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Fuzzy Systems pp 135–195Cite as

Fuzzy Rule Based Modeling as a Universal Approximation Tool

  • Chapter

Part of the book series: The Springer Handbook Series on Fuzzy Sets ((FSHS,volume 2))

Abstract

In some cases, fuzzy rule based model needs tuning. If we have applied some version of fuzzy rule based modeling, and the resulting model is satisfactory, great. But sometimes, the resulting model is not of very high quality:

  • ■ we may have misinterpreted some of the expert’s rules;

  • ■ we may have missed some of the important rules.

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References

Fuzzy rule-based modeling methodology: an additional reference

  1. Larsen, P. M. (1980). Industrial applications of fuzzy logic control, Internat. J. Man-Machine Stud., 12, 3–10.

    Article  Google Scholar 

Universal approximation results for the basic fuzzy rule based modeling methodology: basic results

  1. Buckley, J. J. (1992). Universal Fuzzy Controllers, Automatica, 28, 1245–1248.

    Article  MathSciNet  MATH  Google Scholar 

  2. Buckley, J. J. (1993). Controllable processes and the fuzzy controller, Fuzzy Sets and Systems, 53, 27–31.

    Article  MathSciNet  MATH  Google Scholar 

  3. Buckley, J. J. (1993a). Sugeno Type Controllers are Universal Controllers, Fuzzy Sets and Systems, 53, 299–303.

    Article  MathSciNet  MATH  Google Scholar 

  4. Buckley, J. J. (1993d). Approximation paper: Part I, Proceedings of the Third International Workshop on Neural Networks and Fuzzy Logic, Houston, TX, June 1–3, 1992, NASA, January 1993, I (NASA Conference Publication No. 10111). 170–173.

    Google Scholar 

  5. Buckley, J. J. (1993f). Applicability of the fuzzy controller, In: Wang, P. Z. and Loe, K. F. (eds.), Advances in Fuzzy Systems: Application and Theory, World Scientific, Singapore.

    Google Scholar 

  6. Buckley, J. J. and Czogala, E. (1993e). Fuzzy models, fuzzy controllers, and neural nets, Proc. Polish Academy of Sciences.

    Google Scholar 

  7. Buckley, J. J. and Hayashi, Y. (1993b). Fuzzy input-output controllers are universal approximators, Fuzzy Sets and Systems, 58, 273–278.

    Article  MathSciNet  MATH  Google Scholar 

  8. Buckley, J. J., Hayashi, Y., and Czogala, E. (1992a). On the equivalence of neural nets and fuzzy expert systems, Proc. of Int. Joint Conf. on Neural Networks, June 7-11, Baltimore, MD, 2, 691–695.

    Google Scholar 

  9. Buckley, J. J., Hayashi, Y., and Czogala, E. (1993c). On the equivalence of neural nets and fuzzy expert systems, Fuzzy Sets and Systems, 53, 129–134.

    Article  MathSciNet  MATH  Google Scholar 

  10. Cao, Z. (1990). Mathematical principle of fuzzy reasoning, Proceedings NAFIPS’90, Toronto, Canada, June 1990, 362–365.

    Google Scholar 

  11. Cao, Z., Kandel, A., and Han, J. (1990a). Mechanism of fuzzy logic controller, Proceedings of ISUMA’ 90, Univ. of Maryland, December 1990, 603–607.

    Google Scholar 

  12. Castro, J. L. (1995). Fuzzy logic controllers are universal approximators, IEEE Trans. Syst., Man, Cybern., 25(4), 629–635.

    Article  Google Scholar 

  13. Dubois, D., Grabisch, M., and Prade, H. (1992). Gradual rules and the approximation of functions, Proceedings of the 2nd International Conference on Fuzzy Logic and Neural Networks, Iizuka, Japan, July 17–22, 629–632.

    Google Scholar 

  14. Hayashi, Y., Buckley, J. J., and Czogala, E. (1992). Fuzzy expert systems versus neural networks, Proc. of Int. Joint Conf. on Neural Networks, June 7–11, Baltimore, MD, 2, 720–726.

    Google Scholar 

  15. Hayashi, Y., Buckley, J. J., and Czogala, E. (1992a). Approximations between fuzzy expert systems and neural networks, Proc. of the 2nd Int. Conf. on Fuzzy Logic and Neural Networks, July 17–22, Iizuka, Japan, 135–139.

    Google Scholar 

  16. Jou, C.-C. (1992). On the mapping capabilities of fuzzy inference systems, Proceedings of the International Joint Conference on Neural Networks, Baltimore, Maryland, June 7–11, 2, 708–713.

    Google Scholar 

  17. Kawamoto, S., Tada, K., Onoe, N., Ishigame, A., and Taniguchi, T. (1992). Construction of exact fuzzy system for nonlinear system and its stability analysis, 8th Fuzzy System Symposium, Hiroshima, 517–520 (in Japanese).

    Google Scholar 

  18. Kosko, B. (1991). Neural networks and fuzzy systems: a dynamical systems approach to machine intelligence, Prentice Hall, 1991.

    Google Scholar 

  19. Kosko, B. (1992). Fuzzy Systems as Universal Approximators, IEEE Int. Conf. on Fuzzy Systems, San Diego, CA, March 1992, 1143–1162.

    Google Scholar 

  20. Kosko, B. (1992a). Fuzzy function approximation, Proceedings of the International Joint Conference on Neural Networks, Baltimore, Maryland, June 7–11, 1, 209–213.

    Google Scholar 

  21. Kosko, B. (1994). Fuzzy Systems as Universal Approximators, IEEE Trans. on Computers, 43(11), 1329–1333.

    Article  MATH  Google Scholar 

  22. Kosko, B. (1994a). Optimal fuzzy rules cover extrema, Proceedings of the World Congress on Neural Networks WCNN’94.

    Google Scholar 

  23. Kosko, B. (1995). Optimal fuzzy rules cover extrema, International Journal of Intelligent Systems, 10(2), 249–255.

    Article  Google Scholar 

  24. Kosko, B. (1996a). Additive fuzzy systems: from function approximation to learning, In: C. H. Chen (ed.), Fuzzy Logic and Neural Network Handbook, McGraw-Hill, N.Y., 9-1-9-22.

    Google Scholar 

  25. Kosko, B. and Dickerson, J. A. (1995a). Function approximation with additive fuzzy systems, Chapter 12 in: Nguyen, H. T., Sugeno, M., Tong, R., and Yager, R. (eds.), Theoretical aspects of fuzzy control, J. Wiley, N.Y., 313–347.

    Google Scholar 

  26. Nguyen, H. T. and Kreinovich, V. (1992a). On approximation of controls by fuzzy systems, Technical Report 92-93/302, LIFE Chair of Fuzzy Theory, Tokyo Institute of Technology.

    Google Scholar 

  27. Nguyen, H. T. and Kreinovich, V. (1993). On Approximation of Controls by Fuzzy Systems, Proceedings of Fifth IFSA Congress, Seoul, Korea, 2, 1414–1417.

    Google Scholar 

  28. Wang, L.-X. (1992). Fuzzy Systems Are Universal Approximators, Proceedings of Second IEEE International Conference on Fuzzy Systems, San Diego, CA, March 1992, 1163–1170.

