Publication: IIE Transactions
Publication Date: 01-MAR-07
Delivery: Immediate Online Access
Author: Tseng, Sheng-Tsaing ; Tsung, Fugee ; Liu, Pei-Yun
Article Excerpt
1. Introduction
Statistical Process Control (SPC) and Engineering Process Control (EPC) are two widely used process control techniques. The goal of both of these techniques is to control a process so that it meets a given target with a minimal variability. Box and Luceno (1997) gave a general introduction to both SPC and EPC techniques. SPC mainly uses control charts to monitor process outputs, detect out-of-control signals, and then take corrective action to remove the root cause of any problems. EPC, on the other hand, implements either feedback or feed-forward control schemes to actively adjust the process inputs (Tsung et al., 1998; 1999; Tsung and Shi, 1999). In recent years, many researchers and practitioners have looked into how to integrate and combine the strengths of the SPC and EPC techniques (Tsung and Apley, 2002; Tsung and Tsui, 2003, and references therein). Among them, the Exponentially Weighted Moving Average (EWMA) controller is a well-known model-based run-to-run (R2R) feedback control scheme, which has received a great deal of attention in semiconductor manufacturing (Fan et al., 2002; Del Castillo et al., 2003; Pan and Del Castillo, 2003; Su and Hsu, 2004a, 2004b).
The idea of R2R control is to tune the process output to its desired target by updating the process input variables for every production run. Each adjustment is calculated based on the output deviations from the target of the previous run. Typically, Ingolfsson and Sachs (1993) proposed a single EWMA (sEWMA) controller to adjust a single-input single-output (SISO) process. They investigated the stability and sensitivity of the process output under different process models. Butler and Stefani (1994) showed how a double EWMA (dEWMA) controller can be used to regulate the input variables of a polysilicon gate etch process to compensate for the effect of process drift on the process outputs. Del Castillo (1999) and Tseng et al. (2002a) investigated the stability conditions, the long-run behavior, the transient performance, and the determination of the optimal discount factor of a dEWMA controller.
All the aforementioned models are only valid when the discount factor is a fixed constant. Whilst an EWMA controller with a small discount factor can guarantee long-term stability (under fairly regular conditions), it usually requires a moderately large number of runs to bring the process output to its target value. This can lead to severe consequences particularly for the small batch sizes that are commonly seen in semiconductor manufacture. The reason for this behavior is that the output deviations from the target value are usually very large on the first few production runs and, as a result, the process may be outside the specified units. In order to reduce a high rework rate, a variable discount factor is proposed to overcome this difficulty.
Instead of considering a fixed discount factor, Tseng et al. (2003) proposed a variable sEWMA scheme for a SISO model with no drift. It was demonstrated that the off-target bias can be eliminated efficiently. For a drifted process, obviously, a dEWMA controller with a variable discount factor is an efficient control scheme to eliminate the initial bias. However, due to computational complexity issues, it is not easy to derive an optimal variable dEWMA controller. Focusing on a class of Gradual Change (GC) control schemes (that is a special case of variable control schemes), Su and Hsu (2004a) illustrated that a dEWMA controller with one fixed discount factor and one variable discount factor (which they labeled a Modified Gradual Change (MGC) control scheme) can achieve a better performance than that of a GC control scheme. Using a simulation study, they further demonstrated that the MGC method results in a 10% improvement in MSE over that of the dEWMA scheme with fixed weights. However, the results are restricted to the case of a white noise disturbance and there is no analytical expression for the process output at any time t, which means that the transient behavior of a MGC controller can not be addressed analytically.
The key idea of our approach is based on the fact that the asymptotic expected output of the variable sEWMA controller of Tseng et al. (2003) for a drifted process will be off-target. To compensate for this effect we propose a controller that adds a compensation constant to the variable sEWMA controller. The main advantage of the proposed controller is that it becomes very easy to implement a R2R control scheme. In addition, an analytical expression of the process output of the proposed controller can be derived successfully even if the process disturbance follows a general ARIMA(p, d, q) series. Hence, we can easily compare the process performances of the proposed controller with that of a dEWMA controller. Generally speaking, if the drift rate is properly estimated, our proposed controller will give a better performance than a dEWMA controller. Hence, it provides us with an efficient tool to adjust a drifted R2R process.
This paper is organized as follows. We present a problem description in Section 2. Section 3 states the stability conditions and optimal variable discount factor of the proposed controller. Several illustrative examples are used to demonstrate the advantages of the proposed controller. We show that the proposed controller has a better performance than a conventional dEWMA controller with fixed discount factors. Section 4 deals with the allowable range for the estimated drift rate. We compare the performance of the proposed method with the MGC controller of Su and Hsu (2004a) in Section 5. Conclusions and recommendations for future work are drawn in Section 6.
2. Problem description
2.1. Notation
[Y.sub.t] = the process output for run t, t [greater than or equal to] 1;
[x.sub.t-1] = the input variable (recipe) for run t;
[[eta].sub.t] = the process disturbance for run t;
[kappa] = the drift rate;
[alpha] = the intercept of the linear regression model;
[beta] = the slope of the linear regression model;
[tau] = the desired target value of the process output;
[a.sub.0] = the initial estimate of [alpha];
b = the initial estimate of [beta];
[^.[kappa]] = the estimated drift rate;
[a.sub.t] = the first equation of the dEWMA controller for run t;
[D.sub.t] = the second equation of the dEWMA controller for run t;
[[omega].sub.1] = fixed discount factor for [a.sub.t];
[[omega].sub.2] = fixed discount factor for [D.sub.t];
[[omega]*.sub.1] = optimal fixed discount factor for [a.sub.t];
[[omega]*.sub.2] = optimal fixed discount factor for [D.sub.t];
[[omega].sub.1](t) = variable discount factor of the...
NOTE: All illustrations and photos have been removed from this article.
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