Kernel Methods for the Approximation of Nonlinear Systems

TL;DR

This paper presents a kernel-based data-driven method for reducing the complexity of nonlinear control systems by leveraging high-dimensional feature spaces and balanced truncation techniques.

Contribution

It introduces a novel nonlinear reduction approach using reproducing kernel Hilbert spaces, enabling implicit balanced truncation for nonlinear systems.

Findings

Effective reduction of nonlinear systems demonstrated in simulations

Captures essential input-output behavior with lower model complexity

Utilizes kernel methods for implicit system linearization

Abstract

We introduce a data-driven order reduction method for nonlinear control systems, drawing on recent progress in machine learning and statistical dimensionality reduction. The method rests on the assumption that the nonlinear system behaves linearly when lifted into a high (or infinite) dimensional feature space where balanced truncation may be carried out implicitly. This leads to a nonlinear reduction map which can be combined with a representation of the system belonging to a reproducing kernel Hilbert space to give a closed, reduced order dynamical system which captures the essential input-output characteristics of the original model. Empirical simulations illustrating the approach are also provided.