Stability Analysis and Controller Synthesis Using Single-Hidden-Layer ReLU Neural Networks

This article presents algorithms to solve analysis and controller synthesis problems for dynamical systems modeled as a recurrent single-hidden-layer rectified linear unit neural network (ReLU NN), or equivalently, a piecewise affine dynamical system. Such models are interesting since they may arise through the use of modern machine learning methods for system identification, or as closed-loop solutions in certain classes of model predictive control (MPC) problems. A key idea in the proposed approach is to use piecewise affine Lyapunov functions parametrized as ReLU NNs, and similarly parameterized controllers. This compatible representation between the Lyapunov function and the dynamics simplifies the automation of analysis of and controller synthesis for learned models. Our method of verifying a candidate Lyapunov function is faster than methods based on mixed integer programming. We "learn" controllers and Lyapunov functions using both weight updates and network architecture search, without gradients. We demonstrate the proposed algorithm on examples involving learned models, explicit MPC controllers, and constrained controller synthesis.