Abstract

Fuzzy rule-based systems are one of the most important areas of application of fuzzy sets and fuzzy logic. Constituting an extension of classical rule-based systems, these have been successfully applied to a wide range of problems in different domains for which uncertainty and vagueness emerge in multiple ways. In a broad sense, fuzzy rule-based systems are rule-based systems, where fuzzy sets and fuzzy logic are used as tools for representing different forms of knowledge about the problem at hand, as well as for modeling the interactions and relationships existing between its variables. The use of fuzzy statements as one of the main constituents of the rules allows capturing and handling the potential uncertainty of the represented knowledge. On the other hand, thanks to the use of fuzzy logic, inference methods have become more robust and flexible. This chapter will mainly analyze what is a fuzzy rule-based system (from both conceptual and structural points of view), how is it built, and how can be used. The analysis will start by considering the two main conceptual components of these systems, knowledge, and reasoning, and how they are represented. Then, a review of the main structural approaches to fuzzy rule-based systems will be considered. Hierarchical fuzzy systems will also be analyzed. Once defined the components, structure and approaches to those systems, the question of design will be considered. Finally, some conclusions will be presented.

Keywords

FLare 
CG

center of gravity

DNF

disjunctive normal form

FLC

fuzzy logic controller

FL

fuzzy logic

FRBS

fuzzy rule-based system

FS

fuzzy system

KB

knowledge base

MISO

multiple inputs-single output

MOM

mean of maxima

MV

maximum value

RB

rule base

TSK

Takagi–Sugeno–Kang

References

  1. [13.1]
    A. Bardossy, L. Duckstein: Fuzzy Rule-Based Modeling with Application to Geophysical, Biological and Engineering Systems (CRC, Boca Raton 1995)MATHGoogle Scholar
  2. [13.2]
    Z. Chi, H. Yan, T. Pham: Fuzzy Algorithms: With Applications to Image Processing and Pattern Recognition (World Scientific, Singapore 1996)MATHGoogle Scholar
  3. [13.3]
    K. Hirota: Industrial Applications of Fuzzy Technology (Springer, Berlin, Heidelberg 1993)CrossRefGoogle Scholar
  4. [13.4]
    W. Pedrycz: Fuzzy Modelling: Paradigms and Practice (Kluwer Academic, Dordrecht 1996)CrossRefMATHGoogle Scholar
  5. [13.5]
    R.R. Yager, L.A. Zadeh: An Introduction to Fuzzy Logic Applications in Intelligent Systems (Kluwer Academic, Dordrecht 1992)CrossRefMATHGoogle Scholar
  6. [13.6]
    L.A. Zadeh: The concept of a linguistic variable and its applications to approximate reasoning – Part I, Inf. Sci. 8(3), 199–249 (1975)MathSciNetCrossRefMATHGoogle Scholar
  7. [13.7]
    L.A. Zadeh: The concept of a linguistic variable and its applications to approximate reasoning – Part II, Inf. Sci. 8(4), 301–357 (1975)MathSciNetCrossRefMATHGoogle Scholar
  8. [13.8]
    L.A. Zadeh: The concept of a linguistic variable and its applications to approximate reasoning – Part III, Inf. Sci. 9(1), 43–80 (1975)MathSciNetCrossRefMATHGoogle Scholar
  9. [13.9]
    L.A. Zadeh: Outline of a new approach to the analysis of complex systems and decision processes, IEEE Trans. Syst. Man Cybern. 3, 28–44 (1973)MathSciNetCrossRefMATHGoogle Scholar
  10. [13.10]
    E.H. Mamdani: Applications of fuzzy algorithm for control of simple dynamic plant, Proc. IEE 121(12), 1585–1588 (1974)Google Scholar
  11. [13.11]
    E.H. Mamdani, S. Assilian: An experiment in linguistic synthesis with a fuzzy logic controller, Int. J. Man-Mach. Stud. 7, 1–13 (1975)CrossRefMATHGoogle Scholar
  12. [13.12]
    L.X. Wang: Adaptive Fuzzy Systems and Control: Design and Analysis (Prentice Hall, Englewood Cliffs 1994)Google Scholar
  13. [13.13]
