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.

中文翻译：

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通过快速可靠的基于神经网络的近似实现多面体系统的鲁棒稳定性

我们考虑为具有多面不确定性的线性系统设计快速且可靠的基于神经网络的传统稳定控制器近似，包括具有可变结构的控制律和基于（最小）选择策略的控制律。基于最新设计具有保证结构特性的可靠控制代理的方法，我们开发了一种系统程序，用于在基于训练的修正线性单元 (ReLU) 的近似取代这种传统的线性不确定系统时，验证线性不确定系统的闭环稳定性和性能。控制器。首先，我们提供一个充分条件，其中涉及基于 ReLU 和传统基于控制器的状态到输入映射之间的最坏情况近似误差，确保系统最终限制在一个具有可调整大小和收敛速度的集合内。然后，我们开发了一种基于混合整数优化的离线方法，使我们能够准确计算该数量。