Type-1 Fuzzy Systems: Design Methods and Applications

  • Jerry M. Mendel


This chapter focuses first on what exactly “design of a type-1 fuzzy system” means, and then provides a tabular way for making the choices that are needed in order to fully specify a type-1 fuzzy system, and introduces two approaches to design, the partially dependent approach and the totally independent approach. It then describes six design methods for designing a type-1 fuzzy system, namely: one-pass, least squares, derivative-based, SVD-QR, derivative-free and iterative. It then introduces and covers three case studies (forecasting of time series, knowledge mining using surveys, and fuzzy logic control, all of which are reexamined in Chap.  10), as well as the applications of forecasting of compressed video traffic, and rule-based classification of video traffic. Twelve examples are used to illustrate the chapter’s important concepts.


  1. Adas, A.M. 1998. Using adaptive linear prediction to support real-time VBR video under RCBR network service model. IEEE Transactions on Networking 6: 635–644.Google Scholar
  2. Arabshahi, P., J.J. Choi, R.J. Marks, II, and T.P. Caudell. 1996. Fuzzy parameter adaptation in optimization: Some neural net training examples. IEEE Computational Science & Engineering 57–65 (Spring).Google Scholar
  3. Astrom, K.J., and T. Hagglund. 2001. The future of PID control. Control Engineering Practice 9 (11): 1163–1175.Google Scholar
  4. Astrom, K. J., and T. Hagglund. 2005. Advanced PID control. ISA.Google Scholar
  5. Autonne, L. 1902. Sur Les Groupes Lineaires, Reels et Orthogonaux. Bulletin de la société mathématique de France 30: 121–133.MathSciNetzbMATHGoogle Scholar
  6. Casdagli, M. 1992. A dynamical systems approach to modeling input–output systems. In Nonlinear modeling and forecasting. Vol. XII of SFI studies in the sciences of complexity Process, 265–281. New York: Addison-Wesley.Google Scholar
  7. Castillo, O., and P. Melin. 2012. Optimization of type-2 fuzzy systems based on bio-inspired methods: A concise review. Information Sciences 205: 1–19.Google Scholar
  8. Castillo, O., R.M.-Marroquin, P. Melin, F. Valdez, and J. Soria. 2012. Comparative study of bio-inspired algorithms applied to the optimization of type-1 and type-2 fuzzy controllers for an autonomous mobile robot. Information Sciences 192: 19–38.Google Scholar
  9. Chen, S., S.A. Billings, and W. Luo. 1989. Orthogonal least squares methods and their application to nonlinear system identification. International Journal of Control 50: 1873–1896.MathSciNetzbMATHGoogle Scholar
  10. Chen, S., C.F.N. Cowan, and P.M. Grant. 1991. Orthogonal least squares learning algorithm for radial basis function networks. IEEE Transactions on Neural Networks 2: 302–309.Google Scholar
  11. Chiu, S. 1994. Fuzzy model identification based on cluster estimation. Journal of Intelligent and Fuzzy Systems 2: 267–278.Google Scholar
  12. Chiu, S. 1997a. Extracting fuzzy rules from data for function approximation and pattern classification. In Fuzzy information engineering: A guided tour of applications, ed. D. Dubois, H. Prade and R. Yager, Ch. 9. New York: Wiley.Google Scholar
  13. Chiu, S. 1997b. An efficient method for extracting fuzzy classification rules from high dimensional data. Journal of Advanced Computational Intelligence 1 (1): 1–7.Google Scholar
  14. Chu, P., and J.M. Mendel. 1994. First break refraction event picking using fuzzy logic systems. IEEE Transactions on Fuzzy Systems 2: 255–266.Google Scholar
  15. Cox, E.A. 1995. Fuzzy logic for business and industry. Rockland, MA: Charles River Media.Google Scholar
