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This talk will give a layman's introduction to the topic and explore some of the real world applications in control and human decision making. Examples might include household appliances, control of large industrial plant, and health monitoring systems for the elderly. We will look at where the field might be going over the next ten years, highlighting areas where DMU's specialist expertise drives the way.

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- 1. Introduction Motivation What is Fuzzy Logic? Fuzzy Logic Systems Example Applications Uncertainty and Fuzziness The Future Fuzzy Logic in the Real World Simon Coupland Centre for Computational Intelligence De Montfort University The Gateway Leicester United Kingdom Email: simonc@dmu.ac.uk October 22nd 2009 Fuzzy Logic in the Real World Simon Coupland
- 2. Introduction Motivation What is Fuzzy Logic? Fuzzy Logic Systems Example Applications Uncertainty and Fuzziness The Future Overview • Motivation • What is Fuzzy Logic? • Fuzzy Logic Systems • Example Applications • Uncertainty and Fuzziness • The Future of Fuzzy Systems Fuzzy Logic in the Real World Simon Coupland
- 3. Introduction Motivation What is Fuzzy Logic? Fuzzy Logic Systems Example Applications Uncertainty and Fuzziness The Future Crisp Sets and Logic • A well deﬁned, unordered collection of items which are identiﬁable and distinct • Six nations rugby teams = {England, Scotland, France, Italy, Ireland, Wales} Fuzzy Logic in the Real World Simon Coupland
- 4. Introduction Motivation What is Fuzzy Logic? Fuzzy Logic Systems Example Applications Uncertainty and Fuzziness The Future Crisp Sets and Logic Deﬁnition A crisp set A over the universe for discourse X is subset of the domain X based on some condition(s): A = {x |x meets some condition(s)} A membership function µA is used to map elements of X to their respective membership in A of zero or one: 1 if x ∈ A µA (x ) = 0 if x ∈ A / Fuzzy Logic in the Real World Simon Coupland
- 5. Introduction Motivation What is Fuzzy Logic? Fuzzy Logic Systems Example Applications Uncertainty and Fuzziness The Future Crisp Sets and Logic So a crisp set is a relation from some domain to binary values: A : X × {0, 1} x1 0 x2 x3 1 Fuzzy Logic in the Real World Simon Coupland
- 6. Introduction Motivation What is Fuzzy Logic? Fuzzy Logic Systems Example Applications Uncertainty and Fuzziness The Future The Trouble with Crisp Sets The Sorites Paradox • Premise 1: Consider 100,000 grains of sand to be a heap • Premise 2: A heap of sand minus one grain is still a heap of sand • But at some point it must stop being a heap Fuzzy Logic in the Real World Simon Coupland
- 7. Introduction Motivation What is Fuzzy Logic? Fuzzy Logic Systems Example Applications Uncertainty and Fuzziness The Future The Trouble with Crisp Sets The Sorites Paradox - Bertrand Russell’s view • Person x is tall if their height is 170cm or more • Tall = {person | height(person) ≥ 170} Charles’ Alan’s height Jon’s height height is 170cm is 185cm 169cm Fuzzy Logic in the Real World Simon Coupland
- 8. Introduction Motivation What is Fuzzy Logic? Fuzzy Logic Systems Example Applications Uncertainty and Fuzziness The Future The Trouble with Crisp Sets Perhaps we need: • A softer model • Degrees of set membership • Some conceptual vagueness Fuzzy Logic in the Real World Simon Coupland
- 9. Introduction Motivation What is Fuzzy Logic? Fuzzy Logic Systems Example Applications Uncertainty and Fuzziness The Future Fuzzy Sets Fuzzy sets - proposed by Lotﬁ Zadeh in 1965 • Set membership is graduated • Degrees of belonging measured as real numbers between zero and one • Boundaries of the set are soft, not crisp Fuzzy Logic in the Real World Simon Coupland
- 10. Introduction Motivation What is Fuzzy Logic? Fuzzy Logic Systems Example Applications Uncertainty and Fuzziness The Future Fuzzy Sets Deﬁnition A fuzzy set A over the universe for discourse X is a set of ordered pairs mapping domain elements their respective degrees of belonging measured as a real number between zero and one: A = {(x1 , 0.4), (x2 , 0.3), (x3 , 1), (x4 , 0.6)} Or using Zadeh’s notation: A = {0.4/x1 + 0.3/x2 + 1/x3 + 0.6/x4 } A fuzzy set A is usually expressed in terms of its membership function µA which maps domain elements (x) their respective degrees of of belonging in the interval [0, 1]: A = {(x , µA (x ))|x ∈ X } Fuzzy Logic in the Real World Simon Coupland
