1 Goal Programming and Isoperformance Isoperformance March 29, 2004 Lecture 15 Olivier de Weck ? Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox Engineering Systems Division and Dept. of Aeronautics and Astronautics 2 Why not performance-optimal ? optimal ? “The experience of the 1960’s has shown that for military aircraft the cost of the final increment of performance usually is excessive in terms of other characteristics and that the overall system must be optimized, not just performance” Ref: Current State of the Art of Multidisciplinary Design Optimization (MDO TC) - AIAA White Paper, Jan 15, 1991 TRW Experience Industry designs not for optimal performance, but according to targets specified by a requirements document or contract - thus, optimize design for a set of GOALS. ? Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox Engineering Systems Division and Dept. of Aeronautics and Astronautics 3 Lecture Outline Lecture Outline Motivation - why goal programming ? Example: Goal Seeking in Excel Case 1: Target vector T in Range = Isoperformance Case 2: Target vector T out of Range = Goal Programming Application to Spacecraft Design Stochastic Example: Baseball Forward Perspective Choose x What is J ? Reverse Perspective Choose J What x satisfy this? ? Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox Engineering Systems Division and Dept. of Aeronautics and Astronautics A Domain Range B T f a Jx Target Vector Many-To-One Goal Seeking Goal Seeking max(J) T=J req J x * min(J) * ,iiso x x iLB x , , x min x iUB i max 4 ? Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox Engineering Systems Division and Dept. of Aeronautics and Astronautics 5 Excel: Tools Excel: Tools – Goal Seek Goal Seek Excel - example J=sin(x)/x -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 1.2 -6 -5 .2 -4 .4 - 3. 6 -2 .8 -2 - 1. 2 -0 .4 0. 4 1. 2 2 2. 8 3. 6 4. 4 5. 2 6 x J sin(x)/x - example single variable x no solution if T is out of range For information about 'Goal Seek', consult Microsoft Excel help files. ? 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 – is essentially the same as finding the set of points x that will satisfy the following “soft” equality constraint on the objective: xJ () ? J req Find all x such that ≤ ε J req Target mass Example a m oa 1000kg o sat ? ? Target data rate x Target J ()= R data ? ? ≡ ? 1.5Mbps req ?? ? Vector: Target Cost ? ??15M $ ? ? ? C sc ?? ? Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox Engineering Systems Division and Dept. of Aeronautics and Astronautics 7 Goal Programming vs. IsoperformanceGoal Programming vs. Isoperformance Criterion Space Decision Space (Objective Space) (Design Space) J 2 is not in Z - don’t get a solution - find closest x 2 J 1 S Z c 2 x 1 x 4 x 3 x 2 J 1 J 3 J 2 J 2 The target (goal) vector = Isoperformance T 2 T 1 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 Isoperformance Analogy Isoperformance Analogy Non-Uniqueness of Design if n > z Analogy: Sea Level Pressure [mbar] Chart: 1600 Z, Tue 9 May 2000 2 Performance: Buckling Load cEIπ =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: P REQE = tonsmetric1000 , 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 P E E,I c Bridge-Column Isoperformance Contours = Locus of constant system performance Parameters = e.g. Wheel Imbalance Us, Support Beam I xx , Control Bandwidth ω c 8 ? 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 cx Let c T x be performance objective and k T x a cost objective s tx≤≤x. . x LBUB 1. Optimize for performance c T x* 2. Decide on acceptable performance penalty ε 3. Search for solution on isoperformance hyperplane that minimizes cost k T x* c T x* = c T k c Efficient SolutionI s o p e r f o r m a n c e h y p e r p l a n e x** B (primal feasibility) Performance Optimal Solution c T x*+ε x iso 9 ? 