On the ball and beam problem: regulation with guaranteed transient performance and tracking periodic orbits

Despite the large number of controllers presented in the literature for the stabilization of the ball and beam system, only a few results are available on the transient performance problem. The aim of this paper is to propose an asymptotically stabilizing controller that ensures that, for a well–defined set of initial conditions, the ball remains on the bar during the transient. Tuning the controller, the set can be extended to include any initial position with zero velocity. The controller is a nonlinear static state feedback that is derived using the interconnection and damping assigment energy– shaping controller design methodology. As an extension of the regulation problem we also propose in this paper a controller that forces the ball and beam to oscillate. This is achieved, within the framework of energy–shaping, by assigning an energy function that attains its mimimum along the desired periodic orbit. A key step in our design is the immersion of the oscilator system into the fourth order system dynamics.