On: 27 April 2014, At: 17:12 Publisher: Taylor & Francis
Informa Ltd Registered in England and Wales Registered Number: 1072954 Registered office: Mortimer House, 37-41 Mortimer Street, London W1T 3JH, UK
International Journal of Production
Research
Publication details, including instructions for authors and subscription information:
http://www.tandfonline.com/loi/tprs20
On-line tuning of a single EWMA
controller based on the neural
technique
C.-T. Su a & C.-C. Hsu a a
Department of Industrial Engineering and Management , National Chiao Tung University , 1001 Ta Hsueh Road, Hsinchu, Taiwan, ROC
Published online: 21 Feb 2007.
To cite this article: C.-T. Su & C.-C. Hsu (2004) On-line tuning of a single EWMA controller
based on the neural technique, International Journal of Production Research, 42:11, 2163-2178, DOI: 10.1080/00207540410001661409
To link to this article: http://dx.doi.org/10.1080/00207540410001661409
PLEASE SCROLL DOWN FOR ARTICLE
Taylor & Francis makes every effort to ensure the accuracy of all the information (the “Content”) contained in the publications on our platform. However, Taylor & Francis, our agents, and our licensors make no representations or warranties whatsoever as to the accuracy, completeness, or suitability for any purpose of the Content. Any opinions and views expressed in this publication are the opinions and views of the authors, and are not the views of or endorsed by Taylor & Francis. The accuracy of the Content should not be relied upon and should be independently verified with primary sources of information. Taylor and Francis shall not be liable for any losses, actions, claims, proceedings, demands, costs, expenses, damages, and other liabilities whatsoever or howsoever caused arising directly or indirectly in connection with, in relation to or arising out of the use of the Content.
This article may be used for research, teaching, and private study purposes. Any substantial or systematic reproduction, redistribution, reselling, loan, sub-licensing, systematic supply, or distribution in any form to anyone is expressly forbidden. Terms & Conditions of access and use can be found at http://www.tandfonline.com/ page/terms-and-conditions
On-line tuning of a single EWMA controller based on the neural
technique
C.-T. SU* and C.-C. HSU
The exponentially weighted moving average (EWMA) controller has been proven to be an effective algorithm in the control the modern manufacturing system. The performance of the EWMA controlled process is based on choosing the correct EWMA gain. Most related research has focused on analysing the optimal EWMA gain in the static condition. The objective was to propose an approach based on the neural technique for on-line tuning of the single EWMA gain. The underlying approach indicated that the network learns very quickly when taking auto-correlation function and sample partial autoauto-correlation function patterns as the input features. It is shown that the sequence of the EWMA gains, generated by the proposed adaptive approach, converges close to the optimal controller value under several disturbance models, including IMA(1,1), and step and small ramp disturbances. In addition, the approach possesses a superior controlled output performance compared with the previous adaptive system.
1. Introduction
Statistical process control (SPC) is a well-recognized technique for process monitoring. The main assumption of traditional SPC is that successive quality char-acteristic values should have no correlation with each other. However, modern manufacturing processes exhibit quality data that are serially correlated over time. These traditional control charts will give misleading results in the form of too many false alarms if the data are correlated. To address this problem, an approach called engineering process control (EPC) or automatic process control (APC) is widely used in the chemical and processing industries for variation reduction. The EPC scheme refers to an algorithm that describes how the manipulating variable of a process needs to be adjusted from observation to observation.
Recently, a run-by-run feedback control algorithm called the exponentially weighted moving average (EWMA) controller was popular in semiconductor manufacturing, particularly in the chemical mechanical polishing (CMP) process. The performance of the controlled process output under the EWMA controller depends on setting its gains, which means that incorrectly selecting the weight will have the opposite effect on the controlled process output. When the process environ-ment is static and the system parameters are determined, the perfect controller parameters can be obtained by mathematical modelling technology. Unfortunately, a process environment is usually dynamic in a real manufacturing world. To achieve a better performance in the dynamic system, developing a method of on-line tuning of the EWMA controller parameter is an important issue.
