Fuzzy Systems

1. Slovak University of Technology Faculty of Material Science and Technology in Trnava Fuzzy Systems Introduction to Fuzzy Sets and Systems

2. Introduction to Fuzzy Sets and Systems • The concept of Fuzzy Logic Fuzzy Sets and Fuzzy Systems was conceived by Zadeh, a professor at the University of California at Berkley. It is presented not as a control methodology, but as a way of processing data by allowing partial set membership rather than crisp set membership or non-membership. This approach to set theory was not applied to control systems until the 70's due to insufficient small-computer capability prior to that time. Professor Zadeh reasoned that people do not require precise, numerical information input, and yet they are capable of highly adaptive control. If feedback controllers could be programmed to accept noisy, imprecise input, they would be much more effective and perhaps easier to implement. • The word fuzzy has become common knowledge and comes up in every day conversation. This comes as quite a surprise to those of us who specialize in research into fuzzy systems. Scientific methodology requires strict logic, but one can say that not much effort goes into verification of premises and assumptions. The premises and assumptions that sciences and technology worry so little about are the same axioms in mathematics, and this probably comes about because they are not logical on the whole. At preset, this problems can only be presented through human perception and experience. If premises and assumptions are not thoroughly investigated in technical fields, there is the fear of inviting big mistakes. For example, unexpected accidents in safety systems, nonsensical conclusions in information systems, automation systems that large balance all occur when design premises are far from the actual circumstances. • Science and technology do their best to exclude subjectivity, but discovery and invention originate in right hemisphere activities that are based on subjectivity, and logicizing are no more than secondary processes for gaining the assent of others. The use of subjectivity is even more effective during the process of objectification.

3. Some notations of crisp set theory • If A,B are sets, then A is a subset of B ( AB) if xAxB for all xA • If U is an universal set, we denote by P(U) set of all subset of U, P(U)={A;AU}. P(U) is called potential set of universal se U. • If U is finite and has n elements nN, it is known that P(U) is finite and has 2n elements. • It is patent that P(U) is a Boolean algebra with respect operations union (), intersection () and complement of sets. • Some basic(standard) operation set • A B={xU;xA or xB}={xU;xA  xB} (the union of sets) • A  B={xU;xA and xB}=={xU;xA  xB} (the intersection of sets) • Ac={xU;xA } (the complement of the set) • A - B=A\B={xU;xA and xB}={xU;xA  xB} (the different of sets).

4. AB=(A-B)(B-A)= (A\B)(B\A) (the symmetric different of sets) AB={(x,y);xA and yB}={(x,y);xA  yB} (Cartesian product of sets ). If A,B are sets, we call relation any non empty subset R AB. If R is a relation, then notation (x,y)R is the same as xRy.

5. Some properties of relations • The relation R is • left-total: if for all x in A there exists a y in B such that xRy (this property, although sometimes also referred to as total, is different from the definition of total in the next section). • right-total: if for all y in B there exists an x in A such that xRy. • symmetric, if (x,y)R(y,x)R, • reflexive, if (x,x)R • transitive, if [(x,y)R] [(y,z)R] [(x,z)R] • If R is symmetric, reflexive and transitive then it is relation equivalence. • antisymmetric: iffor all x and y in B it holds that if xRy and yRx then x = y. "Greater than or equal to" is an antisymmetric relation, because if x≥y and y≥x, then x=y. • asymmetric: if for all x and y in A it holds that if xRy then notyRx. "Greater than" is an asymmetric relation, because if x>y then not y>x. • functional (also called right-definite): for all x in X, and y and z in Y it holds that if xRy and xRz then y = z. • funkcional is surjective: if for all y in B there exists an x in X such that xRy. • funkcional is injective: if for all x and z in A and y in B it holds that if xRy and zRy then x = z. • funkcional is bijective: left-total, right-total, functional, and injective.

6. Mapping (function) A onto B. If non empty relation fAB have following properties 1) for all xA there exists yB so (.x,y)f 2) If [(.x,y1)f and (.x,y2)f]y1=y2. then f is also called mapping (function) A onto B. Notations (x,y)f, y=f(x), f:xy are equivalent. The mapping O:AA ...AA is n-aryoperation. If n=2 we have binary operation. If n=1 we have u-nary operation If O(x,y)=O(y,x) then the binary operation is commutative. If O(x, O(y,z))= O(O(x,y), z) then the binary operation is associative

7. If A is a subset of universal set U, then function defined on U as follows Is a characteristic function of subset A. It is easy to show that P(U) and set of all characteristic functions CH(U) are isomorphic (as sets). There exist bijection P(U) onto CH(U) i.e. there exists two maps : P(U) CH(U) and : CH(U) P(U) defined by(A)=A, (A)={xU; A(x)=1}=Athus CH(U) P (U). P(U) is a Boolean algebra with respect operation union, intersection and complement. This means that following eight identities are valid Characteristic function of set

