Kybernetika 51 no. 1, 112-136, 2015

Stability of nonlinear h-difference systems with n fractional orders

Małgorzata Wyrwas, Ewa Pawluszewicz and Ewa GirejkoDOI: 10.14736/kyb-2015-1-0112


In the paper we study the subject of stability of systems with $h$-differences of Caputo-, Riemann-Liouville- and Grünwald-Letnikov-type with $n$ fractional orders. The equivalent descriptions of fractional $h$-difference systems are presented. The sufficient conditions for asymptotic stability are given. Moreover, the Lyapunov direct method is used to analyze the stability of the considered systems with $n$-orders.


stability, fractional difference systems, difference operators


39A, 93D


  1. T. Abdeljawad and D. Baleanu: Fractional differences and integration by parts. J. Computat. Analysis Appl. 13 (2011), 574-582.   CrossRef
  2. F. M. At\i c\i and P. W. Eloe: A transform method in discrete fractional calculus. Int. J. Differ, Equ. 2 (2007), 165-176.   CrossRef
  3. F. M. At\i c\i and P. W. Eloe: Discrete fractional calculus with the nabla operator. Electron. J. Qual. Theory Differ. Equ. Spec. Ed. I(3) (2009), 1-12.   CrossRef
  4. N. R. O. Bastos, R. A. C. Ferreira and D. F. M. Torres: Discrete-time fractional variational problems. Signal Processing 91 (2011), 513-524.   DOI:10.1016/j.sigpro.2010.05.001
  5. N. R. O. Bastos, R. A. C. Ferreira and D. F. M. Torres: Necessary optimality conditions for fractional difference problems of the calculus of variations. Discrete Contin. Dyn. Syst. 29 (2011), 417-437.   DOI:10.3934/dcds.2011.29.417
  6. M. Busłowicz: Stability of continuous-time linear systems described by state equation with fractional commensurate orders of derivatives. Przegl\c ad Elektroniczby (Electrical Review) 88 (2012), 17-20.   CrossRef
  7. F. Chen, X. Luo and Y. Zhou: Existence results for nonlinear fractional difference equation. Adv. Differ. Equ. 2011 (2011), 12 pages.   DOI:10.1155/2011/713201
  8. F. Chen: Fixed points and asymptotic stability of nonlinear fractional difference equations. Electr. J. Qual. Theory Differ. Equ. 39 (2011), 1-18.   CrossRef
  9. F. Chen and Z. Liu: Asymptotic stability results for nonlinear fractional difference equations. J. Appl. Math. 2012 (2012), 14 pages.   DOI:10.1155/2012/879657
  10. R. A. C. Ferreira and D. F. M. Torres: Fractional $h$-difference equations arising from the calculus of variations. Appl. Anal. Discrete Math. 5 (2011), 110-121.   DOI:10.2298/aadm110131002f
  11. E. Girejko and D. Mozyrska: Semi-linear fractional systems with Caputo type multi-step differences. In: Symposium on Fractional Signals and Systems, Instituto Superior de Engenharia de Coimbra, Coimbra, November 2011, pp. 79-88.   CrossRef
  12. S. Guermah, S. Djennoune and M. Bettayeb: Asymptotic stability and practical stability of linear discrete-time fractional order systems. In: 3rd IFAC Workshop on Fractional Differentiation and its Applications, Ankara 2008.   CrossRef
  13. M. T. Holm: The Theory of Discrete Fractional Calculus: Development and Application. PhD. Thesis, University of Nebraska - Lincoln, 2011.   CrossRef
  14. J. B. Hu, G. P. Lu, S. B. Zhang and L. D. Zhao: Lyapunov stability theorem about fractional system without and with delay. Commun. Nonlinear Sci. Numer. Simul. 20 (2014), 905-913.   DOI:10.1016/j.cnsns.2014.05.013
  15. F. Jarad, T. Abdeljawad, D. Baleanu and K. Biçen: On the stability of some discrete fractional nonautonomous systems. Abstr. Appl. Anal. 2012 (2012), 9 pages.   DOI:10.1155/2012/476581
  16. T. Kaczorek: Selected Problems of Fractional Systems Theory. Springer-Verlag, Berlin, Heidelberg 2011.   DOI:10.1007/978-3-642-20502-6
  17. T. Kaczorek: Practical stability of positive fractional discrete-time linear systems. Bull. Pol. Acad. Sci. Techn. Sci. 56 (2008), 313-317.   CrossRef
