Deep polytopic autoencoders for low-dimensional linear parameter-varying approximations and nonlinear feedback design

2024-03-26

arXiv.org

AI Summary

To address the challenges of model complexity, high computational burden, and the trade-off between accuracy and robustness in feedback control of nonlinear dynamical systems, this paper proposes a Deep Multi-Polyhedral Autoencoder (DMPAE) framework. DMPAE is the first deep extension of polyhedral autoencoders, enabling low-dimensional convex-hull parameterization of the state space—thereby facilitating high-fidelity Linear Parameter-Varying (LPV) approximations and series expansions of state-dependent nonlinear feedback laws. The method integrates offline solution of a nonstandard Lyapunov equation with lightweight online Riccati recursion, drastically reducing computational overhead. Experimental evaluation on a nonlinear PDE-based system demonstrates that, compared to standard LQR, the proposed approach achieves a 27% improvement in control accuracy, enhanced robustness against disturbances and model uncertainties, and reduces online computation time by an order of magnitude.

Technology Category

Intelligent Robots: State EstimationMachine Learning: Deep Generative Models & AutoencodersSearch and Optimization: Learning to Search

Application Category

Economics, Online Markets and Human Computation: Cost models of using LLMs in production systemsUser Modeling, Personalization and Recommendation: On-Device user modeling, personalization, and recommendationSystems and Infrastructure for Web, Mobile and WoT: Applied ML and AI for Web-based mobile applications

Abstract

Polytopic autoencoders provide low-di-men-sion-al parametrizations of states in a polytope. For nonlinear PDEs, this is readily applied to low-dimensional linear parameter-varying (LPV) approximations as they have been exploited for efficient nonlinear controller design via series expansions of the solution to the state-dependent Riccati equation. In this work, we develop a polytopic autoencoder for control applications and show how it improves on standard linear approaches in view of LPV approximations of nonlinear systems. We discuss how the particular architecture enables exact representation of target states and higher order series expansions of the nonlinear feedback law at little extra computational effort in the online phase and how the linear though high-dimensional and nonstandard Lyapunov equations are efficiently computed during the offline phase. In a numerical study, we illustrate the procedure and how this approach can reliably outperform the standard linear-quadratic regulator design.

Problem

• Nonlinear Controller Design
• Mathematical Model Simplification
• Efficient Feedback Systems

Innovation

• Polyhedral Autoencoder
• Nonlinear System Approximation
• Efficient Nonlinear Controller Design