在半導體產業中，大部份生產製程的投入-產出模型均屬於MIMO
（Multiple Input-Multiple Output）系統。針對有飄移現象的製程，傳統上使用Double MEWMA 回饋控制器來調整下一批的投入變數之設定值。一般而言，此控制器雖然可以確保產出值收斂至目標值，但由於其收斂速度過於緩慢，對現今IC產業「多樣少量」的生產型態而言，將會造成極高的重製成本 (rework cost)。有鑒於此，本文以變動折扣因子的觀念來調整有飄移現象的 MIMO 系統，在要求有限批量的總均方差 (Total mean square error，TMSE) 極小化的限制下，建構出最適變動折扣因子之控制器。最後，本文將以 tr(TMSE) 為評估準則，討論此最適變動折扣因子控制器與傳統Double MEWMA 控制器在執行效能之優劣比較。由本文研究中可知，本文所提之 Modified VMEWMA 回饋控制器可加快產出值的收斂速度，大幅減少製程初期所造成的偏差。

Many IC manufacturing processes have, by nature, Multiple Input and Multiple Output (MIMO) system. For a drifted MIMO system, the double Multivariate Exponentially Weighted Moving Average (MEWMA) feedback controller is a popular run-to-run (R2R) control scheme. Although the double MEWMA controller can guarantee a long-term stability, it usually requires a moderately large number of runs to bring the output of a process to its target. This is impractical for process with small batches. The reason is that the output deviation is usually large at the beginning of the first few runs. In order to reduce a possibly high rework rate, in this paper we introduce a variable MEWMA scheme to eliminate the initial bias as soon as possible, even though the inaccuracy of model estimation. Using the criterion of minimizing the total mean square error (TMSE), the optimal variable MEWMA controller can be obtained. An example is given to demonstrate the performance. Finally, the comparisons between variable MEWMA and double MEWMA controllers are addressed at the end of this thesis.

[1] Bulter, S.W. and Stefani, J.A., (1994). “Supervisory run-to-run control of a polysilicon gate etch using in situ ellipsometry. ” IEEE Transactions on Semiconductor Manufacturing, 7, 193-201.
[2] Del Castillo, E., and Hurwitz, A. (1997). Run to run process control: A literature review and some extensions. Journal of Quality Technology, 29, 184-196.
[3] Del Castillo, E., (1999). “Long run and transient analysis of a double EWMA feedback controller.” IIE Transactions, 31, 1157-1169.
[4] Ingolfsson, A. and Sachs, E. (1993). “Stability and sensitivity of an EWMA controller.” Journal of Quality Technology, 25, 271-287.
[5] Sachs, E., Hu, A. and Ingolfsson A. (1995). “Run by Run Process Control: Combining SPC and Feedback Control.” IEEE Transactions on Semiconductor Manufacturing, 8, 26-43.
[6] Tseng, S. T., Chou, R.J. and Lee, S. P. (2002a). “A study of multivariate EWMA controller.” IIE Transactions, 34, 541-549.
[7] Tseng, S. T., Chou, R.J. and Lee, S. P. (2002b). “Statistical design of double EWMA controller.” Applied Stochastic Models in Business and Industry, 18, 313-322.
[8] Tseng, S. T., Yeh, A. B, Tseng, F and Chen, T. Y., “A study of Variable EWMA Controller,” Tentatively Accepted by IEEE Transactions on Semiconductor Manufacturing.
[9] 李水彬 “多變量 EWMA 控制器設計之研究”。國立清華大學統計學研究所博士論文，2001。
[10] 劉珮筠 “Modified VEWMA 控制器之研究”。國立清華大學統計學研究所碩士論文，2003。
[11] 吳俊達 “MEWMA 控制器最是變動折扣因子之研究”，國立清華大學統計學研究所碩士論文，2003。