Structure-preserving Model Reduction for Nonlinear Port-Hamiltonian Systems

TL;DR

This paper introduces a structure-preserving model reduction method for large-scale nonlinear port-Hamiltonian systems, ensuring stability and passivity while providing error bounds and efficient computation techniques.

Contribution

It develops a novel reduction approach that maintains port-Hamiltonian structure in nonlinear systems, combining basis construction methods and a modified DEIM for efficiency.

Findings

The reduced models retain stability and passivity.

Error bounds are established for state and output accuracy.

The method is validated on a nonlinear ladder network and a tethered Toda lattice.

Abstract

This paper presents a structure-preserving model reduction approach applicable to large-scale, nonlinear port-Hamiltonian systems. Structure preservation in the reduction step ensures the retention of port-Hamiltonian structure which, in turn, assures the stability and passivity of the reduced model. Our analysis provides a priori error bounds for both state variables and outputs. Three techniques are considered for constructing bases needed for the reduction: one that utilizes proper orthogonal decompositions; one that utilizes $\mathcal{H}_2/\mathcal{H}_{\infty}$-derived optimized bases; and one that is a mixture of the two. The complexity of evaluating the reduced nonlinear term is managed efficiently using a modification of the discrete empirical interpolation method (DEIM) that also preserves port-Hamiltonian structure. The efficiency and accuracy of this model reduction framework are illustrated with two examples: a nonlinear ladder network and a tethered Toda lattice.