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![FUZZY LOGIC Began…
Fuzzy theory proposed making the membership function (or the
values False and True) operate over the range of real numbers
[0.0, 1.0].
Fuzzy logic has been applied to many fields, from Control
Theory to Artificial Intelligence](https://image.slidesharecdn.com/fuzzysettheory-250107154339-e29b0d7c/75/Fuzzy-Logic-Application-and-Fuzzy-Set-Theory-10-2048.jpg)
![Term Definitions:
Distance := {far, medium, close, zero, neg_close}
Angle := {pos_big, pos_small, zero, neg_small, neg_big}
Power := {pos_high, pos_medium, zero, neg_medium, neg_high}
1. Fuzzification:
- Linguistic Variables -
Slide 18
Membership Function Definition:
-90° -45° 0° 45° 90°
0
1
µ
Angle
zero
pos_small
neg_small
neg_big pos_big
4°
0.8
0.2
-10 0 10 20 30
0
1
µ
Distance [yards]
zero close medium far
neg_close
12m
0.9
0.1
The Linguistic
Variables Are the
“Vocabulary” of a
Fuzzy Logic System !](https://image.slidesharecdn.com/fuzzysettheory-250107154339-e29b0d7c/75/Fuzzy-Logic-Application-and-Fuzzy-Set-Theory-18-2048.jpg)
![3. Defuzzification
Slide 23
Finding a Compromise Using “Center-of-Maximum”:
-30 -15 0 15 30
0
1
µ
Power [Kilowatts]
zero
neg_medium
neg_high pos_medium pos_high
6.4 KW
“Balancing” Out
the Result !](https://image.slidesharecdn.com/fuzzysettheory-250107154339-e29b0d7c/75/Fuzzy-Logic-Application-and-Fuzzy-Set-Theory-23-2048.jpg)
Fuzzy Logic Application and Fuzzy Set Theory
- 1. Fuzzy Logic Applications Slide1 Tutorial and Workshop Internet: www.fuzzytech.com https://www.tutorialspoint.com/fuzzy_logic History, Current Level and Further Development of Fuzzy Logic Technologies in the U.S., Japan, and Europe, etc. Types of Uncertainty and the Modeling of Uncertainty The Basic Elements of a Fuzzy Logic System Types of Fuzzy Logic Controllers
- 2. History, State ofthe Art, and Future Development Slide 2 1965 Seminal 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) 1975 Introduction of Fuzzy Logic in Japan 1980 Empirical Verification of Fuzzy Logic in Europe 1985 Broad Application of Fuzzy Logic in Japan 1990 Broad Application of Fuzzy Logic in Europe 1995 Broad Application of Fuzzy Logic in the U.S. 2000 Fuzzy 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 !
- 3. Applications Study ofthe IEEE in 1996 Slide 3 About 1100+ Successful Fuzzy Logic Applications Have Been Published (an estimated 5% of those in existence) Almost All Applications Have Not Involved the Replacement of a Standard Type Controller (PID,..), But Rather Multi-Variable Supervisory Control Applications Range from Embedded Control (28%), Industrial Automation (62%) to Process Control (10%) Of 311 Authors That Answered a Questionnaire, About 90% State That Fuzzy Logic Has Slashed Design Time By More Than Half In This Questionnaire, 97.5% of the Designers Stated That They Will Use Fuzzy Logic Again in Future Applications, If Fuzzy Logic Is Applicable Fuzzy Logic Will Play a Major Role in Control Engineering !
- 4. Stochastic Uncertainty: TheProbability 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. Types of Uncertainty and the Modeling of Uncertainty Slide 4 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!
- 5. “... a personsuffering 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 5 Stochastics and Fuzzy Logic Complement Each Other !
- 6. WHAT IS FUZZYLOGIC? Fuzzy Logic (FL) is a method of reasoning that resembles human reasoning. The approach of FL imitates the way of decision making in humans that involves all intermediate possibilities between digital values YES and NO. The conventional logic block that a computer can understand takes precise input and produces a definite output as TRUE or FALSE, which is equivalent to human’s YES or NO.
