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Approximation method for a fractional order transfer function with zero and pole

Krzysztof Oprzędkiewicz
Archives of Control Sciences | 2014 | No 4 | DOI: 10.2478/acsc-2014-0024
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Krzysztof Oprzędkiewicz

The practical stability of the discrete, fractional order, state space model of the heat transfer process

Krzysztof Oprzędkiewicz Edyta Gawin
Archives of Control Sciences | 2018 | vol. 28 | No 3 | 463–482 | DOI: 10.24425/acs.2018.124712
Keywords discrete non-integer order system heat transfer diffusion equation Grünvald-Letnikov operator practical stability
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Abstract

In the paper the practical stability problem for the discrete, non-integer order model of one dimmensional heat transfer process is discussed. The conditions associating the practical stability to sample time and maximal size of finite-dimensional approximation of heat transfer model are proposed. These conditions are formulated with the use of spectrum decoposition property and practical stability conditions for scalar, positive, fractional order systems. Results are illustrated by a numerical example.

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Authors and Affiliations

Krzysztof Oprzędkiewicz
Edyta Gawin

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

Edyta Gawin Krzysztof Oprzędkiewicz
Archives of Control Sciences | 2016 | No 2 | DOI: 10.1515/acsc-2016-0015
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Abstract

Abstract 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.
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Authors and Affiliations

Edyta Gawin
Krzysztof Oprzędkiewicz

The positivity of the fractional order model of a two-dimensional temperature field

Krzysztof Oprzędkiewicz
Bulletin of the Polish Academy of Sciences Technical Sciences | 2023 | 71 | 4 | e145675 | DOI: 10.24425/bpasts.2023.145675
Keywords noninteger order systems heat transfer equation fractional order state equation Caputo operator positivity thermal camera
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Abstract

The paper presents analysis of the positivity for a two-dimensional temperature field. The process under consideration is described by the linear, infinite-dimensional, noninteger order state equation. It is derived from a two-dimensional parabolic equation with homogenous Neumann boundary conditions along all borders and homogenous initial condition. The form of control and observation operators is determined by the construction of a real system. The internal and external positivity of the model are associated to the localization of heater and measurement. It has been proven that the internal positivity of the considered system can be achieved by the proper selection of attachment of a heater and place of a measurement as well as the dimension of the finite-dimensional approximation of the considered model. Conditions of the internal positivity associated with construction of real experimental system are proposed. The postivity is analysed separately for control and output of the system. This allows one to analyse the positivity of thermal systems without explicit control. Theoretical considerations are numerically verified with the use of experimental data. The proposed results can be applied i.e. to point suitable places for measuring of a temperature using a thermal imaging camera.
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Authors and Affiliations

Krzysztof Oprzędkiewicz
1
ORCID: ORCID
  1. AGH University of Science and Technology, al. A. Mickiewicza 30, 30-059 Kraków, Poland

Fractional order, discrete model of heat transfer process using time and spatial Grünwald-Letnikov operator

Krzysztof Oprzędkiewicz
Bulletin of the Polish Academy of Sciences Technical Sciences | 2021 | 69 | No. 1 | e135843 | DOI: 10.24425/bpasts.2021.135843
Keywords fractional order systems heat transfer equation fractional order state equation Fractional Order Backward Difference Grünwald-Letnikov operator practical stability
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Abstract

In the paper a new, state space, fully discrete, fractional model of a heat transfer process in one dimensional body is addressed. The proposed model derives directly from fractional heat transfer equation. It employes the discrete Grünwald-Letnikov operator to express the fractional order differences along both coordinates: time and space. The practical stability and numerical complexity of the model are analysed. Theoretical results are verified using experimental data.
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Bibliography

