Abstract
In this paper, we propose a method for computing an estimate of the domain of attraction of a system containing time-varying sector nonlinearity together with quadratic and cubic nonlinearities. The method is based on use of non-symmetric Lyapunov functions combined with scanning the phase space in polar coordinates. Examples testifying to the efficiency of the new approach are presented.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Kamenetskii, V.A., Construction of Domains of Attraction by the Method of Lyapunov Functions, Autom. Remote Control, 1994, vol. 55, no. 6, pp. 778–791.
Chesi, G., Garulli, A., Tesi, A., et al., LMI-based Computation of Optimal Quadratic Lyapunov Functions for Odd Polynomial Systems, Int. J. Robust Nonlin. Control, 2005, vol. 15, pp. 35–49.
Valmorbida, G., Tarbouriech, S., and Garcia, G., Region of Attraction Estimates for Polynomial Systems, Proc. Joint 48th IEEE Conf. on Decision and Control and 28th Chinese Control Conf., Shanghai, P.R. China, December 2009, pp. 5947–5952.
Vannelli, A. and Vidyasagar, M., Maximal Lyapunov Functions and Domains of Attraction for Polynomial Differential Equations, Automatica, 1985, vol. 21, no. 1, pp. 69–80.
Barkin, A.I., Estimation of the Domain of Attraction of Systems with Polynomial Right-hand Sides, Inform. Tekhnol. Vychisl. Sist., 2013, no. 4. pp. 3–8.
Barkin, A.I., Evaluation of the Domain of Attraction of Systems with Cubic Nonlinearities, Tr. ISA RAN, 2014, vol. 64, no. 4, pp. 3–6.
Barkin, A.I, Attraction Domains of Systems with Polynomial Nonlinearities, Autom. Remote Control, 2015, vol. 76, no. 7, pp. 1156–1168.
Liberzon, M.R, New Results on Absolute Stability of Nonstationary Controlled Systems (Survey), Autom. Remote Control, 1979, vol. 40, no. 8, pp. 1124–1140.
Barkin, A.I., Attraction Domains of Systems with Sector- and Polynomial-type Nonlinearities, Tr. ISA RAN, 2015, vol. 65, no. 2, pp. 3–7.
Bongiorno, J.J., An Extension of the Nyquist–Barkhausen Stability Criterion to Linear Lumped-Parameter Systems with Time-Varying Elements, IEEE Trans. Automat. Control, 1963, vol. 8, no. 2, pp. 116–170.
Barkin, A.I. and Zelentsovsky, A.L., Absolute Stability Criterion for Nonlinear Control Systems, Autom. Remote Control, 1981, vol. 42, no. 7, pp. 853–857.
Barkin, A.I., Zelentsovsky, A.L., and Pakshin, P.V., Absolyutnaya ustoichivost’ determinirovannykh i stokhasticheskikh sistem upravleniya (Absolute Stability of Deterministic and Stochastic Control Systems), Moscow: Mosk. Aviats. Inst., 1992.
Barkin, A.I. and Rokin, D.N., Modernization of a Criterion for Absolute Stability, Autom. Remote Control, 2008, vol. 69, no 5, pp. 748–757.
Barkin, A.I., Absolyutnaya ustoichivost’ sistem upravleniya (Absolute Stability of Control Systems), Moscow: Nauka, 2012.
Gelig, A.Kh., Leonov, G.A., and Yakubovich, V.A., Ustoichivost’ nelineinykh sistem c needinstvennym sostoyaniem ravnovesiya (Stability of Nonlinear Systems with Non-unique Equilibrium Position), Moscow: Nauka, 1978.
Benner, P., Li, J.-R., and Penzl, T., Numerical Solution of Large-Scale Lyapunov Equations, Riccati Equations, and Linear-Quadratic Optimal Control Problems, Numer. Linear Algebra Appl., 2008, vol. 15, pp. 755–777.
Author information
Authors and Affiliations
Corresponding author
Additional information
Original Russian Text © A.I. Barkin, 2016, published in Avtomatika i Telemekhanika, 2016, No. 4, pp. 24–34.
Rights and permissions
About this article
Cite this article
Barkin, A.I. Evaluation of the domain of attraction of time-varying systems with sector and polynomial nonlinearities. Autom Remote Control 77, 569–577 (2016). https://doi.org/10.1134/S0005117916040032
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S0005117916040032