A time-varying weights tuning method of the double EWMA controller

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Omega 32 (2004) 473–480

www.elsevier.com/locate/dsw

A time-varying weights tuning method of the double

EWMA controller

Chao-Ton Su

∗

_{, Chun-Chin Hsu}

Department of Industrial Engineering and Management, National Chiao Tung University, 1001 Ta Hsueh Road, Hsinchu, Taiwan Received 2 July 2003; accepted 15 March 2004

Abstract

In recent years, the exponential-weighted-moving-average (EWMA) statistic based controllers are popular in semiconductor manufacturing. However, the single EWMA controller is not su3cient for compensating for the wear-out process. Thus, a double EWMA controller was proposed to enhance the capability for controlling the drifting process. In the literatures, in the solution of the double EWMA controller, only the “trade-o5” solution weights are used to tune the controller. However, it is a 7xed weight tuning method, and it is known that a time-varying weight will produce a superior performance over that of a 7xed one (J. Quality Technol. 29 (1997) 184). Therefore, this study aims to develop a heuristic time-varying weights tuning method for the double EWMA controller. The numerical results showed that the proposed time-varying tuning method possesses an improvement of least 10% over that of the 7xed weight scheme.

Keywords: Double EWMA controller; Trade-o5; All-bias; All-variance

1. Introduction

Variation reduction to enhance product quality in a pro-cess is a crucial determinant of competitiveness for the man-ufacturing industries. Statistical process control (SPC) and engineering process control (EPC) are two important tech-niques for process variation reduction. The main objective of SPC is the on-line “monitoring” of the process, and it has been successfully used in discrete parts manufacturing industries. The objective of the EPC technique is process “adjustment”, which means the use of feedback control al-gorithms to maintain the process as close as possible to the desired target value.

Recently, a control algorithm called the exponen-tial-weighted-moving-average (EWMA) controller is in vogue in the semiconductor manufacturing industry, and particularly in the chemical mechanical polishing (CMP) process. In the literature, the use of the single EWMA

∗_{Corresponding author. Tel.: +886-3-573-1857;}

fax: +886-3-572-2392.

E-mail address:ctsu@cc.nctu.edu.tw(C.-T. Su).

controller had been shown to be e5ective for reducing the process variability. However, a new problem then began to emerge. Many processes were starting to exhibit drifts in the performance of the equipment. Such drifts were often caused by worn-out tools or other systematic causes of deterioration. When using the single EWMA controller to compensate for the drifting process, the controlled process output will exhibit considerable o5sets. Butler and Stefani [1] proposed using two EWMA equations to control the drifting process: one equation for estimating the drift, the other for estimating the step change deviation. It can be shown that the double EWMA controller is a minimum mean square error (MMSE) controller when the disturbance model IMA(2, 2) exists in the process [2].

Until recently, the only solution of the double EWMA controller was to use the 7xed weights solution to tune the controller, until Del Castillo [2] proposed an optimization form for solving the problem. The objective of the optimiza-tion form was to compromise on the transient e5ect and the long run variance. He called the solution the “trade-o5” solution weights. Although the trade-o5 solution is shown to be e5ective, it still is a 7xed weight control scheme. In

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the literature, many researches have shown that the EWMA controller with time-varying parameters possesses better performance than the 7xed weight tuning method. How-ever, available researches all focus on the single EWMA controller. Therefore, there is a great need for develop-ing a time-varydevelop-ing weights tundevelop-ing method to enhance the performance of the double EWMA controller.

In this study, an e5ective time-varying weights tuning approach is developed by adding the discount factor into the double EWMA parameters. The advantage of introduc-ing the discount factor is to quickly compensate for the ini-tial transient e5ect. A simulation study demonstrates that the proposed tuning approach signi7cantly outperforms the 7xed “trade-o5” control scheme.

This study is organized as follows: Section2reviews the double EWMA controller. Section3presents the optimiza-tion form for solving the double EWMA controller prob-lem, as proposed by Del Castillo [2]. In Section4, the pro-posed time-varying tuning method is speci7ed. Section 5

implements the proposed tuning method by using the soft-ware of Matlab/Simulink. In addition a comparison between the 7xed trade-o5 weights control scheme and the proposed time-varying weights tuning method is made by running Monte Carlo simulations.

