Minimum-time optimal feedforward control of conductive heating systems for microelectronics processing of silicon wafers and quatz photomasks

Abstract

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 substrate at ambient temperature on a large thermal-mass bake plate at processing temperature. A model-based optimal controller is presented based on minimum time control strategy for minimizing the worst-case deviation from a nominal temperature set-point during the load disturbance condition. This results in a predictive controller that performs a pre-determined heating sequence prior to the arrival of the substrate as part of the resulting feedforward/feedback strategy to eliminate the load disturbance. The controller is easy to design and implement for conventional thermal processing equipment. The minimum time control formulation also makes it more suitable for on-line implementation such as automatic on-line tuning of feedforward controller. 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.