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QUALITY AND RELIABILITY ENGINEERING INTERNATIONAL

Qual. Reliab. Engng. Int. 15: 135–142 (1999)

IMPROVING AUTOMATIC-CONTROLLED PROCESS QUALITY USING ADAPTIVE PRINCIPAL COMPONENT MONITORING FUGEE TSUNG∗

Department of Industrial Engineering and Engineering Management, Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong

SUMMARY Manufacturing quality is an important factor in global market competition. The implementation of statistical process control (SPC) techniques is necessary if a manufacturing firm desires to be competitive in the global market. With advances in sensing and data capture technologies, large volumes of data are routinely being collected in automatic-controlled processes. There is a great need for SPC techniques for variation reduction and quality improvement in these processes. This paper focuses on SPC schemes that are based on a combination of the process outputs and automatic control actions using adaptive principal component monitoring (APCM). These schemes are more efficient in detecting process changes for automatic-controlled processes. Copyright  1999 John Wiley & Sons, Ltd. KEY WORDS : automatic process control; statistical process control; principal component analysis; multivariate analysis; time series

INTRODUCTION The importance of statistical process control (SPC) techniques in quality improvement is well recognized in industry. Currently most competitive manufacturing companies are implementing SPC to varying extents. Although SPC techniques have been implemented, these techniques cannot effectively handle automaticcontrolled processes. This paper develops efficient SPC techniques for quality improvement of automaticcontrolled processes. Many manufacturing processes, such as those in the chemical and process industries, are equipped with automatic process control (APC) or engineering process control (EPC) for short-term variation reduction [1]. However, SPC techniques are still needed to detect the out-of-control conditions and to remove their root causes for long-term process improvement. Most SPC techniques essentially treat the manufacturing processes as black boxes. The techniques are primarily applied to the process outputs after automatic control and are often ineffective, as the information contained in the APC is ignored. In the past, APC and SPC have been developed in parallel, but there has been little interaction between ∗ Correspondence to: F. Tsung, Department of Industrial Engi-

neering and Engineering Management, Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong.

CCC 0748–8017/99/020135–08$17.50 Copyright  1999 John Wiley & Sons, Ltd.

researchers who work on these areas. Although the concept of combining APC and SPC was suggested decades ago, this idea has received very little attention until recently [2,3]. Vander Wiel et al. [4] and Tucker et al. [5] integrated APC and SPC by using both to perform the separate functions of control and monitoring. Their idea is to develop and use appropriate control schemes to do APC and monitor the APC-controlled processes using SPC techniques. If the model is correct and there is no process change, then the resulting automatic-controlled process is close to white noise, so the traditional control charts can be applied. However, as shown by Wardell et al. [6] the application of traditional control charts to forecast errors in a correlated process, which is equivalent to charting the process output after minimum mean squared error (MMSE) control, can perform poorly in detecting process changes. The main reason is that the MMSE control causes the output of the process to adapt to process changes, such as a jump shift in the mean, and there is only a limited ‘window of opportunity’ during which the process change can be detected. All the existing SPC techniques suffer from the problem that they do not consider the information in the automatic control actions and utilize the engineering/physical models in APC. Messina et al. [7] considered the effectiveness of Received 7 February 1998 Revised 29 July 1998

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monitoring the control actions. Tsung et al. [8] proposed joint monitoring schemes based on both the process output and the control actions. They used Hotelling’s and Bonferroni’s techniques to control the overall error probabilities. In this paper, appropriate linear functions of the process output and control actions that are most informative in detecting a process change are determined. This alleviates the ‘window of opportunity’ problem, as there is information in the control action even when the process output adapts to the process change. Principal component analysis (PCA) techniques are used to identify the best linear function of the output and control actions for application of SPC. In the case of process uncertainty and environmental change the PCA model is updated adaptively by monitoring the residual of new input–output observations. Also, a specific procedure is developed to implement the proposed adaptive principal component monitoring (APCM) scheme. It shows that the proposed procedure can be easily extended from single–input/single– output (SISO) systems to multiple-input/multipleoutput (MIMO) systems. SPC FOR THE PROCESS OUTPUT OR INPUT ONLY In many automatic-controlled processes the only monitored variable, if any, is the process output (see e.g. Reference [3]). Monitoring the process outputs alone may not be sufficient, because the process changes can be compensated by the control actions and are therefore hard to detect from the process outputs. On the other hand, monitoring the control actions alone may not be efficient either. This section demonstrates the ineffectiveness of applying conventional Shewhart control charts to the process outputs or to the control actions alone. To evaluate Shewhart charts for automaticcontrolled processes, the properties of Shewhart charts are studied under an autoregressive moving average (ARMA(1,1)) model [9] Dt = φ Dt −1 + at − θ at −1

