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Optimal interval enclosures for fractionally-linear functions, and their application to intelligent control

Оптимальные интервальные включения для дробно-линейных функций и их приложение к интеллектуальному управлению

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Reliable Computing

Abstract

One of the main problems of interval computations is, given a functionf(x 1, ...,x n ) andn intervals x1, ..., x n , to compute the range y=f(x1, ..., x n ). This problem is feasible for linear functionsf, but for generic polynomials, it is known to be computationally intractable. Because of that, traditional interval techniques usually compute theenclosure of y, i.e., an interval that contains y. The closer this enclosure to y, the better. It is desirable to describe cases in which we can compute theoptimal enclosure, i.e., the range itself.

In this paper, we describe a feasible algorithm for computing the optimal enclosure forfractionally linear functionsf. Applications of this result tointelligent control are described.

Abstract

Одна из основных задач интервальных вычислений формулируется следуюим образом: дана функцияf(x 1, ...,x n ) иn интервалов x1, ..., x n ; требуется вычислить множество значений y=f(x1, ..., x n ). Эта задача имеет смысл для линейных функцийf, однако известно, что для обобщенных многочленов она вычислительно неразрецима. Поэтому традиционные интервальные методы, как правило, вычисляютвклчеuе y, т.е. интервал, содержащий в себе y. Чем ближе это включение к y, тем лучще. Желательно найти случаи, в которых возможно вычислитьоnmuмлъое вклчеue, т.е. само множество значений.

В работе описан адгоритм, лопускающий практическую реализацию, для вычисления оптимального включениябробно-лцных функцийf. Описаны приложения этого результата в областннмеллекмуалъно¶rt; управленцн.

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References

  1. Alefeld, G. and Herzberger, J.Introduction to interval computations. Academic Press, N.Y., 1983.

    Google Scholar 

  2. Berenji, H. R.Fuzzy logic controllers. In: Yager, R. R. and Zadeh, L. A. (eds) “An Introduction to Fuzzy Logic Applications in Intelligent Systems”, Kluwer Academic Publ., 1991.

  3. Blin, J. M.Fuzzy relations in group decision theory. J. of Cybernetics4 (1974), pp. 17–22.

    MathSciNet  Google Scholar 

  4. Blin, J. M. and Whinston, A. B.Fuzzy sets and social choice. J. of Cybernetics3 (1973), pp. 28–36.

    MathSciNet  Google Scholar 

  5. Chang, S. S. L. and Zadeh, L. A.On fuzzy mapping and control. IEEE Transactions on Systems, Man and CyberneticsSMC-2 (1972), pp. 30–34.

    MathSciNet  Google Scholar 

  6. Cormen, Th. H., Leiserson, Ch. L., and Rivest, R. L.Introduction to algorithms. MIT Press, Cambridge, MA, 1990.

    Google Scholar 

  7. Dubois, D. and Prade, H.Fuzzy sets and systems: theory and applications. Academic Press, N.Y., London, 1980.

    Google Scholar 

  8. Gananov, A. A.Computational complexity of the range of the polynomial in several variables. Cybernetics, 1985, pp. 418–421.

  9. Gerasimov, A. I. and Kreinovich, V.Piecewise fractionally-linear approximation. Leningrad Polytechnical University and National Research Institute for Scientific and Technical Information (VINITI), 1988 (in Russian).

  10. Gerasimov, A. I. and Kreinovich, V.On the problem of optimal approximation choice for metrological characteristics. Leningrad Polytechnical University and National Research Institute for Scientific and Technical Information (VINITI), 1988 (in Russian).

  11. Hansen, E. R. (ed.)Topics in interval analysis. Oxford University Press, 1969.

  12. Klir, G. J. and Folger, T. A.Fuzzy sets, uncertainty and information. Prentice Hall, Englewood Cliffs, NJ, 1988.

    Google Scholar 

  13. Krotkov, I. N., Kreinovich, V., and Mazin, V. D.Methodology of designing measuring systems, using fractionally linear transformations. In: “Measuring Systems. Theory and Applications. Proceedings of the Novosibirsk Electrical Engineering Institute”, 1986, pp. 5–14 (in Russian).

  14. Krotkov, I. N., Kreinovich, V., and Mazin, V. D.General form of measurement transformations which admit the computational methods of metrological analysis of measuring-testing and measuring-computing systems. Izmeritelnaya Tekhnika10 (1987), pp. 8–10 (in Russian); English translation: Measurement Techniques30 (10) (1987), pp. 936–939.

    Google Scholar 

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

  16. Lea, R. N.Automated space vehicle control for rendezvous proximity operations. Telemechanics and Informatics5 (1988), pp. 179–185.