    Google Scholar 

  29. Wang, L.-X. (1994). Adaptive Fuzzy Systems and Control, Prentice-Hall, Englewood-Cliffs, NJ.

    Google Scholar 

  30. Wang, L.-X. and Mendel, J. M. (1991).} Generating fuzzy rules from numerical data with applications, University of Southern California, Signal and Image Processing Institute, Technical Report USC-SIPI # 169.

    Google Scholar 

  31. Wang, L.-X. and Mendel, J. M. (1992a). Fuzzy basis functions, universal approximation, and orthogonal least-squares learning, IEEE Transactions on Neural Networks, 3, 807–814.

    Article  Google Scholar 

  32. Wang, L.-X. and Mendel, J. M. (1992b). Generating fuzzy rules by learning from examples, IEEE Transactions on Systems, Man, and Cybernetics, 22, 1414–1417.

    Article  MathSciNet  Google Scholar 

  33. Yager, R. R. and Filev, D. P. (1994). Essentials of fuzzy modeling and control, J. Wiley & Sons.

    Google Scholar 

  34. Ying, H. (1994). Sufficient conditions on general fuzzy systems as function approximators, Automatica, 30(3), 521–525.

    Article  MATH  Google Scholar 

  35. Zeng, X.-J. and Singh, M. G. (1994). Approximation theory of fuzzy systems — SISO case, IEEE Trans. on Fuzzy Systems, 2(2), 162–176.

    Article  Google Scholar 

  36. Zeng, X.-J. and Singh, M. G. (1995). Approximation theory of fuzzy systems — MIMO case, IEEE Trans. on Fuzzy Systems, 3, 219–235.

    Article  Google Scholar 

Takagi-Sugeno methodology, its generalizations, and the corresponding universal approximation results

  1. Buckley, J. J. (1993a). Sugeno Type Controllers are Universal Controllers, Fuzzy Sets and Systems, 53, 299–303.

    Article  MathSciNet  MATH  Google Scholar 

  2. Buckley, J. J. (1993d). Approximation paper: Part I, Proceedings of the Third International Workshop on Neural Networks and Fuzzy Logic, Houston, TX, June 1-3, 1992, NASA, January 1993, I (NASA Conference Publication No. 10111). 170–173.

    Google Scholar 

  3. Buckley, J. J. and Hayashi, Y. (1993b). Fuzzy input-output controllers are universal approximators, Fuzzy Sets and Systems, 58, 273–278.

    Article  MathSciNet  MATH  Google Scholar 

  4. Cao, S. G., Rees, N. W., and Feng, G. (1995). Fuzzy modelling and identification for a class of complex dynamic systems, Proceedings of the Pacific-Asia Conference on Expert Systems, Huanshan, China, 212–217.

    Google Scholar 

  5. Cao, S. G., Rees, N. W., and Feng, G. (1996a). Analysis and design of uncertain fuzzy control systems. Part I: Fuzzy modelling and identification, Proceedings of the Fifth IEEE International Conference on Fuzzy Systems FUZZ-IEEE’96, New Orleans, September 8-11, 1, 640–646.

    Article  Google Scholar 

  6. Cao, S. G., Rees, N. W., and Feng, G. (1996b). Analysis and design of uncertain fuzzy control systems. Part II: Fuzzy controller design, Proceedings of the Fifth IEEE International Conference on Fuzzy Systems FUZZ-IEEE’96, New Orleans, September 8-11, 1, 647–653.

    Article  Google Scholar 

  7. Chak, C. K., Feng, G., and Cao, S. G. (1996). Universal fuzzy controllers, Proceedings of the Fifth IEEE International Conference on Fuzzy Systems FUZZ-IEEE’96, New Orleans, September 8–11, 3, 2020–2025.

    Article  Google Scholar 

  8. Takagi, T. and Sugeno, M. (1985). Fuzzy identification of systems and its application to modeling and control, IEEE Transactions on Systems, Man, and Cybernetics, 15, 116–132.

    Article  MATH  Google Scholar 

Universal approximation results for the ellipsoid approach

  1. Dickerson, J. A. and Kosko, B. (1993). Fuzzy function learning with covariant ellipsoids, Proceedings of the IEEE International Conference on Neural Networks, San Francisco, March 28-April 1, 3, 1162–1167.

    Article  Google Scholar 

  2. Dickerson, J. A. and Kosko, B. (1993a). Fuzzy function approximation with supervised ellipsoidal learning, Proceedings of the World Congress on Neural Networks, Portland, Oregon, July 11–15, 2, 9–17.

    Google Scholar 

  3. Dickerson, J. A. and Kosko, B. (1996). Fuzzy function approximation with ellipsoidal rules, IEEE Trans. Syst., Man, Cybern., 26(4), 542–560.

    Article  Google Scholar 

  4. Kim, H. M. and Kosko, B. (1996). Fuzzy prediction and filtering in impulsive noise, Fuzzy Sets and Systems, 77, 15–33.

    Article  Google Scholar 

  5. Kosko, B. (1994). Fuzzy Systems as Universal Approximators, IEEE Trans. on Computers, 43(11), 1329–1333.

    Article  MATH  Google Scholar 

  6. Kosko, B. (1994a). Optimal fuzzy rules cover extrema, Proceedings of the World Congress on Neural Networks WCNN’94.

    Google Scholar 

  7. Kosko, B. (1995). Optimal fuzzy rules cover extrema, International Journal of Intelligent Systems, 10(2), 249–255.

    Article  Google Scholar 

  8. Kosko, B. (1996a). Additive fuzzy systems: from function approximation to learning, In: Chen, C. H. (ed.), Fuzzy Logic and Neural Network Handbook, McGraw-Hill, N.Y., 9-1-9-22.

    Google Scholar 

Universal approximation results for the case when fuzzy if-then rules use a more complicated implication

  1. Dubois, D., Grabisch, M., and Prade, H. (1992). Gradual rules and the approximation of functions, Proceedings of the 2nd International Conference on Fuzzy Logic and Neural Networks, Iizuka, Japan, July 17–22, 629–632.

    Google Scholar 

  2. Dubois, D., Grabisch, M., and Prade, H. (1993). Synthesis of real-valued mappings based on gradual rules and interpolative reasoning, Proc. of the IJCAI’93 Fuzzy Logic in AI Workshop, Chaméry, France, Aug. 28-Sep. 3, 29–40.

    Google Scholar 

  3. Dubois, D., Prade, H., and Grabisch, M. (1995). Gradual rules and the approximation of control laws, In: Nguyen, H. T., Sugeno, M., Tong, R., and Yager, R. (eds.), Theoretical aspects of fuzzy control, J. Wiley, N.Y., 147–181.

    Google Scholar 

Universal approximation results for the case when different grade of truth are assigned to different rules

  1. Dickerson, J. A. and Kosko, B. (1993). Fuzzy function learning with covariant ellipsoids, Proceedings of the IEEE International Conference on Neural Networks, San Francisco, March 28-April 1, 3, 1162–1167.

    Article  Google Scholar 

  2. Dickerson, J. A. and Kosko, B. (1993a). Fuzzy function approximation with supervised ellipsoidal learning, Proceedings of the World Congress on Neural Networks, Portland, Oregon, July 11–15, 2, 9–17.

    Google Scholar 

  3. Dickerson, J. A. and Kosko, B. (1996). Fuzzy function approximation with ellipsoidal rules, IEEE Trans. Syst., Man, Cybern., 26(4), 542–560.

    Article  Google Scholar 

  4. Kosko, B. (1992). Fuzzy Systems as Universal Approximators, IEEE Int. Conf. on Fuzzy Systems, San Diego, CA, March 1992, 1143–1162.