  1.  S. Das, Functional Fractional Calculus for System Identification and Controls, Springer, Berlin, 2010.
  2.  R. Caponetto, G. Dongola, L. Fortuna, and I. Petras, “Fractional order systems: Modeling and Control Applications”, in World Scientific Series on Nonlinear Science, ed. L.O. Chua, pp. 1–178, University of California, Berkeley, 2010.
  3.  A. Dzieliński, D. Sierociuk, and G. Sarwas, “Some applications of fractional order calculus”, Bull. Pol. Ac.: Tech. 58(4), 583– 592 (2010).
  4.  C.G. Gal and M. Warma, “Elliptic and parabolic equations with fractional diffusion and dynamic boundary conditions”, Evol. Equ. Control Theory 5(1), 61–103 (2016).
  5.  E. Popescu, “On the fractional Cauchy problem associated with a feller semigroup”, Math. Rep. 12(2), 81–188 (2010).
  6.  D. Sierociuk et al., “Diffusion process modeling by using fractional-order models”, Appl. Math. Comput. 257(1), 2–11 (2015).
  7.  J.F. Gómez, L. Torres, and R.F. Escobar (eds.), “Fractional derivatives with Mittag-Leffler kernel trends and applications in science and engineering”, in Studies in Systems, Decision and Control, vol. 194, ed. J. Kacprzyk, pp. 1–339. Springer, Switzerland, 2019.
  8.  M. Dlugosz and P. Skruch, “The application of fractional-order models for thermal process modelling inside buildings”, J. Build Phys. 1(1), 1–13 (2015).
  9.  A. Obrączka, Control of heat processes with the use of noninteger models. PhD thesis, AGH University, Krakow, Poland, 2014.
  10.  A. Rauh, L. Senkel, H. Aschemann, V.V. Saurin, and G.V. Kostin, “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).
  11.  T. Kaczorek, “Singular fractional linear systems and electri cal circuits”, Int. J. Appl. Math. Comput. Sci. 21(2), 379–384 (2011).
  12.  T. Kaczorek and K. Rogowski, Fractional Linear Systems and Electrical Circuits, Bialystok University of Technology, Bialystok, 2014.
  13.  I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, 1999.
  14.  B. Bandyopadhyay and S. Kamal, “Solution, stability and realization of fractional order differential equation”, in Stabilization and Control of Fractional Order Systems: A Sliding Mode Approach, Lecture Notes in Electrical Engineering 317, pp. 55–90, Springer, Switzerland, 2015.
  15.  D. Mozyrska, E. Girejko, M. Wyrwas, “Comparison of hdifference fractional operators”, in Advances in the Theory and Applications of Non- integer Order Systems, eds. W. Mitkowski et al., pp. 1–178. Springer, Switzerland, 2013.
  16.  P. Ostalczyk, “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).
  17.  E.F. Anley and Z. Zheng, “Finite difference approximation method for a space fractional convection–diffusion equation with variable coefficients”, Symmetry 12(485), 1–19 (2020).
  18.  P. Ostalczyk, Discrete Fractional Calculus. Applications in Control and Image Processing, World Scientific, New Jersey, London, Singapore, 2016.
  19.  M. Buslowicz and T. Kaczorek, “Simple conditions for practical stability of positive fractional discrete-time linear systems”, Int. J. Appl. Math. Comput. Sci. 19(2), 263–269 (2009).
  20.  R. Brociek and D. Słota, “Implicit finite difference method for the space fractional heat conduction equation with the mixed boundary condition”, Silesian J. Pure Appl. Math. 6(1), 125–136 (2016).
  21.  D. Mozyrska and E. Pawluszewicz, “Fractional discrete-time linear control systems with initialization”, Int. J. Control 1(1), 1–7 (2011).
  22.  K. Oprzędkiewicz, “The interval parabolic system”, Arch. Control Sci. 13(4), 415–430 (2003).
  23.  K. Oprzędkiewicz, “A controllability problem for a class of uncertain parameters linear dynamic systems”, Arch. Control Sci. 14(1), 85–100 (2004).
  24.  K. Oprzędkiewicz, “An observability problem for a class of uncertain-parameter linear dynamic systems”, Int. J. Appl. Math. Comput. Sci. 15(3), 331–338 (2005).
  25.  A. Dzieliński and D. Sierociuk, “Stability of discrete fractional order state-space systems”, in Proc. of the 2nd IFAC Workshop on Fractional Differentiation and its Applications, Porto, Portugal, 2006, pp. 505–510.
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Authors and Affiliations

