Passivation of Underactuated Systems with Physical Damping

In recent works, IDA-PBC has been succesfully applied to mechanical control problems with no physical damping present. In some cases, the friction terms can be obviated without compromising stability in closed loop. However in methods that modify the kinetic energy, a controller designed for stabilizing the undamped system might loose passivity, a key property for stabilization, when damping is introduced. This paper presents a necessary and sufficient condition, namely the dissipation condition, for recovering passivity (and hence stability) in such cases. If the dissipation condition is fulfilled, an IDA-PBC redesign is necessary in general, and with this goal two different methods for passivating the damped system are presented.