Matrix Experiments Using Orthogonal Arrays Robust System Design 16.881 MIT Comments on HW#2 and Quiz #1 Questions on the Reading Quiz Brief Lecture Paper Helicopter Experiment Robust System Design 16.881 MIT Learning Objectives ? Introduce the concept of matrix experiments ? Define the balancing property and orthogonality ? Explain how to analyze data from matrix experiments ? Get some practice conducting a matrix experiment Robust System Design 16.881 MIT Static Parameter Design and the P-Diagram Noise Factors Induce noise Product / Process Response Signal Factor Hold constant Optimize for a “static” experiment Control Factors Vary according to an experimental plan Robust System Design 16.881 MIT Parameter Design Problem ? Define a set of control factors (A,B,C…) ? Each factor has a set of discrete levels ? Some desired response η (A,B,C…) is to be maximized Robust System Design 16.881 MIT Full Factorial Approach ? Try all combinations of all levels of the factors (A 1 B 1 C 1 , A 1 B 1 C 2 ,...) ? If no experimental error, it is guaranteed to find maximum ? If there is experimental error, replications will allow increased certainty ? BUT ... #experiments = #levels #control factors Robust System Design 16.881 MIT Additive Model ? Assume each parameter affects the response independently of the others η( A i , B j , C k , D i ) =μ+ a i + b j + c k + d i + e ? This is similar to a Taylor series expansion ?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 Robust System Design 16.881 MIT 1 1 2 2 3 3 4 4 5 5 6 6 7 7 8 8 9 9 One Factor at a Time Control Factors Expt. No. A C 2 η 1 η 3 η B D 2 2 2 2 2 2 2 2 2 2 η 2 η 2 η 2 2 1 2 2 3 2 1 2 2 η 2 η 2 η 2 3 2 1 2 2 3 2 2 Robust System Design 16.881 MIT 1 1 2 2 3 3 4 4 5 5 6 6 7 7 8 8 9 9 Orthogonal Array Control Factors Expt. No. A C D 1 1 1 η 2 2 2 η 3 3 3 η 1 2 3 η 2 3 1 η 3 1 2 η 1 3 2 η 2 1 3 η 3 2 1 η B 1 1 1 2 2 2 3 3 3 Robust System Design 16.881 MIT Notation for Matrix Experiments Number of experiments L 9 (3 4 ) Number of levels Number of factors 9=(3-1)x4+1 Robust System Design 16.881 MIT Why is this efficient? ? One factor at a time – Estimated response at A 3 is η 3 =μ+a 3 +e 3 ? Orthogonal array – Estimated response at A 3 is η 3 =μ+a 3 +1/3(e 7 + e 7 + e 7 ) – Variance sums for independent errors – Error variance ~ 1/replication number Robust System Design 16.881 MIT Factor Effect Plots ?Which CF levels will you choose? ? What is your scaling factor? Factor Effects on the Mean A1 A2 A3 B1 B2 B3 C1 C2 C3 D1 D2 D3 0 5 10 15 20 A1 A2 A3 B1 B2 B3 C1 C2 C3 D1 D2 D3 T i me ( s ec) Factor Effects on the Variance A1 A2 A3 B1 B2 B3 C1 C2 C3 D2 D3 D1 0 5 10 15 20 25 A1 A2 A3 B1 B2 B3 C1 C2 C3 D1 D2 D3 T i me ( sec ) Robust System Design 16.881 MIT Factor Effects on the Variance A1 A2 A3 B1 B2 B3 C1 C2 C3 D2 D3 D1 0 5 10 15 20 25 A1 A2 A3 B1 B2 B3 C1 C2 C3 D1 D2 D3 T i me ( sec) Prediction Equation μ η( A i , B j , C k , D i ) =μ+ a i + b j + c k + d i + e Robust System Design 16.881 MIT 1 1 2 2 3 3 4 4 5 5 6 6 7 7 8 8 9 9 Inducing Noise 1 η 1 η 1 η 1 1 1 2 2 2 3 3 3 Control Factors Expt. No. A C { B D Noise Factor Expt. No. N 1 2 1 2 2 1 2 2 η 2 η 2 η 1 3 2 2 1 3 3 η 3 η 3 η 3 3 1 3 1 2 1 2 1 Robust System Design 16.881 MIT Analysis of Variance (ANOVA) ? ANOVA helps to resolve the relative magnitude of the factor effects compared to the error variance ? Are the factor effects real or just noise? ? I will cover it in Lecture 7 ? You may want to try the Mathcad “resource center” under the help menu Robust System Design 16.881 MIT