Robust Position Control of a Levitating Ball via a Backstepping Controller

In this paper, a combination of a robust backstepping controller and an integral action for a magnetic levitation system is presented. The mathematical model of the magnetic levitation system containing uncertainties and high-order nonlinear terms has quite a complex structure. The principal aim of this study is to drive the ball position to the desired reference in the presence of a complex structure, parametric uncertainties, and time-varying disturbances. The designed nonlinear controller is based on the robust backstepping technique, in which the robustness is provided via nonlinear damping terms. The boundedness of the tracking error is guaranteed with this method. In order to eliminate steady-state position error caused by the uncertainties and unmodeled dynamics, an integral term is added to the controller structure. After designing the proposed nonlinear controller, the overall closed-loop stability is accordingly analyzed with a Lyapunov-like function. Simulation studies are performed and the results are presented to test the success and the performance of the proposed controller.

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1. L. Yan, “Development and application of the Maglev transportation system,” IEEE Trans. Appl. Supercond., vol. 18, no. 2, pp. 92–99, 2008.
2. P. Samanta and H. Hirani, “Magnetic bearing configurations: Theoretical and experimental studies,” IEEE Trans. Magn., vol. 44, no. 2, pp. 292–300, 2008.
3. M. Tsuda et al., “Vibration transmission characteristics against vertical vibration in magnetic levitation type hts seismic/vibration isolation device,” IEEE Trans. Appl. Supercond., vol. 19, no. 3, pp. 2249–2252, 2009.
4. Z.-J. Yang, K. Miyazaki, S. Kanae, and K. Wada, “Robust position control of a magnetic levitation system via dynamic surface control technique,” IEEE Trans. Ind. Electron., vol. 51, no. 1, pp. 26–34, 2004.
5. A. El Hajjaji and M. Ouladsine, “Modeling and nonlinear control of magnetic levitation systems,” IEEE Trans. Ind. Electron., vol. 48, no. 4, pp. 831–838, 2001.
6. N. I. Mahmoud, “A backstepping design of a control system for a magnetic levitation system,” M.Sc. thesis, Dept. Electr. Eng., Linköping Univ., Linköping, Sweden, 2003.
7. A. V. Duka, “Modelling of an electromagnetic levitation system using a neural network,” in IEEE Int. Conf. Autom. Qual. Test. Robot. (AQTR), 2010, pp. 1–5.
8. Y. Qin, H. Peng, W. Ruan, J. Wu, and J. Gao, “A modeling and control approach to magnetic levitation system based on state-dependent arx model,” J. Process Control, vol. 24, no. 1, pp. 93–112, 2014.
9. M. Golob and B. Tovornik, “Modeling and control of the magnetic suspension system,” ISA Trans., vol. 42, no. 1, pp. 89–100, 2003.
10. A. Javadi and S. Pezeshki, “A new model-free adaptive controller versus non-linear H1 controller for levitation of an electromagnetic system,” Trans. Inst. Meas. Control, vol. 35, no. 3, pp. 321–329, 2013.
11. J. de Jesús Rubio, L. Zhang, E. Lughofer, P. Cruz, A. Alsaedi, and T. Hayat, “Modeling and control with neural networks for a magnetic levitation system,” Neurocomputing, vol. 227, pp. 113–121, 2017.
12. Z. J. Yang and M. Minashima, “Robust nonlinear control of a feedback linearizable voltage-controlled magnetic levitation system,” IEEJ Trans. EIS, vol. 121, no. 7, pp. 1203–1211, 2001.
13. C. Bonivento, L. Gentili, and L. Marconi, “Balanced robust regulation of a magnetic levitation system,” IEEE Trans. Control. Syst. Technol., vol. 13, no. 6, pp. 1036–1044, 2005.
14. H. J. Shieh, J. H. Siao, and Y. C. Liu, “A robust optimal sliding-mode control approach for magnetic levitation systems,” Asian J. Control, vol. 12, no. 4, pp. 480–487, 2010.
15. R. Morales, V. Feliu, and H. Sira-Ramírez, “Nonlinear control for magnetic levitation systems based on fast online algebraic identification of the input gain,” IEEE Trans. Control Syst. Technol., vol. 19, no. 4, pp. 757–771, 2011.
16. F. Adıgüzel, E. Dokumacılar, O. Akbatı, and T. Türker, “Design and implementation of an adaptive backstepping controller for a magnetic levitation system,” Trans. Inst. Meas. Control, vol. 40, no. 8, pp. 2466–2475, 2018.
17. C. M. Lin, M. H. Lin, and C. W. Chen, “SoPC-based adaptive PID control system design for magnetic levitation system,” IEEE Syst. J., vol. 5, no. 2, pp. 278–287, 2011.
18. T. Bächle, S. Hentzelt, and K. Graichen, “Nonlinear model predictive control of a magnetic levitation system,” Control Eng. Pract., vol. 21, no. 9, pp. 1250–1258, 2013.
19. K. Wada, S. Kanae, Y. Fukushima, and Z. -J. Yang, “Robust non-linear output-feedback control of a magnetic levitation system by K-filter approach,” IET Control Theor. Appl., vol. 3, no. 7, pp. 852–864, 2009.
20. R. Morales and H. Sira-Ramirez, “Trajectory tracking for the magnetic ball levitation system via exact feed-forward linearisation and GPI control,” Int. J. Control, vol. 83, no. 6, pp. 1155–1166, 2010.
21. T. Glück, W. Kemmetmüller, C. Tump, and A. Kugi, “A novel robust position estimator for self-sensing magnetic levitation systems based on least squares identification” Control Eng. Pract., vol. 19, no. 2, pp. 146–157, 2011.
22. J. Baranowski and P. Piatek, “Observer-based feedback for the magnetic levitation system,” Trans. Inst. Meas. Control, vol. 34, no. 4, pp. 422–435, 2012.
23. P. Pranayanuntana and V. Riewruja, “Nonlinear backstepping control design applied to magnetic ball control,” in Proc. IEEE TENCON 2000, 2000, vol. 3 pp. 304–307.
24. I. B. Furtat and E. A. Tupichin, “Control of nonlinear plant based on modified robust backstepping algorithm,”, in IEEE Conf. Control Appl. (CCA), 2014, pp. 941–946.
25. Z.-J. Yang and M. Tateishi, “Robust nonlinear control of a magnetic levitation system via backstepping approach,” in SICE’98 – Proc. 37th SICE Ann. Conf., International Session Papers, 1998, pp. 1063–1066.
26. A. S. Al-Araji, “Cognitive non-linear controller design for magnetic levitation system,” Trans. Inst. Meas. Control, vol. 38, no. 2, pp. 215–222, 2016.
27. Z.-J. Yang, K. Kunitoshi, S. Kanae, and K. Wada, “Adaptive robust outputfeedback control of a magnetic levitation system by K-filter approach,” IEEE Trans. Ind. Electron., vol. 55, no. 1, pp. 390–399, 2008.
28. L. Ting, J. Nan, and J. Yuanwei, “Nonlinear large disturbance attenuation controller design based on backstepping method,” in 25th Chinese Control Decision Conf. (CCDC), 2013, pp. 226–230.
29. F. Adıgüzel and T. Türker, “Control of a magnetic levitation system via robust backstepping controller,” in Otomatik Kontrol Türk Milli Komitesi Konferansı (TOK’17), Istanbul, 2017, pp. 79–84.
30. M. Krstic, I. Kanellakopoulos, and P. V. Kokotovic, Nonlinear and Adaptive Control Design. Chichester, England: Wiley, 1995.