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 (a) deterministic I soperformance Approach Jz,req Deterministic System Model Isoperformance Algorithms Design A Design B Des ign 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 Des ign 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 11 Nonlinear Problem Setting Nonlinear Problem Setting “Science Target Observation Mode” White Noise Input Appended LTI System Dynamics d J Disturbances Opto-Structural Plant Control (ACS, FSM) (RWA, Cryo) w u y z Σ Σ Actuator Noise Sensor Noise [A d ,B d ,C d ,D d ] [A p ,B p ,C p ,D p ] [A c ,B c ,C c ,D c ] [A zd , B zd , C zd , D zd ] Variables: x j J = RSS LOS z,2 ? Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox Engineering Systems Division and Dept. of Aeronautics and Astronautics Performances Phasing Pointing z,1 =RMMS WFE z=C zd q zd 12 Problem Statement Problem Statement Given xq+ B zd () xr LTI System Dynamics q = A zd () xd+ B zr () jjj z = C zd () xd+ D ( xr, where j = 1, 2,..., n p xq+ D zd () zr j ) jj And Performance Objectives T T o 1/ 2 §· 1/ 2 J = F () , e.g. J =z zz t= E a z z ? = ¨ ¨ T 1 3 ( ) 2 dt ? RMS ,zz i 2 ? ? ? 0 1 Find Solutions x such that iso J ( x ) ≡ J ? i = 1, 2,..., n zz i iso z req i,,, ?≥1 and x jLB ≤ x j ≤ x j,UB ? j = 1, 2,..., n Assuming nz , Jx () ? J τ ,zz req Subject to a numerical tolerance τ : ≤ = ε J 100 ,zreq ? Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox Engineering Systems Division and Dept. of Aeronautics and Astronautics 13 Bivariate Exhaustive Search (2D) Bivariate Exhaustive Search (2D) “Simple” Start: Bivariate Isoperformance Problem First Algorithm: Exhaustive Search Performance J (xx 2 ) : z = 1 z 1 , coupled with bilinear interpolation Variables xj= 1, 2 : j , n = 2 Number of points along j-th axis: a x ? x j,UB j,LB o x1 x2 n = j ? ? ? ?x ? Can also use standard contouring code like MATLAB contourc.m ? Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox Engineering Systems Division and Dept. of Aeronautics and Astronautics 14 Contour Following (2D) Contour Following (2D) k-th isoperformance point: Taylor series expansion 1 TT Jx ( k ( ) = Jx) +?J x? + ?xH x? + H.O.T. k kz z z x   x 2 first order term second order term T x?J ? ≡ 0 a τ J z k p T α ( a ?J o z ?? ?x J 1 ? ?= ? z ??J ? z ? ? ?x 2 ? ? o 1/ 2 ?1 = ? 2 zreq, tH k t k ) ?kk x ? 100 ? H: Hessian a 0 ?1 o??J k z k t =?? = ? ? 10 ? ? ? n α k : Step size J 1k x ? k x 1k x + k n k t ?J kz p ,zreq k n k t k : tangential ?=α k ? tx step direction k +1 k k+1-th isoperformance point: x = x +?x ? Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox Engineering Systems Division and Dept. of Aeronautics and Astronautics Mass m [kg] Progressive Spline Approximation (III) Approximation (III) First find iso-points on boundary Progressive Spline Approximation For RMS z Then progressive spline approximation via segment-wise bisection Makes use of MATLAB spline toolbox , 5 (b(b) t = 1 3 2 p iso 4 1 t = 0 e.g. function csape.m 4.5 4 t t l () = ? ax ,1 () o a f 1 () o3.5 iso t 6 Pt ? = ??3 t t iso ? x ,2 () ? ? f 2 () ? 2.5 2 1.5 t ∈ [ 0,1 ] 6 Pt () ∈ [ a,b ] 1 l 0.5 0 10 (a) 20 30 40 50 60 70 Disturbance Corner ωd [rad/sec] ? Use cubic ki k ( t ?ζ ) l splines: k=4 f jl () = | t i=1 ( ki ) ! c jlk , t ∈ [ ζ ! ζ l+1 ] , ? , , l 15 ? Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox Engineering Systems Division and Dept. of Aeronautics and Astronautics Bivariate Algorithm Comparison Comparison Metric Exhaustive Search (I) Contour Follow (II) Spline Approx (III) FLOPS 2,140,897 783,761 377,196 CPU time [sec] 1.15 0.55 0.33 Tolerance τ 1.0% 1.0% 1.0% Actual Error γ iso 0.057% 0.347% 0.087% # of isopoints 35 45 7 x 10 -4 Quality of Isoperformance Solution Plot Results for SDOF Problem Conclusions: (I) most general but expensive (II) robust, but requires guesses (III) most efficient, but requires monotonic performance J z ? 