Control barrier functions (CBFs) play a critical role in the design of safe optimization-based controllers for control-affine systems. Given a CBF associated with a given, predefined ``safe'' set, the typical approach consists in embedding CBF-based constraints into the optimization problem defining the control law to enforce forward invariance of the safe set. While this approach effectively guarantees safety for a given CBF, the CBF-based control law can introduce undesirable equilibrium points (i.e., points that are not equilibria of the original system). Given that there exist many different CBFs associated with a given fixed safe set, open questions remain on how the choice of CBF influences the number and locations of undesirable equilibria and, in general, the dynamics of the closed-loop system. This paper investigates how the choice of CBF impacts the dynamics of the closed-loop system and shows that: (i) The choice of CBF does not affect the number, location, and (local) stability properties of the equilibria in the interior of the safe set; (ii) undesirable equilibria only appear on the boundary of the safe set; and, (iii) the number and location of undesirable equilibria for the closed-loop system do not depend of the choice of the CBF. Additionally, for the well-established \textit{safety filters}, we show that the stability properties of equilibria of the closed-loop system are independent of the choice of the CBF and of the associated extended class-K function, provided that the CBFs are chosen from the same equivalence class.
Safe Feedback Optimization through Control Barrier Functions
AI4I institute（AI4I研究所）；Eindhoven University of Technology（埃因霍温理工大学）；California Institute of Technology（加州理工学院）；University of California San Diego（加利福尼亚大学圣地亚哥分校）
Closed-loop vs. Open-loop Kalman Filter Architectures in Airborne Aided Inertial Navigation
Airbus Central Research and Technology（空客中央研究与技术中心）；Department of Engineering Cybernetics, Norwegian University of Science and Technology（挪威科技大学工程 cybernetics 系）
