Robust stabilization of polytopic systems via fast and reliable neural network-based approximations

We consider the design of fast and reliable neural network (NN)-based approximations of traditional stabilizing controllers for linear systems with polytopic uncertainty, including control laws with variable structure and those based on a (minimal) selection policy. Building upon recent approaches for the design of reliable control surrogates with guaranteed structural properties, we develop a systematic procedure to certify the closed-loop stability and performance of a linear uncertain system when a trained rectified linear unit (ReLU)-based approximation replaces such traditional controllers. First, we provide a sufficient condition, which involves the worst-case approximation error between ReLU-based and traditional controller-based state-to-input mappings, ensuring that the system is ultimately bounded within a set with adjustable size and convergence rate. Then, we develop an offline, mixed-integer optimization-based method that allows us to compute that quantity exactly.