Introduction to Intelligent Control Part 3

Transcription

1 ECE Spring 2010 Introduction to Part 3 Prof. Marian S. Stachowicz Laboratory for Intelligent Systems ECE Department, University of Minnesota Duluth January 26-29, 2010

2 Part 1: Outline TYPES OF UNCERTAINTY Fuzzy Sets and Basic Operations on Fuzzy Sets Further Operations on Fuzzy Sets 2

3 References for reading 1. M.S. Stachowicz, Lance Beall, Fuzzy Logic Package, Version-2 for Mathematica 5.1, Wolfram Research, Inc., Demonstration Notebook: 2.1, 2.2, 2.7.1, 2.7.2, 2.8.1, 2.8.2, 2.8.3, 2.8.5, 2.8.6, 2.8.8, Manual: 1.1, 1.3.1, 1.3.2, 1.3.4, 1.4, 1.5 3

4 Randomness versus Fuzziness Randomness refers to an event that my or may not occur. Randomness: frequency of car accidents. Fuzziness refers the boundary of a set that is not precise. Fuzziness: seriousness of a car accident. Prof. George Klir PROFESSOR GEJ. 4

5 TYPES OF UNCERTAINTY STOCHASTIC UNCERTAINTY THE PROBABILITY OF HITTING THE TARGET IS 0.8. LEXICAL UNCERTAINTY WE WILL PROBABLY HAVE A SUCCESFUL FINANCIAL YEAR. 5

6 FUZZY SETS THEORY versus PROBABILITY THEORY Patients suffering from hepatitis show in 60% of all cases high fever, in 45 % of all cases a yellowish colored skin, and in 30% of all cases nausea. 6

7 What are Fuzzy Sets?

8 Fuzzy set Lotfi A. Zadeh[1965] A fuzzy subset A of a universe of discourse U is characterized by a membership function µ A : U [0,1] which associates with each element u of U a number µ A (u) in the interval [0,1], which µ A (u) representing the grade of membership of u in A. 8

9 Fuzzy Sets Fuzzy set A defined in the universal space U is a function defined in U which assumes values in the range [ 0,1 ]. A : U [ 0, 1] 9

10 Characteristic Function A : U {0, 1} Membership Function M : U [0, 1] 10

11 Universal Set U - is the universe of discourse, or universal set, which contains all the possible elements of concern in each particular context of applications. 11

12 Membership function The membership function M maps each element of U to a membership grade ( or membership value) between 0 and 1. 12

13 Presentation of a fuzzy set A fuzzy set M, in the universal set can be presented by: - list form, - rule form, - membership function form. 13

14 List form of a fuzzy set M = {{1,1},{2,1},{3,0.9},{4,0.7},{5,0.3},{6,0.1}, {7,0},{8,0},{9,0},{10,0},{11,0},{12,0}}, where M is the membership function (MF) for fuzzy set M. Note: The list form can be used only for finite sets. 14

15 Fuzzy Logic Package form [M.S. Stachowicz + Lance Beall,1995 & 2003] M={{1,1},{2,1},{3,0.9},{4,0.7},{5,0.3},{6,0.1}} and U={1,2,3,4,5,6,7,8,9,10,11,12} 15

16 Rule form of a fuzzy set M = { u U u meets some conditions}, where symbol denotes the phrase such as. 16

17 Membership form of a fuzzy set Let A 1 be a fuzzy set named numbers closed to zero A 1 (0)=1 A 1 (2)=exp(-4) A 1 (-2)=exp(-4) A 2 1 (u) =exp(-u ) 17

18 Numbers closed to zero

19 Representation of a fuzzy set A fuzzy set A in U may be represented as a set of ordered pairs of generic element u and its membership value A(u), A = {{u, A(u)} u U} 19

20 Representation of a fuzzy set When U is continuous, A is commonly written as: / u A(u) /u where integral sign does not denote integration; it denotes the collection of all points u U with the associated MF A(u). 20

21 Representation of a fuzzy set When U is discrete, A is commonly written as: A(u) /u, where the summation sign does not represent arithmetic addition. 21

22 Example 1 Let U be the integer from 1 to 10, that U={ 1,2,,9,10}. The fuzzy set Several may be defined as using: - the summation notation Several={0.5/ /4 + 1/5 + 1/6 +0.8/ /8} - FLP notation Several={{3.0.5},{4,0.8},{5,1},{6,1},{7,0.8},{8,0.5}} 22

