Safety-critical autonomous systems must satisfy hard state constraints under tight computational and sensing budgets, yet learning-based controllers are often far more complex than safe operation requires. To formalize this gap, we study how many distinct control signals are needed to render a compact set forward invariant under sampled-data control, connecting the question to the information-theoretic notion of invariance entropy. We propose a vector-quantized autoencoder that jointly learns a state-space partition and a finite control codebook, and develop an iterative forward certification algorithm that uses Lipschitz-based reachable-set enclosures and sum-of-squares programming. On a 12-dimensional nonlinear quadrotor model, the learned controller achieves a $157\times$ reduction in codebook size over a uniform grid baseline while preserving invariance, and we empirically characterize the minimum sensing resolution compatible with safe operation.

翻译：安全关键自主系统必须在严格的计算与感知预算下满足硬状态约束，然而基于学习的控制器往往远比安全运行所需的复杂。为量化这一差距，我们研究了在采样数据控制下使紧集保持前向不变性所需的不同控制信号数量，将该问题与信息论意义上的不变熵概念相联系。我们提出了一种向量量化自编码器，它联合学习状态空间划分与有限控制码本，并开发了一种基于Lipschitz可达集包络与平方和规划的迭代前向认证算法。在12维非线性四旋翼模型上，学习得到的控制器在保持不变性的前提下，较均匀网格基线实现了码本大小157倍的缩减，同时我们通过实验表征了与安全运行兼容的最小感知分辨率。