Adaptive Control Barrier Functions with Vanishing Conservativeness Under Persistency of Excitation

This article presents a closed-form adaptive control-barrier-function (CBF) approach for satisfying state constraints in systems with parametric uncertainty. This approach uses a sampled-data recursive-least-squares algorithm to estimate the unknown model parameters and construct a nonincreasing upper bound on the norm of the estimation error. Together, this estimate and upper bound are used to construct a CBF-based constraint that has nonincreasing conservativeness. Furthermore, if a persistency of excitation condition is satisfied, then the CBF-based constraint has vanishing conservativeness in the sense that the CBF-based constraint converges to the ideal constraint corresponding to the case where the uncertainty is known. In addition, the approach incorporates a monotonically improving estimate of the unknown model parameters - thus, this estimate can be effectively incorporated into a desired control law. We demonstrate constraint satisfaction and performance using a nonholonomic robot.