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Fuzzy Control

Fuzzy Control. Lecture 2 Fuzzy Set Basil Hamed Electrical Engineering Islamic University of Gaza. Content. Crisp Sets Fuzzy Sets Set-Theoretic Operations Extension Principle Fuzzy Relations. Introduction. Fuzzy set theory provides a means for representing uncertainties.

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Fuzzy Control

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  1. Fuzzy Control Lecture 2 Fuzzy Set Basil Hamed Electrical Engineering Islamic University of Gaza

  2. Content • Crisp Sets • Fuzzy Sets • Set-Theoretic Operations • Extension Principle • Fuzzy Relations Dr Basil Hamed

  3. Introduction Fuzzy set theory provides a means for representing uncertainties. Natural Language is vague and imprecise. Fuzzy set theory uses Linguistic variables, rather than quantitative variables to represent imprecise concepts. Dr Basil Hamed

  4. Fuzzy Logic Fuzzy Logic is suitable to Very complex models Judgmental Reasoning Perception Decision making Dr Basil Hamed

  5. Crisp Set and Fuzzy Set Dr Basil Hamed

  6. Information World Crisp set has a unique membership function A(x) = 1 x  A 0 x  A A(x)  {0, 1} Fuzzy Set can have an infinite number of membership functions A  [0,1] Dr Basil Hamed

  7. Fuzziness Examples: A number is close to 5 Dr Basil Hamed

  8. Fuzziness Examples: He/she is tall Dr Basil Hamed

  9. Classical Sets Dr Basil Hamed

  10. CLASSICAL SETS Define a universe of discourse, X, as a collection of objects all having the same characteristics. The individual elements in the universe X will be denoted as x. The features of the elements in X can be discrete, or continuous valued quantities on the real line. Examples of elements of various universes might be as follows: • the clock speeds of computer CPUs; • the operating currents of an electronic motor; • the operating temperature of a heat pump; • the integers 1 to 10. Dr Basil Hamed

  11. Operations on Classical Sets Union: A  B = {x | x  A or x  B} Intersection: A  B = {x | x  A and x  B} Complement: A’ = {x | x  A, x  X} X – Universal Set Set Difference: A | B = {x | x  A and x  B} Set difference is also denoted by A - B Dr Basil Hamed

  12. Operations on Classical Sets Union of sets A and B (logical or). Intersection of sets A and B. Dr Basil Hamed

  13. Operations on Classical Sets Complement of set A. Difference operation A|B. Dr Basil Hamed

  14. Properties of Classical Sets A  B = B A A  B = B  A A  (B  C) = (A  B)  C A  (B  C) = (A  B)  C A  (B  C) = (A  B)  (A  C) A  (B  C) = (A  B)  (A  C) A  A = A A  A = A A  X = X A  X = A A   = A A   =  Dr Basil Hamed

  15. Mapping of Classical Sets to Functions Mapping is an important concept in relating set-theoretic forms to function-theoretic representations of information. In its most general form it can be used to map elements or subsets in one universe of discourse to elements or sets in another universe. Dr Basil Hamed

  16. Fuzzy Sets Dr Basil Hamed

  17. Fuzzy Sets • A fuzzy set, is a set containing elements that have varying degrees of membership in the set. • Elements in a fuzzy set, because their membership need not be complete, can also be members of other fuzzy sets on the same universe. • Elements of a fuzzy set are mapped to a universe of membership values using a function-theoretic form. Dr Basil Hamed

  18. Fuzzy Set Theory • An object has a numeric “degree of membership” • Normally, between 0 and 1 (inclusive) • 0 membership means the object is not in the set • 1 membership means the object is fully inside the set • In between means the object is partially in the set Dr Basil Hamed

  19. If U is a collection of objects denoted generically by x, then a fuzzy setA in U is defined as a set of ordered pairs: membership function U : universe of discourse. Dr Basil Hamed

  20. Fuzzy Sets Characteristic function X, indicating the belongingness of x to the set A X(x) = 1 x  A 0 x  A or called membership Hence, A  B  XA  B(x) = XA(x)  XB(x) = max(XA(x),XB(x)) Note:Some books use + for , but still it is not ordinary addition! Dr Basil Hamed

  21. Fuzzy Sets A  B  XA  B(x) = XA(x)  XB(x) = min(XA(x),XB(x)) A’  XA’(x) = 1 – XA(x) A’’ = A Dr Basil Hamed

  22. Fuzzy Set Operations A  B(x) = A(x)  B(x) = max(A(x), B(x)) A  B(x) = A(x)  B(x) = min(A(x), B(x)) A’(x) = 1 - A(x) De Morgan’s Law also holds: (A  B)’ = A’  B’ (A  B)’ = A’  B’ But, in general A  A’ A  A’ Dr Basil Hamed

  23. Fuzzy Set Operations Union of fuzzy sets A and B∼ . Intersection of fuzzy sets Aand B∼ . Dr Basil Hamed

  24. Fuzzy Set Operations Complement of fuzzy set A∼ . Dr Basil Hamed

  25. Operations A B A  B A  B A Dr Basil Hamed

  26. A  A’ = X A  A’ = Ø Excluded middle axioms for crisp sets. (a) Crisp set A and its complement; (b) crisp A ∪ A= X (axiom of excluded middle); and (c) crisp A ∩ A = Ø (axiom of contradiction). Dr Basil Hamed

