Equilibria and Their Stability Do Not Depend on the Control Barrier Function in Safe Optimization-Based Control
<|reference_start|>Equilibria and Their Stability Do Not Depend on the Control Barrier Function in Safe Optimization-Based Control: 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 desired ``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); 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 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 safety filters and controllers based on both CBF and control Lyapunov functions (CLFs), 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.<|reference_end|>
@article{chen2024equilibria,
    title={Equilibria and Their Stability Do Not Depend on the Control Barrier
  Function in Safe Optimization-Based Control},
    author={Yiting Chen, Pol Mestres, Jorge Cortes and Emiliano Dall'Anese},
    journal={arXiv preprint arXiv:2409.06808},
    year={2024},
    archivePrefix={arXiv},
    eprint={2409.06808},
    primaryClass={eess.SY cs.SY}
}