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
1
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
2
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.
3
Motivation of Using APC
y  f (x, e)
Online adjust
X based on e
Offline
fix x=x1
Offline
fix x=x2
 yb x  x1
 yb x  x2
y(x,e)
x=x1
 yb x  x2
x= x2
 ya x  x1
 ya x  x1
 ya x  x2
a
e
b
e e
noise distribution
4
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
Automatic
Process Control
(APC)
Design of
Experiments
(DOE)
DOE-Based
APC
Statistical
Process Control
(SPC)
5
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.
6
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.
7
Methodology Development Procedures
 APC Using Regression Response Models
Based on key
process variable
Based on observation
uncertainty
Based on process
operation constraints
on controller
S1: Conduct DOE and
process modeling
Obtain significant
factors & estimated
process model
S2: Determine APC
control strategy
(considering model errors
Use certainty
equivalence control
or cautious control
S3: Online adjust
controllable factors
Obtain reduced
process variation
S4: Control performance
evaluation
8
1. Process Variable Characterization
Process
Variables
Noise
Factors
Controllable
Factors
Off-line setting
Factors
On-line adjustable
Factors
Observable
Noise Factors
Unobservable
Noise Factors
Y= f (X, U, e, n)
9
2. Control System Framework
Unobservable
Noise Factors (n)
Target Feedforward
Controller
Predicted
Response
Controllable
Factors (x)
yˆ  En [ f (x, e, n | x, eˆ )]
Noise Factors
Manufacturing
Process
Observable
Noise Factors (e)
Response (y)
In-Process
Sensing of e
Observer for
Noise Factors (e)
10
3 Controller Design
3.1 Problem Assumptions
y   0  β1T X  βT2 U  βT3 e  βT4 n  XT B1e  UT B 2e  XT B 3n  UT B 4n  



J APC X, U | eˆ, βˆ  Ee,n ,β , c( y  t ) 2 eˆ , βˆ

•
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.
~
ˆ
• Estimated process parameters denoted by β  β  β,
Cov(βˆ  β)   β~ is estimated from experimental data.
•
Observations of measurable noise factors, denoted by ê , are
unbiased, i.e., e
ˆ e~
e and E eˆ  e | eˆ  0 Cov(eˆ  e | eˆ )   ~e .


11
3 Controller Design
3.2 Objective Function
Objective Function (Quadratic Loss)

J APC ( X, U eˆ ,βˆ )  Ee,n ,β , c( y  t ) 2 eˆ , βˆ

 Ee,n ,β ,
  
y eˆ , βˆ  t
2
   
 
ˆ eˆ  U B
ˆ eˆ 
 ˆ  t  βˆ X  βˆ U  βˆ eˆ  X B
ˆ XB
ˆ U   βˆ  B
ˆ XB
ˆ U
 βˆ  B
ˆ XB
ˆ U   βˆ  B
ˆ XB
ˆ U
 βˆ  B
2
 Ee,n ,β, y eˆ , βˆ  t  Vare,n ,β , y eˆ , βˆ
T
1
0
T
2
3
T
1
T
2
4
T
3
T
4
T
3
T
1
T
~
e
n
2
3
T
1
T
2
4
T
3
T
4
T
2
T
  20  XT  β 3 X  U T  β 2 U  eˆ T  β 3 eˆ
 En (nT  β 4 n)  varβ ( XT B1e)  varβ (U T B 2e)
 En (varβ ( XT B 3n))  En (varβ (U T B 4n))   2
 f ( X, U, eˆ , βˆ , e , β , n )
Optimization Problem
( X* , U * )  arg
min
X  1, U
1

J APC X, U 
12
3 Controller Design
3.3 Control Strategy
( X* , U * )  arg
min
X  1, U  1
J APC X, U 
Procedure for Solving Optimization Problem
Step 1 Closed form solution of U* by solving J APC


U  0
U*  arg min J APC X, U | X, eˆ, βˆ  h(X, eˆ, βˆ , ~e , n β )
U
1

Step 2 obtain X* by solving optimization problem of JAPC
 

X*  arg min Eeˆ J APC X, U* | eˆ, βˆ .
X
1

Process Control Strategy – Two Step Procedure
Step 1 Off-line Controllable Factors Setting
X  X*
Step 2 On-line Automatic Control Law
U  U*  h(X* , eˆ , βˆ , e , n β )
13
4. Case Study :
An Injection Molding Process
Process Description
Response Variable (y):
Percentage Shrinkage of Molded Parts
Process Variables:
14
DOE Modeling
Designed Experiment Result (Engel, 1992)
Reduced DOE Model after Coefficient Significance Tests
y  2.25  0.075 x1  0.063x2  0.231x3  0.425u1  0.281u 2  0.144u 3  0.05n1
 0.588 x 2 e1  0.556u 3 e1  0.063x1 n1  0.125 x 2 n1  0.094u 2 n1  0.106u 3 n1  
Parameter Estimation Error
 ~β  5.5110-4  I 2121
15
Robust Parameter Design
Response Model
y  2.25  0.075 x1  0.063x 2  0.231x3  0.425u1  0.281u 2  0.144u 3  0.05n1
 0.588 x 2 e1  0.556u 3 e1  0.063x1 n1  0.125 x 2 n1  0.094u 2 n1  0.106u 3 n1  
Variance Model
Var ( yˆ )  (0.05  0.0625 x1  0.125 x2  0.0938u2  0.1063u3 ) 2  n21
 (0.5875 x2  0.5563u3 ) 2  e21 .
RPD Settings

