1. Conventional (Boolean) Set Theory: Fuzzy Set Theory © INFORM 1990-1998Slide 1 “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”

2. 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 © INFORM 1990-1998Slide 2 Continuous Definition: No More Artificial Thresholds!

3. ...Terms, Degree of Membership, Membership Function, Base Variable... Linguistic Variable © INFORM 1990-1998Slide 3 … pretty much raised …... but just slightly strong … A Linguistic Variable Defines a Concept of Our Everyday Language!

4. Fuzzification, Fuzzy Inference, Defuzzification: Basic Elements of a Fuzzy Logic System © INFORM 1990-1998Slide 4 Fuzzy Logic Defines the Control Strategy on a Linguistic Level!

5. Container Crane Case Study: Basic Elements of a Fuzzy Logic System © INFORM 1990-1998Slide 5 Two Measured Variables and One Command Variable !

6. Control Loop of the Fuzzy Logic Controlled Container Crane: Basic Elements of a Fuzzy Logic System © INFORM 1990-1998Slide 6 Closing the Loop With Words !

7. 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} 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 - © INFORM 1990-1998Slide 7 Membership Function Definition: 4° 0.8 0.2 12m 0.9 0.1 The Linguistic Variables Are the “Vocabulary” of a Fuzzy Logic System !

8. 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 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 - © INFORM 1990-1998Slide 8 X XAggregation:Computing the “IF”-Part X XComposition: Computing the “THEN”-Part X XAggregation:Computing the “IF”-Part X XComposition: Computing the “THEN”-Part The Rules of the Fuzzy Logic Systems Are the “Laws” It Executes !

9. 2. Fuzzy-Inference: - Aggregation - © INFORM 1990-1998Slide 9 Boolean Logic Only Defines Operators for 0/1: ABAvB 000 010 100 111 Boolean Logic Only Defines Operators for 0/1: ABAvB 000000 010010 100100 111111 Fuzzy Logic Delivers a Continuous Extension: X X AND: µ AvB = min{ µ A ; µ B } X X OR: µ A+B = max{ µ A ; µ B } X X NOT: µ -A = 1 - µ A Fuzzy Logic Delivers a Continuous Extension: X X AND: µ AvB = min{ µ A ; µ B } X X OR: µ A+B = max{ µ A ; µ B } X X NOT: µ -A = 1 - µ A 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 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 !

10. 2. Fuzzy-Inference: Composition © INFORM 1990-1998Slide 10 Result for the Linguistic Variable "Power": pos_highwith 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 Result for the Linguistic Variable "Power": pos_highwith 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 !

11. 3. Defuzzification © INFORM 1990-1998Slide 11 Finding a Compromise Using “Center-of-Maximum”: 6.4 KW “Balancing” Out the Result !

12. Types of Fuzzy Controllers: - Direct Controller - Types of Fuzzy Controllers: - Direct Controller - © INFORM 1990-1998Slide 12 The Outputs of the Fuzzy Logic System Are the Command Variables of the Plant: Fuzzy Rules Output Absolute Values !

13. Types of Fuzzy Controllers: - Supervisory Control - Types of Fuzzy Controllers: - Supervisory Control - © INFORM 1990-1998Slide 13 Fuzzy Logic Controller Outputs Set Values for Underlying PID Controllers: Human Operator Type Control !

14. Types of Fuzzy Controllers: - PID Adaptation - Types of Fuzzy Controllers: - PID Adaptation - © INFORM 1990-1998Slide 14 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 !

15. Types of Fuzzy Controllers: - Fuzzy Intervention - Types of Fuzzy Controllers: - Fuzzy Intervention - © INFORM 1990-1998Slide 15 Fuzzy Logic Controller and PID Controller in Parallel: Intervention of the Fuzzy Logic Controller into Large Disturbances !