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2016 | 26 | 4 | 749-756
Tytuł artykułu

Modeling heat distribution with the use of a non-integer order, state space model

Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
A new, state space, non-integer order model for the heat transfer process is presented. The proposed model is based on a Feller semigroup one, the derivative with respect to time is expressed by the non-integer order Caputo operator, and the derivative with respect to length is described by the non-integer order Riesz operator. Elementary properties of the state operator are proven and a formula for the step response of the system is also given. The proposed model is applied to the modeling of temperature distribution in a one dimensional plant. Results of experiments show that the proposed model is more accurate than the analogical integer order model in the sense of the MSE cost function.
Rocznik
Tom
26
Numer
4
Strony
749-756
Opis fizyczny
Daty
wydano
2016
otrzymano
2015-12-15
poprawiono
2016-04-13
poprawiono
2016-07-11
zaakceptowano
2016-08-28
Twórcy
  • Department of Automatics and Biomedical Engineering, AGH University of Science and Technology, al. A. Mickiewicza 30, 30-079 Kraków, Poland
autor
  • Department of Computer Sciences, High Vocational School in Tarnów, al. A Mickiewicza 8, 33-100 Tarnów, Poland
  • Department of Automatics and Biomedical Engineering, AGH University of Science and Technology, al. A. Mickiewicza 30, 30-079 Kraków, Poland
Bibliografia
  • Almeida, R. and Torres, D.F.M. (2011). Necessary and sufficient conditions for the fractional calculus of variations with Caputo derivatives, Communications in Nonlinear Science and Numerical Simulation 16(3): 1490-1500.
  • Baeumer, B., Kurita, S. and Meerschaert, M. (2005). Inhomogeneous fractional diffusion equations, Fractional Calculus and Applied Analysis 8(4): 371-386.
  • Balachandran, K. and Divya, S. (2014). Controllability of nonlinear implicit fractional integrodifferential systems, International Journal of Applied Mathematics and Computer Science 24(4): 713-722, DOI: 10.2478/amcs-2014-0052.
  • Balachandran, K. and Kokila, J. (2012). On the controllability of fractional dynamical systems, International Journal of Applied Mathematics and Computer Science 22(3): 523-531, doi: 10.2478/v10006-012-0039-0.
  • Bartecki, K. (2013). 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, DOI: 10.2478/amcs-2013-0022.
  • Caponetto, R., Dongola, G., Fortuna, L. and Petras, I. (2010). Fractional, order systems: Modeling and control applications, in L.O. Chua (Ed.), World Scientific Series on Nonlinear Science, University of California, Berkeley, CA, pp. 1-178.
  • Curtain, R.F. and Zwart, H. (1995). An Introduction to InfiniteDimensional Linear Systems Theory, Springer-Verlag, New York, NY.
  • Das, S. (2010). Functional Fractional Calculus for System Identification and Control, Springer, Berlin.
  • Dlugosz, M. and Skruch, P. (2015). The application of fractional-order models for thermal process modelling inside buildings, Journal of Building Physics 1(1): 1-13.
  • Dzielinski, A., Sierociuk, D. and Sarwas, G. (2010). Some applications of fractional order calculus, Bulletin of the Polish Academy of Sciences: Technical Sciences 58(4): 583-592.
  • Evans, K.P. and Jacob, N. (2007). Feller semigroups obtained by variable order subordination, Revista Matematica Complutense 20(2): 293-307.
  • Gal, C. and Warma, M. (2016). Elliptic and parabolic equations with fractional diffusion and dynamic boundary conditions, Evolution Equations and Control Theory 5(1): 61-103.
  • Kaczorek, T. (2011). Selected Problems of Fractional Systems Theory, Springer, Berlin.
  • Kaczorek, T. (2016). Reduced-order fractional descriptor observers for a class of fractional descriptor continuous-time nonlinear systems, International Journal of Applied Mathematics and Computer Science 26(2): 277-283, DOI: 10.1515/amcs-2016-0019.
  • Kaczorek, T. and Rogowski, K. (2014). Fractional Linear Systems and Electrical Circuits, Białystok University of Technology, Białystok.
  • Kochubei, A. (2011). Fractional-parabolic systems, arXiv: 1009.4996 [math.ap], (reprint).
  • Mitkowski, W. (1991). Stabilization of Dynamic Systems, WNT, Warsaw, (in Polish).
  • Mitkowski, W. (2011). Approximation of fractional diffusion-wave equation, Acta Mechanica et Automatica 5(2): 65-68.
  • N'Doye, I., Darouach, M., Voos, H. and Zasadzinski, M. (2013). Design of unknown input fractional-order observers for fractional-order systems, International Journal of Applied Mathematics and Computer Science 23(3): 491-500, DOI: 10.2478/amcs-2013-0037.
  • Obraczka, A. (2014). Control of Heat Processes with the Use of Non-integer Models, Ph.D. thesis, AGH University of Science and Technology, Kraków.
  • Oprzedkiewicz, K. (2003). The interval parabolic system, Archives of Control Sciences 13(4): 415-430.
  • Oprzedkiewicz, K. (2004). A controllability problem for a class of uncertain parameters linear dynamic systems, Archives of Control Sciences 14(1): 85-100.
  • Oprzędkiewicz, K. (2005). An observability problem for a class of uncertain-parameter linear dynamic systems, International Journal of Applied Mathematics and Computer Science 15(3): 331-338.
  • Ostalczyk, P. (2012). Equivalent descriptions of a discrete-time fractional-order linear system and its stability domains, International Journal of Applied Mathematics and Computer Science 22(3): 533-538, DOI: 10.2478/v10006-012-0040-7.
  • Pazy, A. (1983). Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer, New York, NY.
  • Podlubny, I. (1999). Fractional Differential Equations, Academic Press, San Diego, CA.
  • Popescu, E. (2010). On the fractional Cauchy problem associated with a Feller Semigroup, Mathematical Reports 12(2): 181-188.
  • Sierociuk, D., Skovranek, T., Macias, M., Podlubny, I., Petras, I., Dzielinski, A. and Ziubinski, P. (2015). Diffusion process modeling by using fractional-order models, Applied Mathematics and Computation 257(1): 2-11.
  • Szekeres, B.J. and Izsak, F. (2014). Numerical solution of fractional order diffusion problems with Neumann boundary conditions, preprint, arXiv: 1411.1596, [math.NA], (preprint).
  • Yang, Q., Liu, F. and Turner, I. (2010). Numerical methods for fractional partial differential equations with Riesz space fractional derivatives, Applied Mathematical Modelling 34(1): 200-218.
Typ dokumentu
Bibliografia
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