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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

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.

3

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

Oprzedkiewicz K., Gawin E.

Archives of Control Sciences

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2016

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Vol. 26, no. 2

261--275

EN

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.

4

Optymalizacja parametryczna regulatora niecałkowitego rzędu typu PD^alfa

Zagórowska M.

Pomiary Automatyka Robotyka

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2015

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R. 19, nr 4

5--14

PL

W artykule przeanalizowano zachowanie układu całkowitego rzędu ze sprzężeniem zwrotnym niecałkowitego rzędu. Przedstawiono nową metodę doboru optymalnych parametrów dla regulatorów typu PDα we wspomnianych układach z nieskończonym horyzontem. Zaprezentowano wykorzystaną metodę aproksymacji układów niecałkowitego rzędu z wykorzystaniem funkcji Laguerre’a oraz sformułowano problem w postaci zagadnienia minmax. Pokazano również przykładowy przebieg optymalizacji ze względu na parametry związane z typem aproksymacji.

EN

In this paper we analysed the behaviour of an integer order system with non-integer control function. We presented a new method for tuning the non-integer order controllers PDα for use in systems with infinite horizon. An approximation method for non-integer order systems was presented (using Laguerre functions) along with the definition of the issue in form of minmax problem. Finally some examples were analysed with respect to parameters specific for this approximation.

5

A comparison between integer order and non-integer order controllers applied to a water levelling system

Kania B., Dziwiński T., Bauer W., Zagórowska M., Baranowski J.

Automatyka / Automatics

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2014

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Vol. 18, no. 2

71--82

EN

The purpose of this paper is to compare responses of two systems with two types of controllers: integer order PI controller and non-integer order PI controller (FOPI). Comparison is made on the grounds of time domain responses of waters level in tank. In order to design both of the controllers, numerical model is established and described. Synthesis method with use of simulated annealing optimisation method is developed for each controller. In order to do a correct comparison, the two control systems have the same designing parameters. Matlab/Simulink environment and numerical model are used to do the computing. Numerically designed controllers are verified experimentally with tank system. The obtained results show that using non-integer order PIl controller instead of a ”traditional” PI controller in simple integrating systems is useless.

PL

W artykule porównano odpowiedzi układów z dwoma regulatorami: typu PI oraz typu PI niecałkowitego rzędu. W celu syntezy regulatorów, stworzono i opisano model numeryczny, a następnie wykorzystano metodę symulowanego wyzarzania do znalezienia optymalnych parametrów. Tak zaprojektowane regulatory zostały przetestowane na rzeczywistym zbiorniku. W rezultacie okazało się, że zastosowanie regulatora niecałkowitoliczbowego rzędu do układu całkującego nie jest konieczne. Obliczenia były prowadzone w środowisko Matlab/Simulink.