23 Basic concepts and terminology The concepts of support, fuzzy singleton, crossover point, height, normal FS, α-cut, and convex fuzzy set are defined as follows: 23

24 Core, support, and crossover point MF 1.5 α 0 Core Crossover points α -cut Support u 24

25 The support of fuzzy set A The support of a fuzzy set A in the universal set U is a crisp set that contains all the elements of U that have nonzero membership values in A, that is, supp(a)= {u U A(u) > 0} 25

26 Core The core of a fuzzy set A is the set of all points u in U such that A(u) = 1 Core(A) = {u A(u) = 1} 26

27 Normality A fuzzy set A is normal if its core in nonempty. u U A(u) = 1 27

28 Example Several={0.5/ /4 + 1/5 + 1/6 +0.8/ /8} in U={ 1,2,,9,10}. Supp(Several) = {3,4,5,6,7,8} 28

29 Fuzzy singleton A fuzzy singleton is a fuzzy set A(u)=1 whose support is a single point in U. 29

30 Crossover point The crossover point of a fuzzy set is the point in U whose membership value in A equals

31 Height The height of a fuzzy set is the largest membership value attained by any point. If the height of fuzzy set equals one, it is called a normal fuzzy set. 31

32 Alpha Cuts

33 Alpha-cut Alpha-cut of fuzzy set/fuzzy relation is the crisp set that contains all the elements of universal space whose membership grades in set/relation are greater than or equal to the specified value of alpha. Crisp set α A = { x A(x) α} 33

34 Strong alpha-cut Strong alpha-cut of fuzzy set/fuzzy relation is the crisp set that contains all the elements of universal space whose membership grades in set/relation are greater than the specified value of alpha. Crisp set α + A = { x A(x) > α} 34

35 Level set The set of all alpha-cuts of a fuzzy set/fuzzy relation is called a level set of set/relation. L(A) = {α A(x) = αfor some x X 35

36 Level set LevelSet [fs1] L(fs1) = {0.2, 0.4, 0.6, 0.8, 1} 36

37 Alpha-cuts (0.2), {2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16}, (0.4), {4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15}, (0.6), {5, 6, 7, 8, 9, 10, 11, 12, 13, 14}, (0.8), {6, 7, 8, 9, 10, 11, 12, 13}, (1.0), {8, 9, 10, 11, 12}. A if α1< α2 then α1 A α2 A α1 A α2 A = α2 A α1 A α2 A = α1 A 37

38 Decomposition of fuzzy sets For any A R A = α α A(x) for α [ 0,1] 38

39 The relationship between fuzzy set and crisp set Each fuzzy set can be uniquely presented by the family of all its α-cuts. This representation allows extending various properties of crisp set and operations on crisp sets to their fuzzy counterparts. 39

40 Decomposition of fuzzy sets L(fs1) = {0.2, 0.4, 0.6, 0.8, 1} 40

41 Property of alpha-cuts A if α1< α2 then α1 A α2 A α1 A α2 A = α2 A α1 A α2 A = α1 A 41

42 Convex fuzzy set A fuzzy set A is convex if and only if it s α -cuts α A is a convex set for any α in the interval α (0,1]. A[λ x 1 + (1- λ ) x 2 ] min[a(x 1 ),A(x 2 )] for all x 1, x 2 R n and all λ [0,1]. 42

43 ECE Fall 2009 The Mathematics of Fuzzy Systems Part 2 Prof. Marian S. Stachowicz Laboratory for Intelligent Systems ECE Department, University of Minnesota, USA October 1, 2009

44 References for reading 1. W. Pedrycz and F. Gomide, Fuzzy Systems Engineering, J. Wiley & Sons, Ltd, 2007 Chapter 2 and 3 2. M.S. Stachowicz, Lance Beall, Fuzzy Logic Package, Version-2 for Mathematica 5.1, Wolfram Research, Inc., Demonstration Notebook: 2.1, 2.2, 2.7.1, 2.7.2, 2.8.1, 2.8.2, 2.8.3, 2.8.5, 2.8.6, 2.8.8, Manual: 1.1, 1.3.1, 1.3.2, 1.3.4, 1.4, G.J. Klir, Ute H.St. Clair, Bo Yuan, Fuzzy Set Theory, Prentice Hall, 1997 Chapters 1, 2 44

