Multi-Step Model Predictive Safety Filters: Reducing Chattering by
  Increasing the Prediction Horizon
Federico Pizarro Bejarano Lukas Brunke Angela P. Schoellig
Learning-based controllers have demonstrated superior performance compared to
classical controllers in various tasks. However, providing safety guarantees is
not trivial. Safety, the satisfaction of state and input constraints, can be
guaranteed by augmenting the learned control policy with a safety filter. Model
predictive safety filters (MPSFs) are a common safety filtering approach based
on model predictive control (MPC). MPSFs seek to guarantee safety while
minimizing the difference between the proposed and applied inputs in the
immediate next time step. This limited foresight can lead to jerky motions and
undesired oscillations close to constraint boundaries, known as chattering. In
this paper, we reduce chattering by considering input corrections over a longer
horizon. Under the assumption of bounded model uncertainties, we prove
recursive feasibility using techniques from robust MPC. We verified the
proposed approach in both extensive simulation and quadrotor experiments. In
experiments with a Crazyflie 2.0 drone, we show that, in addition to preserving
the desired safety guarantees, the proposed MPSF reduces chattering by more
than a factor of 4 compared to previous MPSF formulations.