    Article  Google Scholar 

  17. Lea, R. N., Jani, Y. K., and Berenji, H.,Fuzzy logic controller with reinforcement learning for proximity operations and docking. In: “Proceedings of the 5th IEEE International Symposium on Intelligent Control”, Vol. 2, 1990, pp. 903–906.

    Google Scholar 

  18. Lea, R. N., Togai, M., Teichrow, J., and Jani, Y.Fuzzy logic approach to combined translational and rotational control of a spacecraft in proximity of the Space Station. In: “Proceedings of the 3rd International Fuzzy Systems Association Congress”, 1989, pp. 23–29.

  19. Lee, C. C.Fuzzy logic in control systems: fuzzy logic controller. IEEE Transactions on Systems, Man and Cybernetics20 (2) (1990), pp. 404–435.

    Article  MATH  Google Scholar 

  20. Mamdani, E. H.Application of fuzzy algorithms for control of simple dynamic plant. Proceedings of the IEEE121 (12) (1974), pp. 1585–1588.

    Google Scholar 

  21. Mazin, V. D. and Kreinovich, V.An explanation of the universal character of the transformation functions y=(a lnx+b)/(c lnx+d). Leningrad Polytechnical Institute and National Research Institute for Scientific and Technical Information (VINITI), 1988 (in Russian).

  22. Mazin, V. D. and Kreinovich, V.An important property of fractional-linear transformation functions. Leningrad Polytechnical Institute and National Research Institute for Scientific and Technical Information (VINITI), 1988 (in Russian).

  23. Moore, R. E.Interval analysis. Prentice Hall, 1966.

  24. Moore, R. E.Methods and applications of interval analysis. SIAM Studies in Applied Mathematics, 1979.

  25. Nickel, K. (ed.)Interval mathematics. Lecture Notes in Computer Science29, Springer-Verlag, Berlin-Heidelberg-N.Y., 1975.

    Google Scholar 

  26. Nickel, K. (ed.)Interval mathematics 1980. Academic Press, N.Y., 1980.

    Google Scholar 

  27. Nickel, K. (ed.)Interval mathematics 1985. Lecture Notes in Computer Science212, Springer-Verlag, Berlin-Heidelberg-N.Y., 1985.

    Google Scholar 

  28. Ratschek, H. and Rokne, J.Computer methods for the range of functions. Ellis Horwood and John Wiley, 1984.

  29. Savage, L. J.The foundations of statistics. Wiley, New York, 1954.

    Google Scholar 

  30. Sugeno, M. (ed.)Industrial applications of fuzzy control. North Holland, Amsterdam, 1985.

    Google Scholar 

  31. Trejo, R. and Gerasimov, A. I.Choosing interval functions to represent measurement inaccuracies: group-theoretic approach. In: “Extended Abstracts of APIC'95: International Workshop on Applications of Interval Computations, El Paso, TX, Febr. 23–25, 1995”, Reliable Computing (1995), Supplement, pp. 207–210.

  32. Yen, J., Pfluger, N., and Langari, R.A defuzzification strategy for a fuzzy logic controller employing prohibitive information in command formulation. In: “Proceedings of IEEE International Conference on Fuzzy Systems”, San Diego, CA, March 1992.

  33. Zadeh, L.Fuzzy sets. Information and Control8 (1965), pp. 338–353.

    Article  MATH  MathSciNet  Google Scholar 

  34. Zadeh, L. A.Towards a theory of fuzzy systems. In: Kalman, R. E. and DecLaris, N. (eds) “Aspects of Network and Systems Theory”, Holt. Rinehart, Winston, 1971.

  35. Zadeh, L. A.Outline of a new approach to the analysis of complex systems and decision processes. IEEE Transactions on Systems, Man and Cybernetics3 (1973), pp. 28–44.

    MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. NASA Johnson Space Center, 77058, Houston, TX, USA

    R. N. Lea

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

    V. Kreinovich

  3. Sistemas de Informacion Division de Ingeneria y Ciencias, ITESM (Instituto Technologico de Monterrey), Campus Estado de Mexico, Apdo. Postal 2 Modulo de Servicio Postal, 52926, Atizapan, Mexico

    R. Trejo

Authors
  1. R. N. Lea
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  2. V. Kreinovich
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  3. R. Trejo
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R. N. Lea, V. Kreinovich, R. Trejo, 1996

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Lea, R.N., Kreinovich, V. & Trejo, R. Optimal interval enclosures for fractionally-linear functions, and their application to intelligent control. Reliable Comput 2, 265–285 (1996). https://doi.org/10.1007/BF02391700

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  • DOI: https://doi.org/10.1007/BF02391700

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