  Model uncertainty poses a significant challenge to the implementation of
safety-critical control systems. With this as motivation, this paper proposes a
safe control design approach that guarantees the robustness of nonlinear
feedback systems in the presence of matched or unmatched unmodelled system
dynamics and external disturbances. Our approach couples control barrier
functions (CBFs) with a new uncertainty/disturbance estimator to ensure robust
safety against input and state-dependent model uncertainties. We prove upper
bounds on the estimator's error and estimated outputs. We use an uncertainty
estimator-based composite feedback control law to adaptively improve robust
control performance under hard safety constraints by compensating for the
matched uncertainty. Then, we robustify existing CBF constraints with this
uncertainty estimate and the estimation error bounds to ensure robust safety
via a quadratic program (CBF-QP). We also extend our method to higher-order
CBFs (HOCBFs) to achieve safety under unmatched uncertainty, which causes
relative degree differences with respect to control input and disturbance. We
assume the relative degree difference is at most one, resulting in a
second-order cone (SOC) condition. The proposed robust HOCBFs method is
demonstrated in a simulation of an uncertain elastic actuator control problem.
Finally, the efficacy of our method is experimentally demonstrated on a tracked
robot with slope-induced matched and unmatched perturbations.
