The Optimal Erection of the Inverted Pendulum

The erection of the inverted pendulum is a classic control problem, which has appeared in several variants. One of the most challenging is the minimum-time erection of a pendulum that is mounted on a moving cart. The aim is to erect the pendulum from the ‘straight-down’ (stable equilibrium) to a ‘straight-up’ (unstable equilibrium) position in minimum time. The swing-up maneuver is usually addressed using a pre-defined control strategy, e.g., energy-based control or selecting the switching times in a bang-bang structure. The aim of this paper is to show that the minimum-time solution may have a singular arc, with the optimal control taking a bang-singular-bang form. The singular arc segment of the control law is a feedback law that is derived herein with the solution discussed. A sensitivity analysis of the solution structure is also performed by varying the model parameters. Finally, the time-optimal solution is compared with that obtained using an energy-based control strategy.