Hopf Bifurcation in PD Controlled Pendulum or Manipulator

Abstract

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.