Abstract
A nonlinear control force is presented to stabilize the under-actuated inverted pendulum mounted on a cart. The control strategy is based on partial feedback linearization, in a first stage, to linearize only the actuated coordinate of the inverted pendulum, and then, a suitable Lyapunov function is formed to obtain a stabilizing feedback controller. The obtained closed-loop system is locally asymptotically stable around its unstable equilibrium point. Additionally, it has a very large attraction domain.
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References
Lozano, R., Fantoni, I., and Block, D. J., ‘Stabilization on the inverted pendulum around its homoclinic orbit’, System & Control Letters (40) 3, 2000, 197–204.
Olfati-Saber, R., ‘Fixed point controllers and stabilization of the cart-pole and rotating pendulum’, in Proceedings of the 38th Conference on Decision and Control, Phoenix, AZ, December 7–10, 1999, pp. 1174–1181.
Bloch, A. M., Leonard, N. E., and Marsden, J. E., ‘Controlled lagrangians and the stabilization of mechanical systems: The first matching theorem’, IEEE Transactions on Systems and Control 45, 2001, 2253–2270.
Furuta, K., Yamakita, M., and Kobayashi, S., ‘Swing up control of inverted pendulum using pseudo-state feedback’, Journal of System and Control Engineering 206, 2001, 263–269.
Asrom, K. J. and Furuta, K., ‘Swinging up a pendulum by energy control’, Automatica 36(2), 2000, 287–295.
Chung, C. C. and Hauser, J., ‘Nonlinear control of a swinging pendulum’, Automatica 31(6), 1995, 851–862.
Jakubczyk, B. and Respondek, W., ‘On the linearization of control systems’, Bulletin of the Academy of Poland Science and Mathematics 28, 1980, 517–522.
Shiriaev, A. S. and Fradkov, A. L., ‘Stabilization of invariant sets of nonlinear systems with applications to control of oscillations’, International Journal of Robust and Nonlinear Control 11, 2001, 215–240.
Kwakernaak, H. and Sivan R., Linear Optimal Control Systems, Wiley, New York, 1972.
Spong, M. W. and Praly, L., ‘Control of underactuated mechanical systems using switching and saturation’, in Proceedings of the Block Island Workshop on Control Using Logic Based Switching, Springer-Verlag, Berlin, 1996.
Spong, M. W., ‘Energy-based control of a class of underactuated mechanical systems’, in IFAC World Congress, San Francisco, CA, July, 1996.
Olfati-Saber, R., ‘Nonlinear control of underactuated mechanical systems with applications to robotic and aerospace vehicles’, PhD Thesis, Department of Electrical Engineering and Computer Science of the Massachusetts Institute of Technology, Cambridge, MA, 2001.
Fantoni, I. and Lozano, R., Nonlinear Control for Underactuated Mechanical System, Springer-Verlag, London, 2002.
Sepulchre, R., Jankovíc, M., and Kokotovíc, P., Constructive Nonlinear Control, Springer-Verlag, London, 1997.
Khalil, H. K., Nonlinear Systems, 2nd edn., Prentice-Hall, Englewoods Cliffs, NJ, 1996.
Bloch, A. M., Leonard, N. E., and Marsden, J. E., ‘Potential shaping and the method of controlled lagrangians’, in Proceedings of the 38th Conference on Decision and Control, Phoenix, AZ, December 7–10, 1999, pp. 1652–1657.
Mazenc, F. and Praly, L., ‘Adding integrations, saturated controls, and stabilization for feedforward systems’, IEEE Transactions on Automation Control 41, 1996, 1559–1578.
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Contributed by Prof. F. Pfeiffer.
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Ibañez, C.A., Frias, O.G. & Castañón, M.S. Lyapunov-Based Controller for the Inverted Pendulum Cart System. Nonlinear Dyn 40, 367–374 (2005). https://doi.org/10.1007/s11071-005-7290-y
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DOI: https://doi.org/10.1007/s11071-005-7290-y