PID Charts for Process Monitoring

Editor’s Note: This article will be presented at the Fall Technical Conference in Wilmington, DE, October 2002.

Proportional Integral Derivative Charts for Process Monitoring Huaiqing Wu

Wei Jiang

Department of Statistics Iowa State University Ames, IA 50011 ([email protected])

INSIGHT, AT&T Morristown, NJ 07960 ([email protected]) Fugee Tsung

Vijayan N. Nair

Department of Industrial Engineering & Engineering Management Hong Kong University of Science & Technology Clear Water Bay, Hong Kong ([email protected])

Department of Statistics University of Michigan Ann Arbor, MI 48109 ([email protected])

Kwok-Leung Tsui School of Industrial & System Engineering Georgia Institute of Technology Atlanta, GA 30332 ([email protected]) We introduce a new class of monitoring procedures based on the relationship between a proportional integral derivative (PID) feedback control scheme and the corresponding prediction scheme. The charts are obtained by applying the PID predictor to the autocorrelated data to get residuals and then monitoring the residuals. This class of procedures includes as special cases several charts that have been recently proposed in the literature and thus provides a unifying framework. The PID charts have three parameters that can be suitably tuned to achieve good average run length (ARL) performance for large or small mean shifts. Methods for determining chart parameters to obtain good ARL performance are discussed. Simulation studies for autoregressive moving average (1
1) models show that PID charts are competitive with the special cause charts of Alwan and Roberts for detecting large shifts and perform better in detecting small to moderate shifts. The effects of model parameter misspecification and bias in estimating the variance of the residuals are investigated in a robustness study. KEY WORDS:

1.

Exponentially weighted moving average; Feedback control; Statistical process control.

INTRODUCTION

Harris and Ross (1991), Superville and Adams (1994), Runger and Willemain (1995), and Runger, Willemain, and Prabhu (1995) investigated process monitoring based on the MMSE prediction errors for autoregressive (AR) (1) models; Wardell, Moskowitz, and Plante (1992, 1994) discussed the performance of the SCC chart for autoregressive moving average (ARMA) (1, 1) and ARMAp
q models; and Vander Wiel (1996) studied the performance of the SCC chart for integrated moving average (IMA) (0, 1, 1) models. These authors compared the performances of the traditional Shewhart, cumulative sum (CUSUM), and exponentially weighted moving average (EWMA) charts applied to the prediction errors.

Recent years have seen considerable interest in the monitoring of industrial processes with serial correlation. Two types of monitoring procedures are common in the literature. The first type uses traditional statistical process control (SPC) charts for independent and identically distributed (iid) data but with modified control limits to account for the autocorrelation. An example is the exponentially weighted moving average chart for stationary processes (EWMAST) studied by Zhang (1998). The second approach uses the special cause chart (SCC) proposed by Alwan and Roberts (1988). Their method “whitens” the autocorrelated data by subtracting their one-step-ahead minimum mean squared error (MMSE) prediction and then monitors the errors from the prediction. If the model and prediction are adequate, then the errors are approximately uncorrelated. Thus traditional SPC techniques for iid data, such as the Shewhart chart, can be applied to these prediction errors. The SCC method has attracted considerable attention and has been further studied by many authors. Among these,

© 2002 American Statistical Association and the American Society for Quality TECHNOMETRICS, AUGUST 2002, VOL. 44, NO. 3 DOI 10.1198/004017002188618392 1

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WEI JIANG ET AL.

The basic idea in the SCC method is to transform the original, autocorrelated data to a set of “residuals” and monitor the residuals. The MMSE predictor used in the SCC chart is optimal for reducing the variance of the residuals but is not necessarily best for the purposes of process monitoring. In fact, as shown by Wardell et al. (1992, 1994), monitoring the original autocorrelated data can outperform the SCC chart when the process is positively autocorrelated. Wardell et al. (1992) gave guidelines for selecting the appropriate chart. Zhang (1998) found that the EWMAST chart performs better than the SCC chart when the process autocorrelation is not very strong and the mean changes are not large. Furthermore, the MMSE predictor is closely tied to a corresponding MMSE scheme in feedback control problems. Despite a huge literature on MMSE-based feedback control, the class of proportional integral derivative (PID) control schemes is more common in industry (Åström and Hägglund 1995). We use an analogous relationship between PID control and the corresponding PID predictor to propose a new class of procedures for process monitoring. As in SCC charts, we transform the autocorrelated data to a set of “residuals” by subtracting the PID predictor and monitoring the residuals. PID charts and control schemes have three parameters corresponding to P, I, and D terms. For feedback control, the parameters are chosen to minimize the variance of the output error (control error) after feedback control. In our case, however, the goal is process monitoring, detection, and average run length (ARL) performance. We discuss the use of two “capability indices” for determining appropriate chart parameters to yield good ARL performance. These capability indices were discussed by Jiang, Tsui, and Woodall (2000) and are based on transient and steady-state levels of the mean shifts for the residual process. Special cases of the PID charts (P, I, D, PI, and PD) are obtained by setting some of the parameter values to 0. We show that the EWMA chart for iid data and the EWMAST chart for correlated data coincide with the P chart. The M-M chart (Montgomery and Mastrangelo 1991) corresponds to the I chart. Thus this new class of procedures provides a unified framework for several existing procedures in the literature. [Readers should keep in mind that the I chart (or control) corresponds to the EWMA predictor, whereas the P chart (P control) corresponds to the EWMA chart. The EWMA chart should not be confused with the EWMA predictor.] Although the PID charts are intended for autocorrelated data, we study their properties briefly for monitoring iid data under level shifts in the mean. We show that the P component is the most important, so that the usual EWMA chart (or P chart in the terminology here) for iid data is close to being optimal within the class of PID charts. The I component adversely affects ARL performance and should not be used with iid data. The D component gives some improvement, so PD charts can outperform EWMA charts in the iid case, but the improvements are small in this (iid) case. For correlated data, we study the performance of the PID charts and compare them to the SCC chart and the EWMAST chart (or P chart) under ARMA(1
1) models. We show that the parameters of the PID charts can be tuned to detect small or large shifts, and that PID charts outperform SCC TECHNOMETRICS, AUGUST 2002, VOL. 44, NO. 3

