FL-BASED MODELING

The random phenomena that represent uncertainty are modeled by probability theory, which is based on crisp (binary) logic. Our interest is to model the uncertainty that abounds in nature and to discuss the need to model uncertainty in science and engineering problems. One way is to use crisp logic:

The crisp or Boolean characteristic function:

fA(x) = 1 if input x is in set A

= 0 if input x is not in set A

In classical set theory, a set consists of a finite/infinite number of elements that belong to some specified set called the universe of discourse. In crisp logic, we have answers like yes or no; 0 or 1; —1 or 1; off or on. Examples are (1) a person is in the room or not in the room, (2) an event A has occurred or not occurred, (3) light is on or off. However, real-life experiences indicate that some extension of crisp logic is definitely necessary. Events or occurrences leading to FL are (1) the light could be dim, (2) the day could be bright with a certain degree of brightness, (3) the day could be cloudy with a certain degree, and (4) the weather could be warm or cold. Thus, FL allows for a degree of uncertainty and gradation. Thus, truth and falsity (1 or 0) become the extremes of a continuous spectrum of uncertainty. This leads to multi­valued logic and the fuzzy set theory [7]. Fuzziness is the theory of sets and a characteristic function is generalized to take an infinite number of values between 0 and 1: e. g., triangular form. Several types/forms of membership function are available in MATLAB FL toolbox.

Fuzzy systems can model any continuous function or system. The quality of approximation depends on the quality of rules that can be formed by experts. Fuzzy engineering is a function approximation (FA) with fuzzy systems. This approxima­tion does not depend on words, cognitive theory, or linguistic paradigm. It rests on mathematics of FA and statistical learning theory (SLT). Much of the mathematics is well known, and as such there is no magic in fuzzy systems. A fuzzy system is a natural way to turn speech and measured action into functions that approximate hard tasks. Words are just a tool or a ladder we climb on to perform the task of FA. Fuzzy language is a means to the end of computing and not the goal. The basic unit of fuzzy FA is the ‘‘If then’’ rule: ‘‘If the wash water is dirty then add more detergent.’’ Thus, a fuzzy system is a set of ‘‘If then’’ rules, which maps input sets like ‘‘dirty wash water’’ to output sets like ‘‘more detergent.’’ Overlapping rules define polynomials/richer functions. A set of possible rules are given below [23]:

Rule 1: if the air is cold then set the motor speed to stop Rule 2: if the air is cool then set the motor speed to slow Rule 3: if the air is just right then set the motor speed to medium Rule 4: if the air is warm then set the motor speed to fast Rule 5: if the air is hot then set the motor speed to blast

This gives the first-cut fuzzy system. More rules can be guessed, formulated, and added by experts and by learning new rules adaptively from training data sets. ANNs can be used to learn the rules from the data. Much of fuzzy engineering deals with tuning these rules and adding new rules or pruning old rules.