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# Fuzzy logic

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### Fuzzy logic

1. 1. Presented by: Bhanu Fix Poudyal 066BEL305 Department Of Electrical Engineering Pulchowk Campus2/20/2012 Institute Of Engineering, Pulchowk Campus
2. 2. Agenda  General Definition  Applications  Formal Definitions  Operations  Rules  Fuzzy Air Conditioner  Controller Structure2/20/2012 Institute Of Engineering, Pulchowk Campus
3. 3. Fuzzy LogicDefinition  Experts rely on common sense when they solve problems.  How can we represent expert knowledge that uses vague and ambiguous terms in a computer?  Fuzzy logic is not logic that is fuzzy, but logic that is used to describe fuzziness. Fuzzy logic is the theory of fuzzy sets, sets that calibrate vagueness.  Fuzzy logic is based on the idea that all things admit of degrees. Temperature, height, speed, distance, beauty – all come on a sliding scale.  The motor is running really hot.  Tom is a very tall guy.2/20/2012 Institute Of Engineering, Pulchowk Campus
4. 4. Fuzzy LogicDefinition  Many decision-making and problem-solving tasks are too complex to be understood quantitatively, however, people succeed by using knowledge that is imprecise rather than precise.  Fuzzy set theory resembles human reasoning in its use of approximate information and uncertainty to generate decisions.  It was specifically designed to mathematically represent uncertainty and vagueness and provide formalized tools for dealing with the imprecision intrinsic to many engineering and decision problems in a more natural way.  Boolean logic uses sharp distinctions. It forces us to draw lines between members of a class and non-members. For instance, we may say, Tom is tall because his height is 181 cm. If we drew a line at 180 cm, we would find that David, who is 179 cm, is small.  Is David really a small man or we have just drawn an arbitrary line in the sand?2/20/2012 Institute Of Engineering, Pulchowk Campus
5. 5. Fuzzy LogicBit of History  Fuzzy, or multi-valued logic, was introduced in the 1930s by Jan Lukasiewicz, a Polish philosopher. While classical logic operates with only two values 1 (true) and 0 (false), Lukasiewicz introduced logic that extended the range of truth values to all real numbers in the interval between 0 and 1.  For example, the possibility that a man 181 cm tall is really tall might be set to a value of 0.86. It is likely that the man is tall. This work led to an inexact reasoning technique often called possibility theory.  In 1965 Lotfi Zadeh, published his famous paper “Fuzzy sets”. Zadeh extended the work on possibility theory into a formal system of mathematical logic, and introduced a new concept for applying natural language terms. This new logic for representing and manipulating fuzzy terms was called fuzzy logic.2/20/2012 Institute Of Engineering, Pulchowk Campus
6. 6. Fuzzy Logic Why Fuzzy Logic? Why fuzzy? 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. Why logic? Fuzziness rests on fuzzy set theory, and fuzzy logic is just a small part of that theory. The term fuzzy logic is used in two senses:  Narrow sense: Fuzzy logic is a branch of fuzzy set theory, which deals (as logical systems do) with the representation and inference from knowledge. Fuzzy logic, unlike other logical systems, deals with imprecise or uncertain knowledge. In this narrow, and perhaps correct sense, fuzzy logic is just one of the branches of fuzzy set theory.  Broad Sense: fuzzy logic synonymously with fuzzy set theory 2/20/2012 Institute Of Engineering, Pulchowk Campus
7. 7. Applications  ABS Brakes  Expert Systems  Control Units  Bullet train between Tokyo and Osaka  Video Cameras  Automatic Transmissions  Washing Machines2/20/2012 Institute Of Engineering, Pulchowk Campus
