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研究生:張源融
研究生(外文):Chang Yuan Jung
論文名稱:AStudyofGradualChangeEWMAControllerforDriftedProcesses
論文名稱(外文):A Study of Gradual Change EWMA Controller for Drifted Processes
指導教授:顏慧顏慧引用關係
學位類別:碩士
校院名稱:南台科技大學
系所名稱:工業管理研究所
學門:商業及管理學門
學類:其他商業及管理學類
論文種類:學術論文
論文出版年:2007
畢業學年度:95
語文別:英文
論文頁數:78
中文關鍵詞:Run to RunEWMA折扣因子
外文關鍵詞:Run to RUnEWMADiscount Factor
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在半導體製程中,EWMA控制器已被證實為一有效的回饋控制系統。在文獻中,當生產製程有老化與飄移現象發生時,使用s-EWMA控制器會有製程輸出值偏離目標值的問題,因此在傳統上大多使用d-EWMA控制器來針對這一類型的生產製程來進行製程線上回饋控制,亦即使用二個折扣因子,利用製程前一批量的產出,來對下一批量投入變數進行調整。然而使用固定折扣因子通常難以兼顧減少製程初始偏差與降低製程輸出值之變異數,為了能同時考慮快速消除初始偏差與製程輸出變異數最小的二個條件,可使用變動(Variable)折扣因子來達此目的。
在本研究中,探討兩個利用隨時間漸近調整(Gradual Change)折扣因子的EWMA控制器,分別為Modified Gradual Change d-EWMA (MGC) 控制器與Modified Variable s-EWMA (MVEWMA) 控制器。MGC控制器利用隨時間漸近變化的方式來調整第二個折扣因子,在此控制器中需要對製程干擾項有所估計。然而,MGC控制器的製程產出公式相當複雜,使得MGC控制器在實際應用上有所困難,只能使用在製程干擾項為白干擾的生產製程。
有鑒於此,Tseng et al. (2007) 提出MVEWMA控制器,由於它只有一個折扣因子,因此它的製程產出公式較MGC控制器來得簡單。它針對其單一折扣因子,利用隨時間漸近調整的方法來進行調整折扣因子。但是,MVEWMA控制器與s-EWMA控制器一樣在應用於有飄移現象的生產製程中,其產出值會偏離目標值。針對此偏離目標值的現象它提出經由截距項的調整來補償飄移量。也就是增加一個常數項來對飄移量做估計。然而在實際應用上,飄移量估計的好壞依然影響著製程產出是否偏離目標值。
本研究因上述的原因,經由推導出變動折扣因子d-EWMA控制器的製程產出公式,除了將MGC控制器應用在更廣泛更實際的半導體製程干擾項上,如:ARIMA(p,d,q)模型,並且進行有效性與穩健性的研究。在有效性方面,在假設製程干擾項為IMA(1,1)與ARIMA(1,1,1)模型下與MVEWMA控制器做執行效能性的分析與比較。在穩健性方面,分別在母體參數錯估與初始偏差較大的兩種情況下與一個相當穩健的變動折扣因子d-EWMA控制器,Self-Turning Discount Factor d-EWMA (STDF )控制器做比較。結果顯示,MGC控制器不論在有效性與穩健性的表現上都比STDF控制器與MVEWMA控制器表現的更佳。
In the semiconductor manufacturing, process shift or drift often exists. The double exponentially weighted moving average (d-EWMA) controller has shown to be able to compensate for such processes much more effectively than the single EWMA (s-EWMA) controller. The effectiveness of a EWMA controller depends heavily on the selection of its weight parameter(s), called “discount factor(s)”. It has also been shown in the recent literature that online tuning discount factor(s) dynamically is superior to using fixed discount factor(s) for processes with sudden shift or unexpected drift.
In this research, two Gradual Change (GC) variable EWMA control schemes in the literature are of our particular interest, including the Modified Gradual Change d-EWMA (MGC) controller and the Modified Variable s-EWMA (MVEWMA) controller. The common characteristic of both controllers is to gradually reduce one discount factor with time (while holding the other discount factor fixed for the MGC controller). The efficiency of the MGC controller has been established, but its application is limited to the simplest linear drifting model with white noise disturbance (referred to as the deterministic trend, DT, model.) The MVEWMA controller improves the inefficient problem of the s-EWMA control scheme for controlling drifted processes by adding a compensation constant to the Variable s-EWMA (VEWMA) controller, but its performance depends on the estimation of the drifting speed.
