A novel approach to optimize the estimated stability region via energy function for a class of nonlinear dynamical systems in technical models

90 views

Authors

  • Ho Thi Thu Thuy Air Defence-Air Force Academy
  • Pham Hong Quan (Corresponding Author) Air Defence-Air Force Academy

DOI:

https://doi.org/10.54939/1859-1043.j.mst.83.2022.82-94

Keywords:

Nonlinear dynamical systems; Stability boundary; Stability region; Optimal energy function.

Abstract

The theory of differential equations has been widely known and developed in recent years. Many researchers have drawn attention to the problem of finding the stability region of a nonlinear dynamical system in technical models, which is a complicated issue in the stability theory of dynamical systems. In this problem, how to construct an optimal energy function is considered an essential step to approximate the stability region of a locally stable equilibrium point. The main purpose of this paper is to give a novel approach to optimize the estimated stability region via energy function for nonlinear dynamical models. This ensures that the stability region estimated is optimal in the sense that this estimated region is the largest one characterized by the energy function, which lies entirely in the stability region. Furthermore, numerical experiments are also conducted to compare the difference between the proposed algorithm.

References

[1]. H. D. Chiang, “Direct methods for stability analysis of electric power systems: theoretical foundation, BCU methodologies, and applications”, John Wiley & Sons, Hoboken, (2011). DOI: https://doi.org/10.1002/9780470872130

[2]. H. D. Chiang, J. S. Thorp, “Stability regions of nonlinear dynamical systems: A constructive methodology”, IEEE Transactions on Automatic Control, Vol. 34, No. 12, pp. 1229-1241, (1989). DOI: https://doi.org/10.1109/9.40768

[3]. H. D. Chiang and L. F. C Alberto, “Stability regions of nonlinear dynamical systems: theory, estimation, and applications”, Cambridge University Press, Cambridge, (2015). DOI: https://doi.org/10.1017/CBO9781139548861

[4]. M. Sassano and A. Astolfi, Dynamic lyapunov functions, Automatica, Vol. 49, No. 4, pp. 1058-1067, (2013). DOI: https://doi.org/10.1016/j.automatica.2013.01.027

[5]. E. D. Ferreira and B. H. Krogh, “Using neural networks to estimate regions of stability”, Proceedings of the 1997 American Control Conference, Vol. 3, pp. 1989-1993, (1997). DOI: https://doi.org/10.1109/ACC.1997.611036

[6]. A. Vannelli and M. Vidyasagar, “Maximal Lyapunov functions and domains of attraction for autonomous nonlinear systems”, Automatica, Vol. 21, No. 1, pp. 69-89, (1985). DOI: https://doi.org/10.1016/0005-1098(85)90099-8

[7]. T. J. Koo, H. Su, “A computational approach for estimating stability regions”, 2006 IEEE International Symposium on Intelligent Control, pp. 62-68, (2006).

[8]. V. I. Zubov, “Mathematical methods for the study of automatic control systems”, PergmSon Press, (1963).

[9]. P. Giesl and S. Hafstein, “Review on computational methods for Lyapunov functions”, Discrete and Continuous Dynamical Systems-Series B, Vol. 20, No. 8, pp. 2291-2331, (2015). DOI: https://doi.org/10.3934/dcdsb.2015.20.2291

[10]. L. G. Matallana, A. M. Blanco and J. A. Bandoni, “Estimation of domains of attraction: A global optimization approach”, Mathematical and Computer Modeling, Vol. 52, No. 3-4, pp. 574-585, (2010). DOI: https://doi.org/10.1016/j.mcm.2010.04.001

[11]. P. Giesl, “Approximation of domains of attraction and Lyapunov functions using radial basis functions”, IFAC Proceedings Volumes, Vol. 37, No. 13, pp. 697-702, (2004). DOI: https://doi.org/10.1016/S1474-6670(17)31306-X

[12]. W. Tan and A. Packard, “Stability region analysis using polynomial and composite polynomial Lyapunov functions and sum of squares programming”, IEEE Transactions on Automatic Control, Vol. 53, No. 2, pp. 565-571, (2008). DOI: https://doi.org/10.1109/TAC.2007.914221

[13]. G. Yuan and Y. Li, “Estimation of the regions of attraction for autonomous nonlinear systems”, Transactions of the Institute of Measurement and Control, Vol. 41, No. 1, pp. 97-106, (2019). DOI: https://doi.org/10.1177/0142331217752799

[14]. M. P. Rezaiee and B. Moghaddasie, “Determination of stability domains for nonlinear dynamical systems using the weighted residuals method”, Civil Engineering Infrastructures Journal, Vol. 46, No. 1, pp. 27-50, (2013).

Downloads

Published

18-11-2022

How to Cite

Ho Thi Thu Thuy, and H. Q. Pham. “A Novel Approach to Optimize the Estimated Stability Region via Energy Function for a Class of Nonlinear Dynamical Systems in Technical Models”. Journal of Military Science and Technology, no. 83, Nov. 2022, pp. 82-94, doi:10.54939/1859-1043.j.mst.83.2022.82-94.

Issue

Section

Research Articles

Categories