  16. Duan, X.-G., H.-X. Li, and H. Deng. 2008. Effective tuning method for fuzzy PID with internal model control. Industrial and Engineering Chemistry Research 47: 8317–8323.Google Scholar
  17. Duda, R.O. 1994. Elements of pattern recognition. In A prelude to neural networks: Adaptive and learning systems, ed. J.M. Mendel, 3–33. Englewood-Cliffs, NJ: Prentice-Hall.Google Scholar
  18. Duda, R.O., P.E. Hart, and D.G. Stork. 2001. Pattern classification, 2nd ed. New York: Wiley.zbMATHGoogle Scholar
  19. Eckart, C., and G. Young. 1939. A principal axis transformation for non-hermitian matrices. Bulletin of the American Mathematical Society 45: 118–121.MathSciNetzbMATHGoogle Scholar
  20. Farmer, J.D. 1982. Chaotic attractors of infinite-dimensional dynamical systems. Physica 4-D: 366–393.Google Scholar
  21. Feng, G. 2006. A survey an analysis and design of model-based fuzzy control systems. IEEE Transactions on Fuzzy Systems 14: 676–697.Google Scholar
  22. Francis, B.A., and W.M. Wonham. 1976. The internal model principle of control theory. Automatica 12: 457–465.MathSciNetzbMATHGoogle Scholar
  23. Galichet, S., and L. Foulloy. 1995. Fuzzy controllers: Synthesis and equivalences. IEEE Transactions on Fuzzy Systems 3: 140–148.zbMATHGoogle Scholar
  24. Golub, G.H., and C.F. Van Loan. 1983. Matrix computations. Baltimore, MD: Johns Hopkins Univ. Press.zbMATHGoogle Scholar
  25. Haykin, S. 1996. Adaptive filter theory, 3rd ed. Upper Saddle River, NJ: Prentice-Hall.zbMATHGoogle Scholar
  26. Hirota, K. 1995. History of industrial applications of fuzzy logic in Japan. In Industrial applications of fuzzy logic and intelligent systems, ed. J. Yen, R. Langari, L.A. Zadeh, 43–54. IEEE Press.Google Scholar
  27. Hohensohn, J., and J.M. Mendel. 1994. Two-pass orthogonal least-squares algorithm to train and reduce fuzzy logic systems. In Proceedings of the third IEEE conference on fuzzy systems, vol. 1, 696–700, Orlando, FL.Google Scholar
  28. Hohensohn, J., and J.M. Mendel. 1996. Two-pass orthogonal least-squares algorithm to train and reduce the complexity of fuzzy logic systems. Journal of Intelligent and Fuzzy Systems 4: 295–308.Google Scholar
  29. Holmblad, L., and I. Ostergaard. 1982. Control of a cement kiln by fuzzy logic. In Fuzzy information and decision-processes, ed. M.M. Gupta and E. Sanchez, 389–399. Amsterdam, The Netherlands: North-Holland.Google Scholar
  30. Horikawa, S., T. Furahashi, and Y. Uchikawa. 1992. On fuzzy modeling using fuzzy neural networks with back-propagation algorithm. IEEE Transactions on Neural Networks 3: 801–806.Google Scholar
  31. Hu, B., G.K.I. Mann, and R.G. Gasine. 2001. A systematic study of fuzzy PID controllers—Function-based evaluation approach. IEEE Transactions on Fuzzy Systems 9 (5): 699–711.Google Scholar
  32. Huang, T.T., H.Y. Chung, and J.J. Lin. 1999. A fuzzy PID controller being like parameter varying PID. In Proceedings of FUZZ-IEEE 1999, 269–275, Seoul, Korea.Google Scholar
  33. Jang, J.-S.R. 1992. Self-learning fuzzy controllers based on temporal back-propagation. IEEE Transactions on Neural Networks 3: 714–723.Google Scholar
  34. Jang, J.-S.R. 1993. ANFIS: Adaptive-network-based fuzzy inference system. IEEE Transactions on Systems, Man and Cybernetics 23: 665–684.Google Scholar
  35. Jang, J.-S.R., and C.-T. Sun. 1995. Neuro-fuzzy modeling and control. Proceedings of the IEEE 83: 378–406.Google Scholar
  36. Jang, J.-S.R., C.-T. Sun, and E. Mizutani. 1997. Neuro-fuzzy and soft-computing. Upper Saddle River, NJ: Prentice-Hall.Google Scholar
  37. 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:; then choose “sipi technical reports/418”.