2. The double EWMA controller

The EWMA statistic, sometimes called a geometric mov-ing average (GMA), was suggested by Roberts [3]. The use of the EWMA statistic has two distinct purposes [4]: as con-trol charts [5–9] and as time series forecasts [10,11]. Re-cently, this statistic has been widely used for process adjust-ment purposes [12–20].

In semiconductor manufacturing, EWMA controllers are sometimes called bias tuning controller. The EWMA-based controllers are used for compensating against disturbances which a5ect the run-by-run variability in quality character-istics [17]. The use of the single EWMA controller is re-lated to the pure-integral (I) control action, which is a part theme of the well-known PID controller. Box and Jenkins [10] showed that the single EWMA statistic is a minimum mean square error (MMSE) controller when the IMA(1,1) disturbance model exists in the process. The IMA(1,1) is an important nonstationary time series model in the study of process regulation and adjustment [7]. Other than the IMA(1,1), the single EWMA controller has been shown to perform e5ectively in some disturbance models such as the step and ramp with slow drift rate disturbance model. How-ever, the single EWMA controller cannot compensate for a ramp disturbance with severe drift speed. That is, a con-siderable o5set will be exhibited in the controlled process output.

Butler and Stefani [1] extended the single EWMA con-troller with another EWMA equation in order to compen-sate for the ramp disturbance model. In this section, we will

brieMy introduce the double EWMA controller that was pro-posed by Butler and Stefani [1]. Note that they do not call it a double EWMA controller, but refer to it as a predictor corrector control (PCC) scheme.

Consider a drifting process model as follows:

yt= + xt−1+ t + t; (1)

where yt is the process output, is the intercept term, is the system gain, xtis the manipulate variable, denotes the drifting speed, and tis the white noise term. A PCC control equation for xtcan be expressed as

xt= T − (a_bt+ Dt); (2)

where T is the target value, b is the estimate of which can be obtained o5-line by using designed experiments in a pre-control phase. at and Dt can be expressed as follows: at= 1(yt− bxt−1) + (1 − 1)at−1; 0 ¡ 16 1; (3) Dt= 2(yt− bxt−1− at−1) + (1 − 2)Dt−1;

0 ¡ 26 1; (4)

where 1 and 2 are the weights for the 7rst and second EWMA equations. Note that if we set 1= 0 or 2= 0, then the double EWMA controller will reduce to a single-EWMA controller. From Eqs. (3) and (4), it is clear that the perfor-mance of a double EWMA controller depends on selecting both parameters of 1 and 2. To appropriately select both

parameters, the stability conditions of controller parameters should be held to as follows [2]:

1 −1₂( 1+ 2) + 1₂ [(( 1+ 2)2− 4 1 2)]¡1; 1 −1₂( 1+ 2) − 1₂ [(( 1+ 2)2− 4 1 2)]¡1; (5) where = =b, is the bias in the gain estimate.

As time approaches in7nity, each at and Dt works as follows:

lim

t→∞E[at] → + (t + 1) − = 1; (6) lim

t→∞E[Dt] → = 1: (7)

We can see that at is an asymptotical estimate of the ramp disturbance with a bias term (= 1), and Dt is the asymp-totical estimate of the bias term. Thus, at + Dt becomes an asymptotically unbiased one-step-ahead estimate of the ramp disturbance model. Another form of Eqs. (3) and (4) can be expressed as follows:

at= 1 1 − (1 − 2)B 1 − 2B + B2 yt; (8) Dt= 2 1 1 − B yt; (9)

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yt filter 1 filter 2 T -+ 1/b xt Unit delay

Double EWMA Filter

intercept Ramp White Noise α 1−(1−λ2)B 1-2B+B2 1 1-B λ2 λ1 βB Σ

Fig. 1. Double EWMA controller.

where B is the backshift operator (Byt= yt−1). We can see that the at term is a second order 7lter ( 2 = 0) and can 7lter out the trend. The term, Dt, is the 7rst order 7lter that can 7lter out the process o5sets. Therefore, the PCC control equations can be expressed as follows:

at+ Dt= 1 1 − (1 − 2)B 1 − 2B + B2 yt+ 2 1 1 − B yt = w1 t i=1 yi+ w2 t i=1 t j=i yj; (10) where, w1= 1+ 2− 1 2and w2= 1 2.

