Computationally Efficient Sampling-Based Algorithm for Stability Analysis of Nonlinear Systems

TL;DR

This paper introduces a scalable, sampling-based algorithm leveraging linear programming and parallelization to efficiently approximate the stability region of nonlinear systems, validated through numerical comparisons.

Contribution

It presents a novel, computationally efficient method combining Lyapunov function parametrization, $\, ext{l}_1$ optimization, and ADMM parallelization for stability analysis.

Findings

Accurately estimates the domain of attraction for nonlinear systems.

Outperforms existing methods in computational efficiency.

Validated on multiple numerical examples.

Abstract

For complex nonlinear systems, it is challenging to design algorithms that are fast, scalable, and give an accurate approximation of the stability region. This paper proposes a sampling-based approach to address these challenges. By extending the parametrization of quadratic Lyapunov functions with the system dynamics and formulating an $\ell_1$ optimization to maximize the invariant set over a grid of the state space, we arrive at a computationally efficient algorithm that estimates the domain of attraction (DOA) of nonlinear systems accurately by using only linear programming. The scalability of the Lyapunov function synthesis is further improved by combining the algorithm with ADMM-based parallelization. To resolve the inherent approximative nature of grid-based techniques, a small-scale nonlinear optimization is proposed. The performance of the algorithm is evaluated and compared to state-of-the-art solutions on several numerical examples.