Krzysztof Oprzędkiewicz
1
ORCID: ORCID
  1. AGH University of Science and Technology, al. A. Mickiewicza 30, 30-059 Kraków, Poland

Fractional discrete-continuous model of heat transfer process

Krzysztof Oprzędkiewicz Klaudia Dziedzic
Archives of Control Sciences | 2021 | vol. 31 | No 2 | 287-306 | DOI: 10.24425/acs.2021.137419
Keywords non integer order systems heat transfer equation finite difference Caputo operator positive systems
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Abstract

The paper proposes a new, state space, finite dimensional, fractional order model of a heat transfer in one dimensional body. The time derivative is described by Caputo operator. The second order central difference describes the derivative along the length. The analytical formulae of the model responses are proved. The stability, convergence, and positivity of the model are also discussed. Theoretical results are verified by experiments.
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Bibliography

[1] R. Almeida and D.F.M. Torres: Necessary and sufficient conditions for the fractional calculus of variations with caputo derivatives. Communications in Nonlinear Science and Numerical Simulation, 16(3), (2011), 1490–1500, DOI: 10.1016/j.cnsns.2010.07.016.
[2] A. Atangana and D. Baleanu: New fractional derivatives with non-local and non-singular kernel: theory and application to heat transfer. Thermal Sciences, 20(2), (2016), 763–769, DOI: 10.2298/TSCI160111018A.
[3] R. Caponetto, G. Dongola, L. Fortuna, and I. Petras: Fractional order systems: Modeling and Control Applications. In: Leon O. Chua, editor, World Scientific 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, 39(5), (2016), 440–451, DOI: 10.1177/1744259115591251.
[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, DOI: 10.2478/v10175-010-0059-6.
[7] C.G. Gal and M. Warma Elliptic and parabolic equations with fractional diffusion and dynamic boundary conditions. Evolution Equations and Control Theory, 5(1), (2016), 61–103, DOI: 10.3934/eect.2016.5.61.
[8] T. Kaczorek Fractional positive contiuous-time linear systems and their reachability. International Journal of Applied Mathematics and Computer Science, 18(2), (2008), 223–228, DOI: 10.2478/v10006-008-0020-0.
[9] T. Kaczorek: Singular fractional linear systems and electrical circuits. International Journal of Applied Mathematics and Computer Science, 21(2), (2011), 379–384, DOI: 10.2478/v10006-011-0028-8.
[10] T. Kaczorek and K. Rogowski: Fractional Linear Systems and Electrical Circuits. Bialystok University of Technology, Bialystok, 2014.
[11] A. Kochubei: Fractional-parabolic systems, preprint, arxiv:1009.4996 [math.ap], 2011.
[12] W. Mitkowski: Approximation of fractional diffusion-wave equation. Acta Mechanica et Automatica, 5(2), (2011), 65–68.
[13] W. Mitkowski: Finite-dimensional approximations of distributed rc networks. Bulletin of the Polish Academy of Sciences. Technical Sciences, 62(2), (2014), 263–269, DOI: 10.2478/bpasts-2014-0026.
[14] W. Mitkowski,W. Bauer, and M. Zagorowska: Rc-ladder networks with supercapacitors. Archives of Electrical Engineering, 67(2), (2018), 377– 389, DOI: 10.24425/119647.
[15] K. Oprzedkiewicz: The discrete-continuous model of heat plant. Automatyka, 2(1), (1998), 35–45 (in Polish).
[16] K. Oprzedkiewicz: The interval parabolic system. Archives of Control Sciences, 13(4), (2003), 415–430.
