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Derivation, interpretation, and analog modelling of fractional variable order derivative definition. (English) Zbl 1443.26003

Summary: The paper presents derivation and interpretation of one type of variable order derivative definitions. For mathematical modelling of considering definition the switching and numerical scheme is given. The paper also introduces a numerical scheme for a variable order derivatives based on matrix approach. Using this approach, the identity of the switching scheme and considered definition is derived. The switching scheme can be used as an interpretation of this type of definition. Paper presents also numerical examples for introduced methods. Finally, the idea and results of analog (electrical) realization of the switching fractional order integrator (of orders 0.5 and 1) are presented and compared with numerical approach.

MSC:

26A33 Fractional derivatives and integrals
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[1] Oldham, K. B.; Spanier, J., The Fractional Calculus, (1974), Academic Press · Zbl 0428.26004
[2] Podlubny, I., Fractional differential equations, (1999), Academic Press · Zbl 0918.34010
[3] Samko, S.; Kilbas, A.; Maritchev, O., Fractional integrals and derivative. theory and applications, (1987), Gordon & Breach Sci. Publishers · Zbl 0617.26004
[4] Monje, C. A.; Chen, Y.; Vinagre, B. M.; Xue, D.; Feliu, V., Fractional-order systems and controls, (2010), Springer · Zbl 1211.93002
[5] Sheng, H.; Chen, Y.; Qiu, T., Signal processing fractional processes and fractional-order signal processing, (2012), Springer London · Zbl 1245.94004
[6] Sayevand, K.; Golbabai, A.; Yildirim, A., Analysis of differential equations of fractional order, Appl. Math. Model., 36, 9, 4356-4364, (2012) · Zbl 1252.34007
[7] Kazem, S.; Abbasbandy, S.; Kumar, S., Fractional-order Legendre functions for solving fractional-order differential equations, Appl. Math. Model., 37, 7, 5498-5510, (2013) · Zbl 1449.33012
[8] A. Dzielinski, W. Malesza, Point to point control of fractional differential linear control systems, Adv. Differ. Equ., http://dx.doi.org/10.1186/1687-1847-2011-13. · Zbl 1268.34110
[9] El Brouji, H.; Vinassa, J.-M.; Briat, O.; Bertrand, N.; Woirgard, E., Ultracapacitors self discharge modelling using a physical description of porous electrode impedance, (Vehicle Power and Propulsion Conference, 2008. VPPC ’08, (2008), IEEE), 1-6
[10] A. Dzielinski, G. Sarwas, D. Sierociuk, Time domain validation of ultracapacitor fractional order model, in: 2010 49th IEEE Conference on Decision and Control (CDC), 2010, pp. 3730-3735. http://dx.doi.org/10.1109/CDC.2010.5717093. · Zbl 1268.34091
[11] A. Dzielinski, G. Sarwas, D. Sierociuk, Comparison and validation of integer and fractional order ultracapacitor models, Adv. Differ. Equ. 2011:11. http://dx.doi.org/10.1186/1687-1847-2011-11. · Zbl 1268.34091
[12] R. Martin, J. Quintana, A. Ramos, I. de la Nuez, Modeling electrochemical double layer capacitor, from classical to fractional impedance, in: The 14th IEEE Mediterranean Electrotechnical Conference, 2008. MELECON 2008. 2008, pp. 61-66. http://dx.doi.org/10.1109/MELCON.2008.4618411.
[13] H. Sheng, H. Sun, C. Coopmans, Y. Chen, G.W. Bohannan, Physical experimental study of variable-order fractional integrator and differentiator, in: Proceedings of The 4th IFAC Workshop Fractional Differentiation and its Applications FDA’10, 2010.
[14] Ramirez, L.; Coimbra, C., On the variable order dynamics of the nonlinear wake caused by a sedimenting particle, Physica D, 240, 13, 1111-1118, (2011) · Zbl 1219.76054
