Efficient Design of a Stable Double Multivariate Exponentially Weighted Moving Average Controller

Abstract

Many semiconductor manufacturing processes have, by nature, multiple-input and multiple-output (MIMO) variables. For the first-order MIMO manufacturing processes with linear drifts, 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. To implement this feedback control scheme, we need to build an input-output (I-O) predicted model at the off-line stage. Recently, Tseng, et al. (2007) presented an explicit formula for determining a minimum sample size (which is needed to construct I-O predicted model) in such a way that the asymptotic stability of dMEWMA controller can be achieved with a guaranteed probability. This formula indicates that two key components on the sample size determination are: the canonical correlation of I-O variables and the condition number of the covariance matrix of input variables. Since this condition number is a nuisance parameter, the problem on how to minimize its effect on the sample size determination is of great practical importance. This paper proposes a stable dMEWMA controller with which the sample size (required at the off-line stage) only depends on the canonical correlation of I-O variables. Hence, the sample size can be reduced significantly. Keywords: Double multivariate exponentially weighted moving average (dMEWMA) controller; Multiple-input and multiple-output 　　　　　(MIMO) system; Run-to-run (R2R) feedback control; Stability conditions; Sample size determination.