  Designing control inputs that satisfy safety requirements is crucial in
safety-critical nonlinear control, and this task becomes particularly
challenging when full-state measurements are unavailable. In this work, we
address the problem of synthesizing safe and stable control for control-affine
systems via output feedback (using an observer) while reducing the estimation
error of the observer. To achieve this, we adapt control Lyapunov function
(CLF) and control barrier function (CBF) techniques to the output feedback
setting. Building upon the existing CLF-CBF-QP (Quadratic Program) and CBF-QP
frameworks, we formulate two confidence-aware optimization problems and
establish the Lipschitz continuity of the obtained solutions. To validate our
approach, we conduct simulation studies on two illustrative examples. The
simulation studies indicate both improvements in the observer's estimation
accuracy and the fulfillment of safety and control requirements.
