A Memory–Efficient Noninteger–Order Discrete–Time State–Space Model of a Heat Transfer Process

Al-Alaoui,M. (1993). Novel digital integrator and differentiator, Electronics Letters 29(4): 376-378.10.1049/el:19930253Search in Google Scholar

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): 1490-1500.10.1016/j.cnsns.2010.07.016Search in Google Scholar

Rauh, A., Senkel, L., Aschemann, H., Saurin, V.V. and Kostin, G.V. (2016). 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, DOI: 10.1515/amcs-2016-0002.10.1515/amcs-2016-0002Open DOISearch in Google Scholar

Baeumer, B., Kurita, S. and Meerschaert, M. (2005). Inhomogeneous fractional diffusion equations, Fractional Calculus and Applied Analysis 8(4): 371-386.Search in Google Scholar

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-2014-0052.10.2478/amcs-2014-0052Open DOISearch in Google Scholar

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/v10006-012-0039-0.10.2478/v10006-012-0039-0Open DOISearch in Google Scholar

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-2013-0022.10.2478/amcs-2013-0022Open DOISearch in Google Scholar

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. 1-178.10.1142/7709Search in Google Scholar

Chen, Y.Q. and Moore, K.L. (2002). Discretization schemes for fractional-order differentiators and integrators, IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications 49(3): 263-269.10.1109/81.989172Search in Google Scholar

Das, S. (2010). Functional Fractional Calculus for System Identification and Control, Springer, Berlin.10.1007/978-3-642-20545-3_10Search in Google Scholar

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.Search in Google Scholar

Dzieliński, A., Sierociuk, D. and Sarwas, G. (2010). Some applications of fractional order calculus, Bulletin of the Polish Academy of Sciences: Technical Sciences 58(4): 583-592.10.2478/v10175-010-0059-6Search in Google Scholar

Gal, C. and Warma, M. (2016). Elliptic and parabolic equations with fractional diffusion and dynamic boundary conditions, Evolution Equations and Control Theory 5(1): 61-103.10.3934/eect.2016.5.61Search in Google Scholar

Kaczorek, T. (2011). Selected Problems of Fractional Systems Theory, Springer, Berlin.10.1007/978-3-642-20502-6Search in Google Scholar

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-2016-0019.10.1515/amcs-2016-0019Open DOISearch in Google Scholar

Kaczorek, T. and Rogowski, K. (2014). Fractional Linear Systems and Electrical Circuits, Bialystok University of Technology, Bialystok.10.1007/978-3-319-11361-6Search in Google Scholar

Kochubei, A. (2011). Fractional-parabolic systems, arXiv:1009.4996 [math.ap].10.1007/s11118-011-9243-zSearch in Google Scholar

Mitkowski, W. (1991). Stabilization of Dynamic Systems, WNT, Warsaw, (in Polish).Search in Google Scholar

Mitkowski, W. (2011). Approximation of fractional diffusion-wave equation, Acta Mechanica et Automatica 5(2): 65-68.Search in Google Scholar

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-2013-0037.10.2478/amcs-2013-0037Open DOISearch in Google Scholar

Obrączka, A. (2014). Control of Heat Processes with the Use of Non-integer Models, PhD thesis, AGH UST, Kraków.Search in Google Scholar

Oprzędkiewicz, K. (2003). The interval parabolic system, Archives of Control Sciences 13(4): 415-430.Search in Google Scholar

Oprzędkiewicz, K. (2004). A controllability problem for a class of uncertain parameters linear dynamic systems, Archives of Control Sciences 14(1): 85-100.Search in Google Scholar

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.Search in Google Scholar

Oprzędkiewicz, K. and Gawin, E. (2016). A noninteger order, state space model for one dimensional heat transfer process, Archives of Control Sciences 26(2): 261-275.10.1515/acsc-2016-0015Search in Google Scholar

Oprzędkiewicz, K., Gawin, E. and Mitkowski, W. (2016a). Modeling heat distribution with the use of a noninteger order, state space model, International Journal of Applied Mathematics and Computer Science 26(4): 749-756, DOI: 10.1515/amcs-2016-0052.10.1515/amcs-2016-0052Open DOISearch in Google Scholar

Oprzędkiewicz, K., Gawin, E. and Mitkowski, W. (2016b). Parameter identification for noninteger order, state space models of heat plant, MMAR 2016: 21st International Conference on Methods and Models in Automation and Robotics, Międzyzdroje, Poland, pp. 184-188.10.1109/MMAR.2016.7575130Search in Google Scholar

Oprzędkiewicz, K., Mitkowski, W. and Gawin, E. (2017a). An accuracy estimation for a noninteger order, discrete, state space model of heat transfer process, Automation 2017: Innovations in Automation, Robotics and Measurement Techniques, Warsaw, Poland, pp. 86-98.10.1007/978-3-319-54042-9_8Search in Google Scholar

Oprzędkiewicz, K., Stanisławski, R., Gawin, E. and Mitkowski, W. (2017b). A new algorithm for a CFE approximated solution of a discrete-time noninteger-order state equation, Bulletin of the Polish Academy of Sciences: Technical Sciences 65(4): 429-437.10.1515/bpasts-2017-0048Search in Google Scholar

Ostalczyk, P. (2016). Discrete Fractional Calculus. Applications in Control and Image Processing, World Scientific, Singapore.10.1142/9833Search in Google Scholar

Pazy, A. (1983). Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer, New York, NY.10.1007/978-1-4612-5561-1Search in Google Scholar

Petras, I. (2009a). Fractional order feedback control of a DC motor, Journal of Electrical Engineering 60(3): 117-128.Search in Google Scholar

Petras, I. (2009b). http://people.tuke.sk/igor.podlubny/USU/matlab/petras/dfod2.m.Search in Google Scholar

Podlubny, I. (1999). Fractional Differential Equations, Academic Press, San Diego, CA.Search in Google Scholar

Popescu, E. (2010). On the fractional Cauchy problem associated with a feller semigroup, Mathematical Reports 12(2): 181-188.Search in Google Scholar

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): 2-11.10.1016/j.amc.2014.11.028Search in Google Scholar

Stanisławski, R. and Latawiec, K. (2013a). Stability analysis for discrete-time fractional-order LTI state-space systems. Part I: New necessary and sufficient conditions for asymptotic stability, Bulletin of the Polish Academy of Sciences: Technical Sciences 61(2): 353-361.10.2478/bpasts-2013-0034Search in Google Scholar

Stanisławski, R. and Latawiec, K. (2013b). Stability analysis for discrete-time fractional-order LTI state-space systems. Part II: Stability criterion for FD-based systems, Bulletin of the Polish Academy of Sciences; Technical Sciences 61(2): 362-370.10.2478/bpasts-2013-0035Search in Google Scholar

Stanisławski, R., Latawiec, K. and Łukaniszyn, M. (2015). A comparative analysis of Laguerre-based approximators to the Grünwald-Letnikov fractional-order difference, Mathematical Problems in Engineering 2015(1): 1-10.10.1155/2015/512104Search in Google Scholar

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): 200-218.10.1016/j.apm.2009.04.006Search in Google Scholar