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

An optimal control scheme is designed to improve repeatability by minimizin g the loading effects induced by the common processing condition of placeme nt of a semiconductor wafer at ambient temperature on a large thermal-mass bake plate at processing temperature. The optimal control strategy is a mod el-based method using linear programming to minimize the worst-case deviati on from a nominal temperature set point during the load disturbance conditi on. This results in a predictive controller that performs a predetermined h eating sequence prior to the arrival of the wafer as part of the resulting feedforward/feedback strategy to eliminate the load disturbance. This proce dure is based on an empirical model generated from data obtained during clo sed-loop operation. It is easy to design and implement for conventional the rmal processing equipment. Experimental results are performed for a commerc ial conventional bake plate and depict an order-of-magnitude improvement in the settling time and the integral-square temperature error between the op timal predictive controller and a feedback controller for a typical load di sturbance.