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

摘要

An optimal control scheme is designed to improve repeatability bynminimizing the loading effects induced by the common processingncondition of placement of a semiconductor wafer at ambient temperaturenon a large thermal-mass bake plate at processing temperature. Thenoptimal control strategy is a model-based method using linearnprogramming to minimize the worst-case deviation from a nominalntemperature set point during the load disturbance condition. Thisnresults in a predictive controller that performs a predetermined heatingnsequence prior to the arrival of the wafer as part of the resultingnfeedforward/feedback strategy to eliminate the load disturbance. Thisnprocedure is based on an empirical model generated from data obtainednduring closed-loop operation. It is easy to design and implement fornconventional thermal processing equipment. Experimental results arenperformed for a commercial conventional bake plate and depict annorder-of-magnitude improvement in the settling time and thenintegral-square temperature error between the optimal predictivencontroller and a feedback controller for a typical loadndisturbance