在本論文中, 我們針對實務上常見的靜態 (static) 與動態 (dynamic) 之MIMO (multiple-input and multiple-output) 生產製程,
討論double MEWMA回饋控制器 (double, multiple exponentially weighted moving average controller)
之穩定性條件, 以及最適折扣矩陣的選取,
更進一步討論其穩定設計之回饋分析。

在靜態且有老化的MIMO製程下, 當製程投入-產出變數的典型相關係數 (canonical correlation coefficient) 已知時,
本文提出一簡便公式來決定線外 (off-line) 預測模型所需之最適樣本數。
由此公式可發現若採用製程投入及產出變數有較高的典型相關係數,
且其投入變數之間的相關程度越低,
將可使樣本數大幅地降低。
其次, 當製程投入-產出變數的典型相關係數未知時,
我們提出一個三階段的方法來決定最適樣本數,
其所求得的最適樣本數仍然可以保證製程產出滿足穩定條件的機率大於所給定的機率值。

在動態且有老化的MIMO製程,
我們推導出double MEWMA回饋控制器之穩定性條件以及 $TMSE$ 的近似公式。
其次, 根據總期望離差平方和 (total mean square error, TMSE) 極小化的準則,
進而求得最佳折扣矩陣。
在一較簡單的二階動態MIMO生產製程下, 本文可進一步地得到Double MEWMA回饋控制器之全域性穩定條件以及線外預測模型所需的最適樣本數。
此外, 在動態參數的敏感度分析部份,
可發現如果生產製程明顯地存在動態參數,
其所需的最適樣本數將遠比靜態模型來的大;
而且其 TMSE 亦會隨之放大。

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.

[1] 莊達人 (1996), VLSI製造技術, 第三版, 高立圖書有限公司.

[2] 李水彬 (2001), ``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.

[3] 吳俊達 (2003), ``MEWMA控制器最適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.

[4] 劉珮筠 (2003), ``Modified VEWMA控制器之研究", 國立清華大學統計學研究所碩士論文.

[5] 林資程 (2004), ``Single EWMA在動態模型下之績效分析", 國立清華大學統計學研究所碩士論文.

[6] 顏慧貞 (2004), ``Modified VMEWMA回饋控制器之研究", 國立清華大學統計學研究所碩士論文.

[7] 陳怡珍 (2005), ``動態漂移模型下Double EWMA控制器之穩定性研究", 國立清華大學統計學研究所碩士論文.

[8] T. W. Anderson (2003),
An Introduction to Multivariate Statistical Analysis, Third Edition,
John Wiley \& Sons Inc., New York.

[9] G. E. P. Box, W. G. Jenkins and G. C. Reinsel (1994),
Time Series Analysis, Forecasting and Control, Third Edition,
Prentice-Hall: Englewood Cliffs.

[10] G. E. P. Box and J. F. MacGregor (1974),
``The analysis of closed-loop dynamic-stochastic systems,"
Technometrics, vol. 16, pp. 391-398.

[11] G. E. P. Box and J. F. MacGregor (1976),
``Parameters estimation with closed-loop operating data,"
Technometrics, vol. 18, pp. 371-380.

[12] S. W. Butler and J. A. Stefani (1994),
``Supervisory run-to-run control of polysilicon gate etch using
in situ ellipsometry,"
IEEE Trans. Semiconduct. Manufact., vol. 7, pp. 193-201.

[13] A. Chen and R. S. Guo (2001),
``Aged-based double EWMA controller and its application to CMP processes,"
IEEE Trans. Semiconduct. Manufact., vol. 14, pp. 11-19.

[14] E. Del Castillo (1999),
``Long run and transient analysis of a double EWMA feedback controller,''
IIE Transactions, vol. 31, pp. 1-13.

[15] E. Del Castillo (2002),
Statistical Process Adjustment for Quality Control,
John Wiley \& Sons Inc.: New York.

[16] E. Del Castillo (2002),
``Closed-loop disturbance identification and controller tunning for discrete
manufacturing processes," Technometrics, vol. 44, pp. 134-141.

[17] E. Del Castillo and A. M. Hurwitz (1997),
``Run-by-run process control: literature review and extensions,''
Journal of Quality Technology, vol. 29, pp. 184-195.

[18] E. Del Castillo and R. Rajagopal (1999),
``A multivariate double EWMA process adjustment scheme for drifting processes ,''
IIE Transactions, vol. 34, pp. 1055-1068.

[19] A. K. Gupta and D. K. Nagar (2000),
Matrix Variate Distributions,
Chapman \& Hall/CRC: New York.

[20] R. A. Horn and C. A. Johnson (1985),
Matrix Analysis,
Cambridge University Press.: New York.

[21] E. S. Hamby, P. T. Kabamba and P. P. Khargonekar (1998),
``A probabilistic approach to run-to-run control,"
IEEE Trans. Semiconduct. Manufact., vol. 11, pp. 654-669.

[22] A. Ingolfsson and E. Sachs (1993),
``Stability and sensitivity of an EWMA controller,"\\
Journal of Quality Technology, vol. 25, pp. 271-287.

[23] J. Moyne, E. Del Castillo and A. M. Hurwitz (2001),
Run-to-Run Control in Semiconductor Manufacturing,
CRC Press LLC: New York.

[24] R. Pan and E. Del Castillo (2001),
``Identification and fine tuning of closed-loop processes under discrete EWMA and PI adjustments,"
Qual. Reliab. Engng. Int., vol. 17, pp. 419-427.

[25] G. C. Reinsel (1994),
Elements of Miltivariate Time Series Analysis,
Springer-Verlag: New York.
[26] E. Sachs, A. Hu and A. Ingolfsson (1995),
``Run by run process control: combining SPC and feedback control,"
IEEE Trans. Semiconduct. Manufact., vol. 8, pp. 26-43.

[27] S. T. Tseng, R. J. Chou, and S. P. Lee (2002a),
``Statistical design of double EWMA controller,"
Applied Stochastic Models in Business and Industry,
vol. 18, no. 3, pp. 313-322.

[28] S. T. Tseng, R. J. Chou, and S. P. Lee (2002b),
``A study of multivariate EWMA controller,"
IIE Transactions,
vol. 34, pp. 541-549.

[29] S. T. Tseng and N. J. Hsu (2005),
``Sample-size determination for achieving asymptotic stability of
a double EWMA control scheme," IEEE Trans. Semiconduct. Manufact.,
vol. 18, pp. 104-111.