charts in detecting small to moderate shifts. The SCC charts generally do well for detecting large shifts, although the parameters of the PID charts can be chosen judiciously to be competitive with the SCC charts for large shifts. This flexibility is an advantage of PID charts. The PID charts outperform EWMAST and M-M charts, indicating that the D component is useful in the autocorrelated case. We also compare ARL performance when the model parameters are wrongly estimated. We show that the SCC chart is very nonrobust when the standard deviation is estimated by a moving average (Adams and Tseng 1998). A suitably modified SCC chart and the PID charts have comparable robustness properties except in cases where the ARMA model is close to being nonstationary. In these cases, the I term in the PID chart becomes important. Of course, in these situations, it is possible to modify the MMSE predictor to include an I term. (See Luceño 1998 for similar ideas for improving the robustness of the MMSE controller.) The article is organized as follows. We formulate and develop the PID charts in Section 2, and illustrate them with an example in Section 3. In Section 4 we describe a heuristic approach for determining appropriate parameters of the PID charts. We investigate ARL performance for monitoring mean shifts under iid data in Section 5, and study performance and comparisons under ARMA(1, 1) processes in Section 6. We provide concluding remarks in Section 7. 2.

THE PROPORTIONAL INTEGRAL DERIVATIVE CHART

Suppose that we have a feedback-controlled process with a disturbance Dt and a pure-gain dynamic model Yt+1 = Ut , where Yt+1 is the process output at time t + 1 from the process dynamics and Ut is the control action at time t. We assume, without loss of generality, that the target value is 0, so the output can also be viewed as the deviation from target (i.e., the output error or control error). Then the control error at time t + 1 is et+1 = Yt+1 + Dt+1 = Ut + Dt+1  (1)
t+1 is an estimate (i.e., a prediction) of Dt+1 Suppose that D made at time t. Then a realizable form of control could be obtained by setting
t+1  Ut = −D (2) Thus the control error at time t + 1 becomes
t+1  et+1 = Dt+1 − D

(3)

The control error (3) equals numerically the prediction error
t+1 to predict Dt+1 . Therefore, a feedback conwhen using D trol and a prediction can have an analogous relationship via (2) and (3). For example, the MMSE control of the process (with the pure-gain dynamic model) corresponds to the MMSE prediction of the disturbance. An alternative to MMSE control is the widely used class of PID control schemes, 1 e − kD 1 − Bet
(4) Ut = −kP et − kI 1−B t
t , B is the backward shift operator defined where et = Dt − D by Bet = et−1 and kP
kI , and kD are constants that determine

PROPORTIONAL INTEGRAL DERIVATIVE CHARTS

the amount of proportional, integral, and derivative control actions (Box, Jenkins, and Reinsel 1994; Åström and Hägglund 1995). It follows from (2) and (4) that the predictor corresponding to PID control is given by
t+1 = D
t + kI et + kP 1 − Bet + kD 1 − B2 et  D

(5)


t given by (5) the PID predictor. It We call the predictor D is a special case of the general linear predictor 
t+1 = D
t + cj et−j
D j≥0

where cj ’s are suitably chosen adjustment parameters. The PID predictor has only three terms; that is, the prediction update is based on only the three most recent error terms. When kD = 0, we have the PI predictor, which corresponds to the proportional integral control scheme commonly used in industry (Box and Luceño 1997). When kP = kD = 0, we have the I predictor, which is the well-known EWMA predictor (Box and Luceño 1997). The PID charts are obtained by subtracting the PID predictor from the original data to yield the PID-based residuals and monitoring the residuals. Because the residuals are correlated, we must take into account the correlation structure when computing the control limits. We can use any of the traditional approaches, such as Shewhart, CUSUM, or EWMA, to monitor the residuals; here we restrict attention to Shewhart charts on the residuals.
t , we have et − et−1 = Dt − Dt−1  − Because et = Dt − D

Dt − Dt−1 . Thus et in (5) can also be written as et = 1 − kI et−1 − kP 1 − Bet−1 − kD 1 − B2 et−1 + Dt − Dt−1 

(6)

It is reasonable to require that et be stationary when the monitored process Dt is stationary. Otherwise, the charting process et would drift even when the monitored process has no mean shift. That requirement leads to the following conditions for the parameters kP
kI
kD  (Tsung and Shi 1999): kP
kI
kD  kI ≥ 0
kP + kI /2 + 2kD < 1
kD  < 1
kP + kD 1 + kP + kI  + 1 > 0

(7)

