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 Signal Factor Response Hold constant Optimize for a “static” Control Factors experiment 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 (A1B1C1, A1B1C2,...) • 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( Ai , B j , Ck , Di ) = µ + ai + b j + ck + di + e • This is similar to a Taylor series expansion ∂f ∂f f (x, y) = f (xo , yo ) + ⋅ (x − xo ) + ⋅ ( y − yo ) + h.o.t ∂x x= x ∂y o y = yo Robust System Design 16.881 MIT One Factor at a Time Control Factors Expt. A B C D No. 1 2 2 2 2 η1 2 1 2 2 2 η2 3 3 2 2 2 η3 4 2 1 2 2 η4 5 2 3 2 2 η5 6 2 2 1 2 η6 7 2 2 3 2 η7 8 2 2 2 1 η8 9 2 2 2 3 η9 Robust System Design 16.881 MIT Orthogonal Array Control Factors Expt. A B C D No. 1 1 1 1 1 η1 2 1 2 2 2 η2 3 1 3 3 3 η3 4 2 1 2 3 η4 5 2 2 3 1 η5 6 2 3 1 2 η6 7 3 1 3 2 η7 8 3 2 1 3 η8 9 3 3 2 1 η9 Robust System Design 16.881 MIT Notation for Matrix Experiments 4 L9 (3 ) Number of experiments 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 A3 is η3=µ+a3+e3 • Orthogonal array – Estimated response at A3 is η3=µ+a3+1/3(e7+ e7+ e7) – Variance sums for independent errors – Error variance ~ 1/replication number Robust System Design 16.881 MIT Factor Effect Plots Factor Effects on the Mean 20 •Which C F A2 A1 B3 C1 C2 D1 D2 15 A3 B2 C3 D3 ec) levels will s 10 B1 me ( i 5 T you choose? 0 • What is your A1 A2 A3 B1 B2 B3 C1 C2 C3 D1 D2 D3 scaling Factor Effects on the Variance 25 A2 factor? ) 20 C1 D1 D2 B1 B2 B3 sec 15 C2 A3 C3 10 A1 D3 me ( i T 5 0 A1 A2 A3 B1 B2 B3 C1 C2 C3 D1 D2 D3 Robust System Design 16.881 MIT Prediction Equation Factor Effects on the Variance 25 A2 20 C1 D1 D2 B1 B2 B3 µ sec) 15 C2 A3 C3 10 A1 D3 me ( i T 5 0 A1 A2 A3 B1 B2 B3 C1 C2 C3 D1 D2 D3 η( Ai , B j , Ck , Di ) = µ + ai + b j + ck + di + e Robust System Design 16.881 MIT Inducing Noise Control Factors Expt. A B C D Noise No. Factor 1 1 1 1 1 η1 { Expt. N 2 1 2 2 2 η2 No. 1 1 3 1 3 3 3 η3 2 2 4 2 1 2 2 η4 5 2 2 3 1 η5 6 2 3 1 2 η6 7 3 1 3 3 η7 8 3 2 1 3 η8 9 3 1 2 1 η9 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.