半導體製程中常使用批次間控制以減少輸出與目標值的偏差，消除擾動對產品品質的影響。然而，製程中隨機性擾動的動態模型和確定性偏移擾動的發生時間都是未知的，且有關批次間控制的研究往往以消除已發生的擾動為目標，並針對特定的動態擾動模型設計控制器，使得實際應用之效果不佳。因此，若能偵測偏移擾動開始與結束的時間，便可以即時地利用批次間控制來消除偏移擾動所造成輸出與目標值的穩態誤差，並降低變異度以維持產品品質。
本論文根據貝氏定理來推導偵測偏移擾動的事後機率，可適用於白噪音模型、自迴歸模型及自迴歸滑動平均模型，再將偏移擾動依據其量值分為較大和較小之擾動，分別統計評估不同判別標準的偵測結果，得到各自最佳的判別標準。為提升偵測偏移擾動的開始和結束之準確性，對偵測方法提出改善方式，先以最佳的判別標準偵測擾動的開始，再使用相同判別標準以連續二次偵測到擾動開始作為進行偵測擾動結束的觸發機制，此方式能夠降低隨機性擾動所造成誤判偏移擾動開始的影響，以改善偵測擾動結束位置的準確性。
為了使此偏移擾動偵測方法更好的應用於批次間控制，本論文提出了以事後機率為基礎的控制方法，依據貝氏推論所得之偏移擾動發生的事後機率，對無擾動發生和有擾動發生的批次間控制器分配不同比重的輸入，提升控制效果的適用性。最後，利用對不同動態模型的鑑別，模擬實際應用之控制結果，再將不同條件的控制結果整合後，得到最佳的控制設定去應對不同的偏移擾動量值和動態模型，以減少輸出值與目標值的差異。

In semiconductor manufacturing, run-to-run control is commonly used to reduce deviations between output and target values, mitigating the impact of disturbances on product quality. However, the dynamic models of stochastic disturbances and the occurrence times of deterministic shift disturbances are often unknown. Research on run-to-run control typically aims to eliminate already-occurred disturbances, designing controllers for specific dynamic disturbance models, leading to unsatisfied practical results. The detection of the start and end times of shift disturbances enables real-time run-to-run control application, eliminating steady-state errors and reducing output variability to maintain product quality.

This paper uses Bayesian theorem to derive the posterior probability of detecting shift disturbances, applicable to various models such as white noise, autoregressive, and autoregressive moving average models. We categorize shift disturbances by magnitude (large or small) and statistically evaluate discrimination criteria to obtain the optimal criteria for each category. To improve the accuracy of detecting the start and end of shift disturbances, a method is proposed. It involves initiating the detection of disturbance onset using the optimal criteria and subsequently using the same criteria to detect the onset of disturbance twice continuously as a trigger mechanism for detecting the end of disturbance. This method eliminates the interference of stochastic disturbances and reduces the impact of misjudging the onset of shift disturbances, improving the accuracy of detecting the end of disturbances.

To enhance the control results in run-to-run control using this shift disturbance detection method, a posterior probability-based control method is proposed. Based on the posterior probability of shift disturbance occurrence obtained from Bayesian inference, different weights are assigned to the inputs of the controller, improving control effectiveness. Finally, by identifying dynamic models without shift disturbances, simulating shift disturbance detection and control results for stochastic dynamic models, and integrating results under different conditions, optimal control settings are obtained to address varying magnitudes of shift disturbances and dynamic models, reducing deviations between output and target values.

