# Fuzzy Logic: An Introductory Course for Engineering Students

# Fuzzy Logic: An Introductory Course for Engineering Students

## Enric Trillas, Luka Eciolaza

Language: English

Pages: 562

ISBN: 3319386433

Format: PDF / Kindle (mobi) / ePub

This book was thought as a non-conventional first course textbook in Fuzzy Logic

for engineers ending with an introduction to one of the most fruitful topics arisen

from it, Fuzzy Control. It is from the teaching’s strategy of the authors, summarized

by “Nothing can substitute the own homework of the student” from which it comes

its non-conventional character, partially manifested by the ‘continuous’ form of

presenting the considered topics by joining theoretical explanations and examples,

and not always following the typically mathematical style of ‘theorem-corolaries’.

Behind this strategy is the opinion that, at the university level, students and

professors ought to learn jointly, students do not wait to receive everything from the

professor’s lectures, but should read more than a single recommended textbook.

Consequently, this book is neither a manual with recipes to be uncritically applied,

nor it is directed to those that can be only interested in mathematical subtleties. The

reader should be aware that fuzzy logic is the study and computational management

of imprecision and non-random uncertainty, both with the highest accuracy and

precision possible at each case, that fuzzy logic is not fuzzy in itself.

Each university course requires a particular teaching tactic that not only depends

on the number of lecturing hours, but on the aim of the course and on the audience’s

characteristics. In particular, additional tutorials supplied by the professor are

essential for a good learning process. Tutorials in which other forms of considering

the course’s topics and more sophisticated problems can be proposed. This is at the

own hands of the professor.

The book just presents some basic mathematical models for fuzzy logic but

without the intention to just subordinate it to mathematics. Fuzzy logic is neither a

part of mathematics, nor even of logic, like Physics is not so. Notwithstanding,

what is paramount is the importance and usefulness of mathematical models in

experimental sciences and technology, as well as in computer science and computer

technology and, in particular, in Soft Computing, where fuzzy logic plays a pivotal

role. But the suitability of such models only can come from the success of its testing

against some reality, for instance, in true applications; applications play in the

techno-scientific world an analogous role to that of experimentation in natural

sciences. For instance, if the branch called ‘Fuzzy Control’ served not as a direct justification of fuzzy logic, the success fuzzy logic has in control applications can be

seen as a kind of experimentation to show its usefulness in the study of dynamical

systems linguistically described by systems of imprecise rules. Fuzzy logic is much

more than what is in this introductory textbook; its applications spread along many

domains of science and technology.

* fetched from springer

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negation is a function such that If , then , for all , or . Notice that is equivalent to , that shows is a continuous function: It is and if it should be since would imply , or . Hence, N is strictly decreasing. Since is continuous, the equation has solutions, but there is only one. Suppose and . Either , or . In the first case, it follows , or , and . In the second case, , or , that is absurd. Then, each strong negation has a single fixed point , in the open interval , since show that 0

With , the equation gives , and . Lemma 2.2.45 If and are, respectively, a t-norm and a continuous t-conorm, it is for all in if and only if . Proof Since , it follows . With , the equation gives , and . Hence,The law holds for all and The law holds for all and The two laws hold jointly if and only if and . 2.2.8.7 The Law of von Neumann With classical sets it always holds the law of von Neumann, or law of the perfect repartition, that follows from , and generalizes that of the

with : , hence, the three cases are W-conditionals. Example 3.2.8 Let us see how is , when is, respectively, the continuous t-norm , , . , ( implication). , (Goguen implication). , (Łukasiewicz implication). Since each is a -conditional, ’s is a min-conditional, Goguen’s are prod-conditionals, and Łukasiewicz’s are W-conditionals. Notice that the S-implications of the form are exactly the Łukasiewicz’s R-implications: the only R-implications that are S-implications are the

not perfectly reflect the primary use of big in . In the same way, the crisp degree does not perfectly reflect the primary use of big when translated into ‘after eight’. Another possible model for is with graphic. All these models are linear, with the exception os , that is quadratic. Another quadratic models are given by , and , with graphics Finally, the following two models are hyperbolic, with graphics. Example 1.1.9 The predicate old, once numerically characterized by , can be

luka.eciolaza@softcomputing.es 2.1 Introduction From now on it will be only considered the case in which , that is, of Zadeh’s fuzzy sets, with predicates in known through a degree , and without knowing, necessarily, its primary use . The set of all fuzzy sets in , , will be also denoted by . In this case, the preorder is linear, or total, since for all in it is either , or , that is, it is either or for all in . Hence, rarely will perfectly reflect the primary use of in , since is usually not