Skip to main content
SpringerLink
Log in
Book cover

Proceedings of the Computational Methods in Systems and Software

CoMeSySo 2021: Data Science and Intelligent Systems pp 18–23Cite as

Value Set-Based Numerical Analysis of Robust Stability for Fractional-Order Retarded Quasi-Polynomials with Uncertain Parameters and Uncertain Fractional Orders

  • Conference paper
  • First Online:

  • 1051 Accesses

Part of the book series: Lecture Notes in Networks and Systems ((LNNS,volume 231))

Abstract

This example-oriented contribution deals with the value set-based numerical analysis of robust stability for the family of fractional-order retarded quasi-polynomials with both uncertain parameters and uncertain fractional orders. The specific investigated feedback control system consists of the fractional-order PID controller and the controlled plant, represented by a heat transfer process described by the linear time-invariant fractional-order time-delay model with parametric uncertainty (with three uncertain parameters, namely, gain, fractional time constant, and fractional time-delay term, and furthermore two fractional orders). The graphical robust stability analysis is based on the numerical calculation of the value sets and the application of the zero exclusion principle.

This is a preview of subscription content, 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   139.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   179.99
Price excludes VAT (USA)
  • Compact, lightweight 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

Learn about institutional subscriptions

References

  1. Bhattacharyya, S.P.: Robust control under parametric uncertainty: an overview and recent results. Annu. Rev. Control. 44, 45–77 (2017)

    Article  Google Scholar 

  2. Barmish, B.R.: New Tools for Robustness of Linear Systems. Macmillan, New York (1994)

    MATH  Google Scholar 

  3. Machado, J.A.T., Kiryakova, V., Mainardi, F.: Recent history of fractional calculus. Commun. Nonlinear Sci. Numer. Simul. 16(3), 1140–1153 (2011)

    Article  MathSciNet  Google Scholar 

  4. Chen, Y.Q., Petráš, I., Xue, D.: Fractional order control – a tutorial. In: Proceedings of the 2009 American Control Conference, St. Louis, MO, USA (2009)

    Google Scholar 

  5. Xue, D.: Fractional-Order Control Systems: Fundamentals and Numerical Implementations. De Gruyter, Berlin, Germany (2017)

    Book  Google Scholar 

  6. Tan, N., Özgüven, Ö.F., Özyetkin, M.M.: Robust stability analysis of fractional order interval polynomials. ISA Trans. 48(2), 166–172 (2009)

    Article  Google Scholar 

  7. Jiao, Z., Zhong, Y.: Robust stability for fractional-order systems with structured and unstructured uncertainties. Comput. Math. Appl. 64(10), 3258–3266 (2012)

    Article  MathSciNet  Google Scholar 

  8. Matušů, R., Şenol, B., Pekař, L.: Robust stability of fractional order polynomials with complicated uncertainty structure. PLOS ONE 12(6), e0180274 (2017)

    Google Scholar 

  9. Busłowicz, M.: Stability of linear continuous-time fractional order systems with delays of the retarded type. Bulletin of the Polish Academy of Sciences: Technical Sciences 56(4), 319–324 (2008)

    Google Scholar 

  10. Petráš, I., Chen, Y.Q., Vinagre, B.M., Podlubný, I.: Stability of linear time invariant systems with interval fractional orders and interval coefficients. In: Second IEEE International Conference on Computational Cybernetics, Vienna, Austria (2004)

    Google Scholar 

  11. Gómez-Aguilar, J.F., Razo-Hernández, R., Granados-Lieberman, D.: A physical interpretation of fractional calculus in observables terms: analysis of the fractional time constant and the transitory response. Rev. Mex. Fís. 60(1), 32–38 (2014)

    MathSciNet  Google Scholar 

  12. Macias, M., Sierociuk, D.: Fractional order calculus for modeling and fractional PID control of the heating process. In: Proceedings of the 13th International Carpathian Control Conference, High Tatras, Podbanské, Slovakia (2012)

    Google Scholar 

  13. Sierociuk, D., Dzieliński, A., Sarwas, G., Petráš, I., Podlubný, I., Škovránek, T.: Modelling heat transfer in heterogeneous media using fractional calculus. Philos. Trans. R. Soc. A, 371(1990), 20120146 (2013)

    Google Scholar 

  14. Matušů, R., Pekař, L.: Robust stability of thermal control systems with uncertain parameters: the graphical analysis examples. Appl. Therm. Eng. 125, 1157–1163 (2017)

    Article  Google Scholar 

  15. Yeroğlu, C., Özyetkin, M.M., Tan, N.: Frequency response computation of fractional order interval transfer functions. Int. J. Control Autom. Syst. 8(5), 1009–1017 (2010)

    Article  Google Scholar 

  16. Şenol, B., Yeroğlu, C.: Computation of the value set of fractional order uncertain polynomials: a 2q convex parpolygonal approach. In: Proceedings of the 2012 IEEE International Conference on Control Applications, Dubrovnik, Croatia (2012)

    Google Scholar 

  17. Yeroğlu, C., Şenol, B.: Investigation of robust stability of fractional order multilinear affine systems: 2q-convex parpolygon approach. Syst. Control Lett. 62(10), 845–855 (2013)

    Article  MathSciNet  Google Scholar 

  18. Labora, D.C., Nieto, J.J., Rodríguez-López, R.: Is it possible to construct a fractional derivative such that the index law holds? Prog. Fract. Differ. Appl. 4(1), 1–3 (2018)

    Article  Google Scholar 

  19. Bhalekar, S., Patil, M.: Can we split fractional derivative while analyzing fractional differential equations? Commun. Nonlinear Sci. Numer. Simul. 76, 12–24 (2019)

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Centre for Security, Information and Advanced Technologies (CEBIA–Tech), Faculty of Applied Informatics, Tomas Bata University in Zlín, Nám. T. G. Masaryka 5555, 76001, Zlín, Czech Republic

    Radek Matušů

  2. Department of Computer Engineering, Faculty of Engineering, Inonu University, 44280, Malatya, Turkey

    Bilal Senol, Baris Baykant Alagoz & Abdullah Ates

Authors
  1. Radek Matušů
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Bilal Senol
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Baris Baykant Alagoz
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Abdullah Ates
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Radek Matušů .

Editor information

Editors and Affiliations

  1. Faculty of Applied Informatics, Tomas Bata University in Zlín, Zlín, Czech Republic

    Radek Silhavy

  2. Faculty of Applied Informatics, Tomas Bata University in Zlín, Zlín, Czech Republic

    Petr Silhavy

  3. Faculty of Applied Informatics, Tomas Bata University in Zlín, Zlín, Czech Republic

    Zdenka Prokopova

Rights and permissions

Reprints and permissions

Copyright information

© 2021 The Author(s), under exclusive license to Springer Nature Switzerland AG

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Matušů, R., Senol, B., Alagoz, B.B., Ates, A. (2021). Value Set-Based Numerical Analysis of Robust Stability for Fractional-Order Retarded Quasi-Polynomials with Uncertain Parameters and Uncertain Fractional Orders. In: Silhavy, R., Silhavy, P., Prokopova, Z. (eds) Data Science and Intelligent Systems. CoMeSySo 2021. Lecture Notes in Networks and Systems, vol 231. Springer, Cham. https://doi.org/10.1007/978-3-030-90321-3_3

Download citation

Publish with us

Policies and ethics

Search

Navigation