  In this paper, we propose a notion of high-order (zeroing) barrier functions
that generalizes the concept of zeroing barrier functions and guarantees set
forward invariance by checking their higher order derivatives. The proposed
formulation guarantees asymptotic stability of the forward invariant set, which
is highly favorable for robustness with respect to model perturbations. No
forward completeness assumption is needed in our setting in contrast to
existing high order barrier function methods. For the case of controlled
dynamical systems, we relax the requirement of uniform relative degree and
propose a singularity-free control scheme that yields a locally Lipschitz
control signal and guarantees safety. Furthermore, the proposed formulation
accounts for "performance-critical" control: it guarantees that a subset of the
forward invariant set will admit any existing, bounded control law, while still
ensuring forward invariance of the set. Finally, a non-trivial case study with
rigid-body attitude dynamics and interconnected cell regions as the safe region
is investigated.
