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Appendix a Basic Concepts of Fuzzy Set Theory A.I Fuzzy Sets ILA{X): X

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Appendix a Basic Concepts of Fuzzy Set Theory A.I Fuzzy Sets ILA{X): X Appendix A Basic Concepts of Fuzzy Set Theory This appendix gives the definitions of the concepts of fuzzy set theory, which are used in this book. For a comprehensive treatment of fuzzy set theory, see, for instance, (Zimmermann, 1996; Klir and Yuan, 1995). A.I Fuzzy Sets Definition A.I (fuzzy set) A fuzzy set A on universe (domain) X is defined by the membership function ILA{X) which is a mapping from the universe X into the unit interval: ILA{X): X -+ [0,1]. (A.l) F{X) denotes the set of all fuzzy sets on X. Fuzzy set theory allows for a partial membership of an element in a set. If the value of the membership function, called the membership degree (grade), equals one, x belongs completely to the fuzzy set. If it equals zero, x does not belong to the set. If the membership degree is between 0 and 1, x is a partial member of the fuzzy set. In the fuzzy set literature, the term crisp is often used to denote nonfuzzy quantities, e.g., a crisp number, a crisp set, etc. A.2 Membership Functions In a discrete set X = {Xi I i = 1,2, ... ,n}, a fuzzy set A may be defined by a list of ordered pairs: membership degree/set element: (A.2) or in the form of two related vectors: In continuous domains, fuzzy sets are defined analytically by their membership func­ tions. In this book, the following forms of membership functions are used: 228 FUZZY MODELING FOR CONTROL Trapezoidal membership function: x-a d-X) /-L(x;a,b,c,d)=max ( O,min(b_a,l'd_c) , (A4) where a, b, c and d are coordinates of the trapezoid apexes. When b = c, a triangular membership function is obtained. Piece-wise exponential membership function: if x < CI, if x > Cn (AS) otherwise, where CI and Cr are the left and right shoulder, respectively, and WI, Wr are the left and right width, respectively. For CI = Cr and WI = W r , the Gaussian membership function is obtained. A.3 Basic Definitions Definition A.2 (a-cut) The a-cut Au of a fuzzy set A is the crisp subset of the universe of discourse X whose elements all have membership degrees greater than or equal to a: Au = {x I /-LA(X) ~ a}, a E [0,1]. (A6) The a-cut operator is also denoted by a-cut(A, a). Definition A.3 (strict a-cut) An a-cut Au is strict if /-LA (x) ¥- a for each x E Au. Definition A.4 (convex fuzzy set) A fuzzy set defined in IR n is convex if each of its a-cuts is a convex set. Definition A.S (support) The support of a fuzzy set A is the crisp subset of X whose elements all have nonzero membership grades: supp(A) = {x I /-LA(X) > O}. (A7) Definition A.6 (core) The core of a fuzzy set A is a crisp subset of X consisting of all elements with membership grades equal to one: core(A) = {x I /-LA(X) = I}. (A8) In the literature, the core is sometimes also denoted as the kernel, ker(A). Definition A.7 (cardinality) Cardinality of a fuzzy set A = {/-LA (Xi)/Xi 1, 2, ... , n} is defined as the sum of the membership degrees: n IAI = L/-LA(Xi). (A9) i=1 APPENDIX A: BASIC CONCEPTS OF FUZZY SET THEORY 229 Definition A.S (height) The height of a fuzzy set A is the supremum of the member­ ship grades of elements in A: hgt(A) = sup JLA(X), (A. 10) xEX Definition A.9 (normal fuzzy set) A fuzzy set A is normal if 3x E X such that JLA(X) = 1. The operator norm(A) denotes normalization of a fuzzy set, i.e., A' = norm(A) ¢:> JLA' (x) = JLA(X)/ hgt(A), Vx. A.4 Operations on Fuzzy Sets Definitions of operations on sets extend from ordinary set theory to fuzzy sets. In most cases, there are various ways to extend these operations. This section presents the basic definitions of fuzzy intersection, union and complement, and definitions of some other operations used in this book. Definition A.10 (intersection offuzzy sets) Let A and B be two fuzzy sets in X. The intersection of A and B is a fuzzy set C, denoted C = A n B, such that for each x EX: JLC(X) = min(JLA(x),JLB(x)). (A.ll) The minimum operator is also denoted by 'N, i.e., JLc{x) = JLA (x) /\ JLB(X), Definition A.ll (union of fuzzy