Continuous Approximations of Projected Dynamical Systems via Control Barrier Functions

• 成果类型: Article
• 署名作者: Delimpaltadakis, Giannis; Cortes, Jorge; Heemels, W. P. M. H.
• 署名单位: Eindhoven University of Technology; University of California System; University of California San Diego
• 刊物名称: IEEE TRANSACTIONS ON AUTOMATIC CONTROL
• ISSN/ISSBN: 0018-9286
• DOI: 10.1109/TAC.2024.3449151
• 发表日期: 2025
• 页码: 681-688
• 关键词: trajectory; Power system dynamics; optimization; Vehicle dynamics; Perturbation methods; dynamical systems; control systems; Control barrier functions; discontinuous dynamical systems; Feedback optimization; nonsmooth dynamics; projected dynamical systems; safety-critical control; synchronverters

摘要

Projected dynamical systems (PDSs) form a class of discontinuous constrained dynamical systems, and have been used widely to solve optimization problems and variational inequalities. Recently, they have also gained significant attention for control purposes, such as high-performance integrators, saturated control, and feedback optimization. In this work, we establish that locally Lipschitz continuous dynamics, involving Control Barrier Functions (CBFs), namely, CBF-based dynamics, approximate PDSs. Specifically, we prove that trajectories of CBF-based dynamics uniformly converge to trajectories of PDSs, as a CBF-parameter approaches infinity. Toward this, we also prove that CBF-based dynamics are perturbations of PDSs, with quantitative bounds on the perturbation. Our results pave the way to implement discontinuous PDS-based controllers in a continuous fashion, employing CBFs. We demonstrate this on an example on synchronverter control. Moreover, our results can be employed to numerically simulate PDSs, overcoming disadvantages of existing discretization schemes, such as computing projections to possibly nonconvex sets. Finally, this bridge between CBFs and PDSs may yield other potential benefits, including novel insights on stability.