(1)

where Dt is the process disturbance, at represents an independent and identically distributed (i.i.d) random variable, i.e. white noise, and the parameters |φ| < 1 and |θ | < 1. This model represents a large number of stationary disturbance processes in industry (see e.g. Reference [10]). Some of the results in this paper can be extended to other disturbance models. For instance, when φ is close to one, Dt is Copyright  1999 John Wiley & Sons, Ltd.

approximately an IMA(0,1,1) model. For simplicity, this paper will focus on ARMA(1,1) disturbance processes. Moreover, it is known that proportional– integral–derivative (PID) control schemes are by far the most common APC schemes [11]. The PID schemes include the proportional–integral (PI) control and integral (I) control as special cases. Here a PID control is used to manipulate the input to minimize the process variation due to process disturbance. Based on this criterion, a design of PID control parameters kP , kI and kD is obtained by Tsung and Shi [12]. The performance of Shewhart control charts for the PIDcontrolled process can be measured by investigating their average run length (ARL) properties. The run length (RL), a standard measure for comparing how quickly control charts detect a process change, is the number of time periods between a process change and the first signal of the control chart, and usually its average, the ARL, is emphasized. In order to make a comparison, the control limits are adjusted so that the ARL is the same for all charts when there is no shift in the mean. In this study the in-control ARL value of 370 is used. The chart with the lowest out-of-control ARL when a shift in the mean has occurred is then considered superior. The ARL is determined via Monte Carlo simulation. The simulation procedure is similar to that reported by Wardell et al. [13]. Disturbance processes are generated according to equation (1), with a step change in the mean introduced at time zero, i.e. Dt = µt − φµt −1 + φ Dt −1 + at − θ at −1

(2)

where the mean shift µt is defined by ( 0, t Qual. Reliab. Engng. Int. 15: 135–142 (1999)

ADAPTIVE PRINCIPAL COMPONENT MONITORING

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Figure 1. ARL of Shewhart charts for process outputs and control actions when mean shifts are 2σ D

10) in the region φ > θ and φ > 0.5. In addition, Figures 1c and 1d show that the Shewhart charts for the control actions do not perform well (i.e. ARL > 10) when both φ and θ are negative or when φ moves close to one. It is obvious that Shewhart charts for the process Copyright  1999 John Wiley & Sons, Ltd.

outputs and for the control actions are ineffective in somewhat complementary areas. This motivates us to monitor an appropriate combination of the process output and control actions to compensate for the poor performance of the separate monitoring of the inputs and outputs in different regions of the parameter space. Qual. Reliab. Engng. Int. 15: 135–142 (1999)

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SPC FOR THE INPUT–OUTPUT COMBINATION USING PCA As the monitoring of the input or the output alone is not sufficient, Tsung et al. [8] used bivariate control charts to monitor the process outputs and the control actions simultaneously. Our APCM scheme extends their idea by monitoring an appropriate and effective input–output combination using PCA. This method is more economical and easier to implement, as it needs only traditional univariate control charts such as Shewhart charts. The use of PCA for process monitoring has been extensively discussed in the literature [14,15]. Here PCA modelling is used to find an effective linear combination of control actions and process outputs to be monitored. In the PCA approach a set of the control action x and process output y data is collected during APC operations where the process is known to operate under normal operating conditions, i.e. an ‘in-control’ condition. The first principal component of the input–output matrix Y = (x, y) is defined as a linear combination T = P T Y that has maximum variance subject to |P| = 1. It can be shown that the principal component loading vector P = (c1 , c2 ) is the first eigenvector of the covariance matrix 6 of Y , and the corresponding largest eigenvalue λ is the variance of the first PC. After building up a PCA model based on closed loop data collected under in-control conditions, future process monitoring can be referenced against this model. Thus an input–output combination To is suggested to be monitored based on the projection defined by P: To = P T Yo = c1 x o + c2 yo

