Type-1 Fuzzy Systems

Chapter

Abstract

This chapter explores many aspects of the type-1 fuzzy system that was introduced in Chap.  1. It provides a very comprehensive and unified description of the two major kinds of type-1 fuzzy systems that are widely used in real-world applications—Mamdani and TSK fuzzy systems. The coverage of this chapter includes rules, singleton, and non-singleton fuzzifiers, input–output formulas for the fuzzy inference engine, type-1 first- and second-order rule partitions, the effects of the two kinds of fuzzifiers on the input–output formulas, combining or not combining fired-rule output sets on the way to defuzzification, defuzzifiers (centroid, height, and center-of-sets), fuzzy basis functions which provide a mathematical description of a fuzzy system from its input to its output, remarks and insights about a type-1 fuzzy system (including layered architecture interpretations for it, universal approximation by it, continuity of it, rule explosion and some ways to control it, and rule interpretability for it). Eighteen examples are used to illustrate the important concepts and there is also a comprehensive numerical example in Sect. 3.7 that is continued in later chapters. Chap.  9 builds upon the material that is in this chapter.

References

  1. Alspach, D. L. and H. W. Sorenson. 1972. Nonlinear bayesian estimation using gaussian sum approximations. IEEE Transaction on Automatic Control. 17: 439–448.Google Scholar
  2. Antonelli, M. D. Bernardo, H. Hagras and F. Marcelloni. 2016. Multi-objective evolutionary optimization of type-2 fuzzy rule-based systems for financial data classification. Accepted for publication in IEEE Transaction on Fuzzy Systems.Google Scholar
  3. Barai, R. K. T. Tjahjowidodo and B. K. Pappachan. 2015. Fuzzy inference system based intelligent sensor fusion for estimation of surface roughness in machine process, In Proceeding of 9th IEEE international conference on sensing technology, pp. 799–802, Aukland, New Zealand.Google Scholar
  4. Blum, E.K., and L.K. Li. 1991. Approximation theory and feedforward networks. Neural Networks 4: 511–515.CrossRefGoogle Scholar
  5. Buckley, J.J. 1992. Universal fuzzy controllers. Automatica 28: 1245–1248.MathSciNetCrossRefMATHGoogle Scholar
  6. Buckley, J.J. 1993. Sugeno-type-controllers are universal controllers. Fuzzy Sets and Systems 25: 299–303.MathSciNetCrossRefMATHGoogle Scholar
  7. Casillas, J., O. Cordon, F. Herrera, and L. Magdalena (eds.). 2003. Interpretability issues in fuzzy modeling. Berlin Heidelberg: Springer-Verlag.MATHGoogle Scholar
  8. Castro, J. L. 1995. Fuzzy logic controllers are universal approximators, In IEEE Transaction Systems on Man Cybernetics 25(4): 629–635.Google Scholar
  9. Combs, W.E., and J.E. Andrews. 1998. Combinatorial rule explosion eliminated by a fuzzy rule configuration. IEEE Transaction on Fuzzy Systems 6 (1): 1–11.CrossRefGoogle Scholar
  10. Cordon, O. 2011. A historical review of evolutionary learning methods for Mamdani-type fuzzy rule-based systems: Designing interpretable genetic fuzzy systems. International Journal on Approximate Reason 52: 894–913.CrossRefGoogle Scholar
  11. Cybenko, G. 1989. Approximation by superpositions of a sigmoidal function. Mathematics of Control: Signals SystemsMATHGoogle Scholar
  12. Driankov, D. H. 1996. Hellendoorn and M. Reinfrank, An Introduction to Fuzzy Control (2nd ed.) Springer-Verlag.Google Scholar
  13. Dubois, D., and H. Prade. 1985. A Review of fuzzy set aggregation connectives. Info Science 36: 85–121.MathSciNetCrossRefMATHGoogle Scholar
  14. Ducange, P., and F. Marcelloni. 2011. Multi-objective evolutionary fuzzy systems”, 9th international workshop on fuzzy logic applications, Italia, 29–31 August. Lecture Notes on ComputerScience 6857: 83–90.CrossRefGoogle Scholar
