Plan for the Session
? Quiz on Constructing Orthogonal Arrays
(10 minutes)
? Complete some advanced topics on OAs
? Lecture on Computer Aided Robust Design
? Recitation on HW#5
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How to Estimate Error variance
in an L
18
? Consider Phadke pg. 89
? How would the two unassigned columns
contribute to error variance?
? Remember L18(21x37)
– Has 1+1*(2-1)+7*(3-1) = 16 DOF
– But 18 rows
– Therefore 2 DOF can be used to estimate the
sum square due to error
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Breakdown of Sum Squares
GTSS
SS due
to mean
Total SS
SS due
to factor A
SS due
to factor B
SS due
to error
etc.
Robust System Design
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MIT
Column Merging
? Can turn 2 two level factors into a 4 level
factor
? Can turn 2 three level factors into a six level
factor
? Need to strike out interaction column
(account for the right number of DOF!)
? Example on an L
8
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1
2
3
4
5
6
7
8
Column Merging in an L
8
? Eliminate the column which is confounded with
interactions
? Create a new four-level column
Control Factors
Exp no. A B C D E F G
η
1 1 1
1 1 2
1 2 2
1 2 1
2 1 2
2 1 1
2 2 1
2 2 2
1 1 1 1
2 2 2 1
2 1 1 2
1 2 2 2
1 2 1 2
2 1 2 2
2 2 1 1
1 1 2 1
Robust System Design
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MIT
Computer Aided Robust Design
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Engineering Simulations
? Many engineering systems can be
modeled accurately by computer
simulations
– Finite Element Analysis
– Digital and analog circuit simulations
– Computational Fluid Dynamics
? Do you use simulations in design &
analysis?
? How accurate & reliable are your
simulations?
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Simulation & Design
Optimization
? Formal mathematical form
minimize y = f (x)
minimize weight
subject to h(x) = 0
subject to height=23”
g(x) ≤ 0
max stress<0.8Y
f(x)
x
Simulation
h(x)
g(x)
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Robust Design Optimization
? Vector of design variables x
– Control factors (discrete vs continuous)
? Objective function f(x)
– S/N ratio (noise must be induced)
? Constraints h(x), g(x)
– Not commonly employed
– Sliding levels may be used to handle equality
constraints in some cases
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MIT
px
Noise Distributions
? Normal
– Arises when many independent random
variables are summed
? Uniform
– Arises when other distributions are truncated
? Lognormal
Lognormal Distribution
– Arises when normally
distributed variables are
()
multiplied or transformed
x
Robust System Design
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MIT
Correlation of Noise Factors
? Covariance
COV ( x, y) = E(( x ? E( x))( y ? E( y))
n m
COV ( x, y) ?
∑∑
( x
i
? x )( y
j
? y)
i=1 j=1
Size error, Pin #1
? Correlation coefficient
k =
) ( ) ( yVARxVAR ?
COV ( x, y)
– What does k=1 imply?
– What does negative k imply?
– What does k=0 imply?
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Size error, Pin #2
Question About Noise
? Does the distribution of noise affect the S/N
ratio of the simulation?
– If so, under what conditions?
? Does correlation of noise factors affect S/N
ratios?
– If so, in what way? (raise / lower)
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Simulating Variation in Noise Factors
? Taylor series expansion
– Linearize the system response
– Apply closed form solutions
? Monte Carlo
– Generate random numbers
– Use as input to the simulation
? Orthogonal array based simulation
– Create an ordered set of test conditions
– Use as input to the simulation
Robust System Design
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Taylor Series Expansion
? Approximate system response
?f ?f
f (x, y) = f (x
o
, y
o
) +
?x
? (x ? x
o
) +
?y
? ( y ? y
o
) + h.o.t
x= x
o y= y
o
? Apply rules of probability
VAR(aX ) = aVAR( X )
VAR( X + Y ) = VAR( X ) + VAR(Y ) iff x, y independent
?To get
n
σ
2
( y) =
∑
?y
σ
2
( X
i
)
i=1
?x
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qn
Local Linearity of the
System Response Surface wrt Noise
n
? Holds for
n
1
f(t)
1
t
2
t
1
f(n)
j
?
1
E(n)
q
1 i
() ? f (t) +
∑
?
?
?q
?
?
?(n
j
? t
j
)
=
?n
nt
?
ij
=
– Machining (most)
–CMMs
? Fails for
– Dimensions
of form
n
2
– Dual head valve grinding
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MIT
Key Limitations of
Taylor Series Expansion
? System response must be approximately
linear w.r.t. noise factors
– Linear over what range?
– What if it isn’t quite linear?
? Noise factors must be statistically
independent
– How common is correlation of noise?
– What happens when you neglect correlation?
Robust System Design
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Monte Carlo Simulation
()
2.01
px1
x1
y
trial
= f ( x
1
, x
2
)
px2()
1.59
1
trials
σ
2
y
?
∑
( y
trial
? y)
2
trials ?1
i=1
x2
Robust System Design
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Monte Carlo Simulation
Pros and Cons
? No assumptions about system response f(x)
? You may simulate correlation among noises
– How can this be accomplished?
? Accuracy not a function of the number of noises
196σ.
95% confidence interval =±
trials
It’s easy too!
? It takes a large number of trials to get very
accurate answers
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Othogonal Array Based
Simulation
? Define noise factors and levels
? Two level factors
– Level 1 = μ
i
-σ
i
Level 2 = μ
i
+σ
i
? Three level factors
–Level 1 = μ
i
- σ
2
3
i
Level 2 = μ
i
Level 3 =μ
i
+ σ
2
3
i
? Choose an appropriate othogonal array
? Use as the outer array to induce noise
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px
px
px
Setting Levels for Lognormal
Distributions
()
x
()
μ
7μ
log
x
μ
0
log(7)
()
3
x
μ
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Using Sliding Levels to Simulate
Correlation
? Try it for RFP
? Mean is defined as RFM
? Tolerance is 2%
? Fill out rows 1 and 19 of the noise array
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Run the Noise Array
? At the baseline control factor settings
? Run the simulation at all of the noise factor
settings
? Find the system response for each row of
the array
? Perform ANOVA on the data
– Percent of total SS is percent contribution to
variance in system response
Robust System Design
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MIT
Othogonal Array
Pros and Cons
? Can handle some degree of non-linearity
? Can accommodate correlation
? Provides a direct evaluation of noise factor
contributions
? Usually requires orders of magnitude fewer
function evaluations than OA simulation
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Optimization
? Choose control factors and levels
? Set up an inner array of control factors
? For each row, induce noise from the outer
array
? Compute mean, variance, and S/N
? Select control factors based on the additive
model
? Run a confirmation experiment
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Next Steps
? Homework #8 due on Lecture 13
? Next session
– Read Phadke Ch. 9 -- “Design of Dynamic
Systems”
– No quiz tomorrow
? Lecture 13- Quiz on Dynamic Systems
Robust System Design
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MIT