- 7. WHAT IS FUZZYLOGIC? Fuzzy logic is a form of many-valued logic (multi-valued logic); it deals with reasoning that is approximate rather than fixed and exact. In contrast with traditional logic theory, where binary sets have two-valued logic: true or false, fuzzy logic variables may have a truth value that ranges in degree between 0 and 1
- 8. WHAT IS FUZZYLOGIC? Fuzzy logic has been extended to handle the concept of partial truth, where the truth value may range between completely true and completely false. Furthermore, when linguistic variables are used, these degrees may be managed by specific functions.
- 9. FUZZY LOGIC Began… Fuzzy logic began with the 1965 proposal of fuzzy set theory by Lotfi Zadeh. Lotfi A. Zadeh, a professor of UC Berkeley in California, known as the founder of fuzzy logic observed that conventional computer logic was incapable of manipulating data representing subjective or vague human ideas such as "an attractive person" . Fuzzy logic, hence was designed to allow computers to determine the distinctions among data with shades of gray, similar to the process of human reasoning.
- 10. FUZZY LOGIC Began… Fuzzy theory proposed making the membership function (or the values False and True) operate over the range of real numbers [0.0, 1.0]. Fuzzy logic has been applied to many fields, from Control Theory to Artificial Intelligence
- 11. Classical Set Theory A set is an unordered collection of different elements. It can be written explicitly by listing its elements using the set bracket. If the order of the elements is changed or any element of a set is repeated, it does not make any changes in the set.
- 12. Conventional (Boolean) SetTheory: Fuzzy Set Theory © INFORM 1990-1998 Slide 12 “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”
- 13. Discrete Definition: µSF (35°C) =0µSF (38°C) = 0.1 µSF (41°C) = 0.9 µSF (36°C) = 0µSF (39°C) = 0.35 µSF (42°C) = 1 µSF (37°C) = 0µSF (40°C) = 0.65 µSF (43°C) = 1 Fuzzy Set Definitions Slide 13 Continuous Definition: 39°C 40°C 41°C 42°C 38°C 37°C 36°C 1 0 µ(x) No More Artificial Thresholds!
- 14. ...Terms, Degree ofMembership, Membership Function, Base Variable... Linguistic Variable Slide 14 39°C 40°C 41°C 42°C 38°C 37°C 36°C 1 0 µ(x) low temp normal raised temperature strong fever … pretty much raised … ... but just slightly strong … A Linguistic Variable Defines a Concept of Our Everyday Language!
- 15. Fuzzification, Fuzzy Inference,Defuzzification: Basic Elements of a Fuzzy Logic System Slide 15 Linguistic Level Numerical Level Measured Variables Measured Variables (Numerical Values) (Linguistic Values) 2. Fuzzy-Inference Command Variables 3. Defuzzification Plant 1. Fuzzification (Linguistic Values) Command Variables (Numerical Values) Fuzzy Logic Defines the Control Strategy on a Linguistic Level!
- 16. Container Crane CaseStudy: Basic Elements of a Fuzzy Logic System Slide 16 Two Measured Variables and One Command Variable !
- 17. Control Loop ofthe Fuzzy Logic Controlled Container Crane: Basic Elements of a Fuzzy Logic System © INFORM 1990-1998 Slide 17 Linguistic Level Numerical Level Angle, Distance Angle, Distance (Numerical Values) (Numerical Values) 2. Fuzzy-Inference Power Power (Numerical Values) (Linguistic Variable) 3. Defuzzification Container Crane 1. Fuzzification Closing the Loop With Words !
- 18. Term Definitions: Distance :={far, medium, close, zero, neg_close} Angle := {pos_big, pos_small, zero, neg_small, neg_big} Power := {pos_high, pos_medium, zero, neg_medium, neg_high} 1. Fuzzification: - Linguistic Variables - Slide 18 Membership Function Definition: -90° -45° 0° 45° 90° 0 1 µ Angle zero pos_small neg_small neg_big pos_big 4° 0.8 0.2 -10 0 10 20 30 0 1 µ Distance [yards] zero close medium far neg_close 12m 0.9 0.1 The Linguistic Variables Are the “Vocabulary” of a Fuzzy Logic System !
