基于学习的动态不确定性下收缩不变管构建方法，保障控制安全
Learning-Based Shrinking Disturbance-Invariant Tubes for State- and Input-Dependent Uncertainty
- 用高斯过程建模不确定性和扰动，生成可信椭球并转为多面体
- 双时间尺度迭代使扰动集收敛到紧致不动点，管体随数据积累而收缩
- 适用于状态与输入相关不确定性的安全强化控制，适合机器人等场景
本文提出一种基于学习的框架，用于在状态和输入相关不确定性下构建收缩的扰动不变管，作为管式模型预测控制（MPC）的基础模块，并通过升维的保序不动点映射实现安全性验证。高斯过程后验转化为(1−α)可信椭球，再外接为多面体以支持确定性集合运算。采用双时间尺度方案：学习阶段冻结多面体，内层迭代从外向内收敛至紧致不动点 $Z^ullet\subseteq\mathcal G$；其状态投影对系统为闭环不变集。随着数据积累，扰动多面体收紧，对应管体单调嵌套，解决了集合验证与扰动模型间的循环依赖问题，同时保持硬约束。双积分器实验表明，在数据丰富的区域管截面持续收缩，仍维持不变性。
原文摘要 · Abstract (English)
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.
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