    Google Scholar 

  5. Kosko, B. (1996a). Additive fuzzy systems: from function approximation to learning, In: Chen, C. H. (ed.), Fuzzy Logic and Neural Network Handbook, McGraw-Hill, N.Y., 9-1-9-22.

    Google Scholar 

  6. Kosko, B. and Dickerson, J. A. (1995a). Function approximation with additive fuzzy systems, Chapter 12 in: Nguyen, H. T., Sugeno, M., Tong, R., and Yager, R. (eds.), Theoretical aspects of fuzzy control, J. Wiley, N.Y., 313–347.

    Google Scholar 

Universal approximation results for hierarchical systems

  1. Buckley, J. J. (1993). Controllable processes and the fuzzy controller, Fuzzy Sets and Systems, 53, 27–31.

    Article  MathSciNet  MATH  Google Scholar 

  2. Buckley, J. J., and Hayashi, Y. (1993b). Fuzzy input-output controllers are universal approximators, Fuzzy Sets and Systems, 58, 273–278.

    Article  MathSciNet  MATH  Google Scholar 

  3. Buckley, J. J., Hayashi, Y., and Czogala, E. (1992a). On the equivalence of neural nets and fuzzy expert systems, Proc. of Int. Joint Conf. on Neural Networks, June 7–11, Baltimore, MD, 2, 691–695.

    Google Scholar 

  4. Buckley, J. J., Hayashi, Y., and Czogala, E. (1993c). On the equivalence of neural nets and fuzzy expert systems, Fuzzy Sets and Systems, 53, 129–134.

    Article  MathSciNet  MATH  Google Scholar 

  5. Hayashi, Y. and Buckley, J. J. (1994). Approximations between fuzzy expert systems and neural networks, International Journal of Approximate Reasoning, 10(1), 63–73.

    Article  MATH  Google Scholar 

  6. Hayashi, Y., Buckley, J. J., and Czogala, E. (1992). Fuzzy expert systems versus neural networks, Proc. of Int. Joint Conf. on Neural Networks, June 7–11, Baltimore, MD, 2, 720–726.

    Google Scholar 

  7. Hayashi, Y., Buckley, J. J., and Czogala, E. (1992a). Approximations between fuzzy expert systems and neural networks, Proc. of the 2nd Int. Conf. on Fuzzy Logic and Neural Networks, July 17–22, Iizuka, Japan, 135–139.

    Google Scholar 

  8. Jang, J.-S. R. and Sun, C.-T. (1993). Functional equivalence between radial basis function networks and fuzzy inference systems, IEEE Transactions on Neural Networks, 4, 156–159.

    Article  Google Scholar 

  9. Kosko, B. (1995b). Combining fuzzy systems, Proceedings of the IEEE International Conference on Fuzzy Systems FUZZ-IEEE/IFES’95, March 1995, 4, 1855–1863.

    Google Scholar 

Universal approximation results for simplified versions of fuzzy rule based modeling methodology

  1. Buckley, J. J. (1993a). Sugeno Type Controllers are Universal Controllers, Fuzzy Sets and Systems, 53, 299–303.

    Article  MathSciNet  MATH  Google Scholar 

  2. Buckley, J. J. (1993d). Approximation paper: Part I, Proceedings of the Third International Workshop on Neural Networks and Fuzzy Logic, Houston, TX, June 1-3, 1992, NASA, January 1993, I (NASA Conference Publication No. 10111). 170–173.

    Google Scholar 

  3. El Hajjaji, A. and Rachid, A. (1994). Explicit formulas for fuzzy controller, Fuzzy Sets and Systems, 62, 135–141.

    Article  MathSciNet  Google Scholar 

  4. Kóczy, L. T. and Tikk, D. (1996). Approximation in rule bases, IPMU’96: Proceedings of the International Conference on Information Processing and Management of Uncertainty in Knowledge-Based Systems, Granada, July 1–5, 1996, 489–494.

    Google Scholar 

  5. Su, C.-Y. and Stepanenko, Y. (1994). Adaptive control of a class of nonlinear systems with fuzzy logic, Proceedings of the Third IEEE International Conference on Fuzzy Systems FUZZ-IEEE’94, Orlando, FL, June 26-29, 2, 779–785.

    Google Scholar 

  6. Su, C.-Y. and Stepanenko, Y. (1994a). Adaptive control of a class of nonlinear systems with fuzzy logic, IEEE Transactions on Fuzzy Systems, 2(4), 285–294.

    Article  Google Scholar 

  7. Tan, S., and Vandewalle, J. (1996). Defuzzification, structure transparency, and fuzzy system learning, Proceedings of the Fifth IEEE International Conference on Fuzzy Systems FUZZ-IEEE’96, New Orleans, September 8–11, 1, 470–474.

    Article  Google Scholar 

  8. Zeng, X.-J. and Singh, M. G. (1996). Approximation accuracy analysis of fuzzy systems as function approximators, IEEE Trans. on Fuzzy Systems, 4, 44–63.

    Article  Google Scholar 

Universal approximation results for distributed systems (and for universal controllers)

  1. Cao, S. G., Rees, N. W., and Feng, G. (1996). Fuzzy control of nonlinear discrete-time systems, Proceedings of the Fifth IEEE International Conference on Fuzzy Systems FUZZ-IEEE’96, New Orleans, September 8’11, 1, 265–271.

    Article  Google Scholar 

  2. Kreinovich, V., Nguyen, H. T., and Sirisaengtaksin, O. (1994a). On approximations of controls in distributed systems by fuzzy controllers, Proceedings of the 5th International Conference on Information Processing and Management of Uncertainty in Knowledge-Based Systems IPMU’94, Paris, July 4–8, 1, 79–83.

    Google Scholar 

  3. Kreinovich, V., Nguyen, H. T., and Sirisaengtaksin, O. (1995a). On approximation of controls in distributed systems by fuzzy controllers, In: Bouchon-Meunier, B., Yager, R. R., and Zadeh, L. A. (eds.), Fuzzy Logic and Soft Computing, World Scientific, 137–145.

    Google Scholar 

  4. Nguyen, H. T., Kreinovich, V., and Sirisaengtaksin, O. (1996). Fuzzy control as a universal control tool, Fuzzy Sets and Systems, 80(1), 71–86.

    Article  MathSciNet  MATH  Google Scholar 

Universal approximation results for discrete systems: Application to expert system design

  1. Garey, M. and Johnson, D. (1979). Computers and intractability: a guide to the theory of NP-completeness, Freeman, San Francisco.

    MATH  Google Scholar 

  2. Kreinovich, V. (1987). Semantics of S. Yu. Maslov’s iterative method, Problems of Cybernetics, Moscow, 1987, 131, 30–62 (in Russian); English translation in: Kreinovich, V. and Mints, G. (eds.), Problems of reducing the exhaustive search, American Mathematical Society, Providence, RI, 1996.

    MathSciNet  Google Scholar 

  3. Kreinovich, V. (1996). S. Maslov’s Iterative Method: 15 Years Later (Freedom of Choice, Neural Networks, Numerical Optimization, Uncertainty Reasoning, and Chemical Computing), a chapter in Kreinovich, V. and Mints, G. (eds.), Problems of reducing the exhaustive search, American Mathematical Society, Providence, RI.

