Singular Port-Hamiltonian Systems Beyond Passivity

Abstract: In this paper, we study a class of port-Hamiltonian systems whose vector fields exhibit singularities. A representative example of this class has recently been employed in the power electronics literature to implement a grid-forming controller. We show that, under certain conditions, these port-Hamiltonian systems, when interconnected with passive systems, converge to a prescribed non-equilibrium steady state. At first glance, the apparently passive nature of the port-Hamiltonian system seems incompatible with the active power injection required to sustain this non-equilibrium condition. However, we demonstrate that the discontinuity inherent in the vector field provides the additional energy needed to maintain this operating point, indicating that the system is not globally passive. Moreover, when the discontinuity is replaced by a continuous approximation, the resulting system becomes cyclo-dissipative while still capable of supplying the required power.