Special cases of the PID charts arise by setting appropriate chart parameters to 0. For example, the P chart corresponds Table 1. Shift ( = /D ) 0 .5 1 2 3 NOTE:

3

to kI = kD = 0; other special cases such as the I, D, PI, and PD charts arise similarly. These coincide with some recently proposed and well-known control charts in the literature:
t = Dt − • The I chart has kP = kD = 0. By (6), D
t−1 ; that is, D
t is et = Dt−1 − 1 − kI et−1 = kI Dt−1 + 1 − kI D an EWMA predictor of Dt (Box and Luceño 1997, p. 95). Hence the I chart is the same as the M-M chart (Montgomery and Mastrangelo 1991). • The P chart has kI = kD = 0, so (6) becomes et = −kP et−1 + Dt . Hence et = 1 + kP B−1 Dt = Dt + −kP Dt−1 + t /, where  = 1 + kP , and D t =  −kP 2 Dt−2 + · · · = D 2 Dt + 1 − Dt−1 + 1 −  Dt−2 + · · ·  is an EWMA of Dt . Thus, when the Shewhart chart is applied to et , the P chart (with −1 < kP ≤ 0) is equivalent to the EWMAST chart (Zhang 1998), provided that  = 1 + kP and that the control limits ±l and ±l are used for the P and EWMAST charts. • When Dt is an iid process, the EWMAST chart becomes the EWMA chart, and the P chart is then equivalent to the EWMA chart. Note that there is no connection between the EWMA chart and the EWMA predictor. As shown earlier, I control leads to the EWMA predictor, and the EWMA prediction-based chart is the I chart (i.e., the M-M chart). 3.

ILLUSTRATIVE EXAMPLE

Consider the following example from Pandit and Wu (1983) and also discussed by Jiang et al. (2000). The data are serially correlated observations from a mechanical vibratory system consisting of a mass, a spring, and a dashpot (see fig. 7.1 of Pandit and Wu 1983). The observations are vertical displacements of the mass. Such a system could represent the damping system of a vehicle. As shown by Jiang et al. (2000), the ARMA(2
1) process Dt = 14385Dt−1 − 6000Dt−2 + at + 5193at−1


(8)

with ˆ D = 9130 and ˆ a = 2212 fitted the first 100 observations well. Suppose now that a mean shift of  = /D occurs after the first 100 observations. Table 1 shows the ARL needed to detect

The ARLs of the SCC, P, PI and PD Charts

Chart Parameters L=

SCC 3.000

P
−8 0 0 2.556

PI
−3 18 0 2.950

PD
−8 0 5 2.474

370
100 297
71 982
21 100
00 100
00

370
100 134
32 411
10 102
03 497
01

370
100 331
75 112
22 100
00 100
00

370
100 112
24 345
08 985
02 515
01

Standard errors are given in parentheses.

TECHNOMETRICS, AUGUST 2002, VOL. 44, NO. 3

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WEI JIANG ET AL.

the shift for various values of  for several PID charts and the SCC chart. These values are based on a simulation with 160,000 replications. The simulation standard errors (shown inside parentheses) are negligible. As seen from Table 1, the SCC chart is very good for detecting large shifts but performs poorly for small to moderate shifts. The EWMAST (or P) chart is better for these cases, but its ARLs are still quite large. Thus we need charts that have better performance with small to moderate shifts. The PD chart in Table 1 can detect shifts of magnitude  = 5
1, and 2 more quickly than the EWMAST chart. There is a reduction in ARLs of about 16% for  = 5 and 1 compared to the EWMAST chart. On the other hand, the PI chart in Table 1 has comparable performance to the SCC chart for detecting large shifts. We provide a detailed explanation of the behavior of the various charts for this example in Section 4.2, and give a summary here. The detection ability and ARLs of the charts depend on the transient and steady-state shift levels (and capabilities) of the corresponding residual processes (see Sec. 4). If the transient capability is high, the chart will signal quickly. For a shift of  = 5, however, all four charts have small transient capability indices. For small shifts, we have to rely on their steady-state capability indices. The parameters of the PID charts can be tuned to optimize this. The PD chart in Table 1 has good steady-state capability, and hence better ARL, for detecting a shift of this magnitude. Similar explanations are provided in Section 4.2 for  = 1. We see that the class of PID charts is flexible because the chart parameters can be tuned to achieve good performance for small or large shifts. The EWMAST and SCC charts do not have this flexibility. The choice of parameter values for the PID charts is discussed in the next section. By (11), tuning of the PID chart requires computing only RE = D2 /e2 , which can be estimated from the sample variances of Dt and et . This example demonstrates that the PID charts are competitive with existing charts such as the SCC and EWMAST charts. A more extensive comparison appears in Sections 5 and 6. Section 6.2 shows that the PI chart (with kI > 0) has good robustness properties when the process Dt is close to being nonstationary.

