1
DOE-based Automatic Process Control with Consideration of Model Uncertainties
Jan Shi and Jing ZhongThe University of Michigan
C. F. Jeff WuGeorgia Institute of Technology
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
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
x=x1
e
noise distribution
y(x,e)
a b
e e
Online adjust X based on e
2byx x
1ayx x
x= x2
2ayx x
2byx x
Offlinefix x=x2
Offlinefix x=x1
1byx x
1ayx x
( , )y f x e
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
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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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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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 controlor cautious control
Obtain reduced process variation
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1. Process Variable Characterization
ProcessVariables
ControllableFactors
NoiseFactors
UnobservableNoise Factors
ObservableNoise Factors
Off-line settingFactors
On-line adjustableFactors
Y= f (X, U, e, n)
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2. Control System Framework
ControllableFactors (x)
ManufacturingProcess
UnobservableNoise Factors (n)
Observable Noise Factors (e)
In-ProcessSensing of e
Response (y)
Observer for Noise Factors (e)
Feedforward Controller
Noise Factors
Predicted Response
ˆˆ [ ( , , | , )]ny E f x e n x e
Target
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• Observations of measurable noise factors, denoted by , are unbiased, i.e., and .
3 Controller Design3.1 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.
βββ~ˆ
βββ ~)ˆ( Cov
eee ~ˆ 0ˆ|ˆ eeeE eeee ~)ˆ|ˆ( Cov
• 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.
nBUnBXeBUeBXnβeβUβXβ 432143210TTTTTTTTy
βeUX ˆ,ˆ|,APCJ βeβneˆ,ˆ)( 2
,,, tycE
e
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3 Controller Design 3.2 Objective Function
)ˆ,ˆ,( βeUXAPCJ
Objective Function (Quadratic Loss)
243
21
2
434434
213~213
2
213210
))((var))((var
)(var)(var)(
ˆˆ
ˆˆˆˆˆˆ
ˆˆˆ ˆˆˆ
ˆˆˆˆˆˆˆˆˆ
4
3230
nBUnBX
eBUeBXnn
eeUUXX
UBXBβUBXBβ
UBXBβUBXBβ
eBUeBXeβUβXβ
βnβn
βββn
βββ
n
e
TT
TTT
TTT
TTTTT
TTTTT
TTTTT
EE
E
t
2,,,ˆ,ˆ tyE βeβne
βeβe βneβneˆ,ˆˆ,ˆ ,,,
2
,,, yVartyE
βeβneˆ,ˆ)( 2
,,, tycE
UXUXUX
,min arg ),( 1,1
**APCJ
Optimization Problem
ˆˆ( , , , , , , )f e nβX U e β
13
.ˆ,ˆ|,minarg *ˆ
1
* βeUXX eX
APCJE
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 JAPC
),,ˆ,ˆ,(ˆ,ˆ,|,minarg ~1
*βne
UβeXβeXUXU
hJ APC
3 Controller Design 3.3 Control Strategy
* * ˆˆ( , , , , )h e n βU U X e β
Step 1 Closed form solution of U* by solving 0U APCJ
Process Control Strategy – Two Step Procedure
*XX
UXUXUX
,min arg ),( 1,1
**APCJ
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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
Reduced DOE Model after Coefficient Significance Tests
Designed Experiment Result (Engel, 1992)
131212111312
1321321
106.0094.0125.0063.0556.0588.0
05.0144.0281.0425.0231.0063.0075.025.2
nununxnxeuex
nuuuxxxy
2121-4~ 105.51 IβParameter Estimation Error
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RPD Settings
Robust Parameter Design
.)5563.05875.0(
)1063.00938.0125.00625.005.0()ˆ(22
32
223221
1
1
e
n
ux
uuxxyVar
Variance Model
Response Model
Tx*3
* 04664.0X Tu 02222.0*1
* U, and
u1 and x3 are adjusted according to target values as in right table
131212111312
1321321
106.0094.0125.0063.0556.0588.0
05.0144.0281.0425.0231.0063.0075.025.2
nununxnxeuex
nuuuxxxy
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Objective Loss Function
UUXXUUXX
UUXX
UBXBβUBXBβUBXBβUBXBβ
eBUeBXeβUβXββeUX
BBBB
ββ
ne
41312341
3230
2221
21
22
221
2
434434213~213
2
2132102
ˆˆ
ˆ
ˆˆˆˆˆˆ ˆˆˆ ˆˆˆ
ˆˆˆˆˆˆˆˆˆ)ˆ,ˆ,(
Tn
Tn
TTn
TT
TTTTTTTTTT
TTTTTAPC
ee
e
tJ
*
342
4*
132~212211
*1310
122
12
42
422~2122122
*
ˆˆˆˆˆˆˆˆˆˆˆˆˆˆˆ
ˆˆˆˆˆˆˆˆˆˆˆ
11
412211
XBβBXBβBBβBXβXβ
BBBBBβBβU BBβ
Tn
Te
TTT
nT
nT
e
T
eeet
eee
Optimal Settings
DOE-Based APC
βUXXX
ˆ,ˆ|,minarg 1*
ˆ1
*
1eJE APCe
where
1
1
2
)(ˆ
2ˆ
1*
1*
ˆ
21
21
1
1
2
1)(ˆ|,
1
2ˆ|,
M
i
ie
e
APCAPCeeeieJ
MeJE
UXUX
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1e ~ )25.0,0(N 1n ~ )25.0,0(NAssuming
Optimal Off-line Setting
Simulation Results
T0.50850.2817-0.5121* X
Comparison of RPD, CE control and Cautious Control
Control Strategy Evaluation
Cautious control law performs much better than RPD
1e ~ (0,0.025)N
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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.
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
1
2
3
4
5
6
7
J CE/J
RD
1 1
2 2/e e
Certainty Equivalence – assume observation perfect
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0 50 100 150-2
0
2
e 0
0 50 100 150-2
-1
0
1
Obe
rver
Noi
se L
evel
0 50 100 1501
1.6
2
Per
cent
age
Shr
inka
ge
Observations
yy
cey
rd
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
1 1 1 1
2 2 2 20.1 1e e e e
CE Control suffers greatly from noise level change
Mean of RPD has deviated from target
* * ˆˆ( , , , , )h e n βU U X e β
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Control strategy with partial sensing failure – 2
255 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
* * ˆˆ( , , , , )h e n βU U X e β
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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
1.011
~ ee
Control Strategy
Switch to RPD setting after the detection of sensor failure
- Actual system will have step response
0 50 100 150 200 250-2
0
2
e 0
0 50 100 150 200 250-2
0
2
e hat
0 50 100 150 200 2501.4
1.6
1.8
2
Per
cent
age
Shr
inka
ge
Observations
yy
cey
rd
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[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
casterin-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, materialcoating 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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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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