Revision received October 2003.
Department of Industrial Engineering and Management, National Chiao Tung University, 1001 Ta Hsueh Road, Hsinchu, Taiwan, ROC.
*To whom correspondence should be addressed. Fax: 00886 35 722 392
International Journal of Production ResearchISSN 0020–7543 print/ISSN 1366–588X online # 2004 Taylor & Francis Ltd
http://www.tandf.co.uk/journals DOI: 10.1080/00207540410001661409
As mentioned above, some previous studies have used statistical techniques such as the least-squares estimate (LSE) (Sastri 1988) or the maximum likelihood estimate (MLE) (Lucen`o 1995) to estimate the parameters recursively. However, these tech-niques assumed that the probability distribution is known in advance. Patel and Jenkins (2000) provided another statistical adaptive algorithm to update the EWMA gain by taking the signal-to-noise (SN) ratio. This involves estimating the mean of the output and the mean square of the output, which implies that additional parameters should be chosen at first to estimate them. The present will introduce their adaptive system in more detail below.
In this study, a simple and efficient method based on the neural technique is suggested to tune adaptively the EWMA gain. This proposed approach takes advan-tage of the fact that no assumption is needed; it constructs input–output relations by learning the historic patterns. The autocorrelation function (ACF) and partial auto-correlation function (PACF) patterns are taken as input features of the training network. The ACF is a well-known tool for identifying the order of the moving average (MA) stochastic process, and the PACF is used to determine the order of the autoregressive (AR) model. Taking another view of both statistics: they are functions of autoregressive moving average (ARMA) parameters, which implies that different combinations of AR and MA parameters will produce distinct ACF/PACF patterns. Based on this idea, a neural network will be trained to estimate on-line the EWMA gain at the next run via ACF/PACF pattern families.
The paper is organized as follows. Section 2 reviews the single EWMA controller, and its effects of incorrectly setting the EWMA gain will be shown. Section 3 intro-duces a recent adaptive algorithm for the EWMA controller. Section 4 introintro-duces neural network techniques. Section 5 presents the structure of the proposed approach. The idea of using the SACF and PACF patterns to estimate the EWMA gain will be demonstrated in more detail. Section 6 specifies the performance of the off-line-trained network and, by using the Simulink toolbox, implements the proposed structure on-line through three examples. Finally, conclusions are drawn, and future research is suggested in section 7.
2. Single EWMA controller
2.1. Brief review of the single EWMA controller
The EWMA statistic was first suggested by Roberts (1959) for process monitoring, but Roberts referred to it as a geometric moving average (GMA). The use of the EWMA statistic has two distinct purposes (Fatin et al. 1990): as control charts (Box and Kramer 1992, Montgomery 1996, Box and Lucen`o 1997, Chen and Elsayed 2002) and as forecasts (Box and Jenkins 1976, Box et al. 1994, Brockwell and Davis, 1996). Recently, the statistic has been used widely for process adjustment purposes (Lucas and Saccucci 1992, Ingolfsson and Sachs 1993, Del Castillo and Hurwitz 1997, Del Castillo 2001, 2002a, Pan and Del Castillo 2001, Fan et al. 2002, O’Shaughnessy and Haugh 2002).
