Tree-based solution representations for quadratic bilinear systems and their consequences in model order reduction

Jul 16, 2026

Investigates quadratic bilinear systems by introducing novel tree-based representations that decompose solutions into sequences of coupled bilinear subsystems with explicit expansions indexed by full binary trees. These representations provide sufficient conditions for global solutions and new output bounds via reachability Gramians. Time-limited and infinite-horizon reachability/observability Gramians are shown to exist under conditions characterized through nonlinear matrix equations. Gramians identify dominant state spaces and enable exact reduced models by removing Gramian kernels, culminating in a balanced truncation method with provable approximation error bounds.

LanguageEnglish

CategoryResearch & Report

FormatPDF

Pages28

Reading Time1 h 11 min

Introduction

Model order reduction background

Limitations for nonlinear balanced truncation

Goal and contributions of the work

arXiv :2607 .07841v1 [math .OC] 8 Jul 2026 Tree-based solution representations for quadratic bilinear systems and their consequences in model order reduction Martin Redmann∗ July 10, 2026 Abstract We investigate quadratic bilinear systems by developing novel tree-based representations of their solutions. The proposed framework decomposes the solution into a sequence of coupled bilinear subsystems whose components admit explicit expansions indexed by full binary trees. These representations yield sufficient conditions for the existence of global solutions and lead to new output bounds in terms of reachability Gramians. Motivated by these estimates, we introduce time-limited and infinite-horizon reachability and observability Gramians, establish sufficient conditions for their existence, and characterize them through nonlinear matrix equations. The associated Gramians are employed to identify dominant state-spaces and to derive exact reduced-order models obtained by removing Gramian kernels. Building on these results, we develop a balanced truncation method for quadratic bilinear systems and prove an error bound for the reduced-order approximation. The proposed framework provides a unified connection between tree-based solution representations, nonlinear Gramian theory, and balanced truncation for quadratic bilinear systems, closing several theoretical gaps in the analysis of Gramian-based model reduction for this class of systems. Keywords: nonlinear systems · tree-based representations · Gramians · model order reduction · error bounds MSC classification: 34A05 · 65L05 · 68Q25 · 93A15 · 93B20 · 93C10 1 Introduction Large-scale dynamical systems arise naturally in many areas of science and applied mathematics. While these models often provide an accurate description of the underlying dynamics, their high dimensionality renders numerical simulation and analysis computationally demanding. Model order reduction (MOR) seeks to overcome this difficulty by replacing the original model with a surrogate of significantly smaller dimension that accurately reproduces the essential features of the original system. Most projection-based MOR techniques achieve this by identifying lowdimensional subspaces that capture the dominant dynamics and projecting the governing equations onto these subspaces. Over the past decades, a variety of MOR methodologies has been developed. Proper orthogonal decomposition [16] constructs reduced spaces from representative solution trajectories and has become one of the standard tools in data-driven model reduction. Projection methods based on interpolation or H2-optimality, such as the iterative rational Krylov algorithm [13], provide another successful class of reduction techniques. Among system-theoretic approaches, balanced truncation [18] occupies a prominent position due to its robustness and strong mathematical foundation. By employing reachability and observability Gramians, balanced truncation identifies state variables that are simultaneously difficult to control and observe. Moreover, for linear deterministic systems, it preserves stability and is accompanied by rigorous a priori error bounds [9, 11] . ∗ University of Rostock, Institute of Mathematics, Ulmenstraße 69, 18057 Rostock, Germany, Email: [martin.redmann@uni-rostock.de](martin.redmann@uni-rostock.de). 2 M. Redmann Extending these favorable properties to nonlinear systems has attracted considerable attention in recent years. Data-driven MOR techniques for nonlinear systems include, for instance, [12, 14, 19], whereas interpolation-and optimization-based methods have been investigated in [1, 3] . Several balancing-related approaches have also been proposed for nonlinear systems [10, 24, 25] . However, unlike the classical linear theory, these methods generally lack rigorous error estimates. Balancing related MOR techniques for nonlinear system with error bounds have been developed in more recent works, see [6, 7, 20, 2

FAQs

What is the main contribution of this work for quadratic bilinear systems?

The work develops a tree-based framework that represents solutions of quadratic bilinear systems using coupled bilinear subsystems indexed by full binary trees, enabling new analysis for global solutions and model reduction.

How are reachability and observability Gramians used in the framework?

It introduces time-limited and infinite-horizon reachability and observability Gramians, proves sufficient conditions for their existence, characterizes them via nonlinear matrix equations, and uses them to identify dominant state-space directions.

What reduced-order modeling method is derived from these results?

Using the Gramian theory, the paper builds a balanced truncation method for quadratic bilinear systems and proves an error bound for the reduced-order approximation.

Jul 16, 2026

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