Stabilization of Underactuated Mechanical Systems via Interconnection and Damping Assignment

Passivity{based control (PBC) is a well{established structure{preserving design methodology which has shown to be very powerful to design robust controllers for physical systems described by Euler{Lagrange equations of motion. For regulation problems of mechanical systems it provides a natural procedure to \shape" the potential energy yielding controllers with a clear physical interpretation. It is well known, however, that to stabilize some underactuated devices it is necessary to modify also the kinetic energy. On the other hand, to be able to apply passivity ideas to electrome-chanical and mass{balance applications, which require shaping of the total energy (or mass), we have introduced in a series of recent papers a new so{called interconnection and damping assignment (IDA) PBC. The present paper is concerned with the application of IDA{PBC to the problem of stabilization of underactuated mechanical systems. Our main contribution is the characterization of a class of systems for which IDA{PBC yields a smooth stabilizing controller. The class is given in terms of solvability of a partial diierential equation. As an illustration we present a dynamic nonlinear output feedback IDA{PBC which \globally" stabilizes (except for a set of initial conditions that has zero measure) the upward position of a novel inverted pendulum. That is, we prove that it is possible to swing up this pendulum without switching nor measurement of velocities.