In semiconductor manufacturing, EWMA controllers are sometimes called bias tuning controllers (Butler and Stefani 1994). The purpose of EWMA-based control-lers is for compensating against disturbances that affect the run-to-run variability in quality characteristics (Del Castillo 2002b). Assume that the relation between the input and output of a manufacturing process can be expressed as follows:
et¼ þ ut1þ"t, ð1Þ
where etis the observed output deviation from target, "tis a white noise stochastic
process and ut is the manipulated variable; the parameter represents the process
offset, is the process gain, and both parameters need to be estimated. Equation (1) implies that all the effects of a change in the compensating variable will be realized at the output, in one time interval. Such a system is called a responsive system (Box and Lucen`o 1997) and is commonly seen in the discrete part manufacturing. Let b repre-sent an estimate of the gain () that can be estimated off-line by fitting the regression model. The single EWMA scheme can be expressed as follows:
ut¼
at
b, ð2Þ
where
at ¼ðetbut1Þ þ ð1 Þat1
¼½etbut1þ ð1 Þðet1but2Þ þ ð1 Þ2ðet2but3Þ þ
ð3Þ is an estimate of the offset and is computed recursively based on the EWMA statistic with the last measurement data. The previous estimate at1and are the controller
parameters, which can be adjusted to achieve a desired output. Substituting equation (3) into (2) leads to:
ut¼ b Xt j¼1 ej: ð4Þ
Therefore, the single EWMA controller is a pure integral (I ) controller with integral constant KI¼ /b, which is a particular case of the well-known PID controller. A
PID control scheme can be expressed as ut¼ kPetkI
P1
j¼0etjkDðetet1Þ,
where kP, KI and KDare constants. The integral action in the EWMA controller can
eliminate offsets or shifts, and provides robustness in the controlled process. 2.2. Effects of incorrectly setting the controller parameter
As mentioned by Smith and Boning (1997), the controlled process output under a higher EWMA weight would return to the target much faster than a lower weight, but it would create more oscillations. Therefore, it is important to select the EWMA parameter carefully. This section will describe in more detail the effect of incorrectly choosing the EWMA gain. Consider a process that can be modelled by:
et¼ þ ut1þNt, ð5Þ
where Ntis the disturbance model one wants to compensate for. Assume it follows
an IMA(1,1) stochastic process as follows:
ð1 BÞNt ¼ ð1 BÞ"t, ð6Þ
where B is the backshift operator (BkNt¼Nt–k) and is the moving average
parameter. An IMA(1,1) model is widely used for modelling the drift in the discrete manufacturing (Lucas and Saccucci 1992, Box et al. 1994). If the single EWMA controller is used to compensate for the disturbance, then the controlled process is as follows:
ð1 ð1 ÞBÞet¼ ð1 BÞ"t, ð7Þ
where ¼ /b is a bias of the gain estimate. It can be seen that the controlled process exhibits an ARMA(1,1) process, and that the stable condition is |1 | 1.
Therefore, the inflation factor of the controlled process will be: e2 2 " ¼1 þð1 Þ 2 1 ð1 Þ2: ð8Þ
Figure 1 shows the inflation factor ð2e="2Þversus and given the process gain is
known ( ¼ 1). Consider that the disturbance model follows a white noise process, which implies ¼ 1 in equation (6) and a full adjustment to the process ( ¼ 1) is used, then the controlled output variance will be inflated twice as much than if there was no adjustment. This is what Deming (1986) meant by ‘tampering with the process’. To achieve the minimum mean square error (MMSE) controlled process output, one should set the controller parameter to be as follows:
¼1 : ð9Þ
Although the actual optimal controller parameter in the above equation was already known, it was only an optimal value in the static sense. In practice, the parameter of the disturbance model changes with time, so it is necessary to develop an approach to adjust the controller parameter dynamically in order to obtain a better performance of the controlled process output. Section 3 introduces an adaptive algorithm recently suggested by Patel and Jenkins (2000).
3. Adaptive algorithm
In the sense of an adaptive system, Sastri (1988) used the theory of LSE for sequential parameter detection and revision of the moving average parameter in the IMA(1,1) time series model. Lucen`o (1995) presented a computer program to choose the EWMA controller parameter in the EPC. Lucen`o’s algorithm was based on the MLE theory. These above-mentioned algorithms all have a common con-straint in that the probability distribution must be known beforehand. Therefore, Smith and Boning (1997) used a neural network as an approximation function to map from the disturbance state (magnitude of linear drift and random noise) of a given process to the corresponding optimal EWMA weights. However, it was a
0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 1 1.5 2 0 0.2 0.4 0.6 0.8 1 2 2 e e
s
s
q
l
Figure 1. Inflation factor versus and .
problem to estimate the slope of the controlled process. Del Castillo and Yeh (1998) presented an adaptive run-to-run multiple-input multiple-output controller for linear and non-linear semiconductor processes. Recently, an adaptive algorithm to estimate the EWMA gain was suggested by Patel and Jenkins (2000). Their objective was to design an automated scheme for optimizing the numerical parameter of the EWMA controller. Figure 2 shows the adaptive EWMA controller block diagram they proposed.
