New Results of Energy Based Swing up Control for a Rotational Pendulum
In this paper, we revisit the energy based swing up control problem for a rotational pendulum. Different from the existing energy based control solution, first, we present a necessary and sufficient condition for avoiding singular points in the control law and carry out a global motion analysis of the pendulum. Next, we remove the previous required constraint on the initial state of the pendulum and the control parameters for preventing the pendulum getting stuck at the downward equilibrium point by revealing that the point is saddle (hyperbolic and unstable) Specifically, we show that the Jacobian matrix evaluated at the point has two and two eigenvalues in the open leftand right half planes, respectively. We prove that the pendulum will eventually be swung up into the basin of attraction of any locally stabilizing controller for all initial conditions with the exception of a set of Lebesgue measure zero. Finally, we validate the presented theoretical results via numerical simulation of two rotational pendulums. Our simulation results show that the swing up control can be achieved quickly under the improved conditions on the control parameters.