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Fuzzy Logic

Fuzzy Logic

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• 1. Fuzzy Logic Presented by: Mahesh Todkar
• 2. Content What is Fuzzy? Sets Theory What is Fuzzy Logic? Why use Fuzzy Logic? Theory of Fuzzy Sets Vocabulary Fuzzy if-then Rules Fuzzy Logic Operations Fuzzy Inference Systems (FIS) Fuzzy Inference Process References
• 3. What is Fuzzy? Fuzzy means not clear, distinct or precise; not crisp (well defined); blurred (with unclear outline).
• 4. Sets Theory Classical Set: An element either belongs or does not belong to a sets that have been defined. Fuzzy Set: An element belongs partially or gradually to the sets that have been defined.
• 5. What is Fuzzy Logic? It has two different meanings as, In narrow sense: Fuzzy logic is a logical system, which is an extension of multi-valued logic. In a wider sense: Fuzzy logic (FL) is almost synonymous with the theory of fuzzy sets, a theory which relates to classes of objects with unsharp boundaries in which membership is a matter of degree. Fuzzy logic (FL) should be interpreted in its wider sense
• 6. What is Fuzzy Logic? A way to represent variation or imprecision in logic A way to make use of natural language in logic Approximate reasoning Definition of Fuzzy Logic: A form of knowledge representation suitable for notions that cannot be defined precisely, but which depend upon their contexts. Superset of conventional (Boolean) logic that has been extended to handle the concept of partial truth - the truth values between "completely true & completely false".
• 7. Why use Fuzzy Logic? Conceptually easy to understand Flexible Tolerant of imprecise data FL can model nonlinear functions of arbitrary complexity FL can be built on top of the experience of experts FL can be blended with conventional control techniques FL is based on natural language
• 8. Theory of Fuzzy Sets Classical Set Fuzzy Set
• 9. Theory of Fuzzy Sets Theory which relates to classes of objects with unsharp boundaries in which membership is matter of degree Thus every problem can be presented in terms of Fuzzy Sets A set without crisp Fuzzy set describes vague concepts Fuzzy set admits the possibility of partial membership in it Degree of an object belongs to Fuzzy Set is denoted by membership value between 0 to 1 Membership Function (MF) associated with a given Fuzzy Set maps an input value to its appropriate membership value
• 10. Vocabulary Linguistic Variable: Variable whose values are words or sentences rather than numbers It represent qualities spanning a particular spectrum Example: Speed, Service, Tip, Temperature, etc. Linguistic Value or Term: Values or Terms used to describe Linguistic Variable Example: For Speed (Slowest, Slow, Fast, Fastest), For Service (Poor, Good, Excellent), For Temperature (Freezing, Cool, Warm, Hot), etc.
• 11. Vocabulary Universe of Discourse or Universe or Input Space (U): Set of all possible elements that can come into consideration, confer the set U in (1). It depends on context. Elements of a fuzzy set are taken from a Universe of Discourse. An application of the universe is to suppress faulty measurement data. Example: Set of x >> 1 could have as a universe of all real numbers, alternatively all positive integer.
• 12. Vocabulary Membership Function (MF) is a curve that defines how each point in the input space is mapped to a membership value between 0 and 1. It is denoted by µ. Membership value is also called as degree of membership or membership grade or degree of truth of proposal. Types of Membership Functions: Piece-wise linear functions Gaussian distribution function Sigmoid curve Quadratic and cubic polynomial curves Singleton Membership Function
• 13. Membership Functions
• 14. Syntax of Fuzzy Set A = {x, µA(x) | x X} Where, A – Fuzzy Set x – Elements of X X – Universe of Discourse µA(x) – Membership Function of x in A
• 15. Fuzzy if-then Rules Statements used to formulate the conditional statements that comprise fuzzy logic Example: if x is A then y is B where, A & B – Linguistic values x – Element of Fuzzy set X y – Element of Fuzzy set Y In above example, Antecedent (or Premise)– if part of rule (i.e. x is A) Consequent (or Conclusion) – then part of rule (i.e. y is B) Antecedent is interpretation & Consequent is assignment
• 16. Fuzzy if-then Rules Antecedent is combination of proposals by AND, OR, NOT operators Consequent is combination of proposals linked by AND operators. OR and NOT operators are not used in consequents as these are cases of uncertainty. Example: If it is early, then John can study. Universe: U = {4,8,12,16,20,24}; time of day Input Fuzzy set: early = {(4,0),(8,1),(12,0.9),(16,0.7),(20,0.5),(24,0.2)} Output Fuzzy set: can study=singleton Fuzzy set (assume) so study =1 i.e. at 20 (8 pm), early (20) = 0.5
• 17. Fuzzy if-then Rules Interpreting if-then rule is a three–part process 1) Fuzzify Input: Resolve all fuzzy statements in the antecedent to a degree of membership between 0 and 1. 2) Apply fuzzy operator to multiple part antecedents: If there are multiple parts to the antecedent, apply fuzzy logic operators and resolve the antecedent to a single number between 0 and 1. 3) Apply implication method: The output fuzzy sets for each rule are aggregated into a single output fuzzy set. Then the resulting output fuzzy set is defuzzified, or resolved to a single number.
