Abstract
The chapter presents the fractional order, state space models of the one dimensional heat transfer process. The proposed models are based on a known semigroup state space model of a parabolic system with distributed control and observation. The first model presented is the time continuous model using the Caputo definition of the fractional derivative over time and the Riesz definition to express the spatial fractional derivative. The second proposed model is the discrete time model, employing the discrete Grünwald-Letnikov operator. The last discrete time fractional order model uses a new, memory-effective method, proposed by the authors as a method of solution for the discrete fractional state equation. The proposed method uses the CFE approximant to express the fractional derivative. Elementary properties (stability, convergence and accuracy) for each proposed model are analysed and results are verified using real experimental data.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Similar content being viewed by others
References
Al-Alaoui, M.A.: Novel digital integrator and differentiator. Electron. Lett. 29(4), 376–378 (1993)
Bartecki, K.: A general transfer function representation for a class of hyperbolic distributed parameter systems. Int. J. Appl. Math. Comput. Sci. 23(2), 291–307 (2013)
Buslowicz, M., Kaczorek, T.: Simple conditions for practical stability of positive fractional discrete-time linear systems. Int. J. Appl. Math. Comput. Sci. 19(2), 263–269 (2009)
Caponetto, R., Dongola, G., Fortuna, L., Petras, I.: Fractional order systems: modeling and control applications. In: Chua, L.O. (ed.) World Scientific Series on Nonlinear Science, pp. 1–178. University of California, Berkeley (2010)
Chen, Y.Q., Moore, K.L.: Discretization schemes for fractional-order differentiators and integrators. IEEE Trans. Circuits Syst. I Fundam. Theory Appl. 49(3), 263–269 (2002)
Curtain, R.F., Zwart, H.: An Introduction to Infinite-Dimensional Linear Systems Theory. Springer, New York (1995)
Das, S.: Functional Fractional Calculus for System Identification and Controls. Springer, Berlin (2010)
Dlugosz, M., Skruch, P.: The application of fractional-order models for thermal process modelling inside buildings. J. Build. Phys. 1(1), 1–13 (2015)
Wyrwas, M., Mozyrska, D., Girejko, E.: Comparison of h-difference fractional operators. In: Mitkowski, W. et al. (ed.) Advances in the Theory and Applications of Non-integer Order Systems, pp. 1–178. Springer, Switzerland (2013)
Dzieliński, A., Sierociuk, D., Sarwas, G.: Some applications of fractional order calculus. Bull. Pol. Acad. Sci. Tech. Sci. 58(4), 583–592 (2010)
Gal, C.G., Warma, M.: Elliptic and parabolic equations with fractional diffusion and dynamic boundary conditions. Evol. Eqn. Control Theory 5(1), 61–103 (2016)
Gómez, J.F., Torres, L., Escobar, R.F. (eds.): Fractional derivatives with Mittag-Leffler kernel. Trends and applications in science and engineering. In: Kacprzyk, J. (ed.) Studies in Systems, Decision and Control, vol. 194, pp. 1–339. Springer, Switzerland (2019)
Kaczorek, T.: Selected Problems of Fractional Systems Theory. Springer, Berlin (2011)
Kaczorek, T.: Singular fractional linear systems and electrical circuits. Int. J. Appl. Math. Comput. Sci. 21(2), 379–384 (2011)
Kaczorek, T., Rogowski, K.: Fractional Linear Systems and Electrical Circuits. Bialystok University of Technology, Bialystok (2014)
Kulczycki, P., Korbicz, J., Kacprzyk, J. (eds.): Control, Robotics and Information Processing (in Polish). PWN, Warsaw (2020)
Mitkowski, W.: Stabilization of Dynamic Systems (in Polish). WNT, Warszawa (1991)
Mitkowski, W.: Approximation of fractional diffusion-wave equation. Acta Mechanica et Automatica 5(2), 65–68 (2011)
Mitkowski, W.: Finite-dimensional approximations of distributed RC networks. Bull. Pol. Acad. Sci. Tech. Sci. 62(2), 263–269 (2014)
Mitkowski, W.: Outline of Control Theory (in Polish). Wydawnictwa AGH, Kraków (2019)
Mitkowski, W., Obraczka, A.: Simple identification of fractional differential equation. Solid State Phenom. 1(180), 331–338 (2012)
Mozyrska, D., Pawluszewicz, E.: Fractional discrete-time linear control systems with initialisation. Int. J. Control 1(1), 1–7 (2011)
Obraczka, A., Mitkowski, W.: The comparison of parameter identification methods for fractional, partial differential equation. Diffusion and defect data – solid state data. Part B. Solid State Phenom. 210(2014), 265–270 (2014)
Obrączka, A.: Control of heat processes with the use of non-integer models. Ph.D. thesis, AGH University, Krakow, Poland (2014)
Oprzędkiewicz, K.: The interval parabolic system. Arch. Control Sci. 13(4), 415–430 (2003)
Oprzędkiewicz, K.: A controllability problem for a class of uncertain parameters linear dynamic systems. Arch. Control Sci. 14(1), 85–100 (2004)
