Singular Port-Hamiltonian Systems Beyond Passivity

TL;DR

This paper explores singular port-Hamiltonian systems, revealing how singularities enable non-passive behavior and steady-state energy supply, challenging traditional passivity assumptions.

Contribution

It demonstrates that singularities induce sliding modes that supply energy, allowing non-equilibrium steady states in port-Hamiltonian systems, and introduces cyclo-passivity in regularized models.

Findings

Singular port-Hamiltonian systems can sustain non-equilibrium steady states.

Singularities induce sliding modes that contribute effective energy.

Regularized systems are cyclo-passive but still supply steady-state power.

Abstract

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.