138 CHAPTER 4. FUZZY CONTROL
control laws using logical rules. That is why we sometimes call this methodology
fuzzy logic control.
The rules are either obtained from experts (human controllers) or generated
from numerical observations. When human experts experienced in controlling a
given system are available, design engineers can interview these experts about
their control strategies and then express the expertsí knowledge as a collection
of control ìIf...then...î rules. Even when the input-output relations in a complex
system are not known precisely, the behavior of the control process provides a
valuable source of information for suggesting control strategies and extracting
fuzzy rules. That is, one can supply numerical inputs and observe numerical
outputs of a control system as a means to obtain linguistic control ìIf...then...î
rules.
It is clear that there isflexibility in choices of membership functions, rules,
fuzzy logic operations, and defuzzification procedures when designing fuzzy con-
trollers. Note that fuzzy systems, or fuzzy models, are actually mathematical
models ñ that is, they are mathematical descriptions of relations between vari-
ables, using membership functions of fuzzy sets. These mathematical models are
flexible, but they are not ìfuzzyî in layman terms. The membership functions
themselves are precise mathematical functions.
Just as in the case of standard control, where existence of control laws should
be guaranteed before proceeding to obtain the control algorithm, we should ask
whether a given linguistic rule base is sufficient to derive a ìsuccessfulî control
law. The rule base is a description of how we should control the plant. Each
rule islocalin nature ñ that is, each rule tells us how we should control the
plant in some small region of the input spaceX. Since we are going to use this
information to derive aglobalcontrol lawφ, a map on the entire input spaceX,
these small regions specified in the rules should cover all points ofXin some
fashion. In other words, the membership functions defined onXshould form a
fuzzy partition ofX.
The methodology of fuzzy control consists of selecting and using- a collection of rules that describe the control strategy,
- membership functions for the linguistic labels in the rules,
- logical connectives for fuzzy relations, and
- a defuzzification method.
At the end, the derived control law is the realization of a functionφfromXto
U, the space of control variables.
The strategy above for deriving control laws is based somewhat on common
sense. However, its successful applications can be explained theoretically in
the context of the theory of approximation of functions. See the statement
of the Stone-Weierstrass theorem, Chapter 3, theorem 3.8, for example. Fuzzy
systems satisfy a universal approximation property [78], which means that there
isasystemthatwilldothetaskyouwanttoaccomplish. However,thisdoesnot
say how to set up or tune such a fuzzy system. Such mathematical justifications
only provide some assurance that, if you work at it hard enough, it is possible
to design successful fuzzy controllers. The capability of fuzzy controllers can