Work

Bilinear Systems with Quadratic Outputs: $\mathcal{H}_2$ Analysis, Optimality Conditions for Model Reduction, and Algorithmic Solutions

Year: 2026

Type: preprint

Abstract: Bilinear systems with quadratic outputs (BQO) have recently emerged as an important system class, arising naturally in applications where both the dynamics and the quantities of interest depend nonlinearly on the state. Despite the growing interest in this class of systems, a systematic $\mathcal{H}_2$ framework for BQO systems has been lacking. In this paper, we develop such a framework by establishing an $\mathcal{H}_2$ inner product and norm for BQO systems, deriving output bounds in terms of the $\mathcal{H}_2$ norm, and obtaining first-order optimality conditions for $\mathcal{H}_2$ optimal model reduction. Building on these theoretical foundations, we propose an algorithm that computes a reduced BQO system satisfying these optimality conditions, and thus generalizing existing $\mathcal{H}_2$ optimal methods for bilinear and linear quadratic output systems. The effectiveness of the proposed framework is demonstrated on two numerical examples.

Source: arXiv (Cornell University)

DOI: 10.48550/arxiv.2607.19507

Authors: Faßbender, Heike, Gugercin, Serkan, Peters, Till

Institutions: Technische Universität Braunschweig, Virginia Tech

Topic: Model Reduction and Neural Networks

Subfield: Statistical and Nonlinear Physics

Field: Physics and Astronomy

Domain: Physical Sciences

Open Access status: green

Funders: National Science Foundation, Deutscher Akademischer Austauschdienst, Division of Civil, Mechanical and Manufacturing Innovation

Awards: G1377434643, Collaborative Research: Nonlinear Balancing: Reduced Models and Control

OpenAlex ID: W7170180445

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