Implicit Predecessor-Based Region of Attraction Estimation and Robust Invariance Analysis for a Two-Wheeled Inverted Pendulum

Lorenzo Fici, Guillaume Ducard

Estimating the region of attraction (RoA) of nonlinear systems is fundamental for assessing closed-loop stability and ensuring safe operation. While Lyapunov-based approaches provide certified stability guarantees, they often yield conservative inner approximations of the RoA. This paper combines a certified Lyapunov-based positively invariant set with a predecessor-based implicit representation to compute a significantly less conservative inner approximation of the RoA while preserving formal stability guarantees. In addition, the robust positive invariance of the initial certified Lyapunov-based invariant set is analyzed under bounded additive input disturbances, providing formal robustness guarantees. The proposed methodology is demonstrated on a nonlinear two-wheeled inverted pendulum stabilized by a saturated linear quadratic regulator. The resulting RoA approximation is compared with the initial Lyapunov-certified invariant set and validated through Monte Carlo simulations and hardware experiments, showing a substantially enlarged certified operating region that matches the empirical closed-loop behavior. These results demonstrate the practical applicability of combining certified Lyapunov analysis with predecessor-based set propagation for RoA approximation and robustness assessment of nonlinear systems.