  38. Kennedy, J., and R. Eberhart. 1995. Particle swarm optimization. In Proceedings of IEEE international conference on neural network, 1942–1948.Google Scholar
  39. Klema, V.C., and A.J. Laub. 1980. The singular-value decomposition: Its computation and some applications. IEEE Transactions on Automatic Control AC-25: 164–176.Google Scholar
  40. Krunz, M., R. Sass, and H. Hughes. 1995. Statistical characteristics and multiplexing of mpeg streams. In Proceedings of IEEE international conference on computer communications, INFOCOM’95, vol. 2, Boston, MA, 455–462.Google Scholar
  41. Kumbasar, T., and H. Hagras. 2015. Interval type-2 fuzzy PID controllers. In Springer handbook of computational intelligence, ed. J. Kacprzyk and W. Pedrycz, Chapter 18, 285–294. New York: Springer.Google Scholar
  42. Kuncheva, L.I. 2000. Fuzzy classifier design. Heidelberg: Physica-Verlag.zbMATHGoogle Scholar
  43. Lapedes, A.S., and R. Farber. 1987. Nonlinear signal processing using neural networks: Prediction and system modeling. Technical report LA-UR-87–2662, Los Alamos National Lab., Los Alamos, NM.Google Scholar
  44. Lee, C.-C. 1990. Fuzzy logic in control systems: Fuzzy logic controller, Part II. IEEE Transactions on Systems, Man, and Cybernetics. SMC-20: 419–435.Google Scholar
  45. Li H.X., and H.B. Gatland. 1996. Conventional fuzzy control and its enhancement. IEEE Transactions on Systems, Man, and Cybernetics—Part B 26(5): 791–797.Google Scholar
  46. Liang, Q., and J.M. Mendel. 2001. MPEG VBR video traffic modeling and classification using fuzzy techniques. IEEE Transactions on Fuzzy Systems 9: 183–193.Google Scholar
  47. 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
  48. MacDuffee, C.C. 1933. The theory of matrices. New York: Springer.zbMATHGoogle Scholar
  49. Mackey, M.C., and L. Glass. 1977. Oscillation and chaos in physiological control systems. Science 197: 287–289.zbMATHGoogle Scholar
  50. Mamdani, E.H. 1994. Fuzzy control—A misconception of theory and application. IEEE Expert-A Fuzzy Logic Symposium 9 (4): 27–28.zbMATHGoogle Scholar
  51. Mamdani, E.H., and S. Assilian. 1975. An experiment in linguistic synthesis with a fuzzy logic controller. International Journal of Man-Machine Studies 7: 1–13.zbMATHGoogle Scholar
  52. Manzoni, P., P. Cremonesi, and G. Serazzi. 1999. Workload models of VBR video traffic and their use in resource allocation policies. IEEE Transactions on Networking 7: 387–397.Google Scholar
  53. Mendel, J.M. 1994. A prelude to neural networks: Adaptive and learning systems. Englewood-Cliffs, NJ: Prentice-Hall.zbMATHGoogle Scholar
  54. Mendel, J.M. 1995a. Fuzzy logic systems for engineering: A tutorial. Proceedings of the IEEE 83: 345–377.Google Scholar
  55. Mendel, J.M. 1995b. Lessons in estimation theory for signal processing, communications and control. Englewood Cliffs, NJ: Prentice-Hall PTR.zbMATHGoogle Scholar
  56. Mendel, J.M. 2001. Introduction to rule-based fuzzy logic systems. Upper Saddle River, NJ: Prentice-Hall.zbMATHGoogle Scholar
  57. Mendel, J.M. 2002. An architecture for making judgments using computing with words. International Journal of Applied Mathematics and Computer Science 12 (3): 325–335.zbMATHGoogle Scholar
  58. Mendel, J.M. 2007. Computing with words: Zadeh, turing, popper and occam. IEEE Computational Intelligence Magazine 2: 10–17.Google Scholar
  59. Mendel, J.M. 2014. General type-2 fuzzy logic systems made simple: A tutorial. IEEE Transactions on Fuzzy Systems 22: 1162–1182.Google Scholar
  60. Mendel, J.M., and K.S. Fu, ed. 1970. Adaptive, learning and pattern recognition systems: Theory and applications. Academic Press, Inc.Google Scholar
  61. Mendel, J.M., and G.C. Mouzouris. 1997. Designing fuzzy logic systems. IEEE Transactions on Circuits and Systems–II: Analog and Digital Signal Processing 44: 885–895.Google Scholar
