Error bounds for port-Hamiltonian model and controller reduction based on system balancing

TL;DR

This paper develops error bounds for port-Hamiltonian model and controller reduction using system balancing, ensuring stability and passivity, and introduces an optimal pH representation to minimize error.

Contribution

It proposes a novel approach to port-Hamiltonian controller reduction with explicit error bounds and an optimal Hamiltonian representation for minimal error.

Findings

The method guarantees stability and passivity of reduced controllers.

An optimal pH representation minimizes the error bound.

Numerical examples validate the theoretical error bounds.

Abstract

We study linear quadratic Gaussian (LQG) control design for linear port-Hamiltonian systems. To this end, we exploit the freedom in choosing the weighting matrices and propose a specific choice which leads to an LQG controller which is port-Hamiltonian and, thus, in particular stable and passive. Furthermore, we construct a reduced-order controller via balancing and subsequent truncation. This approach is closely related to classical LQG balanced truncation and shares a similar a priori error bound with respect to the gap metric. By exploiting the non-uniqueness of the Hamiltonian, we are able to determine an optimal pH representation of the full-order system in the sense that the error bound is minimized. In addition, we discuss consequences for pH-preserving balanced truncation model reduction which results in two different classical H-infinity-error bounds. Finally, we illustrate the theoretical findings by means of two numerical examples.