Advertisement

Fuzzy Modeling for Uncertain Nonlinear Systems Using Fuzzy Equations and Z-Numbers

  • Raheleh Jafari
  • Sina Razvarz
  • Alexander Gegov
  • Satyam Paul
Conference paper
Part of the Advances in Intelligent Systems and Computing book series (AISC, volume 840)

Abstract

In this paper, the uncertainty property is represented by Z-number as the coefficients and variables of the fuzzy equation. This modification for the fuzzy equation is suitable for nonlinear system modeling with uncertain parameters. Here, we use fuzzy equations as the models for the uncertain nonlinear systems. The modeling of the uncertain nonlinear systems is to find the coefficients of the fuzzy equation. However, it is very difficult to obtain Z-number coefficients of the fuzzy equations.

Taking into consideration the modeling case at par with uncertain nonlinear systems, the implementation of neural network technique is contributed in the complex way of dealing the appropriate coefficients of the fuzzy equations. We use the neural network method to approximate Z-number coefficients of the fuzzy equations.

Keywords

Fuzzy modeling Z-number Uncertain nonlinear system 

References

  1. 1.
    Barthelmann, V., Novak, E., Ritter, K.: High dimensional polynomial interpolation on sparse grids. Adv. Comput. Math. 12, 273–288 (2000)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Jafarian, A., Jafari, R., Mohamed Al Qurashi, M., Baleanud, D.: A novel computational approach to approximate fuzzy interpolation polynomials, SpringerPlus 5, 1428 (2016).  https://doi.org/10.1186/s40064-016-3077-5
  3. 3.
    Neidinger, R.D.: Multi variable interpolating polynomials in newton forms. In: Proceedings of the Joint Mathematics Meetings, Washington, DC, USA, pp. 5–8 (2009)Google Scholar
  4. 4.
    Schroeder, H., Murthy, V.K., Krishnamurthy, E.V.: Systolic algorithm for polynomial interpolation and related problems. Parallel Comput. 17, 493–503 (1991)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Zolic, A.: Numerical Mathematics. Faculty of mathematics, Belgrade, pp. 91–97 (2008)Google Scholar
  6. 6.
    Szabados, J., Vertesi, P.: Interpolation of Functions. World Scientific Publishing Co., Singapore (1990)CrossRefGoogle Scholar
  7. 7.
    Xiu, D., Hesthaven, J.S.: High-order collocation methods for differential equations with random inputs. SIAM J. Sci. Comput. 27, 1118–1139 (2005)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Friedman, M., Ming, M., Kandel, A.: Fuzzy linear systems. Fuzzy Sets Syst. 96, 201–209 (1998)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Abbasbandy, S.: The application of homotopy analysis method to nonlinear equations arising in heat transfer. Phys. Lett. A 360, 109–113 (2006)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Abbasbandy, S., Ezzati, R.: Newton’s method for solving a system of fuzzy nonlinear equations. Appl. Math. Comput. 175, 1189–1199 (2006)MathSciNetMATHGoogle Scholar
  11. 11.
    Allahviranloo, T., Ahmadi, N., Ahmadi, E.: Numerical solution of fuzzy differential equations by predictor-corrector method. Inform. Sci. 177, 1633–1647 (2007)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Kajani, M., Asady, B., Vencheh, A.: An iterative method for solving dual fuzzy nonlinear equations. Appl. Math. Comput. 167, 316–323 (2005)MathSciNetMATHGoogle Scholar
  13. 13.
    Waziri, M., Majid, Z.: A new approach for solving dual fuzzy nonlinear equations using Broyden’s and Newton’s methods. Adv. Fuzzy Syst. (2012). Article 682087, 5 pagesGoogle Scholar
  14. 14.
    Pederson, S., Sambandham, M.: The Runge-Kutta method for hybrid fuzzy differential equation. Nonlinear Anal. Hybrid Syst. 2, 626–634 (2008)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Buckley, J., Eslami, E.: Neural net solutions to fuzzy problems: the quadratic equation. Fuzzy Sets Syst. 86, 289–298 (1997)CrossRefGoogle Scholar
  16. 16.
