Richard E. Haskell Oakland University Rochester, MI USA Fuzzy Logic Richard E. Haskell Oakland University Rochester, MI USA

Fuzzy Logic

Normal “Crisp” logic where everything must be either True or False leads to PARADOXES

The sentence on the other side of the line is false The sentence on the other side of the line is false

A barber has a sign that reads: “I shave everyone who does not shave himself” Who shaves the barber?

Fuzzy Logic Lotfi Zadeh - Fuzzy Sets - 1965 Membership functions Degree of membership between 0 and 1 Fuzzy logic operations on fuzzy sets A and B NOT A => 1 - A A AND B => MIN(A,B) A OR B => MAX (A,B)

Membership Functions Young Not Young Age

Membership Functions Not Old Old Age

Membership Functions Not Old Not Young Age Middle Age = Not Old AND Not Young Age

Probabiltiy vs. Fuzziness Probability describes the uncertainty of an event occurrence. Fuzziness describes event ambiguity. Whether an event occurs is RANDOM. To what degree it occurs is FUZZY.

Probability: There is a 50% chance of an apple being in the refrigerator. Fuzzy: There is a half an apple in the refrigerator.

Fuzzy logic acknowledges and exploits the tolerance for uncertainty and imprecision.

Fuzzy Rules IF A AND B THEN L * * Fuzzy Control Inputs Map to Fuzzy Sets get_inputs(); Fuzzy Rules IF A AND B THEN L * * fire_rules(); Defuzzification find_output(); Output

Algorithm for a fuzzy controller do_forever { get_inputs(); fire_rules(); find_output(); }

get_inputs(); Given inputs x1 and x2, find the weight values associated with each input membership function. NM NS Z PS PM 0.7 0.2 X1 W = [0, 0, 0.2, 0.7, 0]

Fuzzy Inference fire_rules(); find_output(); Rule 1: If x1 is A1 and x2 is B1 then y is L1 find_output(); Rule 2: If x1 is A2 and x2 is B2 then y is L2 Given: x1 is a and x2 is b What is y?