4. 4.1

DESIGN OF PROPORTIONAL INTEGRAL DERIVATIVE CHARTS

Capability Indices

This section introduces the transient and steady-state capability indices, briefly discusses their properties, and derives formulas for computing them for a PID chart (under a stationary process) and for the SCC chart [under an ARMA(1, 1) process]. The indices are used to select appropriate parameters of a PID chart, as is discussed in the next section. Consider monitoring a stationary process Dt with mean 0 and variance D2 . The process is said to have a mean shift of  at time t0 if the process is Dt for t < t0 and becomes  + Dt for t ≥ t0 . In this case, assuming et = 0 for t ≤ 0, the mean t of the PID charting process et (prediction error) in (6) is TECHNOMETRICS, AUGUST 2002, VOL. 44, NO. 3

given by t = 0 for t < t0 , t0 = , and t = 1 − kI t−1 − kP 1 − Bt−1 − kD 1 − B2 t−1
for t > t0 

(9)

Then the transient shift, T , is t0 = , the first mean response just after the shift, and the steady-state shift, S , is limt→ t . As shown in the Appendix, we have S = 0 for kI > 0 and S = /1 + kP  for kI = 0. Following the signal-to-noise (SN) ratios introduced by Jiang et al. (2000), we can choose appropriate parameters of a PID chart using the transient capability, CT = T /e , and the steady-state capability, CS = S /e , where e2 is the variance of the charting process et when the monitored process Dt is in control. For the PID chart, we have  √ CT = = RE ·  and e (10)   CT   = if kI = 0 CS = e 1 + kP  1 + kP  0 if k > 0
I

where  = /D and RE = D2 /e2 is the relative efficiency of the corresponding PID feedback control (Tsung and Shi 1999). Note that  represents the magnitude of the shift in terms of D , the standard deviation of the monitored process. In this article, we use  = 1
2, and 3 to represent small, moderate, and large mean shifts. Note that the D term (kD ) of a PID chart does not affect the values of T and S and that it affects the values of CT and CS only through RE. The transient capability, CT , measures the chart’s capability to detect the shift in the first few runs and is more appropriate for large shifts. If the chart fails to signal early, then the steady-state capability, CS , becomes important for detecting the shift efficiently in later runs. Because CS = 0 for kI > 0, the PID chart with kI > 0 (e.g., the I or M-M chart) is generally not good for detecting small shifts. We now consider the details when the disturbance is an ARMA(1
1) process, Dt =

1 − %B a
1 − &B t

(11)

where at is white noise with variance a2 and the parameters & and % satisfy the conditions & < 1 and % < 1. The variance of Dt in (11) is D2 =

1 − 2&% + % 2 2 a  1 − &2

(12)

The variance of the PID residuals from this process, e2 , can be computed in a straightforward manner (see, e.g., Tsung 1999). In general, e2 and D2 can be estimated by the sample variances of et in (6) and Dt . Note that e2 is also used in setting control limits ±Le for the PID chart, which signals when et exceeds the limits. When the SCC chart is used to monitor an ARMA(1, 1) process (11), the charting process is et = %et−1 + Dt − &Dt−1 . Then et = at when the process Dt is in control, and thus e = a . For a mean shift of  that occurs at time t0 , the mean t of et for the SCC chart is given by t = 0 for t < t0 ,

PROPORTIONAL INTEGRAL DERIVATIVE CHARTS

t0 = , and t = %t−1 + 1 − & for t > t0 . Thus T =  and S = 1 − &/1 − %. It then follows from (12) that the two capability indices, CT and CS , for the SCC chart are given by 1/2  1 − 2&% + % 2 CT =  · and 1 − &2 (13)  1−& CS = CT  1−% 4.2

Choosing Proportional Integral Derivative Chart Parameters

A PID chart with large transient capability CT >5 will be able to detect the shift quickly. If such a PID chart cannot be found for a specified shift level to be detected, then a PID chart with a large value of CS can often detect the shift efficiently at the steady state. However, if CS is tuned too high, then it may result in a very small value of CT and make the transition of the shifts t from the transient to the steady state too slow. In this case, a value of CS around 3 is often appropriate for balancing the values of CT and CS . These observations are consistent with those of Jiang et al. (2000) for their ARMA chart. We thus adapt their heuristic algorithm and suggest the following procedure for choosing appropriate parameters of a PID chart: 1. Specify the shift level  to be detected. 2. Compute Max CT , the maximum value of CT for the PID chart, by varying its chart parameters kP
kI
kD . 3. If Max CT > 5, then choose the PID chart with the transient capability equal to Max CT and stop; otherwise, go to the next step. 4. Compute Max CS , the maximum value of CS for the PD chart, by varying its chart parameters kP
0
kD . 5. If Max CS ≤ 35 or if CT ≥ 1 when CS is maximized, then choose the PD chart with the steady-state capability equal to Max CS ; otherwise, choose a PD chart with CS ∈ 25
35 to balance the values of CT and CS . From our experience, Max CT in step 2 is often obtained from a PI chart. Thus we can set kD = 0 there and simplify this step. Note that no closed-form expressions exist for Max CS and Max CT . Hence numerical methods are used for Table 2. kP = Shift
 0 5 1 2 3 NOTE:

