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

TL;DR

This paper extends high-relative-degree control barrier functions to PDE systems, specifically the Stefan model with actuator dynamics, using backstepping and QP control to ensure safety and prevent phase change freezing.

Contribution

It introduces a novel infinite-dimensional CBF framework for PDEs with actuator dynamics, including a backstepping design for relative degree two systems, ensuring safety without initial condition restrictions.

Findings

Safety guarantees in PDE control with actuator dynamics.

Successful simulation in metal additive manufacturing.

Control prevents liquid freezing while obeying operator commands.

Abstract

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.