  Predictive safety filters provide a way of projecting potentially unsafe
inputs, proposed, e.g. by a human or learning-based controller, onto the set of
inputs that guarantee recursive state and input constraint satisfaction by
leveraging model predictive control techniques. In this paper, we extend this
framework such that in addition, robust asymptotic stability of the closed-loop
system can be guaranteed by enforcing a decrease of an implicit Lyapunov
function which is constructed using a predicted system trajectory. Differently
from previous results, we show robust asymptotic stability with respect to a
predefined disturbance set on an extended state consisting of the system state
and a warmstart input sequence. The proposed strategy is applied to an
automotive lane keeping example in simulation.
