Abstract
New sufficient conditions for the global stability of different classes of nonlinear fractional feedback systems are presented. The linear parts of the systems are positive systems with interval state matrices. The nonlinear parts are described by static nonlinear characteristics located in the first and third quarter of the plane. The feedbacks are described in general case by matrices with positive entries. The sufficient conditions for the global stability are given for the following classes of the nonlinear systems: Positive interval continuous-time feedback nonlinear systems; Fractional positive interval continuous-time feedback nonlinear systems; Positive interval discrete-time feedback nonlinear systems; Descriptor nonlinear feedback discrete-time systems and Positive nonlinear electrical circuits. Procedures are given for calculations of gain matrices of the characteristics of nonlinear elements of the systems. The effectiveness of the procedures are demonstrated on numerical examples of nonlinear systems.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Similar content being viewed by others
References
Berman, A., Plemmons, R.J.: Nonnegative Matrices in the Mathematical Sciences. SIAM (1994).
Borawski, K.: Modification of the stability and positivity of standard and descriptor linear electrical circuits by state feedbacks. Electr. Rev. 93(11), 176–180 (2017)
Busłowicz, M., Kaczorek, T.: Simple conditions for practical stability of positive fractional discrete-time linear systems. Int. J. Appl. Math. Comput. Sci. 19(2), 263–169 (2009)
Farina, L., Rinaldi, S.: Positive Linear Systems; Theory and Applications. Wiley, New York (2000)
Kaczorek, T.: Absolute stability of a class of fractional positive nonlinear systems. Int. J. Appl. Math. Comput. Sci. 29(1), 93–98 (2019)
Kaczorek, T.: Analysis of positivity and stability of discrete-time and continuous-time nonlinear systems. Comput. Probl. Electr. Eng. 5(1), 11–16 (2015)
Kaczorek, T.: Analysis of positivity and stability of fractional discrete-time nonlinear systems. Bull. Pol. Acad.: Tech. 64(3), 491–494 (2016)
Kaczorek, T.: Global stability of nonlinear feedback systems with fractional positive linear parts. Int. J. Appl. Math. Comput. Sci. 30(3), 493–500 (2020)
Kaczorek, T.: Global stability of positive standard and fractional nonlinear feedback systems. Bull. Pol. Acad.: Tech. 68(2), 285–288 (2020)
Kaczorek, T.: Global stability of nonlinear feedback systems with positive linear parts, Int. J. Nonlinear Sci. Numer. Simul. 20(5), 575–579 (2019)
Kaczorek, T.: Positivity and stability of descriptor continuous-time linear systems with interval state matrices. In: Nolle, L., Burger, A., Tholen, C., Werner, J., Wellhausen, J., (eds.) Proceedings 32nd European Conference on Modelling and Simulation ©ECMS. ISBN: 978–0–9932440–6–3/ ISBN: 978–0–9932440–7–0 (CD)
Kaczorek, T.: Positivity and stability of descriptor linear systems with interval state matrices. Comput. Probl. Electr. Eng. 8(1), 7–17 (2018)
Kaczorek, T.: Positive 1D and 2D Systems. Springer, London (2002)
Kaczorek, T.: Positive linear systems with different fractional orders. Bull. Pol. Acad.: Tech.58(3), 453–458 (2010)
Kaczorek, T.: Positive linear systems consisting of n subsystems with different fractional orders. IEEE Trans. Circuits Syst. 58(7), 1203–1210 (2011)
Kaczorek, T.: Positive fractional continuous-time linear systems with singular pencils. Bull. Pol. Acad.: Tech. 60(1), 9–12 (2012)
Kaczorek, T.: Selected Problems of Fractional Systems Theory. Springer, Berlin (2011)
Kaczorek, T.: Stability of fractional positive nonlinear systems. Arch. Control Sci. 25(4), 491–496 (2015)
Kaczorek, T., Sajewski, Ł.: The pointwise completeness and the pointwise degeneracy of linear discrete-time different fractional order systems. Bull. Pol. Acad.: Tech. 68(6), 1513–1516 (2020)
Kaczorek, T.: Superstabilization of positive linear electrical circuit by state-feedbacks. Bull. Pol. Acad.: Tech. 65(5), 703–708 (2017)
Kaczorek, T., Borawski, K.: Stability of Positive Nonlinear Systems. In: 22nd International Conference on Methods and Models in Automation and Robotics. Międzyzdroje, Poland (2017)
Kaczorek, T., Rogowski, K.: Fractional Linear Systems and Electrical Circuits. Springer, Cham (2015)
Kaczorek, T., Ruszewski, A.: Global stability of discrete-time nonlinear systems with descriptor standard and fractional positive linear parts and scalar feedbacks. Arch. Control Sci. 30(4), 667–681 (2020)
Kaczorek, T., Ruszewski, A.: Global stability of positive discrete-time standard and fractional nonlinear systems with scalar feedbacks. In: Bartoszewicz, A., Kabziński, J., Kacprzyk, J. (eds.) Advanced, Contemporary Control. Advances in Intelligent Systems and Computing, vol. 1196, pp. 1167–1175. Springer, Cham (2020)
Kaczorek, T., Sajewski, Ł: Pointwise completeness and pointwise degeneracy of fractional standard and descriptor linear continuous-time systems with different fractional orders. Int. J. Appl. Math. Comput. Sci. 30(4), 641–647 (2020)
Kudrewicz, J.: Ustoicivost nieliniejnych sistem z obratnoj swjazju. Avtomatika i Telemechanika 25(8) (1964)
Lyapunov, A.M.: Obscaja zadaca ob ustoicivosti dvizenija. Gostechizdat, Moskwa (1963)
Leipholz, H.: Stability Theory. Academic Press, New York (1970)
Mitkowski, W.: Dynamical properties of Metzler systems. Bull. Pol. Acad.: Tech. 56(4), 309–312 (2008)
Ostalczyk, P.: Discrete Fractional Calculus. World Scientific, River Edgle (2016)
Podlubny, I.: Fractional Differential Equations. Academic Press, San Diego (1999)
Ruszewski, A.: Practical and asymptotic stabilities for a class of delayed fractional discrete-time linear systems. Bull. Pol. Acad.: Tech. 67(3), 509–515 (2019)
Ruszewski, A.: Stability of discrete-time fractional linear systems with delays. Arch. Control Sci. 29(3), 549–567 (2019)
Sajewski, Ł.: Decentralized stabilization of descriptor fractional positive continuous-time linear systems with delays. In: 22nd International Conference on Methods and Models in Automation and Robotics, pp. 482–487. Międzyzdroje, Poland (2017)
Sajewski, Ł.: Stabilization of positive descriptor fractional discrete-time linear systems with two different fractional orders by decentralized controller. Bull. Pol. Acad.: Tech. 65(5), 709–714 (2017)
Acknowledgements
This work was supported by National Science Centre in Poland under work No. 2017/27/B/ST7/02443.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2022 The Author(s), under exclusive license to Springer Nature Switzerland AG
About this chapter
Cite this chapter
Kaczorek, T. (2022). Global Stability of Nonlinear Fractional Dynamical Systems. In: Kulczycki, P., Korbicz, J., Kacprzyk, J. (eds) Fractional Dynamical Systems: Methods, Algorithms and Applications. Studies in Systems, Decision and Control, vol 402. Springer, Cham. https://doi.org/10.1007/978-3-030-89972-1_10
Download citation
DOI: https://doi.org/10.1007/978-3-030-89972-1_10
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-89971-4
Online ISBN: 978-3-030-89972-1
eBook Packages: Intelligent Technologies and RoboticsIntelligent Technologies and Robotics (R0)