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Fuzzy logic provides a method for representing analog processes in a digital framework. Processes that are implemented through fuzzy logic are often not easily separated into discrete segments and may be difficult to model with conventional mathematical or rule-based paradigms that require hard boundaries or decisions, i.e., binary logic where elements are a member of a given set or they are not. Consequently, fuzzy logic is valuable where the boundaries between sets of values are not sharply defined or there is partial occurrence of an event. In fuzzy set theory, an element's membership in a set is a matter of degree. This chapter describes the concepts inherent in fuzzy set theory and applies them to the solution of the inverted pendulum problem and a Kalman filter problem. Fuzzy and artificial neural network concepts may be combined to form adaptive fuzzy neural systems where either the weights and∕or the input signals are fuzzy sets. Fuzzy set theory may be extended to fuse information from multiple sensors as discussed in the concluding section. 9.1 Conditions under which fuzzy logic provides an appropriate solution Lotfi Zadeh originally developed fuzzy set theory in 1965. Zadeh reasoned that the rigidity of conventional set theory made it impossible to account for vagueness, imprecision, and shades of gray that are commonplace in real-world events. By establishing rules and fuzzy sets, fuzzy logic creates a control surface that allows designers to build a control system even when their understanding of the mathematical behavior of the system is incomplete. Fuzzy logic permits the incorporation of the concept of vagueness into decision theory. For example, an observer may say that a person is “short” without specifying their actual height as a number. One may postulate that a reasonable specification for an adult of short stature is anyone less than 5 feet. However, other observers may declare 5 feet-2 inches or 5 feet-3 inches the cutoff between average and short height. Other examples of vagueness abound. An object may be said to be “near” or “far” from the observer, or that an automobile is traveling “faster” than the speed limit. In these examples, there is a range of values that satisfies the subjective term in quote marks.
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