AI Fuzzy Systems

1. AIFuzzy Systems

2. History, State of the Art, and Future Development 1965Seminal Paper “Fuzzy Logic” by Prof. Lotfi Zadeh, Faculty in Electrical Engineering, U.C. Berkeley, Sets the Foundation of the “Fuzzy Set Theory” 1970 First Application of Fuzzy Logic in Control Engineering (Europe) 1975Introduction of Fuzzy Logic in Japan 1980Empirical Verification of Fuzzy Logic in Europe 1985Broad Application of Fuzzy Logic in Japan 1990Broad Application of Fuzzy Logic in Europe 1995Broad Application of Fuzzy Logic in the U.S. 2000Fuzzy Logic Becomes a Standard Technology and Is Also Applied in Data and Sensor Signal Analysis. Application of Fuzzy Logic in Business and Finance. Today, Fuzzy Logic Has Already Become the Standard Technique for Multi-Variable Control ! Sde 2

3. Types of Uncertainty and the Modeling of Uncertainty • Stochastic Uncertainty: • The Probability of Hitting the Target Is 0.8 • Lexical Uncertainty: • "Tall Men", "Hot Days", or "Stable Currencies" • We Will Probably Have a Successful Business Year. • The Experience of Expert A Shows That B Is Likely to Occur. However, Expert C Is Convinced This Is Not True. Most Words and Evaluations We Use in Our Daily Reasoning Are Not Clearly Defined in a Mathematical Manner. This Allows Humans to Reason on an Abstract Level! Slide 3

4. Probability and Uncertainty “... a person suffering from hepatitis shows in 60% of all cases a strong fever, in 45% of all cases yellowish colored skin, and in 30% of all cases suffers from nausea ...” Stochastics and Fuzzy Logic Complement Each Other ! Slide 4

5. “Strong Fever” Fuzzy Set Theory Conventional (Boolean) Set Theory: 38.7°C 38°C 40.1°C 41.4°C Fuzzy Set Theory: 42°C 39.3°C 38.7°C 38°C 37.2°C 40.1°C 41.4°C 42°C 39.3°C “Strong Fever” “More-or-Less” Rather Than “Either-Or” ! 37.2°C Slide 5

6. Fuzzy Sets... Representing crisp and fuzzy sets as subsets of a domain (universe) U".

7. Fuzziness versus probability Probability density function for throwing a dice and the membership functions of the concepts "Small" number, "Medium", "Big".

8. Conceptualising in fuzzy terms... One representation for the fuzzy number "about 600".

9. Conceptualising in fuzzy terms... Representing truthfulness (certainty) of events as fuzzy sets over the [0,1] domain.

10. “Strong Fever” Strong Fever Revisited Conventional (Boolean) Set Theory: 38.7°C 38°C 40.1°C 41.4°C Fuzzy Set Theory: 42°C 39.3°C 38.7°C 38°C 37.2°C 40.1°C 41.4°C 42°C 39.3°C “Strong Fever” 37.2°C Slide 10

11. Why fuzzy? As Zadeh said, the term is concrete, immediate and descriptive; we all know what it means. However, many people in the West were repelled by the word fuzzy, because it is usually used in a negative sense. • Why logic? Fuzziness rests on fuzzy set theory, and fuzzy logic is just a small part of that theory.

12. . Range of logical values in Boolean and fuzzy logic

13. The classical example in fuzzy sets is tall men. The elements of the fuzzy set “tall men” are all men, but their degrees of membership depend on their height.

14. Fuzzy Set Definitions Discrete Definition: µSF(35°C) = 0 µSF(38°C) = 0.1 µSF(41°C) = 0.9 µSF(36°C) = 0 µSF(39°C) = 0.35 µSF(42°C) = 1 µSF(37°C) = 0 µSF(40°C) = 0.65 µSF(43°C) = 1 Continuous Definition: No More Artificial Thresholds! Slide 14

15. Representation of hedges in fuzzy logic

16. Representation of hedges in fuzzy logic (continued)

17. Linguistic Variable ...Terms, Degree of Membership, Membership Function, Base Variable... … pretty much raised … A Linguistic Variable Defines a Concept of Our Everyday Language! ... but just slightly strong … Slide 17

18. Fuzzy Sets • Formal definition: A fuzzy set A in X is expressed as a set of ordered pairs: Membership function (MF) Universe or universe of discourse Fuzzy set A fuzzy set is totally characterized by a membership function (MF).

