PID control of laser surface hardening of steel

We discuss control strategies for the laser surface hardening of steel. The goal is to achieve a prescribed hardening depth avoiding surface melting. Our mathematical model consists of a system of ordinary differential equations (ODEs) for the phase volume fractions coupled with the heat equation. The system is solved semi-implicitely using the finite element method. To obtain a uniform hardening depth the first attempt is to use proportional integral differential (PID) control to achieve a constant temperature in the hot spot of the laser beam on the surface. However, the numerical results prove that this is not sufficient. We show that the best strategy is to control the temperature close to the lower boundary of the hardening zone. Then one can compute the optimal temperature in the hot spot of the beam and use it as the setpoint for the pyrometer control of the real process