  62. Mendel, J.M., and D. Wu. 2010. Perceptual computing: Aiding people in making subjective judgments. Hoboken, NJ: Wiley and IEEE Press.Google Scholar
  63. Mendel, J.M., S. Murphy, L.C. Miller, M. Martin, and N. Karnik. 1999. The fuzzy logic advisor for social judgments. In Computing with words in information/intelligent systems, ed. L.A. Zadeh and J. Kacprzyk, 459–483. Physica-Verlag.Google Scholar
  64. Mendel, J.M., H. Hagras, W.-W. Tan, W.W. Melek, and H. Ying. 2014. Introduction to type-2 fuzzy logic control. Hoboken, NJ: John Wiley and IEEE Press.zbMATHGoogle Scholar
  65. Moody, J. 1989. Fast learning in multi-resolution hierarchies. In Advances in neural information processing systems I, ed. D.S. Touretzky, Chapter 1, 29–39. San Mateo, CA: Morgan Kaufman.Google Scholar
  66. Moody, J., and C.J. Darken. 1989. Fast learning in networks of locally-tuned processing units. Neural Computation 1: 281–294.Google Scholar
  67. Moon, B.S. 1995. Equivalence between fuzzy logic controllers and PI controllers for single input systems. Fuzzy Sets and Systems 69: 105–113.MathSciNetGoogle Scholar
  68. Mouzouris, G.C., and J.M. Mendel. 1996. Designing fuzzy logic systems for uncertain environments using a singular-value–QR decomposition method. In Proceedings of the fifth IEEE international conference on fuzzy systems, New Orleans, LA.Google Scholar
  69. Mouzouris, G.C., and J.M. Mendel. 1997. A singular-value–QR decomposition based method for training fuzzy logic systems in uncertain environments. Journal of Intelligent and Fuzzy Systems 5: 367–374.Google Scholar
  70. Munakata, T., and Y. Jani. 1994. Fuzzy systems: An overview. Communications of the ACM 37: 69–96.Google Scholar
  71. Palm, R. 1992. Sliding mode fuzzy control. In Proceedings of 1992 IEEE international conference on fuzzy systems, 519–526, San Diego, CA.Google Scholar
  72. Quinney, D. 1985. An introduction to the numerical solution of differential equations. England: Research Studies Press.zbMATHGoogle Scholar
  73. Qiao, W.Z., and M. Mizumoto. 1996. PID type fuzzy controller and parameters adaptive method. Fuzzy Sets and Systems 78: 23–35.MathSciNetzbMATHGoogle Scholar
  74. Rasband, S.N. 1990. Chaotic dynamics of non-linear systems. New York: Wiley.zbMATHGoogle Scholar
  75. Rickard, J.T., J. Aisbett, R. Yager, and G. Gibbon. 2011. Linguistic weighted power means: Comparison with the linguistic weighted average. In Proceedings of FUZZ-IEEE 2011, 2011 World Congress on Computational Intelligence, 2185–2192, Taipei, Taiwan.Google Scholar
  76. Rickard, J.T., J. Aisbett, R.R. Yager, and G. Gibbon. 2013. Computing with words using weighted power mean aggregation operators. In Soft computing: State of the art theory and novel applications, ed. R.R. Yager, A.M. Abbasov, M.Z. Reformat, and S. Shahbazova, 145–160. New York: Springer.Google Scholar
  77. Rose, O. 1995. Statistical properties of MPEG video traffic and their impact on traffic modeling in ATM systems. Univ. of Wurzburg, Institute of Computer Science, Research Report 101.Google Scholar
  78. Rutkowski, L. 2004. Flexible neuro-fuzzy systems: Structures, learning and performance evaluation. Boston: Kluwer.zbMATHGoogle Scholar
  79. Sanger, T.D. 1991. A tree-structured adaptive network for function approximation in high-dimensional spaces. IEEE Transaction on Neural Networks 2: 285–293.Google Scholar
  80. Seborg, D.E., F.E. Thomas, and A.M. Duncan. 2004. Process dynamics and control, 2nd ed. New York: Wiley.Google Scholar
  81. Setnes, M., and H. Hellendoorn. 2000. Orthogonal transforms for ordering and reduction of fuzzy rules. In Proceedings of FUZZ-IEEE’00, 700–705, San Antonio, TX.Google Scholar
  82. Simon, D. 2013. Evolutionary optimization algorithms. Hoboken, NJ: Wiley.Google Scholar
