On a generalized port-Hamiltonian representation for the control of damped underactuated mechanical systems

A well-known problem in controller design for underactuated mechanical systems using the Interconnection and Damping Assignment (IDA-PBC) technique is friction in unactuated degrees of freedom. For certain equilibria the definiteness requirements on the virtual energy of the port-Hamiltonian (pH) target system and the closed-loop dissipation matrix can not be satisfied simultaneously. In this contribution a modification of the pH target system is proposed, where particularly the total energy function is augmented by a cross term between coordinates and momenta. The approach stems from the fact that, although IDA-PBC may fail, unstable equilibria of underactuated mechanical systems are stabilized by linear state feedback, if the linearization is stabilizable. Then the solution of a Lyapunov equation for the linearized closed-loop system is not block diagonal, which gives rise to the proposed structure of the energy.