Practical Deployment of Spectral Submanifold Reduction for Optimal Control of High-Dimensional Systems

Abstract

Real-time optimal control of high-dimensional, nonlinear systems remains a challenging task due to the computational intractability of their models. While several model-reduction and learning-based approaches for constructing low-dimensional surrogates of the original system have been proposed in the literature, these approaches suffer from fundamental issues which limit their application in real-world scenarios. Namely, they typically lack generalizability to different control tasks, ability to trade dimensionality for accuracy, and ability to preserve the structure of the dynamics. Recently, we proposed to extract low-dimensional dynamics on Spectral Submanifolds (SSMs) to overcome these issues and validated our approach in a highly accurate simulation environment. In this manuscript, we extend our framework to a real-world setting by employing time-delay embeddings to embed SSMs in an observable space of appropriate dimension. This allows us to learn highly accurate, low-dimensional dynamics purely from observational data. We show that these innovations extend Spectral Submanifold Reduction (SSMR) to real-world applications and showcase the effectiveness of SSMR on a soft robotic system.