We develop a learning-based framework for constructing shrinking disturbance-invariant tubes under state- and input-dependent uncertainty, intended as a building block for tube Model Predictive Control (MPC), and certify safety via a lifted, isotone (order-preserving) fixed-point map. Gaussian Process (GP) posteriors become $(1-α)$ credible ellipsoids, then polytopic outer sets for deterministic set operations. A two-time-scale scheme separates learning epochs, where these polytopes are frozen, from an inner, outside-in iteration that converges to a compact fixed point $Z^\star\!\subseteq\!\mathcal G$; its state projection is RPI for the plant. As data accumulate, disturbance polytopes tighten, and the associated tubes nest monotonically, resolving the circular dependence between the set to be verified and the disturbance model while preserving hard constraints. A double-integrator study illustrates shrinking tube cross-sections in data-rich regions while maintaining invariance.

翻译：本文提出一种基于学习的框架，用于在状态与输入依赖的不确定性下构建收缩扰动不变管，旨在作为管式模型预测控制（MPC）的构建模块，并通过一个提升的、保序的固定点映射来验证安全性。高斯过程（GP）后验被转化为$(1-α)$可信椭球，进而通过确定性集合运算得到多面体外近似集。采用双时间尺度方案：在学习阶段冻结这些多面体，内部则通过由外至内的迭代收敛至一个紧致固定点$Z^\star\!\subseteq\!\mathcal G$；其状态投影构成被控对象的鲁棒正不变集。随着数据积累，扰动多面体逐渐收紧，相应的不变管序列单调嵌套，从而在保持硬约束的同时，解决了待验证集合与扰动模型之间的循环依赖问题。以双积分器为例的研究表明，在数据丰富区域不变管截面持续收缩，同时保持不变性。