Abstract
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 (double MEWMA) 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 before and is the focus of this dissertation. 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 dissertation we first derive a formula for the adequate sample size required to achieve stability for the closed-loop MIMO system with a guaranteed probability when the canonical correlation coefficient of I-O variables is known. It is shown that the sample size depends not only on the canonical correlation coefficient between process I-O variables, but also on the eigen-structure of input variables of the MIMO process. In practice, the canonical correlation coefficient of I-O variables is always unknown, we then propose a 3-step procedure to obtain the minimum sample size when the process parameters are estimated so the desired stability probability can still be guaranteed. In practical R2R controls, the effect of the controllable factors on the response can be carried over several periods. For a 2-order dynamic and drifted MIMO process, the stability conditions of a double MEWMA controller shall be taken into a serious consideration. Assuming that the process disturbance follows a general ARIMA (p,d,q) series, we propose a systematic approach to address this R2R control problem. We first investigate the long-term stability conditions of a double MEWMA controller. The stability conditions are expressed in terms of the dynamic terms and the predicted model based on DOE/RSM. Based on the criterion of minimizing the trace of TMSE (total mean square error), we can obtain the optimal discount matrices by using an approximation of TMSE. Finally, focusing on a special case of 2-order dynamic and drifted MIMO process, we can derive the global stability conditions of a double MEWMA controller. Consequently, the adequate sample size required to achieve stability for this MIMO process with a guaranteed probability is easily obtained.