Neural Lyapunov Control for Discrete-Time Systems

While ensuring stability for linear systems is well understood, it remains a major challenge for systems with nonlinear dynamics. A general approach in such cases is to leverage Lyapunov stability theory to compute a combination of a Lyapunov control function and an associated control policy. However, finding Lyapunov functions for general nonlinear systems is a challenging task. To address this challenge, several methods have been recently proposed that represent Lyapunov functions using neural networks. However, such approaches have been designed exclusively for continuous-time systems. We propose the first approach for learning neural Lyapunov control in discrete-time systems. Three key ingredients enable us to effectively learn provably stable control policies. The first is a novel mixed-integer linear programming approach for verifying the stability conditions in discrete-time systems. The second is a novel approach for computing sub-level sets which characterize the region of attraction. Finally, we rely on a heuristic gradient-based approach for quickly finding counterexamples to significantly speed up Lyapunov function learning. Our experiments on four standard benchmarks demonstrate that our approach significantly outperforms state-of-the-art baselines. For example, on the path tracking benchmark, we outperform recent neural Lyapunov control baselines by an order of magnitude in both running time and the size of the region of attraction, and on two of the four benchmarks (cartpole and PVTOL), ours is the first automated approach to return a provably stable controller.