CHAPTER 3

MAMDANI FUZZY SYSTEMS

A system is a combination of components that as a whole operate on a vector of input functions of time x(t) ∈ ^{n} for each t to produce a vector of output functions of time y(t) ∈ ^{m} for each t. A fuzzy system is a system that uses fuzzy logic to operate on the input x(t) to produce the crisp output y(t). It can be shown that a fuzzy system with n inputs and m outputs (i.e., a multi-input multi-output, or MIMO fuzzy system) is equivalent to m fuzzy systems, each with n inputs and one output (i.e., m multi-input single-output, or MISO fuzzy systems). Therefore, we will study only MISO fuzzy systems in this book. All fuzzy controllers and identifiers are fuzzy systems.

3.1 IF-THEN RULES AND RULE BASE

Much of human decision making occurs in the form of “if-then” rules. The four forms of if-then rules in classical logic are called modus ponendo ponens (Latin for “mode that affirms by affirming”) or simply modus ponens, modus tollendo tollens (Latin for “mode that denies by denying”), or simply modus tollens, modus ponendo tollens (Latin for “mode that denies by affirming”), and modus tollendo ponens (Latin for “mode that affirms by denying”).

For examples of these, consider the problem of steering a vehicle toward a target in the presence of an obstacle. Possible ...