From Eq. (10), we can see that the double EWMA con-troller is not a discrete PID form, but an integral-double-integral (I–II) form [21]. It can be shown that the I–II controller is an MMSE controller when the IMA(2, 2) disturbance model a5ects the process. Compared to the non-stationary IMA(1,1) process (i.e. drift in any direction with equal probability), the IMA(2, 2) noise model can be interpreted as a process that experiences random changes in the drifting speed in Eq. (1) [2]. Fig. 1shows a block diagram of the double EWMA controller model when a ramp disturbance model exists in the process.

3. Trade-o tuning method

The control strategies of EWMA controllers can simply be divided into time-invariant and time-varying weights control schemes. The time-invariant control scheme means that the EWMA weights do not change with time, but that the weights are 7xed to control the process. Del Castillo [15] presented a solution of balancing the adjustment and output variances to control the single-EWMA controller. He also designed a trade-o5 solution of transient and steady-state performance to control the double-EWMA controller (1999). The time-varying control scheme is sometimes called a self-tuning or adaptive control, because

the EWMA gains change with time. Smith and Boning [22] used the neural technique to self-tune the EWMA controller. Del Castillo and Hurwitz [14] used the recursive least squares (RLS) theory to continuously estimate the process parameters. Patel and Jenkins [23] proposed an adaptive EWMA control algorithm by taking the signal-to-noise ratio (SNR) into consideration. The above-mentioned adaptive algorithms all have one point in common, they self-tune the single-EWMA controller, but not the double-EWMA controller. Up till now, when it came to the topic of the double-EWMA control scheme, only Del Castillo [2] has presented an optimization form for solving the double-EWMA gains. This algorithm is introduced below. AVAR(yt) = lim t→∞Var(yt) = 2 1 +₍ 1 1− 2)2 1 22+ 1( 1− 2)2 2 − 1 + 12 2+ 2( 1− 2)2 2 − 2 ; (11) where 2

represents the variance of the white noise term. The transient e5ect is measured by averaging the mean square deviation up to run m, and can be expressed as follows: MSD = 1_m m t=1 E(yt)2 =_m( 1 1− 2)2 ( − 2)2[1 − (1 − 2)2(m+1)] 1 − (1 − 2)2 +2(− 2)( 1₁₋₍₁₋−)[1−(1− 2)m+1(1− 1)m+1] 1)(1− 2) + ( − 1_{1 − (1 −})2[1 − (1 − 1)2(m+1)] 1)2 : (12)

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-6 -4 -2 0 2 4 Controlled Output Run 0 20 40 60 80 100 120 140 160 180 200

Fig. 2. Change control scheme at run 100.

Therefore, the optimization form can be modeled as follows: min

1; 2 k1AVAR(yt) + k2MSD

S:T 0 ¡ 16 1 0 ¡ 26 1;

(13) where the parameters (k1; k2) are determined by the engi-neers. For (k1; k2) = (0; 1), it is an all-bias solution. For (k1; k2) = (1; 0), it becomes an all-variance solution. If we set k1= k2= 1, then a trade-o5 solution will be obtained. Del Castillo [2] suggested that keeping the trade-o5 solution weights to control the process would provide adequate per-formance than a variety case. For example: if we 7rst use the all-bias solution to cancel out the transient e5ect, and then abruptly change it to the all-variance solution at a speci7c time, then the controlled process will incur a new transient at that speci7c time. Fig.2shows the above condition in which, we simulate the process with a drift rate of 0.5. By solving Eq. (13), we obtain the all-bias solution (0.1811, 0.7917) and the all-variance solution (0.0173, 0.1067). Sup-pose we change the control scheme at run 100. Clearly, a new transient occurs at run 100.

4. Proposed time-varying weights tuning method In order to enhance the performance of the double EWMA controller, a simple but e5ective time-varying weights tun-ing algorithm will be proposed in this section. We will 7rst present a preliminary model of the time-varying con-trol scheme and then modify it to be our proposed tuning method.

It was intuition to initially use the all-bias solution to bring the process on target, and then use the all-variance so-lution to reduce the process oscillations around the target. However, it has shown to be an ine3cient tuning method from the viewpoint of “abrupt change”. Therefore, we are

attempting to use the viewpoint of “gradual change”, which means using higher weights 7rst and then “gradually” re-ducing them to the all-variance solution weights. We call this control strategy a GC (gradual change) control scheme, and it can be expressed as follows:

1(t) = 1;+ (f1)t; (14)

2(t) = 2;+ (f2)t; (15)

where f1(0 6 f1¡ 1) and f2(0 6 f2¡ 1) denote the dis-count factor. 1;and 2;individually represent the at’s and Dt’s all-variance solution weight. From Eqs. (14) and (15),

we can see that 1(t) and 2(t) approach the all-variance

solution as time approaches to in7nity.

