H2 optimal model reduction of linear dynamical systems with quadratic output by the Riemannian BFGS method

Abstract

This paper considers the $H_2$ optimal model reduction problem of linear dynamical systems with quadratic output on the Riemannian manifolds. A one-sided projection is used to reduce the state equation, while a suitable symmetric matrix is chosen to determine the output equation of the reduced system. Since the projection matrix is an orthonormal matrix, it can be seen as a point on the Stiefel manifold. Because symmetric matrices of the same dimension allow a manifold structure, it is used to define a product manifold combined with the Stiefel manifold. The $H_2$ error between the original system and the reduced system is treated as a function defined on the product manifold. Then, the $H_2$ optimal model reduction problem is formulated as an unconstrained optimization problem defined on the product manifold. Concerning the symmetric matrix, the $H_2$ error is proved to be convex. In terms of the orthonormal matrix and the symmetric matrix, the gradients of the $H_2$ error are derived respectively. Then, the Riemannian BFGS method is used to obtain the orthonormal matrix, and the symmetric matrix is calculated by the convexity and the related gradient. By introducing the Riemannian manifolds to the $H_2$ optimal model reduction problem, the constrained optimization problem in the Euclidean space is transformed into an unconstrained optimization problem on the manifolds, and the gradients of the $H_2$ error are equipped with relatively concise formulas. Finally, numerical results illustrate the performance of the proposed model reduction method.