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1 DOE-based Automatic Process Control with Consideration of Model Uncertainties Jan Shi and Jing Zhong The University of Michigan C. F. Jeff Wu Georgia Institute of Technology

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2 Outline Introduction DOE-based Automatic Process Control with Consideration of Model Uncertainty –Process model –Control objective function –Controller design strategies Simulation and case study Summary

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3 Problem Statement Process variation is mainly caused by the change of unavoidable noise factors. Process variation reduction is critical for process quality improvement. Offline Robust Parameter Design (RPD) used at the design stage –To set an optimal constant level for controllable factors that can ensure noise factors have a minimal influence on process responses –Based on the noise distribution but not requiring online observations of noise factors Online Automatic Process Control (APC) during production –With the increasing usage of in-process sensing of noise factors, it will provide an opportunity to online adjust control factors to compensate the change of noise factors, which is expected to achieve a better performance than offline RPD.

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4 Motivation of Using APC x=x 1 e noise distribution y(x,e) aa bb Online adjust X based on e x= x 2 Offline fix x=x 2 Offline fix x=x 1

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5 The Objective and Focus DOE-Based APC Design of Experiments (DOE) Automatic Process Control (APC) Statistical Process Control (SPC) The research focuses on the development of automatic process control (APC) methodologies based on DOE regression models and real-time measurement or estimation of noise factors for complex mfg processes

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6 Literature Review For complex discrete manufacturing processes, the relationship between the responses (outputs) and process variables (inputs) are obtained by DOE using a response surface model, rather than using dynamic differential/difference equations –offline robust parameter design (RPD) (Taguchi, 1986) –Improve robust parameter design based on the exact level of the observed uncontrollable noise factors (Pledger,1996) Existing APC literature are mainly for automatic control of dynamic systems that are described by dynamic differential/difference equations. –Certainty Equivalence Control (CEC) (Stengel, 1986): The controller design and state estimator design are conducted separately (The uncertainty of system states is not considered in the controller design) –Cautious Control (CC) (Astrom and Wittenmark, 1995): The controller is designed by considering the system state estimation uncertainty, which is extremely difficult for a complex nonlinear dynamic system. Jin and Ding (2005) proposed Doe-Based APC concepts: –considering on-line control with estimation of some noise factors. –No interaction terms between noise and control factors in their model.

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7 Objective Develop a general methodology for controller design based on a regression model with interaction terms. Investigate a new control law considering model parameter estimation uncertainties Compare the performances of CC, CEC, and RPD, as well as performance with sensing uncertainties.

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8 Methodology Development Procedures APC Using Regression Response Models Based on key process variable S1: Conduct DOE and process modeling Obtain significant factors & estimated process model S2: Determine APC control strategy (considering model errors S3: Online adjust controllable factors S4: Control performance evaluation Based on observation uncertainty Based on process operation constraints on controller Use certainty equivalence control or cautious control Obtain reduced process variation

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9 1. Process Variable Characterization Process Variables Controllable Factors Noise Factors Unobservable Noise Factors Observable Noise Factors Off-line setting Factors On-line adjustable Factors Y= f (X, U, e, n)

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10 2. Control System Framework Controllable Factors (x) Manufacturing Process Unobservable Noise Factors (n) Observable Noise Factors (e) In-Process Sensing of e Response (y) Observer for Noise Factors (e) Feedforward Controller Noise Factors Predicted Response Target

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11 Observations of measurable noise factors, denoted by, are unbiased, i.e., and. 3 Controller Design 3.1 Problem Assumptions The manufacturing process is static with smoothly changing variables over time – Parameter Stability Estimated process parameters denoted by, is estimated from experimental data. e, n and ε are independent, with E(e)=0, Cov(e)=Σ e, E(n)=0, Cov(n)=Σ n, E(ε)=0, Cov(ε)=Σ ε. ε are i.i.d.

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12 3 Controller Design 3.2 Objective Function Objective Function (Quadratic Loss) Optimization Problem

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13 Step 1 Off-line Controllable Factors Setting Step 2 On-line Automatic Control Law Procedure for Solving Optimization Problem Step 2 obtain X * by solving optimization problem of J APC 3 Controller Design 3.3 Control Strategy Step 1 Closed form solution of U * by solving Process Control Strategy – Two Step Procedure

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14 4. Case Study : An Injection Molding Process Process Description Response Variable (y): Percentage Shrinkage of Molded Parts Process Variables:

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15 DOE Modeling Reduced DOE Model after Coefficient Significance Tests Designed Experiment Result (Engel, 1992) Parameter Estimation Error

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16 RPD Settings Robust Parameter Design Variance Model Response Model, and u 1 and x 3 are adjusted according to target values as in right table

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17 Objective Loss Function Optimal Settings DOE-Based APC where

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18 ~ ~ Assuming Optimal Off-line Setting Simulation Results Comparison of RPD, CE control and Cautious Control Control Strategy Evaluation Cautious control law performs much better than RPD ~

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19 Simulation Results - 2 CE controller performs much better than RD when the measurement is perfect, but its advantage decreases when the measurement is not perfect, and will cause a larger quality loss than RPD controller under high measurement uncertainty. Certainty Equivalence – assume observation perfect

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20 Control strategy with partial sensing failure – 1 Sensor noise level change – no modeling error 150 observations, sensor noise level increased from point 51 to 100, then restored. t=1.6 CE Control suffers greatly from noise level change Mean of RPD has deviated from target

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21 Control strategy with partial sensing failure – observations, sensor noise level increased from point 101 to 200, then restored Sensor noise level change Overall J/J_ce=16.8%. APC performance is steady over different noise levels. – APC considering modeling error

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22 Control strategy with partial sensing failure – 3 Sensor failure - Assume no modeling error, observations, sensor failed from point 51 to 150, then repaired Control Strategy Switch to RPD setting after the detection of sensor failure - Actual system will have step response

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23 [2] In-process sensing variables: tonnage signal, shut height, vibration, punch speed, temperature [3] In-process part sensing: surface and dimension measurements [1] Controllable variables: shut height, punch speed, temperature, binding force caster in-process part forming Formed part DOE-Based APC Estimable noise factors: material properties (hardness, thickness), gib conditions, die/tool wear Inestimable noise factors: distribution of lubrication, material coating properties, die set-up variation Process change detection and on-line estimation of estimable noise factors Industrial Collaboration with OG Technologies: DOE-Based APC Test bed in Hot Deformation Processes

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24 Summary DOE-Based APC performs better than RPD when measurable noise factors are present with not too large measurement uncertainty. RPD should be employed in case of too large measurement uncertainty or there are no observable noise factors. Cautious control considering measurable noise factors and model estimation uncertainty performs better than RPD and CE strategy. Model updating and adaptive control with supervision are promising or the future study.

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25 Impacts Expanding the DOE from off-line design and analysis to on-line APC applications, and investigates the associated issues in the DOE test design and analysis; Developing a new theory and strategy to achieve APC by using DOE-based models including on-line DOE model updating, cautious control, and supervision.

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