- 11. Introduction Motivation What is Fuzzy Logic? Fuzzy Logic Systems Example Applications Uncertainty and Fuzziness The Future Fuzzy Sets A Graphical Comparison with Crisp Sets So a fuzzy set is a relation from some domain to real numbers: A : X × {0, 1} x1 0.4 x3 1 x2 0.3 x4 0.6 Fuzzy Logic in the Real World Simon Coupland
- 12. Introduction Motivation What is Fuzzy Logic? Fuzzy Logic Systems Example Applications Uncertainty and Fuzziness The Future Fuzzy Sets A Graphical Comparison with Crisp Sets The Membership Function of the Crisp Set Tall The Membership Function of the Fuzzy Set Tall µ µ 1 1 0 0 160 165 170 175 180 185 160 165 170 175 180 185 Height (cm) Height (cm) Fuzzy Logic in the Real World Simon Coupland
- 13. Introduction Motivation What is Fuzzy Logic? Fuzzy Logic Systems Example Applications Uncertainty and Fuzziness The Future Fuzzy Sets Implementation Reality The Membership Function of the Fuzzy Set Tall µ 1 • Computers don’t like continuous functions • Instead use discrete approximations • A number of ordered pairs mapping x values to the µ 0 160 165 170 175 180 185 Height (cm) Fuzzy Logic in the Real World Simon Coupland
- 14. Introduction Motivation What is Fuzzy Logic? Fuzzy Logic Systems Example Applications Uncertainty and Fuzziness The Future Fuzzy Sets and Probability A Cautionary Tale • Quite different meanings • Example - bottles of liquid: Fuzzy Bottle Probabilistic Bottle 0.7 Drinkable 0.7 Drinkable Fuzzy Logic in the Real World Simon Coupland
- 15. Introduction Motivation What is Fuzzy Logic? Fuzzy Logic Systems Example Applications Uncertainty and Fuzziness The Future Fuzzy Logic Operators • Logical operations of fuzzy sets are well deﬁned • Together these form fuzzy logic: • AND • OR • NOT • IMPLIES • Crucial for rule based fuzzy logic systems Fuzzy Logic in the Real World Simon Coupland
- 16. Introduction Motivation What is Fuzzy Logic? Fuzzy Logic Systems Example Applications Uncertainty and Fuzziness The Future Fuzzy Logic Operators Logical AND • Deﬁned for each point in the membership function • Extends Boolean AND • Any t-norm but usually minimum: µA AND B (x ) = µA (x ) ∧ µB (x ) µ A µ B µ A AND B 1 1 1 X X X Fuzzy Logic in the Real World Simon Coupland
- 17. Introduction Motivation What is Fuzzy Logic? Fuzzy Logic Systems Example Applications Uncertainty and Fuzziness The Future Fuzzy Logic Operators Logical OR • Deﬁned for each point in the membership function • Extends Boolean OR • Any t-norm but usually maximum: µA AND B (x ) = µA (x ) ∨ µB (x ) µ A µ B µ A OR B 1 1 1 X X X Fuzzy Logic in the Real World Simon Coupland
- 18. Introduction Motivation What is Fuzzy Logic? Fuzzy Logic Systems Example Applications Uncertainty and Fuzziness The Future Fuzzy Logic Operators Logical NOT • Deﬁned for each point in the membership function • Extends Boolean NOT: ¬µA (x ) = 1 − µA (x ) µ A µ NOT A 1 1 X X Fuzzy Logic in the Real World Simon Coupland
- 19. Introduction Motivation What is Fuzzy Logic? Fuzzy Logic Systems Example Applications Uncertainty and Fuzziness The Future Fuzzy Logic Operators Logical IMPLIES • Deﬁned for each point in the membership function • A variety of operators • Most commonly used is generalised modus ponens: • Modus ponens: If X THEN Y . X, therefore Y • Generalised modus ponens: If X THEN Y . X to degree 0.6, therefore Y to degree 0.6 • Any t-norm but usually minimum: µα =⇒ A (x ) = α ∨ µA (x ) µ A µ α =⇒ A 1 1 α X X Fuzzy Logic in the Real World Simon Coupland
- 20. Introduction Motivation What is Fuzzy Logic? Fuzzy Logic Systems Example Applications Uncertainty and Fuzziness The Future Fuzzy Logic Operators Fuzziﬁcation • The process of ﬁnding the membership grade of an input: µ 1 0.5 µTall (170) = 0.5 0 160 165 170 175 180 185 Alan’s height (170 cm) Fuzzy Logic in the Real World Simon Coupland
- 21. Introduction Motivation What is Fuzzy Logic? Fuzzy Logic Systems Example Applications Uncertainty and Fuzziness The Future Fuzzy Logic Operators Defuzziﬁcation • The process of reducing a fuzzy set to a single crisp value • Centre of area is most commonly used: ∑ µA (x )x CA = ∑ µA (x ) µ Tall 1 0.13 × 166.25 + . . . + 1 × 185 CTall = = 177.78 0.13 + . . . + 1 0 160 165 170 175 180 185 177.78cm Fuzzy Logic in the Real World Simon Coupland
