Open-loop resonance-cancellation control for a base-excited pendulum

A new open-loop control strategy applied to a planar pendulum subjected to the most severe combination of base excitations, horizontal motion at the primary-resonance frequency, and vertical motion at the principal-parametric resonance frequency is developed. The control action is typical of many single-input control systems; the control authority in one direction (horizontal) is high, and the control authority in the orthogonal direction (vertical) is zero in a linear sense. Although the action of the controller is linearly orthogonal to part (vertical or parametric) of the system excitation, effects are transferred to this direction through nonlinear actuator action. Proper enhancement of the control input allows the nonlinear action to provide control authority over the parametric excitation. The dynamics at reduced orders, determined by a multiple-scales perturbation analysis, suggest the appropriate form for the control enhancement. The normal form of the system provides information about how each parameter of the enhancement affects the steady-state pendulation of the system. In this case, heuristic arguments are used to reduce the dimension of the unknown enhancement design parameters to a manageable size. The maximum pendulation angle of the steady-state motion of the system is one of the appropriate metrics for the system response; here it is used as the cost function for evaluation of the optimal enhancement gains. Because the size of the design parameter set has been reduced, a simple grid search is employed to find the optimal control. Relative to simple linear cancellation of the disturbance, the proposed approach reduces the response of the system by an order of magnitude for typical disturbance cases.