Exponential Stability for a Class of Fractional Order Dynamic Systems

  • Krzysztof OprzędkiewiczEmail author
  • Wojciech Mitkowski
Conference paper
Part of the Lecture Notes in Electrical Engineering book series (LNEE, volume 559)


The paper presents a comparinson of exponential, Mittag-Leffler and generalized Mittag-Leffler stability problems for a class of fractional order dynamical systems. The considered system is described by state equation with diagonal state matrix, the spectrum of the system contains single, separated, real, decreasing eigenvalues. An example of such a system is a heat object described by a fractional order state equation. The fractional order derivative is described by Caputo and Caputo-Fabrizio operators. For the considered system the simple conditions of approximated equivalence of the all discussed stabilities are proposed. Results are illustrated by the numerical example.


Fractional order systems Fractional order state equation Caputo operator Caputo-Fabrizio operator Exponential stability Mittag-Leffler stability Generalized Mittag-Leffler stability 



This paper was sponsored by AGH UST project no


  1. 1.
    Caputo, M., Fabrizio, M.: A new definition of fractional derivative without singular kernel. Prog. Fractiona Differ. Appl. 1(2), 1–13 (2015)Google Scholar
  2. 2.
    Kaczorek, T.: Selected Problems of Fractional Systems Theory. Springer, Berlin (2011). Scholar
  3. 3.
    Kaczorek, T.: Singular fractional linear systems and electrical circuits. Int. J. Appl. Math. Comput. Sci. 21(2), 379–384 (2011)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Kaczorek, T.: Reduced-order fractional descriptor observers for a class of fractional descriptor continuous-time nonlinear systems. Int. J. Appl. Math. Comput. Sci. 26(2), 277–283 (2016)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Kaczorek, T., Borawski, K.: Fractional descriptor continuous-time linear systems described by the Caputo-Fabrizio derivative. Int. J. Appl. Math. Comput. Sci. 26(3), 533–541 (2016)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Kaczorek, T., Rogowski, K.: Fractional Linear Systems and Electrical Circuits. Bialystok University of Technology, Bialystok (2014)zbMATHGoogle Scholar
  7. 7.
    Kamocki, R., Pajek, K.: On the existence of optimal solutions for optimal control problems involving the Caputo fractional derivatives with nonsingular kernels. In: MMAR 2018: 23rd International Conference on Methods and Models in Automation and Robotics, Miedzyzdroje, Poland, 27–30 August 2018 (2018)Google Scholar
  8. 8.
    Li, Y., Chen, Y.Q., Podlubny, I.: Mittag-Leffler stability of fractional order nonlinear dynamic systems. Automatica 45(4), 1965–1969 (2009)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Li, Y., Chen, Y.Q., Podlubny, I.: Stability of fractional order nonlinear dynamic systems: Lyapunov direct method and generalized Mittag-Leffler stability. Comput. Math. Appl. 59(2010), 1810–1821 (2010)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Murray, R.M., Li, Z., Sastry, S.S.: A Mathematical Introduction to Robotic Manipulation. CRC Press, Boca Raton (1993)zbMATHGoogle Scholar
  11. 11.
    Oprzedkiewicz, K., Gawin, E.: A non-integer order, state space model for one dimensional heat transfer process. Arch. Control. Sci. 26(2), 261–275 (2016)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Oprzedkiewicz, K., Gawin, E., Mitkowski, W.: Modeling heat distribution with the use of a non-integer order, state space model. Int. J. Appl. Math. Comput. Sci. 26(4), 749–756 (2016). Scholar
  13. 13.
    Oprzedkiewicz, K., Mitkowski, W., Gawin, E.: Parameter identification for non integer order, state space models of heat plant. In: 21st International Conference on Methods and Models in Automation and Robotics, MMAR 2016, Miedzyzdroje, Poland, 29 August–01 September 2016, pp. 184–188 (2016)Google Scholar
  14. 14.
    Sadati, S.J., Baleanu, D., Ranjbar, A., Ghaderi, R., Abdeljawad (Maraaba), T.: Mittag-Leffler stability theorem for fractional nonlinear systems with delay. In: Abstract and Applied Analysis, no. 1, pp. 1–7 (2010)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Wong, R., Zhao, Y.-Q.: Exponential asymptotics of the Mittag-Leffler function. Constr. Approx. 18(1), 355–385 (2002)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Wyrwas, M., Mozyrska, D.: On Mittag-Leffler stability of fractional order difference systems. In: Latawiec, K.J., et al. (eds.) Advances in Modeling and Control of Non-integer Order Systems. Lecture Notes in Electrical Engineering, vol. 320, pp. 209–220. Springer, Switzerland (2015)Google Scholar
  17. 17.
    Yu, J., Hu, H., Zhou, S., Lin, X.: Generalized Mittag-Leffler stability of multi-variables fractional order nonlinear systems. Automatica 49(1), 1798–1803 (2013)MathSciNetCrossRefGoogle Scholar

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© Springer Nature Switzerland AG 2020

Authors and Affiliations

  1. 1.AGH UniversityKrakowPoland

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