  83. Skogestad, S. 2003. Simple analytic rules for model reduction and PID controller tuning. Journal of Process Control 13 (4): 291–309.MathSciNetGoogle Scholar
  84. Stewart, G.W. 1973. Introduction to matrix computations. New York: Academic Press.zbMATHGoogle Scholar
  85. Sugeno, M. 1985. An introductory survey of fuzzy control. Information Sciences 36: 59–83.Google Scholar
  86. Sugeno, M., and T. Yasukawa. 1993. A fuzzy-logic-based approach to qualitative modeling. IEEE Transactions on Fuzzy Systems 1: 7–31.Google Scholar
  87. Sun, J., X. Wu, V. Palade, W. Fang, C.H. Lai, and W. Xu. 2012. Convergence analysis and improvements of quantum-behaved particle swarm optimization. Information Sciences 193: 81–103.MathSciNetGoogle Scholar
  88. Vaccaro, R. (ed.). 1991. SVD and signal processing algorithms, II, algorithms, analysis and applications. New York: Elsevier.Google Scholar
  89. Wang, L.-X. 1992. Analysis and design of fuzzy systems. Ph.D. dissertation, University of Southern California, Los Angeles, CA.Google Scholar
  90. Wang, L.-X. 1994. Adaptive fuzzy systems and control: Design and stability analysis. Englewood Cliffs, NJ: PTR Prentice-Hall.Google Scholar
  91. Wang, L.-X. 2003. The WM method completed: A flexible fuzzy system approach to data mining. IEEE Transactions on Fuzzy Systems 11 (6): 768–782.Google Scholar
  92. Wang, L.-X., and J.M. Mendel. 1991. Generating fuzzy rules from numerical data, with applications. USC-SIPI Report #169, January 1991.Google Scholar
  93. Wang, L.-X., and J.M. Mendel. 1992a. Fuzzy basis functions, universal approximation, and orthogonal least squares learning. IEEE Transactions on Neural Networks 3: 807–813.Google Scholar
  94. Wang, L.-X., and J.M. Mendel. 1992b. Back-propagation of fuzzy systems as non-linear dynamic system identifiers. In Proceedings of IEEE international conference on fuzzy systems, 1409–1418, San Diego, CA.Google Scholar
  95. Wang, L.-X., and J.M. Mendel. 1992c. Generating fuzzy rules by learning from examples. IEEE Transactions on Systems, Man and Cybernetics 22: 1414–1427.Google Scholar
  96. Wang, X., Y. He, L. Dong, and H. Zhao. 2011. Particle swarm optimization for determining fuzzy measures from data. Information Sciences 181: 4230–4252.zbMATHGoogle Scholar
  97. Wei, F., S. Jun, X.Z.-Ping, and W.-B. Xu. 2010. Convergence analysis of quantum-behaved particle swarm optimization algorithm and study on its control parameter. Acta Physica Sinica 59(6): 3686–3694.Google Scholar
  98. Wu, D., and J.M. Mendel. 2007. Aggregation using the linguistic weighted average and interval type-2 fuzzy sets. IEEE Transactions on Fuzzy Systems 15 (6): 1145–1161.Google Scholar
  99. Yager, R., and D. Filev. 1994. Generation of fuzzy rules by mountain clustering. Journal of Intelligent and Fuzzy Systems 2: 209–219.Google Scholar
  100. Yasunobu, S., and S. Miyamoto. 1985. Automatic train operation system by predictive fuzzy control. In Industrial applications of fuzzy control, ed. M. Sugeno. North-Holland, Amsterdam, The Netherlands: Elsevier Science.Google Scholar
  101. Yen, J., and R. Langari. 1999. Fuzzy logic: Intelligence, control, and information. Upper Saddle River, NJ: Prentice-Hall.Google Scholar
  102. 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
  103. Yen, J., and L. Wang. 1999. Simplifying fuzzy rule-based models using orthogonal transformations. IEEE Transactions on Systems, Man and Cybernetics 29, Part B.Google Scholar
  104. Ying, H. 2000. Fuzzy control and modeling: Analytical foundations and applications. Piscataway, NJ: IEEE Press.Google Scholar
  105. Zadeh, L.A. 1996. Fuzzy logic = computing with words. IEEE Transactions on Fuzzy Systems 4: 103–111.Google Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.University of Southern CaliforniaLos AngelesUSA

Personalised recommendations