  27. A  A’ A  A’ Excluded middle axioms for fuzzy sets are not valid. (a) Fuzzy set A∼ and its complement; (b) fuzzy A∪ A∼ = X (axiom of excluded middle); and (c) fuzzy A ∩ A = Ø (axiom of contradiction). Dr Basil Hamed

  28. Set-Theoretic Operations Dr Basil Hamed

  29. Examples of Fuzzy Set Operations • Fuzzy union (): the union of two fuzzy sets is the maximum (MAX) of each element from two sets. • E.g. • A = {1.0, 0.20, 0.75} • B = {0.2, 0.45, 0.50} • A  B = {MAX(1.0, 0.2), MAX(0.20, 0.45), MAX(0.75, 0.50)} = {1.0, 0.45, 0.75} Dr Basil Hamed

  30. Examples of Fuzzy Set Operations • Fuzzy intersection (): the intersection of two fuzzy sets is just the MIN of each element from the two sets. • E.g. • A  B = {MIN(1.0, 0.2), MIN(0.20, 0.45), MIN(0.75, 0.50)} = {0.2, 0.20, 0.50} Dr Basil Hamed

  31. Examples of Fuzzy Set Operations A = {1/a, 0.3/b, 0.2/c 0.8/d, 0/e} B = {0.6/a, 0.9/b, 0.1/c, 0.3/d, 0.2/e} Complement: = {0/a, 0.7/b, 0.8/c 0.2/d, 1/e} Union: A B = {1/a, 0.9/b, 0.2/c, 0.8/d, 0.2/e} Intersection: A B = {0.6/a, 0.3/b, 0.1/c, 0.3/d, 0/e} Dr Basil Hamed

  32. Properties of Fuzzy Sets A  B = B A A  B = B  A A  (B  C) = (A  B)  C A  (B  C) = (A  B)  C A  (B  C) = (A  B)  (A  C) A  (B  C) = (A  B)  (A  C) A  A = A A  A = A A  X = X A  X = A A   = A A   =  If A  B  C, then A  C A’’ = A Dr Basil Hamed

  33. Fuzzy Sets Note (x)  [0,1] not {0,1} like Crisp set A = {A(x1) / x1 + A(x2) / x2 + …} = { A(xi) / xi} Note: ‘+’  add ‘/ ’  divide Only for representing element and its membership. Also some books use (x) for Crisp Sets too. Dr Basil Hamed

  34. # courses a student may take in a semester. appropriate # courses taken 1 0.5 0 2 4 6 8 x : # courses Example (Discrete Universe) Dr Basil Hamed

  35. # courses a student may take in a semester. appropriate # courses taken Example (Discrete Universe) Alternative Representation: Dr Basil Hamed

  36. possible ages x : age Example (Continuous Universe) U : the set of positive real numbers about 50 years old Alternative Representation: Dr Basil Hamed

  37. Alternative Notation U: discrete universe U: continuous universe Note that and integral signs stand for the union of membership grades; “ / ” stands for a marker and does not imply division. Dr Basil Hamed

  38. Fuzzy Disjunction • AB max(A, B) • AB = C "Quality C is the disjunction of Quality A and B" • (AB = C)  (C = 0.75) Dr Basil Hamed

  39. Fuzzy Conjunction • AB min(A, B) • AB = C "Quality C is the conjunction of Quality A and B" • (AB = C)  (C = 0.375) Dr Basil Hamed

  40. Example: Fuzzy Conjunction Calculate AB given that A is .4 and B is 20 Dr Basil Hamed

  41. Example: Fuzzy Conjunction Calculate AB given that A is .4 and B is 20 • Determine degrees of membership: Dr Basil Hamed

  42. Example: Fuzzy Conjunction Calculate AB given that A is .4 and B is 20 0.7 • Determine degrees of membership: • A = 0.7 Dr Basil Hamed

  43. Example: Fuzzy Conjunction Calculate AB given that A is .4 and B is 20 0.9 0.7 • Determine degrees of membership: • A = 0.7 B = 0.9 Dr Basil Hamed

  44. Example: Fuzzy Conjunction Calculate AB given that A is .4 and B is 20 0.9 0.7 • Determine degrees of membership: • A = 0.7 B = 0.9 • Apply Fuzzy AND • AB = min(A, B) = 0.7 Dr Basil Hamed

  45. Generalized Union/Intersection • Generalized Union Or called triangular norm. • Generalized Intersection t-norm t-conorm Or called s-norm. Dr Basil Hamed

  46. T-norms and S-norms • And/OR definitions are called T-norms (S-norms) • Duals of one another • A definition of one defines the other implicitly • Many different ones have been proposed • Min/Max, Product/Bounded-Sum, etc. • Tons of theoretical literature • We will not go into this. Dr Basil Hamed

  47. Examples: T-Norm & T-Conorm • Minimum/Maximum: • Lukasiewicz: Dr Basil Hamed

  48. Classical Logic &Fuzzy Logic Hypothesis : Engineers are mathematicians. Logical thinkers do not believe in magic. Mathematicians are logical thinkers. Conclusion :Engineers do not believe in magic. Let us decompose this information into individual propositions P: a person is an engineer Q: a person is a mathematician R: a person is a logical thinker S: a person believes in magic The statements can now be expressed as algebraic propositions as ((PQ)(RS)(QR))(PS) Dr Basil Hamed

  49. Fuzzy Relations … Dr Basil Hamed

  50. b1 a1 b2 A B a2 b3 a3 b4 a4 b5 Crisp Relation (R) Dr Basil Hamed

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