Frequency-limited H$_2$ Model Order Reduction Based on Relative Error

TL;DR

This paper introduces a novel frequency-limited H2 model order reduction method based on relative error, which avoids large-scale Lyapunov equations by solving Sylvester equations, leading to efficient computation and improved controller stability.

Contribution

The paper derives optimality conditions for frequency-limited H2 relative error reduction and proposes an efficient oblique projection algorithm that outperforms existing methods in controller design.

Findings

The proposed algorithm achieves smaller relative errors within the frequency range.

Reduced-order controllers from the new method ensure better robust stability.

Numerical results show superior performance compared to existing techniques.

Abstract

Frequency-limited model order reduction aims to approximate a high-order model with a reduced-order model that maintains high fidelity within a specific frequency range. Beyond this range, a decrease in accuracy is acceptable due to the nature of the problem. The quality of the reduced-order model is typically evaluated using absolute or relative measures of approximation error. Relative error, which represents the percentage error, becomes particularly relevant when reducing a plant model for the purpose of designing a reduced-order controller. This paper derives the necessary conditions for achieving a local optimum of the frequency-limited H2 norm for the relative error system. Based on these optimality conditions, an oblique projection algorithm is proposed to ensure a small relative error within the desired frequency interval. Unlike existing algorithms, the proposed approach does not necessitate solving large-scale Lyapunov and Ricatti equations. Instead, the proposed algorithm relies on solving sparse-dense Sylvester equations, which typically emerge in the majority of H2 model order reduction algorithms, but can be efficiently solved. To evaluate the performance of the proposed algorithm, a comparison is conducted with three existing techniques: frequency-limited balanced truncation, frequency-limited balanced stochastic truncation, and frequency-limited iterative Rational Krylov algorithm. The comparative analysis focuses on designing reduced-order controllers for high-order plants. Numerical results confirm that the reduced-order controllers obtained using the proposed algorithm ensure superior robust closed-loop stability.