Equilibria and Their Stability Do Not Depend on the Control Barrier Function in Safe Optimization-Based Control
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866912021980643328 |
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| author | Chen, Yiting Mestres, Pol Cortes, Jorge Dall'Anese, Emiliano |
| author_facet | Chen, Yiting Mestres, Pol Cortes, Jorge Dall'Anese, Emiliano |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2409_06808 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Equilibria and Their Stability Do Not Depend on the Control Barrier Function in Safe Optimization-Based Control Chen, Yiting Mestres, Pol Cortes, Jorge Dall'Anese, Emiliano Systems and 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. |
| title | Equilibria and Their Stability Do Not Depend on the Control Barrier Function in Safe Optimization-Based Control |
| topic | Systems and Control |
| url | https://arxiv.org/abs/2409.06808 |