On the Equivalence Between Prescribed Performance Control and Control Barrier Functions

In this paper, we show that Prescribed Performance Control (PPC) is a model-free Control Barrier Function (CBF)-based control approach. Specifically, we establish that a function utilized in the PPC design is a Time-Varying Reciprocal Control Barrier Function (TVRCBF). We demonstrate that PPC satisfies the same gradient condition that is well-known in the CBF literature, ensuring forward invariance. As a result, the control inputs generated by the PPC law belong to the input set characterized by the TVRCBF. Apart from assuming a certain controllability property, no further knowledge on the system dynamics is required. Our theoretical findings improve the understanding of the relationship between PPC and other CBF-based controllers. The theoretical results are validated through numerical simulations.