• 18. Fuzzy if-then Rules Interpreting if-then rule is a three–part process:
• 19. Fuzzy Logic Operations Fuzzy Logic Operators are used to write logic combinations between fuzzy notions (i.e. to perform computations on degree of membership) Zadeh operators 1) Intersection: The logic operator corresponding to the intersection of sets is AND. (A AND B) = MIN( (A), (B)) 2) Union: The logic operator corresponding to the union of sets is OR. (A OR B) = MAX( (A), (B)) 3) Negation: The logic operator corresponding to the complement of a set is the negation. (NOT A) =1- (A)
• 20. Fuzzy Logic Operations
• 21. Fuzzy Inference Systems (FIS) Fuzzy Inference is the process of formulating the mapping from a given input to an output using fuzzy logic. Process of fuzzy inference involves Membership Functions (MF), Logical Operations and If-Then Rules. FIS having multidisciplinary nature, so cab called as fuzzy-rule-based systems, fuzzy expert systems, fuzzy modeling, fuzzy associative memory, fuzzy logic controllers, and simply (and ambiguously) fuzzy systems. Types of FIS: 1) Mamdani-type: Most commonly used. Expects the output MF’s to be fuzzy sets. 2) Sugeno-type: Output MF’s are either linear or constant.
• 22. Fuzzy Inference Process To describe the fuzzy inference process, lets consider the example of two-input, one-output, two-rule valve control problem.
• 23. Fuzzy Inference Process Step 1: Fuzzify Input (Fuzzification) Take the inputs and determine the degree to which they belong to each of the appropriate fuzzy sets via membership functions. Input is always a crisp numerical value limited to the universe of discourse of the input variable. Output is a fuzzy degree of membership in the qualifying linguistic set. Each input is fuzzified over all the qualifying membership functions required by the rules.
• 24. Fuzzy Inference Process Step 1: Fuzzify Input (Fuzzification)
• 25. Fuzzy Inference Process Step 2 : Apply Fuzzy Operator If the antecedent of a given rule has more than one part, the fuzzy operator is applied to obtain one number that represents the result of the antecedent for that rule. The input to the fuzzy operator is two or more membership values from fuzzified input variables. The output is a single truth value.
• 26. Fuzzy Inference Process Step 2 : Apply Fuzzy Operator
• 27. Fuzzy Inference Process Step 3: Apply Implication Method First must determine the rule’s weight. Operation in which the result of fuzzy operator is used to determine the conclusion of the rule is called as implication. The input for the implication process is a single number given by the antecedent. The output of the implication process is a fuzzy set. Implication is implemented for each rule.
• 28. Fuzzy Inference Process Step 3: Apply Implication Method Antecedent Consequent
• 29. Fuzzy Inference Process Step 4 : Aggregate All Outputs Aggregation is the process by which the fuzzy sets that represent the outputs of each rule are combined into a single fuzzy set. Aggregation only occurs once for each output variable. The input of the aggregation process is the list of truncated output functions returned by the implication process for each rule. The output of the aggregation process is one fuzzy set for each output variable.
• 30. Fuzzy Inference Process Step 4 : Aggregate All Outputs
• 31. Fuzzy Inference Process Step 5: Defuzzify Move from the “fuzzy world” to the “real world” is known as defuzzification. The input for the defuzzification process is a fuzzy set. The output is a single number. The most popular defuzzification method is the centroid calculation, which returns the center of area under the curve Other methods are bisector, middle of maximum (the average of the maximum value of the output set), largest of maximum, and smallest of maximum.
• 32. Fuzzy Inference Process Step 5: Defuzzify
• 33. References Fuzzy Logic Toolbox™ 2 User’s Guide Tutorial On Fuzzy Logic by Jan Jantzen Fuzzy Logic by Cahier Technique Schneider
• 34. Thank You…
• 35. Any Questions?