PID Charts for Process Monitoring. Wei Jiang INSIGHT, AT&T. Huaiqing Wu Iowa State University

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1 PID Charts for Process Monitoring Wei Jiang INSIGHT, AT&T Huaiqing Wu Iowa State University Fugee Tsung Hong Kong University of Science & Technology Vijayan N. Nair University of Michigan Kwok-Leung Tsui Georgia Institute of Technology

2 Outline The Shewhart x Chart Monitoring Charts for Correlated Data The PID Chart (Proportional Integral Derivative Chart) An Illustrative Example Design of PID Charts Average Run Length Performance Robustness to Parameter Misspecification Concluding Remarks

3 The Shewhart x Chart Shewhart s partition of process variation (Shewhart 1931): Baseline variation (common causes) Variation that can be eliminated (assignable causes) The Shewhart x chart for iid data based on samples of size m A plot of sample mean measurements, x i, against sample number, i Center line: process target mean (µ) Control limits: µ ± 3 σ m (σ: baseline variation) When a point plotted falls outside the control limits, an out-of-control alarm is triggered.

4 Monitoring Charts for Correlated Data Special cause charts (SCC) of Alwan and Roberts (1988): First whiten the autocorrelated data by subtracting their one-step-ahead minimum mean squared error (MMSE) prediction Then monitor the errors from the prediction using a traditional control chart for iid data Traditional control charts for iid data with modified control limits to account for autocorrelation among data EWMAST charts of Zhang (1998) (exponentially weighted moving average charts for stationary processes) Autoregressive moving average (ARMA) charts of Jiang, Tsui, and Woodall (2000): Monitor the ARMA statistic of the underlying process

5 Special Cause Charts (SCC) SCC methods applied to MMSE prediction errors: For AR(1) models: Harris and Ross (1991), Superville and Adams (1994), Runger and Willemain (1995), and Runger, Willemain, and Prabhu (1995) For ARMA(1,1) and ARMA(p, q) models: Wardell, Moskowitz, and Plante (1992, 1994) For IMA(0,1,1) models: Vander Wiel (1996) The MMSE prediction: not necessarily best for process monitoring Monitoring the original autocorrelated data can outperform the SCC chart (Wardell, Moskowitz, and Plante 1992, 1994; Zhang 1998) Idea: other prediction methods may work better

6 Prediction and Feedback Control Discrete feedback control: D t : process disturbance U t : control action Y t : output from process dynamics Y t+1 = U t : pure-gain dynamic model e t+1 = Y t+1 + D t+1 : control error at time (t + 1) (assuming the target value is 0) Prediction of D t+1 Feedback control (e t+1 = D t+1 D t+1, by setting U t = D t+1 ) MMSE prediction of D t+1 MMSE Feedback Control

7 PID Control and Prediction PID control: U t = k P e t k I 1 1 B e t k D (1 B)e t e t = D t D t B: backward shift operator (Be t = e t 1 ) k P, k I, and k D : constants (the amount of proportional, integral, and derivative control actions) (Box, Jenkins, and Reinsel 1994; Åström and Hägglund 1995) PID predictor: D t+1 = D t +k I e t +k P (1 B)e t +k D (1 B) 2 e t

8 The PID Chart First subtract the PID predictor from the original data to yield the PID-based residuals Then monitor the residuals using a traditional control chart and taking account of the correlation structure of the residuals in computing the control limits Special cases: P, I, D, PI, and PD charts (e.g., the P chart corresponds to k I = k D = 0) The I chart: same as the M-M chart (Montgomery and Mastrangelo 1991) The P chart: equivalent to the EWMAST chart (Zhang 1998)

9 An Illustrative Example Data from a mechanical vibratory system consisting of a mass, a spring, and a dashpot (Pandit and Wu 1983): vertical displacements of the mass An ARMA(2,1) process D t = D t D t 2 +a t a t 1, with σ D = and σ a = 2.212, fitted the first 100 observations well, where a t is white noise. Suppose that a mean shift of = µ/σ D occurred after the first 100 observations. A simulation with 160,000 replications was done to obtain the average run length (ARL) needed to detect the shift for various values of for several PID charts and the SCC chart.

