Comparative analysis of sum-of-squares optimization and Neural Network Lyapunov Functions for region of attraction estimation

Abstract

Region of Attraction (ROA) estimation with accuracy is crucial for effective control design in nonlinear dynamical systems. Sum-of-squares (SOS) optimization refines Lyapunov function representations and expands ROA estimation for polynomial dynamical systems. However, traditional SOS methods tend to be overly conservative. In contrast, deep learning has emerged as a powerful tool in robotics, enabling data-driven ROA estimation. While deep learning offers strong empirical performance, its lack of stability guarantees remains challenging in safety-critical applications. This paper compares two ROA estimation methods: Sum-of-squares (SOS) optimization and Neural Network Lyapunov Functions (NLFs). We examine their effectiveness in approximating Lyapunov functions for stability assessment, highlighting their strengths, limitations, and practical relevance. Using the Van der Pol oscillator as a benchmark, we conduct detailed simulations to evaluate each method's performance. We provide a comprehensive comparison, considering computational efficiency, accuracy, and scalability, offering insights into their applicability in real-world aerospace systems.