    Google Scholar 

  4. Kreinovich, V. and Fuentes, L. O. (1991). Simulation of chemical kinetics — a promising approach to inference engines, in: Liebowitz. J. (ed.), Proceedings of the World Congress on Expert Systems, Orlando, Florida, 1991, Pergamon Press, N.Y., 3, 1510–1517.

    Google Scholar 

  5. Kreinovich, V. and Mints, G. (eds.) (1996a). Problems of reducing the exhaustive search, American Mathematical Society (AMS Translations — Series 2), 178, Providence, RI.

    Google Scholar 

  6. Maslov, S. Yu. (1981). Iterative methods in intractable problems as a model of intuitive methods, Abstracts of the 9th All-Union Symposium on Cybernetics, 52–56 (in Russian).

    Google Scholar 

  7. Maslov, S. Yu. (1983). Asymmetry of cognitive mechanisms and its implications, Semiotika i Informatisa, 20, 3–31 (in Russian).

    MATH  Google Scholar 

  8. Maslov, S. Yu. (1987). Theory of Deductive Systems and its Applications, MIT Press, Cambridge, MA.

    MATH  Google Scholar 

  9. Sirisaengtaksin, O., Fuentes, L. O., and Kreinovich, V. (1995). Non-traditional neural networks that solve one more intractable problem: propositional satisfiability, Proceedings of the First International Conference on Neural, Parallel, and Scientific Computations, Atlanta, GA, May 28-31,1995,1, 427–430.

    Google Scholar 

Universal approximation results that describe the possibility of approximating a smooth system by a smooth one, and a simple system by a simple one

  1. Fantuzzi, C. and Rovatti, R. (1996). On the approximation capabilities of the homogeneous Takagi-Sugeno model, Proceedings of the Fifth IEEE International Conference on Fuzzy Systems FUZZ-IEEE’96, New Orleans, September 8–11, 2, 1067–1072.

    Article  Google Scholar 

  2. Galichet, S. and Foulloy, L. (1993). Fuzzy equivalence of classical controllers, Proceedings of the 1st European Congress on Fuzzy and Intelligent Technologies, Aachen, Germany, 3, 1567–1573.

    Google Scholar 

  3. Gordon, W. J. and Riesenfeld, R. F. (1974). B-spline curves and surfaces, In: Barnhill, R. E. and Riesenfeld, R. F. (eds.), Computer Aided Geometric Design, Academic Press, N.Y.

    Google Scholar 

  4. Kovalerchuk, B. (1996). Second interpolation in fuzzy control, Proceedings of the Fifth IEEE International Conference on Fuzzy Systems FUZZ-IEEE’96, New Orleans, September 8–11, 1, 150–155.

    Article  Google Scholar 

  5. Kovalerchuk, B. (1996a). Fuzzy control with the second interpolation, Working paper, Louisiana State University, Baton Rouge, LA, 1996.

    Google Scholar 

  6. Kovalerchuk, B. and Yusupov, H. (1993). Fuzzy control as interpolation, Proceedings of the 5th IFSA Congress, Seoul, Korea, July 4–9, 1151–1154.

    Google Scholar 

  7. Kovalerchuk, B., Yusupov, H., and Kovalerchuk, N. (1994). Comparison of interpolations in fuzzy control, Proceedings of the 2nd IFAC Workshop on Computer Software Structures Integrating AI/KBS Systems in Process Control, Lund, Sweden, 76–81.

    Google Scholar 

  8. Kreinovich, V., Quintana, C, Lea, R., Fuentes, O., Lokshin, A., Kumar, S., Boricheva, I., and Reznik, L. (1992). What non-linearity to choose? Mathematical foundations of fuzzy control, Proceedings of the 1992 International Conference on Fuzzy Systems and Intelligent Control, Louisville, KY, 349–412.

    Google Scholar 

  9. Meyer-Gramann, K. D. (1993). Easy implementation of fuzzy controller with a smooth control surface, Proceedings of the 1st European Congress on Fuzzy and Intelligent Technologies, Aachen, Germany, 1, 117–123.

    Google Scholar 

  10. Mohler, R. R. (1991). Nonlinear systems. 1. Dynamics and control, Prentice Hall, Englewood Cliff, NJ.

    Google Scholar 

  11. Raymond, C., Boverie, S., and Le, J. M. (1993). Practical realization of fuzzy controllers; comparison with conventional methods, Proceedings of the 1st European Congress on Fuzzy and Intelligent Technologies, Aachen, Germany, 1, 149–155.

    Google Scholar 

  12. Rovatti, R. (1996). Takagi-Sugeno models as approximators in Sobolev norms: the SISO case, Proceedings of the Fifth IEEE International Conference on Fuzzy Systems FUZZ-IEEW’96, New Orleans, September 8–11, 2, 1060–1066.

    Article  Google Scholar 

  13. Runkler, T. A. and Glesner, M. (1993). A new approach to fuzzy logic controller realizations using B-splines, Proceedings of the International Workshop on Current Issues in Fuzzy technology, Trento, Italy.

    Google Scholar 

  14. Runkler, T. A. and Glesner, M. (1993). Approximative Synthese von Fuzzy-Controllern in Fuzzy Logic, In: Reusch, B. (ed.), Fuzzy Logic — Theorie und Praxis, Springer-Verlag, Berlin, 22–31.

    Google Scholar 

  15. Tilli, T. A. (1993). Practical tools for simulation and optimization fuzzy systems with various operators and defuzzification methods, Proceedings of the 1st European Congress on Fuzzy and Intelligent Technologies, Aachen, Germany, 1, 256–262.

    Google Scholar 

  16. Zhang, J. and Knoll, A. (1996). Constructing fuzzy controllers with B-spline models, Proceedings of the Fifth IEEE International Conference on Fuzzy Systems FUZZ-IEEE’96, New Orleans, September 8–11, 1, 416–421.

    Article  Google Scholar 

  17. Zhang, J., Raczkowsky, J., and Herp, A. (1994). Emulation of spline curves and its application in robot motion control, Proceedings of the IEEE International Conference on Fuzzy Systems FUZZ-IEEE’91 Orlando, FL, 831–836.

    Google Scholar 

  18. Zhang, J., Wille, F., and Knoll, A. (1996a). Modular design of fuzzy controller integrating deliberate and reactive strategies, Proceedings of the IEEE International Conference on Robotics and Automation.

    Google Scholar 

Universal approximation results and stability issues

  1. Buckley, J. J. (1992). Universal Fuzzy Controllers, Automatica, 28, 1245–1248.

    Article  MathSciNet  MATH  Google Scholar 

  2. Buckley, J. J. (1993). Controllable processes and the fuzzy controller, Fuzzy Sets and Systems, 53, 27–31.

    Article  MathSciNet  MATH  Google Scholar 

  3. Buckley, J. J. (1993f). Applicability of the fuzzy controller, In: Wang, P.-Z. and Loe, K. F. (eds.), Advances in Fuzzy Systems: Application and Theory, World Scientific, Singapore.

    Google Scholar 

  4. Buckley, J. J. (1995). System stability and the fuzzy controller, In: Nguyen, H. T., Sugeno, M., Tong, R., and Yager, R. (eds.), Theoretical aspects of fuzzy control, J. Wiley and Sons, N.Y., 51–63.

    Google Scholar 

  5. Cao, S. G., Rees, N. W., and Feng, G. (1996). Fuzzy control of nonlinear discrete-time systems, Proceedings of the Fifth IEEE International Conference on Fuzzy Systems FUZZ-IEEE’96, New Orleans, September 8–11, 1, 265–271.

