Discrete-time port-Hamiltonian systems: A definition based on symplectic integration
Provides a rigorous definition for discrete-time port-Hamiltonian systems, benefiting researchers in structure-preserving simulation of open physical systems.
This paper defines discrete-time port-Hamiltonian systems via structure-preserving discretization using symplectic integration, preserving a discrete-time energy balance. Numerical experiments confirm that energy approximation errors match the order of the integration scheme.
We introduce a new definition of discrete-time port-Hamiltonian systems (PHS), which results from structure-preserving discretization of explicit PHS in time. We discretize the underlying continuous-time Dirac structure with the collocation method and add discrete-time dynamics by the use of symplectic numerical integration schemes. The conservation of a discrete-time energy balance - expressed in terms of the discrete-time Dirac structure - extends the notion of symplecticity of geometric integration schemes to open systems. We discuss the energy approximation errors in the context of the presented definition and show that their order is consistent with the order of the numerical integration scheme. Implicit Gauss-Legendre methods and Lobatto IIIA/IIIB pairs for partitioned systems are examples for integration schemes that are covered by our definition. The statements on the numerical energy errors are illustrated by elementary numerical experiments.