The Patel–Jenkins adaptive algorithm can simply be described as the following system: tþ1¼tþtðetþ1tÞ tþ1¼ tþtðe2tþ1 tÞ t ¼ 2þ42t þ 2 t þ t , ð10Þ
where {t} is the estimate of the mean of the output and { t} is the estimate of the
mean square of the output. Initial conditions of (0, 0) satisfy 0 20 0. is a
constant with a very small value that satisfies 0<<1, and {t} is a sequence such
that 0 t<1 and satisfies (I) limt!1t ¼0 (II)
P1 t¼0t¼ 1and (III) P1 t¼0 2 t < 1.
The form of tin equation (10) intuitively provides a measure of the SN ratio that
satisfies 0 t2. According to the above-mentioned adaptive system, the EWMA
control equation can be updated dynamically as follows: ut ¼ t b Xt j¼1 ej, ð11Þ
where the adaptive parameter tfollows the system in equation (10).
EWMA controller
System gain
∑
One-step delay
Disturbance
tu
tLinear Filter
e
t+1e
t tTuner
el
Figure 2. Structure of the adaptive tuning controller.
4. Neural network techniques
Neural networks are of particular interest because they offer a means of model-ling large and complex problems efficiently in which there might be hundreds of predictor variables with many interactions. Neural nets have been used widely in pattern recognition (Su et al. 2002), function approximation (Smith and Boning 1997), optimization (Sjoberg and Agarwal 2002) and data clustering (Andrews and Geva 2002). In general, neural networks can be classified into two different categories: feed-forward and feedback (Cheng and Titterington 1994). The present study used the feed-forward network because it is an effective system for learning distinguishing patterns from a body of examples.
The back-propagation learning algorithm is the most commonly used algorithm to train multilayer feed-forward networks by implementing a local gradient-search to minimize the square error between realized and desired outputs. A typical back-propagation neural network always has an input layer, an output layer and at least one hidden layer. There is no theoretical limit on the number of hidden layers, but typically, there will be one or two. Figure 3 shows a three-layer network. Each layer is fully connected to the succeeding layer. The back-propagation algo-rithm involves forward and backward passes. The purpose of the forward pass is to obtain the activation value; the purpose of the backward pass is to adjust weights according to the difference between the desired and actual network outputs. The above statement can be explained by the following equations.
4.1. Forward pass
The net input to node i for pattern p is: netpi¼ X j wijapjþbi ð12Þ apj¼ 1 1 þ enetpj, ð13Þ
where wijis the weight from unit j to unit i, biis a basis associated with unit i, and apj
is the activation value of unit j with sigmoid function for pattern p.
Hidden Layer
Output Pattern
Input Pattern
Input Layer
Output Layer
Figure 3. Backpropagation neural network structure.
4.2. Backward pass
The sum of the squares error function is as follows: Ep¼ 1 2 tpop 2 2, ð14Þ
where tpis the target output for the pth pattern and opis the actual output for the pth
pattern. By minimizing the errors Epusing the gradient decent method, the weights
can be updated using the following equation:
pwij¼ piapj, ð15Þ
where
pi¼
tpiopi
opi1 opi if unit i is an output unit opi1 opi P
k
pkwki
if unit i is a hidden unit 8
<
: ð16Þ
and is the learning rate. In general, a larger learning rate will increase the training speed. However, it may oscillate widely. One way to increase the learning rate with-out oscillating is to modify equation (15) to the following:
pwij¼ piapjþmp1wij, ð17Þ
where m is the momentum coefficient (m 2 ½0, 1) that determines the effect of past weight changes on the current direction of movement in weight space. There is no principle to determine the parameters of and m; they are chosen by the neural network trainer via the trial-and-error approach. Concerning the model selection, one of the most useful methods in selecting problems is the cross-validation (CV) method. Breiman and Spector (1992) found 10- and fivefold CV to work better than the leave-one-out method for choosing subsets of inputs in linear regression. Zhang (1993) showed that the delete-d multifold CV (MVC) criterion is asymptotically equivalent to the well-known FPE criterion under a regression model. For a detailed discussion about the CV method, see Witten and Frank (2001).
