Continuous Approximations of Projected Dynamical Systems via Control Barrier Functions

Abstract

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. Towards 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 numerical examples on feedback optimization and synchronverter control. Moreover, our results can be employed to numerically simulate PDSs, overcoming disadvantages of existing discretization schemes, such as computing projections to possibly non-convex sets. Finally, this bridge between CBFs and PDSs may yield other potential benefits, including novel insights on stability.

Figures (4)

• Figure 1: Evolution of the control input $\xi(t)$ and states $\zeta_1(t)$ and $\zeta_2(t)$ of system \ref{['eq:feedbck_opti_sys']}, when the controller has PDS dynamics and CBF-based dynamics, for different values of $\alpha$.
• Figure 2: Trajectories $(\zeta(t),\xi(t))$ of system \ref{['eq:feedbck_opti_sys']} coupled with dynamic controller, when the controller has PDS dynamics and CBF-based dynamics, for different values of $\alpha$.
• Figure 3: Evolution of the state $\zeta(t)$ and outputs $P,Q$, for controllers with PDS-based dynamics and CBF-based ones, for different $\alpha$.
• Figure 4: Trajectories $\xi(t)$ of the dynamic controller with PDS-based dynamics and CBF-based ones, for different $\alpha$.

Theorems & Definitions (34)

• Definition 2.1: Tangent Cone rockafellar2009variational
• Definition 2.2: Clarke regularity rockafellar2009variational
• Definition 2.3: Normal Cone rockafellar2009variational
• Definition 2.4: Prox-regularity adly2016preservation
• Definition 2.5: $\sigma$-perturbation hybrid_book
• Theorem 3.1: hauswirth2021projectedhauswirth2020anti
• Remark 1
• Proposition 4.1
• Remark 2
• Corollary 4.1