    Jafarian, A., Jafari, R., Khalili, A., Baleanud, D.: Solving fully fuzzy polynomials using feed-back neural networks. Int. J. Comput. Math. 92(4), 742–755 (2015)CrossRefGoogle Scholar
  17. 17.
    Jafarian, A., Jafari, R.: Approximate solutions of dual fuzzy polynomials by feed-back neural networks. J. Soft Comput. Appl. (2012).  https://doi.org/10.5899/2012/jsca-00005CrossRefGoogle Scholar
  18. 18.
    Mosleh, M.: Evaluation of fully fuzzy matrix equations by fuzzy neural network. Appl. Math. Model. 37, 6364–6376 (2013)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Allahviranloo, T., Otadi, M., Mosleh, M.: Iterative method for fuzzy equations. Soft. Comput. 12, 935–939 (2007)CrossRefGoogle Scholar
  20. 20.
    Zadeh, L.A.: Toward a generalized theory of uncertainty (GTU) an outline. Inform. Sci. 172, 1–40 (2005)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Gardashova, L.A.: Application of operational approaches to solving decision making problem using Z-Numbers. J. Appl. Math. 5, 1323–1334 (2014)CrossRefGoogle Scholar
  22. 22.
    Aliev, R.A., Alizadeh, A.V., Huseynov, O.H.: The arithmetic of discrete Z-numbers. Inform. Sci. 290, 134–155 (2015)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Kang, B., Wei, D., Li, Y., Deng, Y.: Decision making using Z-Numbers under uncertain environment. J. Comput. Inf. Syst. 8, 2807–2814 (2012)Google Scholar
  24. 24.
    Kang, B., Wei, D., Li, Y., Deng, Y.: A method of converting Z-number to classical fuzzy number. J. Inf. Comput. Sci. 9, 703–709 (2012)Google Scholar
  25. 25.
    Zadeh, L.A.: A note on Z-numbers. Inf. Sci. 181, 2923–2932 (2011)CrossRefGoogle Scholar
  26. 26.
    Jafari, R., Yu, W.: Fuzzy control for uncertainty nonlinear systems with dual fuzzy equations. J. Intell. Fuzzy Syst. 29, 1229–1240 (2015)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Jafari, R., Yu, W.: Uncertainty nonlinear systems modeling with fuzzy equations. In: Proceedings of the 16th IEEE International Conference on Information Reuse and Integration, San Francisco, Calif, USA, pp. 182–188, August 2015Google Scholar
  28. 28.
    Jafari, R., Yu, W.: Uncertainty nonlinear systems control with fuzzy equations. In: IEEE International Conference on Systems, Man, and Cybernetics, pp. 2885–2890 (2015)Google Scholar
  29. 29.
    Razvarz, S., Jafari, R., Granmo, O.Ch., Gegov, A.: Solution of dual fuzzy equations using a new iterative method. In: Asian Conference on Intelligent Information and Database Systems, pp. 245–255 (2018)Google Scholar
  30. 30.
    Aliev, R.A., Pedryczb, W., Kreinovich, V., Huseynov, O.H.: The general theory of decisions. Inform. Sci. 327, 125–148 (2016)MathSciNetCrossRefGoogle Scholar
  31. 31.
    Jafari, R., Yu, W., Li, X.: Solving fuzzy differential equation with Bernstein neural networks. In: IEEE International Conference on Systems, Man, and Cybernetics, Budapest, Hungary, pp. 1245–1250 (2016)Google Scholar
  32. 32.
    Jafari, R., Yu, W., Li, X., Razvarz, S.: Numerical solution of fuzzy differential equations with Z-numbers using Bernstein neural networks. Int. J. Comput. Intell. Syst. 10, 1226–1237 (2017)CrossRefGoogle Scholar
  33. 33.
    Suykens, J.A.K., Brabanter, JDe, Lukas, L., Vandewalle, J.: Weighted least squares support vector machines: robustness and sparse approximation. Neurocomputing 48, 85–105 (2002)CrossRefGoogle Scholar
  34. 34.
    Bede, B., Stefanini, L.: Generalized differentiability of fuzzy-valued functions. Fuzzy Sets Syst. 230, 119–141 (2013)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Centre for Artificial Intelligence Research (CAIR)University of AgderGrimstadNorway
  2. 2.Departamento de Control AutomáticoCINVESTAV-IPN (National Polytechnic Institute)Mexico CityMexico
  3. 3.School of ComputingUniversity of PortsmouthPortsmouthUK
  4. 4.School of Engineering and SciencesTecnológico de MonterreyMonterreyMexico

Personalised recommendations