    Article  Google Scholar 

  6. Chak, C. K., Feng, G., and Cao, S. G. (1996). Universal fuzzy controllers, Proceedings of the Fifth IEEE International Conference on Fuzzy Systems FUZZ-IEEE’96, New Orleans, September 8–11, 3, 2020–2025.

    Article  Google Scholar 

  7. Jagannathan, S. (1996). Adaptive discrete-time fuzzy logic control of feedback linearizable nonlinear systems, Proceedings of the Fifth IEEE International Conference on Fuzzy Systems FUZZ-IEEE’96, New Orleans, September 8–11, 2, 1273–1278.

    Article  Google Scholar 

  8. Lei, S. and Langari, R. (1996). An approach to synthesis and approximation of stable fuzzy logic controllers, Proceedings of the Fifth IEEE International Conference on Fuzzy Systems FUZZ-IEEE’96, New Orleans, September 8–11, 2, 1446–1452.

    Article  Google Scholar 

  9. Wang, L.-X. (1993). Stable adaptive fuzzy control of nonlinear systems, IEEE Trans. Fuzzy Syst., 1, 146–155.

    Article  Google Scholar 

Universal approximation results for the case when we have both fuzzy if-then rules describing a model and fuzzy if-then rules describing control

  1. Abello, J., Kreinovich, V., Nguyen, H. T., Sudarsky, S., and Yen, J. (1994). Computing an appropriate control strategy based only on a given plant’s rule-based model is NP-hard, In: Hall, L., Ying, H., Langari, R., and Yen, J. (eds.), NAFIPS/IFIS/NASA’ 94, Proceedings of the First International Joint Conference of The North American Fuzzy Information Processing Society Biannual Conference, The Industrial Fuzzy Control and Intelligent Systems Conference, and The NASA Joint Technology Workshop on Neural Networks and Fuzzy Logic, San Antonio, December 18–21, 1994, IEEE, Piscataway, NJ, 331–332.

    Google Scholar 

  2. Bouchon-Meunier, B., Kreinovich, V., Lokshin, A., and Nguyen, H. T. (1996). On the formulation of optimization under elastic constraints (with control in mind). Fuzzy Sets and Systems, 81(1), 5–29.

    Article  MathSciNet  MATH  Google Scholar 

  3. Chen, G., Pham, T. T., and Weiss, J. J. (1995). Fuzzy modeling of control systems, IEEE Transactions on Aerospace and Electronic Systems, 31(1), 414–429.

    Article  Google Scholar 

  4. Nguyen, H. T., Kreinovich, V., and Sirisaengtaksin, O. (1996). Fuzzy control as a universal control tool, Fuzzy Sets and Systems, 80(1), 71–86.

    Article  MathSciNet  MATH  Google Scholar 

  5. Pham, T. T., Weiss, J. J., and Chen, G. R. (1992). Optimal fuzzy logic control for docking a boat, Proc. of the Second International Workshop on Industrial Fuzzy Control and Intelligent Systems, Dec. 2–4, 1992, College Station, TX, 66–73.

    Google Scholar 

Universal approximation results for the case when expert rules use unusual logical connectives

  1. Kreinovich, V., Nguyen, H. T., and Walker, E. A. (1996b). Maximum entropy (MaxEnt) method in expert systems and intelligent control: new possibilities and limitations, In: Hanson, K. M. and Silver, R. N. (eds.), Maximum Entropy and Bayesian Methods, Kluwer Academic Publishers, Dordrecht, 93–100.

    Chapter  Google Scholar 

  2. Nguyen, H. T., and Kreinovich, V. (1997). Kolmogorov’s Theorem and its impact on soft computing, In: Yager, R. R. and Kacprzyk, J. The Ordered Weighted Averaging Operators: Theory, Methodology, and Applications, Kluwer, Norwell, MA, 3–17.

    Chapter  Google Scholar 

  3. Nguyen, H. T., Kreinovich, V., and Sprecher, D. (1996a). Normal forms for fuzzy logic — an application of Kolmogorov’s theorem International Journal on Uncertainty, Fuzziness, and Knowledge-Based Systems, 4(4), 331–349.

    Article  MathSciNet  MATH  Google Scholar 

  4. Türkşen, I. B. and Kreinovich, V. (1996). Fuzzy implication revisited: a new type of fuzzy implication explains Yager’s implication operation, In: Dimitrov, V. and Dimitrov, J. (eds.), Fuzzy Logic and the Management of Complexity (Proceedings of the 1996 International Discourse), UTS Publ., Sydney, Australia, 3, 292–295.

    Google Scholar 

Universal approximation results for neural networks

  1. Blum, E. K. and Li, L. K. (1991). Approximation Theory and feedforward networks, Neural Networks, 4, 511–515.

    Article  Google Scholar 

  2. Cotter, N. E. (1990). The Stone-Weierstrass Theorem and Its Applications to Neural Networks, IEEE Trans. on Neural Networks, 1(4), 290–295.

    Article  MathSciNet  Google Scholar 

  3. Cybenko, G. (1989). Approximation by superpositions of a sigmoidal function, Math, of Control, Signals, and Systems, 2, 303-314, 1989.

    MathSciNet  Google Scholar 

  4. Funahashi, K. (1989). On the Approximate Realization of Continuous Mappings by Neural Networks, Neural Networks, 2, 183–192.

    Article  Google Scholar 

  5. Funahashi, K. and Nakamura, Y. (1993). Approximation of Dynamical Systems by continuous time recurrent neural networks, Neural Networks, 6, 801–806.

    Article  Google Scholar 

  6. Haykin, S. (1994). Neural Networks: A Comprehensive Foundation, Macmillan College Publishing Company, New York, NY.

    MATH  Google Scholar 

  7. Hecht-Nielsen, R. (1987). Kolmogorov’s Mapping Neural Network Existence Theorem, Proceedings of First IEEE International Conference on Neural Networks, 11–14, San Diego, CA.

    Google Scholar 

  8. Hornik, K., Stinchcombe, M., and White, H. (1989). Multilayer Feedforward Neural Networks Are Universal Approximators, Neural Networks, 2, 359–366.

    Article  Google Scholar 

  9. Hornik, K. (1991). Approximation Capabilities of Multilayer Feedforward Neural Networks, Neural Networks, 4, 251–257.

    Article  Google Scholar 

  10. Ito, Y. (1991). Approximation of functions on a compact set by finite sums of a sigmoid function without scaling, Neural Networks, 4, 817–826.

    Article  Google Scholar 

  11. Ito, Y. (1992). Approximation of continuous functions on R d by linear combinations of shifted rotations of a sigmoid function with and without scaling, Neural Networks, 5, 105–115.

    Article  Google Scholar 

  12. Kosko, B. (1992b). Neural Networks and Fuzzy Systems, Prentice Hall, Englewood Cliffs, NJ.

    Google Scholar 

  13. Kreinovich, V. (1991b). Arbitrary Nonlinearity is Sufficient to Represent All Functions by Neural Networks: A Theorem, Neural Networks, 4, 381–383.

    Article  Google Scholar 

  14. Kreinovich, V. and Quintana, C. (1991a). Neural networks: what nonlinearity to choose?, Proceedings of the 4th University of New Brunswick Artificial Intelligence Workshop, Fredericton, N.B., Canada, 627–637.

