Initiation and directional control of period-1 rotation for a parametric pendulum

We study a time-delayed feedback control for initiating period-1 rotations of a vertically excited parametric pendulum from arbitrary initial conditions. The possibility of controlling the direction of rotation has also been explored. We start with a simple linear time-delayed control for which the control gain corresponding to the most stable period-1 rotation has been obtained using the Floquet theory. This control increases the basins of attraction of rotations, but they do not encompass the full initial condition space. We modify our control law by using a switched control gain that destabilizes all the oscillatory solutions, and the entire initial condition space becomes the basin of attraction of either the clockwise or the anticlockwise rotation. By a suitable modification of the switching condition, we can choose a preferential stable direction of rotation. Hence, we can initiate either clockwise or anticlockwise rotation for a parametric pendulum from arbitrary initial conditions. Performance of our controller in achieving this objective has been demonstrated for different sets of parameters to establish its effectiveness.