kI = L= e /a =

5

the maximization in steps 2 and 4 over the parameter spaces of kP
kI
0 and kP
0
kD . The PID chart parameters in Table 1 were obtained using this procedure. We can explain the results in Table 1 using the transient and steady-state capability indices CT and CS . First, it is easy to see that the values of CT and CS for the SCC chart for model (8) are given by   D CT =  and = e a (14) 1 − 14385 − −6000 CS = CT  1 − −5193 Recall from Section 3 that D was estimated to be 9130. The estimated values of e were 2212
30233
2799, and 25621 for the SCC, P (EWMAST), PI, and PD charts. For detecting a shift of  = 5, the two indices CT and CS  of the SCC, P, PI, and PD charts are 206
22
15
76
163
0, and 18
89. Because these CT values are not high enough, the charts cannot detect the shift quickly. Then the values of CS become important for detecting the shift. In this case, the PD and P charts have the highest steady-state capabilities of .89 and .76 and thus the shortest ARLs of 112 and 134. The same is true for detecting a shift of  = 1. In this case, CT for the SCC chart is quite high (4.13), but its CS value is much smaller than those for the PD and P charts (.44 versus 1.78 and 1.51). Thus its ARL is 98.2, compared to 34.5 for the PD chart and 41.1 for the P chart. For detecting a shift of  = 3, the two indices (CT and CS ) of the SCC, P, PI, and PD charts are 124
132
91
453
979
0, and 107
535. In this case, the SCC and PI charts have very high transient capability values (124 and 979); thus they can detect the shift very quickly, with ARLs of about 1. These observations also show that the two capability indices are indeed good indicators of the performance of a PID chart. Of course, the properties of the charts depend on how the transient capability changes, not just the initial value. 5.

INDEPENDENT AND IDENTICALLY DISTRIBUTED DATA

Although the PID charts are intended for autocorrelated data, we briefly study their performance under the iid case. Tables 2 and 3 present simulation results in terms of ARL

ARLs of the PI Chart for an iid Process With a Mean Shift

0

−8

.8

0 3000 1000

1 3000 1026

2 2998 1054

0 2864 1667

1 2882 1690

2 2889 1715

0 2862 1667

1 2808 1925

2 2717 2357

370
100 155
31 439
09 631
01 200
00

371
100 364
98 341
75 206
42 383
08

370
100 365
98 353
75 274
45 114
24

371
100 357
08 965
02 354
01 228
00

370
100 360
97 322
72 146
32 190
04

370
100 366
98 347
74 249
58 100
24

369
100 313
72 215
51 910
21 385
07

370
100 369
100 364
99 342
87 302
82

370
100 369
100 366
99 356
92 335
85

Standard errors are given in parentheses.

TECHNOMETRICS, AUGUST 2002, VOL. 44, NO. 3

6

WEI JIANG ET AL. Table 3. kP = kD = L= e /a =

Shift
 0 5 1 2 3 NOTE:

ARLs of the PD Chart for an iid Process With a Mean Shift 0

−8

−2 2996 1035

−1 2999 1009

0 3000 1000

1 2999 1011

2 2993 1054

−2 2855 1846

−1 2860 1748

0 2864 1667

1 2866 1599

2 2871 1543

370
100 162
37 473
09 648
01 195
00

370
100 156
37 445
09 624
01 195
00

370
100 155
37 439
09 631
01 200
00

370
100 157
37 454
09 670
01 214
00

370
100 166
37 504
10 785
02 245
00

370
100 403
08 990
02 345
01 229
00

371
100 375
08 975
02 350
01 227
00

371
100 357
08 965
02 354
01 228
00

370
100 342
08 957
02 361
01 229
00

370
100 332
07 959
02 368
01 232
00

Standard errors are given in parentheses.

based on 160,000 replications for the PI and PD charts for detecting various magnitudes of mean shifts for an iid process Dt = at . Several conclusions can be drawn from these tables. The Shewhart chart corresponds to kP = kI = kD = 0, whereas the P chart has kI = kD = 0. Table 2 shows that the P charts with −1 < kP ≤ 0 perform much better than the PI charts with kI > 0. In particular, the Shewhart chart is the best for detecting large mean shifts, and the P chart with kP = −8 is the best for detecting small and moderate mean shifts. This latter observation is consistent with the results of Lucas and Saccucci (1990), because for the iid process, the P chart is equivalent to the EWMA chart with  = 1 + kP , as shown in Section 2. Note that CT = CS = /a =  for the Shewhart chart and CT = /e < /a and CS = 0 for the PID chart with kI > 0, because e > a = D for an iid process. Thus the Shewhart chart has higher capability indices and is expected to perform better than the PID charts with kI > 0. Therefore, for an iid process, we need to focus on only the PD charts. As shown in Table 2, the P charts with kP > 0 do not perform well. Thus we consider only the PD charts with kP ≤ 0. Table 3 shows that the D chart with kD = −1 is the best (and better than the Shewhart chart) for detecting large mean shifts, and the PD chart with kP = −8 and kD = 2 is the best (and better than the EWMA chart with  = 2) for detecting small mean shifts. The table also indicates that for detecting both small and large shifts, the ARL of the PD chart depends mainly on kP (−1 < kP ≤ 0), although for fixed kP , the ARL varies slightly (quadratically) as kD changes. 6. 6.1

AUTOCORRELATED DATA

ARL Performance

We now compare the ARL performance of the PID and SCC charts under ARMA(1
1) models. We also study the special case of the EWMAST chart with  = 2 (equivalently, the P chart with kP = −8), which has been proposed for monitoring autocorrelated data (Zhang 1998). The capability indices were used to design two PID charts (denoted by PID1 and PID2 ) aimed at detecting small ( = 1) and large ( = 3) shifts for the various scenarios. A simulation study with 160,000 TECHNOMETRICS, AUGUST 2002, VOL. 44, NO. 3