    Google Scholar 

  15. Kreinovich, V. and Sirisaengtaksin, O. (1993). 3-layer neural networks are universal approximators for functionals and for control strategies, Neural, Parallel, and Scientific Computations, 1, 325–346.

    MathSciNet  MATH  Google Scholar 

  16. Kreinovich, V., Sirisaengtaksin, O., and Nguyen, N. T. (1995). Sigmoid neurons are the safest against additive errors, Proceedings of the First International Conference on Neural, Parallel, and Scientific Computations, Atlanta, GA, May 28–31, 1, 419–423.

    MathSciNet  Google Scholar 

  17. Kůrková, V. (1991). Kolmogorov’s Theorem is Relevant, Neural Computation, 3, 617–622.

    Article  Google Scholar 

  18. Kurkova, V. (1992). Kolmogorov’s Theorem and Multilayer Neural Networks, Neural Networks, 5, 501–506.

    Article  Google Scholar 

  19. Kurkova, V. (1992a). Universal approximation using feedforward neural networks with Gaussian bar units, Proceedings of ECAI’92, European Conference on Artificial Intelligence, Wiley, Chichester, 193–197.

    Google Scholar 

  20. Leshno, M., Lin, V. Ya., Pinkus, A., and Schocken, S. (1993). Multilayer Feedforward Networks With a Nonpolynomial Activation Function Can Approximate Any Function, Neural Networks, 6, 861–867.

    Article  Google Scholar 

  21. Nakamura, M., Mines, R., and Kreinovich, V. (1993). Guaranteed intervals for Kolmogorov’s theorem (and their possible relation to neural networks). Interval Computations, No. 3, 183–199.

    Google Scholar 

  22. Park, J. and Sandberg, J. W. (1991). Universal approximation using radial basis function networks, Neural Computation, 3, 246-257.

    Google Scholar 

  23. Sirisaengtaksin, O. and Kreinovich, V. (1993). Neural networks that are not sensitive to the imprecision of hardware neurons, Interval Computations, No. 4, p. 100–113.

    Google Scholar 

  24. Wray, J. and Green, G. G. R. (1995). Neural networks, approximation theory, and finite precision computation, Neural Networks, 8(1), 31–37.

    Article  Google Scholar 

Universal approximation results for fuzzy neural networks

  1. Buckley, J. J. and Hayashi, Y. (1994). Can fuzzy neural networks approximate continuous fuzzy functions?, Fuzzy Sets and Systems, 61, 43–52.

    Article  MathSciNet  MATH  Google Scholar 

  2. Buckley, J. J. and Hayashi, Y. (1994a). Hybrid fuzzy neural networks are universal approximators, Proceedings of the Third IEEE International Conference on Fuzzy Systems FUZZ-IEEE’94, Orlando, FL, June 26-29, 1, 238–243.

    Article  Google Scholar 

  3. Feuring, T. and Lippe, W.-M. (1995). Fuzzy neural networks are universal approximators, Proceedings of the IFSA World Congress, San Paulo, Brazil, 659–662.

    Google Scholar 

  4. Hayashi, Y. and Buckley, J. J. (1994). Approximations between fuzzy expert systems and neural networks, International Journal of Approximate Reasoning, 10(1), 63–73.

    Article  MATH  Google Scholar 

Universal approximation results for systems with fuzzy inputs and interval inputs

  1. Buckley, J. J. (1991). Fuzzy I/O controller, Fuzzy Sets and Systems, 43, 127–137.

    Article  MathSciNet  MATH  Google Scholar 

  2. Buckley, J. J. and Hayashi, Y. (1993b). Fuzzy input-output controllers are universal approximators, Fuzzy Sets and Systems, 58, 273–278.

    Article  MathSciNet  MATH  Google Scholar 

  3. Kreinovich, V. and Nguyen, H. T. (1994b). Applications of fuzzy intervals: a skeletal outline, In: Hall, L., Ying, H., Langari, R., and Yen, J. (eds.), NAFIPS/IFIS/NASA’ 94, Proceedings of the First International Joint Conference of The North American Fuzzy Information Processing Society Biannual Conference, The Industrial Fuzzy Control and Intelligent Systems Conference, and The NASA Joint Technology Workshop on Neural Networks and Fuzzy Logic, San Antonio, December 18–21, 1994, IEEE, Piscataway, NJ, 461–463.

    Google Scholar 

  4. Kreinovich, V. and Nguyen, H. T. (1995b). On Hilbert’s Thirteenth Problem for Soft Computing, Proceedings of the Joint 4th IEEE Conference on Fuzzy Systems and 2nd IFES, Yokohama, Japan, March 20–24, 1995, IV, 2089–2094.

    Google Scholar 

  5. Lea, R., Kreinovich, V., and Trejo, R. (1996). Optimal interval enclosures for fractionally-linear functions, and their application to intelligent control, Reliable Computing, 2(3), 265–286.

    Article  MathSciNet  MATH  Google Scholar 

  6. Mouzouris, G. C. and Mendel, J. M. (1994). Non-Singleton Fuzzy Logic Systems, Proceedings of Third IEEE International Conference on Fuzzy Systems, June 1994, 1, 456–461.

    Article  Google Scholar 

  7. Mouzouris, G. C. and Mendel, J. M. (1994a). Non-Singleton Fuzzy Logic Systems: Theory and Application, University of Southern California, Signal and Image Processing Institute, Report No. 262.

    Google Scholar 

  8. Mouzouris, G. C. and Mendel, J. M. (1995). Nonlinear Time Series Analysis with Non-Singleton Fuzzy Logic Systems, Proceedings IEEE/IAFE Conference on Computational Intelligence for Financial Engineering, New York, NY, April 1995, 47–56.

    Google Scholar 

  9. Mouzouris, G. C. and Mendel, J. M. (1996). Nonlinear predictive modeling using dynamic non-singleton fuzzy logic systems, Proceedings of the Fifth IEEE International Conference on Fuzzy Systems FUZZ-IEEE’96, New Orleans, September 8–11, 2, 1217–1223.

    Article  Google Scholar 

  10. Mouzouris, G. C. and Mendel, J. M. (1996a). Designing fuzzy logic systems for uncertain environments using a singular-value-QR decomposition method, Proceedings of the Fifth IEEE International Conference on Fuzzy Systems FUZZ-IEEE’96, New Orleans, September 8–11, 1, 295–301.

    Article  Google Scholar 

  11. Mouzouris, G. C. and Mendel, J. M. (1997). Non-Singleton Fuzzy Logic Systems: Theory and Application, IEEE Transactions on Fuzzy Systems, 5(1), 56–71.

    Article  Google Scholar 

  12. Nakamura, M. (1994). What fuzzy interval operations should be hardware supported?, In: Hall, L., Ying, H., Langari, R., and Yen, J. (eds.), NAFIPS/IFIS/NASA’ 94, Proceedings of the First International Joint Conference of The North American Fuzzy Information Processing Society Biannual Conference, The Industrial Fuzzy Control and Intelligent Systems Conference, and The NASA Joint Technology Workshop on Neural Networks and Fuzzy Logic, San Antonio, December 18-21, 1994, IEEE, Piscataway, NJ, 393–397.

    Google Scholar 

  13. Nguyen, H. T. and Kreinovich, V. (1995). Towards theoretical foundations of soft computing applications (1995). International Journal on Uncertainty, Fuzziness, and Knowledge-Based Systems, 3(3), 341–373.