  15. Fazzolari, M., R. Alcala, Y. Nojima, H. Ishibuchi, and F. Herrera. 2013. A review of the application of multi-objective evolutionary fuzzy systems: Current issues and further directions. IEEE Transaction on Fuzzy Systems 21 (1): 45–65.CrossRefGoogle Scholar
  16. Gacto, M.J., R. Alcala, and F. Herrera. 2011. Interpretability of linguistic fuzzy rule-based systems: an overview of interpretability measures. Infomation Science 181: 4340–4360.CrossRefGoogle Scholar
  17. Galende-Hernandez, M., G.I. Sainz-Palmero, and M.J. Fuente-Aparicio. 2012. Complexity reduction and interpretability improvement for fuzzy rule systems based on simple interpretability measures and indices by bi-objective evolutionary rule selection. Soft Computing 16: 451–470.CrossRefGoogle Scholar
  18. Garcia, D., J.C. Gamez, A. Gonzalez, and R. Perez. 2015. An interpretability improvement for fuzzy rule bases obtained by the iterative learning approach. International Journal on Approximate Reason 67: 37–58.MathSciNetCrossRefMATHGoogle Scholar
  19. Haykin, S. 1996. Adaptive filter theory, 3rd ed. Upper Saddle River, NJ: Prentice-Hall.MATHGoogle Scholar
  20. Herrera, F., and L. Martinez. 2000. A 2-tuple fuzzy linguistic representation model for computing with words. IEEE Transcation on Fuzzy Systems 8 (6): 746–752.CrossRefGoogle Scholar
  21. Herrera, F. and L. Martinez. 2001. A model based on linguistic 2-tuples for dealing with multigranular hierarchical linguistic contexts in multi-expert decision-making, IEEE Trans on Systems, Man, and Cybernetics, Part B (Cybernetics) 31(2): 227–234.Google Scholar
  22. Hornik, K. 1993. Some results on neural network approximation. Neural Networks 6: 1069–1072.CrossRefGoogle Scholar
  23. Hornik, K., M. Stinchcombe, and H. White. 1989. Multilayer feedforward networks are universal approximators. Neural Networks 2: 359–366.CrossRefMATHGoogle Scholar
  24. Hüllermeier, E. 2015. Does machine learning need fuzzy logic? Fuzzy Sets and Systems 281: 292–299.MathSciNetCrossRefGoogle Scholar
  25. Ishibuchi, H. 2007. Multiobjective genetic fuzzy systems: Review and future research directions. In Proceedings of 2007 IEEE international conference on fuzzy systems (FUZZ-IEEE 2007), London, UK pp. 1–6.Google Scholar
  26. Ishibuchi, H., T. Nakashima, and M. Nii. 2004. Classification and modeling with linguistic information granules: Advanced approaches to linguistic data mining. Berlin: Springer-Verlag.MATHGoogle Scholar
  27. Ishibuchi, H., and T. Yamamoto. 2003. Interpretability issues in fuzzy genetic-based machine learning for linguistic modeling. Modeling With Words, Lecture Notes in Computer Sciences 2873: 209–228.CrossRefGoogle Scholar
  28. Jamshidi, M. 1997. Large Scale Systems: Modeling, Control and Fuzzy Logic, Prentice-Hall, PTR, Upper Saddle River, NJ, (Section 8.3.2, Rule-base reduction).Google Scholar
  29. Jang, J.-S. R. 1993. ANFIS: Adaptive-network-based fuzzy inference system. IEEE Transaction on Systems, Man and Cybernetics 23: 665–684.Google Scholar
  30. Juang, C.-F. 2002. A TSK-type recurrent fuzzy network for dynamic systems processing by neural network and gnetic algorithm. IEEE Transaction on Fuzzy System 10(2): 155–170.Google Scholar
  31. Juang, C.-F. and J.-S. Chen. 2006. Water bath temperature control by a recurrent fuzzy controller and its FPGA implementation. IEEE Transaction on Industrial Electronics 53(3): 941–949.Google Scholar
  32. Juang, C.-F. and C.-T. Lin. 1999. A recurrent self-organizing neural fuzzy inference network. IEEE Transcation on Neural Networks, 10(4): 828–845.Google Scholar
  33. Karnik, N. N. and J. M. Mendel. 1998. An Introduction to Type-2 Fuzzy Logic Systems, USC-SIPI Report #418, Univ. of Southern Calif., Los Angeles, CA, June 1998. This can be accessed at: http://sipi.usc.edu/research; then choose “sipi technical reports/418”.