Oprzędkiewicz, K.: An observability problem for a class of uncertain-parameter linear dynamic systems. Int. J. Appl. Math. Comput. Sci. 15(3), 331–338 (2005)
Oprzędkiewicz, K.: Non integer order, discrete, state space model of heat transfer process using grünwald-letnikov operator. Bull. Pol. Acad. Sci. Tech. Sci. 66(3), 249–255 (2018)
Oprzędkiewicz, K., Dziedzic, K., Więckowski, L.: Non integer order, discrete, state space model of heat transfer process using Grünwald-Letnikov operator. Bull. Pol. Acad. Sci. Tech. Sci. 67, ?? (2019)
Oprzędkiewicz, K., Gawin, E.: The practical stability of the discrete, fractional order, state space model of the heat transfer process. Arch. Control Sci. 28(3), 463–482 (2018)
Oprzędkiewicz, K., Gawin, E., Mitkowski, W.: Modeling heat distribution with the use of a non-integer order, state space model. Int. J. Appl. Math. Comput. Sci. 26(4), 749–756 (2016)
Oprzędkiewicz, K., Gawin, E., Mitkowski, W.: 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–01 September 2016, Miȩdzyzdroje, Poland, pp. 184–188 (2016)
Oprzędkiewicz, K., Mitkowski, W.: A memory efficient non integer order discrete time state space model of a heat transfer process. Int. J. Appl. Math. Comput. Sci. 28(4), 649–659 (2018)
Oprzędkiewicz, K., Mitkowski, W., Gawin, E.: An accuracy estimation for a non integer order, discrete, state space model of heat transfer process. In: Automation 2017 : Innovations in Automation, Robotics and Measurement Techniques : 15–17 March, Warsaw, Poland, pp. 86–98 (2017)
Oprzędkiewicz, K., Stanisławski, R., Gawin, E., Mitkowski, W.: A new algorithm for a CFE approximated solution of a discrete-time non integer-order state equation. Bull. Pol. Acad. Sci. Tech. Sci. 65(4), 429–437 (2017)
Ostalczyk, P.: Equivalent descriptions of a discrete-time fractional-order linear system and its stability domains. Int. J. Appl. Math. Comput. Sci. 22(3), 533–538 (2012)
Ostalczyk, P.: Discrete Fractional Calculus. Applications in Control and Image Processing. World Scientific, New Jersey, London, Singapore (2016)
Pazy, A.: Semigroups of Linear Operators and Applications to Partial Differential Equations. Springer, New York (1983)
Petras, I.: Fractional order feedback control of a DC motor. J. Electr. Eng. 60(3), 117–128 (2009)
Petras, I.: http://people.tuke.sk/igor.podlubny/usu/matlab/petras/dfod2.m (2009)
Podlubny, I.: Fractional Differential Equations. Academic Press, San Diego (1999)
Popescu, E.: On the fractional Cauchy problem associated with a feller semigroup. Math. Rep. 12(2), 181–188 (2010)
Rauh, A., Senkel, L., Aschemann, H., Saurin, V.V., Kostin, G.V.: An integrodifferential approach to modeling, control, state estimation and optimization for heat transfer systems. Int. J. Appl. Math. Comput. Sci. 26(1), 15–30 (2016)
Sierociuk, D., Skovranek, T., Macias, M., Podlubny, I., Petras, I., Dzielinski, A., Ziubinski, P.: Diffusion process modeling by using fractional-order models. Appl. Math. Comput. 257(1), 2–11 (2015)
Stanisławski, R., Latawiec, K.: Stability analysis for discrete-time fractional-order LTI state-space systems. Part i: new necessary and sufficient conditions for asymptotic stability. Bull. Pol. Acad. Sci. Tech. Sci. 61(2), 353–361 (2013)
Stanisławski, R., Latawiec, K.: Stability analysis for discrete-time fractional-order LTI state-space systems. Part ii: stability criterion for FD-based systems. Bull. Pol. Acad. Sci. Tech. Sci. 61(2), 362–370 (2013)
Stanisławski, R., Latawiec, K., Łukaniszyn, M.: A comparative analysis of Laguerre-based approximators to the Grünwald-Letnikov fractional-order difference. Hindawi Publ. Corpor. Math. Prob. Eng. 2015(1), 1–10 (2015)
Yang, Q., Liu, F., Turner, I.: Numerical methods for fractional partial differential equations with Riesz space fractional derivatives. Appl. Math. Model. 34(1), 200–218 (2010)
Acknowledgements
This chapter is sponsored by AGH project no 16.16.120.773.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2022 The Author(s), under exclusive license to Springer Nature Switzerland AG
About this chapter
Cite this chapter
Oprzȩdkiewicz, K., Mitkowski, W. (2022). Fractional Order State Space Models of the One-Dimensional Heat Transfer Process. In: Kulczycki, P., Korbicz, J., Kacprzyk, J. (eds) Fractional Dynamical Systems: Methods, Algorithms and Applications. Studies in Systems, Decision and Control, vol 402. Springer, Cham. https://doi.org/10.1007/978-3-030-89972-1_13
Download citation
DOI: https://doi.org/10.1007/978-3-030-89972-1_13
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-89971-4
Online ISBN: 978-3-030-89972-1
eBook Packages: Intelligent Technologies and RoboticsIntelligent Technologies and Robotics (R0)