5. Proposed approach
A methodology for tuning the EWMA controller on-line based on a neural technique is developed in this section. The input features of the neural structure are sample autocorrelation function (SACF) and sample partial autocorrelation function (SPACF). The output unit is an estimate of the EWMA controller param-eter at run t. The theoretical autocorrelation function (ACF) at lag h is defined as follows:
ðhÞ ¼Covðet, etþhÞ ðetÞ ðetþhÞ
: ð18Þ
Equation (18) is estimated by the SACF as follows:
^ ðhÞ ¼ P nh i¼1
ðeiee Þðeiþhee Þ Pn
i¼1
ðeiee Þ2
, ð19Þ
where n is the sample size and ee is the sample mean. Also, the theoretical partial autocorrelation function (PACF) is defined as follows:
hh¼
Covðet, etþhetþ1, etþ2, . . . , etþh1Þ
ðetÞðetþhÞ
, ð20Þ
which can be estimated by the SPACF as follows:
^ hh¼ ^ h P h1 j¼1 ^ h1, j ^hj 1 P h1 j¼1 ^ h1, j ^j , h ¼3, . . . ð21Þ where ^ hj¼ ^h1, j ^hh ^h1, hj h ¼2, . . . , j ¼1, 2, . . . , h 1 ð22Þ and ^ 11¼ ^1, ^ 22 ¼ ð ^2 ^21Þ=ð1 ^ 21Þ.
Opinions of selecting SACF and SPACF statistics to be input features can be described simply from figure 4. Figure 4(a) shows the family of SACF/SPACF (denoted ) patterns under the condition of a perfect controlled process up to run t,
0 5 10 15 20 -0.5 -0.4 -0.3 -0.2 -0.1 0 0 0.1 -0.5 -0.4 -0.3 -0.2 -0.1 0.1 Lag 0 5 10 15 20 Lag
(a)
(b)
r
r
Figure 4. Family of SACF/SPACF patterns: (a) perfect controlled and (b) ¼ 1; ¼ 1.
given the disturbance model obeys equation (6) and the controller parameter follows equation (9). If this type of SACF/SPACF patterns is received, then the controller parameter at next run (t þ 1) will be set to be the same as the previous run. Figure 4(b) shows the simulated family of SACF/SPACF patterns with parameter ¼ 1 (white noise process) in equation (6) and the full adjustment ( ¼ 1) in equation (9) up to run t. Obviously, figure 4(b) behaves in a more non-stationary manner than figure 4(a) because of the incorrect way of choosing the controller parameter. If the neural network receives the types of patterns such as shown in figure 4(b) at run t, then it will respond by setting the controller parameter to be zero at the next run to meet the optimal condition. As per the above, the objective is to estimate the con-troller parameter adaptively through pattern recognition on the SACF and SPACF patterns.
The structure of the proposed adaptive neural network (NN)-based single EWMA controller is shown in figure 5. At first, the controlled quality characteristic was sent to SACF and SPACF blocks to calculate the statistic individually. The combined (denoted as the black bar) SACF/SPACF pattern was then sent to the trained NN model block to estimate the controller parameter for the next run. After estimating the parameter, one should update the single EWMA controller parameter dynamically to provide a better control performance. The proposed methodology was implemented in Section 7 and compared with the method of Patel and Jenkins (2000) introduced in Section 3.
