Balanced truncation for model order reduction of linear dynamical systems with quadratic outputs

TL;DR

This paper develops a balanced truncation method for reducing the complexity of linear dynamical systems with quadratic outputs by transforming them into quadratic-bilinear systems with a single output, simplifying the MOR process.

Contribution

It introduces a novel quadratic-bilinear system formulation for quadratic output systems, enabling efficient balanced truncation with fewer Lyapunov equations.

Findings

The new approach reduces computational complexity in MOR.

Numerical results demonstrate comparable accuracy to traditional methods.

The method effectively stabilizes the reduced systems.

Abstract

We investigate model order reduction (MOR) for linear dynamical systems, where a quadratic output is defined as a quantity of interest. The system can be transformed into a linear dynamical system with many linear outputs. MOR is feasible by the method of balanced truncation, but suffers from the large number of outputs in approximate methods. To ameliorate this shortcoming we derive an equivalent quadratic-bilinear system with a single linear output and analyze the properties of this system. We examine MOR for this system via the technique of balanced truncation, which requires a stabilization of the system. Therein, the solution of two quadratic Lyapunov equations is traced back to the solution of just two linear Lyapunov equations. We present numerical results for several test examples comparing the two MOR approaches.