Daniele Romano - pro enbis

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Dec 13, 2004... applied statistical methods in. European business and industry”. Daniele Romano. University of Cagliari e.mail: [email protected] ...
austriamicrosystems AG – Graz (Austria) December 13, 2004

Robust Design Workshop

Designing for Robustness and Cost Saving Daniele Romano University of Cagliari e.mail: [email protected]

The workshop is supported by the Pro-ENBIS European network whose mission is

“to promote the widespread use of sound science driven, applied statistical methods in European business and industry”

Schedule of the presentation

• What Robust Design can do • How it works • Some applications

Why is Robust Design a sound methodology for industry? Because it positively affects:

the customer, providing an improved product/process (with performance on the target and with minimal variability)

the company, providing a systematic way to improve quality with a minimum increase of cost or, possibly, with a reduction of cost

Brief history of Robust Design 1980-1990 :

• Genichi Taguchi introduces Robust Parameter Design and Robust Tolerance Design • the «Taguchi method» is applied in industry (USA and Japan) and lively debated in the scientific community

1990-2000 :

• The Taguchi concept is largely accepted and appreciated but a more sound and efficient statistical analysis (based on Response Surface Methodology) receives more recognition than the Taguchi method

2000-2010:

• Technical enhancement of Robust Design (multiple responses, dynamic responses, new optimisation algorithms,…) • Expansion of the potential of Robust Design due to possibility to simulate the product/process on the computer (Computer Experiments) • …

What is Robust Design ? It is a statistical methodology for engineering design aimed at finding optimal setting of design parameters of a product (process) in order to minimise, at the lowest cost, undesired variation in product (process) performance caused by variation of uncontrolled parameters during both manufacturing and usage of the product (process) while keeping average performance on the desired target. Design parameters: control factors Uncontrolled parameters: noise factors

Type of noise factors External noise factors: random variations of environmental (temperature, pressure, humidity, radiations, vibrations, supply voltage etc.) and use conditions. Totally uncontrollable. Internal noise factors: random deviations of parameters from their nominal value. Partially controllable by choosing appropriate manufacturing processes (Tolerance Design).

Examples Noise type

Product/process

Noise factors

Ignition system

Temperature

Environmental (external)

Paper-traction system of a printer

Error in feeding paper

Condition of use (external)

Shirt

Manufacturing errors in shirt components

internal

External noise factors Z2 Z1 Z3

Z4

Performance

Control factors x1

x5

y1

Product/Process

x2 x3

y2

x4

y3 y4

Z6 Z5

Z7

Internal noise factors

We get this result Performance before RD Probability density function

Performance after RD

• mean is off-target • variability is large

Probability density function

• mean is on target • variability is small

bias

Y

mean target

mean=target

by selecting • a different setting of the design factors

x1 = 5,9 x2 = 20,3 x3 = 0,6

x1 = 6,7 x2 = 18,5 x3 = 0,8

• a different setting of the variability of internal noise factors

σ1 = 1,0 σ2 = 4,1 σ3 = 0,03

σ1 = 1,5 σ2 = 3,0 σ3 = 0,02

Y

Concept of Robustness x is a design factor, controllable by the designer ze is an external noise factor, uncontrollable y, performance

x and ze have an interaction effect on the performance x=a The design solution x=b is robust as it reduces random variability in the performance

target x=b

Cost of the design is unchanged

zmin

zmax

ze

Where does the cost come from? Once the design of a product/process has been delivered, the cost comes from the quality of the components (tolerances) that the designer has specified, i.e. on the manufacturing processes that are necessary to produce components with the specified quality Cost, €

Example: the resistance value of a resistor (10 kΩ ± 0,6) part-to-part variability of the resistance value

σ = 0,2 kΩ

2€

0,6

9,4 kΩ

Tolerance on the resistance, kΩ Ω

10 kΩ

2.0,6 kΩ = 6 σ

10,6 kΩ

Concept of Cost Saving (1) x is a design factor, controllable by the designer zi is an internal noise factor, controllable in dispersion y, performance

x and zi have an interaction effect x=a

cost

cd ce cf d

f

σz

The design solution σz=d is robust as it reduces random variability in the performance but cost of the design is increased to cd

target x=b

but

σz = d

σz = f

e

σz = e zi

The design solution x=b and σz=f is equally robust and cost of the design is decreased to cf

