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Remarks on five equivalent forms of the fractional – order backward – difference

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EN
Abstrakty
EN
In this paper a fractional-order backward-difference/sum (FOBD/S) equivalent formulae are considered. From the Gr¨unwald- Letnikov (GL - FOBD) definition formula and its Horner equivalent form one derives the Riemann-Liouville FOBD (RL - FOBD). Also the Caputo and polynomial-like forms are defined. All forms may be useful in real-time calculations (in the evaluation of digital control strategies) due to the reduction of fractional orders. The investigations are illustrated by a numerical example.
Rocznik
Strony
271--278
Opis fizyczny
Bibliogr. 16 poz., wykr.
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autor
  • Institute of Applied Computer Science, Lodz University of Technology, 18/ 22 Stefanowskiego St., 90-924 Łodź, Poland
Bibliografia
  • [1] K. Miller and B. Ross, An Introduction to Fractional Calculus and Fractional Differential Equations, Wiley, New York, 1993.
  • [2] A. Oustaloup, La D`erivation non Enti´ere – Theorie, Synthese et Applications, Hermes, Paris, 1995.
  • [3] I. Podlubny, Fractional Differential Equations, Academic Press, New York, 1999.
  • [4] S. Samko, A. Kilbas, and O. Marichev, Fractional Integrals and Derivatives: Theory and Applications, Gordon and Breach, London, 1993.
  • [5] P. Ostalczyk, “The non-integer difference of the discrete-time function and its application to the control system synthesis”, Int. J. System Science 31, 1551–1561 (2000).
  • [6] T. Kaczorek, Selected Issues on Theory of Non-integer Order Systems, Publishing House of Białystok University of Technology, Białystok, 2009, (in Polish).
  • [7] T. Kailath, Linear Systems, Prentice-Hall, New York, 1980.
  • [8] M. Busłowicz, “Robust stability of convex combination of two fractional degree characteristic polynomials”, Acta Mechanica et Automatica 2 (2), 5–10 (2008).
  • [9] D. Sierociuk, “Estimation and control of discrete dynamical systems of fractal order described in a state space”, Doctoral Thesis, PW, Warszawa, 2007, (in Polish).
  • [10] T. Kaczorek, “Reachability and controllability to zero of cone fractional linear systems”, Archives Control Sciences 17 (3), 357–367, (2007).
  • [11] T. Kaczorek, “Reachability and minimum energy control of positive 2D systems with delays”, Control and Cybernetics 34 (2), 411–423, (2005).
  • [12] J. Klamka, “Controllability and minimum energy control of fractional discrete-time systems”, 3rd Conf. on Fractional Differentials and Applications 1, CD-ROM (2008).
  • [13] J. Klamka, “Controllability and minimum energy control problem of fractional discrete-time systems”, in New Trends in Nanotechnology and Fractional Calculus, Springer, Berlin, 2010.
  • [14] J. Klamka, “Controllability and minimum energy control problem of infinite-dimensional fractional discrete-time systems”, Proc. 1st Asian Conf. on Intelligent Information and Data Base Systems ACIIDS, CD-ROM (2009).
  • [15] J. Klamka, “Controllability of fractional discrete-time systems with delay”, 16 Conf. on Discrete Processes Automation ADP-2008, CD-ROM (2008).
  • [16] P. Ostalczyk, Discrete-variable Functions, Monographs of Lodz Technical University, Łódź, 2001.
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Bibliografia
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bwmeta1.element.baztech-950324e5-c820-43a2-a09d-63128eac9f0b
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