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**The University of Michigan Georgia Institute of Technology**

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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**Outline Introduction Simulation and case study Summary**

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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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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**Motivation of Using APC**

Online adjust X based on e Offline fix x=x1 x= x2 Offline fix x=x2 y(x,e) x=x1 a b e noise distribution

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**The Objective and Focus**

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 DOE-Based APC Design of Experiments (DOE) Automatic Process Control (APC) Statistical (SPC)

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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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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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**S2: Determine APC control strategy (considering model errors**

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

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**1. Process Variable Characterization**

Variables Controllable Factors Noise Factors Off-line setting Factors On-line adjustable Factors Observable Noise Factors Unobservable Noise Factors Y= f (X, U, e, n)

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**2. Control System Framework**

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

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**3 Controller Design 3.1 Problem Assumptions**

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

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**3 Controller Design 3.2 Objective Function**

Objective Function (Quadratic Loss) Optimization Problem

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**3 Controller Design 3.3 Control Strategy**

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

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**4. Case Study : An Injection Molding Process**

Process Description Response Variable (y): Percentage Shrinkage of Molded Parts Process Variables:

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**DOE Modeling Designed Experiment Result (Engel, 1992)**

Reduced DOE Model after Coefficient Significance Tests Parameter Estimation Error

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**Robust Parameter Design**

Response Model Variance Model RPD Settings , and u1 and x3 are adjusted according to target values as in right table

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DOE-Based APC Objective Loss Function Optimal Settings where

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**Simulation Results Comparison of RPD, CE control and Cautious Control**

Assuming ~ ~ ~ Optimal Off-line Setting Cautious control law performs much better than RPD Control Strategy Evaluation

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Simulation Results - 2 Certainty Equivalence – assume observation perfect 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.

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**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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**Control strategy with partial sensing failure – 2**

Sensor noise level change – APC considering modeling error 255 observations, sensor noise level increased from point 101 to 200, then restored Overall J/J_ce=16.8%. APC performance is steady over different noise levels.

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**Control strategy with partial sensing failure – 3**

Sensor failure - Assume no modeling error, - 250 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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**Industrial Collaboration with OG Technologies: DOE-Based APC Test bed in Hot Deformation Processes**

[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

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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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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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