Abstract
The dynamical behaviour of a simple underactuated mechanical system with a bounded continuous control law is analyzed. The system consists of a pendulum with an inertia disk mounted on its free extreme. It is driven applying torques to the inertia disk by means of a DC motor. The closed-loop system exhibits a rich and complex dynamic when a control parameter is varied. A numerical analysis reveals Hopf, fold and homoclinic bifurcations as the main phenomena. It is shown that the pendulum can be stabilized in its inverted position with zero velocity of the disk if the controller’s gains are appropriately chosen.
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D. M. Alonso, E. E. Paolini, and J. L. Moiola, “On anticontrol of Hopf bifurcations.” Proc. 39th IEEE Conference on Decision and Control, Sydney, Australia, pp. 2072–2077, Dec. 12–15, 2000.
J. Aracil, K. J. Åström and D. Pagano, “Global bifurcations in the Furuta pendulum.” Proc. IFAC Nonlinear Control System Design, Enschede, The Netherlands, pp. 37–41, 1998.
J. Aracil, E. Ponce and T. Álamo, “A frequency-domain approach to bifurcations in control systems with saturations.” Int. J. of Systems Science 31, pp. 1261–1271, 2000.
K. J. Åström and K. Furuta, “Swinging up a pendulum by energy control.” Proc. 13th IFAC World Congress, San Francisco, USA, 1996.
K. Furuta, M. Yamakita, and S. Kobayashi, “Swing-up control of inverted pendulum using pseudo-state feedback.” Proc. Instn. Mech. Engrs. 206, pp. 263–269, 1992.
P. Glendinning, J. Abshagen and T. Mullin, “Imperfect homoclinic bifurcations.” Preprint, available on line at: http://arxiv.org/abs/nlin/0103054, March 2001.
J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields. Applied Mathematical Sciences 42, Springer Verlag, New York, 1983.
E. A. Jackson, Perspectives of Nonlinear Dynamics 2. Cambridge University Press, 1990.
R. Lozano, I. Fantoni and D. Block, “Stabilization of the inverted pendulum around its homoclinic orbit.” Systems & Control Letters 40, pp. 197–204, 2000.
J. L. Moiola and G. Chen, Hopf Bifurcation Analysis. A Frequency Domain Approach. World Scientific Pub., Singapore, 1996.
M. G. Ortega, J. Aracil, F. Gordillo and F. Rubio, “Bifurcation analysis of a feedback system with dead zone and saturation.” IEEE Control Systems Magazine 20, pp. 91–101, Aug. 2000.
R. Ortega and M. Spong, “Stabilization of underactuated mechanical systems via interconnection and damping assignment.” IFAC Workshop on Lagrangian and Hamiltonian Methods in Nonlinear Systems, Princeton, NJ, March 16–18, 2000.
L. Praly, R. Ortega and G. Kaliora, “Stabilization of nonlinear systems via forwarding mod{L g V}.” 2nd NCN Workshop, Paris, June 5–9, 2000.
A. Shiriaev, A. Pogromsky, H. Ludvigsen and O. Egeland, “On global properties of passivity-based control of an inverted pendulum.” Int. J. Robust Nonlinear Control 10, pp. 283–300, 2000.
M. W. Spong, “The swing up control problem for the acrobot.” IEEE Control Systems Magazine 15, pp. 49–55, Feb. 1995.
T. L. Vincent, “Utilizing chaos in control system design.” In: G. Chen (Ed.) Controlling Chaos and Bifurcations in Engineering Systems, ch. 5, CRC Press, 1999.
Q. Wei, W. P. Dayawansa, and W. S. Levine, “Nonlinear controller for an inverted pendulum having restricted travel.” Automatica 316, pp. 841–850, 1995.
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Alonso, D.M., Paolini, E.E., Moiola, J.L. (2002). Controlling an Inverted Pendulum with Bounded Controls. In: Colonius, F., Grüne, L. (eds) Dynamics, Bifurcations, and Control. Lecture Notes in Control and Information Sciences, vol 273. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-45606-6_1
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DOI: https://doi.org/10.1007/3-540-45606-6_1
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