Deterministic Non-Smooth Safety via Dual-Algebraic Control Barrier Functions
Abstract
This paper presents a dual-algebraic framework for control barrier functions (CBFs) that guarantees deterministic execution using exclusively elementary arithmetic. We develop this deterministic approach to solve a fundamental bottleneck in safety-critical control: pointwise minima compose intersecting safe sets, but generate non-smooth boundaries where standard Lie derivatives fail. Existing mathematical workarounds inject approximation bias, probabilistic non-determinism, or combinatorial execution delays that impede hard real-time hardware certification. By embedding the system state and vector field into the dual-number ring, our method extracts both the composite barrier value and its exact directional derivative in a single evaluation. The standard floating-point minimum deterministically isolates a single vertex of the Clarke generalized gradient for the quadratic-program solver. We prove this selected vertex constitutes a Clarke subgradient and the resulting simultaneous-enforcement safety filter guarantees forward invariance. The arithmetic overhead remains a fixed constant factor, independent of state dimension and constraint count. We extend this framework to finite min/max Boolean compositions, for which enforcement of the routed constraint of each δ-active clause guarantees forward invariance, and to systems of higher relative degree, for which a bivariate truncated-dual evaluation extracts the control coupling without symbolic differentiation. Three numerical examples illustrate the computational scaling.
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BibTeX
@misc{kamaldar2026deterministic,
title = {{Deterministic Non-Smooth Safety via Dual-Algebraic Control Barrier Functions}},
author = {Mohammadreza Kamaldar},
howpublished = {arXiv},
year = {2026},
month = aug,
doi = {10.48550/arxiv.2606.25333},
eprint = {2606.25333},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2606.25333},
}
RIS
TY - UNPB
TI - Deterministic Non-Smooth Safety via Dual-Algebraic Control Barrier Functions
AU - Mohammadreza Kamaldar
PY - 2026
DA - 2026/08/31
PB - arXiv
DO - 10.48550/arxiv.2606.25333
UR - https://arxiv.org/abs/2606.25333
AB - This paper presents a dual-algebraic framework for control barrier functions (CBFs) that guarantees deterministic execution using exclusively elementary arithmetic. We develop this deterministic approach to solve a fundamental bottleneck in safety-critical control: pointwise minima compose intersecting safe sets, but generate non-smooth boundaries where standard Lie derivatives fail. Existing mathematical workarounds inject approximation bias, probabilistic non-determinism, or combinatorial execution delays that impede hard real-time hardware certification. By embedding the system state and vector field into the dual-number ring, our method extracts both the composite barrier value and its exact directional derivative in a single evaluation. The standard floating-point minimum deterministically isolates a single vertex of the Clarke generalized gradient for the quadratic-program solver. We prove this selected vertex constitutes a Clarke subgradient and the resulting simultaneous-enforcement safety filter guarantees forward invariance. The arithmetic overhead remains a fixed constant factor, independent of state dimension and constraint count. We extend this framework to finite min/max Boolean compositions, for which enforcement of the routed constraint of each δ-active clause guarantees forward invariance, and to systems of higher relative degree, for which a bivariate truncated-dual evaluation extracts the control coupling without symbolic differentiation. Three numerical examples illustrate the computational scaling.
ER -