Ch. 2: The Mathematics of Fuzzy Control

The Mathematics of Fuzzy Control

• Introduction: Fuzzy Sets• Fuzzy Relations• Approximate Reasoning• Representation of a Set of Rules

Introduction: Fuzzy Sets

• Vagueness• Fuzzy Set Theory Versus Probability

Theory• Classical Set Theory• Fuzzy Sets• Properties of Fuzzy Sets• Operations on Fuzzy Sets

VaguenessHow do we get computers to handle vagueness, like

“Is person A tall?”

What does it mean to be “tall”?Male/female?

Geography/subpopulations?Canary Islanders as Spanish colonists

thresholders estimators

conservatives

Fuzzy logic grows out of the estimators approach to vagueness.

Fuzziness is due to lack of well defined boundaries.•Universe of discourse•precise membership degrees do not exist by themselves, but are only tendency indices that are subjectively assigned by an individual or a group of individuals.

•Membership degree is an ordering.•Membership degrees are context dependent.•Fuzziness is not imprecision. We might agree on precisely how tall someone is, but disagree on the person’s tallness

Homework

• Come up with 3 vague concepts, for each – define the universe of discourse for two

different contexts

Fuzzy Set Theory vs Probability

Fuzzy set theory is not probability theory.

Enough said.

Classical Set Theory• Set operations

– Intersection, Union, Complement, inference– and, or, not, if-then statements– Venn Diagrams

• Logic– truth tables

• Mathematization of Logic– Boolean algebra– characteristic function

• Table 2.2, properties of classical set operations

Fuzzy Sets• Characteristic function to membership

function– Expensive cars– natural numbers close to 6– formulas vs /notation

• Bell, S, Z membership functions• triangular (lambda), trapezoidal, S, Z

membership functions (pg. 51)

Properties of fuzzy sets

• Support• width• nucleus• height• convexity

Operations on Fuzzy Sets

• Equality• inclusion• union• intersection• complement

Axiomatics (pg. 57)

• Triangular norm (general intersection)– Archimedean property

• Triangular co-norm (general union)• Complement• pp. 58-61 other norms and co-norms

Properties of Fuzzy Sets

Operations on Fuzzy sets

Fuzzy Relations

• Classical Relations• Fuzzy Relations• Operations on Fuzzy Relations• The Extension Principle

Classical Relations

Fuzzy Relations

Operations on Fuzzy Relations

The extension principle

Approximate Reasoning

• Introduction• Linguistic Variables• Fuzzy Propositions• Fuzzy If-Then statements• Inference Rules• The compositional Rule of Inference• Representing the Meaning of If-Then Rules

Intro to approx reasoning

Linguistic variables

Fuzzy Propositions

Fuzzy If-Then statements

Inference rules

The compositional Rule of Inference

Representing the Meaning of If-Then Rules

Representing a Set of Rules

• Mamdani versus Godel• Properties of a Set of Rules• Completeness of a set of rules• Consistency of a set of rules• Continuity of a set of rules• Interaction of a set of rules

Mamdani vs Godel

Properties of a set of Rules

Completeness of a set of rules

Consistency of a set of rules

Continuity of a set of rules

Interaction of a set of rules

Chapter 2 Homework