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Global Stability of Nonlinear Fractional Dynamical Systems

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Fractional Dynamical Systems: Methods, Algorithms and Applications

Part of the book series: Studies in Systems, Decision and Control ((SSDC,volume 402))

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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.

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References

  1. Berman, A., Plemmons, R.J.: Nonnegative Matrices in the Mathematical Sciences. SIAM (1994).

    Google Scholar 

  2. 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)

    Google Scholar 

  3. 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)

    Article  MathSciNet  Google Scholar 

  4. Farina, L., Rinaldi, S.: Positive Linear Systems; Theory and Applications. Wiley, New York (2000)

    Book  Google Scholar 

  5. Kaczorek, T.: Absolute stability of a class of fractional positive nonlinear systems. Int. J. Appl. Math. Comput. Sci. 29(1), 93–98 (2019)

    Article  MathSciNet  Google Scholar 

  6. Kaczorek, T.: Analysis of positivity and stability of discrete-time and continuous-time nonlinear systems. Comput. Probl. Electr. Eng. 5(1), 11–16 (2015)

    Google Scholar 

  7. Kaczorek, T.: Analysis of positivity and stability of fractional discrete-time nonlinear systems. Bull. Pol. Acad.: Tech. 64(3), 491–494 (2016)

    Google Scholar 

  8. Kaczorek, T.: Global stability of nonlinear feedback systems with fractional positive linear parts. Int. J. Appl. Math. Comput. Sci. 30(3), 493–500 (2020)

    MathSciNet  MATH  Google Scholar 

  9. Kaczorek, T.: Global stability of positive standard and fractional nonlinear feedback systems. Bull. Pol. Acad.: Tech. 68(2), 285–288 (2020)

    Google Scholar 

  10. Kaczorek, T.: Global stability of nonlinear feedback systems with positive linear parts, Int. J. Nonlinear Sci. Numer. Simul. 20(5), 575–579 (2019)

    Google Scholar 

  11. 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)

    Google Scholar 

  12. Kaczorek, T.: Positivity and stability of descriptor linear systems with interval state matrices. Comput. Probl. Electr. Eng. 8(1), 7–17 (2018)

    Google Scholar 

  13. Kaczorek, T.: Positive 1D and 2D Systems. Springer, London (2002)

    Book  Google Scholar 

  14. Kaczorek, T.: Positive linear systems with different fractional orders. Bull. Pol. Acad.: Tech.58(3), 453–458 (2010)

    Google Scholar 

  15. Kaczorek, T.: Positive linear systems consisting of n subsystems with different fractional orders. IEEE Trans. Circuits Syst. 58(7), 1203–1210 (2011)

    Article  MathSciNet  Google Scholar 

  16. Kaczorek, T.: Positive fractional continuous-time linear systems with singular pencils. Bull. Pol. Acad.: Tech. 60(1), 9–12 (2012)

    Google Scholar 

  17. Kaczorek, T.: Selected Problems of Fractional Systems Theory. Springer, Berlin (2011)

    Book  Google Scholar 

  18. Kaczorek, T.: Stability of fractional positive nonlinear systems. Arch. Control Sci. 25(4), 491–496 (2015)

    Article  MathSciNet  Google Scholar 

  19. 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)

    Google Scholar 

  20. Kaczorek, T.: Superstabilization of positive linear electrical circuit by state-feedbacks. Bull. Pol. Acad.: Tech. 65(5), 703–708 (2017)

    Google Scholar 

  21. Kaczorek, T., Borawski, K.: Stability of Positive Nonlinear Systems. In: 22nd International Conference on Methods and Models in Automation and Robotics. Międzyzdroje, Poland (2017)

    Google Scholar 

  22. Kaczorek, T., Rogowski, K.: Fractional Linear Systems and Electrical Circuits. Springer, Cham (2015)

    Book  Google Scholar 

  23. 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)

    MathSciNet  MATH  Google Scholar 

  24. 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)

    Chapter  Google Scholar 

  25. 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)

    MathSciNet  MATH  Google Scholar 

  26. Kudrewicz, J.: Ustoicivost nieliniejnych sistem z obratnoj swjazju. Avtomatika i Telemechanika 25(8) (1964)

    Google Scholar 

  27. Lyapunov, A.M.: Obscaja zadaca ob ustoicivosti dvizenija. Gostechizdat, Moskwa (1963)

    Google Scholar 

  28. Leipholz, H.: Stability Theory. Academic Press, New York (1970)

    Google Scholar 

  29. Mitkowski, W.: Dynamical properties of Metzler systems. Bull. Pol. Acad.: Tech. 56(4), 309–312 (2008)

    Google Scholar 

  30. Ostalczyk, P.: Discrete Fractional Calculus. World Scientific, River Edgle (2016)

    Book  Google Scholar 

  31. Podlubny, I.: Fractional Differential Equations. Academic Press, San Diego (1999)

    MATH  Google Scholar 

  32. 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)

    Google Scholar 

  33. Ruszewski, A.: Stability of discrete-time fractional linear systems with delays. Arch. Control Sci. 29(3), 549–567 (2019)

    MathSciNet  MATH  Google Scholar 

  34. 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)

    Google Scholar 

  35. 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)

    Google Scholar 

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Acknowledgements

This work was supported by National Science Centre in Poland under work No. 2017/27/B/ST7/02443.

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Authors and Affiliations

  1. Faculty of Electrical Engineering, Białystok University of Technology Bialystok, Wiejska 45D, 15-351, Białystok, Poland

    Tadeusz Kaczorek

Authors
  1. Tadeusz Kaczorek
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Correspondence to Tadeusz Kaczorek .

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Editors and Affiliations

  1. Systems Research Institute, Polish Academy of Sciences, Warsaw, Poland

    Piotr Kulczycki

  2. Institute of Control and Computation Engineering, University of Zielona Góra, Zielona Góra, Poland

    Józef Korbicz

  3. Systems Research Institute, Polish Academy of Sciences, Warsaw, Poland

    Janusz Kacprzyk

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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

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