Anisotropic Template Ansätze for Robust Positive Invariance under State-Dependent Uncertainty

• URL: http://arxiv.org/abs/2606.16068v1
• Date: Sun, 14 Jun 2026 23:52:29 GMT
• Title: Anisotropic Template Ansätze for Robust Positive Invariance under State-Dependent Uncertainty
• Abstract summary: We map a fixed ellipsoidal template through a GP-derived positive-definite matrix field.<n>The resulting LMI conditions couple the learned field to Schur-stable dynamics.<n> Quadrotor simulations show a $195times$ reduction in 3D velocity-tube volume.
• Score: 1.2234742322758416
• License: http://creativecommons.org/licenses/by-nc-sa/4.0/
• Abstract: 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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