Stabilization of the Acrobot system using the IDA-PBC approach

From a theorical point of view, underactuated mechanical systems can be controlled by means of the so called IDA-PBC method. In this method, in order to achieve the control objective, the stabilization mechanism follows two basic stages: (1) energy holding stage, which consists on shaping the total energy function of the system in order to assign the desired equilibrium state, and (2) damping introduction stage, necessary to achieve asymptotic stability. In order to mantain an energy approach of the stabilization process, it is necessary to obtain a port controlled Hamiltonian form for the closed loop system. However, the theoretical development faces practical dificulties when applied to real systems, due to the difficult solution to the partial differential equations envolved. The Acrobot system is a prototype of a underactuated mechanical system widely studied by the non linear control community. In this paper, a controller is designed taking into account the port controlled Hamiltonian approach based on the total energy of the system, considered as the sum of kinetic and potencial energies. The controller stabilizes globally and asymptotically the equilibrium point, showing an excellent preformance. The numerical simulations confirm this appreciation.