Information Technology Reference
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The processing unit cares for the compatibility of data (input and output interfaces)
and for the execution of the rules (inference engine), which is mostly done through
pointwise numerical calculations of implications.
As it has been mentioned in previous chapters, the basic rules of reasoning used
in classical logic are the modus ponens for forward reasoning and modus tollens for
backwards reasoning. The symbolic expression of the modus ponens,
A
B
A
B
,
means that if the rule “if
A
then
B
” is given and the event A is observed, then the
event B should also be observed.
These processes are referred to in the literature as
inference
. As it has been
presented in this topic, in the case of fuzzy logic, a generalization of modus ponens
is used, based on fuzzy sets. Given a universal set
X
, a fuzzy set
A
on
X
is defined by
its membership function
gives the degree
of membership of
x
to
A
, or the degree with which
x
fulfills the concept represented
by
A
. Fuzzy sets have a semantic role: they represent the way in which a statement
“
x
is
A
” is used in a given context.
The generalized modus ponens used in fuzzy control may be given in its simplest
expression as
μ
A
:
X
ₒ[
0
,
1
]
and for all
x
∈
X
,
μ
A
(
x
)
A
B
A
B
,
where
A
,
A
∗
,
B
and
B
∗
are fuzzy sets,
A
and
A
∗
are defined on a same universe, but
they are not necessarily equal. Similarly for
B
and
B
∗
. The meaning in this case is
the following: given a rule “if
A
then
B
” and observing an event
A
∗
which is similar
to
A
, an event
B
∗
is expected, which should also be similar to
B
.
To allow more specified situations to define a rule base in a multivariate system,
in the expression
A
B
,
A
may stand for a set of conditions that have to be fulfilled
at the same time. The formal representation is a conjunction of fuzzy premises.
For the computation of conjunctions, operations belonging to the class of trian-
gular norms, or simply t-norms, are used. If
T
ₒ
2
is a t-norm, then it
is non-decreasing, associative, commutative and has 1 as identity. The other needed
operation is the “then” in “if
A
then
B
”. In the case of fuzzy logic this operation is
not only an extension of the classical implication and its characterization has been
thoroughly studied.
:[
0
,
1
]
ₒ[
0
,
1
]
Example 8.1.1
The following example is presented at a “phenomenological” level
to allow an intuitive understanding of the main issues. In this case a heating system
has to be controlled. The system is based on warmwater circulating at constant speed
and passing through well distributed heating panels. The water temperature should
be proportional to the heating demand. The following rules reflect the knowledge of
an experienced operator of the heating system:
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