replications was carried out for different ARMA(1
1) models; the results are given in Table 4. The ARL values for the shift level of  are denoted by ARL in the table. Without loss of generality, we used a = 1 in the simulation. For a fair comparison, the values of L (for the control limits ±Le ) were chosen so that the in-control ARLs for all of the charts were around 370. For each PID chart, the chart parameters kP
kI
kD , the standard deviation of the charting process e , and the values of L are given in the table. The CT and CS values for the shift of  = 1 are also given. Multiplying these numbers by  gives the CT and CS values for other shift levels. Table 4 shows that the PID1 charts perform best for detecting small shifts of  = 5 and 1 for most of the ARMA(1
1) models. For example, for detecting a small mean shift of  = 1 for the AR(1) model with & = 475, the ARL for the PID1 chart is 19.6, whereas the ARLs for the EWMAST, PID2 , and SCC charts are 224
740, and 118. Again, this can be explained by the capability indices. Note that the values of CT are all relatively small. Hence the ARLs are determined mainly by the CS values, which are 3.60, 2.01, .80, and .60 for the PID1 , EWMAST, PID2 , and SCC charts. For detecting a mean shift of  = 5 for this model, the ARLs for the PID1 , EWMAST, PID2 , and SCC charts are 557
833
203, and 253. The table also shows that the SCC charts perform best for detecting a large mean shift of  = 3 for most of the models. However, the PID2 charts perform competitively with the SCC charts. For example, for the AR(1) model with & = 475, the ARL for the PID2 chart is 27, which is smaller than 4.2, the ARL for the SCC chart. In contrast, for the ARMA(1, 1) model with & = 95 and % = 45, the ARLs for the PID2 and SCC charts are 26 and 13, respectively. For other models in the table, the ARLs for the PID2 and SCC charts are fairly close. For detecting a moderate mean shift of  = 2, there are no clear winners, with each of the four charts performing well under one or more cases. Note that the PID charts in all but the third and last cases in Table 4 are PD charts with the I action absent. The last case in Table 4 deserves some elaboration. In this situation, the tuning process for detecting small and large shifts led to the same PID chart, so PID1 and PID2 are the same. In fact, this is an I chart (M-M chart). The ARL performance of this chart is essentially the same as that of the SCC chart. However, as

PROPORTIONAL INTEGRAL DERIVATIVE CHARTS Table 4.

ARL Performance of the PID1 , PID2 , P (EWMAST), and SCC Charts for ARMA(121) Processes





Chart

kP

475

00

−9 4 −8

000

45

950

45

PID1 PID2 P SCC PID1 PID2 P SCC PID1 PID2 P SCC PID1 PID2 P SCC PID1 PID2 P SCC PID1 PID2 P SCC PID1 PID2 P SCC PID1 PID2 P SCC PID1 PID2 P SCC PID1 PID2 P SCC

950

00

475

−90

000

−90

−475

00

−475

−90

−95

−45

950

−90

7

kI

kD

CT

CS

e

L

ARL1

ARL2

ARL3

0 0 0

8 −1 0

36 111 40 114

360 80 201 60

316 102 283 100

253 300 273 300

557 2030 833 2530

196 740 224 1180

79 122 62 226

48 27 34 42

−8 −5 −8

0 0 0

−2 0 0

88 109 95 110

442 219 474 199

124 100 116 100

296 300 297 300

126 375 124 456

43 69 46 80

22 22 22 22

16 14 16 14

−9 0 −8

0

5 0 0

14 169 24 189

142 0 118 17

1332 111 802 100

200 298 225 300

2150 3600 2260 3500

921 3100 975 2750

253 798 219 435

113 26 72 13

−8 9 −8

0 0 0

7 0 0

24 156 22 320

120 82 111 16

1340 205 1445 100

210 274 220 300

2320 2530 2370 3310

1050 1170 1080 1390

267 147 257 11

101 14 83 10

−9 8 −8

0 0 0

8 0 0

32 157 35 186

317 87 176 51

584 118 527 100

241 298 269 300

691 1910 1050 1090

237 631 298 228

90 56 77 28

54 11 40 10

−8 8 −8

0 0 0

5 0 0

55 133 45 135

274 74 224 71

245 101 301 100

274 300 278 300

542 2150 659 2210

156 803 172 829

54 95 52 81

32 15 30 14

−8 −3 −8

0 0 0

−4 −2 0

75 113 90 114

373 161 448 168

152 101 127 100

291 300 294 300

170 704 147 655

46 122 50 114

23 22 23 22

18 14 16 14

−8 3 −8

0 0 0

2 2 0

57 109 53 111

286 84 264 86

193 102 211 100

283 300 283 300

430 1900 470 1850

119 634 123 601

43 93 42 84

26 23 26 21

−8 −3 −8

0 0 0

−3 −4 0

115 149 120 189

575 213 599 254

164 127 158 100

292 293 289 300

75 402 83 242

30 45 33 34

18 15 18 12

13 11 12 10

0 0 −8

19 19 0

0 0 0

593 593 22 601

0 0 110 16

101 101 2742 100

300 300 219 300

426 426 2400 428

10 10 1100 10

10 10 264 10

10 10 85 10

0

2

ARL05

NOTE: The CT and CS values given are for the shift of  = 1. Multiplying these numbers by  gives the CT and CS values for other shift levels. Standard errors are approximately 1/400 times the corresponding ARL values.

