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 Robust System Design 16.881 Session #11 MIT 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 Robust System Design 16.881 Session #11 MIT 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 16.881 Session #11 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 Robust System Design 16.881 Session #11 MIT 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 16.881 Session #11 MIT Computer Aided Robust Design Robust System Design 16.881 Session #11 MIT 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? Robust System Design 16.881 Session #11 MIT 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) Robust System Design 16.881 Session #11 MIT 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 Robust System Design 16.881 Session #11 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 16.881 Session #11 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? Robust System Design 16.881 Session #11 MIT 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) Robust System Design 16.881 Session #11 MIT 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 16.881 Session #11 MIT 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 Robust System Design 16.881 Session #11 MIT 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 Robust System Design 16.881 Session #11 15 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 16.881 Session #11 MIT 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 16.881 Session #11 MIT 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 Robust System Design 16.881 Session #11 MIT 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 Robust System Design 16.881 Session #11 MIT px px px Setting Levels for Lognormal Distributions () x () μ 7μ log x μ 0 log(7) () 3 x μ Robust System Design 16.881 Session #11 MIT 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 Robust System Design 16.881 Session #11 MIT 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 16.881 Session #11 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 Robust System Design 16.881 Session #11 MIT 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 Robust System Design 16.881 Session #11 MIT 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 16.881 Session #11 MIT