Robust Stability Check of Fractional Discrete-Time Linear Systems with Interval Uncertainties

  • Mikołaj Busłowicz
  • Andrzej Ruszewski
Part of the Lecture Notes in Electrical Engineering book series (LNEE, volume 320)


The paper presents the problems of robust practical stability and robust asymptotic stability of fractional-order discrete-time linear systems with uncertainty. It is supposed that the system matrix is the interval matrix which elements are the convex combinations of the elements of specified bounded matrices and the fractional order α satisfies 0 < α < 1. Using the matrix measure the robust stability conditions are given. The considerations are illustrated by numerical examples.


Linear system discrete-time fractional-order robust stability interval matrix 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Ahn, H.-S., Chen, Y.Q.: Necessary and Sufficient Stability Condition of Fractional-order Interval Linear Systems. Automatica 44, 2985–2988 (2008)CrossRefzbMATHMathSciNetGoogle Scholar
  2. 2.
    Busłowicz, M.: Asymptotic Stability of Dynamical Interval Systems with Pure Delay. Scientific Journal Bialystok University of Technology, Technical Sciences 83, Electricity 11, 61–77 (1992)Google Scholar
  3. 3.
    Busłowicz, M.: Robust Stability of Positive Discrete-time Linear Systems of Fractional Order. Bulletin of the Polish Academy of Sciences, Technical Sciences 58, 567–572 (2010)zbMATHGoogle Scholar
  4. 4.
    Busłowicz, M.: Stability of State-space Models of Linear Continuous-time Fractional Order Systems. Acta Mechanica et Automatica 5, 15–22 (2011)Google Scholar
  5. 5.
    Busłowicz, M.: Stability Analysis of Continuous-time Linear Systems Consisting of n Subsystems with Different Fractional Orders. Bulletin of the Polish Academy of Sciences, Technical Sciences 60, 279–284 (2012)Google Scholar
  6. 6.
    Busłowicz, M.: Simple Analytic Conditions for Stability of Fractional Discrete-time Linear Systems with Diagonal State Matrix. Bulletin of the Polish Academy of Sciences, Technical Sciences 60, 809–814 (2012)Google Scholar
  7. 7.
    Busłowicz, M., Kaczorek, T.: Simple Conditions for Practical Stability of Linear Positive Fractional Discrete-time Linear Systems. International Journal of Applied Mathematics and Computer Science 19, 263–269 (2009)zbMATHMathSciNetGoogle Scholar
  8. 8.
    Busłowicz, M., Ruszewski, A.: Necessary and Sufficient Conditions for Stability of Fractional Discrete-time Linear State-space Systems. Bulletin of the Polish Academy of Sciences, Technical Sciences 61, 779–786 (2013)Google Scholar
  9. 9.
    Chen, Y.Q., Ahn, H.-S., Podlubny, I.: Robust Stability Check of Fractional Order Linear Time Invariant Systems with Interval Uncertainties. Signal Processing 86, 2611–2618 (2006)CrossRefzbMATHGoogle Scholar
  10. 10.
    Desoer, C.A., Vidyasagar, M.: Feedback Systems: Input-output Properties. Acad. Press, New York (1975)zbMATHGoogle Scholar
  11. 11.
    Dzieliński, A., Sierociuk, D.: Stability of Discrete Fractional State-space Systems. Journal of Vibration and Control 14, 1543–1556 (2008)CrossRefzbMATHMathSciNetGoogle Scholar
  12. 12.
    Kaczorek, T.: Practical Stability of Positive Fractional Discrete-time Systems. Bulletin of the Polish Academy of Sciences, Technical Sciences 56, 313–317 (2008)Google Scholar
  13. 13.
    Kaczorek, T.: Selected Problems of Fractional Systems Theory. LNCIS, vol. 411. Springer, Berlin (2011)zbMATHGoogle Scholar
  14. 14.
    Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations. Elsevier, Amsterdam (2006)zbMATHGoogle Scholar
  15. 15.
    Liao, Z., Peng, C., Li, W., Wang, Y.: Robust Stability Analysis for a Class of Fractional Order Systems with Uncertain Parameters. Journal of the Franklin Institute 348, 1101–1113 (2011)CrossRefzbMATHMathSciNetGoogle Scholar
  16. 16.
    Monje, C., Chen, Y., Vinagre, B., Xue, D., Feliu, V.: Fractional-order Systems and Controls. Springer, London (2010)CrossRefzbMATHGoogle Scholar
  17. 17.
    Ostalczyk, P.: Equivalent Descriptions of a Discrete-time Fractional-order Linear System and its Stability Domains. Int. J. Applied Mathematics and Computer Science 22, 533–538 (2012)MathSciNetGoogle Scholar
  18. 18.
    Petras, I.: Stability of Fractional-order Systems with Rational Orders: A Survey. Fractional Calculus & Applied Analysis. An International Journal for Theory and Applications 12, 269–298 (2009)zbMATHMathSciNetGoogle Scholar
  19. 19.
    Podlubny, I.: Fractional Differential Equations. Academic Press, San Diego (1999)zbMATHGoogle Scholar
  20. 20.
    Ruszewski, A.: Practical Stability and Asymptotic Stability of Interval Fractional Discrete-time Linear State-space System. In: Szewczyk, R., Zieliński, C., Kaliczyńska, M. (eds.) Recent Advances in Automation, Robotics and Measuring Techniques. AISC, vol. 267, pp. 217–228. Springer, Heidelberg (2014)CrossRefGoogle Scholar
  21. 21.
    Sabatier, J., Agrawal, O.P., Machado, J.A.T. (eds.): Advances in Fractional Calculus, Theoretical Developments and Applications in Physics and Engineering. Springer, London (2007)zbMATHGoogle Scholar
  22. 22.
    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. Bulletin of the Polish Academy of Sciences 61, 353–361 (2013)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.Faculty of Electrical EngineeringBialystok University of TechnologyBiałystokPoland

Personalised recommendations