1. 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

2. 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

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.

4. 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

5. 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)

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.

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.

8. 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

9. 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)

10. 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)

11. 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

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

13. 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

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

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

16. Robust Parameter Design
Response Model
Variance Model
RPD Settings
, and
u1 and x3 are adjusted according to target values as in right table

17. DOE-Based APC
Objective Loss Function
Optimal Settings
where

18. 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

19. 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.

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

21. 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.

22. 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

23. 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

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.

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.