In this brief the dynamic behavior of a parametrically forced manipulator, or pendulum, system with PD control is examined. For an excitation of sufficient amplitude or frequency a Hopf bifurcation to a steady-state limit cycle is shown to result, appearing as a precursor to instability. The parameter space is mapped in order to illustrate regions where control failure will likely occur, even in the strongly damped case. For weakly damped systems, the Hopf bifurcation can additionally exhibit a dependence on initial conditions. The resulting case of competing point and periodic attractors is discussed.
Hopf Bifurcation in PD Controlled Pendulum or Manipulator
Contributed by the Dynamic Systems and Control Division of THE AMERICAN SOCIETY OF MECHANICAL ENGINEERS. Manuscript Received by the Dynamics Systems and Control Division April 13, 2000. Associate Editor: C. Rahn.
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Bucklaew, T., and Liu, C. (May 10, 2002). "Hopf Bifurcation in PD Controlled Pendulum or Manipulator." ASME. J. Dyn. Sys., Meas., Control. June 2002; 124(2): 327–332. https://doi.org/10.1115/1.1455025
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