  34. Keller, J.M., R.R. Yager, and H. Tahani. 1992. Neural network implementation of fuzzy logic. Fuzzy Sets and Systems 45: 1–12.MathSciNetCrossRefMATHGoogle Scholar
  35. Kim, Y. M. and J. M. Mendel. 1995. Fuzzy basis functions: Comparisons with other basis functions. IEEE Transcation on Fuzzy Systems 3: 158–168.Google Scholar
  36. Kiska, J. B. M. E. Kochanska and D. S. Sliwinska. 1985a. The influence of some parameters on the accuracy of fuzzy model. In Industrial applications of fuzzy control (M. Sugeno, Ed.), North Holland, Amsterdam, pp. 187–230.Google Scholar
  37. Kiska, J.B., M.E. Kochanska, and D.S. Sliwinska. 1985b. The influence of some parameters on the accuracy of a fuzzy model—part I. Fuzzy Sets and Systems 15: 111–128.MathSciNetCrossRefMATHGoogle Scholar
  38. Kiska, J.B., M.E. Kochanska, and D.S. Sliwinska. 1985c. The influence of some parameters on the accuracy of a fuzzy model—part II. Fuzzy Sets and Systems 15: 223–240.MathSciNetCrossRefMATHGoogle Scholar
  39. Kóczy, L. T, and K. Hirota. 1997. Size reduction by interpolation in fuzzy rule bases, IEEE Transcation on Systems and Man Cybernetics—Part B: Cybernetics 27: 14–25.Google Scholar
  40. Kosko, B. 1992. Fuzzy Systems as Universal Approximators. IEEE International Conference on Fuzzy Systems, pp. 1153–1162.Google Scholar
  41. Kosko, B. 1994. Fuzzy systems as universal approximators. IEEE Transcation on Computers 43(11): 1329–1333.Google Scholar
  42. Kosko, B. 1997. Fuzzy engineering. Upper Saddle River, NJ: Prentice Hall.MATHGoogle Scholar
  43. Kovács, S. 2009. Fuzzy Rule Interpolation. In Encyclopedia of Artificial Intelligence, ed. Juan Ramón Rabuñal Dopico, Julian Dorado and Alejandro Pazos, pp. 728–733, 2009, accessed January 15, 2016. doi: 10.4018/978-1-59904-849-9.ch108.
  44. Kreinovich, V., G.C. Mouzouris, and H.T. Nguyen. 1998. Fuzzy rule based modeling as a universal approximation tool. In Fuzzy systems, modeling and control, ed. T. Nguyen, and M. Sugeno, 135–195. Boston: H Kluwer Ac. Publ.Google Scholar
  45. Kreinovich, V., H.T. Nguyen, and Y. Yam. 1999. Fuzzy systems are universal approximators for a smooth function and its derivatives. International Journal of Intelligent Systems 15 (6): 565–574.CrossRefMATHGoogle Scholar
  46. Lee, C.-C. 1990. Fuzzy Logic in Control Systems: Fuzzy Logic Controller, Part II. IEEE Transcation on System and Man Cybernetics 20: 419–435.Google Scholar
  47. Lee, C.-H. and C.-C. Teng. 2000. Identification and control of dynamic systems using recurrent fuzzy neural networks. IEEE Transaction on Fuzzy Systems 8(4): 349–366.Google Scholar
  48. Lin, C.-J and C.-C. Chin. 2004. Prediction and identification using wavelet-based recurrent fuzzy neural networks. IEEE Transcation Systems and Man Cybernetics Part B: Cybernetics34(5): 2144–2154.Google Scholar
  49. Lin, C.-T. and C. S. G. Lee. 1991. Neural-network-based fuzzy logic control and decision system. IEEE Transcation on Computers 40(12): 1320–1336.Google Scholar
  50. Lin, C.-T., and C.S.G. Lee. 1996. Neural fuzzy systems: A neuro-fuzzy synergism to intelligent systems. Upper Saddle River, NJ: Prentice-Hall PTR.Google Scholar
  51. Mendel, J. M. 1995. Fuzzy logic systems for engineering: A tutorial. IEEE Proceedings 83: 345–377.Google Scholar
  52. Mendel, J.M. 2001. Introduction to rule-based fuzzy logic systems. Upper Saddle River, NJ: Prentice-Hall.MATHGoogle Scholar
  53. Mendel, J. M. and Q. Liang. 1999. Comments on combinatorial rule explosion eliminated by a fuzzy rule configuration. by W. E. Combs and J. E. Andrews, IEEE Transaction on Fuzzy Systems, vol. 7, pp. 369–373, June 1999.Google Scholar
  54. Mendel, J.M., and D. Wu. 2010. Perceptual computing: aiding people in making subjective judgments. Hoboken, NJ: Wiley and IEEE Press.CrossRefGoogle Scholar
  55. Mizumoto, M. 1987. Comparison of various fuzzy reasoning methods. Proceeding of 2nd IFSA Congress Tokyo, Japan, pp. 2–7.Google Scholar
