Structure-preserving $H^2$ optimal model reduction based on Riemannian trust-region method

TL;DR

This paper introduces a Riemannian trust-region method for structure-preserving $H^2$ optimal model reduction of linear systems, ensuring stability and symmetry are maintained, with demonstrated advantages over traditional balanced truncation.

Contribution

It formulates the model reduction as a nonlinear optimization on a product manifold and derives efficient gradient and Hessian computations, especially for gradient systems.

Findings

The proposed method preserves system structure in reduced models.

Solutions to the optimization problem are generally non-unique.

Numerical experiments confirm the method's effectiveness in structure preservation.

Abstract

This paper studies stability and symmetry preserving $H^2$ optimal model reduction problems of linear systems which include linear gradient systems as a special case. The problem is formulated as a nonlinear optimization problem on the product manifold of the manifold of symmetric positive definite matrices and the Euclidean spaces. To solve the problem by using the trust-region method, the gradient and Hessian of the objective function are derived. Furthermore, it is shown that if we restrict our systems to gradient systems, the gradient and Hessian can be obtained more efficiently. More concretely, by symmetry, we can reduce linear matrix equations to be solved. In addition, by a simple example, we show that the solutions to our problem and a similar problem in some literatures are not unique and the solution sets of both problems do not contain each other in general. Also, it is revealed that the attained optimal values do not coincide. Numerical experiments show that the proposed method gives a reduced system with the same structure with the original system although the balanced truncation method does not.