Neural Lyapunov Control of Unknown Nonlinear Systems with Stability Guarantees

Abstract

Learning for control of dynamical systems with formal guarantees remains a challenging task. This paper proposes a learning framework to simultaneously stabilize an unknown nonlinear system with a neural controller and learn a neural Lyapunov function to certify a region of attraction (ROA) for the closed-loop system. The algorithmic structure consists of two neural networks and a satisfiability modulo theories (SMT) solver. The first neural network is responsible for learning the unknown dynamics. The second neural network aims to identify a valid Lyapunov function and a provably stabilizing nonlinear controller. The SMT solver then verifies that the candidate Lyapunov function indeed satisfies the Lyapunov conditions. We provide theoretical guarantees of the proposed learning framework in terms of the closed-loop stability for the unknown nonlinear system. We illustrate the effectiveness of the approach with a set of numerical experiments.Preprint. Under review.

Stability and Safety Learning Methods for Legged Robots

“…This is used to generate a set of counterexamples that are included in the original data set D to obtain an augmented data set D * . If the Lyapunov function conditions are not satisfied, as described in [32,34], the training is terminated when the maximum number of iterations is reached without a solution being found, while the algorithm in [30,31] offers the possibility to reduce the search domain and repeat the learning procedure.…”