6. Implementation
6.1. Off-line training the neural network
The training data sets were generated by simulating different combinations of the disturbance and controller parameters. There were 121 data sets (i.e. 2 ½0, 1, 2 ½0, 1), which implied that there were 121 SACF/SPACF patterns. Thirty data sets were used as the testing data; the remainder was to be training data. A useful
Dynamic Plant
Process
Linear Filter
∑
N
tu
tSPACF
e
t tλ
ˆ
Trained
NN Model
tε
EWMA algorithm
SACF
Figure 5. NN-based adaptive EWMA controller.
guide was provided by Box and Jenkins (1976: 33), who suggested that the size of time series (t) be at least 50 and lags (h) to analyse the series at most t/4. Thus, 50 runs were simulated at each simulation, and 12 lags in each SACF and SPCAF pattern were taken.
The considered case included 24 nodes at the input layer and one node at the output layer. The problem in a full-connected neural network was to determine the number of neurons in the hidden layer. A trial-and-error approach was used to determine that a single hidden layer with 22 neurons formed the required structure for the considered problem. To improve the network performance, a 22 factorial design was used to find the learning rate and momentum constants (table 1). The factors involved in this design were the learning rate and momentum constants, and the response variable was the number of epochs used to achieve the desired level (0.01) of the root-mean-square (RMS) error. Note that the best network perfor-mance (with the smallest number of epochs) was achieved when the learning rate was 0.15 and the momentum constant was 0.85. Figure 6 shows the learning behaviour versus iterations of the selected network structure; it indicates that the network learns very quickly and only required 3393 iterations (about 28 epochs). Thus, the 24-22-1 network structure will be used to implement the NN-based EWMA controller on-line in the following examples.
6.2. Examples of on-line implementation
The toolbox of the Matlab/Simulink version 4.1.1 was used to implement the Patel–Jenkins method and the proposed NN-based adaptive EWMA controller, and then to make a comparison between them. Three examples under different distur-bance models, which are commonly encountered in practice, will also be shown, including step IMA(1,1) and the trend disturbance models.
6.2.1. Example 1. Step disturbance model
The step disturbance model can be expressed as follows: Nt¼
L t ts
0 t < ts
: ð23Þ
where L is the level of the step change disturbance and ts is the time of the
disturbance introduced into the process. The tuner parameters in the Patel– Jenkins system (equation 10) were set to be the same as their simulation example:
2"¼1, 0¼0.1, 0¼1, ¼ 104, ¼ 0.005, and the step disturbance was
intro-duced at run 50 with L ¼ 10. Figure 7(a) shows the controlled process output under the Patel–Jenkins approach; figure 7(b) plots the EWMA gain tthrough 800 runs.
The off-line trained network was implemented on-line to tune the EWMA controller gain under the step disturbance, which was also introduced at run 50
Run Learning rate Momentum constant Epoch 1 0.35 0.85 54 2 0.15 0.85 28 3 0.35 0.95 51 4 0.15 0.95 37
Table 1. Experimental design results.
Figure 6. Neural network learning behaviour.
with magnitude 10. Figure 8(a) shows the NN-based controlled process output; figure 8(b) plots the NN-based EWMA gain t through 800 runs. As expected, t
increased on a shift and decreased to a small number with time. The performance of the uncontrolled process measured in the inflation factor ( ^e2= ^2") was 5.2795; the controlled inflation factor under the Patel–Jenkins method was 1.8521, and 1.3321 in the NN-based EWMA controller. Therefore, the NN-based adaptive EWMA controller had a superior performance.
6.2.2. Example 2. IMA(1,1) disturbance model
The IMA(1,1) disturbance model is now considered. The moving average parameter ¼ 0.2 and "2¼1 was used to simulate the problem. From equation
(9), it is known that the optimal controller parameter was ¼1 . Assume that the disturbance was introduced at run 50 over 800 runs. Figure 9 shows the EWMA gain under the NN-based adaptive controller. The value of toscillated along with
the optimal controller value (say 0.8). Taking the sample mean of the last 200 runs, it tended to 0.8078, which was close to the optimal controller parameter. The inflation factor of the controlled process output approximated to 1, which implied that the
0 100 200 300 400 500 600 700 800 -4 -2 0 2 4 6 8 10 12 0 100 200 300 400 500 600 700 800 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 t λ
(a)
(b)
Controlled Run No. (t) Run No. (t)Figure 7. Patel–Jenkins adaptive method: (a) controlled output and (b) EWMA gain.
