Model Reduction of Homogeneous Polynomial Dynamical Systems via Tensor Decomposition

TL;DR

This paper introduces a tensor decomposition-based model reduction technique for homogeneous polynomial dynamical systems, preserving key properties and demonstrated through numerical examples, advancing nonlinear system control methods.

Contribution

It presents a novel reduction method for HPDSs using tensor decomposition, maintaining stability, controllability, and observability, which was previously challenging for nonlinear systems.

Findings

Reduced models preserve stability, controllability, and observability.

Numerical examples demonstrate the effectiveness of the method.

Tensor structure enables efficient model reduction for nonlinear systems.

Abstract

Model reduction plays a critical role in system control, with established methods such as balanced truncation widely used for linear systems. However, extending these methods to nonlinear settings, particularly polynomial dynamical systems that are often used to model higher-order interactions in physics, biology, and ecology, remains a significant challenge. In this article, we develop a novel model reduction method for homogeneous polynomial dynamical systems (HPDSs) with linear input and output grounded in tensor decomposition. Leveraging the inherent tensor structure of HPDSs, we construct reduced models by extracting dominant mode subspaces via higher-order singular value decomposition. Notably, we establish that key system-theoretic properties, including stability, controllability, and observability, are preserved in the reduced model. We demonstrate the effectiveness of our method using numerical examples.