8. 1) AB=BA, AB=BA (commutavity) 2) (AB)C=A(BC), (AB)C=A(BC) (associativity) 3) (AB)C=(AC|(BC), A (BC) = (AB)(AC) (distributivity) 4) AA=A, AA=A (idenpotency) 5) A(AB)=A, A(AB)=A (absorption) 6) A=A, A=, UU=U, UA=A 7) 8) , , (´)(x)=max{(x), ´(x)} (´)(x)=min {(x), ´(x)} ´(x)=1-(x) Propperties of set operations

9. Definition of fuzzy set • Definition 1.1: Definition of fuzzy set: Let U is an universal set and . A fuzzy set is a pair {U,}. A function  we call the membership function. • The value of membership function is a degree of membership of x as an element of set. • The membership function is a graphical representation of the magnitude of participation of each input. It associates a weighting with each of the inputs that are processed, define functional overlap between inputs, and ultimately determines an output response. • The rules use the input membership values as weighting factors to determine their influence on the fuzzy output sets of the final output conclusion. Once the functions are inferred, scaled, and combined, they are defuzzified into a crisp output which drives the system. There are different memberships functions associated with each input and output response.

10. Example Error Membership Function Example Membership Function • Figure illustrates the features of the triangular membership function which is used in this example because of its mathematical simplicity. Other shapes can be used but the triangular shape lends itself to this illustration. • The degree of membership is determined by plugging the selected input parameter (error or error-dot) into the horizontal axis and projecting vertically to the upper boundary of the membership function(s).

11. Difference between crisp set (a) and fuzzy set (b)

12. Some notations of fuzzy set • Let is a fuzzy set. Then • a support of fuzzy set is Supp A =; • if support of fuzzy set is finite then is discreet; • -cut of fuzzy set is A =; • -level of fuzzy setis A =; • a kernel of fuzzy set is Ker A =; • if Ker A then is normal else is subnormal; • a height of fuzzy set is ; • a singleton of fuzzy set is the set with one element; • if then the fuzzy set is crisp(conventional

13. Some notations on fuzzy sets

14. Representation theorem of fuzzy set

15. Example

16. Slovak University of Technology Faculty of Material Science and Technology in Trnava Fuzzy Systems Measures on Fuzzy Sets

17. The cardinality of fuzzy set

19. Center of fuzzy set

20. Measure of uncertainty

21. Definition of measure of uncertainty

22. Measure of uncertainty of discrete fuzzy set

23. Euclid’s measure of uncertainty

24. Entropic measure of uncertainty of fuzzy set

25. An example

26. An example 2

27. Measure of uncertainty of fuzzy set if U=R

29. Fuzzy complement. Membership function

30. Membership function of fuzzy intersection

31. Membership function of fuzzy union

32. Fuzzy complements

33. Fuzzy intersections

34. Fuzzy intersections. An example • An example of fuzzy fuzzy intersection is • µ*(= µAB(x)= min {µA(x),µB(x)} • To prove that we show that it satisfy axioms i1-i6. • Axiom i1.: µ* (a, 1) = a. Let us compute µ* (a, 1)= µ*(= µAU(x)= • =min {µA(x),1 }= µA(x), • Axiom i2. If a=µA(x), µB(x)=b ≤ d= µC(x) implies µ* (a, b) ≤ µ* (a, d). • Let us compute µ* (a, b)= µ*(= µAB(x)= min {µA(x), µB(x) }≤ min {µA(x), µC(x) } • Axiom i3: Commutativity µ* (a, b) = µ* (b, a). Let us compute µ* (a, b) = • =min {µA(x), µB(x) }=min {µB(x), µA(x) }=µ* (b, a) • Axiom i4. Associativity µ* (a, µ* (b, d)) = min {µA(x), min {µB(x), µC(x) } }= • =min{min{ µA(x), µB(x)}, µC(x) }= µ* (µ* (a, b), d) • Axiom i5. Continuity: µ*(= µAB(x)= min {µA(x), µB(x)}=min{u,v} is continues • function • Axiom i6. Subidempotency µ* (a, a) = min {µA(x), µA(x) }= µA(x)≤ µA(x)=a

35. Fuzzy unions

36. An example of fuzzy union operation

37. Aggregation operations

38. T-norm

39. Motivations and applicationsof T-norm

40. Slovak University of Technology Faculty of Material Science and Technology in Trnava Fuzzy Systems Classification of t-normsand conorms

41. Classification of t-normsProminent examples

44. Properties of continuous t-norms

46. T-conorms

47. Examples of t-conorms

48. Probabilistic sum and bounded sum

49. Drastic t-conorm and Nilpotent maximum