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# Lecture 32 fuzzy systems

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### Lecture 32 fuzzy systems

1. 1. FUZZY LOGICFuzzy Set (Value)Let X be a universe of discourse of a fuzzy variable and x be its elementsOne or more fuzzy sets (or values) Ai can be defined over XExample: Fuzzy variable: Age Universe of discourse: 0 – 120 years Fuzzy values: Child, Young, OldA fuzzy set A is characterized by a membership function µA(x) that associates each element x with a degree of membership value in AThe value of membership is between 0 and 1 and it represents the degree to which an element x belongs to the fuzzy set A
2. 2. FUZZY LOGICFuzzy Set RepresentationFuzzy Set A = (a1, a2, … an) ai = µA(xi) xi = an element of X X = universe of discourseFor clearer representation A = (a1/x1, a2/x2, …, an/xn)Example: Tall = (0/5’, 0.25/5.5’, 0.9/5.75’, 1/6’, 1/7’, …)
3. 3. FUZZY LOGICFuzzy Sets OperationsIntersection (A  B)In classical set theory the intersection of two sets containsthose elements that are common to bothIn fuzzy set theory, the value of those elements in theintersection: µA  B(x) = min [µA(x), µB(x)]e.g. Tall = (0/5, 0.1/5.25, 0.5/5.5, 0.8/5.75, 1/6) Short = (1/5, 0.8/5.25, 0.5/5.5, 0.1/5.75, 0/6) Tall  Short = (0/5, 0.1/5.25, 0.5/5.5, 0.1/5.75, 0/6) = Medium
4. 4. FUZZY LOGICFuzzy Sets OperationsUnion (A  B)In classical set theory the union of two sets contains thoseelements that are in any one of the two setsIn fuzzy set theory, the value of those elements in the union: µA  B(x) = max [µA(x), µB(x)]e.g. Tall = (0/5, 0.1/5.25, 0.5/5.5, 0.8/5.75, 1/6) Short = (1/5, 0.8/5.25, 0.5/5.5, 0.1/5.75) Tall  Short = (1/5, 0.8/5.25, 0.5/5.5, 0.8/5.75, 1/6) = not Medium
5. 5. FUZZY LOGICFuzzy Sets OperationsComplement (A)In fuzzy set theory, the value of complement of A is: µ  A(x) = 1 - µA(x)e.g. Tall = (0/5, 0.1/5.25, 0.5/5.5, 0.8/5.75, 1/6)  Tall = (1/5, 0.9/5.25, 0.5/5.5, 0.2/5.75, 0/6)
6. 6. FUZZY LOGICFuzzy RelationsFuzzy relation between two universes U and V is defined as: µR (u, v) = µAxB (u, v) = min [µA (u), µB (v)]i.e. we take the minimum of the memberships of the twoelements which are to be related
7. 7. FUZZY RULESApproximate ReasoningExample: Let there be a fuzzy associative matrix M for therule: if A then Be.g. If Temperature is normal then Speed is mediumLet A = [0/100, 0.5/125, 1/150, 0.5/175, 0/200] B = [0/10, 0.6/20, 1/30, 0.6/40, 0/50]
8. 8. FUZZY RULESApproximate Reasoning: Max-Min InferenceLet A = [0/100, 0.5/125, 1/150, 0.5/175, 0/200] B = [0/10, 0.6/20, 1/30, 0.6/40, 0/50]then M= (0, 0) (0, 0.6) . . . (0.5, 0) . . . . . . = 0 0 0 0 0 0 0.5 0.5 0.5 0 by taking the minimum 0 0.6 1 0.6 0 of each pair 0 0.5 0.5 0.5 0 0 0 0 0 0
9. 9. FUZZY LOGICComposition of Fuzzy RelationsNow we need a operator which allows us to infer somethingabout B, given Acurrent“Composition” is such an operator
10. 10. FUZZY LOGICComposition of Fuzzy RelationsLet there be three universes U, V and WLet R be the relation that relates elements from U to V e.g. R= 0.6 0.8 0.7 0.9And let S be the relation between V and W e.g. S= 0.3 0.1 0.2 0.8
11. 11. FUZZY LOGICComposition of Fuzzy RelationsWith the help of an operation called “composition” we can findthe relation T that maps elements of U to WBy max-min rule T = R  S = maxvV { min(R(u, v), S(v, w)) } 0.6 0.8 0.3 0.1 = 0.3 0.8 0.7 0.9  0.2 0.8 0.3 0.8Where element (1,1) is obtained by max{min(0.6, 0.3), min(0.8, 0.2)} = 0.3Note that S  R = 0.3 0.3  R  S 0.7 0.8
12. 12. FUZZY LOGICComposition of Fuzzy RelationsR = R(u, v) v1 v2 u1 0.6 0.8 u2 0.7 0.9 V v2 0.9 0.8 v1 0.6 0.7 u1 u2 U
13. 13. FUZZY LOGICComposition of Fuzzy RelationsS = S(v, w) w1 w2 v1 0.3 0.1 v2 0.2 0.8 V v2 0.8 0.2 v1 0.1 0.3 w1 w2 W
14. 14. FUZZY LOGICComposition of Fuzzy RelationsT = R  S = maxvV { min(R(u, v), S(v, w)) } 0.6 0.8 0.3 0.1 = 0.3 0.8 0.7 0.9  0.2 0.8 0.3 0.8 V v2 0.9 0.8 0.2 v1 0.8 0.1 0.3 0.6 0.7 u1 u2 U w1 w2 W
15. 15. FUZZY LOGICComposition of Fuzzy RelationsT = R  S = maxvV { min(R(u, v), S(v, w)) } 0.6 0.8 0.3 0.1 = 0.3 0.8 0.7 0.9  0.2 0.8 0.3 0.8 V v2 v1 u1 u2 w1 0.3 0.3 U 0.8 w2 0.8 W
16. 16. Reading Assignment & ReferencesEngelbrecht Chapter 18 & 19