Skip to main content
SpringerLink
Log in

Robust Stability by Path-Connectivity of Fractional Order Polynomials

  • Published:
Acta Applicandae Mathematicae Aims and scope Submit manuscript

Abstract

Given any pair of stable fractional order polynomials with fixed commensurate order \(\alpha \), for \(0<\alpha <2\), a stable path joining them is provided by mean of convex combinations in the arguments and magnitudes of their roots. It is shown that any pair of paths obtained with this technique are homotopically equivalent. Consequently, a dense trajectory in the stable fractional order polynomials set has been exhibited. Finally, a stabilizing feedback control in terms of the continuous coefficients of the path has been designed and an illustrative example is given.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1

Similar content being viewed by others

References

  1. Caputo, M.: Elasticita e Dissipazione. Zanichelli, Bologna (1969)

    Google Scholar 

  2. Caputo, M.: Mean fractional-order-derivatives differential equations and filters. Ann. Univ. Ferrara 41, 73–84 (1995)

    MathSciNet  MATH  Google Scholar 

  3. Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations. Math. Studies, vol. 204. Elsevier/North-Holland, Amsterdam (2006)

    Book  MATH  Google Scholar 

  4. Podlubny, I.: Fractional Differential Equations. Academic Press, San Diego (1999)

    MATH  Google Scholar 

  5. Chen, Y., Petras, I., Xue, D.: Fractional order control—a tutorial. In: American Control Conference, St. Louis, MO, USA (2009)

    Google Scholar 

  6. Petras, I.: Stability of fractional-order systems with rational orders: A survey. Fract. Calc. Appl. Anal. 12(3), 269–298 (2009)

    MathSciNet  MATH  Google Scholar 

  7. Petras, I.: Fractional-Order Nonlinear Systems: Modeling, Analysis and Simulation. Higher Education Press/Springer, Beijing/Berlin–Heidelberg (2011)

    Book  MATH  Google Scholar 

  8. Matignon, D.: Stability results on fractional differential equations with applications to control processing. In: Proceedings of IMACS-SMC, Lille, France, pp. 963–968 (1996)

    Google Scholar 

  9. Matignon, D.: Stability properties for generalized fractional differential systems. ESAIM Proc. 5, 145–158 (1998). doi:10.1051/proc:1998004

    Article  MathSciNet  MATH  Google Scholar 

  10. Sabatier, J., Moze, M., Farges, C.: LMI stability conditions for fractional order systems. Comput. Math. Appl. 59, 1594–1609 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  11. Ahn, H-S., Chen, Y-Q., Podlubny, I.: Robust stability test of a class of linear time-invariant interval fractional-order system using Lyapunov inequality. Appl. Math. Comput. 187, 27–34 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  12. Ahn, H.S., Chen, Y.: Necessary and sufficient stability condition of fractional-order interval linear systems. Automatica 44(11), 2985–2988 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  13. Aguirre-Hernández, B., Frías-Armenta, M., Verduzco, F.: On differential structures of polynomial spaces in control theory. J. Syst. Sci. Syst. Eng. 21(3), 372–382 (2012)

    Article  Google Scholar 

  14. Aguirre, B., Ibarra, B., Suárez, R.: Sufficient algebraic conditions for stability of cones of polynomials. Syst. Control Lett. 46, 255–263 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  15. Aguirre, B., Suárez, R.: Algebraic test for the Hurwitz stability of a given segment of polynomials. Bol. Soc. Mat. Mexicana 3(12), 261–275 (2006)

    MathSciNet  MATH  Google Scholar 

  16. Bhattacharyya, S.P., Chapellat, H., Keel, L.H.: Robust Control: The Parametric Approach. Prentice Hall, New York (1995)

    MATH  Google Scholar 

  17. Hinrichsen, D., Pritchard, J.: Mathematical Systems Theory I: Modeling, State Space Analysis, Stability and Robustness. Texts in Applied Mathematics, vol. 48. Springer, Berlin (2005)

    MATH  Google Scholar 

  18. López-Rentería, J.A., Aguirre-Hernández, B., Verduzco, F.: The boundary crossing theorem and the maximal stability interval. Math. Probl. Eng. 2011, 123403 (2011). doi:10.1155/2011/123403

    Article  MathSciNet  MATH  Google Scholar 

  19. López-Rentería, J.A., Aguirre-Hernández, B., Verduzco, F.: On Hurwitz and Schur connecting-curves and dense trajectories. AIP Conf. Proc. 1368, 271 (2011). doi:10.1063/1.3663511

    Google Scholar 

  20. Aguirre-Hernández, B., Campos-Cantón, E., López-Rentería, J.A., Díaz González, E.C.: A polynomial approach for generating a monoparametric family of chaotic attractors via switched linear systems. Chaos Solitons Fractals 71, 100–106 (2015)

    Article  MathSciNet  Google Scholar 

  21. López-Rentería, J.A., Aguirre-Hernández, B., Verduzco, F.: Control of the Hopf bifurcation by a linear feedback control. Int. J. Bifurc. Chaos Appl. Sci. Eng. 25, 1550006 (2015). doi:10.1142/S0218127415500066

    Article  MathSciNet  MATH  Google Scholar 

  22. Lu, J., Ma, Y., Chen, W.: Maximal perturbation bounds for robust stabilizability of fractional-order systems with norm bounded perturbations. J. Franklin Inst. 350, 3365–3383 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  23. Buslowicz, M.: Robust stability of convex combination of two fractional degree characteristic polynomials. Acta Mech. Autom. 2(2), 5–10 (2008)

    Google Scholar 

  24. Munkres, J.R.: Topology. Prentice Hall, New York (2000)

    MATH  Google Scholar 

  25. Herstein, I.N., Kaplansky, I.: Matters Mathematical. Chelsea, New York (1978)

    MATH  Google Scholar 

Download references

Acknowledgements

First author wants to thank Iberoamerican University for the support in the realization of this paper.

Author information

Authors and Affiliations

  1. Departamento de Matemáticas, Universidad de Sonora, Blvd. Rosales y Luis Encinas s/n, Col. Centro, 83000, Hermosillo, Sonora, Mexico

    J. A. López-Renteria

  2. Departamento de Física y Matemáticas, Universidad Iberoamericana, Prolongación Paseo de la Reforma 880, México, DF, 01219, Mexico

    G. Fernández-Anaya

Authors
  1. J. A. López-Renteria
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. G. Fernández-Anaya
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to J. A. López-Renteria.

Additional information

Research supported by CONACYT with the postdoctoral grant number 290941-UIA.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

López-Renteria, J.A., Fernández-Anaya, G. Robust Stability by Path-Connectivity of Fractional Order Polynomials. Acta Appl Math 145, 1–14 (2016). https://doi.org/10.1007/s10440-016-0047-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10440-016-0047-4

Keywords

Search

Navigation