Fuzzy Logic
Dave Saad
CS498
Origin
Proposed as a mathematical model
similar to traditional set theory but with
the possibility of partial set membership
by Lotfi A. Zadeh in 1965.
Allows for ambiguity or “fuzzy”
boundaries of set membership as opposed
to “crisp” membership of “in” or “out” of a
set
Useful when given inputs are subjective.
Initially Rejected by
Math and Science
Communities
While initially rejected in the western world, the
idea of embracing ambiguity in a control
system was quickly adopted by Japanese
Industry and successfully turned into many
varied products from rice cookers to subway
train controllers.
American Industry followed slowly.
Fuzzy Logic
▸ Based on Linguistic Rules of Inference
Modus Ponens
Modus Tollens
▸ Rules of logic developed by ancient Greeks
that became the basis of (Aristotelean)
western philosophy.
▸ Eastern philosophical thought is more
amenable to ambiguity which may explain the
early adoption of Fuzzy Logic
Modus Ponens
A conditional Statement and its antecedent:
If P then Q
P is TRUE
Therefore Q is also TRUE
The consequent (Q) is inferred to be TRUE from
the truth value of the antecedent (P)
Modus Tollens
A conditional Statement and its antecedent:
If P then Q
NOT P (or P is FALSE)
Therefore Q is also FALSE
The consequent (Q) is inferred to be FALSE
from the truth value of the antecedent (P)
Logical Operators
▸ NOT (negation
Defined as 1-A
▸ AND (intersection)
Defined as MIN(A,B)
▸ OR (union)
Defined as MAX(A,B)
Other operators are possible but rarely used
Partial Membership
NS
Z
V
Rules of Inference
These rules are often based on human instinct and experience.
•If
Temperature is COLD then fan speed is STOP
•If
Temperature is COOL then fan speed is SLOW
•If
Temperature is JUST RIGHT then fan speed is MEDIUM
•If
Temperature is WARM then fan speed is FAST
•If
Temperature is HOT then fan speed is TORNADO
Execute the Rules
•Rules all fire all the time
•Rules all fire in parallel
•All rules fire to some degree
•Most rules fire to zero degree
•The result is a union of fuzzy results from each
rule
Defuzzify The Result
Set
•Centroid or area under the output set curve
•Computationally
intensive
•Mean of Maximum
•Max-membership (height method)
•Weighted Average
•This
produces results very close to the
COA method and is less computationally
intensive
Defuzzification
The best method is problem dependent..
Four criteria against which to measure the methods:
1. Continuity. A small change in input should produce a
small change in the output
2. Disambiguity. Output should resolve to a unique value.
3. Plausibility. Crisp output value should have a high
degree of membership.
4. Computational Simplicity. Integrals are hard to do on a
8-bit microcontroller.
Summary
1. Fuzzification of inputs
2. Choose linguistic variables and terms and
associated fuzzy sets
3. Build rules of inference
4. Defuzzification