we show in the next section, the PID chart is more robust to model misspecification than the SCC chart near this region of the parameter space. Note also that the EWMAST chart does poorly in this case. In summary, none of the charts performs uniformly better than any of the other charts, but the PID1 chart generally performs best for detecting small shifts of  = 5 and 1. The advantage of the PID charts is that their relative ARL performance can be predicted based on the two capability indices computed from the charting parameters. Consequently, the charts can be tuned to have good performance for situations of interest. As shown in Table 4, such well-tuned PID charts perform competitively with the better one of the SCC and EWMAST charts. Note that similar conclusions seem to hold for more general situations such as the ARMA(2
1) case in Section 3, but this deserves further study. 6.2

Robustness to Parameter Misspecification

Adams and Tseng (1998) defined the robustness of a control chart by how its ARL changes when the process

model is misspecified. In this section we investigate the robustness of PID and SCC charts to misspecification of the parameters in the ARMA(1
1) model. Specifically, we assume that the true ARMA parameters &
% are wrongly estimated or assumed to be &∗
% ∗ . The ARL∗ values in Table 5 show the performance of the charts based on the wrong model. The true parameters &
% are assumed to be in the neighborhood of &∗ ± 05
% ∗ ± 05. Robustness is studied under four assumed ARMA(1
1) processes with &∗
% ∗  = 475
0
0
45
−475
−9, and 95
−9. Note that the last assumed model is very close to a nonstationary process. For these values, we computed the control limits ±L∗ · e∗ from the assumed model, where e∗ is the standard deviation of the charting process when the charts are applied to data from the assumed model. For example, L∗ = 3 and e∗ = a for the SCC chart, and L∗ = 2534 and e∗ = 3156a for the PID1 chart when the underlying process is assumed to have an ARMA(1, 1) model with parameters 475
0 (see Table 4). To separate different sources of errors, we also examined TECHNOMETRICS, AUGUST 2002, VOL. 44, NO. 3

8

WEI JIANG ET AL. Table 5.

ARL Performance of the PID and SCC Charts When the Model Parameters Incorrectly Estimated Shift

∗

∗

(   )
475 0


045

( )





425 05

0


525 −05

1 2 3 0 5 1 2 3

793 600 193 773 480 199 512 199 801 491

367 198 701 109 241 373 214 814 143 327

1 2 3 0 5 1 2 3

351 105 393 207 156 394 156 488 233 172

688 126 426 218 163 208 125 440 220 163

0 5 1 2 3 0 5 1 2 3

362 382 108 397 246 377 485 133 462 280


9 −85

0


999 −95

1 2 3 0 5 1 2 3

372 598 100 100 100 370 342 100 100 100


−05 5


−525 −85

5

0

5

5

ARL

ARL

332 431 159 664 416 415 723 247 941 572

SCC ARL


−425 −95


95 −9

PID2 ∗

ARL


05 4


−475 −9

PID1

∗

SCCMR ∗

ARL∗∗

ARL

ARL

430 222 793 118 258 289 166 666 122 295

370 254 118 225 394 370 256 120 238 443

354 238 113 218 385 352 241 116 228 426

595 398 178 325 542 223 158 784 164 321

369 367 664 216 140 371 398 744 225 142

372 367 664 215 140 333 363 719 221 140

370 456 786 222 140 370 471 851 229 141

354 439 766 222 139 339 452 843 221 140

595 483 737 223 143 223 264 578 202 132

585 474 121 426 260 249 392 118 431 265

370 192 650 957 233 368 191 653 964 234

339 174 602 909 227 339 174 602 909 227

371 189 620 881 218 370 188 621 880 218

327 165 558 817 210 334 168 570 832 211

669 319 987 124 265 247 131 454 701 193

307 412 100 100 100 393 360 100 100 100

372 598 100 100 100 370 342 100 100 100

307 412 100 100 100 393 360 100 100 100

372 581 100 100 100 2841 2265 422 100 100

327 442 100 100 100 164 160 100 100 100

601 129 100 100 100 137 975 100 100 100

NOTE: Standard errors are approximately 1/400 times the corresponding ARL (or ARL∗ and ARL∗∗ ) values. The ARL, ARL∗ , and ARL∗∗ values are obtained by setting control limits at ±L∗ e , ±L∗ e∗ , and ±L∗ ˆ MR . Here the values of L∗ are the corresponding L values in Table 4 for the assumed models, e and e∗ are defined in the second paragraph of this section 6.2, and ˆ MR is the moving range estimator of e .

the ARLs with control limits ±L∗ · e , where e is the true standard deviation of the charted process et . This is relevant because the model may be identified incorrectly but we may still be able to estimate the underlying variance unbiasedly using in-control data. These simulation results are denoted by ARL in Table 5. Adams and Tseng (1998) used a moving range estimator, ˆ MR , to estimate e , the true standard deviation of the residuals in the SCC chart when the underlying process is in control. The SCC chart with control limits ±3ˆ MR are denoted by SCCMR , and the corresponding ARLs are denoted by ARL∗∗ in Table 5. Based on their simulation study, Adams and Tseng (1998) found that this SCC chart is not robust when the model parameters are wrongly estimated; however, this is mainly due to the fact that the moving range estimator is biased. Note that

Eˆ MR  = e 1 − )e TECHNOMETRICS, AUGUST 2002, VOL. 44, NO. 3

(15)

(Cryer and Ryan 1990), where )e is the first-lag autocorrelation coefficient of et . The least robust cases (with the ARLs most different from their assumed values) always occur when &
% are at the corners of the neighborhood; that is, when &
% = &∗ ± 05
% ∗ ± 05. Table 5 reports the two least robust cases for each assumed model. The results are based on 160,000 replications. The robustness problem of the SCC chart reported by Adams and Tseng (1998) is caused mainly by the biased estimate of e . This is clear because the SCCMR chart is not robust, whereas the SCC chart with control limits ±3e is quite robust (see the ARL column for the SCC chart in Table 5). One exception is the last case with &∗
% ∗  = 95
−9 and &
% = 999
−95, for which the in-control ARL becomes extremely large. Thus use of the moving range estimate should be avoided.