(3)

where Yo = (x o , yo ) is the new observed control action and process output data set. A conventional SPC chart such as a Shewhart chart can then be applied to To . The control limits (CLT ) are set at some constants Z multiplied by the standard deviations of T (σT ) above and below the centre lines: √ (4) CLT = ±Z σT = ±Z λ √ where σT is estimated by λ. Here Z is based on the desired type I error α; that is, if the process is in control and approximately normally distributed, we would expect 100(1 − α) per cent of To to fall between the control limits. The decision rule suggests that the process is out of control and needs investigation when To is outside the control limits. For MIMO systems the input and output observations (x, y) can be a high-dimensional matrix. In Copyright  1999 John Wiley & Sons, Ltd.

practice, most of the variability in the data are captured in the first few PCs (say, two or three). The number of PCs that provide an adequate description of the process data can be assessed using a number of methods such as significance tests and cross-validation [16].

ADAPTIVE PC UPDATING Applying SPC for the input–output combination To may not be sufficient to indicate some events that cannot be characterized by the current PCA model. It is possible that the underlying covariance structure, i.e. the relationship between the control action and the process output, may change with time; so too may its associated PCA model. For instance, some system parameters such as the automatic control parameters may be changed by the process engineer but remain unknown to the quality engineer. Hence the current PCA model and the corresponding SPC scheme are not valid. This situation may also be caused by a new type of special event that was not presented in the reference data used to develop the PCA model. Thus new PCs will appear and the new observations Yo will deviate from the designed input–output combination direction. A solution for this problem is to update the PCA model adaptively by monitoring the residual (R), i.e. the squared prediction error of new observations, R = (x o − xô )2 + (yo − yô )2

(5)

with the prediction ( xô , yô ) = PTo . This is also referred to as a Q-statistic by Jackson [16]. The R-statistic checks if the distance of the new observations from the projection space is within acceptable limits. When the process model is valid, the value of R should be small. Again, a conventional Shewhart chart can be applied to R. The upper control limit for R (CL R ) is set at √ (6) CL R = (σx2 + σ y2 − λ)( 2 Z /3 + 7/9)3 using the approximation method of Jackson and Mudholkar [17]. Here σx and σ y are obtained during the estimation of 6, and the constant Z is based on the desired type I error. A more general equation of CL R for MIMO systems can be found in Reference [16]. When an unusual event occurs including intended system and controller change, it results in a change in the covariance structure between the control action and process output. This will be detected by a high value of R outside its control limit and a PC model update and validation will follow. Qual. Reliab. Engng. Int. 15: 135–142 (1999)


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Figure 3. Sketch of parts made on automatic screw machine

Figure 2. Guideline for implementation of APCM

IMPLEMENTATION PROCEDURE In particular, the APCM scheme described in the previous sections can be implemented by the following five steps: (1) APC design, (2) PCA modelling, (3) SPC design, (4) combined input–output and R monitoring and (5) process diagnosis and improvement. A guideline for the implementation of this scheme is shown in Figure 2 and described as follows. First, system and automatic controller parameters are designed in step (1) based on data collection and engineering knowledge. For instance, an APC design of a PID-controlled process is described. In step (2) the relationship between the control action and process output is modelled based on data collection during APC operations using statistical methods such as PCA. The detail of PCA modelling is also described. The purpose of PCA modelling is to obtain an effective input–output combination To to monitor. In step (3), SPC control charts are designed based on the input–output covariance structure and its PCA model. Specific design of the control limit settings can be obtained by equations (4) and (6). The SPC schemes will be applied to To and to R in step (4). In step (4a) we monitor the input–output combination To to see if there is any process change, e.g. a process mean shift such as a step, ramping or cyclical change. Copyright  1999 John Wiley & Sons, Ltd.