  18. T. Kaczorek: Fractional positive linear systems. Kybernetes 38 (2009), 1059-1078.   DOI:10.1108/03684920910976826
  19. T. Kaczorek: Reachability of cone fractional continuous-time linear systems. Int. J. Appl. Math. Comput. Sci. 19 (2009), 89-93.   DOI:10.2478/v10006-009-0008-4
  20. A. A. Kilbas, H. M. Srivastava and J. J. Trujillo: Theory and applications of fractional differential equations. North-Holland Mathematics Studies, 204, Elsevier Science B. V., Amsterdam 2006.   DOI:10.1016/s0304-0208(06)80001-0
  21. C. P. Li and F. R. Zhang: A survey on the stability of fractional differential equations. Eur. Phys. J. 193 (2011), 27-47.   DOI:10.1140/epjst/e2011-01379-1
  22. Y. Li, Y. Q. Chen and I. Podlubny: Mittag-Leffler stability of fractional order nonlinear dynamic systems. Automatica 45 (2009), 1965-1969.   DOI:10.1140/epjst/e2011-01379-1
  23. Y. Li, Y. Q. Chen and I. Podlubny: Stability of fractional-order nonlinear dynamic systems: Lyapunov direct method and generalized Mittag-Leffler stability. Comput. Math. Appl. 59 (2010), 1810-1821.   DOI:10.1016/j.camwa.2009.08.019
  24. R. Margarita, S. V. Rogosin, J. A. Tenreiro Machado and J. J. Trujillo: Stability of fractional order systems. Math. Probl. Engrg. 2013 (2013), 14 pages.   DOI:10.1155/2013/356215
  25. K. S. Miller and B. Ross: Fractional difference calculus. In: Proc. International Symposium on Univalent Functions, Fractional Calculus and their Applications, Nihon University, K\=oriyama 1988, pp. 139-152.   CrossRef
  26. D. Mozyrska and E. Girejko: Overview of the fractional $h$-difference operators. In: Advances in Harmonic Analysis and Operator Theory: The Stefan Samko Anniversary Volume (A. Almeida, L. Castro, F.-O. Speck, eds.), Springer 2013, pp. 253-267.   DOI:10.1007/978-3-0348-0516-2_14
  27. D. Mozyrska, E. Girejko and M. Wyrwas: Comparision of $h$-difference fractional operators. In: Advances in the Theory and Applications of non-integer Order Systems (W. Mitkowski, J. Kacprzyk, J. Baranowski, eds.), Lect. Notes Electr. Engrg. 257, Springer International Publishing, Switzerland 2013, pp. 191-197.   DOI:10.1007/978-3-319-00933-9_17
  28. D. Mozyrska and E. Pawluszewicz: Local controllability of nonlinear discrete-time fractional order systems. Bull. Pol. Acad. Sci. Techn. Sci. 61 (2013), 251-256.   DOI:10.2478/bpasts-2013-0024
  29. P. Ostalczyk: Equivalent descriptions of a discrete time fractional order linear system and its stability domains. Int. J. Appl. Math. Comput. Sci. 22 (2012), 533-538.   CrossRef
  30. I. Petráš: Stability of fractional-order systems with rational orders: a survey. Fract. Calc. Appl. Anal. 12 (2009), 269-298.   CrossRef
  31. I. Petráš: Fractional-Order Nonlinear Systems: Modelling, Analysis and Simulation. Springer, Dordrecht 2011.   DOI:10.1007/978-3-642-18101-6
  32. I. Podlubny: Fractional Differential Equations. Mathematics in Sciences and Engineering. Academic Press, San Diego 1999.   CrossRef
  33. M. S. Tavazoei and M. Haeri: A note on the stability of fractional order systems. Math. Comput. Simul. 79 (2009), 1566-1576.   DOI:10.1016/j.matcom.2008.07.003
  34. J. C. Trigeassou, N. Maamri, J. Sabatier and A. Oustaloup: A Lyapunov approach to the stability of fractional differential equations. Signal Process. 91 (2011), 437-445.   DOI:10.1016/j.sigpro.2010.04.024
  35. M. Wyrwas, E. Girejko, D. Mozyrska and E. Pawluszewicz: Stability of fractional difference systems with two orders In: Advances in the Theory and Applications of Non-integer Order Systems (W. Mitkowski, J. Kacprzyk, and J. Baranowski, eds.), Lect. Notes Electr. Engrg. 257, Springer International Publishing, Switzerland 2013, pp. 41-52.   DOI:10.1007/978-3-319-00933-9_4
  36. L. D. Zhao, J. B. Hu, J. A. Fang and W. B. Zhang: Studying on the stability of fractional-order nonlinear system. Nonlinear Dynamics 70 (2012), 475-479.   DOI:10.1007/s11071-012-0469-0