Gramians, Energy Functionals and Balanced Truncation for Linear Dynamical Systems with Quadratic Outputs

TL;DR

This paper introduces a novel balancing-based model order reduction method for large-scale linear systems with quadratic outputs, utilizing new algebraic Gramians linked to energy functionals to improve reduction accuracy.

Contribution

It proposes a new algebraic observability Gramian for systems with quadratic outputs, connecting it to energy functionals and deriving error bounds based on $\\mathcal H_2$ energy considerations.

Findings

Demonstrates efficiency on semi-discretized PDEs

Provides error bounds depending on neglected singular values

Compared favorably with existing reduction techniques

Abstract

Model order reduction is a technique that is used to construct low-order approximations of large-scale dynamical systems. In this paper, we investigate a balancing based model order reduction method for dynamical systems with a linear dynamical equation and a quadratic output function. To this aim, we propose a new algebraic observability Gramian for the system based on Hilbert space adjoint theory. We then show the proposed Gramians satisfy a particular type of generalized Lyapunov equations and we investigate their connections to energy functionals, namely, the controllability and observability. This allows us to find the states that are hard to control and hard to observe via an appropriate balancing transformation. Truncation of such states yields reduced-order systems. Finally, based on $\mathcal H_2$ energy considerations, we, furthermore, derive error bounds, depending on the neglected singular values. The efficiency of the proposed method is demonstrated by means of two semi-discretized partial differential equations and is compared with the existing model reduction techniques in the literature.