45 Operations on fuzzy sets Inclusion Equality Standard Complement Standard Union Standard Intersection 45

46 Inclusion Let X and Y be fuzzy sets defined in the same universal space U. We say that the fuzzy set X is included in the fuzzy set Y if and only if: for every u in the set U we have X(u) Y(u) 46

47 Subset and proper subset X is a subset of Y, or is smaller than or equal to Y if and only if X(u) Y(u) for all u. X Y X is a proper subset of Y, or is smaller than Y if and only if X(u) < Y(u) for all u. X Y 47

48 Equality Let X and Y be fuzzy sets defined in the same universal space U. We say that sets X and Y are equal, which is denoted X = Y if and only if for all u in the set U, X(u) = Y(u). 48

49 Standard complement Let X be fuzzy sets defined in the universal space U. We say that the fuzzy set Y is a complement of the fuzzy set X, if and only if, for all u in the set U, Y(u) = 1 - X(u). 49

50 Standard union Let X and Y be fuzzy sets defined in the space U. We define the union of those sets as the smallest (in the sense of the inclusion) fuzzy set that contains both X and Y. u U, (X Y)(u) = Max(X(u), Y(u)). 50

51 Standard union u U, (X Y)(u) = Max(X(u), Y(u)). 51

52 Standard intersection Let X and Y be fuzzy sets defined in the space U. We define the intersection of those sets as the greatest (in the sense of the inclusion) fuzzy set that included both in X and Y. u U, (X Y)(u) = Min(X(u), Y(u)). 52

53 Standard intersection u U, (X Y)(u) = Min(X(u), Y(u)). 53

54 Properties of crisp set operations [A B = B A Associativity (A U B) U C = A U (B U C) (A B) C = A (B C) Distributivity A (B U C) = (A B) U (A C) A U (B C) = (A U B) (A U C) Idempotents ` A U A = A A A = A 54

55 Properties of fuzzy set operations Involution Commutativity (A ) = A A U B = B U A A B = B A Associativity (A U B) U C = A U (B U C) (A B) C = A (B C) Distributivity A (B U C) = (A B) U (A C) A U (B C) = (A U B) (A U C) Idempotents A U A = A A A = A 55

56 Properties of crisp set operations Absorption Absorption by X and Identity A U (A B) = A A (A U B) = A A U X = X A = A U = A A X = A De Morgan s laws (A B) = A U B (A U B) = A B 56

57 Properties of fuzzy set operations Absorption Absorption by X and Identity A U (A B) = A A (A U B) = A A U X = X A = A U = A A X = A De Morgan s laws (A B) = A U B (A U B) = A B 57

58 Properties of crisp set operations Law of contradiction A A = Law of excluded middle A U A = X 58

59 Properties of fuzzy set operations Law of contradiction A A Law of excluded middle A U A X 59

60 Law of excluded middle A U A X FuzzyPlot[MEDIUM U Complement[MEDIUM]] 60

61 Law of contradiction A A FuzzyPlot[MEDIUM Complement[MEDIUM]] 61

62 A general principle of duality For each valid equation in set theory that is based on the union and intersection operations, there corresponds a dual equation, also valid, that is obtained by replacing, U, and with X,, and U, respectively, and vice versa. 62

63 Standard operators 1 When the range of grade of membership is restricted to the set {0,1}, these functions perform like the corresponding operators for Cantor's sets. 63

64 Standard operators 2 If any error e is associated with the grade of membership A(u) and B(u), then the maximum error associated with the grade of membership of u in A', Union[A, B], and Intersection[A, B] remains e. 64

65 Characteristic function Ch. de la Valle Poussin [1950], Integrales de Lebesque, fonction d'ensemble, classes de Baire, 2-e ed., Paris, Gauthier-Villars. 65

66 Membership function L. A. Zadeh [1965], Fuzzy sets, Information and Control, volume 8, pp

67 Goguen, J.A.[1967] L-fuzzy sets, J. of Math Analysis and Applications, 18(1), pp

68 M. S. Stachowicz and M. E. Kochanska [1982], Graphic interpretation of fuzzy sets and fuzzy relations, Mathematics at the Service of Man. Edited by A. Ballester, D. Cardus, and E. Trillas, based on materials of Second World Conference, Universidad Politecnica Las Palmas, Spain. 68

