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Non integer order, discrete, state space model of heat transfer process using Grünwald-Letnikov operator

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The paper is intented to show a new, state space, discrete, non integer order model of a one-dimensional heat transfer process. The proposed model derives directly from time continuous, state space model and it uses the discrete Grünwald-Letnikov operator to express the fractional order difference with respect to time. Stability and spectrum decomposition for the proposed model are recalled, the accuracy and convergence are analyzed too. The convergence of the proposed model does not depend on parameters of heater and measuring sensors. The dimension of the model assuring stability and predefined rate of convergence and stability is estimated. Analytical results are confirmed by experiments.
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Bibliogr. 31 poz., rys., tab.
  • AGH University of Science and Technology, al. A. Mickiewicza 30, 30-059 Kraków, Poland
  • AGH University of Science and Technology, al. A. Mickiewicza 30, 30-059 Kraków, Poland
  • AGH University of Science and Technology, al. A. Mickiewicza 30, 30-059 Kraków, Poland
  • [1] B. Bandyopadhyay and S. Kamal, Solution, stability and real-ization of fractional order differential equation. In Stabiliza-tion and Control of Fractional Order Systems: A Sliding Mode Approach, Lecture Notes in Electrical Engineering317, pages 55–90, Springer, Switzerland, 2015.
  • [2] K. Bartecki, A general transfer function representation for a class of hyperbolic distributed parameter systems, International Journal of Applied Mathematics and Computer Science 23(2), 291–307, 2013.
  • [3] M. Buslowicz and T. Kaczorek, Simple conditions for practical stability of positive fractional discrete-time linear systems, Inter-national Journal of Applied Mathematics and Computer Science19(2), 263–269, 2009.
  • [4] R. Caponetto, G. Dongola, L. Fortuna, and I. Petras, Fractional order systems: Modeling and Control Applications, In Leon O. Chua, editor, World Scientific Series on Nonlinear Science, pages 1–178. University of California, Berkeley, 2010.
  • [5] S. Das, Functional Fractional Calculus for System Identification and Controls, Springer, Berlin, 2010.
  • [6] M. Dlugosz and P. Skruch, The application of fractional-order models for thermal process modelling inside buildings, Journal of Building Physics 1(1), 1–13, 2015.
  • [7] M. Wyrwas, D. Mozyrska, and E.Girejko, Comparison of hdifference fractional operators. In W. Mitkowski et al, editor, Advances in the Theory and Applications of Non-integer Order Systems, pages 1–178. Springer, Switzerland, 2013.
  • [8] A. Dzieliński, D. Sierociuk, and G. Sarwas. Some applications of fractional order calculus, Bull. Pol. Ac.: Tech. 58(4), 583–592, 2010.
  • [9] C.G. Gal and M. Warma, Elliptic and parabolic equations with fractional diffusion and dynamic boundary conditions, Evolution Equations and Control Theory 5(1), 61–103, 2016.
  • [10] J.F. Gómez, L. Torres, and R.F. Escobar (Eds), Fractional derivatives with Mittag-Leffler kernel. trends and applications in science and engineering. In J. Kacprzyk, editor, Studies in Systems, Decision and Control 194, pages 1–339, Springer, Switzerland, 2019.
  • [11] T. Kaczorek, Singular fractional linear systems and electrical circuits, International Journal of Applied Mathematics and Computer Science, 21(2), 379–384, 2011.
  • [12] T. Kaczorek and K. Rogowski, Fractional Linear Systems and Electrical Circuits, Bialystok University of Technology, Bialy-stok, 2014.
  • [13] W. Mitkowski, Stabilization of dynamic systems (in Polish), WNT, Warszawa, 1991.
  • [14] D. Mozyrska and E. Pawluszewicz, Fractional discrete-time linear control systems with initialisation, International Journal of Control 1(1), 1–7, 2011.
  • [15] A. Obrączka, Control of heat processes with the use of noninteger models, PhD thesis, AGH University, Krakow, Poland, 2014
  • [16] K. Oprzędkiewicz, The interval parabolic system, Archives of Control Sciences 13(4), 415–430, 2003.
  • [17] K. Oprzędkiewicz, A controllability problem for a class of uncertain parameters linear dynamic systems, Archives of Control Sciences 14(1), 85–100, 2004.
  • [18] K. Oprzędkiewicz, An observability problem for a class of uncertain-parameter linear dynamic systems, International Journal of Applied Mathematics and Computer Science 15(3), 331–338, 2005.
  • [19] K. Oprzędkiewicz and E. Gawin, A non integer order, state space model for one dimensional heat transfer process. Archives of Control Sciences 26(2), 261–275, 2016.
  • [20] K. Oprzędkiewicz and E. Gawin, The practical stability of the discrete, fractional order, state space model of the heat transfer process, Archives of Control Sciences 28(3), 463–482, 2018.
  • [21] K. Oprzędkiewicz, E. Gawin, and W. Mitkowski, Modeling heat distribution with the use of a non-integer order, state space model, International Journal of Applied Mathematics and Computer Science 26(4), 749–756, 2016.
  • [22] K. Oprzędkiewicz, E. Gawin, and W. Mitkowski, Parameter identification for non integer order, state space models of heat plant. In MMAR 2016, 21th international conference on Methods and Models in Automation and Robotics, 29 August–1 September 2016, Międzyzdroje, Poland, pages 184–188, 2016.
  • [23] K. Oprzędkiewicz and W. Mitkowski, A memory efficient non integer order discrete time state space model of a heat transfer process, International Journal of Applied Mathematics and Computer Science 28(4), 649–659, 2018.
  • [24] K. Oprzędkiewicz, R. Stanisławski, E. Gawin, and W. Mitkow-ski, A new algorithm for a cfe approximated solution of a dis-cretetime non integer-order state equation, Bull. Pol. Ac.: Tech. 65(4), 429–437, 2017.
  • [25] P. Ostalczyk, Equivalent descriptions of a discrete-time fraction-al-order linear system and its stability domains, International Journal of Applied Mathematics and Computer Science 22(3), 533–538, 2012.
  • [26] P. Ostalczyk, Discrete Fractional Calculus. Applications in Control and Image Processing, World Scientific, New Jersey, London, Singapore, 2016.
  • [27] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations. Springer, New York, 1983.
  • [28] I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, 1999.
  • [29] E. Popescu, On the fractional cauchy problem associated with a feller semigroup. Mathematical Reports 12(2), 181–188, 2010.
  • [30] A. Rauh, L. Senkel, H. Aschemann, V.V. Saurin, and G.V. Kostin, An integrodifferential approach to modeling, control, state estimation and optimization for heat transfer systems. International Journal of Applied Mathematics and Computer Science 26(1), 15–30, 2016.
  • [31] D. Sierociuk, T. Skovranek, M. Macias, I. Podlubny, I. Petras, A. Dzielinski, and P. Ziubinski, Diffusion process modeling by using fractional-order models. Applied Mathematics and Computation 257(1), 2–11, 2015
Opracowanie rekordu ze środków MNiSW, umowa Nr 461252 w ramach programu "Społeczna odpowiedzialność nauki" - moduł: Popularyzacja nauki i promocja sportu (2020).
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