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

18

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.