0 100 200 300 400 500 600 700 800 -4 -2 0 2 4 6 8 10 12 0 100 200 300 400 500 600 700 800 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 t λ
(a)
(b)
Controlled Run No. (t) Run No. (t)Figure 8. NN-based adaptive method: (a) controlled output and (b) EWMA gain.
0 100 200 300 400 500 600 700 800 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 t λ
Figure 9. NN-based adaptive EWMA gain under IMA (1,1).
proposed adaptive controller had the ability to produce the minimum mean-square-error (MMSE) process output.
6.2.3. Example 3. Ramp disturbance model with small slope
This example considered the ramp disturbance model, which can be expressed as: Nt¼
Sðt tsÞ t ts
0 t < ts
; ð24Þ
where S is the trend rate (slope). The optimal EWMA controller parameter under the trend disturbance can be obtained by solving the following equation:
2"3S22þ4S2 4S2 ¼0 ð25Þ
Following the example of Patel–Jenkins, the trend disturbance was introduced, with S ¼ 0.1, 2" ¼1 at run 50. Figure 10 shows the EWMA gain under the
NN-based adaptive method. The value of toscillated along with the optimal controller
value (say 0.3061). The inflation factor under Patel–Jenkins was 2.0914, but only 1.3825 under the proposed method.
7. Conclusion
A method for dynamically tuning the single EWMA controller was proposed. The proposed methodology was based on the pattern recognition of the SACF/ SPACF patterns by training the neural network. The behaviour of the off-line-trained network showed that the network learns quickly with input features being SACF/SPACF patterns. It has been shown that the proposed approach has a superior performance over the Patel–Jenkins adaptive algorithm through three implementations. From Example 1, as expected, the NN-based EWMA gain increased on a transient period and then decreased to a small number on a long run. Example 2 showed that the NN-based adaptive EWMA controller approxi-mated to the MMSE controlled performance under the IMA(1,1) stochastic process. From Example 3, it was observed that the EWMA gain behaved close to the optimal controller parameter when the ramp disturbance with a lower slope existed in the process. Although the proposed methodology was implemented via simulation, it is nevertheless anticipated to improve the performance of the EWMA controller in an actual process. 0 100 200 300 400 500 600 700 800 -0.2 -0.1 0 0.1 0.2 0.3 0.4 t λ
Figure 10. NN-based adaptive EWMA gain under trend disturbance.
Further research can use the proposed techniques properly to pick the controller parameters dynamically for the double or triple EWMA controller, which can com-pensate for the server slopes of the trend disturbance model. Also, similar research can be extended to the multiple-input multiple-output system.
References
ANDREWS, R. and GEVA, S., 2002, Rule extraction from local cluster neural nets.
Neurocomputing, 47, 1–20.
BOX, G. E. P. and JENKINS, G. M., 1976, Time Series Analysis, Forecasting and Control, 2nd
edn (San Francisco: Holden-Day).
BOX, G. E. P., JENKINS, G. M. and REINSEL, G., 1994, Time Series Analysis, Forecasting and
Control, 3rd edn (Englewood Cliffs, NJ: Prentice-Hall).
BOX, G. E. P. and KRAMER, T., 1992, Statistical process monitoring and feedback
adjustment—a discussion. Technometrics, 34, 251–285.
BOX, G. E. P. and LUCEN`O, A., 1997, Statistical Control—By Monitoring and Feedback Adjustment(New York: Wiley).
BREIMAN, L. and SPECTOR, P., 1992, Submodel selection and evaluation in regression: the
X-random case. International Statistics Review, 60, 291–319.
BROCKWELL, P. J. and DAVIS, R. A., 1996, Introduction to Time Series and Forecasting
(New York: Springer).
BUTLER, S. W. and STEFANI, J. A., 1994, Supervisory run-to-run control of a polysilicon gate
etch using in situ ellipsometry. IEEE Transactions on Semiconductor Manufacturing, 7, 193–201.
