Riemannian optimal reduction for linear systems with quadratic outputs

TL;DR

This paper introduces a Riemannian optimization-based H2-optimal model reduction method for linear systems with quadratic outputs, directly optimizing coefficient matrices on a product manifold.

Contribution

It formulates the model reduction as a Riemannian optimization problem on a product manifold, reducing variables and improving accuracy over projection-based methods.

Findings

Reduced models closely approximate original systems.

The method achieves superior H2 error performance.

Numerical simulations confirm effectiveness of the approach.

Abstract

This paper presents an H2-optimal model order reduction (MOR) method for linear systems with quadratic outputs based on Riemannian optimization. The H2-optimal MOR is formulated as an optimization problem in which the optimization variables are selected directly as the coefficient matrices of reduced models. The product manifold is defined properly to impose the stability condition for reduced models. By exploiting the geometric properties of the product manifold, we derive an explicit formula for Riemannian gradient of the objective function, and then a limited-memory Riemannian BFGS method is adopted to solve the resulting optimization problem iteratively. In contrast to selecting projection matrices, optimizing coefficient matrices of reduced models reduces the amount of variables dramatically. Numerical simulation results demonstrate that reduced models accurately approximate the original system and exhibit superior performance in terms of H2 error, which confirms the effectiveness of the proposed algorithm.