  To bring complex systems into real world environments in a safe manner, they
will have to be robust to uncertainties - both in the environment and the
system. This paper investigates the safety of control systems under input
disturbances, wherein the disturbances can capture uncertainties in the system.
Safety, framed as forward invariance of sets in the state space, is ensured
with the framework of control barrier functions (CBFs). Concretely, the
definition of input to state safety (ISSf) is generalized to allow the
synthesis of non-conservative, tunable controllers that are provably safe under
varying disturbances. This is achieved by formulating the concept of tunable
input to state safe control barrier functions (TISSf-CBFs) which guarantee
safety for disturbances that vary with state and, therefore, provide less
conservative means of accommodating uncertainty. The theoretical results are
demonstrated with a simple control system with input disturbance and also
applied to design a safe connected cruise controller for a heavy duty truck.
