$\mathcal{H}_2$ optimal model reduction of linear systems with multiple quadratic outputs

TL;DR

This paper develops a new $ abla_{H_2}$ optimal model reduction method for linear systems with quadratic outputs, deriving necessary conditions and proposing an iterative algorithm with demonstrated numerical effectiveness.

Contribution

It introduces a Gramian-based $ abla_{H_2}$ optimality framework for systems with quadratic outputs and proposes an iterative Petrov-Galerkin projection algorithm for model reduction.

Findings

The proposed LQO-TSIA algorithm effectively reduces model complexity.

Numerical results show improved accuracy over existing methods.

The framework generalizes linear system optimality conditions to quadratic output systems.

Abstract

In this work, we consider the $\mathcal{H}_2$ optimal model reduction of dynamical systems that are linear in the state equation and up to quadratic nonlinearity in the output equation. As our primary theoretical contributions, we derive gradients of the squared $\mathcal{H}_2$ system error with respect to the reduced model quantities and, from the stationary points of these gradients, introduce Gramian-based first-order necessary conditions for the $\mathcal{H}_2$ optimal approximation of a linear quadratic output (LQO) system. The resulting $\mathcal{H}_2$ optimality framework neatly generalizes the analogous Gramian-based optimality framework for purely linear systems. Computationally, we show how to enforce the necessary optimality conditions using Petrov-Galerkin projection; the corresponding projection matrices are obtained from a pair of Sylvester equations. Based on this result, we propose an iteratively corrected algorithm for the $\mathcal{H}_2$ model reduction of LQO systems, which we refer to as LQO-TSIA (linear quadratic output two-sided iteration algorithm). Numerical examples are included to illustrate the effectiveness of the proposed computational method against other existing approaches.