  Recent work showed that stabilizing affine control systems to desired (sets
of) states while optimizing quadratic costs and observing state and control
constraints can be reduced to quadratic programs (QP) by using control barrier
functions (CBF) and control Lyapunov functions. In our own recent work, we
defined high order CBFs (HOCBFs) to accommodating systems and constraints with
arbitrary relative degrees, and a penalty method to increase the feasibility of
the corresponding QPs. In this paper, we introduce adaptive CBF (AdaCBFs) that
can accommodate time-varying control bounds and dynamics noise, and also
address the feasibility problem. Central to our approach is the introduction of
penalty functions in the definition of an AdaCBF and the definition of
auxiliary dynamics for these penalty functions that are HOCBFs and are
stabilized by CLFs. We demonstrate the advantages of the proposed method by
applying it to a cruise control problem with different road surfaces, tires
slipping, and dynamics noise.
