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.

翻译：我们建立了在状态与输入依赖扰动（具有各向异性协方差结构）下实现鲁棒正不变性的充分条件。所提出的方法通过高斯过程导出的正定矩阵场映射固定椭球模板，在保留有限图基验证能力的同时，囊括了标量位似缩放。由此产生的线性矩阵不等式条件将学习到的场与舒尔稳定动力学耦合；采用膨胀因子$r=1/(1-γ_{\mathrm{cl}})$的各向同性回退方案被证明是可行的。在每个学习周期中，该场被冻结，因此在线管道评估仅需一次高斯过程协方差查询和一个小型矩阵平方根运算，无需在线集合迭代或线性矩阵不等式求解。四旋翼仿真实验表明：与未自适应的位似基准方法相比，三维速度管道体积减小$195\times$，联合7维速度-控制子空间体积减小$2.1\times10^5$倍。本扩展版本补充了完整证明、分离式离线/在线复杂度分析，以及控制器扫描、收缩性和投影面积研究。