  In this paper, we analyze the system behavior for general nonlinear
control-affine systems when a control barrier function-induced quadratic
program-based controller is employed for feedback. In particular, we
characterize the existence and locations of possible equilibrium points of the
closed-loop system and also provide analytical results on how design parameters
affect them. Based on this analysis, a simple modification on the existing
quadratic program-based controller is provided, which, without any assumptions
other than those taken in the original program, inherits the safety set forward
invariance property, and further guarantees the complete elimination of
undesired equilibrium points in the interior of the safety set as well as one
type of boundary equilibrium points, and local asymptotic stability of the
origin. Numerical examples are given alongside the theoretical discussions.

Systems and Control;Systems and Control;Optimization and Control