    Article  MathSciNet  MATH  Google Scholar 

  14. Nguyen, H. T., Kreinovich, V., Nesterov, V., and Nakamura, M. (1997a). On hardware support for interval computations and for soft computing: a theorem, IEEE Transactions on Fuzzy Systems, 5(1), 108–127.

    Article  Google Scholar 

  15. Wu, K. C. (1994). A robot must be better than a human driver: an application of fuzzy intervals. In: Hall, L., Ying, H., Langari, R., and Yen, J. (eds.), NAFIPS/IFIS/NASA’ 94, Proceedings of the First International Joint Conference of The North American Fuzzy Information Processing Society Biannual Conference, The Industrial Fuzzy Control and Intelligent Systems Conference, and The NASA Joint Technology Workshop on Neural Networks and Fuzzy Logic, San Antonio, December 18–21, 1994, IEEE, Piscataway, NJ, 171–174.

    Google Scholar 

  16. Wu, K. C. (1996). Fuzzy interval control of mobile robots, Computers and Electrical Engineering, 22(3), 211–229.

    Article  Google Scholar 

Approximation with fewer rules

  1. Barron, A. R. (1993). Universal approximation bounds for superpositions of a sigmoidal function, IEEE Transactions on Information Theory, 39, 930–945.

    Article  MathSciNet  MATH  Google Scholar 

  2. Bauer, P., Klement, E. P., Moser, B., and Leikermoser, A. (1995). Modeling of fuzzy functions by fuzzy controllers, In: Nguyen, H. T., Sugeno, M., Tong, R., and Yager, R. (eds.), Theoretical aspects of fuzzy control, J. Wiley, N.Y.

    Google Scholar 

  3. DeVore, R., Howard, R., and Micchelli, C. A. (1989). Optimal nonlinear approximation, Manuscripta Mathematica, 63, 469–478.

    Article  MathSciNet  Google Scholar 

  4. Dickerson, J. A., and Kosko, B. (1996). Fuzzy function approximation with ellipsoidal rules, IEEE Trans. Syst, Man, Cybern., 26(4), 542–560.

    Article  Google Scholar 

  5. Foo, N. (1994). New perspectives and old problems …, Proceedings of the Soft Computing Symposium, University of New South Wales.

    Google Scholar 

  6. Jones, L. K. (1992). A simple lemma on greedy approximation in Hilbert space and convergence rates for projection pursuit regression and neural network training, The Annals of Statistics, 20, 601–613.

    Article  Google Scholar 

  7. Kóczy, L. T. and Tikk, D. (1996). Approximation in rule bases, IPMU’96: Proceedings of the International Conference on Information Processing and Management of Uncertainty in Knowledge-Based Systems, Granada, July 1–5, 1996, 489–494.

    Google Scholar 

  8. Kosko, B. (1994). Fuzzy Systems as Universal Approximators, IEEE Trans. on Computers, 43(11), 1329–1333.

    Article  MATH  Google Scholar 

  9. Kosko, B. (1994a). Optimal fuzzy rules cover extrema, Proceedings of the World Congress on Neural Networks WCNN’94.

    Google Scholar 

  10. Kosko, B. (1995). Optimal fuzzy rules cover extrema, International Journal of Intelligent Systems, 10(2), 249–255.

    Article  Google Scholar 

  11. Kosko, B. (1996). Asymptotic stability of feedback additive fuzzy systems, Proceedings of the World Congress on Neural Networks WCNN’96, September 1996.

    Google Scholar 

  12. Kosko, B. (1996a). Additive fuzzy systems: from function approximation to learning, In: Chen, C. H. (ed.), Fuzzy Logic and Neural Network Handbook, McGraw-Hill, N.Y., 9-1-9-22.

    Google Scholar 

  13. Kosko, B. and Dickerson, J. A. (1995a). Function approximation with additive fuzzy systems, Chapter 12 in: Nguyen, H. T., Sugeno, M., Tong, R., and Yager, R. (eds.), Theoretical aspects of fuzzy control, J. Wiley, N.Y., 313–347.

    Google Scholar 

  14. Kreinovich, V., Sirisaengtaksin, O., and Cabrera, S. (1994c). Wavelet neural networks are optimal approximators for functions of one variable, Proceedings of the IEEE International Conference on Neural Networks, Orlando, FL, July 1994, 1, 299–303.

    Google Scholar 

  15. Kůrková, V., Kainen, P. C., and Kreinovich, V. (1995). Dimension-independent rates of approximation by neural networks and variation with respect to half-spaces, Proceedings of World Congress on Neural Networks, WCNN’95, Washington, DC, July 1995, INNS Press, NJ, I, 54–57.

    Google Scholar 

  16. Kůrková, V., Kainen, P. C., and Kreinovich, V. (1997). Estimates of the Number of Hidden Units and Variation with Respect to Half-spaces, Neural Networks, 10(6), 1061–1068.

    Article  Google Scholar 

  17. Lee, J. and Chae, S. (1993). Completeness of fuzzy controller carrying a mapping f: R → R, Proceedings of the IEEE International Conference on Fuzzy Systems FUZZ-IEEE’93, 1, 231–235.

    Google Scholar 

  18. Lee, J. and Chae, S. (1993a). Analysis of function duplicating capabilities of fuzzy controllers, Fuzzy Sets and Systems, 56, 127–143.

    Article  MathSciNet  MATH  Google Scholar 

  19. Lorentz, G. G. (1966). Approximation of Functions, Holt, Reinhart and Winston, New York, 1966.

    MATH  Google Scholar 

  20. Watkins, F. A. (1994). Fuzzy Engineering, Ph.D. Dissertation, Department of Electrical and Computer Engineering, University of California at Irvine.

    Google Scholar 

  21. Watkins, F. A. (1995). The representation problem for additive fuzzy systems, Proceedings of the IEEE International Conference on Fuzzy Systems FUZZ-IEEE/IFES’95, March 1995, 1, 117–122.

    Google Scholar 

  22. Ying, H. (1994). Sufficient conditions on general fuzzy systems as function approximators, Automatica, 30(3), 521–525.

    Article  MATH  Google Scholar 

  23. Zeng, X.-J. and Singh, M. G. (1994). Approximation theory of fuzzy systems — SISO case, IEEE Trans. on Fuzzy Systems, 2(2), 162–176.

    Article  Google Scholar 

  24. Zeng, X. J. and Singh, M. G. (1995a). Approximation accuracy analysis of fuzzy systems with the center-average defuzzifier, Proceedings Fourth IEEE Int. Conf. on Fuzzy Systems, 109–116, Yokohama, March 1995.

    Google Scholar 

  25. Zeng, X.-J. and Singh, M. G. (1996). Approximation accuracy analysis of fuzzy systems as function approximators, IEEE Trans. on Fuzzy Systems, 4, 44–63.

    Article  Google Scholar 

How to choose the best variant of fuzzy ruled based modeling methodology

  1. Bouchon-Meunier, B., Kreinovich, V., Lokshin, A., and Nguyen, H. T. (1996). On the formulation of optimization under elastic constraints (with control in mind). Fuzzy Sets and Systems, 81(1), 5–29.

    Article  MathSciNet  MATH  Google Scholar 

  2. Kosheleva, O. (1996). Why Sinc? Signal Processing Helps Fuzzy Control, SC-COSMIC, South Central Computational Sciences in Minority Institutions Consortium, Second Student Conference in Computational Sciences, October 25-27, El Paso, TX, Abstracts, 19–21.

