13. Introduction to Fuzzy Logic and Fuzzy Systems

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2 13. Introduction to and Fuzzy Systems Petr Pošík Czech Technical University in Prague Faculty of Electrical Engineering Department of Cybernetics P. Pošík c 2007 Soft Computing 1 / 26

3 Contents P. Pošík c 2007 Soft Computing 2 / 26

4 Quotations, sayings, etc. What can be accomplished by fuzzy systems? What is (FL)? Soft computing Successful applications FL: Pros and cons P. Pošík c 2007 Soft Computing 3 / 26

5 Quotations, sayings, etc. Quotations, sayings, etc. What can be accomplished by fuzzy systems? What is (FL)? Soft computing Successful applications FL: Pros and cons How important is to know the right and precise answer, when approximately correct estimate is sufficient? Precision is not truth. Henri Matisse Sometimes the more measurable drives out the most important. René Dubos So far as the laws of mathematics refer to reality, they are not certain. And so far as they are certain, they do not refer to reality. Albert Einstein As complexity rises, precise statements lose meaning and meaningful statements lose precision. Lotfi Zadeh Don t lose sight of the forest for the trees. Don t be penny wise and pound foolish. P. Pošík c 2007 Soft Computing 4 / 26

6 What can be accomplished by fuzzy systems? Quotations, sayings, etc. What can be accomplished by fuzzy systems? What is (FL)? Soft computing Successful applications FL: Pros and cons All that can be accomplished by assigning the right outputs to certain inputs. You ll tell me how good the service in the restaurant was, I ll tell you how big tip you should give them. You ll tell me how warm the water should be, I ll set the water cocks for you. You ll tell me how far the object being photographed is, I ll focus the objective for you. You ll tell me how fast the car goes and what the engine load is, and I ll shift into the right gear for you. There are many methods that allow us to assign outputs to inputs. Why is fuzzy better? Since fuzzy allows us to do it naturally, by using (almost) natural language! P. Pošík c 2007 Soft Computing 5 / 26

7 What is (FL)? Quotations, sayings, etc. What can be accomplished by fuzzy systems? What is (FL)? Soft computing Successful applications FL: Pros and cons Methodology for computing with words words are less precise then numbers, but are more familiar to people tolerant to imprecision reduces the solution costs Linguistic variable concept: values are not numbers, but words Logic system extending the logic with multiple values (not just true and false), Synonym for the theory of fuzzy sets, i.e. theory related to the classes of objects with non-sharp bounds; the class membership is not crisp (yes/no), but is rather a question of degree with which a particular object belongs to a particular class Fuzzy if-then rules (or simply fuzzy rules) rule-based systems a long history in AI, but they miss the methodology for working with fuzzy antecedents and consequents Fuzzy-rule calculus forms a basis for the fuzzy dependency and command language (FDCL) P. Pošík c 2007 Soft Computing 6 / 26

8 Soft computing Quotations, sayings, etc. What can be accomplished by fuzzy systems? What is (FL)? Soft computing Successful applications FL: Pros and cons Trend: use FL in combination with neural networks (NN) and evolutionary algorithms (EA). Soft computing Main components: FL, NN, GA, probability Soft vs. hard computing: soft computing contains models and methods that are tolerant to imprecision and vagueness Combination of FL and NN: neuro-fuzzy system important role when inducing the rules from a set of observations Dr. Roger Jang: (Adaptive Neuro-Fuzzy Inference System) efficient method for fuzzy rule induction P. Pošík c 2007 Soft Computing 7 / 26