Concept of Cost Saving (2) zi is an internal noise factor, controllable in dispersion y, performance

zi has no effect on the performance

target

The design solution σz=f is equally robust and reduces the cost σz = e

σz = f zi

General procedure for Robust Design implementation Select the problem variables

Problem definition

• Control factors, x • Noise factors, z • Response, y

Set the design scope

• Parameter design • Tolerance design • Parameter + Tolerance

Experimentation

The experiment is a set of physical (simulated) runs where the response is measured (computed) for given combinations of values of the input factors (control + noise). Noise factors have fixed levels at this stage.

Statistical Analysis

• Fit of a prediction model for the response • From that prediction model two models are obtained, one for the mean and one for the variance of the response for the in-process conditions (noise factors are random).

Optimisation

Optimisation needs the definition of a robustness criterion. It may involve only the response variance or the response variance and mean. In the case of tolerance design also production cost enters the criterion.

How to obtain models for mean and variance of the performance for in-process conditions • By calculating analytically the mean and the variance, over all the noise factors, on the prediction model of the response in experimental conditions (where noise factors are deterministic). Unfortunately this can be done in a few simple cases (response model with simple polynomial form and noise with Normal distribution). • By propagating numerically (via Monte Carlo simulation) the distributions of the noise factors through the prediction model of the response for a combination of settings of the control factors and of the variance of the internal random noise.

Robustness criteria The user can select a customised criterion by mediating among three objectives: • Mean performance is on target (no bias) • Performance variance is minimum (robustness) • Production cost is minimum (when tolerance design is present) Some useful cases 4 Minimising variance with a constraint on maximum bias

Probability density function

4 Minimising bias with a constraint on maximum variance 4 Minimise a loss function 4 Minimising product cost with constraints on maximum bias and maximum variance 4 Minimising total cost

bias

σ2

mean

target

y

L = Ez [ k (y - t)2 ] = k[ (µ – t)2 + σ2] Ctol = ∑ Ci (ti )

Ci

i

L + Ctol ti

How much does Robust Design cost ?  Cost of Robust Design is only the cost of experimentation but it increases rapidly with the number of factors and factor levels  Since noise factors must be controlled in the experiment and they are often difficult to control, the experiment becomes easily too expensive as the number of noise factor increases BUT USING COMPUTER EXPERIMENTS  Cost of experimentation is dramatically reduced (large experiments with several noise factors possible)

Example of innovative design by Robust Design with Computer Experiments b

locking device

Elastic element of a force transducer Barbato, Levi, Romano (1997)

Coop. Bilanciai, Modena (Italy) • Performance: measurement uncertainty • Control factors: geometrical parameters • Internal noise: error in parameters and positioning error of the strain gauges

Measurement uncertainty cut by 50%

ϑ

a

body

h0

Pneumatic climbing robot Manuello, Romano, Ruggiu (2003)

Dept. of Mech. Eng., Cagliari (Italy) • Performance: ability to climb • Control factors: geometrical and mechanical parameters • External noise: friction between the rings and the post • Internal noise: error in parameters

Climbing possible for a wide range of friction conditions

In both cases innovative findings (patented) came from extensive numerical experimentation

Another application: the design of a profilometer Numerical processing

Physical set-up MIRROR

PIEZOELECTRIC TRANSDUCER

TESTPIECE POLARIZED BEAM SPLITTER

 sen(π ⋅ x )  2  Sinc ( π ⋅ x ) =   (π ⋅ x )  1

CCD BEAM SPLITTER POLARIZER

0.8

Measurement result: abscissa of maximum on the interferometric curve

POLARIZER PHASE SHIFTER

2

0.6

LENS 2 IRIS

0.4

LENS 1 0.2

WHITE LIGHT -1.5

-1

-0.5

0.5

1

1.5

Dept. of Mech. Eng., Cagliari (Italy) • Control factors: mirror displacement step, interpolating curve • External noise: reflectivity of the testpiece surface, phase of the first sampled point on the interferometric curve • Internal noise: errors in the coordinates of the sampled points (dependent on the quality of optical components) Measurement uncertainty cut by nearly 80%