When an ‘out-of-control’ signal occurs, we move to step (5) to investigate the process (we will discuss step (5) in detail later). Along with step (4a) we monitor the R-statistic in step (4b). If an ‘out-ofcontrol’ signal is triggered by monitoring R, which is due to recent APC change, the implementation cycle returns to step (2) to update the PCA model, as the relationship between the control action and process output has been changed. If no recent APC change is found, we move to step (5) to investigate the APC system as well as the process. Step (5) is process diagnosis and improvement. It is used to diagnose possible root causes of process change while the information is still fresh and to correct such causes properly for process improvement. The diagnosis after an SPC signal can be done using a number of techniques such as those in References [18,19]. The implementation cycle then returns to step (1), as this is a continuous improvement procedure. AN AUTOMATIC MACHINING PROCESS An automatic machining process is used to demonstrate the applicability and efficiency of the proposed SPC scheme. Here a machining process is considered in which a single-spindle automatic screw machine is used. A bar of SAE 1045 steel of 1/4 inch diameter and 10 ft. length is used as raw material, and parts are made at a rate of 80 per hour. A key quality characteristic, the diameter of the part, is measured after the machining process is finished. A sketch of parts made by this automatic machining process is shown in Figure 3. The objective of this study is to minimize the diameter variation from a target value of 7/32 inch. Wu and Dalal [20] provide additional background and analysis of this process. Simulated data are generated for this study using the disturbance model ARMA(1,1) in equation (1) with φ = 0.9 and θ = 0.4, and a step mean shift 2σ D introduced at time zero. A step-by-step procedure to implement the APCM is illustrated as follows: Step (1): APC design This machining process is equipped with a PID control with control parameters kP = 0.15, kI = 0.44 Qual. Reliab. Engng. Int. 15: 135–142 (1999)

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Figure 4. SPC for automatic machining process: (a) monitoring input–output combination To ; (b) monitoring R; (c) monitoring process output only; (d) monitoring control action only

Copyright  1999 John Wiley & Sons, Ltd.


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and kD = 0.06 (see Reference [12] for the design of PID control).

Step (2): PCA modelling Here the process output y is measured by the diameter reading and the control action x is measured by the cutting tool setpoint. Based on data collection and statistical modelling, the covariance matrix between x and y is obtained as 2 σx σx,y 1.736 −0.326 = 6= −0.326 1.040 σx,y σ y2 Its corresponding first principal component is P = (−0.930, 0.367) with eigenvalue λ = 1.865. Thus the suggested input–output combination to monitor based on PCA is To = −0.930x o + 0.367yo where x o and yo are the new observed control action and process output data.

Step (3): SPC design To design a Shewhart chart for To , we may choose Z = 3 and set the control limits from equation (4) at √ CLT = ±3 1.865 = ±4.097 These are typically called ‘three-sigma’ control limits [21]. This control chart indicates that the process is out of control and needs investigation when To is outside the control limits. Step (4a): PC monitoring In this case the process mean shift 2σ D can be detected within eight observations by monitoring the combination To (see Figure 4a). On the other hand, by monitoring the process output only, which is common practice, it takes 37 observations to signal the outof-control condition (Figure 4c). For the purpose of comparison, control action monitoring is also studied, although it is not used in common practice. Here it takes nine observations to hit the control limits by monitoring the control action only (Figure 4d). Thus in this case the PCA monitoring is shown to be more efficient than individual monitoring of the process outputs or control actions. The follow-up process diagnosis and improvement in step (5) are not discussed in this paper. Copyright  1999 John Wiley & Sons, Ltd.

Step (4b): R monitoring Here R is also monitored to ensure that the PCA model and the associated SPC scheme are valid. We may choose Z = 3 and set the upper control limit for R by equation (6) at √ CL R = (1.736 + 1.040 − 1.865)( 2 + 7/9)3 = 9.595 When a system or a controller changes, a change in the covariance structure between the input and output is a result. This will be detected by a high value of R outside its control limit. In this case no unusual events are detected during the monitoring of R (Figure 4b), which is consistent with a true situation. CONCLUSION In this paper an APCM scheme is proposed that is based on an effective combination of the process outputs and control actions using PCA and adaptive updating by monitoring R. A five-step implementation procedure with an automatic machining process example is presented. This scheme is shown to be more efficient than individual monitoring of the process outputs or control actions in detecting process changes for automatic-controlled processes. This research is expected to have an impact on manufacturing industries through implementation of the proposed SPC scheme as an effective quality control method for automatic-controlled processes.

ACKNOWLEDGEMENTS

I thank Professor Jeff Wu and Professor Jan Shi for their guidance and the referees for their helpful comments. I have benefited from valuable discussions with Professor Vijay Nair and Professor Huiqing Wu.

