Bilinear Quadratic Output Systems and Balanced Truncation

TL;DR

This paper introduces a balanced truncation method for bilinear dynamical systems with quadratic outputs, involving new Gramian definitions and generalized Lyapunov equations, demonstrated through numerical examples.

Contribution

It develops primal-dual formulations and defines system Gramians for bilinear systems with quadratic outputs, enabling efficient model reduction.

Findings

New Gramian-based balanced truncation framework

Conditions for Gramian existence and uniqueness

Numerical examples demonstrating effectiveness

Abstract

Dynamical systems with quadratic outputs have recently attracted significant attention. In this paper, we consider bilinear dynamical systems, a special class of weakly nonlinear systems, with a quadratic output. We develop various primal-dual formulations for these systems and define the corresponding system Gramians. Conditions for the existence and uniqueness of these Gramians are established, and the generalized Lyapunov equations they satisfy are derived. Using these Gramians and their truncated versions, which are computationally more efficient, we construct a balanced truncation framework for bilinear systems with quadratic outputs. The proposed approach is demonstrated through two numerical examples.