    T.J. Procyk, E.H. Mamdani: A linguistic self-organizing process controller, Automatica 15(1), 15–30 (1979)CrossRefMATHGoogle Scholar
  14. [13.14]
    L. Magdalena: Adapting the gain of an FLC with genetic algorithms, Int. J. Approx. Reas. 17(4), 327–349 (1997)CrossRefMATHGoogle Scholar
  15. [13.15]
    G.C. Mouzouris, J.M. Mendel: Nonsingleton fuzzy logic systems: Theory and application, IEEE Trans. Fuzzy Syst. 5, 56–71 (1997)CrossRefGoogle Scholar
  16. [13.16]
    F. Klawonn, R. Kruse: Equality relations as a basis for fuzzy control, Fuzzy Sets Syst. 54(2), 147–156 (1993)MathSciNetCrossRefMATHGoogle Scholar
  17. [13.17]
    J.M. Mendel: Fuzzy logic systems for engineering: A tutorial, Proc. IEEE 83(3), 345–377 (1995)CrossRefGoogle Scholar
  18. [13.18]
    D. Dubois, H. Prade: What are fuzzy rules and how to use them, Fuzzy Sets Syst. 84, 169–185 (1996)MathSciNetCrossRefMATHGoogle Scholar
  19. [13.19]
    O. Cordón, F. Herrera, A. Peregrín: Applicability of the fuzzy operators in the design of fuzzy logic controllers, Fuzzy Sets Syst. 86, 15–41 (1997)CrossRefMATHGoogle Scholar
  20. [13.20]
    D. Driankov, H. Hellendoorn, M. Reinfrank: An Introduction to Fuzzy Control (Springer, Berlin, Heidelberg 1993)CrossRefMATHGoogle Scholar
  21. [13.21]
    M. Sugeno, T. Yasukawa: A fuzzy-logic-based approach to qualitative modeling, IEEE Trans. Fuzzy Syst. 1(1), 7–31 (1993)CrossRefGoogle Scholar
  22. [13.22]
    C.C. Lee: Fuzzy logic in control systems: Fuzzy logic controller – Part I, IEEE Trans. Syst. Man Cybern. 20(2), 404–418 (1990)CrossRefMATHGoogle Scholar
  23. [13.23]
    C.C. Lee: Fuzzy logic in control systems: Fuzzy logic controller – Part II, IEEE Trans. Syst. Man Cybern. 20(2), 419–435 (1990)CrossRefMATHGoogle Scholar
  24. [13.24]
    A. Bastian: How to handle the flexibility of linguistic variables with applications, Int. J. Uncertain. Fuzziness Knowl.-Based Syst. 3(4), 463–484 (1994)MathSciNetCrossRefMATHGoogle Scholar
  25. [13.25]
    B. Carse, T.C. Fogarty, A. Munro: Evolving fuzzy rule based controllers using genetic algorithms, Fuzzy Sets Syst. 80, 273–294 (1996)CrossRefGoogle Scholar
  26. [13.26]
    A. González, R. Pèrez, J.L. Verdegay: Learning the structure of a fuzzy rule: A genetic approach, Fuzzy Syst. Artif. Intell. 3(1), 57–70 (1994)Google Scholar
  27. [13.27]
    L. Magdalena, F. Monasterio: A fuzzy logic controller with learning through the evolution of its knowledge base, Int. J. Approx. Reas. 16(3/4), 335–358 (1997)CrossRefMATHGoogle Scholar
  28. [13.28]
    R. Alcalá, J. Casillas, O. Cordón, F. Herrera: Building fuzzy graphs: Features and taxonomy of learning for non-grid-oriented fuzzy rule-based systems, J. Intell. Fuzzy Syst. 11(3/4), 99–119 (2001)Google Scholar
  29. [13.29]
    O. Cordón, F. Herrera: A three-stage evolutionary process for learning descriptive and approximate fuzzy logic controller knowledge bases from examples, Int. J. Approx. Reas. 17(4), 369–407 (1997)CrossRefMATHGoogle Scholar
  30. [13.30]
    L. Koczy: Fuzzy if $\ldots$ then rule models and their transformation into one another, IEEE Trans. Syst. Man Cybern. 26(5), 621–637 (1996)CrossRefGoogle Scholar
  31. [13.31]
    T. Takagi, M. Sugeno: Fuzzy identification of systems and its application to modeling and control, IEEE Trans. Syst. Man Cybern. 15(1), 116–132 (1985)CrossRefMATHGoogle Scholar
  32. [13.32]
    M. Sugeno, G.T. Kang: Structure identification of fuzzy model, Fuzzy Sets Syst. 28(1), 15–33 (1988)MathSciNetCrossRefMATHGoogle Scholar
  33. [13.33]
    R. Palm, D. Driankov, H. Hellendoorn: Model Based Fuzzy Control (Springer, Berlin, Heidelberg 1997)CrossRefMATHGoogle Scholar
  34. [13.34]
    H. Ishibuchi, T. Nakashima: Effect of rule weights in fuzzy rule-based classification systems, IEEE Trans. Fuzzy Syst. 9, 506–515 (2001)CrossRefGoogle Scholar
  35. [13.35]
