Guess-Free Pseudospectral Solution for Time Optimal Swing-Up of a Rotary Inverted Pendulum

Control of the inverted pendulum is a canonical problem in nonlinear and optimal control. Over the years, many workers have developed solutions for inverting the pendulum link (swing-up phase) and for maintaining the pendulum link upright (stabilization/disturbance rejection). In this paper, the time-optimal swing-up of a rotary inverted pendulum is studied. Previous solutions to this problem have required that the original time-optimal problem formulation be transformed to a more computationally tractable form. For example, one transformation is to a fixed-time problem with bounds on the control. Other approaches involve guessing the switching structure in order to construct a candidate solution. Advances in computational optimal control theory, particularly pseudospectral optimal control, allow the original time-optimal problem to be solved directly, and without the need for a guess. One such solution is presented in this paper. It is shown that the result adheres to the conditions of Pontryagin’s minimum principle. An experimental implementation of the solution illustrates its feasibility in practice.