CHEN, A. and ELSAYED, E. A., 2002, Design and performance analysis of the exponentially
weighted moving average mean estimate for processes subject to random step changes. Technometrics, 44, 379–389.
CHENG, B. and TITTERINGTON, D. M., 1994, Neural networks: a review from a statistical
perspective. Statistical Science, 9, 2–54.
DELCASTILLO, E., 2001, Some properties of EWMA feedback quality adjustment schemes for
drifting disturbance. Journal of Quality Technology, 33, 153–166.
DEL CASTILLO, E., 2002a, Closed-loop disturbance identification and controller tuning for
discrete manufacturing processes. Technometrics, 44, 134–141.
DEL CASTILLO, E., 2002b, Statistical Process Adjustment for Quality Control (New York: Wiley).
DELCASTILLO, E. and HURWITZ, A., 1997, Run-to-run process control: literature review and
extensions. Journal of Quality Technology, 29, 184–196.
DELCASTILLO, E. and YEH, J. Y., 1998, An adaptive run-to-run optimizing controller for linear
and nonlinear semiconductor processes. IEEE Transactions on Semiconductor Manufacturing, 11, 285–295.
DEMING, W. E., 1986, Out of Crisis (Cambridge, MA: Massachusetts Institute of Technology,
Center for Advanced Engineering Studies).
FAN, S.-K. S., JIANG, B. C., JEN, C. H. and WANG, C. C., 2002, SISO run-to-run feedback
controller using triple EWMA smoothing for semiconductor manufacturing processes. International Journal of Production Research, 40, 3093–3120.
FATIN, W. F., HAHN, G. J. and TUCKER, W. T., 1990, Discussion. Technometrics, 32, 1–29.
INGOLFSSON, A. and SACHS, E., 1993, Stability and sensitivity of an EWMA controller. Journal
of Quality Technology, 25, 271–287.
LUCAS, J. M. and SACCUCCI, M. S., 1992, Exponentially weighted moving average control
schemes: properties and enhancements. Technometrics, 32, 1–29.
LUCEN`O, A., 1995, Choosing the EWMA parameter in engineering process control. Journal of
Quality Technology, 27, 162–168.
MONTGOMERY, D. C., 1996, Introduction to Statistical Quality Control, 3rd edn (New York:
Wiley).
O’SHAUGHNESSY, P. and HAUGH, L., 2002, EWMA-based bounded adjustment scheme with
adaptive noise variance estimation. Journal of Quality Technology, 34, 327–339.
PAN, R. and DELCASTILLO, E., 2001, Identification and fine tuning of closed-loop processes
under discrete EWMA and PI adjustments. Quality and Reliability Engineering International, 17, 419–427.
PATEL, N. and JENKINS, S., 2000, Adaptive optimization of run-to-run controllers: the EWMA example. IEEE Transactions on Semiconductor Manufacturing, 13, 97–107.
ROBERTS, S. W., 1959, Control chart tests based on geometric moving averages. Technometrics,
1, 239–250.
SASTRI, T., 1988, An adaptive estimation. IIE Transactions, 20, 176–185.
SJOBERG, J. and AGARWAL, M., 2002, Trajectory tracking in batch processes using neural
controllers. Engineering Applications of Artificial Intelligence, 15, 41–51.
SMITH, T. and BONING, D., 1997, A self-tuning EWMA controller utilizing artificial neural
network function approximation techniques. IEEE Transactions on Components, Packaging, and Manufacturing Technology—Part C, 20, 121–132.
SU, C.-T., YANG, T. and KE, C.-M., 2002, A neural-network approach for semiconductor
wafer post-sawing inspection. IEEE Transactions on Semiconductor Manufacturing, 15, 260–266.
WITTEN, I. H. and FRANK, E., 2001, Data Mining: Practical Machine Learning Tools and
Techniques with Java Implementations(San Francisco: Morgan Kaufmann).
ZHANG, P., 1993, Model selection via multifold cross-validation. Annals of Statistics, 21,
299–311.
