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

Krzysztof Oprzędkiewicz; Edyta Gawin; Wojciech Mitkowski

International Journal of Applied Mathematics and Computer Science (2016)

  • Volume: 26, Issue: 4, page 749-756
  • ISSN: 1641-876X

Abstract

top
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.

How to cite

top

Krzysztof Oprzędkiewicz, Edyta Gawin, and Wojciech 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 (2016): 749-756. <http://eudml.org/doc/287171>.

@article{KrzysztofOprzędkiewicz2016,
abstract = {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.},
author = {Krzysztof Oprzędkiewicz, Edyta Gawin, Wojciech Mitkowski},
journal = {International Journal of Applied Mathematics and Computer Science},
keywords = {non-integer order systems; heat transfer equation; infinite dimensional systems; Feller semigroups},
language = {eng},
number = {4},
pages = {749-756},
title = {Modeling heat distribution with the use of a non-integer order, state space model},
url = {http://eudml.org/doc/287171},
volume = {26},
year = {2016},
}

TY - JOUR
AU - Krzysztof Oprzędkiewicz
AU - Edyta Gawin
AU - Wojciech Mitkowski
TI - Modeling heat distribution with the use of a non-integer order, state space model
JO - International Journal of Applied Mathematics and Computer Science
PY - 2016
VL - 26
IS - 4
SP - 749
EP - 756
AB - 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.
LA - eng
KW - non-integer order systems; heat transfer equation; infinite dimensional systems; Feller semigroups
UR - http://eudml.org/doc/287171
ER -

References

top
  1. 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. Zbl1221.49038
  2. Baeumer, B., Kurita, S. and Meerschaert, M. (2005). Inhomogeneous fractional diffusion equations, Fractional Calculus and Applied Analysis 8(4): 371-386. Zbl1202.86005
  3. 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. Zbl1309.93025
  4. 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. Zbl1302.93042
  5. 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. Zbl1282.93182
  6. 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. 
  7. Curtain, R.F. and Zwart, H. (1995). An Introduction to InfiniteDimensional Linear Systems Theory, Springer-Verlag, New York, NY. Zbl0839.93001
  8. Das, S. (2010). Functional Fractional Calculus for System Identification and Control, Springer, Berlin. 
  9. 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. 
  10. 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. Zbl1220.80006
  11. Evans, K.P. and Jacob, N. (2007). Feller semigroups obtained by variable order subordination, Revista Matematica Complutense 20(2): 293-307. Zbl1153.47033
  12. 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. Zbl1349.35412
  13. Kaczorek, T. (2011). Selected Problems of Fractional Systems Theory, Springer, Berlin. Zbl1221.93002
  14. 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. Zbl1347.93062
  15. Kaczorek, T. and Rogowski, K. (2014). Fractional Linear Systems and Electrical Circuits, Białystok University of Technology, Białystok. Zbl06385236
  16. Kochubei, A. (2011). Fractional-parabolic systems, arXiv: 1009.4996 [math.ap], (reprint). 
  17. Mitkowski, W. (1991). Stabilization of Dynamic Systems, WNT, Warsaw, (in Polish). Zbl0686.93072
  18. Mitkowski, W. (2011). Approximation of fractional diffusion-wave equation, Acta Mechanica et Automatica 5(2): 65-68. 
  19. 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. Zbl1279.93027
  20. Obraczka, A. (2014). Control of Heat Processes with the Use of Non-integer Models, Ph.D. thesis, AGH University of Science and Technology, Kraków. 
  21. Oprzedkiewicz, K. (2003). The interval parabolic system, Archives of Control Sciences 13(4): 415-430. Zbl1151.93368
  22. Oprzedkiewicz, K. (2004). A controllability problem for a class of uncertain parameters linear dynamic systems, Archives of Control Sciences 14(1): 85-100. Zbl1151.93317
  23. 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. Zbl1169.93313
  24. 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. Zbl1302.93140
  25. Pazy, A. (1983). Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer, New York, NY. Zbl0516.47023
  26. Podlubny, I. (1999). Fractional Differential Equations, Academic Press, San Diego, CA. Zbl0924.34008
  27. Popescu, E. (2010). On the fractional Cauchy problem associated with a Feller Semigroup, Mathematical Reports 12(2): 181-188. Zbl1224.26023
  28. 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. 
  29. 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). 
  30. 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. Zbl1185.65200

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.