An Introduction to Fuzzy Control - Fuzzy Logic: An Introductory Course for Engineering Students - page 175

8.1.1 Note

This chapter is divided in two main parts that on the one hand pretends to put fuzzy
control in the context of the topic, and on the other hand introduces the present
situation in the area of fuzzy control.
The first part makes a revision of the fundamentals of approximated reasoning
explained throughout the topic. A condensed view of how to apply the presented
theory from a fuzzy control perspective is given.
The second part outlines a glimpse of the state of the art of fuzzy control, where
fuzzy plant models and the procedures to obtain them are introduced. In the area
of fuzzy control there has been a shift from its original motivation of interpretable
fuzzy systems (systems that emulate human control strategy and are easy to use and
understand) towards a much more rigorous analysis where mainstream (nonlinear)
control criteria are considered. The reader must be aware that in order to fully under-
stand this part, he/she should probably refer to more specialized topics in the topic
as this section only pretends to present a very general view of the problem.

8.2 Revising Conditional and Implications in Fuzzy Control

The success of fuzzy logic mainly resides in the representation of elementary
statements “ x is P ”( x

∈

X and P a precise or imprecise predicate or linguistic

μ P : ₒ[ , ] μ P ( )

label on X ) by a function is the

degree up to which “ x is P ”, or x verifies the property named P . In the same view,
arule“If x is P , then y is Q ”( x

X 0 1 , in the hypothesis that x

∈ X , y ∈ Y ) is represented by a function of the

variables μ P and μ Q ,by

R (μ P , μ Q )( x , y ),

a number in once P , Q , x and y are fixed.

Functions R can be or can be not functionally expressible, that is, it can exist or
it cannot exist a numerical function J

[ 0 , 1 ]

:[ 0 , 1 ]×[ 0 , 1 ]ₒ[ 0 , 1 ] such that

R (μ P , μ Q )( x , y ) = J (μ P ( x ), μ Q ( y ))

for all x Y . Fuzzy logic works mainly within the positive supposition,

and several families of such functions J exist. Such numerical functions are called
conditional functions, and their diverse types are derived from the linguistic meaning
of the conditional phrase (the rule) “If x is P , then y is Q ”, that is, from its use in
the universe X

∈ X , y ∈

Y at each particular problem.
Even if it is some repetition of what was earlier presented, let's summarize the
steps that are necessary.

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