Many semiconductor manufacturing processes have, by nature, multiple-input and 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 yet been thoroughly investigated before and is the focus of our study. The stability of this MIMO system using a controller with estimated process parameters depends on whether an initial input-output (I-O) predicted model can be obtained successfully in advance (i.e., off-line) to accurately and precisely estimate these process parameters. However, the predicted model is typically constructed during an off-line DOE/RSM stage, based on a random sample from the I-O variables, and therefore, the sample size and the strength of linear relationship between I-O variables play major roles in determining the stability of the process. In this talk we first obtain the stability conditions for the MIMO process. Obviously these conditions depend on the estimated controller, and consequently, one of the conditions is expressed in terms of a probability. We then derive a formula for the adequate sample size required to achieve stability for the closed-loop MIMO system with a guaranteed probability. Two practical MIMO examples are given to illustrate the proposed procedure. Finally, a study is given to demonstrate the impacts of the covariance structure of I-O variables on the required sample size.

Dr. Tseng received the B. S. degree in Business Mathematics from Soochow University, the M. S. degree in Applied Mathematics from Tsing-Hua University, and the Ph.D. degree in Management Science from Tamkang University, Taiwan. Currently, he is a professor in the Institute of Statistics at Tsing-Hua University, Taiwan. His research interests include quality & productivity improvement, reliability lifetime analysis, and Industrial statistics. His articles have appeared in Technometrics, Journal of Quality Technology, IEEE Transactions on Reliability, IIE Transactions, Naval Research Logistics, European Journal of Operational Research, International Journal of Production Research, International Journal of Production Economics, IEEE Transactions on Semiconductor Manufacturing, Statistica Sinica, Reliability Engineering and System Safety and other technical journals. One of the most significant research honors of Prof. Tseng is that he had received three times of outstanding research award from the National Science Council (NSC) of ROC, Taiwan in the Years of 1993, 1999, and 2004. Currently, Prof. Tseng served as an Associate Editor of Journal of Statistical Planning & Inference. He is an elected member of ISI.