麻省理工学院：《Robust System Design》Plan for the session

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 mit 人 16881 Session #11

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 li8(21X37 Hasl+1*(2-1)+7*(3-1)=16DOF But i8 rows Therefore 2 dof can be used to estimate the sum square due to error Robust System Design 16881 mit 人 Session #11

How to Estimate Error variance in an L18 • 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 Total ss to mean sS due sS due etc ss due to factor A to factor B to error Robust System Design 16881 mit 人 Session #11

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 Lg Robust System Design 16881 mit 人 Session #11

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 L8 Robust System Design 16.881 Session #11 MIT

Column Merging in an Lg Eliminate the column which is confounded with Interactions Create a new four-level column Control factors Exp nO. ABCDEFG 234 2 12 2222 2122 22121 6 222 212 22 82212112 Robust System Design mit 人 16881 Session #11

1 2 3 4 5 6 7 8 Column Merging in an L8 • 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 1 2 2 2 2 1 1 2 2 2 2 1 2 1 2 1 2 2 1 2 1 1 2 2 1 2 1 1 Robust System Design 16.881 Session #11 MIT

Computer Aided robust design Robust System Design 16881 mit 人 Session #11

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 mit 人 16881 Session #11

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 desigr Optimization Formal mathematical form minimize y=f(x) minimize weight subject to h(x)=0 subject to height=23 8(X)≤0 max stress<o8Y fx Simulation h(x) g(x) Robust System Design mit 人 16881 Session #11

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(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 mit 人 16881 Session #11

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

Noise distributions Normal arises when many independent random variables are summed · Uniform Arises when other distributions are truncated · Lognormal ognormal distribution Arises when normall distributed variables are p(x) multiplied or transformed Robust System Design mit 人 16881 Session #11

p x 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