Time Optimal Swing-Up Control of Single Pendulum1

Abstract

Swing-up of a rotating type pendulum from the pendant to the inverted state is known to be one of most difficult control problems, since the system is nonlinear, underactuated, and has uncontrollable states. This paper studies a time optimal swing-up control of the pendulum using bounded input. Time optimal control of a nonlinear system can be formulated by Pontryagin’s Maximum Principle, which is, however, hard to compute practically. In this paper, a new computational approach is presented to attain a numerical solution of the time optimal swing-up problem. Time optimal control problem is described as minimization of the achievable time to attain the terminal state under the bounded input amplitude, although algorithms to solve this problem are known to be complicated. Therefore, in this paper, it is shown how the optimal time swing-up control is formulated as an auxiliary problem in that the minimal input amplitude is searched so that the terminal state satisfies a specification at a given time. Through the proposed approach, time optimal control can be solved by nonlinear optimization. Its approach is evaluated by numerical simulations of a simplified pendulum model, is checked satisfying the necessary condition of Maximum Principle, and is experimentally verified using the rotating type pendulum.