Physical and computer experiments were combined

Case-study Step 1: Problem definition Subject:

Design and manufacturing of a cyclone, a device used in chemical engineering to separate solid mass and gaseous mass (from Mori, 1985, used also by Li and Wu, 1999)

Scope of the design:

Select optimal settings of seven design variables and the related tolerances  Parameter + Tolerance design

Control factors: design parameters x1, x2, x3, x4, x5, x6 , x7 Internal noise factors: error in parameters z1, z2, z3, z4, z5, z6 , z7 Response:

critical parameter of particles to be separated, y, with target t

[

20 − 0.56 x1  x3  1 − 2.62 1 − 0.36(x4 x2 )  y = 174.42  x5  x2 − x1  x6 x7 17

]

32

( x4

x2 )

1.16

errors zi are independent, zero mean, Normal variables ti = 3 σzi

zi

Step 2: Experimentation • First design (for screening only): 16-run Plackett-Burman • Second design (for accurate modelling): 143-run CCD (128-run resolution IV two-level fractional + STAR points only for the control factors)

Step 3: Statistical Analysis 3.1: prediction model for response y in experimental conditions (noise factors are deterministic) • A fully quadratic model with squares only for the mean factors is estimated • Goodness of the model is tested via a confirmatory experiment 7

8

6

7

in the original response scale after log transformation

6

5

fitted

fitted

5

4

PRESS=9,21914

3

4

PRESS=0,030873

3

2

2

1

1

0

0

0

1

2

3

4

5

6

actual

original response

7

8

0

1

2

3

4

5

6

7

actual

log-transformed response

8

3.2: Models for mean and variance of the in-process response

µ ln(y) = 0,545 + 0,382 x1 - 0,265 x2 - 0,256 x5 + 0,218 x3 - 0,128 x6 - 0,124 x7 - 0,0788 x4 + - 0,0489 x1x2 + 0,0447 x2x4 + 0,0351 x22 + 0,0100 x4x7 - 0,00675 x2x7 + + 0,0307 x52 - 0,0290 x32 - 0,0272 x12 + 0,0145 x62 + 0,0145 x72

σ2ln(y) = (0.0809 - 0.0104 x2 - 0.00911 x1)2 σz12 + (-0.0560 + 0.0143 x2 - 0.0133 x1 + 0.0132 x4)2 σz22 + (0.0463 - 0.0116 x3 - 0.00966 x6)2 σz32 + (-0.0169)2 σz42 + (-0.0545 + 0.0137 x5)2 σz52 + (-0.0272 + 0.00682 x6)2 σz62 + (-0.0273 + 0.00683 x7)2 σz72 µy = exp (µ ln(y) + 0.5 σ2ln(y) )

Since y has Log-Normal distribution:

σ2y = exp (2 (µ ln(y) + σ2ln(y) )) - exp (2 µ ln(y) + σ2ln(y) ) Test on goodness of the two models via a confirmatory experiment Εˆ ( y)



Var ( y )

Ε( y )

Var ( y )

Step 4: Optimisation Cost, Ci, yen 140

Objective:

120

Minimise total cost = L + Ctol L = Ez [ k (y -

t)2 ]

= k[ (µy –

t)2

+ σy

100

2]

Ctol = ∑ Ci (ti )

80 60

i

40

Design solution:

x1 = 0.075 x2 = 0.375 x3 = 0.125 x4 = 0.125 x5 = 1.125 x6 = 20.00 x7 = 0.594

t1 = 8.52 % x1 t2 = 19.30 % x2 t3 = 16.39 % x3 t4 = 19.31 % x4 t5 = 18.27 % x5 t6 = 15.41 % x6 t7 = 15.23 % x7

Total cost reduced by a factor 5

20 0 -20 0

0.05

0.1

0.15

0.2

ti xi

0.25

Some remarks on the austriamicrosystem data Step 1: Problem definition Subject: Scope of the design:

Improvement of an Operational Amplifier (OPSE02)

Select optimal setting of one design variable  Parameter design

Control factors: design parameter lpoly (length of compensation resistance ) External noise factors: temperature, supply Internal noise factors: error in 3 component parameters: resistors, NMOS, PMOS Response:

