$\mathcal{H}_\infty$ model order reduction for quadratic output systems

TL;DR

This paper introduces an $ ext{H}_ olinebreak_ olinebreak ext{infty}$-norm for LTI quadratic output systems and develops an algorithm to produce reduced models that closely approximate the original in this norm, applicable to various nonlinear system classes.

Contribution

It proposes a novel $ ext{H}_ olinebreak_ olinebreak ext{infty}$-norm for LTI quadratic output systems and an algorithm for $ ext{H}_ olinebreak_ olinebreak ext{infty}$-optimal model reduction, extending existing methods to nonlinear regimes.

Findings

The algorithm effectively produces reduced models with small $ ext{H}_ olinebreak_ olinebreak ext{infty}$-norm error.

Compared to balanced truncation and $ ext{H}_2$-methods, the new approach yields competitive or improved results.

Numerical examples demonstrate the applicability to systems like port-Hamiltonian and stochastic models.

Abstract

Linear time-invariant quadratic output (LTIQO) systems generalize linear time-invariant systems to nonlinear regimes. Problems of this class occur in multiple applications naturally, such as port-Hamiltonian systems, optimal control, and stochastical problems. We introduce an $\mathcal{H}_\infty$-norm for LTIQO systems with one or multiple outputs and propose an algorithm to optimize a reduced order model (ROM) to be close in the $\mathcal{H}_\infty$-norm to a given full order model. We illustrate the applicability and the performance with an established numerical example and compare the resulting ROMs with results from balanced truncation and $\mathcal{H}_2$-focussed algorithms.