8. 8. Formal Definitions Definition 1: Let X be some set of objects, with elements noted as x. X = {x}. Definition 2: A fuzzy set A in X is characterized by a membership function mA(x) which maps each point in X onto the real interval [0.0, 1.0]. As mA(x) approaches 1.0, the "grade of membership" of x in A increases. Definition 3: A is EMPTY iff for all x, mA(x) = 0.0. Definition 4: A = B iff for all x: mA(x) = mB(x) [or, mA = mB]. Definition 5: mA = 1 - mA. Definition 6: A is CONTAINED in B iff mA mB. Definition 7: C = A UNION B, where: mC(x) = MAX(mA(x), mB(x)). Definition 8: C = A INTERSECTION B where: mC(x) = MIN(mA(x), mB(x)). 2/20/2012 Institute Of Engineering, Pulchowk Campus
9. 9. Fuzzy Logic Operators  Fuzzy Logic:  NOT (A) = 1 - A  A AND B = min( A, B)  A OR B = max( A, B)2/20/2012 Institute Of Engineering, Pulchowk Campus
10. 10. Operations A B A B A B A2/20/2012 Institute Of Engineering, Pulchowk Campus
11. 11. Fuzzy Logic NOT2/20/2012 Institute Of Engineering, Pulchowk Campus
12. 12. Fuzzy Logic AND2/20/2012 Institute Of Engineering, Pulchowk Campus
13. 13. Fuzzy Logic OR2/20/2012 Institute Of Engineering, Pulchowk Campus
14. 14. Fuzzy Controllers  Used to control a physical system2/20/2012 Institute Of Engineering, Pulchowk Campus
15. 15. Controller Structure  Fuzzification  Scales and maps input variables to fuzzy sets  Inference Mechanism  Approximate reasoning  Deduces the control action  Defuzzification  Convert fuzzy output values to control signals2/20/2012 Institute Of Engineering, Pulchowk Campus
16. 16. Structure of a Fuzzy Controller2/20/2012 Institute Of Engineering, Pulchowk Campus
17. 17. Fuzzification  Conversion of real input to fuzzy set values  e.g. Medium ( x ) = {  0 if x >= 1.90 or x < 1.70,  (1.90 - x)/0.1 if x >= 1.80 and x < 1.90,  (x- 1.70)/0.1 if x >= 1.70 and x < 1.80 }2/20/2012 Institute Of Engineering, Pulchowk Campus
18. 18. Inference Engine  Fuzzy rules  based on fuzzy premises and fuzzy consequences  e.g.  If height is Short and weight is Light then feet are Small  Short( height) AND Light(weight) => Small(feet)2/20/2012 Institute Of Engineering, Pulchowk Campus
19. 19. Fuzzification & Inference Example  If height is 1.7m and weight is 55kg  what is the value of Size(feet)2/20/2012 Institute Of Engineering, Pulchowk Campus
20. 20. Defuzzification  Rule base has many rules  so some of the output fuzzy sets will have membership value > 0  Defuzzify to get a real value from the fuzzy outputs  One approach is to use a centre of gravity method2/20/2012 Institute Of Engineering, Pulchowk Campus
21. 21. Defuzzification Example  Imagine we have output fuzzy set values  Small membership value = 0.5  Medium membership value = 0.25  Large membership value = 0.0  What is the deffuzzified value2/20/2012 Institute Of Engineering, Pulchowk Campus
22. 22. Fuzzy Control Example2/20/2012 Institute Of Engineering, Pulchowk Campus
23. 23. Rule BaseAir Temperature Fan Speed Set cold {50, 0, 0} • Set stop {0, 0, 0} Set cool {65, 55, 45} • Set slow {50, 30, 10} Set just right {70, 65, 60} • Set medium {60, 50, 40} Set warm {85, 75, 65} • Set fast {90, 70, 50} Set hot { , 90, 80} • Set blast { , 100, 80}2/20/2012 Institute Of Engineering, Pulchowk Campus
24. 24. default: Membership function isThe truth of anystatement is a a curve of the degree of truth of a given Rulesmatter of degree input value Air Conditioning Controller Example:  IF Cold then Stop  If Cool then Slow  If OK then Medium  If Warm then Fast  IF Hot then Blast 2/20/2012 Institute Of Engineering, Pulchowk Campus
25. 25. Fuzzy Air Conditioner 0100 s t If Hot90 Bla then Blast80 Fa st If Warm then70 Fast60 Med If Just Right ium then50 Medium40 IF Cool Sl ow then30 Slow if Cold20 then Stop10 St o p0 1 m t ol Ho ar Co W Co Rig t ht Jus ld 0 2/20/2012 Institute Of Engineering, Pulchowk Campus 45 50 55 60 65 70 75 80 85 90
26. 26. Mapping Inputs to Outputs 1 0 100 s t 90 Bla t 80 Fa st 70 60 Med iu m 50 40 Sl ow 30 20 10 St op 0 1 m t ol Ho ar Co W Co Rig t ht Jus ld 0 45 50 55 60 65 70 75 80 85 902/20/2012 Institute Of Engineering, Pulchowk Campus