AI Fuzzy Systems

History, State of the Art, and Future Development Sde Seminal Paper “Fuzzy Logic” by Prof. Lotfi Zadeh, Faculty in Electrical Engineering, U.C. Berkeley, Sets the Foundation of the “Fuzzy Set Theory” 1970First 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. 1965Seminal Paper “Fuzzy Logic” by Prof. Lotfi Zadeh, Faculty in Electrical Engineering, U.C. Berkeley, Sets the Foundation of the “Fuzzy Set Theory” 1970First 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 !

Stochastic Uncertainty: XThe Probability of Hitting the Target Is 0.8 Lexical Uncertainty: X"Tall Men", "Hot Days", or "Stable Currencies" XWe Will Probably Have a Successful Business Year. XThe Experience of Expert A Shows That B Is Likely to Occur. However, Expert C Is Convinced This Is Not True. Stochastic Uncertainty: XThe Probability of Hitting the Target Is 0.8 Lexical Uncertainty: X"Tall Men", "Hot Days", or "Stable Currencies" XWe Will Probably Have a Successful Business Year. XThe Experience of Expert A Shows That B Is Likely to Occur. However, Expert C Is Convinced This Is Not True. Types of Uncertainty and the Modeling of Uncertainty Slide 3 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!

“... 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...” Probability and Uncertainty Slide 4 Stochastics and Fuzzy Logic Complement Each Other !

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

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

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

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

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

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

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. Fuzziness rests on fuzzy set theory, and fuzzy logic is just a small part of that theory. Fuzziness rests on fuzzy set theory, and fuzzy logic is just a small part of that theory. Why fuzzy? Why fuzzy? Why logic? Why logic?

Range of logical values in Boolean and fuzzy logic.

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. 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.

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

Representation of hedges in fuzzy logic

Representation of hedges in fuzzy logic (continued)

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

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

Cantor’s sets

Operations of fuzzy sets

Complement 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  A can be found as follows:  A ( x ) = 1   A ( x )

Containment 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.

Intersection 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:  A  B (x) = min [  A (x),  B (x)] =  A (x)   B (x), where x  X

Union 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:  A  B (x) = max [  A (x),  B (x)] =  A (x)   B (x), where x  X

Set-Theoretic Operations Subset: Complement: Union: Intersection:

What is the difference between classical and fuzzy rules? Rule: 1Rule: 2 IF speed is > 100IF speed is 100IF speed is < 40 THEN stopping_distance is longTHEN 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. A classical IF-THEN rule uses binary logic, for example,

We can also represent the stopping distance rules in a fuzzy form: Rule: 1Rule: 2 IF speed is fastIF speed is slow THEN stopping_distance is longTHEN 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 and long.

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

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

Types of Fuzzy Controllers: - Direct Controller - 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 !

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

Types of Fuzzy Controllers: - PID Adaptation - 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 !