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Uncertain Rule-Based Fuzzy Systems pp 1–23Cite as

Introduction

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Abstract

This chapter describes what this book is about. It explains four kinds of uncertainty partitions —crisp, first-order , second-order with uniform weighting , and second-order with nonuniform weighting —and that they can be respectively mathematically modeled using classical (crisp) set theory, classical (type-1) fuzzy set theory, interval type-2 fuzzy set theory, and general type-2 fuzzy set theory; provides the structure of a rule-based fuzzy system , and explains its four components—rules, fuzzifier , inference , and output processor ; explains why type-2 fuzzy sets are a new direction for fuzzy systems ; states and explains the fundamental design requirement of a type-2 fuzzy system; provides an impressionistic brief history of type-1 fuzzy sets and fuzzy logic ; reviews the early literature (1975–1992) about type-2 fuzzy sets and systems (the literature that was heavily used when the first edition of this book was written) , and some literature about applications of type-2 fuzzy set and systems; and provides a brief summary of what is covered in Chaps. 2–11, a very short statement about the applicability of the book’s coverage outside of the field of rule-based fuzzy systems , and a list of sources that are available for software that can be used to implement much of what is in this book.

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Notes

  1. 1.

    It is conceivable that uncertainty about the filling of the FOU could lead to higher than second-order uncertainty about the FOU.

  2. 2.

    [Dick (2005)] and [Chen et al. (2010)] develop fuzzy systems for complex numbers, but such systems are beyond the scope of this book.

  3. 3.

    Stating that the outputs of the inference engine are fuzzy sets is very general, and is meant to include everything from numbers, to intervals, to type-1 fuzzy sets, to interval type-2 fuzzy sets, and to general type-2 fuzzy sets. This will be clarified in Chaps. 3, 9, and 11.

  4. 4.

    The unmodified fuzzy logic inference mechanisms are still being used, e.g., in approximate reasoning applications, but their use is outside of the scope of this book.

  5. 5.

    An excellent historical view of type-2 fuzzy sets and systems is John and Coupland (2007). It includes a figure with the number of type-2 related publications over time from 1976 through 2006 as well as a figure that depicts a time line of the historical development of type-2 fuzzy sets and systems.

  6. 6.

    This material about IVFSs was written in French, apparently never published in a refereed journal, and so it was not, and still is not available in English to the general scientific community.

  7. 7.

    These are Mamdani type-2 fuzzy systems. The two most popular fuzzy systems used by engineers are the Mamdani and Takagi-Sugeno-Kang (TSK) systems (see Chap. 3). Both are characterized by IF–THEN rules and have the same antecedent structures. They differ in the structures of the consequents. The consequent of a rule in a Mamdani fuzzy system is a fuzzy set, whereas the consequent of a rule in a TSK fuzzy system is a mathematical function.

  8. 8.

    This list is in alphabetical order by application.

  9. 9.

    There also is other proprietary software that is being used by researchers, but, even though it is used, mentioned, described, and referenced in articles, it is not available to others.

  10. 10.

    MATLAB and SIMULINK are registered trademarks of The MathWorks.

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  1. University of Southern California, Los Angeles, CA, USA

    Jerry M. Mendel

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Mendel, J.M. (2017). Introduction. In: Uncertain Rule-Based Fuzzy Systems. Springer, Cham. https://doi.org/10.1007/978-3-319-51370-6_1

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