Optimal time control to swing-up the inverted pendulum-cart in open-loop form

This work deals with simulation on an Inverted Pendulum (IP). The control strategy of an IP is split into two main control phases: (i) swing-up control to bring back the pendulum from the downward position to the upward one, and (ii) upright stabilization control to maintain the pendulum to the upright vertical position. In the case (ii), a feedback or a neuro-fuzzy controller is used to stabilize the pendulum cart, while in the first case (i), a non-linear controller based on the energy of the pendulum is used in order to reach the desired performance with a minimum number of swings. Our contribution is to present a simulation using MatLab of time-optimal control system for swinging-up the pendulum, with a single control law in an open-loop form. From the bang-bang structure of the time-optimal control resulting from the necessary condition of the Pontryagin Maximum Principle, the solution obtained from direct discretization method is adjusted by using Newton based method.