Sum-of-Squares Program and Safe Learning On Maximizing the Region of Attraction of Partially Unknown Systems

TL;DR

This paper combines Gaussian Processes, Chebyshev interpolants, and Sum-of-Squares Programming to safely maximize the region of attraction in partially unknown nonlinear systems, ensuring stability and safety.

Contribution

It introduces a novel approach that integrates learning-based system modeling with SOS-based control synthesis for safety and stability guarantees.

Findings

Improves extrapolation performance of learned models.

Generates significantly larger estimated regions of attraction.

Ensures safety and stability in partially unknown systems.

Abstract

Recent advances in learning techniques have enabled the modelling of unknown dynamical systems directly from data. However, in many contexts, these learning-based methods are short of safety guarantee and strict stability verification. To address this issue, this paper first approximates the partially unknown nonlinear systems by using a learned state space with Gaussian Processes and Chebyshev interpolants. A Sum-of-Squares Programming based approach is then proposed to synthesize a controller by searching an optimal control Lyapunov Barrier function. In this way, we maximize the estimated region of attraction of partially unknown nonlinear systems, while guaranteeing both safety and stability. It is shown that the proposed method improves the extrapolation performance, and at the same time, generates a significantly larger estimated region of attraction.