1 / 35

Ch. 2: The Mathematics of Fuzzy Control

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

lalo
Télécharger la présentation

Ch. 2: The Mathematics of Fuzzy Control

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Ch. 2: The Mathematics of Fuzzy Control

  2. The Mathematics of Fuzzy Control • Introduction: Fuzzy Sets • Fuzzy Relations • Approximate Reasoning • Representation of a Set of Rules

  3. Introduction: Fuzzy Sets • Vagueness • Fuzzy Set Theory Versus Probability Theory • Classical Set Theory • Fuzzy Sets • Properties of Fuzzy Sets • Operations on Fuzzy Sets

  4. How 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 Vagueness Fuzzy logic grows out of the estimators approach to vagueness.

  5. 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

  6. Homework • Come up with 3 vague concepts, for each • define the universe of discourse for two different contexts

  7. Fuzzy Set Theory vs Probability Fuzzy set theory is not probability theory. Enough said.

  8. 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

  9. 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)

  10. Properties of fuzzy sets • Support • width • nucleus • height • convexity

  11. Operations on Fuzzy Sets • Equality • inclusion • union • intersection • complement

  12. Axiomatics (pg. 57) • Triangular norm (general intersection) • Archimedean property • Triangular co-norm (general union) • Complement • pp. 58-61 other norms and co-norms

  13. Properties of Fuzzy Sets

  14. Operations on Fuzzy sets

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

  16. Classical Relations

  17. Fuzzy Relations

  18. Operations on Fuzzy Relations

  19. The extension principle

  20. 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

  21. Intro to approx reasoning

  22. Linguistic variables

  23. Fuzzy Propositions

  24. Fuzzy If-Then statements

  25. Inference rules

  26. The compositional Rule of Inference

  27. Representing the Meaning of If-Then Rules

  28. 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

  29. Mamdani vs Godel

  30. Properties of a set of Rules

  31. Completeness of a set of rules

  32. Consistency of a set of rules

  33. Continuity of a set of rules

  34. Interaction of a set of rules

  35. Chapter 2 Homework

More Related