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

Abstract We consider the design of fast and reliable neural network‐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.