Our paper on generalized Barrier function conditions [1] has been accepted to the 60th Allerton Conference. Congrats to Yue for leading this work!
[Bibtex] [Abstract] [Download PDF]
Barrier functions constitute an effective tool for assessing and enforcing safety-critical constraints on dynamical systems. To this end, one is required to find a function $h$ that satisfies a Lyapunov-like differential condition, thereby ensuring the invariance of its zero super-level set $h_\ge 0$. This methodology, however, does not prescribe a general method for finding the function $h$ that satisfies such differential conditions, which, in general, can be a daunting task. In this paper, we seek to overcome this limitation by developing a generalized barrier condition that makes the search for $h$ easier. We do this in two steps. First, we develop integral barrier conditions that reveal equivalent asymptotic behavior to the differential ones, but without requiring differentiability of $h$. Subsequently, we further replace the stringent invariance requirement on $h≥0$ with a more flexible concept known as recurrence. A set is ($τ$-)recurrent if every trajectory that starts in the set returns to it (within $τ$ seconds) infinitely often. We show that, under mild conditions, a simple sign distance function can satisfy our relaxed condition and that the ($τ$-)recurrence of the super-level set $h_≥ 0$ is sufficient to guarantee the system’s safety.
@inproceedings{ssm2024allerton,
abstract = {Barrier functions constitute an effective tool for assessing and enforcing safety-critical constraints on dynamical systems. To this end, one is required to find a function $h$ that satisfies a Lyapunov-like differential condition, thereby ensuring the invariance of its zero super-level set $h_\ge 0$. This methodology, however, does not prescribe a general method for finding the function $h$ that satisfies such differential conditions, which, in general, can be a daunting task. In this paper, we seek to overcome this limitation by developing a generalized barrier condition that makes the search for $h$ easier. We do this in two steps. First, we develop integral barrier conditions that reveal equivalent asymptotic behavior to the differential ones, but without requiring differentiability of $h$. Subsequently, we further replace the stringent invariance requirement on $h≥0$ with a more flexible concept known as recurrence. A set is ($τ$-)recurrent if every trajectory that starts in the set returns to it (within $τ$ seconds) infinitely often. We show that, under mild conditions, a simple sign distance function can satisfy our relaxed condition and that the ($τ$-)recurrence of the super-level set $h_≥ 0$ is sufficient to guarantee the system's safety.},
author = {Shen, Yue and Sibai, Hussein and Mallada, Enrique},
booktitle = {60th Allerton Conference on Communication, Control, and Computing},
grants = {CPS-2136324, Global-Centers-2330450},
keywords = {Barrier Functions},
month = {07},
pages = {1-8},
pubstate = {accepted, submitted Jul 2024},
record = {submitted Jul 2024},
title = {Generalized Barrier Functions: Integral Conditions & Recurrent Relaxations},
url = {https://mallada.ece.jhu.edu/pubs/2024-Preprint-SSM.pdf},
year = {2024}
}