Skip to main content
SpringerLink
Log in

Stability Analysis of Fractional Discrete-Time Linear Scalar Systems with Pure Delay

  • Conference paper
  • First Online:
Non-Integer Order Calculus and its Applications (RRNR 2017)

Part of the book series: Lecture Notes in Electrical Engineering ((LNEE,volume 496))

Included in the following conference series:

Abstract

In the paper the stability problems of fractional discrete-time linear scalar systems with pure delay are considered. Using the classical D-partition method, the necessary and sufficient condition for practical and asymptotic stability are given. The considerations are illustrated by numerical examples.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 129.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 169.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 169.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Busłowicz, M.: Simple analytic conditions for stability of fractional discrete-time linear systems with diagonal state matrix. Bull. Pol. Acad. Sci. Tech. Sci. 60, 809–814 (2012)

    Google Scholar 

  2. Busłowicz, M.: Stability conditions for linear continuous-time fractional-order state-delayed systems. Bull. Pol. Acad. Sci. Tech. Sci. 64, 3–7 (2016)

    Google Scholar 

  3. Busłowicz, M., Kaczorek, T.: Simple conditions for practical stability of linear positive fractional discrete-time linear systems. Int. J. Appl. Math. Comput. Sci. 19, 263–269 (2009)

    MathSciNet  MATH  Google Scholar 

  4. Busłowicz, M., Ruszewski, A.: Necessary and sufficient conditions for stability of fractional discrete-time linear state-space systems. Bull. Pol. Acad. Sci. Tech. Sci. 61, 779–786 (2013)

    Google Scholar 

  5. Dzieliński, A., Sierociuk, D.: Stability of discrete fractional state-space systems. J. Vibr. Control 14, 1543–1556 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  6. Gryazina, E.N., Polyak, B.T., Tremba, A.A.: D-decomposition technique state-of-the-art. Autom. Remote Control 69(12), 1991–2026 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  7. Kaczorek, T.: Practical stability of positive fractional discrete-time systems. Bull. Pol. Acad. Sci. Tech. Sci. 56, 313–317 (2008)

    Google Scholar 

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

    Book  MATH  Google Scholar 

  9. Kaczorek, T.: A new approach to the realization problem for fractional discrete-time linear systems. Bull. Pol. Acad. Sci. 64, 9–14 (2016)

    Google Scholar 

  10. Kaczorek, T., Ostalczyk, P.: Responses comparison of the two discrete-time linear fractional state-space models. Fractional Calc. Appl. Anal. 19, 789–805 (2016)

    MathSciNet  MATH  Google Scholar 

  11. Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations. Elsevier, Amsterdam (2006)

    MATH  Google Scholar 

  12. Monje, C., Chen, Y., Vinagre, B., Xue, D., Feliu, V.: Fractional-Order Systems and Controls. Springer, London (2010)

    Book  MATH  Google Scholar 

  13. Ostalczyk, P.: Equivalent descriptions of a discrete-time fractional-order linear system and its stability domains. Int. J. Appl. Math. Comput. Sci. 22, 533–538 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  14. Ostalczyk, P.: Discrete Fractional Calculus: Applications in Control and Image Processing. Series in Computer Vision. World Scientific Publishing, Singapore (2016)

    Book  MATH  Google Scholar 

  15. Ruszewski, R.: Stability conditions of fractional discrete-time scalar systems with pure delay. Pomiary Automatyka Robotyka 17, 340–344 (2013)

    MATH  Google Scholar 

  16. Ruszewski, R.: Practical and asymptotic stability of fractional discrete-time scalar systems described by a new model. Arch. Control Sci. 17, 340–344 (2016)

    Google Scholar 

  17. Ruszewski, R.: 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)

    Google Scholar 

  18. Ruszewski, R., 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 

  19. 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)

    Google Scholar 

  20. 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. Sci. 61, 353–361 (2013)

    Google Scholar 

  21. Stanisławski, R.: New results in stability analysis for LTI SISO systems modeled by GL-discretized fractional-order transfer functions. Fractional Calc. Appl. Anal. 20, 243–259 (2017)

    MathSciNet  MATH  Google Scholar 

Download references

Acknowledgement

This work was supported by the National Science Centre in Poland under the work No. 2014/13/B/ST7/03467.

Author information

Authors and Affiliations

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

    Andrzej Ruszewski

Authors
  1. Andrzej Ruszewski
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Andrzej Ruszewski .

Editor information

Editors and Affiliations

  1. Institute of Applied Computer Science, Lodz University of Technology, Lodz, Poland

    Piotr Ostalczyk

  2. Institute of Applied Computer Science, Lodz University of Technology, Lodz, Poland

    Dominik Sankowski

  3. Institute of Applied Computer Science, Lodz University of Technology, Lodz, Poland

    Jacek Nowakowski

Rights and permissions

Reprints and permissions

Copyright information

© 2019 Springer International Publishing AG, part of Springer Nature

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Ruszewski, A. (2019). Stability Analysis of Fractional Discrete-Time Linear Scalar Systems with Pure Delay. In: Ostalczyk, P., Sankowski, D., Nowakowski, J. (eds) Non-Integer Order Calculus and its Applications. RRNR 2017. Lecture Notes in Electrical Engineering, vol 496. Springer, Cham. https://doi.org/10.1007/978-3-319-78458-8_8

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-78458-8_8

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-78457-1

  • Online ISBN: 978-3-319-78458-8

  • eBook Packages: EngineeringEngineering (R0)

Publish with us

Policies and ethics

Search

Navigation