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Global Stability of Positive Discrete-Time Standard and Fractional Nonlinear Systems with Scalar Feedbacks

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Advanced, Contemporary Control

Part of the book series: Advances in Intelligent Systems and Computing ((AISC,volume 1196))

Abstract

The global stability of positive discrete-time standard and fractional orders nonlinear systems with scalar feedbacks is investigated. New sufficient conditions for the global stability of these classes of positive nonlinear systems are established. The effectiveness of these new stability conditions is demonstrated on simple examples of positive nonlinear discrete-time systems with scalar feedbacks.

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References

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

    Book  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). https://doi.org/10.15199/48.2017.11.36

    Article  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), 263–269 (2009). https://doi.org/10.2478/v10006-009-0022-6

    Article  MathSciNet  MATH  Google Scholar 

  4. Grabowski, P.: Absolute stability criteria for infinite-dimensional discrete Lur’e systems with application to loaded electric distortionless RLCG -transmission line. J. Differ. Equ. Appl. 19, 304–331 (2013). https://doi.org/10.1080/10236198.2011.639366

    Article  MATH  Google Scholar 

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

    Book  Google Scholar 

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

    Book  Google Scholar 

  7. Kaczorek, T.: Positive linear systems with different fractional orders. Bull. Pol. Acad. Sci. Techn. 58(3), 453–458 (2010). https://doi.org/10.2478/v10175-010-0043-1

    Article  MATH  Google Scholar 

  8. Kaczorek, T.: Positive linear systems consisting of n subsystems with different fractional orders. IEEE Trans. Circuits Syst. 58(6), 1203–1210 (2011). https://doi.org/10.1109/TCSI.2010.2096111

    Article  MathSciNet  Google Scholar 

  9. Kaczorek, T.: Selected Problems of Fractional Systems Theory. Springer, Heidelberg (2011)

    Book  Google Scholar 

  10. 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 

  11. Kaczorek, T.: Stability of fractional positive nonlinear systems. Arch. Control Sci. 25(4), 491–496 (2015). https://doi.org/10.1515/acsc-2015-0031

    Article  MathSciNet  Google Scholar 

  12. Kaczorek, T.: Analysis of positivity and stability of fractional discrete-time nonlinear systems. Bull. Pol. Acad. Sci. Techn. 64(3), 491–494 (2016). https://doi.org/10.1515/bpasts-2016-0054

    Article  Google Scholar 

  13. Kaczorek, T.: Superstabilization of positive linear electrical circuit by state-feedbacks. Bull. Pol. Acad. Sci. Techn. 65(5), 703–708 (2017). https://doi.org/10.1515/bpasts-2017-0075

    Article  Google Scholar 

  14. Kaczorek, T.: Absolute stability of a class of fractional positive nonlinear systems. Int. J. Appl. Math. Comput. Sci. 29(1), 93–98 (2019). https://doi.org/10.2478/amcs-2019-0007

    Article  MathSciNet  MATH  Google Scholar 

  15. Kaczorek, T.: Global stability of nonlinear feedback systems with positive linear parts. Int. J. Nonlinear Sci. Numer. Simul. 20(5), 575–579 (2019). https://doi.org/10.1515/ijnsns-2018-0189

    Article  MathSciNet  MATH  Google Scholar 

  16. Kaczorek, T.: Global stability of positive standard and fractional nonlinear feedback systems. Bull. Pol. Acad. Sci. Techn. 68(2), 285–288 (2020). https://doi.org/10.24425/bpasts.2020.133112

    Article  Google Scholar 

  17. Kaczorek, T., Borawski, K.: Stability of positive nonlinear systems. In: 22nd International Conference Methods and Models in Automation and Robotics, Poland, pp. 564–569 (2017). https://doi.org/10.1109/MMAR.2017.8046890

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

    Book  Google Scholar 

  19. Lyapunov, A.M.: General Problem of Stable Movement. Gostechizdat, Moscow (1963). (in Russian)

    Google Scholar 

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

    MATH  Google Scholar 

  21. Ostalczyk, P.: Discrete Fractional Calculus. World Scientific, River Edge (2016)

    Book  Google Scholar 

  22. Ruszewski, A.: Stability of discrete-time fractional linear systems with delays. Arch. Control Sci. 29(3), 549–567 (2019). https://doi.org/10.24425/acs.2019.130205

    Article  MathSciNet  MATH  Google Scholar 

  23. Ruszewski, A.: Practical and asymptotic stabilities for a class of delayed fractional discrete-time linear systems. Bull. Pol. Acad. Sci. Techn. 67(3), 509–515 (2019). https://doi.org/10.24425/bpasts.2019.128426

    Article  MathSciNet  Google Scholar 

  24. Sabatier, J., Agrawal, O.P., Machado, J.A.T.: Advances in Fractional Calculus, Theoretical Developments and Applications in Physics and Engineering. Springer, London (2007)

    MATH  Google Scholar 

  25. 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, Poland, pp. 482–487 (2017). https://doi.org/10.1109/MMAR.2017.8046875

  26. Sajewski, Ł.: Stabilization of positive descriptor fractional discrete-time linear systems with two different fractional orders by decentralized controller. Bull. Pol. Acad. Sci. Techn. 65(5), 709–714 (2017). https://doi.org/10.1515/bpasts-2017-0076

    Article  Google Scholar 

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Acknowledgment

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, Bialystok University of Technology, Wiejska 45D, 15-351, Białystok, Poland

    Tadeusz Kaczorek & Andrzej Ruszewski

Authors
  1. Tadeusz Kaczorek
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  2. Andrzej Ruszewski
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Corresponding author

Correspondence to Andrzej Ruszewski .

Editor information

Editors and Affiliations

  1. Institute of Automatic Control, Lodz University of Technology, Lodz, Poland

    Andrzej Bartoszewicz

  2. Institute of Automatic Control, Lodz University of Technology, Lodz, Poland

    Jacek Kabziński

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

    Janusz Kacprzyk

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Kaczorek, T., Ruszewski, A. (2020). 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. Springer, Cham. https://doi.org/10.1007/978-3-030-50936-1_97

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