Sample size determination for achieving stability of double multivariate exponentially weighted moving average controller

摘要

To derive a formula for determining the suitable sample size that will guarantee a certain probability of asymptotic process stability under a double multivariate exponentially weighted moving average (dMEWMA) controller. Many semiconductor manufacturing processes have by nature multiple-output (MIMO) variables. For the first-order MIMO process with a linear drift, the double multivariate exponentially weighted moving average (dMEWMA) controller is a popular run-to-run (R2R) controller for adjusting the process mean to a desired target. The long-term stability of this closed-loop MIMO system has not been thoroughly investigated and is the focus of the authors' present study. The stability of this MIMO system using a controller with estimated process parameters depends on whether an initial input-output (1- 0) predicted model can be obtained successfully in advance (Le. offline) to accurately and precisely estimate these process parameters. However, the predicted model is typically constructed during an offline DOEI RSM stage, based on a random sample from the 1-0 variables, and thus the sample size and the strength of the linear relationship between 1-0 variables play major roles in determining the stability of the process.