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Article Excerpt
1. Introduction
Run-to-run (R2R) process control techniques have been extensively studied and are widely utilized in semiconductor manufacturing operations (Sachs et al., 1995; Hamby et al., 1998; Chen and Guo, 2001; Moyne et al., 2001). In particular the model-based Exponentially Weighted Moving Average (EWMA) R2R control scheme has received considerable attention. The technique consists of the process Input-Output (I-O) model, the predicted model and the EWMA control scheme. In the initial stage, Design of Experiment (DoE) and Response Surface Methodology (RSM) are used to construct the predicted model from measured data. At the end of the production run, the EWMA control scheme is used to create the input condition values for the next run by comparing output and target parameter values. The idea of R2R control is to adjust the process output to attain the target value.
Considering a first-order Single-Input and Single-Output (SISO) system, Ingolfsson and Sachs (1993) proposed a procedure for adjusting a process using a single EWMA controller. The stability conditions of the single EWMA controller were derived and they further demonstrated that the expected process output could not converge to a desired target value if the process had a linear drift. To overcome this weakness, Butler and Stefani (1994) proposed a double EWMA controller that enabled the offset term (of process output) to be eliminated efficiently. Recently, Del Castillo and Hurwitz (1997) and Tseng et al. (2002) investigated the stability conditions, long-term behavior and transient performance and also determined the optimal discount factor for a double EWMA controller. However, the results of the above-mentioned studies were obtained for the situation where the predicted model and process parameters are precisely known. The predicted model is typically constructed during an off-line DoE/RSM stage, based on a random sample of the I-O variables. Therefore, the sample size and the level of correlation between I-O variables play major roles in determining the stability of the process. Recently, Tseng and Hsu (2005) derived a formula for the minimum sample size (which is explicitly expressed in terms of the correlation structure of the process I-O variables) so that the desired stability probability can be guaranteed.
All the aforementioned models are restricted to the case in which the process I-O relationship follows a static model. In practical applications, the effects of the input recipe on the output response can be carried over several periods. Pan and Del Castillo (2001) investigated the identification and fine-tuning of closed-loop dynamic models under discrete EWMA and PI adjustments. The stability conditions of a single EWMA controller were derived and the optimal discount factor obtained by using asymptotic mean square deviation criteria after identification and estimation of the unknown parameters in the controlled output model. The results are practical and interesting. However, this model is restricted to the case where the process disturbance follows an IMA(1, 1) series. For a general ARIMA process disturbance, we are more interested in knowing whether or not the process output of a single EWMA controller will be stable.
Focusing on a first-order SISO dynamic model and assuming that the process disturbance follows a general ARIMA series, we first derive the stability conditions of a single EWMA controller. Furthermore, when the process parameters are unknown, we propose a formula to determine an adequate sample size (during an off-line DoE stage) so that the resulting EWMA controller will be asymptotically stable with a guaranteed probability. The rest of this paper is organized as follows. First, Section 2 states the problem formulation, and Sections 3 and 4 state the stability conditions and global stability of single EWMA controller for a first-order dynamic model respectively. Section 5 shows the minimum sample size determination that ensures the resulting EWMA controller will be stable with a guaranteed probability followed by an illustration of the proposed procedure via an example and discussion of the sensitivity analysis in Section 6. Finally, some concluding remarks are addressed at the end of this paper.
2. Problem formulation
In a R2R feedback control, assume that [Y.sub.t] denotes the response of the process quality characteristic (output) for run number t and [x.sub.t - 1] denotes the input variables chosen at the end of production run t - 1. A first-order static SISO model (with no drift) can be described as
[Y.sub.t] = [alpha] + [beta][x.sub.t - 1] + [[eta].sub.t], (1)
for t = 1, 2, ..., where {[[eta].sub.t]} is the process disturbance, and [alpha] and [beta] are unknown parameters to be estimated from data.
To implement an on-line feedback control, we first need to construct the predicted model for the process I-O model. At the beginning (say t = 1), a regression model is used to predict the process output:
[[^.Y].sub.1] = [a.sub.0] + b[x.sub.0], (2)
where [a.sub.0] and b denote the initial estimates of [alpha] and [beta], respectively. We set [x.sub.0] = ([tau] - [a.sub.0])/b, where [tau] is the desired target value of the process output. Ingolfsson and Sachs (1993) proposed a single EWMA controller to recursively update the intercept parameter [alpha] and input recipe [x.sub.t]. A single EWMA controller is expressed as follows:
For t = 1, 2, ...
[x.sub.t] = [tau] - [a.sub.t]/b, (3)
and
[a.sub.t] = [omega]([Y.sub.t] - b[x.sub.t - 1]) + (1 - [omega])[a.sub.t-1]. (4)
Note that [less...
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