Riemannian optimization for model order reduction of linear systems with quadratic outputs

Abstract

This paper investigates the optimal $H_2$ model order reduction for linear systems with quadratic outputs. In the framework of Galerkin projection, we first formulate the optimal $H_2$ MOR as an unconstrained Riemannian optimization problem on the Stiefel manifold. The Riemannian gradient of the specific cost function is derived with the aid of Gramians of systems, and the Dai-Yuan-type Riemannian conjugate gradient method is adopted to generate structure-preserving reduced models. We also consider the optimal $H_2$ MOR based on the product manifold, where some coefficient matrices of reduced models are determined directly via the iteration of optimization problem, instead of the Galerkin projection method. In addition, we provide a scheme to compute low-rank approximate solutions of Sylvester equations based on the truncated polynomial expansions, which fully exploits the specific structure of Sylvester equations in the optimization problems, and enables an efficient execution of our approach. Finally, two numerical examples are simulated to demonstrate the efficiency of our methods.