Fuzzy Logic 11/6/2001

Agenda General Definition Applications Formal Definitions Operations Rules Fuzzy Air Conditioner Controller Structure

General Definition Fuzzy Logic Lotfi Zadeh, Berkely superset of conventional (Boolean) logic that has been extended to handle the concept of partial truth central notion of fuzzy systems is that truth values (in fuzzy logic) or membership values (in fuzzy sets) are indicated by a value on the range [0.0, 1.0], with 0.0 representing absolute Falseness and 1.0 representing absolute Truth. deals with real world vagueness

Applications ABS Brakes Expert Systems Control Units Bullet train between Tokyo and Osaka Video Cameras Automatic Transmissions

Formal Definitions Definition 1: Let X be some set of objects, with elements noted as x. X = {x}. Definition 2: A fuzzy set A in X is characterized by a membership function mA(x) which maps each point in X onto the real interval [0.0, 1.0]. As mA(x) approaches 1.0, the "grade of membership" of x in A increases. Definition 3: A is EMPTY iff for all x, mA(x) = 0.0. Definition 4: A = B iff for all x: mA(x) = mB(x) [or, mA = mB]. Definition 5: mA' = 1 - mA. Definition 6: A is CONTAINED in B iff mA mB. Definition 7: C = A UNION B, where: mC(x) = MAX(mA(x), mB(x)). Definition 8: C = A INTERSECTION B where: mC(x) = MIN(mA(x), mB(x)).

Operations A B A B A B A

Controller Structure Fuzzification –Scales and maps input variables to fuzzy sets Inference Mechanism –Approximate reasoning –Deduces the control action Defuzzification –Convert fuzzy output values to control signals

Rule Base Air Temperature Set cold {50, 0, 0} Set cool {65, 55, 45} Set just right {70, 65, 60} Set warm {85, 75, 65} Set hot {, 90, 80} Fan Speed Set stop {0, 0, 0} Set slow {50, 30, 10} Set medium {60, 50, 40} Set fast {90, 70, 50} Set blast {, 100, 80}

Rules Air Conditioning Controller Example: IF Cold then Stop If Cool then Slow If OK then Medium If Warm then Fast IF Hot then Blast default: The truth of any statement is a matter of degree default: The truth of any statement is a matter of degree Membership function is a curve of the degree of truth of a given input value

Fuzzy Air Conditioner

Mapping Inputs to Outputs