9 Successful applications Quotations, sayings, etc. What can be accomplished by fuzzy systems? What is (FL)? Soft computing Successful applications FL: Pros and cons Fuzzy control in Japanese undergroud automatic train control increased precision of stopping, smoother breaking, lower energy consumption Camera with automatic setting of the focus point (Minolta) ABS, engine control, air condition (Honda, Nissan, Sabaru) Control of elevators (Mitsubishi) Error correction in the melting devices (Omron) 3.5" floppy drives (30% improvement in time of head positioning) Palmtop Kanji, hand-written character recognition Speech recognition Fuzzy SQL (Omron) Criminal suspect identification supplemental system (tall, not to heavy, more or less old,... ) Wash machines, refridgerators, dryers settings for the cooling power, or water amount, in relation with the amount of food or clothes P. Pošík c 2007 Soft Computing 8 / 26

10 FL: Pros and cons Quotations, sayings, etc. What can be accomplished by fuzzy systems? What is (FL)? Soft computing Successful applications FL: Pros and cons Advantages: easily understandable flexible imprecision tolerant ability to model nonlinear dependencies easy representation of expert knowledge easily coupled with usual methods of automatic control based on native language Do not use when the problem cannot be formulated as input output mapping simpler solution exists FL is common sense codification, use it and you ll get the right 1 solution. 1 The word right is a fuzzy set here. P. Pošík c 2007 Soft Computing 9 / 26

11 Crisp vs. Fuzzy Set Membership function Types of membership functions Logical operations Additional logical operators Fuzzy rules Fuzzy rule processing P. Pošík c 2007 Soft Computing 10 / 26

12 Crisp vs. Fuzzy Set Crisp vs. Fuzzy Set Membership function Types of membership functions Logical operations Additional logical operators Fuzzy rules Fuzzy rule processing Crisp set (usual set) set of weekdays: D = {Mo, Tu, We, Th, Fr, Sa, Su} D {Freedom, Butter, Dog, TV-set,...} = Aristotle: x A x / A Fuzzy set no clear boundaries set of weekend days: V = {So, Ne}... is it all? P. Pošík c 2007 Soft Computing 11 / 26

13 Crisp vs. Fuzzy Set Crisp vs. Fuzzy Set Membership function Types of membership functions Logical operations Additional logical operators Fuzzy rules Fuzzy rule processing Crisp set (usual set) set of weekdays: D = {Mo, Tu, We, Th, Fr, Sa, Su} D {Freedom, Butter, Dog, TV-set,...} = Aristotle: x A x / A Fuzzy set no clear boundaries set of weekend days: V = {So, Ne}... is it all? For me, friday is a weekend day as well, since I can go out in the evening. Friday is and is not a weekend day in the same time, it belongs to both sets. P. Pošík c 2007 Soft Computing 11 / 26

14 Crisp vs. Fuzzy Set Crisp vs. Fuzzy Set Membership function Types of membership functions Logical operations Additional logical operators Fuzzy rules Fuzzy rule processing Crisp set (usual set) set of weekdays: D = {Mo, Tu, We, Th, Fr, Sa, Su} D {Freedom, Butter, Dog, TV-set,...} = Aristotle: x A x / A Fuzzy set no clear boundaries set of weekend days: V = {So, Ne}... is it all? For me, friday is a weekend day as well, since I can go out in the evening. Friday is and is not a weekend day in the same time, it belongs to both sets. Wikipedia: The weekend is a part of the week lasting one or two days in which most paid workers do not work. This is a time for leisure and recreation, and/or for religious activities. In historically Christian countries the weekend typically covers Saturday and Sunday, while in Muslim countries it is Friday and Saturday,... Sometimes, the weekend is thought of as including the evening of the preceding work day. two-valued logic and its crisp boundaries often are not useful FL is a tool allowing us to answer a simple yes/no question in an ambiguous way, something that people do all the time. P. Pošík c 2007 Soft Computing 11 / 26

