多變量EWMA控制器之績效及穩定性研究

Performance and Stability Analysis of Multivariate EWMA Controllers

林建華(Chien-Hua Lin)

指導教授 ： 曾勝滄 ； 唐正

國立清華大學/理學院/統計學研究所/博士(2006年)

https://doi.org/10.6843/NTHU.2006.00653

摘要

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。

關鍵字

批次控制 ； 多投入多產出 ； Double MEWMA回饋控制器 ； 典型相關係數 ； 穩定性條件 ； 全域穩定條件 ； 動態模型 ； 靜態模型

並列摘要

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.

並列關鍵字

Run-to-Run Process Control ； MIMO ； Multiple-Input and Multiple-Output ； Double MEMWA Controller ； Double, Multivariate Exponentially Weighted Moving Average Controller ； Canonical Correlation Coefficient ； Stability Conditions ； Global Stability ； Dynamic I-O Process ； Static I-O Process

參考文獻

``A probabilistic approach to run-to-run control,"

[8] T. W. Anderson (2003),

An Introduction to Multivariate Statistical Analysis, Third Edition,

Time Series Analysis, Forecasting and Control, Third Edition,

Technometrics, vol. 16, pp. 391-398.