An Optimal Projection Framework for Structure-Preserving Model Reduction of Linear Systems

TL;DR

This paper introduces a structure-preserving model reduction method for linear systems that optimizes the $ ext{H}_2$ norm while maintaining key structural properties like dissipativity and passivity, using gradient-based algorithms.

Contribution

It formulates the model reduction as a nonconvex optimization on a Stiefel manifold and provides explicit gradient expressions and convergence analysis for the proposed algorithms.

Findings

Effective preservation of structural features in reduced models.

Demonstrated convergence of gradient descent algorithms.

Successful numerical examples on stability and passivity preservation.

Abstract

This paper presents a structure-preserving model reduction framework for linear systems, in which the $\mathcal{H}_2$ optimization is incorporated with the Petrov-Galerkin projection to preserve structural features of interest, including dissipativity, passivity, and bounded realness. The model reduction problem is formulated in a nonconvex optimization setting on a noncompact Stiefel manifold, aiming to minimize the $\mathcal{H}_2$ norm of the approximation error between the full-order and reduced-order models. The explicit expression for the gradient of the objective function is derived, and two gradient descent procedures are applied to seek for a (local) minimum, followed by a theoretical analysis on the convergence properties of the algorithms. Finally, the performance of the proposed method is demonstrated by two numerical examples which consider stability-preserving and passivity-preserving model reduction problems, respectively.