69 M.S. STACHOWICZ and L. BEALL [1995, 2003] 69

70 ECE Fall 2009 Individual decision making Prof. Marian S. Stachowicz Laboratory for Intelligent Systems ECE Department, University of Minnesota, USA October 10, 2009

71 References for reading 1. W. Pedrycz and F. Gomide, Fuzzy Systems Engineering, J. Wiley & Sons, Ltd, 2007 Chapter 2 and 3 2. M.S. Stachowicz, Lance Beall, Fuzzy Logic Package, Version-2 for Mathematica 5.1, Wolfram Research, Inc., Demonstration Notebook: 2.1, 2.2, 2.7.1, 2.7.2, 2.8.1, 2.8.2, 2.8.3, 2.8.5, 2.8.6, 2.8.8, Manual: 1.1, 1.3.1, 1.3.2, 1.3.4, 1.4, G.J. Klir, Ute H.St. Clair, Bo Yuan, Fuzzy Set Theory, Prentice Hall, 1997 Chapters 1, 2 71

72 Individual decision making A decision is characterized by components: Universal space U of possible actions; a set of goals G i (i N n ) defined on U; a set constraints C j (j N n ) defined on U. Decision is determined by an aggregation operator. 72

73 AG-H - Faculty of Management, May,

74 Example: Job Selection Suppose that Sebastian from AG-H needs to decide which of the four possible jobs, say g(a1) = $40,000 g(a2) = $45,000 g(a3) = $50,000 g(a4) = $60,000 to choose. 74

75 The constraints His goal is to choose a job that offers a high salary under the constraints that the job is interesting and within close driving distance. 75

76 In this case, the goal and constraints are all uncertain concepts and we need to use fuzzy sets to represent these concepts. 76

77 Goal: High salary-indirect form CF = FuzzyTrapezoid[37, 64, 75, 75, UniversalSpace -> {0, 80, 5}] FuzzySet [{{40, 0.11}, {45, 0.3}, {50, 0.48}, {55, 0.67}, {60, 0.85}, {65, 1}, {70, 1}, {75, 1}} 77

78 Fuzzy goal G: High salary-direct form G = FuzzySet[{{1,.11}, {2,.3}, {3,.48}, {4,.85}}] 78

79 Constraint C1: Interesting job C1 = FuzzySet[{{1,.4}, {2,.6}, {3,.2}, {4,.2}}] 79

80 Constraint C2: Close driving C2 = FuzzySet[{{1,.1}, {2,.9}, {3,.7}, {4, 1}}] 80

81 Concept of desirable job D = Intersection[G, C1, C2] = FuzzySet[{{1, 0.1}, {2, 0.3}, {3, 0.2}, {4, 0.2}}] 81

82 For Sebastian from AG-H $ 45,000. The final result from the above analysis is a2, which is the most desirable job among the four available jobs under the given goal G and constraints C1 and C2. g(a2) = $ 45,000 with D(a2) =

83 Example: Optimal dividends The board of directors of a company needs to determine the optimal dividend to paid to the shareholders. For financial reasons, the dividend should be attractive (goal G); for reasons of wage negotiations, it should be modest (constraint C). The U is a set of possible dividends actions. 83

84 In this case, the goal and constraint are both uncertain concepts and we need to use fuzzy sets to represent these concepts. 84

85 Example: Optimal dividends C : modest G : attractive û : optimal dividends 85

86 Optimal dividends OptimalDividends = Core[Normalize[Intersection [modest, attractive]]] 86

87 Comments Since this method ignores information concerning any of the other alternatives, it may not be desirable in all situations. An averaging operator may be used to reflect a some degree of positive compensation exists among goals and constrains. When U is defined on R, it is preferable to determine û by appropriate defuzzification method. 87

88 Several experts There are five springboard divers: 1, 2, 3, 4, 5. There are ten referees: R1, R2,, R10. We need to determine a membership function A that will capture the linguistic term excellent diver. 88

89 The Diving Survey Alice Bonnie Cathy Dina Eva R R R R R R R R R R

90 For every diver we calculate the membership grade of belonging to the fuzzy sets A by taking the ratio of the total number of favorable answers to the total number of referees. A = {{1, 0.3}, {2, 0.4}, {3, 0.6}, {4, 0.9}, {5, 0.6}} 90

91 Several experts FS1 = FuzzySet[{{1,.3}, {2,.4}, {3,.6}, {4,.9}, {5,.6}}, UniversalSpace -> {1, 5, 1}]; 91

92 Questions?