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The present study concerns the analysis of using the integro-differential fractional operators in the process of modelling of electrical systems. As the object of study, the RC circuit with the ultracapacitor was used. A mathematical model of the super capacitor has been introduced, based on the integro-differential fractional order operator. Throughout the modelling it was assumed that the ultracapacitor was ideal. The main goal of the work was to carry out the experimental tests. Static and dynamic characteristics of the RC circuit with the ultracapacitor were determined with a prepared laboratory test rig. The obtained experimental results were compared with simulation tests of the ultracapacitor dynamics. The analysis, modelling and the obtained results allowed for the assessment of applicability of the fractional order operators in modelling of electrical systems. The proposed fractional order model yielded more precise description of dynamic properties of the system.
Słowa kluczowe
Czasopismo
Rocznik
Tom
Strony
275--288
Opis fizyczny
Bibliogr. 18 poz., rys.
Twórcy
autor
- Bialystok University of Technology, Faculty of Mechanical Engineering, Department of Automatic Control and Robotics, Wiejska 45 C, 15-351 Bialystok, Poland
autor
- Bialystok University of Technology, Faculty of Mechanical Engineering, Department of Automatic Control and Robotics, Wiejska 45 C, 15-351 Bialystok, Poland
autor
- Bialystok University of Technology, Faculty of Mechanical Engineering, Department of Automatic Control and Robotics, Wiejska 45 C, 15-351 Bialystok, Poland
Bibliografia
- [1] Buslowicz, M. and Ruszewski, A. (2013) Necessary and sufficient conditions for stability of fractional discrete-time linear state-space systems. Bulletin of the Polish Academy of Sciences, Technical Sciences 61, 779–786.
- [2] Chen, Y. Q, Petr´ aˇ s, I. and Xue, D. (2009) Fractional order control – a tutorial. Proceedings of American Control Conference, June 10-12, St. Louis, MO, USA, 1397-1411.
- [3] Das, S. (2008) Functional Fractional Calculus for System Identification and Controls. Springer, Berlin.
- [4] Dzielinski, A., Sarwas, G. and Sierociuk, D. (2011) Comparison and validation of integer and fractional order ultracapacitor models. Advances in Difference Equations 11(1), 1-15.
- [5] Ferreira, R. A. C. and Torres, D. F. M. (2011) Fractional h-difference equations arising from the calculus of variations. Applicable Analysis and Discrete Mathematics 5(1), 110–121.
- [6] Kaczorek, T. (2011) Selected Problems of Fractional Systems Theory. Springer, Berlin.
- [7] Matasu, R. (2012) Application of fractional order calculus to control theory. International Journal of Mathematical Models and Methods in Applied Sciences 5(7), 1162-1169.
- [8] Mozyrska, D. and Pawluszewicz, E. (2013) Local controllability of nonlinear discrete-time fractional order systems. Bulletin of the Polish Academy of Sciences, Technical Sciences 61(1), 251-256.
- [9] Mozyrska, D. Girejko, E. and Wyrwas, M. (2013) Comparison of hdifference fractional operators. In: W. Mitkowski, J. Kacprzyk and J. Baranowski, eds., Advances in the Theory and Applications of Noninteger Order Systems, 257, Springer, 191–197.
- [10] Mozyrska, D. and Wyrwas, M. (2015) The Z-transform method and delta type fractional difference operators. Discrete Dynamics in Nature and Society 2015, 1-13.
- [11] Ortigueira, M. D. (2011) Fractional Calculus for Scientists and Engineers. Lecture Notes in Electrical Engineering, 84, Springer, Netherlands.
- [12] Ostalczyk, P. (2008) Epitome of the fractional calculus. Theory and its applications in automatics. Publishing House of Lodz University of Technology, Łódź (In Polish).
- [13] Podlubny, I., Dorcak, L. and Misanek, J. (1995) Application of fractional order derivatives to calculation of heat load intensity change in blast furnace walls. Transactions of Technical University of Kosice 5, 137–144.
- [14] Podlubny, I. (1999) Fractional Differential Equations. Academic Press, San Diego, California.
- [15] Podlubny, I. (2002) Geometric and physical interpretation of fractional integration and fractional differentiation. Fractional Calculus and Applied Analysis 5(4), 367-386.
- [16] Sierociuk, D., Podlubny, I. and Petras, I. (2013a) Experimental evidence of variable-order behavior of ladders and nested ladders. IEEE Transactions on Control Systems Technology 21(2), 459-466.
- [17] Sierociuk, D., Dzieli´nski, A., Sarwas, G., Petras, I., Podlubny, I. and Skovranek, T. (2013b) Modelling heat transfer in heterogenous media using fractional calculus. Philosophical Transaction of the Royal Society A-Mathematical, Physical and Engineering Sciences 371, 1-10.
- [18] Wang, J. C. (1987) Realizations of generalized Warburg impedance with RC ladder networks and transmission lines. Journal of Electrochemical Society 134(8), 1915–1920.
Uwagi
PL
Opracowanie ze środków MNiSW w ramach umowy 812/P-DUN/2016 na działalność upowszechniającą naukę (zadania 2017).
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Bibliografia
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