Sample-Size Determination for Achieving Asymptotic Stability of a Double EWMA Control Scheme

Abstract

The double exponentially weighted moving average (dEWMA) feedback control scheme, a conventional run-to-run control scheme, can adjust certain semiconductor manufacturing processes with a linear drift. The long-term stability conditions for this closed-loop system have received considerable attention in literature. These stability conditions can be expressed in terms of the predicted model assuming that an initial process input-output (I-O) predicted model can be obtained successfully in advance. However, the predicted model is constructed by a random sample of I-O variables, and therefore the strength of the linear relationship between I-O variables plays a major role in determining the validation of these stability conditions. In order to design a stable dEWMA control scheme, the covariance (or correlation) structure of I-O variables and the number of experiments should be simultaneously considered. By controlling a guaranteed probability of stability, this study first derives the formula for an adequate sample size required to construct the predicted model in the case of single-input single-output and multiple-input single-output systems. Illustrative examples demonstrate the effectiveness of the covariance structure of I-O variables in determining the sample size. Implications for research on multiple-input multiple-output systems are also addressed.