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Stability of fractional difference systems with two orders. (English) Zbl 1271.93129

Mitkowski, Wojciech (ed.) et al., Advances in the theory and applications of non-integer order systems. 5th conference on non-integer order calculus and its applications, Cracow, Poland, July 4–5, 2013. Cham: Springer (ISBN 978-3-319-00932-2/hbk; 978-3-319-00933-9/ebook). Lecture Notes in Electrical Engineering 257, 41-52 (2013).
Summary: In the paper we study the stability of nonlinear systems with the Caputo fractional difference with two orders. The Lyapunov direct method is used to analyze the stability of a system. Sufficient conditions for uniform stability and uniform asymptotic stability are presented.
For the entire collection see [Zbl 1271.93002].

MSC:

93D20 Asymptotic stability in control theory
93C10 Nonlinear systems in control theory
93D05 Lyapunov and other classical stabilities (Lagrange, Poisson, \(L^p, l^p\), etc.) in control theory
93C55 Discrete-time control/observation systems
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References:

[1] Abdeljawad, T.; Baleanu, D., Fractional differences and integration by parts, Journal of Computational Analysis and Applications, 13, 3, 574-582 (2011) · Zbl 1225.39008
[2] Atici, F. M.; Eloe, P. W., A Transform Method in Discrete Fractional Calculus, International Journal of Difference Equations, 2, 165-176 (2007)
[3] Axtell, M., Bise, E.M.: Fractional calculus applications in control systems. In: Proc. of the IEE 1990 Int. Aerospace and Electronics Conf., New York, pp. 536-566 (1990)
[4] Busłowicz, M.: Stability of continuous-time linear systems described by state equation with fractional commensurate orders of derivatives. Przegląd Elektroniczby (Electrical Review), ISSN 0033-2097, R. 88 NR 4b/2012 · Zbl 1271.93115
[5] Elaydi, S. N., An introduction to difference equations (1967), New York: Springer, New York
[6] Ferreira, R. A.C.; Torres, D. F.M., Fractional h-difference equations arising from the calculus of variations, Appl. Anal. Discrete Math., 5, 1, 110-121 (2011) · Zbl 1289.39007 · doi:10.2298/AADM110131002F
[7] Guermah, S., Djennoune, S., Bettayeb, M.: Asymptotic stability and practical stability of linear discrete-time fractional order systems. In: 3rd IFAC Workshop on Fractional Differentiation and its Applications, Ankara, Turkey (2008) · Zbl 1234.93014
[8] Holm, M. T., The theory of discrete fractional calculus: Development and application (2011), Lincoln: University of Nebraska, Lincoln
[9] Jarad, F.; Abdeljawad, T.; Baleanu, D.; Biçen, K., On the stability of some discrete fractional nonautonomous systems, Abstract and Applied Analysis, 2012, 1-9 (2012) · Zbl 1235.93206
[10] Kaczorek, T., Fractional positive continuous-time linear systems and their reachability, Int. J. Appl. Math. Comput. Sci., 18, 2, 223-228 (2008) · Zbl 1235.34019 · doi:10.2478/v10006-008-0020-0
[11] Kaczorek, T.: Practical stability of positive fractional discrete-time linear systems. Bulletin of the Polish Academy of Sciences. Technical Sciences 56(4) (2008)
[12] Li, Y.; Chen, Y. Q.; Podlubny, I., Stability of fractional-order nonlinear dynamic systems: Lyapunov direct method and generalized Mittag-Leffler stability, Computers and Mathematics with Applications, 59, 1810-1821 (2010) · Zbl 1189.34015 · doi:10.1016/j.camwa.2009.08.019
[13] Li, Y.; Chen, Y. Q.; Podlubny, I., Mittag-Leffler stability of fractional order nonlinear dynamic systems, Automatica, 45, 1965-1969 (1965) · Zbl 1185.93062 · doi:10.1016/j.automatica.2009.04.003
[14] Matignon, D.: Stability results on fractional differential with application to control processing. In: Proc. of the IAMCS, IEEE SMC Conf., Lille France, pp. 963-968 (1996)
[15] Miller, K. S.; Ross, B., Fractional difference calculus, Proceedings of the International Symposium on Univalent Functions, Fractional Calculus and their Applications, 139-152 (1988), Kōriyama: Nihon University, Kōriyama · Zbl 0693.39002
[16] Ostalczyk, P., Equivalent Descriptions of a Discrete-Time Fractional-Order Linear System and its Stability Domains, Int. J. Appl. Math. Comput. Sci., 22, 3, 533-538 (2012) · Zbl 1302.93140
[17] Sadati, S. J.; Baleanu, D.; Ranjbar, A.; Ghaderi, R.; Abdeljawad, T., Mittag-Leffler stability theorem of fractional nonlinear systems with delay, Abstract and Applied Analysis, 2010, 1-7 (2010) · Zbl 1195.34013 · doi:10.1155/2010/108651
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