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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

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[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.
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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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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
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Abstract

In the paper a new, state space, non integer order model of an one-dimensional heat transfer process is proposed. The model uses a new operator with Mittag-Leffler kernel, proposed by Atangana and Beleanu. The non integer order spatial derivative is expressed by Riesz operator. Analytical formula of the step response is given, the convergence of the model is discussed too. Theoretical results are verified by experiments.

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Authors and Affiliations

K. Oprzędkiewicz

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