M esd Goal Seeking and Equality Constraints ESD.11 Goal Seeking- is essentially the same as finding the set of points x that will satisfy the following"soft equality constraint on the objective Find all x such that 网≤E req Example 1000kg I arget mass Target Jre (x)= Rara=1.5 Mbps Target data rate ector 15M$ Target Cost o Massachusetts Institute of Technology -Prof de Weck and Prof Willcox Engineering Systems Division and Dept of Aeronautics and Astronautics
6 Goal Seeking and Equality Constraints Goal Seeking and Equality Constraints • Goal Seeking – is essentially the same as finding the set of points x that will satisfy the following “soft” equality constraint on the objective: J ( ) x − Jreq Find all x such that ≤ ε Jreq Target mass Example ª msat º ª 1000kg º « » Target J ( ) x = R Target data rate data » » ≡ « 1.5Mbps req «« » Vector: Target Cost ¬ » « 15M $ » ¼ « Csc ¼ ¬ © Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox Engineering Systems Division and Dept. of Aeronautics and Astronautics
M esd Goal Programming vS Isoperformance ESD. 77 Criterion space Decision space (Design Space) (Objective Space) 2 X 2 Case 1: The target (goal) vector is in Z- usually get non-unique solutions Isoperformance Case 2: The target (goal)vector is not in Z-don't get a solution -find closest Goal Programming o Massachusetts Institute of Technology-Prof de Weck and Prof Willcox Engineering Systems Division and Dept of Aeronautics and Astronautics
7 Goal Programming vs. Isoperformance Goal Programming vs. Isoperformance Criterion Space Decision Space (Objective Space) (Design Space) J2 is not in Z - don’t get a solution - find closest x2 J1 c S Z 2 x1 x4 x3 x2 J1 J3 J2 J2 The target (goal) vector = Isoperformance T2 T1 The target (goal) vector Case 1: is in Z - usually get non-unique solutions Case 2: = Goal Programming © Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox Engineering Systems Division and Dept. of Aeronautics and Astronautics
M 168g esd Isoperformance Analogy ESD.J7 Non-Uniqueness of Design if n>z Analogy: Sea Level Pressure [mbar Chart: 1600 Z, Tue 9 May 2000 Performance: Buckling load Constants: 1=15 m), c=2.05 P- CTT EI Isobars= Contours of Equal Pressure Parameters Longitude and latitude Variable Parameters: E,I(r) Requirement L E, REQ 1000 metric tons L Solution 1: V2 ,[=10cm,E=19.1e+10 008 Solution 2 ,r=128cm,E=7.1e+10 012 d 2r EI Isoperformance Contours= Locus of constant system performance Bridge- Column多 Parameters=e.g Wheel Imbalance Us Support Beam Ix, Control Bandwidth o o Massachusetts Institute of Technology -Prof de Weck and Prof Willcox Engineering Systems Division and Dept of Aeronautics and Astronautics
Isoperformance Analogy Isoperformance Analogy Non-Uniqueness of Design if n > z Analogy: Sea Level Pressure [mbar] Chart: 1600 Z, Tue 9 May 2000 Performance: Buckling Load 2 c EI π P = Isobars = Contours of Equal Pressure Constants: l=15 [m], c=2.05 E l 2 Parameters = Longitude and Latitude Variable Parameters: E, I(r) Requirement: PE REQ = 1000 metric tons , Solution 1: V2A steel, r=10 cm , E=19.1e+10 Solution 2: Al(99.9%), r=12.8 cm, E=7.1e+10 L L L H 1008 1008 1012 1008 1008 1008 1012 1016 1012 1012 1012 1016 1012 1004 1016 1012 1012 l 2r PE E,I c Bridge-Column Isoperformance Contours = Locus of constant system performance Parameters = e.g. Wheel Imbalance Us, Support Beam Ixx, Control Bandwidth ω c 8 © Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox Engineering Systems Division and Dept. of Aeronautics and Astronautics
M 168g esd Isoperformance and LP ESD.J7 In LP the isoperformance surfaces are hyperplanes min C x Let c'x be performance objective and kx a cost objective S.t. XLB Sxsxur B(primal feasibility) 1. Optimize for performance C X 2. Decide on acceptable performance penalty e 3. Search for solution on isoperformance Efficient hyperplane that Solution minimizes cost kx C X Performance c xtE= C Optimal Solution o Massachusetts Institute of Technology -Prof de Weck and Prof Willcox Engineering Systems Division and Dept of Aeronautics and Astronautics
Isoperformance and LP Isoperformance and LP T • In LP the isoperformance surfaces are hyperplanes min c x • Let cTx be performance objective and kTx a cost objective s. . t x ≤ ≤x x LB UB 1. Optimize for performance cTx* 2. Decide on acceptable performance penalty ε 3. Search for solution on isoperformance hyperplane that minimizes cost kTx* cTx* = cT k c Efficient Is Solution operformance hyperplane x** B (primal feasibility) Performance cTx*+ Optimal Solution ε xiso 9 © Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox Engineering Systems Division and Dept. of Aeronautics and Astronautics
M esd Isoperformance Approaches 16888 ESD.J7 (a)deterministic I so performance Approach Deterministic Isoperformance Model Algorithms Jz, reg- (b)sto cha stic Iso performance Approach Design Space Ind x ●80% k Isoperformance Empirical 10.7592117.34 Algorithms System Model 20913118343 50% Statistical data o Massachusetts Institute of Technology -Prof de Weck and Prof Willcox Engineering Systems Division and Dept of Aeronautics and Astronautics
Isoperformance Algorithms Empirical System Model 10 Isoperformance Approaches Isoperformance Approaches (a) deterministic I soperformance Approach Jz,req Deterministic System Model Isoperformance Algorithms Design A Design B Design C Jz,req Design Space (b) stocha stic I soperformance Approach Ind x y Jz 1 0.75 9.21 17.34 2 0.91 3.11 8.343 3 ...... ...... ...... Statistical Data Design A Design B 50% 80% 90% Jz,req Empirical System Model Isoperformance Algorithms Jz,req P(Jz) © Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox Engineering Systems Division and Dept. of Aeronautics and Astronautics