[17] K. Oprzedkiewicz:Acontrollability 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 uncertainparameter linear dynamic systems. International Journal of Applied Mathematics and Computer Science, 15(3), (2005), 331–338.
[19] K. Oprzedkiewicz:Non integer order, state space model of heat transfer process using Caputo-Fabrizio operator. Bulletin of the Polish Academy of Sciences. Technical Sciences, 66(3), (2018), 249–255, DOI: 10.24425/122105.
[20] K. Oprzedkiewicz: Non integer order, state space model of heat transfer process using Atangana-Baleanu operator. Bulletin of the Polish Academy of Sciences. Technical Sciences, 68(1), (2020), 43–50, DOI: 10.24425/bpasts.2020.131828.
[21] K. Oprzedkiewicz: Positivity problem for the one dimensional heat transfer process. ISA Transactions, 112, (2021), 281-291 DOI: .
[22] K. Oprzedkiewicz: Fractional order, discrete model of heat transfer process using time and spatial Grünwald-Letnikov operator. Bulletin of the Polish Academy of Sciences. Technical Sciences, 69(1), (2021), 1–10, DOI: 10.24425/bpasts.2021.135843.
[23] K. Oprzedkiewicz, K. Dziedzic, and Ł. Wi˛ eckowski: Non integer order, discrete, state space model of heat transfer process using Grünwald-Letnikov operator. Bulletin of the Polish Academy of Sciences. Technical Sciences, 67(5), (2019), 905–914, DOI: 10.24425/bpasts.2019.130873.
[24] K. Oprzedkiewicz and E. Gawin: A non-integer order, state space model for one dimensional heat transfer process. Archives of Control Sciences, 26(2), (2016), 261–275, DOI: 10.1515/acsc-2016-0015.
[25] K. Oprzedkiewicz, E. Gawin, and W. 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, DOI: 10.1515/amcs-2016-0052.
[26] K. Oprzedkiewicz and W. Mitkowski: A memory-efficient nonintegerorder discrete-time state-space model of a heat transfer process. International Journal of Applied Mathematics and Computer Science, 28(4), (2018), 649–659, DOI: 10.2478/amcs-2018-0050.
[27] K. Oprzedkiewicz,W. Mitkowski, E.Gawin, and K. Dziedzic: The Caputo vs. Caputo-Fabrizio operators in modeling of heat transfer process. Bulletin of the Polish Academy of Sciences. Technical Sciences, 66(4), (2018), 501– 507, DOI: 10.24425/124267.
[28] K. Oprzedkiewicz, E. Gawin, and W. Mitkowski: 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, Międzyzdroje, Poland, pages 184– 188, 2016.
[29] P. Ostalczyk: Discrete Fractional Calculus. Applications in Control and Image Processing. Worlsd Scientific Publishing, Singapore, 2016.
[30] I. Podlubny: Fractional Differential Equations. Academic Press, San Diego, 1999.
[31] G. Recktenwald: Finite-difference approximations to the heat equation. 2011.
[32] M. Rozanski: Determinants of two kinds of matrices whose elements involve sine functions. Open Mathematics, 17(1), (2019), 1332–1339, DOI: 10.1515/math-2019-0096.
[33] N. Al Salti, E. Karimov, and S. Kerbal: Boundary-value problems for fractional heat equation involving caputo-fabrizio derivative. New Trends in Mathematical Sciences, 4(4), (2016), 79–89, arXiv:1603.09471.
[34] 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, DOI: 10.1016/j.amc.2014.11.028.
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Authors and Affiliations

Krzysztof Oprzędkiewicz
1
ORCID: ORCID
Klaudia Dziedzic
1
  1. AGH University of Science and Technology in Krakow, Faculty of Electrical Engineering, Automatics, Computer Science and Robotics, Department of Automatics and Biomedical Engineering, Kraków, Poland

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