1. FUZZY SYSTEMS

2. Fuzzy Systems Fuzzy Sets – To quantify and reason about fuzzy or vague terms of natural language – Example: hot, cold temperature small, medium, tall height creeping, slow, fast speed Fuzzy Variable – A concept that usually has vague (or fuzzy) values – Example: age, temperature, height, speed

3. Fuzzy Systems Universe of Discourse – Range of possible values of a fuzzy variable – Example: Speed: 0 to 100 mph

4. Fuzzy Systems Fuzzy Set (Value) – Let X be a universe of discourse of a fuzzy variable and x be its elements – One or more fuzzy sets (or values) Ai can be defined over X – Example: Fuzzy variable: Age – Universe of discourse: 0 – 120 years – Fuzzy values: Child, Young, Old

5. Fuzzy Systems A fuzzy set A is characterized by a membership function μA(x) that associates each element x with a degree of membership value in A The value of membership is between 0 and 1 and it represents the degree to which an element x belongs to the fuzzy set A

6. Fuzzy Systems Fuzzy Set (Value) – In traditional set theory, an object is either in a set or not in a set (0 or 1), and there are no partial memberships – Such sets are called “crisp sets”

7. Fuzzy Systems Fuzzy Set Representation – Fuzzy Set A = (a1, a2, … an), where ai = μA(xi) xi = an element of X X = universe of discourse – For 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’, …)

8. Fuzzy Systems Intersection (A ∧ B) – In classical set theory the intersection of two sets contains those elements that are common to both – The value of the elements in the intersection: μA ∧ B(x) = min [μA(x), μB(x)] – Example: 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

9. Fuzzy Systems Union (A ∨ B) – In classical set theory the union of two sets contains those elements that are in any one of the two sets – The value of the elements in the union: μA ∨ B(x) = max [μA(x), μB(x)] – Example: 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

10. Fuzzy Systems Complement (¬A) The value of complement of A is: – μ ¬ A(x) = 1 - μA(x) – Example: 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)

11. Fuzzy Inference: Fuzzy Associative Matrix Fuzzy Relations – Classical relation between two universes U = {1, 2} and V = {a, b, c} is defined as:

12. Fuzzy Systems Fuzzy Relations – Fuzzy 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 two elements which are to be related Example: – Determine fuzzy relation between A 1 and A 2 A 1 = 0.2/x 1 + 0.9/x 2 A 2 = 0.3/y 1 + 0.5/y 2 + 1/y 3

13. Fuzzy Systems Fuzzy Relations – Example: The fuzzy relation R is

14. Fuzzy Systems Fuzzy Rules – Relates two or more fuzzy propositions – If X is A then Y is B e.g. if height is tall then weight is heavy – X and Y are fuzzy variables – A and B are fuzzy sets

15. Fuzzy Systems Fuzzy Associative Matrix So for the fuzzy rule: – If X is A then Y is B – We can define a matrix M (nxp) which relates A to B – M = A x B – It maps fuzzy set A to fuzzy set B and is used in the fuzzy inference process

16. Fuzzy Systems Concept behind M If a1 is true then b1 is true; and so on

17. Fuzzy Systems Composition of Fuzzy Relations – Let there be three universes U, V and W – Let R be the relation that relates elements from U to V – And let S be the relation between V and W

18. Fuzzy Systems

19. Example: – If Temperature is normal then Speed is medium – Let 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]