用可变椭球模板提升不确定系统稳定性，计算更快更准。
Anisotropic Template Ansätze for Robust Positive Invariance under State-Dependent Uncertainty
- 通过高斯过程生成的矩阵场变形椭球模板，自适应调整保护区域。
- 仿真显示速度管体积缩小195倍，7维状态-控制空间压缩超20万倍。
- 适合需要实时保障安全性的飞行器等复杂系统控制器设计。
本文建立了在状态与输入相关扰动下具有各向异性协方差结构时，保证鲁棒正不变性的充分条件。提出的假设方法将固定椭球模板通过高斯过程推导出的正定矩阵场进行映射，既涵盖标量同质缩放，又保留有限图基验证能力。所得LMI条件将学习到的场与舒尔稳定动力学耦合；一种各向同性退化方案，以膨胀因子 $r=1/(1-γ_{ ext{cl}})$ 证明了可行性。每个学习周期中，该场被冻结，因此在线管路评估仅需一次GP协方差查询和小矩阵平方根计算，无需在线集迭代或LMI求解。四旋翼仿真表明，3维速度管体积减少195倍，7维速度-控制联合子空间缩减达 $2.1\times10^5$ 倍，优于非自适应同质基准。此扩展版本补充完整证明、独立离线/在线复杂度分析，以及控制器扫描、收缩性和投影面积研究。
原文摘要 · Abstract (English)
We establish sufficient conditions for robust positive invariance under state- and input-dependent disturbances with anisotropic covariance structure. The proposed ansatz maps a fixed ellipsoidal template through a GP-derived positive-definite matrix field, subsuming scalar homothetic scaling while retaining finite graph-based verification. The resulting LMI conditions couple the learned field to Schur-stable dynamics; an isotropic fallback with inflation factor $r=1/(1-γ_{\mathrm{cl}})$ proves admissibility. During each learning epoch the field is frozen, so online tube evaluation is one GP covariance query and a small matrix square root, with no online set iteration or LMI solve. Quadrotor simulations show a $195\times$ reduction in 3D velocity-tube volume and a $2.1{\times}10^5$ reduction in the joint 7D velocity-control subspace relative to a non-adaptive homothetic baseline. This extended version adds full proofs, a separated offline/online complexity analysis, and controller-sweep, contraction, and projection-area studies.
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