Fractional order modeling of a nonlinear electromechanical system
DOI:
https://doi.org/10.29019/enfoqueute.v9n4.398Keywords:
Fractional Calculus, Dynamical Electromechanical System, Fractional Parameters Estimation, Extended Kalman FilterAbstract
This paper presents a novel modeling technique for a VTOL electromechanical nonlinear dynamical system, based on fractional order derivatives. The proposed method evaluates the possible fractional differential equations of the electromechanical system model by a comparison against actual measurements and in order to estimate the optimal fractional parameters for the differential operators of the model, an extended Kalman filter was implemented. The main advantages of the fractional model over the classical model are the simultaneous representation of the nonlinear slow dynamics of the system due to the mechanical components and the nonlinear fast dynamics of the electrical components.
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References
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Chen, X., Chen, Y., Zhang, B., and Qiu, D. (2017). A Modeling and Analysis Method for Fractional-Order DC-DC Converters. IEEE Transactions on Power Electronics, 32(9), 7034-7044.
Gómez, J.F., Yépez, H., Escobar, R.F., Astorga, C.M., and Reyes, J. (2016). Analytical and numerical solutions of electrical circuits described by fractional derivatives. Applied Mathematical Modelling, 40(21), 9079-9094.
Lazarevi, M.P., Mandi, P.D., Cvetkovi, B., ekara, T.B., and Lutovac, B. (2016). Some electromechanical systems and analogies of mem-systems integer and fractional order. In 2016 5th Mediterranean Conference on Embedded Computing (MECO), 230-233.
Li, C. and Ma, Y. (2013). Fractional dynamical system and its linearization theorem. Nonlinear Dynamics, 71(4), 621-633.
Miller, K.s. and Ross, B. (1993). An introduction to the fractional calculus and fractional differential equations. Wiley-Interscience, 1 edition.
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Petrás, I. (2011). Fractional-Order Nonlinear Systems. Nonlinear Physical Science. Springer Berlin Heidelberg, Berlin, Heidelberg.
Podlubny, I. (1999). Fractional-order systems and PIλ Dμ-controllers. IEEE Transactions on Automatic Control, 44(1), 208-214.
Podlubny, I. (1998). Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications. Academic Press.
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Rendón, J. (2018). Aplicaciones del cálculo fraccional en modelamiento y control de sistemas dinámicos electromecánicos. Master's thesis, Universidad Nacional de Colombia, Medellín, Colombia.
Schäfer, I. and Krüger, K. (2006). Modelling of coils using fractional derivatives. Journal of Magnetism and Magnetic Materials, 307(1), 91-98.
Sierociuk, D. and Dzieliski, A. (2006). Fractional kalman filter algorithm for the states parameters and order of fractional system estimation. International Journal of Applied Mathematics and Computer Science, 16(1), 129-140.
Swain, S.K., Sain, D., Mishra, S.K., and Ghosh, S. (2017). Real time implementation of fractional order PID controllers for a magnetic levitation plant. AEU – International Journal of Electronics and Communications, 78, 141-156.
Tenreiro, J.A., Silva, M.F., Barbosa, R.S., Jesus, I.S., Reis, C., M, L., Marcos, M.G., and Galhano, A.F. (2010). Some Applications of Fractional Calculus in Engineering. Mathematical Problems in Engineering vol. 2010, article ID 639801. http://dx.doi.org/10.1155/2010/639801
Chen, X., Chen, Y., Zhang, B., and Qiu, D. (2017). A Modeling and Analysis Method for Fractional-Order DC-DC Converters. IEEE Transactions on Power Electronics, 32(9), 7034-7044.
Gómez, J.F., Yépez, H., Escobar, R.F., Astorga, C.M., and Reyes, J. (2016). Analytical and numerical solutions of electrical circuits described by fractional derivatives. Applied Mathematical Modelling, 40(21), 9079-9094.
Lazarevi, M.P., Mandi, P.D., Cvetkovi, B., ekara, T.B., and Lutovac, B. (2016). Some electromechanical systems and analogies of mem-systems integer and fractional order. In 2016 5th Mediterranean Conference on Embedded Computing (MECO), 230-233.
Li, C. and Ma, Y. (2013). Fractional dynamical system and its linearization theorem. Nonlinear Dynamics, 71(4), 621-633.
Miller, K.s. and Ross, B. (1993). An introduction to the fractional calculus and fractional differential equations. Wiley-Interscience, 1 edition.
Özkan, B. (2014). Control of an Electromechanical Control Actuation System Using a Fractional Order Proportional, Integral, and Derivative-Type Controller. IFAC Proceedings Volumes, 47(3), 4493-4498.
Petrás, I. (2011). Fractional-Order Nonlinear Systems. Nonlinear Physical Science. Springer Berlin Heidelberg, Berlin, Heidelberg.
Podlubny, I. (1999). Fractional-order systems and PIλ Dμ-controllers. IEEE Transactions on Automatic Control, 44(1), 208-214.
Podlubny, I. (1998). Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications. Academic Press.
Rahimy, M. (2010). Applications of fractional differential equations. Applied Mathematica Sciences 4(50): 2453-2461.
Rendón, J. (2018). Aplicaciones del cálculo fraccional en modelamiento y control de sistemas dinámicos electromecánicos. Master's thesis, Universidad Nacional de Colombia, Medellín, Colombia.
Schäfer, I. and Krüger, K. (2006). Modelling of coils using fractional derivatives. Journal of Magnetism and Magnetic Materials, 307(1), 91-98.
Sierociuk, D. and Dzieliski, A. (2006). Fractional kalman filter algorithm for the states parameters and order of fractional system estimation. International Journal of Applied Mathematics and Computer Science, 16(1), 129-140.
Swain, S.K., Sain, D., Mishra, S.K., and Ghosh, S. (2017). Real time implementation of fractional order PID controllers for a magnetic levitation plant. AEU – International Journal of Electronics and Communications, 78, 141-156.
Tenreiro, J.A., Silva, M.F., Barbosa, R.S., Jesus, I.S., Reis, C., M, L., Marcos, M.G., and Galhano, A.F. (2010). Some Applications of Fractional Calculus in Engineering. Mathematical Problems in Engineering vol. 2010, article ID 639801. http://dx.doi.org/10.1155/2010/639801
Published
2018-12-21
Issue
Section
Automation and Control, Mechatronics, Electromechanics, Automotive
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How to Cite
Fractional order modeling of a nonlinear electromechanical system. (2018). Enfoque UTE, 9(4), pp. 45 - 56. https://doi.org/10.29019/enfoqueute.v9n4.398
