Singular Port-Hamiltonian Systems Beyond Passivity
This work provides new insights into the energetic properties of port-Hamiltonian systems, addressing a theoretical gap in control theory for researchers in dynamical systems and nonlinear control.
The paper tackles the problem of port-Hamiltonian systems with singular vector fields, showing that their interconnection with passive systems can achieve convergence to a non-equilibrium steady state, resolving an apparent paradox by demonstrating that singularities induce sliding modes that supply energy, making the system non-passive.
In this paper, we investigate a class of port-Hamiltonian systems with singular vector fields. We show that, under suitable conditions, their interconnection with passive systems ensures convergence to a prescribed non-equilibrium steady state. At first glance, this behavior appears to contradict the seemingly passive structure of port-Hamiltonian systems, since sustaining a non-equilibrium steady state requires continuous power injection. We resolve this apparent paradox by showing that the singularity in the vector field induces a sliding mode that contributes effective energy, enabling maintenance of the steady state and demonstrating that the system is not passive. Furthermore, we consider regularizations of the singular dynamics and show that the resulting systems are cyclo-passive, while still capable of supplying the required steady-state power. These results clarify the role of singularities in port-Hamiltonian systems and provide new insight into their energetic properties.