    J.M. Mendel: Type-2 fuzzy sets and systems: An overview, Comput. Intell. Mag. IEEE 2(1), 20–29 (2007)MathSciNetCrossRefGoogle Scholar
  36. [13.36]
    H. Jones, B. Charnomordic, D. Dubois, S. Guillaume: Practical inference with systems of gradual implicative rules, IEEE Trans. Fuzzy Syst. 17, 61–78 (2009)CrossRefGoogle Scholar
  37. [13.37]
    V. Torra: A review of the construction of hierarchical fuzzy systems, Int. J. Intell. Syst. 17(5), 531–543 (2002)MathSciNetCrossRefMATHGoogle Scholar
  38. [13.38]
    R.R. Yager: On a hierarchical structure for fuzzy modeling and control, IEEE Trans. Syst. Man Cybern. 23(4), 1189–1197 (1993)CrossRefGoogle Scholar
  39. [13.39]
    R.R. Yager: On the construction of hierarchical fuzzy systems models, IEEE Trans. Syst. Man Cybern. C 28(1), 55–66 (1998)CrossRefGoogle Scholar
  40. [13.40]
    O. Cordón, F. Herrera, I. Zwir: Linguistic modeling by hierarchical systems of linguistic rules, IEEE Trans. Fuzzy Syst. 10, 2–20 (2002)CrossRefMATHGoogle Scholar
  41. [13.41]
    E. D'Andrea, B. Lazzerini: A hierarchical approach to multi-class fuzzy classifiers, Exp. Syst. Appl. 40(9), 3828–3840 (2013)CrossRefGoogle Scholar
  42. [13.42]
    H. Takagi, N. Suzuki, T. Koda, Y. Kojima: Neural networks designed on approximate reasoning architecture and their applications, IEEE Trans. Neural Netw. 3(5), 752–760 (1992)CrossRefGoogle Scholar
  43. [13.43]
    G.V.S. Raju, J. Zhou, R.A. Kisner: Hierarchical fuzzy control, Int. J. Control 54(5), 1201–1216 (1991)MathSciNetCrossRefMATHGoogle Scholar
  44. [13.44]
    G.A. Miller: The magical number seven, plus or minus two: Some limits on our capacity for processing information, Psychol. Rev. 63, 81–97 (1956)CrossRefGoogle Scholar
  45. [13.45]
    K. Michels, F. Klawonn, R. Kruse, A. Nürnberger: Fuzzy Control: Fundamentals, Stability and Design of Fuzzy Controllers (Springer, Berlin, Heidelberg 2006)MATHGoogle Scholar
  46. [13.46]
    L.-X. Wang, J.M. Mendel: Fuzzy basis functions, universal approximation, and orthogonal least-squares learning, IEEE Trans. Neural Netw. 3(5), 807–813 (1992)CrossRefGoogle Scholar
  47. [13.47]
    B. Kosko: Fuzzy systems as universal approximators, IEEE Trans. Comput. 43(11), 1329–1333 (1994)CrossRefMATHGoogle Scholar
  48. [13.48]
    J.L. Castro: Fuzzy logic controllers are universal approximators, IEEE Trans. Syst. Man Cybern. 25(4), 629–635 (1995)CrossRefGoogle Scholar
  49. [13.49]
    R. Krishnapuram: Membership function elicitation and learning. In: Handbook of Fuzzy Computation, ed. by E.H. Ruspini, P.P. Bonissone, W. Pedrycz (IOP Publ., Bristol 1998) pp. 349–368Google Scholar
  50. [13.50]
    J.M. Alonso, L. Magdalena: HILK++: An interpretability-guided fuzzy modeling methodology for learning readable and comprehensible fuzzy rule-based classifiers, Soft Comput. 15(10), 1959–1980 (2011)CrossRefGoogle Scholar
  51. [13.51]
    N.R. Pal, S. Chakraborty: Fuzzy rule extraction from id3-type decision trees for real data, IEEE Trans. Syst. Man Cybern. B 31(5), 745–754 (2001)CrossRefGoogle Scholar
  52. [13.52]
    M. Delgado, A.F. Gómez-Skarmeta, F. Martín: A fuzzy clustering-based rapid prototyping for fuzzy rule-based modeling, IEEE Trans. Fuzzy Syst. 5, 223–233 (1997)CrossRefGoogle Scholar
  53. [13.53]
    O. Cordón, F. Herrera, F. Hoffmann, L. Magdalena: Genetic Fuzzy Systems: Evolutionary Tuning and Learning of Fuzzy Knowledge Bases (World Scientific, Singapore 2001)CrossRefMATHGoogle Scholar
  54. [13.54]
    D.D. Nauck, A. Nürnberger: Neuro-fuzzy systems: A short historical review. In: Computational Intelligence in Intelligent Data Analysis, ed. by C. Moewes, A. Nürnberger (Springer, Berlin, Heidelberg 2013) pp. 91–109CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  1. 1.European Centre for Soft ComputingMieresSpain

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