[1] D.S. Boning, W.P. Moyne, T.H. Smith, J. Moyne, R. Telfeyan, A. Hurwitz, S. Shellman, and J. Tayor, “Run by run control of chemical-mechanical polishing,” IEEE Transactions on Components, Packaging, and Manufacturing Technology: Part C, vol. 19, no. 4, pp. 307-314, 1996.
[2] J. R. Moyne, R. Telfeyan, A. Hunvitz, and J. Taylor, "A process-independent run-to-run controller and its application to chemical-mechanical planarization," Proceedings of SEMI Advanced Semiconductor Manufacturing Conference and Workshop, pp. 194-200, 1995.
[3] A. -C. Lee, Y. -R. Pan, and M. -T. Hsieh, "Output disturbance observer structure applied to run-to-run control for semiconductor manufacturing," IEEE Transactions on Semiconductor Manufacturing, vol. 24, no. 1, pp. 27-43, 2011.
[4] C. Zhang, H. Deng, and J. S. Baras, "Performance evaluation of run-to-run control methods in semiconductor processes," Proceedings of 2003 IEEE Conference on Control Applications, vol.2, pp. 841-848, 2003
[5] S. V. Crowder, “An SPC model for short production runs: minimizing expected cost,” Technometrics, 34:1, pp. 64-73, 1992.
[6] R. van de Schoot, S. Depaoli, R. King, B. Kramer, K. Martens, M. G. Tadesse, M. Vannucci, A. Gelman, D. Veen, J, Willemsen, and C. Yau, “Bayesian statistics and modelling,” Nature Reviews Methods Primers, no. 1, pp. 1–26, 2021.
[7] J. Wang and Q. P. He, "An overlapping receding horizon approach to reduce delay of disturbance detection and classification using Bayesian statistics," IEEE International Symposium on Semiconductor Manufacturing, pp. 402-405, 2005.
[8] S. N. Kavuri and V. Venkatasubramanian, “Neural network decomposition strategies for large scale fault diagnosis,” International Journal of Control, vol. 59, pp. 767–792, 1994.
[9] Y. Huang, G. V. Reklaitis, and V. Venkatasubramanian, “Dynamic optimization based fault accommodation,” Computers & Chemical Engineering, vol. 24, pp. 439–444, 2000.
[10] J. Berger, Statistical Decision Theory and Bayesian Analysis. New York: Springer-Verlag, 1985.
[11] J. C. Spall, Bayesian Analysis of Time Series and Dynamic Models. New York: Marcel Dekker, 1988.
[12] M.West and J. Harrison, Bayesian Forcasting and Dynamic Models. New York: Springer-Verlag, 1989.
[13] M. N. Nounou, B. R. Bakshi, P. K. Goel, and X. Shen, “Process modeling by Bayesian latent variable regression,” AIChE Journal, vol. 48, no. 8, pp. 1775–1793, 2002.
[14] A. Gelman, J. B. Carlin, H. S. Stern, and D. Rubin, Bayesian Data Analysis. London, U.K.: Chapman and Hall, 1995.
[15] D. C. Montgomery, Introduction to Statistical Quality Control. New York: Wiley, 1985.
[16] V. Venkatasubramanian, R. Rengaswamy, K. Yin, and S. N. Kavuri, “Review of process fault diagnosis—Part I: quantitative model-based methods,” Computers & Chemical Engineering, vol. 27, no. 3, pp. 293–311, 2003.
[17] V. Venkatasubramanian, R. Rengaswamy, K. Yin, and S. N. Kavuri, “Review of process fault diagnosis–Part II: qualitative models and search strategies,” Computers & Chemical Engineering, vol. 27, no. 3, pp. 313–326, 2003.
[18] V. Venkatasubramanian, R. Rengaswamy, S. N. Kavuri, and K. Yin, “Review of process fault diagnosis–Part III: process history based methods,” Computers & Chemical Engineering, vol. 27, no. 3, pp. 327–346, 2003.
[19] M. A. Girshik and H. Rubin, “A Bayes approach to a quality control model,” The Annals of Mathematical Statistics, vol. 23, pp. 114–125, 1952.
[20] A. N. Shiryaev, “The problem of the most rapid detection of a disturbance in a stationary process,” Soviet Mathematics - Doklady, vol. 2, pp. 795–799, 1961.
[21] A. N. Shiryaev, “On optimum methods in quickest detection problems,” Theory of Probability & Its Applications, vol. 8, pp. 22–46, 1963.
[22] A. N. Shiryaev, “Some exact formulas in a disorder problem,” Theory of Probability & Its Applications, vol. 10, pp. 348–354, 1965.
[23] M. Basseville and I. V. Nikiforov, Detection of Abrupt Changes—Theory and Application. Englewood Cliffs, NJ: Prentice-Hall, 1993.
[24] A. Hu, “An optimal Bayesian process control for flexible manufacturing processes,” Ph.D. dissertation, Massachusetts Institute of Technology, Cambridge, MA, 1992.
[25] E. Sachs, A. Hu, and A. Ingolfsson, “Run by run process control: Combining SPC and feedback control,” IEEE Transactions on Semiconductor Manufacturing, vol. 8, no. 1, pp. 26–43, 1995.
[26] E. Sachs, R. S. Guo, S. Ha, and A. Hu, “Process control system for VLSI fabrication,” IEEE Transactions on Semiconductor Manufacturing, vol. 4, no. 2, pp. 134–144, 1991.
[27] A. Ingolfsson and E. Sachs, “Stability and sensitivity of an EWMA controller,” Journal of Quality Technology, vol. 25, no. 4, pp. 271–287, 1993.
[28] W. Tian, H. You, C. Zhang, S. Kang, X. Jia, and W. -T. K. Chien, "Statistical process control for monitoring the particles with excess zero counts in semiconductor manufacturing," IEEE Transactions on Semiconductor Manufacturing, vol. 32, no. 1, pp. 93-103, 2019.
[29] J. Wang and Q. P. He, "A Bayesian approach for disturbance detection and classification and its application to state estimation in run-to-run control," IEEE Transactions on Semiconductor Manufacturing, vol. 20, no. 2, pp. 126-136, 2007.
[30] S.-H. Hwang, J.-C. Lin, and H.-C. Wang, "Robustness diagrams based optimal design of run-to-run control subject to deterministic and stochastic disturbances", Journal of Process Control, vol. 63, pp. 47-64, 2018.
[31] 謝承儒, 基於穩定性圖形之強健最佳批次間控制器設計, 成功大學化學工程學系學位論文, 成功大學, 2019.