6 performance variables: A0 (dB), BW(MHz), EIVN (nV/sqrt(Hz)), PM (deg), THD (dBc), Pdiss (uW)

Step 2: Experimentation (on a simulator) • 26-1 fractional factorial design + center point (32+1=33 runs)

Step 3: Statistical Analysis 3.1: prediction model for response y in experimental conditions (noise factors are deterministic) A0(dB) = 96,3 - 0,882 NMOS - 0,565 PMOS - 0,497 temp + 0,428 supply + 0,217 NMOS*PMOS - 0,324 NMOS*TEMP - 0,0453 NMOS*SUPPLY

R-Sq(adj) = 97,4%

BW (MHz) = 143 + 15,8 NMOS + 18,9 PMOS - 0,976 temp - 6,27 supply - 1,99 NMOS*PMOS + 5,02 NMOS*TEMP - 1,28 PMOS*SUPPLY

R-Sq(adj) = 98,1%

EIVN (nV/sqrt(Hz)) = 19,7 - 0,125 NMOS - 1,15 PMOS + 1,82 temp - 0,0427 supply 0,107 PMOS*TEMP

R-Sq(adj) = 99,8%

PM (deg.) = 67,4 - 5,28 Resistors + 2,93 NMOS - 1,03 PMOS - 1,73 temp + 2,51 Lpoly - 0,516 RES*TEMP*LPOLY + 0,431 RES*PMOS*LPOLY + 0,199 RES*NMOS*LPOLY + 1,40 RES*LPOLY - 0,359 RES*TEMP + 0,345 RES*PMOS - 0,319 PMOS*LPOLY + 0,318 TEMP*LPOLY 0,210 NMOS*TEMP

R-Sq(adj) = 98,9%

THD (dBc) = - 121 - 6,74 PMOS - 3,45 temp - 8,14 supply + 2,17 PMOS*TEMP + 6,73 PMOS*SUPPLY + 1,31 NMOS*TEMP + 2,15 TEMP*SUPPLY + 0,405 NMOS*PMOS - 1,18 PMOS*TEMP*SUPPLY + ,609 NMOS*PMOS*SUPPLY + 0,844 NMOS*PMOS*TEMP

R-Sq(adj) = 98,8%

Pdiss (uW) = 217 + 1,04 PMOS + 0,319 temp + 20,4 supply + 0,115 PMOS*SUPPLY

R-Sq(adj) = 100,0%

Lpoly

Interaction Plot for PM (deg.)

1

1 -1

94

95

96

Lpoly

97

Lpoly

98

70

-1

A0(dB) 1

65 -1

95

105

115

125

135

Lpoly

145

155

165

175

60

185

BW (MHz) 1

-1

Resistors

1

Remarks:

-1

17

18

19

20

21

22

23

EIVN (nV/sqrt(Hz))

Lpoly 1

-1

60

70

80

PM (deg.)

Lpoly 1

-1

-130

-120

Lpoly

-110

-100

-90

THD (dBc) 1

-1

200

210

220

Pdiss (uW)

230

240

• Except for Pdiss, other responses have too large variability for fulfilling specifications, unless the nominal value of resistors, PMOS and NMOS is changed • Only response PM can be made more robust (as it is affected by an interaction between resistor and lpoly) by using a higher value of lpoly • Since “resistor” does not affect the other 5 responses its tolerance could be increased without lowering overall performance

Another remark In every prediction model for responses there is lack-of-fit, as the observation at the centre of the design space (the typical operating condition) exhibits a very large residual for all responses

A0 (dB) BW(MHz) EIVN (nV/sqrt(Hz)) PM (deg) THD (dBc) Pdiss (uW)

4,86 -4,81 -5,07 3,67 -4,27 -4,98

Normal Probability Plot of the Residuals (response is Pdiss (u) 2

1

Normal Score

response

ratio between residual at the centre of the design and the standard deviation of residuals

0

-1

-2 -0,3

-0,2

-0,1

0,0

Residual

It is likeky that square terms are needed in the models. A Response Surface design (3 levels per factor) is advisable. For example a BoxBenhken with 49 runs, otherwise, or the previous design can be augmented to a CCD with only 12 additional runs.