# Structure-preserving discretization for port-Hamiltonian descriptor systems

## TL;DR

This paper extends port-Hamiltonian descriptor systems to more general forms, explores structure-preserving discretization methods, and demonstrates their effectiveness through numerical examples, enhancing modeling and control capabilities.

## Contribution

It introduces a generalized framework for port-Hamiltonian systems, including discretization techniques that preserve their energy structure and stability properties.

## Key findings

- Structure-preserving discretization schemes maintain energy properties.
- Port-Hamiltonian systems are invariant under nonlinear transformations.
- Numerical examples confirm the effectiveness of the proposed methods.

## Abstract

We extend the modeling framework of port-Hamiltonian descriptor systems to include under- and over-determined systems and arbitrary differentiable Hamiltonian functions. This structure is associated with a Dirac structure that encloses its energy balance properties. In particular, port-Hamiltonian systems are naturally passive and Lyapunov stable, because the Hamiltonian defines a Lyapunov function. The explicit representation of input and dissipation in the structure make these systems particularly suitable for output feedback control. It is shown that this structure is invariant under a wide class of nonlinear transformations, and that it can be naturally modularized, making it adequate for automated modeling. We investigate then the application of time-discretization schemes to these systems and we show that, under certain assumptions on the Hamiltonian, structure preservation is achieved for some methods. Numerical examples are provided.