Convex Safety Filtering via Spectral Selection for Nonconvex Safe Sets
Abstract
The discrete-time control barrier function condition for a safe set that is a union of convex sets is nonconvex in the control input. For safe sets defined by a matrix concave function through the number of its nonnegative eigenvalues, this paper shows that the set is a union of convex sets, and that the nonconvexity arises because at least one member of the union must hold, not every member. Selecting eigenvectors of the matrix function at the current state yields a convex input constraint that acts only on the eigenvalues determining membership in the set. This constraint is implied by the matrix-wide condition of prior work, and it guarantees a geometric lower bound on each of those eigenvalues along closed-loop trajectories. A double-integrator simulation with a polytope obstacle and a spectrahedron obstacle compares the proposed filter with the matrix-wide condition.
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BibTeX
@misc{salazar2026convex,
title = {{Convex Safety Filtering via Spectral Selection for Nonconvex Safe Sets}},
author = {Juan Augusto Paredes Salazar and James Usevitch and Ankit Goel},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.12324},
eprint = {2610.12324},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.12324},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Convex Safety Filtering via Spectral Selection for Nonconvex Safe Sets
AU - Juan Augusto Paredes Salazar
AU - James Usevitch
AU - Ankit Goel
PY - 2026
DA - 2026/10/08
PB - arXiv
DO - 10.48550/arxiv.2610.12324
UR - https://arxiv.org/abs/2610.12324
AB - The discrete-time control barrier function condition for a safe set that is a union of convex sets is nonconvex in the control input. For safe sets defined by a matrix concave function through the number of its nonnegative eigenvalues, this paper shows that the set is a union of convex sets, and that the nonconvexity arises because at least one member of the union must hold, not every member. Selecting eigenvectors of the matrix function at the current state yields a convex input constraint that acts only on the eigenvalues determining membership in the set. This constraint is implied by the matrix-wide condition of prior work, and it guarantees a geometric lower bound on each of those eigenvalues along closed-loop trajectories. A double-integrator simulation with a polytope obstacle and a spectrahedron obstacle compares the proposed filter with the matrix-wide condition.
ER -