The inverted pendulum: a benchmark in nonlinear control

This paper aims to present a complete solution to the problem of swinging up and stabilizing control of the invested pendulum with a single control law. Currently, both problems and solved independently and a hybrid controller merges both solutions. In this paper, a new approach to the problem is introduced which loads to a family of feedback laws producing the desired behavior. The laws resulting from the approach have two parts: first, an energy-shaping law is able to swing the pendulum up and to maintain it. It is based on choosing an appropriate desired energy function and a camping term that changes signs, giving rise alternatively to damping and to energy injection depending on the need. Once the problem has been solved for the simple pendulum, the second part introduces additional control to stop the cart and it is based on forwarding control with bounded input. The resulting control law is the sum of both parts and does not commute between different laws. The last part of the paper is devoted to the problem of designing controllers for generating autonomous oscillations in the inverted pendulum.