1. Topic 2 Fuzzy Logic Control

2. Ming-Feng Yeh2-2 Outlines Basic concepts of fuzzy set theory Fuzzy relations Fuzzy logic control General Fuzzy System R.R. Yager and D.P. Filev, Essentials of fuzzy modeling and control, John Wiley & Sons. Inc., 1994 L.X. Wang, A Course in Fuzzy Systems and Control, Prentice-Hall, 1997. K.M. Passino and S. Yurkovich, Fuzzy Control, Addison Wesley, 1998.

3. Ming-Feng Yeh2-3 1. Introduction The concept of a fuzzy subset was originally introduced by L.A. Zadeh in 1965 as a generalization of the idea of an crisp set. A fuzzy subset whose truth values are drawn from the unit interval [0, 1] rather than the set {0, 1}. The fuzzy subset has as its underlying logic a multivalued logic.

4. Ming-Feng Yeh2-4 Classical Set Theory Two-valued logic: {0,1}, i.e., Characteristic function: Intersection: Union: Difference: Complement:

5. Ming-Feng Yeh2-5 Fuzzy Set Theory Fuzzy logic deals with problems that have vagueness, uncertainty, or imprecision, and uses membership functions with values in [0,1]. Membership function: X: universe of discourse

6. Ming-Feng Yeh2-6 Definition Assume X is a set serving as the universe of discourse. A fuzzy subset A of X is a associated with a characteristic function: A(x) or  A (x) Membership function: The relationship between variables, labels and fuzzy sets.

7. Ming-Feng Yeh2-7 Fuzzy Set Representation X: universe of discourse (all the possible values that a variable can assume) A: a subset of X Discrete: Continue:

8. Ming-Feng Yeh2-8 Membership Functions Four most commonly used membership functions:

9. Ming-Feng Yeh2-9 Fundamental Concepts: 1 Assume A is a fuzzy subset of X Normal: If these exists at least one element such that A(x)=1. A fuzzy subset that is NOT normal is called subnormal. Height: The largest membership grade of any element in A. That is, height(A) = max A(x). Crisp sets are special cases of fuzzy sets in which the membership grades are just either zero or one.

10. Ming-Feng Yeh2-10 Fundamental Concepts: 2 Assume A is a fuzzy subset of X Support of A: all elements of A have nonzero membership grades. Core of A: all element of A with membership grade one.

11. Ming-Feng Yeh2-11 Example 2-1 Assume let A is a normal fuzzy subset and B is a subnormal fuzzy subset of X. Height(A)=1 and Height(B)=0.9 Supp(A)={a,b,c,d} and Supp(B)={a,b,c,d,e} Core(A)={a} and Core(B)= 

12. Ming-Feng Yeh2-12 Fundamental Concepts: 3 Assume A and B are two fuzzy subsets of X Contain: A is said to be a subset of B, if Equal: A and B are said to be equal, if and That is, Null fuzzy subset:

13. Ming-Feng Yeh2-13 Operations on Fuzzy Sets Assume A and B are two fuzzy subsets of X Intersection : Union : Complement :

14. Ming-Feng Yeh2-14 Example 2-2 Assume

15. Ming-Feng Yeh2-15 Logical Operations: AND Fuzzy intersection (AND): The intersection of fuzzy sets A and B, which are defined on the universe of discourse X, is a fuzzy set denoted by, with a membership function as Minimum: Algebraic product: Triangular norm “  ”: x  y = min{x,y} or x  y = xy AND:  A (x)   B (x)

16. Ming-Feng Yeh2-16 Logical Operations: OR Fuzzy union (OR): The union of fuzzy sets A and B, which are defined on the universe of discourse X, is a fuzzy set denoted by, with a membership function as Maximum: Algebraic sum: Triangular co-norm “  ”: x  y = max{x,y} or x  y = x+ y – xy OR:  A (x)   B (x)

17. Ming-Feng Yeh2-17  -level Set Assume A is a fuzzy subset of X, the  -level set of A, denoted by A , is the crisp subset of X consisting of all elements in X for which Any fuzzy subset A of X can be written as Let