- 19. Rules :- Fuzzylogic usually uses IF-THEN rules, or constructs that are equivalent. -IF variable IS property THEN action Example:- A simple temperature regulator that uses a fan might look like this: IF temperature IS very cold THEN stop fan IF temperature IS cold THEN turn down fan IF temperature IS normal THEN maintain level IF temperature IS hot THEN speed up fan 2. Fuzzy-Inference: - “IF-THEN”-Rules -
- 20. Computation of the“IF-THEN”-Rules: #1: IF Distance = medium AND Angle = pos_small THEN Power = pos_medium #2: IF Distance = medium AND Angle = zero THEN Power = zero #3: IF Distance = far AND Angle = zero THEN Power = pos_medium 2. Fuzzy-Inference: - “IF-THEN”-Rules - Slide 20 Aggregation: Computing the “IF”-Part Composition: Computing the “THEN”-Part The Rules of the Fuzzy Logic Systems Are the “Laws” It Executes !
- 21. 2. Fuzzy-Inference: - Aggregation- Slide 21 Boolean Logic Only Defines Operators for 0/1: A B AvB 0 0 0 0 1 0 1 0 0 1 1 1 Fuzzy Logic Delivers a Continuous Extension: AND: µAvB = min{ µA; µB } OR: µA+B = max{ µA; µB } NOT: µ-A = 1 - µA Aggregation of the “IF”-Part: #1: min{ 0.9, 0.8 } = 0.8 #2: min{ 0.9, 0.2 } = 0.2 #3: min{ 0.1, 0.2 } = 0.1 Aggregation Computes How “Appropriate” Each Rule Is for the Current Situation !
- 22. 2. Fuzzy-Inference: Composition Slide 22 Resultfor the Linguistic Variable "Power": pos_high with the degree 0.0 pos_medium with the degree 0.8 ( = max{ 0.8, 0.1 } ) zero with the degree 0.2 neg_medium with the degree 0.0 neg_high with the degree 0.0 Composition Computes How Each Rule Influences the Output Variables !
- 23. 3. Defuzzification Slide 23 Findinga Compromise Using “Center-of-Maximum”: -30 -15 0 15 30 0 1 µ Power [Kilowatts] zero neg_medium neg_high pos_medium pos_high 6.4 KW “Balancing” Out the Result !
- 24. Types of FuzzyControllers: - Direct Controller - Slide 24 The Outputs of the Fuzzy Logic System Are the Command Variables of the Plant: Fuzzification Inference Defuzzification IF temp=low AND P=high THEN A=med IF ... Variables Measured Variables Plant Command Fuzzy Rules Output Absolute Values !
- 25. Types of FuzzyControllers: - Supervisory Control - Slide 25 Fuzzy Logic Controller Outputs Set Values for Underlying PID Controllers: Fuzzification Inference Defuzzification IF temp=low AND P=high THEN A=med IF ... Set Values Measured Variables Plant PID PID PID Human Operator Type Control !
- 26. Types of FuzzyControllers: - PID Adaptation - Slide 26 Fuzzy Logic Controller Adapts the P, I, and D Parameter of a Conventional PID Controller: Fuzzification Inference Defuzzification IF temp=low AND P=high THEN A=med IF ... P Measured Variable Plant PID I D Set Point Variable Command Variable The Fuzzy Logic System Analyzes the Performance of the PID Controller and Optimizes It !
- 27. Types of FuzzyControllers: - Fuzzy Intervention - Slide 27 Fuzzy Logic Controller and PID Controller in Parallel: Fuzzification Inference Defuzzification IF temp=low AND P=high THEN A=med IF ... Measured Variable Plant PID Set Point Variable Command Variable Intervention of the Fuzzy Logic Controller into Large Disturbances !