However, there is a problem for adding a discount factor in Eq. (14). Consider a drifting process in Eq. (1). Fig.3

shows the results of the controlled process (without noise term) when the discount value (f1) is large, moderate and 0.

We can see that a large discount value makes it di3cult for the process to converge to the target value. On the contrary, when the discount value is 0, the process converges quickly to the target. Therefore, we are tempering it by setting the discount value in Eq. (14) to be 0. That is

1(t) = 1;; (16)

2(t) = 2;+ (f)t; (17)

where f (0 6 f ¡ 1) expresses the discount factor. We call Eqs. (16) and (17) the MGC (modi7ed gradual change) control scheme.

From Eq. (10), it can be shown that if the double EWMA controller with the 7xed weights control scheme, then the following equation holds as:

( 1; 2) = (max{ 1; 2}; min{ 1; 2})

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0 0 1 2 3 4 5 6 7 Controlled output Run : f1 = 0 f1 = 0.5 f1 = 0.9 ---- : - : 20 40 60 80 100 120 160 180 200

Fig. 3. Controlled output with f1being large, moderate and 0.

20 40 60 80 100 120 140 160 180 200 0 1 2 3 4 5 6 -

:

MGC-1 ----

:

MGC-2 Run Controlled output

Fig. 4. MGC-1 versus MGC-2 tuning method.

But, for the time-varying weights, the above equation does not hold. Therefore, there are two cases in the MGC control scheme, they are MGC-1:

1(t) = min{ 1;; 2;}; (18)

2(t) = max{ 1;; 2;} + (f)t (19) and MGC-2:

1(t) = max{ 1;; 2;}; (20)

2(t) = min{ 1;; 2;} + (f)t: (21) In order to compare the tuning method between MGC-1 and MGC-2, we assume the discount factor f to be 7xed as a constant. Fig.4shows the controlled process under MGC-1 versus MGC-2 control scheme. It shows that both control schemes converge to the target at almost the same time, but the controlled output under the MGC-2 tuning method shows much smaller o5sets from the target. Thus, we will adopt the MGC-2 to be our proposed time-varying weighs tuning method for the EWMA controller.

The advantage of adding a discount factor in our proposed tuning method is the quick response to the initial transient

e5ect. In Fig.5, the dash and solid line individually repre-sent the controlled output ‘with’ and ‘without’ adding the discount value. For the case of f = 0, the double EWMA controller compensates for the initial transient e5ect more quickly than in the case of f = 0. From the proposed tuning equations, we know that the performance of the controlled process output depends on setting the discount factor (f). Even though a larger discount value can quickly compen-sate for the initial transient e5ect, it may cause oscillations. Therefore, we will present how to determine the discount parameter in the following section.

5. Simulation results

In this section, we 7rst present how to determine the dis-count factor, and then we implement the proposed tuning method by using the software of Matlab/Simulink version 4.1. Finally, we will make a comparison between the 7xed weight trade-o5 solution and the proposed time-varying tun-ing method.

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20 40 60 80 100 120 140 160 180 200 0 1 2 3 4 5 6 7 Controlled output Run - : f =0 ---- : f ≠ 0 ( f = 0.9 )

Fig. 5. Controlled output with and without discount factor.

0.7 0.75 0.8 0.85 0.9 0.95 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 Discount factor (f) MSE / σε 2 Fig. 6. MSE=2

versus discount factor.

5.1. Implementation

For the control performance characterization, the normal-ized mean square error (MSE=2

) is used as the performance measure. The prediction MSE is de7ned as follows:

[ MSE =

_m

t=1(yt− T)2

m :

Therefore, the normalized mean square error is the measure of inMation for the controlled process produced against the natural disturbance (t). To implement the proposed tuning method, the 7rst task is to determine the discount factor. The objective of the chosen discount factor is to minimize the normalized mean square error. Consider a ramp process model in Eq. (1) with = 0, T = 0, = =b = 1, = 1, 2

= 1 and m = 200. By solving Eq. (13), we obtain the

all-variance solution weight ( 1;; 2;) = (0:0247; 0:1486).

