Bang-Bang Free Boundary Control of a Stefan Problem for Metallurgical Length Maintenance

In continuous casting of steel, metallurgical length (ML) is the distance between the exit from the mold and the point of full solidification of a steel slab. This work explores the potential of using the open-loop spray-cooling control to minimize ML deviations from the desired location during casting speed changes under spray flow rate constraints. This objective essentially reduces to motion planning, i.e. apriori generation of spray flow rate commands, which when applied to the process make the latter execute the motion that reduces ML deviations from the setpoint in the shortest time possible. The existence and uniqueness of the solution of the single-phase one-dimensional (1D) Stefan solidification model and its two-dimensional (2D) extension representing the solidifying slab cross-section under bounded bang-bang control and some simplifying but practically justified assumptions are proved. The general synthesis setting for bang-bang control of the single-phase 1D Stefan problem and its 2D extension under boundary flux input constraints is formulated. Then, the bang-bang control for the minimization of the ML deviation from the desired value after the casting speed increase is heuristically found for the 2D slab model through trial-and-error. The simulation results of bang-bang ML control are provided.