    Google Scholar 

  3. Kreinovich, V. (1997). Random sets unify, explain, and aid known uncertainty methods in expert systems, in Goutsias, J., Mahler, R. P. S., and Nguyen, H. T. (eds.), Random Sets: Theory and Applications, Springer-Verlag, N.Y., 321–345.

    Google Scholar 

  4. Kreinovich, V. (1996c). Maximum entropy and interval computations, Reliable Computing, 2(1), 63–79.

    Article  MathSciNet  MATH  Google Scholar 

  5. Kreinovich, V., Nguyen, H. T., and Walker, E. A. (1996b). Maximum entropy (MaxEnt) method in expert systems and intelligent control: new possibilities and limitations, In: Hanson K. M. and Silver, R. N. (eds.), Maximum Entropy and Bayesian Methods, Kluwer Academic Publishers, Dordrecht, 93–100.

    Chapter  Google Scholar 

  6. Kreinovich, V., Quintana, C., and Lea, R. (1991c). What procedure to choose while designing a fuzzy control? Towards mathematical foundations of fuzzy control, Working Notes of the 1st International Workshop on Industrial Applications of Fuzzy Control and Intelligent Systems, College Station, TX, 123–130.

    Google Scholar 

  7. Kreinovich, V., Quintana, C., Lea, R., Fuentes, O., Lokshin, A., Kumar, S., Boricheva, I., and Reznik, L. (1992). What non-linearity to choose? Mathematical foundations of fuzzy control, Proceedings of the 1992 International Conference on Fuzzy Systems and Intelligent Control, Louisville, KY, 349–412.

    Google Scholar 

  8. Kreinovich, V. and Tolbert, D. (1994d). Minimizing computational complexity as a criterion for choosing fuzzy rules and neural activation functions in intelligent control. In: Jamshidi, M., Nguyen, C., Lumia, R., and Yuh, J. (eds.), Intelligent Automation and Soft Computing. Trends in Research, Development, and Applications. Proceedings of the First World Automation Congress (WAC’94). August 14-17, 1994, Maui, Hawaii, TSI Press, Albuquerque, NM, 1, 545–550.

    Google Scholar 

  9. Mitaim, S. and Kosko, B. (1996). What is the best shape for a fuzzy set in function approximation?, Proceedings of the Fifth IEEE International Conference on Fuzzy Systems FUZZ-IEEE’96, New Orleans, September 8-11, 2, 1237–1243.

    Article  Google Scholar 

  10. Mitaim, S. and Kosko, B. (1996a). Fuzzy function approximation and the shape of fuzzy sets, Proceedings of the World Congress on Neural Networks WCNN’96, September 1996.

    Google Scholar 

  11. Nguyen, H. T. and Kreinovich, V. (1995). Towards theoretical foundations of soft computing applications (1995). International Journal on Uncertainty, Fuzziness, and Knowledge-Based Systems, 3(3), 341–373.

    Article  MathSciNet  MATH  Google Scholar 

  12. Nguyen, H. T., Kreinovich, V., Lea, B., and Tolbert, D. (1992). How to control if even experts are not sure: robust fuzzy control, Proceedings of the Second International Workshop on Industrial Applications of Fuzzy Control and Intelligent Systems, College Station, December 2–4, 153–162.

    Google Scholar 

  13. Nguyen, H. T., Kreinovich, V., Lea, B., and Tolbert, D. (1995a). Interpolation that leads to the narrowest intervals, and its application to expert systems and intelligent control, Reliable Computing, 1995, 3(1), 299–316.

    Article  MathSciNet  Google Scholar 

  14. Nguyen, H. T., Kreinovich, V., and Tolbert, D. (1993a). On robustness of fuzzy logics. Proceedings of IEEE-FUZZ International Conference, San Francisco, CA, March 1993, 1, 543–547.

    Google Scholar 

  15. Nguyen, H. T., Kreinovich, V., and Tolbert, D. (1994). A measure of average sensitivity for fuzzy logics, International Journal on Uncertainty, Fuzziness, and Knowledge-Based Systems, 2(4), 361–375.

    Article  MathSciNet  MATH  Google Scholar 

  16. Nguyen, H. T. and Walker, E. A. (1997b). A first course in fuzzy logic, CRC Press, Boca Raton, FL.

    MATH  Google Scholar 

  17. Ramer, A. and Kreinovich, V. (1992). Maximum entropy approach to fuzzy control, Proceedings of the Second International Workshop on Industrial Applications of Fuzzy Control and Intelligent Systems, College Station, December 2–4, 113–117.

    Google Scholar 

  18. Ramer, A. and Kreinovich, V. (1994). Information complexity and fuzzy control, Chapter 4 in: Kandel, A. and Langholtz, G. (eds.), Fuzzy Control Systems, CRC Press, Boca Raton, FL, 75–97.

    Google Scholar 

  19. Ramer, A. and Kreinovich, V. (1994a). Maximum entropy approach to fuzzy control, Information Sciences, 81(3–4), 235–260.

    Article  MathSciNet  MATH  Google Scholar 

  20. Smith, M. H. and Kreinovich, V. (1995). Optimal strategy of switching reasoning methods in fuzzy control, Chapter 6 in Nguyen, H. T., Sugeno, M., Tong, R., and Yager, R. (eds.), Theoretical aspects of fuzzy control, J. Wiley, N.Y., 117–146.

    Google Scholar 

  21. Zeng, X.-J. and Singh, M. G. (1996). Approximation accuracy analysis of fuzzy systems as function approximators, IEEE Trans. on Fuzzy Systems, 4, 44–63.

    Article  Google Scholar 

Comparison between fuzzy rules based modeling methodology and neural network modeling methodology

  1. Kreinovich, V. and Bernat, A. (1994). Parallel algorithms for interval computations: an introduction, Interval Computations, No. 3, 6–62.

    Google Scholar 

  2. Lea, R. N. and Kreinovich, V. (1995). Intelligent Control Makes Sense Even Without Expert Knowledge: an Explanation, Reliable Computing, 1995, Supplement (Extended Abstracts of APIC’95: International Workshop on Applications of Interval Computations, El Paso, TX, Febr. 23-25, 1995), 140–145.

    Google Scholar 

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Authors and Affiliations

  1. Department of Computer Science, University of Texas at El Paso, El Paso, TX, 79968, USA

    Vladik Kreinovich

  2. Signal and Image Processing Institute, Department of Electrical Engineering — Systems, University of Southern California, Los Angeles, CA, 90089-2564, USA

    George C. Mouzouris

  3. Department of Mathematical Sciences, New Mexico State University, Las Cruces, NM, 88003-8001, USA

    Hung T. Nguyen

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  1. Vladik Kreinovich
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  2. George C. Mouzouris
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  3. Hung T. Nguyen
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Editors and Affiliations

  1. New Mexico State University, USA

    Hung T. Nguyen

  2. Tokyo Institute of Technology, Japan

    Michio Sugeno

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Kreinovich, V., Mouzouris, G.C., Nguyen, H.T. (1998). Fuzzy Rule Based Modeling as a Universal Approximation Tool. In: Nguyen, H.T., Sugeno, M. (eds) Fuzzy Systems. The Springer Handbook Series on Fuzzy Sets, vol 2. Springer, Boston, MA. https://doi.org/10.1007/978-1-4615-5505-6_5

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