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.
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