Stability Analysis for a Class of Fractional Discrete-Time Linear Scalar Systems with Multiple Delays in State

  • Andrzej RuszewskiEmail author
Conference paper
Part of the Lecture Notes in Electrical Engineering book series (LNEE, volume 559)


The fractional discrete-time linear scalar systems with multiple delays described by the model without a time shift in the difference are addressed. The practical stability and the asymptotic stability of the systems are considered. New stability conditions in terms of intervals of parameter values are given.


Fractional Linear system Discrete-time Stability Time-delays 



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


  1. 1.
    Das, S.: Functional Fractional Calculus for System Identification and Controls. Springer, Berlin (2008)zbMATHGoogle Scholar
  2. 2.
    Busłowicz, M., Ruszewski, A.: Necessary and sufficient conditions for stability of fractional discrete-time linear state-space systems. Bull. Pol. Acad. Tech. 61(4), 779–786 (2013). Scholar
  3. 3.
    Dzieliński, A., Sierociuk, D.: Stability of discrete fractional state-space systems. J. Vibr. Control 14, 1543–1556 (2008). Scholar
  4. 4.
    Gryazina, E.N., Polyak, B.T., Tremba, A.A.: D-decomposition technique state-of-the-art. Autom. Remote Control 69(12), 1991–2026 (2008). Scholar
  5. 5.
    Kaczorek, T.: Practical stability of positive fractional discrete-time systems. Bull. Pol. Acad. Tech. 56(4), 313–317 (2008)Google Scholar
  6. 6.
    Kaczorek, T.: Selected Problems of Fractional Systems Theory. Springer, Berlin (2011)CrossRefGoogle Scholar
  7. 7.
    Kaczorek, T.: A new approach to the realization problem for fractional discrete-time linear systems. Bull. Pol. Acad. Tech. 64(1), 9–14 (2016). Scholar
  8. 8.
    Kaczorek, T., Ostalczyk, P.: Responses comparison of the two discrete-time linear fractional state-space models. Fract. Calc. Appl. Anal. 19(4), 789–805 (2016). Scholar
  9. 9.
    Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations. Elsevier, Amsterdam (2006)zbMATHGoogle Scholar
  10. 10.
    Monje, C., Chen, Y., Vinagre, B., Xue, D., Feliu, V.: Fractional-Order Systems and Controls. Springer, London (2010)CrossRefGoogle Scholar
  11. 11.
    Mozyrska, D., Ostalczyk, P., Wyrwas, M.: Stability conditions for fractional-order linear equations with delays. Bull. Pol. Acad. Tech. 61(4), 449–454 (2018). Scholar
  12. 12.
    Ostalczyk, P.: Equivalent descriptions of a discrete-time fractional-order linear system and its stability domains. J. Appl. Math. Comput. Sci. 22(3), 533–538 (2012). Scholar
  13. 13.
    Ostalczyk, P.: Discrete Fractional Calculus: Applications in Control and Image Processing, Series in Computer Vision. World Scientific Publishing, Singapore (2016)CrossRefGoogle Scholar
  14. 14.
    Ruszewski, A.: Practical and asymptotic stability of fractional discrete-time scalar systems described by a new model. Arch. Control Sci. 26(4), 441–452 (2016). Scholar
  15. 15.
    Ruszewski, A.: Stability analysis for the new model of fractional discrete-time linear state-space systems. In: Babiarz, A., et al. (eds.) Theory and Applications of Non-integer Order Systems. Lecture Notes in Electrical Engineering, vol. 407, pp. 381–389. Springer, Heidelberg (2017). Scholar
  16. 16.
    Ruszewski, A.: Stability analysis of fractional discrete-time linear scalar systems with pure delay. In: Ostalczyk, P., et al. (eds.) Non-integer Order Calculus and Its Applications. Lecture Notes in Electrical Engineering, vol. 496, pp. 84–91. Springer, Heidelberg (2019). Scholar
  17. 17.
    Ruszewski, A., Busłowicz, M.: Practical and asymptotic stability of fractional discrete-time scalar systems with multiple delays. In: Malinowski, K., et al. (eds.) Recent Advances in Control and Automation, pp. 183–192. Academic Publishing House Exit, Warsaw (2014)Google Scholar
  18. 18.
    Sabatier, J., Agrawal, O.P., Machado, J.A.T.: Advances in Fractional Calculus, Theoretical Developments and Applications in Physics and Engineering. Springer, London (2007)zbMATHGoogle Scholar
  19. 19.
    Stanisławski, R., Latawiec, K.J.: Stability analysis for discrete-time fractional-order LTI state-space systems. Part I: new necessary and sufficient conditions for asymptotic stability. Bull. Pol. Acad. Tech. 61(2), 353–361 (2013). Scholar
  20. 20.
    Stanisławski, R.: New results in stability analysis for LTI SISO systems modeled by GL-discretized fractional-order transfer functions. Fract. Calc. Appl. Anal. 20(1), 243–259 (2017). Scholar

Copyright information

© Springer Nature Switzerland AG 2020

Authors and Affiliations

  1. 1.Faculty of Electrical EngineeringBialystok University of TechnologyBiałystokPoland

Personalised recommendations