# 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)
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).
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)
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)
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)
8. 8.
Friedman, M., Ming, M., Kandel, A.: Fuzzy linear systems. Fuzzy Sets Syst. 96, 201–209 (1998)
9. 9.
Abbasbandy, S.: The application of homotopy analysis method to nonlinear equations arising in heat transfer. Phys. Lett. A 360, 109–113 (2006)
10. 10.
Abbasbandy, S., Ezzati, R.: Newton’s method for solving a system of fuzzy nonlinear equations. Appl. Math. Comput. 175, 1189–1199 (2006)
11. 11.
Allahviranloo, T., Ahmadi, N., Ahmadi, E.: Numerical solution of fuzzy differential equations by predictor-corrector method. Inform. Sci. 177, 1633–1647 (2007)
12. 12.
Kajani, M., Asady, B., Vencheh, A.: An iterative method for solving dual fuzzy nonlinear equations. Appl. Math. Comput. 167, 316–323 (2005)
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)
15. 15.
Buckley, J., Eslami, E.: Neural net solutions to fuzzy problems: the quadratic equation. Fuzzy Sets Syst. 86, 289–298 (1997)
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)
17. 17.
Jafarian, A., Jafari, R.: Approximate solutions of dual fuzzy polynomials by feed-back neural networks. J. Soft Comput. Appl. (2012).
18. 18.
Mosleh, M.: Evaluation of fully fuzzy matrix equations by fuzzy neural network. Appl. Math. Model. 37, 6364–6376 (2013)
19. 19.
Allahviranloo, T., Otadi, M., Mosleh, M.: Iterative method for fuzzy equations. Soft. Comput. 12, 935–939 (2007)
20. 20.
Zadeh, L.A.: Toward a generalized theory of uncertainty (GTU) an outline. Inform. Sci. 172, 1–40 (2005)
21. 21.
Gardashova, L.A.: Application of operational approaches to solving decision making problem using Z-Numbers. J. Appl. Math. 5, 1323–1334 (2014)
22. 22.
Aliev, R.A., Alizadeh, A.V., Huseynov, O.H.: The arithmetic of discrete Z-numbers. Inform. Sci. 290, 134–155 (2015)
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)
26. 26.
Jafari, R., Yu, W.: Fuzzy control for uncertainty nonlinear systems with dual fuzzy equations. J. Intell. Fuzzy Syst. 29, 1229–1240 (2015)
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)
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)
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)
34. 34.
Bede, B., Stefanini, L.: Generalized differentiability of fuzzy-valued functions. Fuzzy Sets Syst. 230, 119–141 (2013)

© Springer Nature Switzerland AG 2019

## Authors and Affiliations

• Raheleh Jafari
• 1
• Sina Razvarz
• 2
• Alexander Gegov
• 3
• Satyam Paul
• 4
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