PROPORTIONAL INTEGRAL DERIVATIVE CHARTS Table 6.

Values of R for the PID and SCC Charts When the Model Parameters are Incorrectly Estimated R

(∗   ∗ )

( )

PID1

PID2

SCC

SCCMR


475 0


425 05
525 −05
−05 5
05 4
−525 −85
−425 −95
9 −85
999 −95

114 88 106 93 105 94 98 101

101 97 100 99 99 99 98 101

99 99 99 99 99 99 99 66

105 95 105 95 107 97 106 64


0 45
−475 −9
95 −9

Table 5 also shows that the performance of the PID2 chart is similar to that of the SCC chart except for the last case, in which the PID2 chart (with kP = kD = 0 and kI = 19) is more robust. This is because the PID chart includes a nonzero I term (see Table 4), which is appropriate when the process is close to being nonstationary. Of course, one could also make the SCC chart more robust by adding a similar term in the MMSE predictor (see Luceño 1998 for similar ideas for improving the robustness of the MMSE controller). The PID1 chart is less robust than the PID2 and SCC charts. This is due mainly to the fact that the PID1 chart is very sensitive to small mean shifts but the other two charts are not. Because the estimation errors of the model parameters can be viewed as some kind of process “shift,” a control chart that is sensitive to small shifts will also be sensitive to the estimation errors and thus is less robust. In general, it is difficult to design a control chart that is both sensitive to small shifts and robust to model misspecification. The ARL∗ columns in Table 5 show that the PID1 , PID2 , and SCC charts with control limits L∗ · e∗ are less robust than the corresponding charts with control limits L∗ · e . This implies that the estimation error of e has a big impact on the robustness of a control chart. Table 6 gives the values of R = e∗ /e for the PID1 , PID2 , and SCC charts and R = Eˆ MR /e for the SCCMR chart. Tables 5 and 6 show that a relative error R − 1 > 03 for estimating e can make a control chart nonrobust; see, for example, the SCCMR chart. 7.

CONCLUDING REMARKS

We have proposed a new class of monitoring procedures based on the relationship between widely used PID controllers and their corresponding predictors. However, the rationales for the controller/predictor and the charts are quite different. Choosing appropriate parameters for the PID chart requires balancing output variance and shift dynamics to detect the shift as quickly as possible. The PID charts present a unified framework and include several existing charts in the literature as special cases. An advantage of the PID chart is that it can be tuned to have good performance for small or large shifts. We have seen that the PID charts do better than the EWMAST, M-M, and SCC charts for detecting small to moderate shifts. They can be designed to have performance comparable to SCC charts for large shifts. The robustness study also shows that incorrect

9

estimation of the underlying process error e can have a big impact on all of the charts for correlated data. We have restricted attention here only to the use of Shewhart charts on the PID residuals. Extensions to CUSUM and related procedures may provide further improvements in performance. ACKNOWLEDGMENTS The authors very grateful to the editor, an associate editor, and a referee for extensive comments that have improved the article substantially. In particular, the derivation in the Appendix is based on the referee’s comments. We also thank Hancong Liu for his help in the simulation studies. The research of W. Jiang, F. Tsung, and K.-L. Tsui was supported by RGC Competitive Earmarked Research grant HKUST6134/98E. K.-L. Tsui’s research was also partially supported by National Science Foundation grant DMI9908032 and the Logistics Institute Asia Pacific in Singapore. V. N. Nair’s research was supported in part by National Science Foundation grant DMS-9803281 and AFOSR/ARPA MURI grant F49620-95-1-05. APPENDIX: DERIVATION OF STEADY-STATE SHIFTS The z-transform and its properties (see Ogata 1995, chap. 2) are used here to derive the steady-state shift S from (9). First, note that (9) can be rewritten as t+3 − 1 − kP − kI − kD t+2 − kP + 2kD t+1 + kD t = 0
for t ≥ t0 − 2 Then, for the z-transform z = from (A.1) that



k=0 t0 −2+k z

−k

(A.1)

, we obtain

z3 z − z3 t0 −2 − z2 t0 −1 − zt0 − 1 − kP − kI − kD  × z2 z − z2 t0 −2 − zt0 −1  − kP + 2kD  × zz − zt0 −2  + kD z = 0 Substituting the initial conditions that t0 −2 = t0 −1 = 0 and t0 =  and solving for z, we have z =

z−2   1 − 1 − kI − kP − kD z−1 − kP + 2kD z−2 + kD z−3 (A.2)

Applying the final value theorem of z-transforms (Ogata 1995, chap. 2), we have that S = lim t = lim1 − z−1 z t→

z→1

(A.3)

It then follows from (A.2) and (A.3) that S = 0 for kI > 0 and S = /1 + kP  for kI = 0. [Received February 2000. Revised October 2001.] TECHNOMETRICS, AUGUST 2002, VOL. 44, NO. 3

10

WEI JIANG ET AL.

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