15 Membership function Crisp vs. Fuzzy Set Membership function Types of membership functions Logical operations Additional logical operators Fuzzy rules Fuzzy rule processing How to describe the weekend days? Is Saturday a weekend day? 1 (Yes, true) Is Tuesday a weekend day? 0 (No, false) Is Friday a weekend day? 0.8 (To a great extent yes, but not completely) Is Sunday a weekend day? 0.95 (Basically yes, bot not as much as Saturday) Crisp set: A = {x x > 6} Fuzzy set: A = {x, µ A x X} Fuzzy sets describe vague concepts (warm weather, tall man, precise result) Membership functions expres partial membership (wheather is rather warm, the man is pretty tall,... ) P. Pošík c 2007 Soft Computing 12 / 26

16 Types of membership functions 1 Piecewise linear Gaussian Sigmoidal Polynomial trimf(x, [2 7 9]) gauss(x, [2 5]) sigmf(x, [2 4]) zmf(x, [3 7]) trapmf(x, [ ]) 1 gauss2mf(x, [ ]) 1 dsigmf(x, [ ]) 1 pimf(x, [ ]) gbellmf(x, [2 4 6]) 1 psigmf(x, [ ]) 1 smf(x, [1 8]) P. Pošík c 2007 Soft Computing 13 / 26

17 Logical operations In Boolean logic: Crisp vs. Fuzzy Set Membership function Types of membership functions Logical operations Additional logical operators Fuzzy rules Fuzzy rule processing AND A B A B OR A B A B NOT A A P. Pošík c 2007 Soft Computing 14 / 26

18 Logical operations Crisp vs. Fuzzy Set Membership function Types of membership functions Logical operations Additional logical operators Fuzzy rules Fuzzy rule processing In Boolean logic: AND A B A B OR A B A B NOT A A Possible definition in fuzzy logic: AND A B min(a, B) OR A B max(a, B) A NOT 1 A P. Pošík c 2007 Soft Computing 14 / 26

19 Additional logical operators Crisp vs. Fuzzy Set Membership function Types of membership functions Logical operations Additional logical operators Fuzzy rules Fuzzy rule processing There are more ways to construct the fuzzy equivalents of AND, OR and NOT. Operators on previous slide are called classical operators. Intersection of two fuzzy sets can be in general defined using so called T-norm (triangular norm): µ A B (x) = T(µ A (x), µ B (x)) e.g. µ A B (x) = µ A (x) µ B (x) (1) boundary conditions: T(0, 0) = 0, T(a, 1) = T(1, a) = a monotonicity: T(a, b) T(c, d), if a c a b d commutativity: T(a, b) = T(b, a) associativity: T(a, T(b, c)) = T(T(a, b), c) Union of two fuzzy sets can be in general defined using so called S-norm (T-conorm): µ A B (x) = S(µ A (x), µ B (x)) e.g. µ A B (x) = µ A (x)+µ B (x) µ A (x) µ B (x) (2) boundary conditions: S(1, 1) = 1, S(a, 0) = S(0, a) = a monotonicity: S(a, b) S(c, d), if a c a b d commutativity: S(a, b) = S(b, a) associativity: S(a, S(b, c)) = S(S(a, b), c) P. Pošík c 2007 Soft Computing 15 / 26

20 Fuzzy rules Crisp vs. Fuzzy Set Membership function Types of membership functions Logical operations Additional logical operators Fuzzy rules Fuzzy rule processing If x is A, then y is B If service is Good, then tip is average The word is has different meanings in antecedent and consequent: If x == A, then y = B Fuzzy rules interpretation: Antecendent evaluation (Fuzzification of inputs, application of fuzzy operators) Application of the result to the consequent (implication) Když částečně platí antecedent, pak částečně platí konsekvent (0.5p 0.5q) Antecedent can have more parts: If sky is Cloudy and wind is Strong and pressure Drops, then... All the parts are evaluated resulting in a single number Consequent can have more parts as well: If temperature is Low, then warm water cock is Open and cold water cock is Closed. All parts of the consequent are influenced in the same way P. Pošík c 2007 Soft Computing 16 / 26