REFERENCES 1. G. E. P. Box and A. Luceno, Statistical Control by Monitoring and Feedback Adjustment, Wiley, New York, 1997. 2. G. E. P. Box and T. Kramer, ‘Statistical process monitoring and feedback adjustment—a discussion’, Technometrics, 34, 251–285 (1992). 3. D. C. Montgomery, J. B. Keats, G. C. Runger and W. S. Messina, ‘Integrating statistical process control and engineering process control’, J. Qual. Technol., 26, 79–87 (1994). 4. S. A. Vander Wiel, W. T. Tucker, F. W. Faltin and N. Doganaksoy, ‘Algorithmic statistical process control: concepts and an application’, Technometrics, 34, 286–297 (1992).


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5. W. T. Tucker, F. W. Faltin and S. A. Vander Wiel, ‘Algorithmic statistical process control: an elaboration’, Technometrics, 35, 363–375 (1993). 6. D. G. Wardell, H. Moskowitz and R. D. Plante, ‘Run-length distributions of special-cause control charts for correlated observations’, Technometrics, 36, 3–17 (1994). 7. W. S. Messina, D. C. Montgomery, J. B. Keats and G. C. Runger, ‘Strategies for statistical monitoring of integral control for the continuous process industries’, in J. B. Keats and D. C. Montgomery (eds), Statistical Applications in Process Control, Marcel Dekker, New York, 1996, pp. 193– 215. 8. F. Tsung, J. Shi and C. F. J. Wu, ‘Joint monitoring of PID controlled processes’, J. Qual. Technol., in press (1998). 9. G. E. P. Box, G. M. Jenkins and G. C. Reinsel, Time Series Analysis Forecasting and Control, 3rd edn, Prentice-Hall, Englewood Cliffs, NJ, 1994. 10. F. Tsung, H. Wu and V. N. Nair, ‘On the efficiency and robustness of discrete proportional–integral control schemes’, Technometrics, 40, 214–222 (1998). 11. K. J. Astrom, Automatic Tuning of PID Controllers, Instrument Society of America, Research Triangle Park, NC, 1988. 12. F. Tsung and J. Shi, ‘Integrated design of run-to-run PID controller and SPC monitoring for process disturbance rejection’, IIE Trans., in press (1998). 13. D. G. Wardell, H. Moskowitz and R. D. Plante, ‘Control charts in the presence of data correlation’, Manag. Sci., 38, 1084– 1105 (1992). 14. C. M. Mastrangelo, G. C. Runger and D. C. Montgomery, ‘Statistical process monitoring with principal components’, Qual. Reliab. Engng. Int., 12, 203–210 (1996). 15. G. C. Runger and D. C. Montgomery, ‘Multivariate and univariate process control: geometry and shift directions’, Qual. Reliab. Engng. Int., 13, 153–158 (1997).

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16. J. E. Jackson, A User’s Guide to Principal Components, Wiley, New York, 1991. 17. J. E. Jackson and G. S. Mudholkar, ‘Control procedures for residuals associated with principal component analysis’, Technometrics, 21, 341–349 (1979). 18. S. Hu and S. M. Wu, ‘Identifying root causes of variation in automobile body assembly using principle component analysis’, Trans. NAMRI/SME, XX, 311–316 (1992). 19. D. Ceglarek and J. Shi, ‘Fixture failure diagnosis for autobody assembly using pattern recognition’, ASME J. Engng. Ind., 118, 55–66 (1996). 20. S. M. Wu and J. G. Dalal, ‘Stochastic model for machining processes—optimal decision-making and control’, Trans. ASME. J. Engng. Ind., 93, 593–602 (1971). 21. D. C. Montgomery, Introduction to Statistical Quality Control, 3rd edn, Wiley, New York, 1996.

Author’s biography: Fugee Tsung is an Assistant Professor of Industrial Engineering and Engineering Management at the Hong Kong University of Science and Technology. His research interests are model-based quality improvement, process monitoring and diagnosis, industrial statistics and robust design. He holds a BSc in mechanical engineering from National Taiwan University and an MS and PhD in industrial and operations engineering from the University of Michigan, Ann Arbor. He has worked with Ford Motor Company and Rockwell International and has done post-doctoral research with Chrysler Corporation. He is a Member of the ASQ, where he is a certified quality engineer in training. He is also a Member of the IIE and INFORMS.
