A non integer order, state space model for one dimensional heat transfer proces

• Autorzy: Oprzedkiewicz K. , Gawin E.

Identyfikatory

DOI

10.1515/acsc-2016-0015

• Języki publikacji: EN

Abstrakty

EN

In the paper a new, state space, non integer order model for one dimensional heat transfer process is presented. The model is based on known semigroup model. The derivative with respect to time is described by the non integer order Caputo operator, the spatial derivative is described by integer order operator. The elementary properties of the state operator are proven. The solution of state equation is calculated with the use of Laplace transform. Results of experiments show, that the proposed model is more accurate than analogical integer order model in the sense of square cost function.

Słowa kluczowe

EN

non-integer order systems; heat transfer process; diffusion equation; infinite dimensional systems

• Wydawca: Polish Academy of Sciences, Committee of Automatic Control and Robotics
• Czasopismo: Archives of Control Sciences
• Rocznik: 2016
• Tom: Vol. 26, no. 2
• Strony: 261--275
• Opis fizyczny: Bibliogr. 23 poz., rys., tab., wykr., wzory

Twórcy

autor

Oprzedkiewicz K.

• AGH University of Science and Technology in Krakow, Poland (Faculty of Electrical Engineering, Automatics, Computer Science and Biomedical Engineering, Dept. of Automatics and Biomedical Engineering

autor

Gawin E.

• High Vocational School in Tarnów, Polytechnic Institute, Polan

Bibliografia

• [1] K. Balachandran and J. Kokila: On the controllability of fractional dynamical systems. Int. J. of Applied Mathematics and Computer Science, 22(3), (2012), 523-531.
• [2] K. Bartecki: A general transfer function representation for a class of hyperbolic distributed parameter systems. Int. J. of Applied Mathematics and Computer Science, 23(2), (2013), 291-307.
• [3] R. Caponetto, G. Dongola, L. Fortuna and I. Petr: Fractional order systems: Modeling and control applications. In L. O. Chua, editor, Series on Nonlinear Science, pages 1-178. University of California, Berkeley, 2010.
• [4] S. Das: Functional Fractional Calculus for System Identyfication and Control. Springer, Berlin, 2010.
• [5] M. Dlugosz and P. Skruch: The application of fractional-order models for thermal process modelling inside buildings. Journal of Building Physics, 1(1), (2015), 1-13.
• [6] A. Dzielinski, D. Sierociuk and G. Sarwas: Some applications of fractional order calculus. Bulletin of the Polish Academy of Sciences, Technical Sciences, 58(4), (2010), 583-592.
• [7] K. P. Evans and N. Jacob: Feller semigroups obtained by variable order subordination. Revista Matematica Complutense, 20(2), (2007), 293-307.
• [8] T. Kaczorek: Reachability of cone fractional continuous time linear systems. Int. J. of Applied Mathematics and Computer Science, 19(1), (2009), 89-93.
• [9] T. Kaczorek: Selected Problems of Fractional Systems Theory. Springer, Berlin, 2011.
• [10] T. Kaczorek: Singular fractional linear systems and electrical circuits. Int. J. of Applied Mathematics and Computer Science, 21(2), (2011), 379-384.
• [11] T. Kaczorek and K. Rogowski: Fractional Linear Systems and Electrical Circuits. Bialystok University of Technology, Bialystok, 2014.
• [12] S. L. Levarge: Semigroups of linear operators, 2003.
• [13] W. Mitkowski: Stabilization of dynamic systems (in Polish). WNT, Warszawa, 1991.
• [14] W. Mitkowski: Approximation of fractional diffusion-wave equation. Acta Mechanica et Automatica, 5(2), (2011), 65-68.
• [15] A. Obraczka: Control of heat processes with the use of non-integer models. PhD thesis, AGH University, Krakow, Poland, 2014.
• [16] K. Oprzedkiewicz: The interval parabolic system. Archives of Control Sciences, 13(4), (2003), 415-430.
• [17] K. Oprzedkiewicz: A controllability problem for a class of uncertain parameters linear dynamic systems. Archives of Control Sciences, 14(1), (2004), 85-100.
• [18] K. Oprzedkiewicz: An observability problem for a class of uncertain-parameter linear dynamic systems. Int. J. of Applied Mathematics and Computer Science, 15(3), (2005), 331-338.
• [19] P. Ostalczyk: Equivalent descriptions of a discrete-time fractional-order linear system and its stability domains. Int. J. of Applied Mathematics and Computer Science, 22(3), (2012), 533-538.
• [20] I. Podlubny: Fractional Differential Equations. Academic Press, San Diego, 1991.
• [21] E. Popescu: On the fractional cauchy problem associated with a feller semigroup. Mathematical Reports, 12(2), (2010), 181-188.
• [22] 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), (2015), 2-11.
• [23] B. J. Szekeres and F. Izsak: Numerical solution of fractional order diffusion problems with neumann boundary conditions, preprint,arxiv:1411.1596 [math.na], 2014.

Uwagi

PL

Opracowanie ze środków MNiSW w ramach umowy 812/P-DUN/2016 na działalność upowszechniającą naukę