  56. Moody, J., and C.J. Darken. 1989. Fast learning in networks of locally-tuned processing units. Neural Computing 1: 281–294.CrossRefGoogle Scholar
  57. Mouzouris, G.C., and J.M. Mendel. 1996. In Proceedings on fifth IEEE international conference on fuzzy systems designing fuzzy logic systems for uncertain environments using a singular-value–QR decomposition method. New Orleans, LA:CrossRefGoogle Scholar
  58. Mouzouris, G. C. and J. M. Mendel. 1997. Non-singleton fuzzy logic systems: Theory and applications. IEEE Transaction on Fuzzy Systems 5: 56–71.Google Scholar
  59. Mouzouris, G.C., and J.M. Mendel. 1997b. A singular-value–QR decomposition based method for training fuzzy logic systems in uncertain environments. Journal on Intelligent and Fuzzy Systems 5: 367–374.Google Scholar
  60. Poggio, T. and F. Girosi. 1990. Networks for approximation and learning. Proceeding of IEEE, vol. 78, pp. 1481–1497.Google Scholar
  61. Raha, S., N. R. Pal and K. S. Ray. 2002. Similarity based approximate reasoning: methodology and application. IEEE Transaction on Systems, Man and Cybernetics A 32(2): 541–547.Google Scholar
  62. Raha, S., A. Hossain, and S. Ghosh. 2008. Similarity based approximate reasoning: Fuzzy control. Journal on Applied Logic 6: 47–71.MathSciNetCrossRefMATHGoogle Scholar
  63. Ross, T.J. 2004. Fuzzy logic with engineering applications, 2nd ed. Boston: Wiley.MATHGoogle Scholar
  64. Ruspini, E. 1969. A new approach to clustering. Info Control 15: 22–32.CrossRefMATHGoogle Scholar
  65. Setnes, M., and H. Roubos. 2000. GA-fuzzy modeling and classification: Complexity and performance. IEEE Transaction on Fuzzy System 8 (5): 509–522.CrossRefGoogle Scholar
  66. Specht, D.F. 1991. A general regression neural network. IEEE Transcation on Neural Networks 2: 568–576.CrossRefGoogle Scholar
  67. Stachowicz, M. S. and M. E. Kochanska. 1987. Fuzzy Modeling of the Process. In Proceedings on 2nd IFSA congress, Tokyo, Japan, pp. 86–89.Google Scholar
  68. Stover, J. A., D. L. Hall and R. E. Gibson. 1996. A fuzzy-logic architecture for autonomous multi-sensor data fusion. IEEE Transaction on Industrial Electronics 43: 403–410.Google Scholar
  69. Sugeno, M., and G.T. Kang. 1988. Structure identification of fuzzy model. Fuzzy Sets and Systems 28: 15–33.MathSciNetCrossRefMATHGoogle Scholar
  70. Sugeno, M., and T. Yasukawa. 1993. A fuzzy-logic-based approach to qualitative modeling. IEEE Transaction on Fuzzy System 1: 7–31.CrossRefGoogle Scholar
  71. Takagi, T. and M. Sugeno. 1985. Fuzzy identification of systems and its application to modeling and control. IEEE Transcation on Systems Man Cybernetics, 15: 116–132.Google Scholar
  72. Tanaka, K., M. Sano, and H. Watanabe. 1995. Modeling and control of carbon monoxide concentration using a neuro-fuzzy technique. IEEE Transaction on Fuzzy System 3: 271–279.CrossRefGoogle Scholar
  73. Tanaka, K., and M. Sugeno. 1998. Introduction to fuzzy modeling. In Fuzzy systems modeling and control, ed. T. Nguyen, and M. Sugeno, 63–89. Boston, MA: H Kluwer Academic Publ.Google Scholar
  74. Tanaka, K. and H. O. Wang. 2001. Fuzzy control systems design and analysis: A linear matrix inequality approach, Wiley Interscience.Google Scholar
  75. Tao, K. M. 1993. A closer look at the radial basis function (RBF) networks. In Proceeding of 27th Asilomar conference on signals, systems and computers, Pacific Grove, CA.Google Scholar
  76. Theocharis, J.B. 2006. A high-order recurrent neuro-fuzzy system with internal dynamics: Application to adaptive noise cancellation. Fuzzy Sets and Systems 157 (4): 471–500.MathSciNetCrossRefGoogle Scholar
  77. Tsang, F.C.C., J.W.T. Lee, and D.S. Yeung. 1995. Similarity based fuzzy reasoning methods for fuzzy production rules in Procedings, 157–160. World Congress: IFSA.Google Scholar
