Fuzzy - A Brief Introduction to Genetic Algorithms,...

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Unformatted text preview: A Brief Introduction to Genetic Algorithms, Artificial Neural Networks and Fuzzy Logic Networks Donato da Silva Filho Visiting Scholar Prof. Daniel Pete Loucks School of Civil and Environmental Engineering CEE 593 - Engineering Management Methods I CORNELL UNIVERSITY General Overview s The tutorial is divided in three main topics: c Genetic Algorithms. c Artificial Neural Networks. – Fuzzy Logic. s The Fuzzy Logic tutorial is organized as follows: – Brief theoretical background. – Simple numerical examples. – Engineering application. Examples s Fuzzy Logic: Dealing with imprecise information. c Robots and autonomous systems. c Control of automatic processes. c Stock markets. c Linguistic interpretation of complex systems. c Inference systems. History s Fuzzy Sets were introduced in 1965 by Prof. Lotfi Zadeh, from the University of California, Berkeley. The main aim of Fuzzy Sets is to deal with imprecise information. Let us see some examples. s s Example of imprecise information s Driver and the traffic light: – Start braking the car 70 feet far from the corner because the car has to stop before the crosswalk. – Start braking the car in a reasonable distance from the corner because the car has to stop before the crosswalk. Other examples s s s “Go to bed early!” “Do not watch too much TV!” “How do you want your beef prepared?” (well done → medium well → medium → medium rare → rare) s s s s “Sally is very tall.” “John is young.” “The temperature is high.” (In a car:) “Please, can you drive slower?” Illustration of the process There is a process to be modeled or to be controlled. 1 The process can be described by “imprecise” (fuzzy) statements. 2 We can develop a fuzzy controller for the process (i.e, a fuzzy model). 3 Needs for a fuzzy model First: we need to know how to represent imprecise pieces of information. s s Second: we need to know how to manipulate the representation in order to create models and control processes. Representation Set of numbers “close” to seven. s Conventional Set Theory: Fuzzy Set Theory: H 6 1 7 8 H m 0 6 7 8 Fuzzy x Crisp representation C r is p Set U n iv e rse o f D is c o u r s e F u zzy Set U n iv e rse o f D is c o u r s e Operations with Fuzzy Sets 1. 2. 3. 4. 5. Equality: A = B ⇔ mA (x) = mB (x). Pertinence: A B ⇔ mA (x) mB (x). Complementariety: mA (x) = 1 - mA (x). Intersection: mA B(x) = min { mA (x),mB (x)}. Union: mA B(x) = max { mA (x),mB (x)} ⊂≤ ∩ ∪ 111 mm AA m A > m B m A mA m m Ba x ( m , m m ) A mB A B 000 m B m in ( m A,m B) xxxx Meaning of the membership functions (1) m (x ) 1 6 /7 m m B A 5 /8 3 /7 0 3 6 7 11 x Linguistic variables s A Linguistic Variable is a variable characterized by words or by sentences expressed in natural language. – Example: age. M e m b e r s h ip D e g re e 1 very young young not young o ld v e r y o ld 0 21 32 56 x Approximate reasoning If (antecedent) than (consequent). s Example: s s s Rule: If (X is A) than (Y is B) Information: X is A’ Conclusion: Y is B’ Inference System (1) Rule 1: If x is close to zero, then y is close to zero. Rule 2: If x is close to one, then y is close to one. Rule 3: If x is close to two, then y is close to four. Rule 4: If x is close to three, then y is close to nine. 1. Linguistic Variable x: • Close to zero; • Close to one; • Close to two; • Close to three; 2. Linguistic Variable y: • Close to zero; • Close to one; • Close to four; • Close to nine; m (x ) m (x ) 1 1 0 x 2 3 Figure 1 - Terms of Variable x. 1 0 1 2 3 4 5 6 7 8 Figure 2 - Terms of Variable y. 9 x Inference System (2) •Aggregation: the aggregation operation is used to calculate the degree of fulfillment α the rule k. k of Given a value to x, the antecedent of a rule, rule 2 for instance, will generate a fuzzy membership value m1(x). For example, for x = 0.6, the antecedent of rule 1 is m1(x) = 0.4. If there are n antecedents, the degree of fulfillment of the of the k-rule will be given by: αmin(m1(x), m2(x), …, mN(x)) k= Activation: the • activation of a rule is the deduction of its conclusion, possibly reduced by its firing strength. Accumulation: all activated conclusions are accumulated, using the max operation in • this case. Tank example V a lv e T a rg e t le v e l (v a ria b le ) Three linguistic variables: level, level rate and the opening of the valve. Fuzzy Logic x Probability • You are lost in the middle of a desert and find the two bottles bellow. Which one would you drink? CC Soda AA S u lfu r ic A c id m L Pm ( LC P ) (=C 0) ,=9 01 , 9 1 P (PA ( A L P L) =P ) 0= 0 , 9 1 A Brief Introduction to Genetic Algorithms, Artificial Neural Networks and Fuzzy Logic Networks Donato da Silva Filho Visiting Scholar Prof. Daniel Pete Loucks School of Civil and Environmental Engineering CEE 593 - Engineering Management Methods I CORNELL UNIVERSITY ...
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This note was uploaded on 03/29/2009 for the course CEE 5930 taught by Professor Loucks during the Fall '00 term at Cornell University (Engineering School).

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