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1

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

Oprzędkiewicz Krzysztof

Bulletin of the Polish Academy of Sciences. Technical Sciences

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2023

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Vol. 71, nr 4

art. no. e145675

EN

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.

2

Fractional discrete-continuous model of heat transfer process

Oprzędkiewicz Krzysztof, Dziedzic Klaudia

Archives of Control Sciences

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2021

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Vol. 31, No. 2

287--306

EN

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.

3

Non integer order, state space model of heat transfer process using Atangana-Baleanu operator

Oprzędkiewicz K.

Bulletin of the Polish Academy of Sciences. Technical Sciences

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2020

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Vol. 68, nr 1

43--50

EN

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.

4

Accuracy estimation of the approximated Atangana-Baleanu operator

Oprzędkiewicz Krzysztof, Mitkowski Wojciech

Journal of Applied Mathematics and Computational Mechanics

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2019

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Vol. 18, nr 4

53--62

EN

In the paper, the accuracy analysis of the approximation of the Atangana-Baleanu (AB) operator is presented. The AB operator is the nonsingular kernel operator proposed by Atangana and Baleanu. It is obtained by replacing the exponential function in the Caputo-Fabrizio operator by the Mittag-Leffler function. The Laplace transform of the AB operator requires approximating the factor sa. This is done using the well-known Oustaloup Recursive Approximaion (ORA) approximation. The step and frequency responses of the approximation are compared to the analytical responses. As the cost function, the FIT function available in MATLAB was applied. Results of simulations show that the use of ORA allows us to obtain the accurate approximant of the AB operator.

5

The Caputo vs. Caputo-Fabrizio operators in modeling of heat transfer process

Oprzędkiewicz K., Mitkowski W., Gawin E., Dziedzic K.

Bulletin of the Polish Academy of Sciences. Technical Sciences

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2018

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Vol. 66, nr 4

501--507

EN

In the paper two non-integer order, state space models of heat transfer process are compared. The first uses a known Caputo operator and the second – a new operator proposed by Caputo and Fabrizio in 2015. Both discussed models are modifications of a known, integer order, state space, semigroup model of heat transfer process. Parameters of both models were identified by means of optimization of MSE cost function with the use of simplex method, available in MATLAB. Both proposed models have been compared in the aspect of accuracy and convergence. Analytical and numerical results show that the Caputo-Fabrizio model is faster convergent and easier to implement than the Caputo model. However, its accuracy in the sense of MSE cost function is worse.

6

Non integer order, state space model of heat transfer process using Caputo-Fabrizio operator

Oprzędkiewicz K.

Bulletin of the Polish Academy of Sciences. Technical Sciences

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2018

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Vol. 66, nr 3

249--255

EN

The paper is intended to show a new state space, non integer order model of an one-dimensional heat transfer process. The proposed model derives directly from time continuous, state space semigroup model. The fractional order derivative with respect to time is by a new operator proposed by Caputo and Fabrizio, the non integer order spatial derivative is expressed by Riesz operator. The Caputo-Fabrizio operator can be directly implementated using MATLAB, because it does not require us to apply any approximation. Analytical formulae of step response are given, the system decomposition was discussed also. Main results from the paper show that the use of Caputo Fabrizio operator allows us to obtain the simple in implementation and analysis model of the considered heat transfer process. The accuracy of the proposed model in the sense of a MSE cost function is satisfying.