On the stabilization of the ball and beam system using a direct Lyapunov method

I present the stabilization of the ball on the actuated beam, whether the damping force acts or not in the non-actuated coordinate. The control law is based on the application of the traditional Lyapunov method. The balance strategy consists of shaping a convenient energy function that allow us to derive the required asymptotic stabilizing controller. To this end, we subtle introduce an auxiliary control variable in order to algebraically obtain the kinetic metric and the potential energy of the whole system. The formed positive energy function in conjunction with the proposed feedback controller assures that the whole system be a dissipative one.