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Dynamics of Simple Balancing Models with Time-Delayed Switching Feedback Control

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Abstract

Time-delayed control in a balancing problem may be a nonsmooth function for a variety of reasons. In this paper we study a simple model of the control of an inverted pendulum by either a connected movable cart or an applied torque for which the control is turned off when the state of the pendulum is located within certain regions of phase space. Without applying a small angle approximation for deviations about the vertical position, we see structurally stable periodic orbits which may be attracting or repelling. Due to the nonsmooth nature of the control, these periodic orbits originate in various discontinuity-induced bifurcations. We also show that a coincidence of switching events can produce complicated periodic and aperiodic solutions.

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Authors and Affiliations

  1. Department of Mathematics, University of British Columbia, Vancouver, BC, V6T1Z2, Canada

    D. J. W. Simpson, R. Kuske & Y.-X. Li

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  1. D. J. W. Simpson
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  2. R. Kuske
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  3. Y.-X. Li
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Correspondence to D. J. W. Simpson.

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Communicated by Sue Ann Campbell.

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Simpson, D.J.W., Kuske, R. & Li, YX. Dynamics of Simple Balancing Models with Time-Delayed Switching Feedback Control. J Nonlinear Sci 22, 135–167 (2012). https://doi.org/10.1007/s00332-011-9111-4

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  • DOI: https://doi.org/10.1007/s00332-011-9111-4

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