47]), con-
tinuous functions, and integrable functions. In particular, the case of discrete time
random sequences is considered since they represent the preferential treatment for
soft sensor models adopted in the literature and the corresponding applications [1].
Due to the stochastic characteristics of the variables involved, results turn out to
be in terms of expected values. As a consequence, the problem of estimating them
by time averages of plant measurements appears, since the probability functions for
calculating expected values are usually not available. The modeling problem and the
estimation of expected values come together in the problem of finding the optimal
parameter vector for a model which is linear in the parameters, although it may be
nonlinear with respect to measured plant variables.
4.3.1 Modeling From the Inner Product Spaces Point of View
A number of modeling problems may be approached using the properties of inner
product spaces, of which the familiar Euclidean space ℜ n is just a particular case.
Inner product spaces are linear spaces endowed with an inner product. Other lin-
ear spaces are also special cases of inner product spaces, e.g. , spaces of random
variables, spaces of random functions (stochastic processes), spaces of random se-
quences, spaces of continuous functions, spaces of integrable functions [63, 64]. If
an inner product space has the property of completeness it is called a Hilbert Space,
in which problems concerning limits may be covered.
The fact that inner product spaces share the same axioms with Euclidean spaces
allows one to resort to geometrical images in the solution of problems, in particular,
in the case of modeling. Indeed, the Projection Theorem provides this geometrical
insight for finding a model for a variable y (primary variable in the case of soft