[15] Sheng, H.; Sun, H.; Chen, Y.; Qiu, T., Synthesis of multifractional Gaussian noises based on variable-order fractional operators, Signal Process., 91, 7, 1645-1650, (2011) · Zbl 1213.94049
[16] Sierociuk, D.; Ziubinski, P., Fractional order estimation schemes for fractional and integer order systems with constant and variable fractional order colored noise, Circuits Syst. Signal Process., 33, 12, 3861-3882, (2014) · Zbl 1342.93109
[17] Ostalczyk, P.; Rybicki, T., Variable-fractional-order dead-beat control of an electromagnetic servo, J. Vib. Control, 14, 9-10, 1457-1471, (2008)
[18] Ostalczyk, P., Stability analysis of a discrete-time system with a variable-, fractional-order controller, Bull. Pol. Acad. Sci. Tech. Sci., 58, 4, 613-619, (2010) · Zbl 1219.93100
[19] Sierociuk, D.; Twardy, M., Duality of variable fractional order difference operators and its application to identification, Bull. Pol. Acad. Sci. Tech. Sci., 62, 4, 809-815, (2014)
[20] D. Sierociuk, System properties of fractional variable order discrete state-space system, in: Proceedings of the 13th International Carpathian Control Conference (ICCC), 2012, 2012, pp. 643-648. http://dx.doi.org/10.1109/CarpathianCC.2012.6228725.
[21] Sierociuk, D.; Podlubny, I.; Petras, I., Experimental evidence of variable-order behavior of ladders and nested ladders, IEEE Trans. Control Syst. Technol., 21, 2, 459-466, (2013)
[22] C.-C. Tseng, Design and application of variable fractional order differentiator, in: Proceedings of The 2004 IEEE Asia-Pacific Conference on Circuits and Systems, vol. 1, 2004, pp. 405-408.
[23] C.-C. Tseng, S.-L. Lee, Design of variable fractional order differentiator using infinite product expansion, in: Proceedings of 20th European Conference on Circuit Theory and Design (ECCTD), 2011, pp. 17-20.
[24] Lorenzo, C.; Hartley, T., Variable order and distributed order fractional operators, Nonlinear Dyn., 29, 1-4, 57-98, (2002) · Zbl 1018.93007
[25] Valerio, D.; da Costa, J. S., Variable-order fractional derivatives and their numerical approximations, Signal Processing, 91, 3, SI, 470-483, (2011) · Zbl 1203.94060
[26] Podlubny, I., Matrix approach to discrete fractional calculus, Fract. Calculus Appl. Anal., 3, 359-386, (2000) · Zbl 1030.26011
[27] Podlubny, I.; Chechkin, A.; Skovranek, T.; Chen, Y.; Vinagre Jara, B. M., Partial fractional differential equations, J. Comput. Phys., 228, 8, 3137-3153, (2009) · Zbl 1160.65308
[28] D. Sierociuk, Fractional Variable Order Derivative Simulink Toolkit, http://www.mathworks.com/matlabcentral/fileexchange/38801-fractional-variable-order-derivative-simulink-toolkit, 2012.
[29] W. Malesza, D. Sierociuk, Analytical description and equivalence of additive-switching scheme for fractional variable-order continuous differ-integrals, in: Proceedings of International Conference on Fractional Differentiation and its Applications, Catania, Itally, 2014. http://dx.doi.org/10.1109/ICFDA.2014.6967453.
[30] D. Sierociuk, A. Dzielinski, New method of fractional order integrator analog modeling for orders 0.5 and 0.25, in: Proc. of the 16th International Conference on Methods and Models in Automation and Robotics (MMAR), Miedzyzdroje, Poland, 2011, pp. 137-141. http://dx.doi.org/10.1109/MMAR.2011.6031332.
[31] Petras, I.; Sierociuk, D.; Podlubny, I., Identification of parameters of a half-order system, IEEE Trans. Signal Process., 60, 10, 5561-5566, (2012) · Zbl 1393.94966
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