The aim of this research is to extend the application of the MGC controller from the DT model to the more general ARIMA(p, d, q) disturbance models. An explicit expression for the process output is established; following that, the transient behavior and the asymptotic stability condition of a MGC controller can then be addressed analytically. The efficiency of the MGC controller under the ARIMA(p, d, q) disturbance models is compared to the MVEWMA controller. The robustness of the MGC controller is evaluated via the comparison with a nonparametric based self-tuning discount factor (STDF) d-EWMA controller (which online tunes the first discount factor based on the ratio of adjacent process output deviations at each run and therefore requires no distributional assumption about the disturbance model).
ABSTRACT i
摘要 ii
CONTENTS iii
LIST OF TABLES v
LIST OF FIGURES vi
Chapter 1 INTRODUCTION 1
1.1 Research Motivation 3
1.2 Research Objective 5
1.3 Research Limitation 6
1.4 Thesis Framework 6
Chapter 2 A REVIEW OF EWMA CONTROLLERS 7
2.1 The Basics of the S-EWMA Controller 7
2.2 The Basics of the D-EWMA Controller 11
2.3 The STDF Controller 13
2.4 The MGC Controller 14
2.5 The MVEWMA Controller 17
Chapter 3 METHODOLOGY 19
3.1 The AVAR Solution of the D-EWMA Controller 19
3.1.1 The IMA (1, 1) Model 20
3.1.2 The ARIMA (1, 1, 1) Model 27
3.2 The MGC Controller for ARIMA(p, d, q) Models 37
3.2.1 An Analytical Expression of the Process Output 37
3.2.2 Stability Condition 40
3.2.3 The Optimal Value 41
3.2.4 The IMA(1, 1) Model 42
3.2.5 The ARIMA(1, 1, 1) Model 44
Chapter 4 SIMULATION RESULTS 50
4.1 Results for the IMA (1, 1) Model 50
4.1.1 All Variance Solutions 50
4.1.2 Optimal Values 51
4.1.3 Efficiency Results 52
4.1.4 Robustness Results 53
4.2 Results for the ARIMA(1,1,1) Model 56
4.2.1 All Variance Solutions 56
4.2.2 Optimal Values 58
4.2.3 Efficiency Results 60
4.2.4 Robustness Results 61
Chapter 5 CONCLUSIONS 66
5.1 Research Contributions 67
5.2 Future Research Work 68
REFERENCES 69
APPENDIXS 70
An Analytical Output Expression of a MVEWMA Controlled Process 70
[1]Butler, S.W. and Stefani, J.A. (1994) Supervisor run-to-run control a polysilicon gate etch using in situ Ellipsometry. IEEE Transactions on Semiconductor Manufacturing, 7: 193-201.
[2]Del Castillo, E. (1999) Long run and transient analysis of a double EWMA feedback controller, IIE Transactions, 31: 1157-1169.
[3]Del Castillo, E. (2002) Statistical Process Adjustment for Quality Control, New York: Wiley.
[4]He, R.J. (2000) A Study of Dynamical Discount Factors EWMA Controller, Master’s Thesis, The Institute of Statistics, National Tsing-Hua University, TW.
[5]Ingolfsson, A. and Sachs, E. (1993) Stability and sensitivity of an EWMA controller. Journal of Quality Technology, 25: 271-287.
[6]Su, C.T. and Hsu, C.C. (2004) A time-varying weights tuning method of the double EWMA controller, The International Journal of Management Science, 32: 473-480.
[7]Tseng, S.T., Chou, R.J. and Lee, S.P. (2002) Statistical design of double EWMA controller, Applied Stochastic Models in Business and Industry, 18: 313-322.
[8]Tseng, S.T., Yeh, A.B., Tsung, F. and Chan, Y.Y. (2003) A Study of Variable EWMA Controller, IEEE Transactions on Semiconductor Manufacturing, 16(4): 633-642.
[9]Tseng, S.T., Tsung, F. and Liu, P.Y. (2007) Variable EWMA run to run controller for drifted process IIE Transactions, 39: 291-301.
[10]Yan, H. and Chiu, C.H. (2006) A Study of Dynamical Double EWMA Controllers, International Conference on Knowledge-Based Economy and Global Management, 314-331.
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