Kernel Machine to Estimate a Lyapunov Function and Region of Attraction (ROA) for Nonlinear Systems

The Lyapunov theory establishes the fundamental details to determine the stability and the Lyapunov function associated with an equilibrium point of a nonlinear system. Sum-of-square (SOS) optimization is a well-known technique to estimate the region of attraction (ROA) from the Lyapunov function. But if some of the system’s parameters and the controller are unknown, the Lyapunov function becomes unknown, and the ROA estimation becomes quite impossible. In this article, a methodology is proposed to use Machine Learning Kernel Method to parameterize Lyapunov function candidates, and a two steps optimization approach is adopted to determine the Lyapunov function and the system’s parameter. The objective function of the optimization is set so that (i) the Lyapunov function satisfies all the fundamental properties of the Lyapunov theory and (ii) it learns the unknown parameters of the system. The proposed methodology also finds a larger ROA than the existing methods. We provide numerical experiments on how the optimization technique works and how it can obtain high-quality solutions for challenging control problems.