X *  0.4664 0
x3*

T

*
*
, and U  u1

 0.2222 0
T
u1 and x3 are adjusted
according to target
values as in right table
16
DOE-Based APC
Objective Loss Function


2
J APC ( X, U eˆ ,βˆ )   2  ˆ 0  t  βˆ 1T X  βˆ T2 U  βˆ T3 eˆ  X T Bˆ 1eˆ  U T Bˆ 2 eˆ



 
 
T
T
 βˆ 3  Bˆ 1T X  Bˆ T2 U  ~e βˆ 3  Bˆ 1T X  Bˆ T2 U  βˆ 4  Bˆ T3 X  Bˆ T4 U  n βˆ 4  Bˆ T3 X  Bˆ T4 U

  20  X T  β 3 X  U T  β 2 U  eˆ12 23
  n21  24  eˆ12 X T  B3 X  eˆ12 U T  B 2 U   n21 X T  B3 X   n21 U T  B 4 U
Optimal Settings
 
X*  arg min Eeˆ1 J APC X, U* | eˆ1 , βˆ
X 1
where


Eeˆ1 J APC



X, U | eˆ1 
*


M 1


2
1
*
ˆ
J
X
,
U
|
e
(
i
)

e
 APC
1
2
M  1 i 1
2eˆ1

eˆ1 ( i ) 2
2 eˆ21
ˆ eˆ βˆ  B
ˆ eˆ  B
ˆ  B
ˆ B
ˆ  B
ˆ   eˆ   
U   βˆ 2  B
2 1
2
2 1
2
4
B2
B4
*
ˆ
0
T

2
~
e1
T
2

2
n1
T
4

2
β2
2
1

2
n1


1
ˆ eˆ βˆ  B
ˆ eˆ  B
ˆ  ~2 βˆ  B
ˆ T X*  B
ˆ  2 βˆ  B
ˆ T X*
 t  βˆ 1T X  βˆ T3 eˆ1  X*T B
1 1
2
2 1
2 e1
3
1
4 n1
4
3

17
Simulation Results
Comparison of RPD, CE control and Cautious Control
Assuming e1 ~ N (0,0.25) n1 ~ N (0,0.25) e1 ~ N (0, 0.025)
Optimal Off-line Setting
X *  0.5121 - 0.2817 0.5085
T
Cautious control law
performs much
better than RPD
0.5
0.45
0.4
0.35
0.3
0.25
0.2
0.15
0.1
0.05
0
APC-considering modeling error
APC-w/o modeling error
RPD
1
1.25 1.5 1.75
2
2.25 2.5 2.75
3
3.25 3.5
Target (Percent Shrinkage)
Quadratic Loss (e -3)
Quadratic Loss
Control Strategy Evaluation
4
APC-consider modeling error
3.5
APC - w/o modeling error
3
2.5
2
1.5
1
0.5
0
1.4
1.65
1.9
2.15
2.4
2.65
2.9
Target (Percent Shrinkage)
18
Simulation Results - 2
Certainty Equivalence – assume observation perfect
7
6
J /J
CE RD
5
4
3
2
1
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
 e2 /  e2
1
1
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.
19
Control strategy with partial sensing failure
–1
• Sensor noise level change – no modeling error
 e2  e2  0.1   e2  e2  1
1
1
1
1
CE Control suffers
greatly from noise
level change
e0
2
Percentage Shrinkage
Oberver Noise Level
150 observations, sensor
noise level increased
from point 51 to 100,
then restored. t=1.6
0
-2
0
1
50
100
150
50
100
150
0
-1
-2
0
2
1.6
Mean of RPD has
deviated from target
y
y ce
y rd
1
0
50
Observations
U  U*  h(X* , eˆ , βˆ , e , n β )
100
150
20
Control strategy with partial sensing
failure – 2
• Sensor noise level change – APC considering modeling error
Percent Shrinkage
255 observations, sensor noise level increased
from point 101 to 200, then restored U  U*  h(X* , eˆ, βˆ , e , n β )
2.2
y
y_ce
2
1.8
1.6
1.4
1.2
1
51
101
151
201
251
Observations
Overall J/J_ce=16.8%. APC performance is steady
over different noise levels.
21
Control strategy with partial sensing failure – 3
• Sensor failure
- Assume no modeling
error,  ~e  e  0.1
- 250 observations,
sensor failed from point
51 to 150, then repaired
0
1
ehat
-2
0
2
0
Percentage Shrinkage
1
e0
2
-2
0
2
50
100
150
200
250
50
100
150
200
250
Control Strategy
Switch to RPD setting after the detection
of sensor failure
- Actual system will have step response
y
y ce
1.8
y rd
1.6
1.4
0
50
100
150
200
250
Observations
22
Industrial Collaboration with OG Technologies:
DOE-Based APC Test bed in Hot Deformation Processes
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
forming
caster
[1] Controllable variables:
shut height, punch speed,
temperature, binding force
DOE-Based APC
in-process part
[2] In-process sensing variables:
tonnage signal, shut height, vibration,
punch speed, temperature
Formed part
[3] In-process part sensing:
surface and dimension
measurements
Process change detection and
on-line estimation of
estimable noise factors
23
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.
24
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.
25