- 22. Introduction Motivation What is Fuzzy Logic? Fuzzy Logic Systems Example Applications Uncertainty and Fuzziness The Future Fuzzy Logic Systems Fitting it all Together • Typically rule-based: IF age is Young AND wealth is Rich THEN disposition is Very Happy • Combine fuzzy sets with logical operators • Crisp inputs, often crisp outputs: Rule Base Inputs Fuzziﬁer Defuzziﬁer Outputs Inference Engine Fuzzy Logic System Fuzzy Logic in the Real World Simon Coupland
- 23. Introduction Motivation What is Fuzzy Logic? Fuzzy Logic Systems Example Applications Uncertainty and Fuzziness The Future Fuzzy Logic Systems Fitting it all Together µ Young µ Rich Min µ Happy 1 1 1 0 0 0 0 100 0 £100K 0 10 µ Older µ Doing OK Min µ Cheery 1 1 1 0 0 0 0 100 0 £100K 0 10 Max µ Computed Disposition 1 0 0 10 Fuzzy Logic in the Real World Simon Coupland
- 24. Introduction Motivation What is Fuzzy Logic? Fuzzy Logic Systems Example Applications Uncertainty and Fuzziness The Future Fuzzy Logic Systems Fitting it all Together µ Young µ Rich Min µ Happy 1 1 1 0 0 0 0 100 0 £100K 0 10 µ Older µ Doing OK Min µ Cheery 1 1 1 0 0 0 0 100 0 £100K 0 10 Max Age = 45 Income = £65k µ Computed Disposition 1 0 0 10 6.23 Fuzzy Logic in the Real World Simon Coupland
- 25. Introduction Motivation What is Fuzzy Logic? Fuzzy Logic Systems Example Applications Uncertainty and Fuzziness The Future Fuzzy Logic Systems What I haven’t told you Many other approaches: • Logical operator choices • Neuro-fuzzy systems • Defuzziﬁcation operator choices • Adaptive systems Fuzzy Logic in the Real World Simon Coupland
- 26. Introduction Motivation What is Fuzzy Logic? Fuzzy Logic Systems Example Applications Uncertainty and Fuzziness The Future Application Areas Applied to a wide range of problems including: • Industrial control • Human decision making • Image processing Fuzzy Logic in the Real World Simon Coupland
- 27. Introduction Motivation What is Fuzzy Logic? Fuzzy Logic Systems Example Applications Uncertainty and Fuzziness The Future Industrial Control Control of marine diesel engines • MAN 9000kW Cathedral Engines • Low overshoot tolerance • Highly dynamic and uncertain environments • Require robust and accurate control Fuzzy Logic in the Real World Simon Coupland
- 28. Introduction Motivation What is Fuzzy Logic? Fuzzy Logic Systems Example Applications Uncertainty and Fuzziness The Future Industrial Control Control of marine diesel engines • Typically three engines • Two drive props and generators • One solely for power generation From Lynch, C. et al, Using Uncertainty Bounds in the Design of an Embedded Real-Time Type-2 Neuro-Fuzzy Speed Controller for Marine Diesel Engines, FUZZ-IEEE 2006. Fuzzy Logic in the Real World Simon Coupland
- 29. Introduction Motivation What is Fuzzy Logic? Fuzzy Logic Systems Example Applications Uncertainty and Fuzziness The Future Industrial Control Control of marine diesel engines • VK25 is the current control system • T2NFC and RT2NFC are both fuzzy • Type-2 fuzzy sets • Different defuzziﬁcation techniques From Lynch, C. et al, Using Uncertainty Bounds in the Design of an Embedded Real-Time Type-2 Neuro-Fuzzy Speed Controller for Marine Diesel Engines, FUZZ-IEEE 2006. Fuzzy Logic in the Real World Simon Coupland
- 30. Introduction Motivation What is Fuzzy Logic? Fuzzy Logic Systems Example Applications Uncertainty and Fuzziness The Future Human Decision Making Volkswagen Direct-Shift Gearbox • Automatic gear selection behaviour • Gear choice can be inferred from sensor readings • Need to account for human factor Fuzzy Logic in the Real World Simon Coupland
- 31. Introduction Motivation What is Fuzzy Logic? Fuzzy Logic Systems Example Applications Uncertainty and Fuzziness The Future Human Decision Making Volkswagen Direct-Shift Gearbox • Two fuzzy systems are used: Vehicle speed Change in speed Fuzzy classiﬁer Slope resistance • Infer driving style Accelerator • Select gear Driving style • Gear selection based on: • Sensor data Control • Fuzzy judgement of current system Gear Car driving style Throttle Vehicle speed Engine speed Engine load Fuzzy Logic in the Real World Simon Coupland