sets) Let A and B be two fuzzy sets in X. The union of A and B is a fuzzy set C, denoted C = A u B, such that for each x EX: JLC(X) = max(JLA(x),JLB(X)). (A. 12) The maximum operator is also denoted by 'y', i.e., JLc(x) = JLA(X) Y JLB(X), Fuzzy intersection of two fuzzy sets can be specified in a more general way by a binary operation on the unit interval, i.e., a function of the form: i: [0, 1) x [0, l)-t [0,1). (A. 13) In order for a function i to qualify as a fuzzy intersection, it must have appropriate prop­ erties. Functions known as t-norms posses the properties required for the intersection. Similarly, functions called t-conorms can be used for the fuzzy union. Definition A.12 (t-normlfuzzy intersection) At-norm i is a binary operation on the unit interval that satisfies at least the following axioms for all a, b, c E [0,1) (Klir and Yuan, 1995): i(a, 1) = a (boundary condition), (A. 14a) b:::; c implies i(a, b) :::; i(a, c) (monotonicity), (A. 14b) i(a, b) = i(b, a) (commutativity), (A. 14c) i(a, i(b, c)) = i(i(a,b),c) (associativity). (A. 14d) Some frequently used t-norms are: 230 FUZZY MODELING FOR CONTROL standard intersection: i(a, b) = min(a, b) ( algebraic) product: i(a, b) = ab bold (Lukasiewicz) intersection: i(a, b) = max(O, a + b - 1) Definition A.13 (t-conormlfuzzy union) A t-conorm u is a binary operation on the unit interval that satisfies at least the following axioms for all a, b, c E [0,1] (Klir and Yuan, 1995): u(a, 0) = a (boundary condition), (A 15a) b::; c implies u(a, b) ::; u(a, c) (monotonicity), (AI5b) u(a, b) = u(b, a) (commutativity), (A 15c) u(a,u(b,c)) = u(u(a,b),c) (associativity). (AI5d) Some frequently used t-conorms are: standard union: u(a, b) = max(a, b) (algebraic) sum: u(a, b) = a + b - ab bold (Lukasiewicz) union: u(a, b) = min(I, a + b) Definition A.14 (complementofa fuzzy set) Let A be a fuzzy sets in X. The com­ plement of A is a fuzzy set, denoted 1, such that for each x EX: (AI6) A.S Fuzzy Relations Definition A.IS (fuzzy relation) An n-ary fuzzy relation is a mapping R:Xl x X 2 x ... X Xn -+ [0,1]' (AI7) which assigns membership grades to all n-tuples (Xl, X2, .•. ,xn ) from the Cartesian product Xl x X2 X ... X X n . A fuzzy relation is, in fact, a fuzzy set in the Cartesian product Xl X X 2 X ••. X X n . The membership grades represent the degree of association (correlation) among the elements of the different domains Xi. For computer implementations, R is conveniently represented as an n-dimensional array: R = [ril,i2, ... ,in]. Example: Consider a fuzzy relation R describing the relationship X ~ Y ("x is approximately equal to y") by means of the following membership function ILR( x, y) = e-(x-y)2. Figure Al shows a mesh plot of this relation. A.6 Projections and Cylindrical Extensions Definitions in this section are adopted from (Kruse, et al., 1994). Definition A.16 (n-dimensional universe) A family U = (X(i))iENn of non-empty domains X (i) , i = 1,2, ... , n, n E N, is called a universe of dimension n. Nn = {I, 2, ... ,n} is the index set related to this universe. For any non-empty index subset I ~ Nn the product space is defined by: I - X(i) X - x.tEl APPENDIX B: BASIC CONCEPTS OF FUZZY SET THEORY 231 0.5 -0.5 o --0.5 y -1 -1 x Figure A.1. Fuzzy relation J-LR(X, y) = e-(x-y)2. The product space XNn is denoted by X for short. A fuzzy set defined on a multidi­ mensional universe is called a multidimensional fuzzy set. Definition A.17 (point-wise projection) Let U = (X(i))iENn be a universe of di­ mension n and let C, Sand T be index subsets of Nn which satisfy the conditions T = SUe, S n C = 0 and S i= 0. Point-wise projection of X T onto X S is the mapping redI: XT ---+ X S defined by: (A I 8) Definition A.1S (projection of a fuzzy set) Let U = (X(i))iENn be a universe of dimension n, M an index set with 0 i= M ~ Nn . The projection of A onto X M is the mapping projM: F(X) ---+ F(XM) defined by projM (J-L(x)) = sup {J-L(X') I x' EX 1\ x = red~(x')} (A19) An example of projection from lR 2 to lR is given in Figure A2. Definition A.19 (cylindrical extension of a fuzzy set) LetU = (X(i))iENn be a uni­ verse of dimension n.
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