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

2026-06-14

AI Summary

This work addresses the challenge of constructing robust positively invariant sets under state- and input-dependent disturbances, particularly those with anisotropic covariance. The authors propose an anisotropic ellipsoidal template method grounded in Gaussian process learning, which replaces conventional scalar scaling with a learned positive-definite matrix field to non-uniformly map a fixed ellipsoid. This approach preserves finite-horizon verifiability while coupling with Schur-stable dynamics, avoids online set propagation or LMI solving for computational efficiency, and incorporates an isotropic fallback mechanism to guarantee feasibility. Simulations on a quadrotor demonstrate substantial reductions in tube volume: the 3D velocity tube is shrunk by a factor of 195, and the 7D joint velocity–control subspace volume is reduced by a factor of 2.1×10⁵ compared to a non-adaptive, isotropic baseline.

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.

Problem

• robust positive invariance
• state-dependent uncertainty
• anisotropic covariance
• disturbance
• invariant sets

Innovation

• anisotropic template
• robust positive invariance
• Gaussian process
• state-dependent uncertainty
• LMI conditions