- 32. Introduction Motivation What is Fuzzy Logic? Fuzzy Logic Systems Example Applications Uncertainty and Fuzziness The Future Human Decision Making Volkswagen Direct-Shift Gearbox • Adaptive fuzzy systems • Gradually adjusts the fuzzy sets • Tailored to suit your personal driving style Fuzzy Logic in the Real World Simon Coupland
- 33. Introduction Motivation What is Fuzzy Logic? Fuzzy Logic Systems Example Applications Uncertainty and Fuzziness The Future Image Processing Segmentation of Histopathology Images • Identify regions of the image as: • Nuclei • Lumen • Cytoplasm • Classify tissue Fuzzy Logic in the Real World Simon Coupland
- 34. Introduction Motivation What is Fuzzy Logic? Fuzzy Logic Systems Example Applications Uncertainty and Fuzziness The Future Image Processing Segmentation of Histopathology Images 1 Set the number of classes n (3) 2 Initialise a fuzzy description of each 3 Find the set of fuzzy descriptions of n with the lowest overlap Fuzzy Logic in the Real World Simon Coupland
- 35. Introduction Motivation What is Fuzzy Logic? Fuzzy Logic Systems Example Applications Uncertainty and Fuzziness The Future Image Processing Segmentation of Histopathology Images Nuclei in red and black, lumen in green and cytoplasm in yellow From Adel Haﬁane et al, Lecture Notes in Computer Science. 5259: 903914 (2008) Fuzzy Logic in the Real World Simon Coupland
- 36. Introduction Motivation What is Fuzzy Logic? Fuzzy Logic Systems Example Applications Uncertainty and Fuzziness The Future Image Processing Segmentation of Histopathology Images Nuclei in red and black, lumen in green and cytoplasm in yellow From Adel Haﬁane et al, Lecture Notes in Computer Science. 5259: 903914 (2008) Fuzzy Logic in the Real World Simon Coupland
- 37. Introduction Motivation What is Fuzzy Logic? Fuzzy Logic Systems Example Applications Uncertainty and Fuzziness The Future Application of Fuzzy Methods • Useful wherever vagueness of uncertainty exists • Relatively simple paradigm • Not a panacea - good science is still the key • Areas not mentioned: • White goods - fridges, freezers, washing machines • Camera anti-shake - Minolta and Canon • Scheduling - Seattle trafﬁc light control system Fuzzy Logic in the Real World Simon Coupland
- 38. Introduction Motivation What is Fuzzy Logic? Fuzzy Logic Systems Example Applications Uncertainty and Fuzziness The Future Uncertainty and Vagueness The Trouble with (Type-1) Fuzzy Sets Fuzzy sets and systems: About 0.5 • Vagueness µ 1 • Partial truth • Degrees of set membership But what about uncertainty? • Alan is 0.5 Tall • 0.5 is crisp! • Alan is about 0.5 Tall 0 0 0.25 0.5 0.75 1 Fuzzy Logic in the Real World Simon Coupland
- 39. Introduction Motivation What is Fuzzy Logic? Fuzzy Logic Systems Example Applications Uncertainty and Fuzziness The Future Uncertainty and Vagueness The Trouble with (Type-1) Fuzzy Sets Type-2 Fuzzy Sets: • Set membership measured as a fuzzy number • Alan is about 0.5 Tall • Where about 0.5 is a fuzzy set (number) • DMU lead the world in this ﬁeld • Example type-2 fuzzy set - run program Fuzzy Logic in the Real World Simon Coupland
- 40. Introduction Motivation What is Fuzzy Logic? Fuzzy Logic Systems Example Applications Uncertainty and Fuzziness The Future The Future of Fuzzy Systems A Personal View • Uncertainly management is key • Type-2 fuzzy systems have a big role to play • Other extensions will also be important • Computing with Words has potential • Worth measured by applications Fuzzy Logic in the Real World Simon Coupland
- 41. Introduction Motivation What is Fuzzy Logic? Fuzzy Logic Systems Example Applications Uncertainty and Fuzziness The Future Summary • Fuzzy sets are sets with soft boundaries • Fuzzy logic performs inference on fuzzy sets • Applied in a variety of areas • Future developments are likely to be concerned with uncertainty models Fuzzy Logic in the Real World Simon Coupland
- 42. Introduction Motivation What is Fuzzy Logic? Fuzzy Logic Systems Example Applications Uncertainty and Fuzziness The Future Fuzzy Logic in the Real World Simon Coupland

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