Analytic Estimation of Region of Attraction of an LQR Controller for Torque Limited Simple Pendulum

TL;DR

This paper introduces an analytic method to estimate the region of attraction for an LQR controller applied to a torque-limited simple pendulum, offering faster computation with comparable accuracy to existing sampling methods.

Contribution

The paper presents a novel analytic approach for ROA estimation of LQR-controlled nonlinear systems, specifically applied to a torque-limited simple pendulum, improving speed over sampling methods.

Findings

Faster ROA estimation compared to Lyapunov-sampling baseline.

ROA estimations have similar phase space area to baseline.

Validated through simulation and physical experiments.

Abstract

Linear-quadratic regulators (LQR) are a well known and widely used tool in control theory for both linear and nonlinear dynamics. For nonlinear problems, an LQR-based controller is usually only locally viable, thus, raising the problem of estimating the region of attraction (ROA). The need for good ROA estimations becomes especially pressing for underactuated systems, as a failure of controls might lead to unsafe and unrecoverable system states. Known approaches based on optimization or sampling, while working well, might be too slow in time critical applications and are hard to verify formally. In this work, we propose a novel approach to estimate the ROA based on the analytic solutions to linear ODEs for the torque limited simple pendulum. In simulation and physical experiments, we compared our approach to a Lyapunov-sampling baseline approach and found that our approach was faster to compute, while yielding ROA estimations of similar phase space area.