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Fuzzy Logic

Fuzzy Logic 11
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Published Date:14-07-2017
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Fuzzy Logic Alexander Mathews Yuki Osada Perry Brown 1 Outline  Fundamental fuzzy concepts  Fuzzy propositional and predicate logic  Fuzzification  Defuzzification  Fuzzy control systems  Types of fuzzy algorithms  Applications of fuzzy logic 2 Introduction  Fuzzy concepts first introduced by Zadeh in the 1960s and 70s  Traditional computational logic and set theory is all about  true or false  zero or one  in or out (in terms of set membership)  black or white (no grey)  Not the case with fuzzy logic and fuzzy sets 3 Basic Concepts  Approximation (“granulation”)  A colour can be described precisely using RGB values, or it can be approximately described as “red”, “blue”, etc.  Degree (“graduation”)  Two different colours may both be described as “red”, but one is considered to be more red than the other  Fuzzy logic attempts to reflect the human way of thinking 4 Terminology  Fuzzy set  A set X in which each element y has a grade of membership µ (y) in the range 0 X to 1, i.e. set membership may be partial e.g. if cold is a fuzzy set, exact temperature values might be mapped to the fuzzy set as follows:  15 degrees → 0.2 (slightly cold)  10 degrees → 0.5 (quite cold)  0 degrees → 1 (totally cold)  Fuzzy relation  Relationships can also be expressed on a scale of 0 to 1 e.g. degree of resemblance between two people 5 Terminology (cont’d)  Fuzzy variable  Variable with (labels of) fuzzy sets as its values  Linguistic variable  Fuzzy variable with values that are words or sentences in a language e.g. variable colour with values red, blue, yellow, green…  Linguistic hedge  Term used as a modifier for basic terms in linguistic values e.g. words such as very, a bit, rather, somewhat, etc. 6 Formal Fuzzy Logic  Fuzzy logic can be seen as an extension of ordinary logic, where the main difference is that we use fuzzy sets for the membership of a variable  We can have fuzzy propositional logic and fuzzy predicate logic  Fuzzy logic can have many advantages over ordinary logic in areas like artificial intelligence where a simple true/false statement is insufficient 7 Traditional Logic  Propositional logic:  Propositional logic is a formal system that uses true statements to form or prove other true statements  There are two types of sentences: simple sentences and compound sentences  Simple sentences are propositional constants; statements that are either true or false  Compound sentences are formed from simpler sentences by using negations ¬, conjunctions ∧, disjunctions ∨, implications ⇒, reductions ⇐, and equivalences ⇔  Predicate logic:  Onto propositional logic, this adds the ability to quantify variables, so we can manipulate statements about all or some things  Two common quantifiers are the existential ∃ and universal ∀ quantifiers 8 Formal Fuzzy Logic  Fuzzy Propositional Logic  Like ordinary propositional logic, we introduce propositional variables, truth- functional connectives, and a propositional constant 0  Some of these include:  Monoidal t-norm-based propositional fuzzy logic  Basic propositional fuzzy logic  Łukasiewicz fuzzy logic  Gödel fuzzy logic  Product fuzzy logic  Rational Pavelka logic  Fuzzy Predicate Logic  These extend fuzzy propositional logic by adding universal and existential quantifiers in a manner similar to the way that predicate logic is created from propositional logic 9 Simple Fuzzy Operators  As described by Zadeh (1973)…  NOT X = 1 - µ (y) X  e.g. 0.8 cold → (1 – 0.8) = 0.2 NOT cold  X OR Y (union) = max(µ (y), µ (y)) X Y  e.g. 0.8 cold, 0.5 rainy → 0.8 cold OR rainy  X AND Y (intersection) = min(µ (y), µ (y)) X Y  e.g. 0.9 hot, 0.7 humid → 0.7 hot AND humid 10 Alternative Interpretations of AND and OR  Zadeh’s definition of AND used the Gödel t-norm, but other definitions are possible using different t-norms  Common examples:  Product t-norm: µ (y) µ (y) X Y e.g. 0.9 hot, 0.7 humid → 0.63 hot AND humid  Lukasiewicz t-norm: max(µ (y) + µ (y) - 1, 0) X Y e.g. 0.9 hot, 0.7 humid → 0.6 hot AND humid  Similar possibilities for OR using corresponding t-conorms:  Product t-conorm: µ (y) + µ (y) - µ (y) µ (y) X Y X Y e.g. 0.8 cold, 0.5 rainy → 0.9 cold OR rainy  Lukasiewicz t-conorm: min(µ (y) + µ (y), 1) X Y e.g. 0.8 cold, 0.5 rainy → 1 cold OR rainy 11 Fuzzy System Overview  When making inferences, we want to clump the continuous numerical values into sets  Unlike Boolean logic, fuzzy logic uses fuzzy sets rather than crisp sets to determine the membership of a variable  This allows values to have a degree of membership with a set, which denotes the extent to which a proposition is true  The membership function may be triangular, trapezoidal, Gaussian or any other shape 12 Fuzzification  To apply fuzzy inference, we need our input to be in linguistic values  These linguistic values are represented by the degree of membership in the fuzzy sets  The process of translating the measured numerical values into fuzzy linguistic values is called fuzzification  In other words, fuzzification is where membership functions are applied, and the degree of membership is determined 13 Membership Functions  There are largely three types of fuzzifiers:  singleton fuzzifier,  Gaussian fuzzifier, and  trapezoidal or triangular fuzzifier Gaussian Trapezoidal 14 Defuzzification  Defuzzification is the process of producing a quantifiable result in fuzzy logic  The fuzzy inference will output a fuzzy result, described in terms of degrees of membership of the fuzzy sets  Defuzzification interprets the membership degrees in the fuzzy sets into a specific action or real-value 15 Methods of Defuzzification  There are many methods for defuzzification  One of the more common types of defuzzification technique is the maximum defuzzification techniques. These select the output with the highest membership function  They include:  First of maximum  Middle of maximum  Last of maximum  Mean of maxima  Random choice of maximum 16 Methods of Defuzzification  Given the fuzzy output:  The first of maximum, middle of maximum, and last of maximum would be -2, -5, and -8 respectively as seen in the diagram  The mean would give the same result as middle unless there is more than one plateau with the maximum value Image Source: 17 logic/examples.html?file=/products/demos/shipping/fuzzy/defuzzdm.html Methods of Defuzzification  Two other common methods are:  Centre of gravity:  Calculates the centre of gravity for the area under the curve  Bisector method:  Finds the value where the area on one side of that value is equal to the area on the other side Image Source: 18 logic/examples.html?file=/products/demos/shipping/fuzzy/defuzzdm.html Fuzzy Control Systems  A simple example application: an automated cooling fan that can adjust its speed according to the room temperature  Temperature values are mapped to the fuzzy sets cold, warm and hot based on the following mapping functions for each: Image: 19 Fuzzy Control Systems  The inference engine in a fuzzy system consists of linguistic rules  The linguistic rules consist of two parts:  an antecedent block (the conditions), which consists of the linguistic variables  a consequent block (the output)  Fuzzy algorithm  Algorithm that includes at least some fuzzy instructions, such as conditional or unconditional action statements  Fuzzy conditional statement (A → B)  Conditional statement in which A and/or B are fuzzy sets e.g. IF temperature is hot THEN fan speed is high  Defined in terms of a fuzzy relation between the respective “universes of discourse” of A and B (compositional rule of inference) e.g. relation between temperature groupings and fan speeds 20