Therefore, our proposed tuning method can be expressed as

follows: ∗

1 = max{0:0247; 0:1486}; (22)

∗

2(t) = min{0:0247; 0:1486} + (f)t: (23) A plot of the normalized mean square errors of the controlled process output, obtained from a series of trial values of the discount factor, is shown in Fig.6. The minimum value of

[ MSE=2

appears at f∗= 0:92. Table1shows a similar

pro-cess for determining the discount factor under other drifting speeds. We can see that whatever the drifting speed is, the optimal discount factor always appears at f∗_{= 0:92. Thus,} it shows the robustness for determining the discount factor under our proposed tuning method. Substituting f∗_{= 0:92} into Eq. (23), Fig.7shows the controlled process output, and it shows that observations wander around the target value (T = 0) with [MSE=2

= 1:2481. In addition, if we use the

7xed trade-o5 solution weights to control the process, then [

MSE=2

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Fig. 7. Controlled process output under the proposed time-varying tuning method.

Table 1

Optimal discount factors under various drifting speeds

1; 2; Optimal discount factor 0.1 0.0067 0.0485 0.92 0.5 0.0173 0.1067 0.92 1.0 0.0247 0.1486 0.92 1.5 0.0303 0.1799 0.92 2.0 0.0351 0.2057 0.92 2.5 0.0393 0.2281 0.92 3.0 0.0432 0.2480 0.92

tuning method is 22.88% better than the 7xed weights con-trol scheme. The following section will make a more detailed comparison between the trade-o5 solution, and the proposed tuning method, by using the Monte Carlo simulations. 5.2. Comparisons

In order to validate the e5ectiveness of the proposed time-varying weight tuning method, Monte Carlo simula-tions are performed under various random seeds. In these Monte Carlo simulations, we assume the target value T = 0, = 0, = =b = 1 and t∼ N(0; 1). At each simulation, the performance index, [MSE=2

, is calculated based on the sim-ulation results of 200 runs (m) and 200 simsim-ulation cycles (initial seed from 0 to 199).

Table2shows the comparison results between the 7xed trade-o5 solution weights, and the proposed time-varying tuning methods under various drifting speeds. The estimated standard deviation errors are shown in the parentheses. We can see that the performance between the two control schemes is not signi7cantly di5erent with the slow drifting speed (say = 0:1). But, when the drifting speed is moder-ate to large (say ¿ 0:5), then the proposed time-varying tuning method is much better than the 7xed trade-o5

Table 2

Comparison results

Trade-o5 Proposed Improvement

(MGC-2) (%) over trade-o5 0.1 1.1309 (0.1124) 1.1322 (0.1153) — 0.5 1.3315 (0.1303) 1.1458 (0.1167) 13.95 1.0 1.5096 (0.1447) 1.1955 (0.1214) 20.81 1.5 1.6581 (0.1564) 1.2454 (0.1254) 24.89 2.0 1.7939 (0.1667) 1.2957 (0.1292) 27.77 2.5 1.9222 (0.1763) 1.3469 (0.1329) 29.93 3.0 2.0456 (0.1855) 1.3995 (0.1366) 31.58

control scheme. The last column of Table 2 shows the percent improvement of the proposed time-varying tuning method over the 7xed trade-o5 control scheme. It shows that the larger the drifting speed, the more improved the performance. Thus, it is recommended that when a drifting disturbance model exists in the process, using the proposed method to tune the double EWMA controller will produce a satisfactory performance.

6. Conclusion

In order to enhance the performance of the double EWMA controller, a simple but e5ective time-varying weights tun-ing method was developed. From this study, we know that adding a discount factor into the 7rst EWMA controller equation will decrease the converging speed. But, by adding the discount factor into the second EWMA controller equa-tion, one can quickly compensate for the initial transient e5ect. We also have shown the robustness of determining the discount factor under the proposed tuning method. Fi-nally, through Monte Carlo simulations, we have shown that the proposed time-varying weights tuning method possesses a signi7cant improvement over the 7xed trade-o5 solution

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weights control scheme, especially for the process with a moderate to large drifting rate.

Although the proposed tuning method was implemented via simulation, it is nevertheless anticipated to improve the performance of the double EWMA controller in an actual process. Further research could extend the proposed time-varying tuning technique to the multiple inputs multi-ple outputs (MIMO) system. In addition, the determination of the stability conditions and proving the robustness of the time-varying weights tuning strategy are also important issues.

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