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.

翻译：本文考虑针对具有多胞不确定性的线性系统，设计基于神经网络的快速可靠近似控制器，以替代包括变结构控制律及基于（最小）选择策略的传统镇定控制器。基于近期具有保证结构特性的可靠控制代理设计方法，我们发展了一套系统化流程，用于验证当采用训练后的修正线性单元（ReLU）神经网络近似替代此类传统控制器时，线性不确定系统的闭环稳定性与性能。首先，我们提出一个充分条件，该条件涉及ReLU控制器与传统控制器的状态-输入映射之间的最差近似误差，确保系统最终有界于一个规模与收敛速度可调的集合。随后，我们开发了一种离线混合整数优化方法，用于精确计算该最差近似误差。