  This paper considers the safety-critical control design problem with output
measurements. An observer-based safety control framework that integrates the
estimation error quantified observer and the control barrier function (CBF)
approach is proposed. The function approximation technique is employed to
approximate the uncertainties introduced by the state estimation error, and an
adaptive CBF approach is proposed to design the safe controller which is
obtained by solving a convex quadratic program (QP). Theoretical results for
CBFs with a relative degree 1 and a higher relative degree are given
individually. The effectiveness of the proposed control approach is
demonstrated by two numerical examples.
