A Semialgebraic Framework for Verification and Synthesis of Control Barrier Functions

Andrew Clark

Research output: Contribution to journalArticlepeer-review

2 Scopus citations

Abstract

Safety is a critical property for control systems in medicine, transportation, manufacturing, and other applications, and can be defined as ensuring positive invariance of a predefined safe set. This article investigates the problems of verifying positive invariance of a semialgebraic set as well as synthesizing sets that can be made positive invariant through control barrier function (CBF)-based control. The key to our approach consists of mapping conditions for positive invariance to sum-of-squares constraints via the Positivstellensatz from real algebraic geometry. Based on these conditions, we propose a framework for verifying safety of CBF-based control including single CBFs, high-order CBFs, multi-CBFs, and systems with trigonometric dynamics and actuation constraints. In the area of synthesis, we propose algorithms for constructing CBFs, namely, an alternating-descent approach and a local CBF approach. We evaluate our approach through case studies on quadrotor UAV and power converter test systems.

Original languageEnglish
Pages (from-to)3101-3116
Number of pages16
JournalIEEE Transactions on Automatic Control
Volume70
Issue number5
DOIs
StatePublished - 2025

Keywords

  • Control barrier function (CBF)
  • safety
  • sum-of-squares (SOS) optimization

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