  This paper studies the problem of utilizing data-driven adaptive control
techniques to guarantee stability and safety of uncertain nonlinear systems
with high relative degree. We first introduce the notion of a High Order Robust
Adaptive Control Barrier Function (HO-RaCBF) as a means to compute control
policies guaranteeing satisfaction of high relative degree safety constraints
in the face of parametric model uncertainty. The developed approach guarantees
safety by initially accounting for all possible parameter realizations but
adaptively reduces uncertainty in the parameter estimates leveraging data
recorded online. We then introduce the notion of an Exponentially Stabilizing
Adaptive Control Lyapunov Function (ES-aCLF) that leverages the same data as
the HO-RaCBF controller to guarantee exponential convergence of the system
trajectory. The developed HO-RaCBF and ES-aCLF are unified in a quadratic
programming framework, whose efficacy is showcased via two numerical examples
that, to our knowledge, cannot be addressed by existing adaptive control
barrier function techniques.
