Using Spectral Submanifolds for Nonlinear Periodic Control

TL;DR

This paper introduces a novel approach using Spectral Submanifolds to reduce high-dimensional nonlinear systems for efficient feedback control synthesis, addressing the challenge of preserving nonlinear dynamics in reduced models.

Contribution

It applies recent spectral submanifold theory to nonlinear model reduction, facilitating control design for complex high-dimensional systems.

Findings

Enables structure-preserving model reduction for nonlinear systems

Facilitates efficient feedback controller synthesis

Addresses computational challenges in high-dimensional control

Abstract

Very high dimensional nonlinear systems arise in many engineering problems due to semi-discretization of the governing partial differential equations, e.g. through finite element methods. The complexity of these systems present computational challenges for direct application to automatic control. While model reduction has seen ubiquitous applications in control, the use of nonlinear model reduction methods in this setting remains difficult. The problem lies in preserving the structure of the nonlinear dynamics in the reduced order model for high-fidelity control. In this work, we leverage recent advances in Spectral Submanifold (SSM) theory to enable model reduction under well-defined assumptions for the purpose of efficiently synthesizing feedback controllers.