19. Cantor’s sets

20. Operations of fuzzy sets

21. Complement Crisp Sets: Who does not belong to the set? Fuzzy Sets: How much do elements not belong to the set? The complement of a set is an opposite of this set. For example, if we have the set of tall men, its complement is the set of NOT tall men. When we remove the tall men set from the universe of discourse, we obtain the complement. If A is the fuzzy set, its complement ØA can be found as follows: mØA(x) = 1 -mA(x)

22. Containment • Crisp Sets: Which sets belong to which other sets? • Fuzzy Sets: Which sets belong to other sets? • Similar to a Chinese box, a set can contain other • sets. The smaller set is called the subset. For • example, the set of tall men contains all tall men; • very tall men is a subset of tall men. However, the • tall men set is just a subset of the set of men. In • crisp sets, all elements of a subset entirely belong to • a larger set. In fuzzy sets, however, each element • can belong less to the subset than to the larger set. • Elements of the fuzzy subset have smaller • memberships in it than in the larger set.

23. Intersection Crisp Sets: Which element belongs to both sets? Fuzzy Sets: How much of the element is in both sets? In classical set theory, an intersection between two sets contains the elements shared by these sets. For example, the intersection of the set of tall men and the set of fat men is the area where these sets overlap. In fuzzy sets, an element may partly belong to both sets with different memberships. A fuzzy intersection is the lower membership in both sets of each element. The fuzzy intersection of two fuzzy sets A and B on universe of discourse X: mAÇB(x) = min [mA(x), mB(x)] = mA(x) ÇmB(x), where xÎX

24. Union • Crisp Sets: Which element belongs to either set? • Fuzzy Sets: How much of the element is in either set? • The union of two crisp sets consists of every element • that falls into either set. For example, the union of • tall men and fat men contains all men who are tall • OR fat. In fuzzy sets, the union is the reverse of the • intersection. That is, the union is the largest • membership value of the element in either set. The • fuzzy operation for forming the union of two fuzzy • sets A and B on universe X can be given as: mAÈB(x) = max [mA(x), mB(x)] = mA (x) ÈmB(x), where xÎX

25. Set-Theoretic Operations • Subset: • Complement: • Union: • Intersection:

26. What is the difference between classical and fuzzy rules? A classical IF-THEN rule uses binary logic, for example, Rule: 1 Rule: 2 IF speed is > 100 IF speed is < 40 THEN stopping_distance is long THEN stopping_distance is short The variable speed can have any numerical value between 0 and 220 km/h, but the linguistic variable stopping_distance can take either value long or short. In other words, classical rules are expressed in the black-and-white language of Boolean logic.

27. We can also represent the stopping distance rules in a fuzzy form: Rule: 1 Rule: 2 IF speed is fast IF speed is slow THEN stopping_distance is long THEN stopping_distance is short In fuzzy rules, the linguistic variable speed also has the range (the universe of discourse) between 0 and 220 km/h, but this range includes fuzzy sets, such as slow, medium and fast. The universe of discourse of the linguistic variable stopping_distance can be between 0 and 300 m and may include such fuzzy sets as short, medium andlong.

28. Basic Elements of a Fuzzy Logic System Fuzzy Logic Defines the Control Strategy on a Linguistic Level! Fuzzification, Fuzzy Inference, Defuzzification:

29. Basic Elements of a Fuzzy Logic System Closing the Loop With Words ! Control Loop of the Fuzzy Logic Controlled Container Crane:

30. Types of Fuzzy Controllers: - Direct Controller - The Outputs of the Fuzzy Logic System Are the Command Variables of the Plant: Fuzzy Rules Output Absolute Values !

31. Types of Fuzzy Controllers: - Supervisory Control - Fuzzy Logic Controller Outputs Set Values for Underlying PID Controllers: Human Operator Type Control !

32. Types of Fuzzy Controllers: - PID Adaptation - Fuzzy Logic Controller Adapts the P, I, and D Parameter of a Conventional PID Controller: The Fuzzy Logic System Analyzes the Performance of the PID Controller and Optimizes It !