Optimal predictive control with constraints for the processing of semiconductor wafers on bake plates

An optimal control scheme is designed to improve repeatability by minimizing the loading effects induced by the common processing condition of placement of a semiconductor wafer at ambient temperature on a large thermal-mass bake plate at processing temperature. The optimal control strategy is a model-based method using linear programming to minimize the worst-case deviation from a nominal temperature set point during the load disturbance condition. This results in a predictive controller that performs a predetermined heating sequence prior to the arrival of the wafer as part of the resulting feedforward/feedback strategy to eliminate the load disturbance. This procedure is based on an empirical model generated from data obtained during closed-loop operation. It is easy to design and implement for conventional thermal processing equipment. Experimental results are performed for a commercial conventional bake plate and depict an order-of-magnitude improvement in the settling time and the integral-square temperature error between the optimal predictive controller and a feedback controller for a typical load disturbance