  Control Lyapunov functions (CLFs) and Control Barrier Functions (CBFs) have
been used to develop provably safe controllers by means of quadratic programs
(QPs). This framework guarantees safety in the form of trajectory invariance
with respect to a given set, but it can introduce undesirable equilibrium
points to the closed loop system, which can be asymptotically stable. In this
work, we present a detailed study of the formation and stability of equilibrium
points with the QP framework for a class of nonlinear systems. We introduce the
useful concept of compatibility between a CLF and a family of CBFs, regarding
the number of stable equilibrium points other than the CLF minimum. Using this
concept, we derive a set of compatibility conditions on the parameters of a
quadratic CLF and a family of quadratic CBFs that guarantee that all
undesirable equilibrium points are not attractive. Furthermore, we propose an
extension to the QP-based controller that dynamically modifies the CLF geometry
in order to satisfy the compatibility conditions, guaranteeing safety and
quasi-global convergence of the system state to the CLF minimum. Numeric
simulations illustrate the applicability of the proposed method for
safety-critical, deadlock-free robotic navigation tasks.
