Safe PDE Backstepping QP Control with High Relative Degree CBFs: Stefan Model with Actuator Dynamics

High-relative-degree control barrier functions (hi-rel-deg CBFs) play a prominent role in automotive safety and in robotics. In this paper we launch a generalization of this concept for PDE control, treating a specific, physically-relevant model of thermal dynamics where the boundary of the PDE moves due to a liquid-solid phase change -- the so-called Stefan model. The familiar QP design is employed to ensure safety but with CBFs that are infinite-dimensional (including one control barrier "functional") and with safe sets that are infinite-dimensional as well. Since, in the presence of actuator dynamics, at the boundary of the Stefan system, this system's main CBF is of relative degree two, an additional CBF is constructed, by backstepping design, which ensures the positivity of all the CBFs without any additional restrictions on the initial conditions. It is shown that the "safety filter" designed in the paper guarantees safety in the presence of an arbitrary operator input. This is similar to an automotive system in which a safety feedback law overrides -- but only when necessary -- the possibly unsafe steering, acceleration, or braking by a vigorous but inexperienced driver. Simulations have been performed for a process in metal additive manufacturing, which show that the operator's heat-and-cool commands to the Stefan model are being obeyed but without the liquid ever freezing.