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Soft Computing. Per Printz Madsen Section of Automation and Control E-mail: ppm@es.aau.dk

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Soft Computing. 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. - A. Einstein As complexity rises, precise statements lose meaning and meaningful statements lose precision. - Lotfi Zadeh

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What Is Soft Computing? A Definition of Soft Computing - adapted from L.A. Zadeh Soft computing differs from conventional (hard) computing in that, unlike hard computing, it is tolerant of imprecision, uncertainty, partial truth, and approximation. In effect, the role model for soft computing is the human mind. The principal types of Soft Computing: – Fuzzy Logic (FL), – Neural Computing (NC), – Evolutionary Computation (EC) – Machine Learning (ML) – Probabilistic Reasoning (PR) Some important journals and links: – Journal of Soft Computing Journal of Soft Computing – Applied Soft Computing Applied Soft Computing – World Conference on Soft Computing World Conference on Soft Computing – The IEEE Computational Intelligence Society The IEEE Computational Intelligence Society

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Fuzzy logic Fuzzy logic is a form of multi-valued logic that deals with reasoning that is approximate rather than accurate. In contrast with traditionally logic "crisp logic", where binary sets have binary logic (true or false). Fuzzy logic value are truth to a certain degree e.g. between 0 and 1. Fuzzy logic use linguistic variables for describing the logic in a natural way. 0 1 1.8 m Crisp logic: PersonHeight= tall 0 1 1.7 m Fuzzy logic: PersonHeight= tall 1.9 m

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crisp logic X: The universe of discourse X: The set including all possible members. A, B: Sub sets of X with some specific features descript be two membership function. and A B

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Fuzzy logic 1 0 102030400 Age Young Matlab memberskip functions

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Fuzzy set operations Consider three fuzzy set A, B and C in the same universe of discourse: X The empty set: Ø The set A equal to set B The set A is a part of set B The set A is a true part of set B

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Fuzzy set operations The set A is the complement of B: A= not B The set C is the union of A and B: C= A or B AB CC x

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Fuzzy set operations The set C is the intersection of A and B: C = A and B AB CC

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Linguistic variables A linguistic variable is a variable whose values are not numbers but words or sentences in a natural or artificial language (Zadeh, 1975a, p. 201). The variable: Rum temperature. The values: Hot, Cold, Comfortable, to_cold, to_hot, … Hedges or linguistic modifiers: Very, More_or_less, extremely,.. Very Hot, extremely Cold, More_or_less Comfortable,… More formally, a linguistic variable is characterized by: [x, T(x), X, M], x is the name of the variable, T(x) is the value set of x. Each value is a part of the universe of discourse X. M is a set of semantic rule for associating each member in T(x) with its meaning. This meaning is defined by the membership functions for each value in T(x).

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Linguistic variables Ex: The lingustuic variable age. [x, T(x), X, M], x = Age, T(x)= { young, old}, X all persons in Denmark, M= { young ~ young old ~ old ‘’Per is young’’ is true to the degree: young (Age of Per). ‘’Bent is old’’ is true to the degree: old (Age of Bent). ‘’Per is young and Bent is old’’: the intersection of young (Age of Per) and old (Age of Bent): min( young (Age of Per), old (Age of Bent)). ‘’Per is young or Bent is old’’: The union max( young (Age of Per), old (Age of Bent)).

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Linguistic variables ‘’Per is not young or Bent is not old’’: max(1 - young (Age of Per), 1 - old (Age of Bent)). Hedges or linguistic modifiers: Very, More_or_less, extremely,.. The set C = Very A The set C = More_or_less A ‘’Per is not Very young and Bent is More_or_less old’’:

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Fuzzy implication (if-then) IF Per is young THEN Bent is old Definition: A and B are two fuzzy sets. Given by A (x) and B (y). Then A implicate B is given by:

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Fuzzy implication (if-then) IF water_temp is cold and wind is stormy THEN swimming is bad ColdOKWarmnormalstormyaveragegoodBad Measured temperatureMeasured wind speed y

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IF water_temp is cold AND wind is stormy THEN swimming is bad IF water_temp is warm AND wind is stormy THEN swimming is average IF water_temp is ok AND wind is normal THEN swimming is average IF water_temp is warm AND wind is normal THEN swimming is good averagegoodBad CoG: Center of gravity MoM: Mean of Max LoM: Left of Max RoM: Right of Max CoG LoM MoM RoM out x out Fuzzy Aggregation

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Defuzzyfication averagegoodBad CoG: Center of gravity CoG LoM MoM RoM out x out Matlab defuzzyfication

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Introduction to Fuzzy Control Lecture 10.1 Appendix E.

Introduction to Fuzzy Control Lecture 10.1 Appendix E.

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