10 ARLs of the SCC, P, PI and PD Charts Shift SCC PI P PD Chart parameters (k P, k I, k D ) P: (.8, 0, 0) PI: (.3, 1.8, 0) PD: (.8, 0,.5) Control limits: ±Lσ e, where L = 3, 2.556, 2.95, and for the SCC, P, PI, and PD charts.

11 Design of PID Charts Capability Indices/Signal-to-Noise Ratios (Jiang, Tsui, and Woodall 2000): Consider monitoring a stationary process D t with mean 0 and variance σd 2. Suppose a mean shift of µ occurs at time t 0. Let µ t be the mean of the PID charting process e t. Let σe 2 be the variance of e t when the monitored process D t is in control. Transient capability: C T = µ/σ e Capability of the chart to detect the shift in the first few runs Steady-state capability: C S = lim t µ t /σ e Important for detecting the shift efficiently in later runs if the chart fails to signal early

12 Choosing PID Chart Parameters 1. Specify the shift level = µ/σ D to be detected. 2. Compute Max C T, the maximum value of C T, for the PID chart by varying its chart parameters (k P, k I, k D ). 3. If Max C T > 5, choose the PID chart with the transient capability equal to Max C T and stop; otherwise, go to the next step. 4. Compute Max C S, the maximum value of C S, for the PD chart by varying its chart parameters (k P, 0, k D ). 5. If Max C S 3.5 or if C T 1 when C S is maximized, choose the PD chart with the steady-state capability equal to Max C S ; otherwise, choose a PD chart with C S [2.5, 3.5] to balance the values of C T and C S.

13 ARL Performance Simulation (with 160,000 replications) done for ARMA(1,1) processes D t = 1 θb 1 φb a t for PID 1, PID 2, P (EWMAST) and SCC charts. φ θ Chart k P k I k D L PID PID P SCC PID PID P SCC PID PID P SCC 3.00

14 ARL Performance (Continued) φ θ Chart ARL 0.5 ARL 1 ARL 2 ARL PID PID P SCC PID PID P SCC PID PID P SCC

15 Robustness to Parameter Misspecification Control limits consist of two components: L and σ e. The robustness of a control chart is mainly affected by the estimation of σ e. For Shewhart chart with iid data or SCC chart with correlated data, moving range estimator σ MR is traditionally recommended. However, σ MR is biased when correlation exists in e t (Cryer and Ryan 1990): E( σ MR ) = σ e 1 ρe. Adams and Tseng (1998) reported the nonrobustness of the SCC chart when using σ MR. Thus the moving range estimator should be avoided in practice.

16 Robustness (Continued) When model parameters are misspecified, L is set as L under the misspecified model. σ e may be estimated correctly (σ e ) from the actual residual or incorrectly (σe) from the misspecified model. Two sets of ARL s (ARL and ARL ) are derived to separate the sources of errors.

17 Robustness: SCC Chart (φ, θ ) Shift SCC (φ, θ) ( ) ARL ARL SCC MR ARL (.475, 0) (.425,.05) (.525,.05) (.95,.9) (.9,.85) (.999,.95)

18 Robustness: SCC Chart (Continued) MR estimator for standard deviation causes the nonrobustness of the SCC chart. σ e has to be estimated from the residual rather than from the specified model. An estimation error of more than 3% can make a control chart nonrobust. SCC chart with correct standard deviation of residuals is very robust in both ARL 0 and ARL 1 in most stationary cases. SCC chart becomes nonrobust when the estimated model approaches nonstationary.

19 Sensitivity vs. Robustness: PID Charts Shift PID 1 PID 2 (φ, θ ) (φ, θ) ( ) ARL ARL ARL ARL (.475, 0) (.425,.05) (.525,.05) (.95,.9) (.9,.85) (.999,.95)

20 Sensitivity vs. Robustness (Continued) P ID 1 chart is not robust since it is too sensitive to small shifts. Nonzero I term helps to make PID charts robust for nearly nonstationary models.

21 Concluding Remarks A new class of control charts is proposed based on the relationship between PID controllers and their corresponding predictors. It includes the EWMA and M-M charts as special cases. PID charts can be tuned to have better performance than existing charts for correlated processes. Initial robustness study shows that an estimation error of more than 3% can make a control chart fail. This is critical and needs further investigation.