21 Fuzzy rule processing Crisp vs. Fuzzy Set Membership function Types of membership functions Logical operations Additional logical operators Fuzzy rules Fuzzy rule processing P. Pošík c 2007 Soft Computing 17 / 26

22 (FIS) Example: Determining the tip to the waiter Example: In Detail Sugeno Inference Example: Sugeno FIS Mamdani vs. Sugeno P. Pošík c 2007 Soft Computing 18 / 26

23 (FIS) (FIS) Example: Determining the tip to the waiter Example: In Detail Sugeno Inference Example: Sugeno FIS Mamdani vs. Sugeno Interesting things happen when we have more rules with outputs that influence each other 2 bysic types of FIS: Mamdani and Sugeno (or Takagi-Sugeno) they differ in the way used to determine outputs Mamdani inference: In 1975, attempt to control a boiler and a steam engine using FIS Linguistic control rules obtained from operators and encoded in the form of fuzzy rules The output of the fuzzy rules are fuzzy sets After their aggregation, we get a fuzzy set for each output variable Defuzzification is used to get one crisp value for each variable (we usualy search for the centroid of the output membership functions) Often, the output fuzzy sets are first transformed into crisp values, the final output is then their weighted average. P. Pošík c 2007 Soft Computing 19 / 26

24 Example: Determining the tip to the waiter (FIS) Example: Determining the tip to the waiter Example: In Detail Sugeno Inference Example: Sugeno FIS Mamdani vs. Sugeno If service is Poor or food is Rancid, then tip is Cheap If service is Good, then tip is Average If service is Excellent or food is Delicious, then tip is Generous P. Pošík c 2007 Soft Computing 20 / 26

25 Example: In Detail (FIS) Example: Determining the tip to the waiter Example: In Detail Sugeno Inference Example: Sugeno FIS Mamdani vs. Sugeno P. Pošík c 2007 Soft Computing 21 / 26

26 Sugeno Inference (FIS) Example: Determining the tip to the waiter Example: In Detail Sugeno Inference Example: Sugeno FIS Mamdani vs. Sugeno Also called Takagi-Sugeno, or Takagi-Sugeno-Kang In many respects similar to Mamdani Fuzzification and application of fuzzy operators are the same The output of a rule is not a fuzzy set, but a crisp value given by a constant or a linear function If service is Good and food is Excellent, then tip= a service+b food+c Each such rule produces the weight w of the rule (evaluated antecedent) the output value z (evaluated consequent) P. Pošík c 2007 Soft Computing 22 / 26

27 Example: Sugeno FIS (FIS) Example: Determining the tip to the waiter Example: In Detail Sugeno Inference Example: Sugeno FIS Mamdani vs. Sugeno The output is then given as z c = N i=1 w iz i N i=1 w i (3) P. Pošík c 2007 Soft Computing 23 / 26

28 Mamdani vs. Sugeno (FIS) Example: Determining the tip to the waiter Example: In Detail Sugeno Inference Example: Sugeno FIS Mamdani vs. Sugeno Advantages of Mamdani method Intuitive Broadly accepted Suitable for human inputs Advantages Sugeno method Computationaly more efficient Suitable for use with linear control techniques (e.g. PID control) Suitable for adaptive control Continuous output surface Suitable for mathematical analysis P. Pošík c 2007 Soft Computing 24 / 26

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30 Now we know how FIS works, i.e. how it propagates the inputs to the outputs. How can we instantiate FIS? Manually, using domain knowledge Automatically, e.g. using, based on observed input-output pairs, Adaptive Neuro-Fuzzy Inference System structure of Sugeno FIS is in fact a neural network with a firm structure parameters can be learned from the data and then used to interpret the meaning of the data structure Limitations (according to MATLAB): FIS must be Sugeno system of 0th or 1st order All output functions must be of the same type FIS must have only one output given by weighted average The number of output functions must be equal to the number of rules All rules must have unit weight P. Pošík c 2007 Soft Computing 26 / 26