  78. Türksen, I. B. and Z. Zhong. 1988. An approximate analogical reasoning approach based on similarity measures. IEEE Transcation on Systems, Man and Cybernetics 18(6): 1049–1056.Google Scholar
  79. Vadiee, N. and M. Jamshidi. 1993.A tutorial on fuzzy rule-based expert systems (FRBES) models. 1: Mathematical foundations. Journal on Intelligent and Fuzzy Systems 1: 171–188.Google Scholar
  80. Wagner, C., A. Pourabdollah, J. McCulloch, R. John and J. Garibaldi. 2016. A similarity-based inference engine for non-singleton fuzzy logic systems. In Proceeding of IEEE international conference on fuzzy systems, Vancouver, Canada, pp. 316–323.Google Scholar
  81. Wang, K. Y., D. Tikk, T. D. Gedeon and L. T. Koczy. 2005. Fuzzy rule interpolation for multidimensional spaces with applications: A case study. IEEE Transaction on Fuzzy System 13(6): 809–819.Google Scholar
  82. Wang, L.-X. 1992a. Analysis and design of fuzzy systems, Ph.D. Dissertation, University of Southern California, Los Angeles, CA.Google Scholar
  83. Wang, L.-X. 1992b. Fuzzy systems are universal approximators. In Proceedings on IEEE international conference on fuzzy systems San Diego, CA.CrossRefGoogle Scholar
  84. Wang, L.-X. 1994. Adaptive fuzzy systems and control: Design and stability analysis. Englewood Cliffs, NJ: PTR Prentice-Hall.Google Scholar
  85. Wang, L.-X. 1997. A course in fuzzy systems and control. Upper Saddle River, NJ: Prentice-Hall.MATHGoogle Scholar
  86. Wang L.-X. 1999. Analysis and design of hierarchical fuzzy systems. IEEE Transaction on Fuzzy System 7: 617–624.Google Scholar
  87. Wang, L.-X. and J. M. Mendel. 1992a. Fuzzy basis functions, universal approximation, and orthogonal least squares learning. IEEE Transcation on Neural Networks 3: 807–813.Google Scholar
  88. Wang, L.-X. and J. M. Mendel. 1992b Generating fuzzy rules by learning from examples. IEEE Transcation on Systems and Man Cybernetics 22: 1414–1427.Google Scholar
  89. Wang, Y.-C., C.-J. Chien, and C.-C. Teng. 2004. Direct adaptive iterative learning control of nonlinear systems using an output-recurrent fuzzy neural network. IEEE Transcation on Systems and Man Cybernetics-B: Cybernetics, 34(3): 1348–1359.Google Scholar
  90. Weinschenk, J.J., W.E. Combs, and R.J. Marks II. 2003. In Proceedings on IEEE international conference on fuzzy systems avoidance of rule explosion by mapping fuzzy systems to a union rule configuration , 43–48. St. Louis MO.Google Scholar
  91. Wu, D. and J. M. Mendel. 2009. Perceptual reasoning for perceptual computing: A similarity-based approach. IEEE Transactions Fuzzy Systems 17: 1397–1411.Google Scholar
  92. Wu, D., and J.M. Mendel. 2011. On the continuity of type-1 and interval type-2 fuzzy logic systems. IEEE Transaction on Fuzzy System 19 (1): 179–192.CrossRefGoogle Scholar
  93. Yam, Y. P. Baranyi and C.-T. Yang. 1999. Reduction of fuzzy rule base via singular value decomposition. IEEE Transaction on Fuzzy System 7: 120–132.Google Scholar
  94. Yen, J., and R. Langari. 1999. Fuzzy logic: Intelligence, control, and information. Upper Saddle River, NJ: Prentice-Hall.Google Scholar
  95. Yen, J., and L. Wang. 1996. An SVD-based fuzzy model reduction strategy. In Proceedings of the fifth international conference on fuzzy systems, 835–841. New Orleans, LA.Google Scholar
  96. Yen, J. and L. Wang. 1999. Simplifying fuzzy rule-based models using orthogonal transformations. IEEE Transaction Systems Man and Cybernetics.Google Scholar
  97. Zadeh, L. A. 1973. Outline of a new approach to the analysis of complex systems and decision processes. IEEE Transaction Systems Man Cybernetics vol. SMC-3, pp. 28–44.Google Scholar
  98. Zhang, J. and A. J. Morris. 1999. Recurrent neuro-fuzzy networks for nonlinear process modeling. IEEE Transaction Neural Networks 10(2): 313–326.Google Scholar

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Authors and Affiliations

  1. 1.University of Southern CaliforniaLos AngelesUSA

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