Document Type : Research Article
Author
Departement of Applied Mathematics, Shahrood University of Technology, Shahrood, Iran
Abstract
In this paper, first the space of hyperbolic tangent functions is introduced and then the universal
approximator property of this space is proved. In fact, by using this space, any nonlinear continuous function can be uniformly approximated with any degree of accuracy. Also, as an application, this space of functions is utilized to design feedback control for a nonlinear dynamical system.
Keywords
Main Subjects
bibitem{Byrnes} Byrnes C. I., Isidori A. (1991). `` Asymptotic stabilization of minimum phase nonlinear systems ", IEEE Transactions on Automatic Control, 36, 112-117.
bibitem{Cao} Cao S.G., Rees N.W., Feng G. (1997). `` Analysis and design for a class of complex control systems-Part II: Fuzzy controller design ", Automatica, 33, 1029-1039.
bibitem{zhang004}Cui Y., Zhang H.G., Wang Y., Gao W. (2016). `` Adaptive control for a class of uncertain strict-feedback nonlinear systems based on a generalized fuzzy hyperbolic model ", Fuzzy Sets and Systems, 302, 52-64.
bibitem{Fontes} Fontes, F. A. C. C. (2001). `` A general framework to design stabilizing nonlinear model predictive controllers ", Systems & Control Letters, 42(2), 127-143.
bibitem{Nilpotent} Hermes H. (1991). `` Nilpotent and high-order approximations of vector field systems ", SIAM Review ", 33, 238-264.
bibitem{Isidori2} Isidori A. (1999). `` Nonlinear Control Systems II ", New York: Springer-Verlag.
bibitem{Kawski} Kawski M. (1990). `` Homogeneous stabilizing feedback laws ", Control theory and advanced technology, 6, 497-516.
bibitem{Khalil} Khalil H. K. (1996). `` Nonlinear Systems ", Prentice-Hall, Upper Saddle River, NJ .
bibitem{zhang002} Liu S., Guo Z., Zhang H.G. (2008). `` Fuzzy hyperbolic neural network model and its application in $H_infty$ filter design ", In: Proceedings of the 5th international symposium on Neural Networks: Advances in Neural Networks, F. Sun et al. (Eds.): ISNN 2008, Part I, LNCS 5263, Springer-Verlag Berlin Heidelberg, 222-230.
bibitem{Mayne3} Mayne D. Q., Rawlings J. B., Rao C. V., Scokaert P. O. M. (2000). `` Constrained model predictive control: Stability and optimality ", Automatica, 36, 798-814.
bibitem{Nersesov} Nersesov S. G., Haddad W. M. (2006). `` On the stability and control of nonlinear dynamical systems via vector Lyapunov functions", IEEE Transactions on Automatic Control, 51(2), 203-215.
bibitem{noori} Noori Skandari M. H. (2015). `` On the stability of a class of nonlinear control systems ", Nonlinear Dynamics, 80(3), 1245-1256.
bibitem{Noori1} Noori Skandari M. H., Ghaznavi M., Abedian M. (2019). `` Stabilizer control design for nonlinear systems based on the hyperbolic modelling ", Applied Mathematical Modelling, 67, 413-429.
bibitem{Rudin} Rudin W. (1976). `` Principles of mathematical analysis ", New York, McGraw-Hill, Inc.
bibitem{Tanaka} Tanaka K., Sugeno M. (1992). `` Stability analysis and design of fuzzy control systems ", Fuzzy Sets and Systems, 45, 135-156.
bibitem{Zhang1} Zhang H. G., Yongbing Q. (2001). `` Modeling, identification, and control of a class of nonlinear systems ", IEEE Transactions on Fuzzy Systems, 9(2), 349-354.
bibitem{Zhang4} Zhang H. G., Wang Z., Liu D. (2003). `` Chaotifying fuzzy hyperbolic model using adaptive inverse optimal control approach ", International Journal of Bifurcation and Chaos, 12, 32-43.
bibitem{zhang005} Zhang H. G., Wang Z., Liu D. (2005). `` Chaotifying fuzzy hyperbolic model using impulsive and nonlinear feedback control approaches ", International Journal of Bifurcation and Chaos, 15(8), 2603-2610.
bibitem{Zhang3} Zhang H. G., Wang Z. L., Li M., Quan B., Zhang M. G. (2004). `` Generalized Fuzzy Hyperbolic Model: A Universal Approximator ", Acta Automatica Sinica, 30(3), 416-422.
bibitem{Zhang2} Zhang M., Zhang H., Liu D. (2005). `` A generalized fuzzy hyperbolic modeling and control scheme ", 2004 IEEE International Conference on Fuzzy Systems.
bibitem{zhang001} Zhang M., Zhang H. (2006). `` Robust adaptive fuzzy control scheme for nonlinear system with uncertainty ", Journal of Control Theory and Applications, 4(2), 209-216.
bibitem{zhang003} Zhang J. L., Zhang H. G., Luo Y. H., Liang H. J. (2013). `` Nearly optimal control scheme using adaptive dynamic programming based on generalized fuzzy hyperbolic model ", Acta Automatica Sinica, 39(2), 142-148.
)