ABSTRACT
In this chapter, we set forth the basic mathematical ideas used in fuzzy control. These ideas are illustrated here with simple examples. Their applications will be expanded upon in later chapters.
There is an inherent impreciseness present in our natural language when we describe phenomena that do not have sharply deÞned boundaries. Such statements as “Mary is smart” and “Martha is young” are simple examples. Fuzzy sets are mathematical objects modeling this impreciseness. Our main concern is representing, manipulating, and drawing inferences
from such imprecise statements. Fuzzy set theory provides mathematical tools for carrying out approximate reasoning processes when available information is uncertain, incomplete, imprecise, or vague. By using the concept of degrees of membership to give a mathematical deÞnition of fuzzy sets, we increase the number of circumstances encountered in human reasoning that can be subjected to scientiÞc investigation. Humans do many things that can be classiÞed as control. Examples include
riding a bicycle, hitting a ball with a bat, and kicking a football through the goalposts. How do we do these things? We do not have the beneÞt of precise measurements, or a system of differential equations, to tell us how to control our motion, but humans can nevertheless become very skillful at carrying out very complicated tasks. One explanation is that we learn through experience, common sense, and coaching to follow an untold number of basic rules of the form “If...then...”:
If the bicycle leans to the right, then turn the wheel to the right.
