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

Open access

Abstract

A new, state space, discrete-time, and memory-efficient model of a one-dimensional heat transfer process is proposed. The model is derived directly from a time-continuous, state-space semigroup one. Its discrete version is obtained via a continuous fraction expansion method applied to the solution of the state equation. Fundamental properties of the proposed model, such as decomposition, stability, accuracy and convergence, are also discussed. Results of experiments show that the model yields good accuracy in the sense of the mean square error, and its size is significantly smaller than that of the model employing the well-known power series expansion approximation.

Al-Alaoui,M. (1993). Novel digital integrator and differentiator, Electronics Letters 29(4): 376-378.

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.

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.

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.

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.

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.

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.

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, Bialystok University of Technology, Bialystok.

Kochubei, A. (2011). Fractional-parabolic systems, arXiv:1009.4996 [math.ap].

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.

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

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

Oprzędkiewicz, 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.

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.

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.

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.

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.

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.

Ostalczyk, P. (2016). Discrete Fractional Calculus. Applications in Control and Image Processing, World Scientific, Singapore.

Pazy, A. (1983). Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer, New York, NY.

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

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

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.

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.

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.

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.

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.

International Journal of Applied Mathematics and Computer Science

Journal of the University of Zielona Góra

Journal Information


IMPACT FACTOR 2017: 1.694
5-year IMPACT FACTOR: 1.712

CiteScore 2017: 2.20

SCImago Journal Rank (SJR) 2017: 0.729
Source Normalized Impact per Paper (SNIP) 2017: 1.604

Mathematical Citation Quotient (MCQ) 2017: 0.13

Metrics

All Time Past Year Past 30 Days
Abstract Views 0 0 0
Full Text Views 123 123 20
PDF Downloads 71 71 10