$\mathcal{H}_2$-optimal Model Reduction of Linear Quadratic Output Systems in Finite Frequency Range

TL;DR

This paper introduces a novel $\\mathcal{H}_2$-optimal model reduction method for linear quadratic output systems within a finite frequency range, featuring an efficient algorithm that avoids complex matrix logarithms and is validated on large-scale systems.

Contribution

It derives a new frequency-limited $\mathcal{H}_2$ norm, establishes optimality conditions, and proposes an efficient stationary point iteration algorithm for model reduction within specified frequency ranges.

Findings

The algorithm effectively approximates high-order models within the desired frequency range.

It avoids computationally intensive matrix logarithm calculations.

Successfully reduces a system of order one million, demonstrating high efficiency.

Abstract

In frequency-limited model order reduction, the objective is to maintain the frequency response of the original system within a specified frequency range in the reduced-order model. In this paper, a mathematical expression for the frequency-limited $\mathcal{H}_2$ norm is derived, which quantifies the error within the desired frequency interval. Subsequently, the necessary conditions for a local optimum of the frequency-limited $\mathcal{H}_2$ norm of the error are derived. The inherent difficulty in satisfying these conditions within a Petrov-Galerkin projection framework is also discussed. Using the optimality conditions and the Petrov-Galerkin projection, a stationary point iteration algorithm is proposed, which approximately satisfies these optimality conditions upon convergence. The main computational effort in the proposed algorithm involves solving sparse-dense Sylvester equations. These equations are frequently encountered in $\mathcal{H}_2$ model order reduction algorithms and can be solved efficiently. Moreover, the algorithm bypasses the requirement of matrix logarithm computation, which is typically necessary for most frequency-limited reduction methods and can be computationally demanding for high-order systems. An illustrative example is provided to numerically validate the developed theory. The proposed algorithm's effectiveness in accurately approximating the original high-order model within the specified frequency range is demonstrated through the reduction of an advection-diffusion equation-based model, commonly used in model reduction literature for testing algorithms. Additionally, the algorithm's computational efficiency is highlighted by successfully reducing a flexible space structure model of order one million.