7.8 ? ?? Normalized Error : 0.34685 [%] Allowable Error: 1 [%] correction step 0 5 10 15 20 25 30 35 40 45 Isoperformance Solution Number 16 ? Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox Engineering Systems Division and Dept. of Aeronautics and Astronautics Isoperformance Quality Metric 8.2 Per f or manc e R M S z m 8.15 8.1 “Normalized Error” 8.05 ao 1/ 2 n iso 2 ( ,, ( Jx iso k ) z req ) 8 | ? J 7.95 ?? z 100 ?= iso r=1 7.9 J 7.85 ,z req ? n iso ? 17 Multivariable Branch-and-Bound Bound Exhaustive Search requires n p -nested loops -> NP-cost: e.g. p n a x ? x o UB, jLB, j N = ∏? j=1 ? ? ?x j ? ? ? Branch-and-Bound only retains points/branches which meet the condition: a J () ≥ J , ≥ J () o ∪ a J () ≤ J , ≤ J z () ? xx ?? xx z i zreq z j z i zreq j Expensive for small tolerance τ Need initial branches to be fine enough ? Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox Engineering Systems Division and Dept. of Aeronautics and Astronautics 18 Tangential Front Following Tangential Front Following UV T =?J T a ?J ?J o Σ 1 z " o z U = a ? u 1 u n z ?  ? ?x ?x ? ? 11 zxz ? JΣ= a diag ( σ " σ n ) 0 ( p ?n z ) ? o ?=? #%#? 1 nx n z z ?  z zxn ? ?J 1 ?J z ? ?? ao V = ? v " v z z+1 " v ? V 1V 2 ? ??x n ?x ? ? 1 n ? ?   ? column space nullspace ? ?= ? ( β v +! β ? v ) = αV βx α 1 z+1 nz n t SVD of Jacobian provides V-matrix V-matrix contains the orthonormal vectors of the nullspace. Isoperformance set I is obtained by following the nullspace of the Jacobian ! ? Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox Engineering Systems Division and Dept. of Aeronautics and Astronautics 19 Vector Spline Approximation Vector Spline Approximation Tangential front following is more efficient than branch-and-bound but can still be expensive for n p large. Vector Spline Approximation of Isoperformance Set 10 20 30 40 50 60 1 2 3 4 5 Isoperformance mass disturbance corner Isoperformance Centroid A B Idea: Find a representative subset off all isoperformance points, which are different 600 from other. 500 c ontr ol c o r ner 400 300 200 100 0 “Frame-but-not-panels” analogy in construction Algorithm: 1. Find Boundary (Edge) Points 2. Approximate Boundary curves 3. Find Centroid point 4. Approximate Internal curves Boundary Curves Boundary Points ? Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox Engineering Systems Division and Dept. of Aeronautics and Astronautics Multivariable Algorithm Comparison Comparison Challenges if n p > 2 Computational complexity as a function of [ n z n d n p n s ] Visualization of isoperformance set in n p -space Table: Multivariable Algorithm Comparison for SDOF (n p =3) Problem Size: Metric Exhaustive Branch-and- Tang Front V- Spline z = # of Search Bound Following Approx performances MFLOPS 6,163.72 891.35 106.04 1.49 CPU [sec] 5078.19 498.56 69.59 4.45 d = # of Error Y iso 0.87 % 2.43% 0.22% 0.42% disturbances # of points 2073 7421 4999 20 n = # of From Complexity Theory: Asymptotic Cost [FLOPS] variables Exhaustive Search: log ( J ) → n logα + 3 log n + c exs p s n s = # of Branch-and-Bound: log ( J ) → n ( n log 2 + log β ) + 3log n + c bab g p s states Tang Front Follow: log ( J ) → ( n ? n ) logγ + log 1 + n ) + 3log n + c ( ztff p z s V-Spline Approx: log ( J ) → n log 2 + 3log n + log(n +1)+c vsa p s z Conclusion: Isoperformance problem is non-polynomial in n p ? Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox Engineering Systems Division and Dept. of Aeronautics and Astronautics 20 21 Graphics: Radar Plots For n p >3 Cross Orthogonality Matrix , , , , (, ) iso i iso j iso i iso j p p COM i j p p ? = ? 6.2832 21.3705 5.0000 0.5000 186.5751 628.3185 Disturbance corner ω d Oscillator mass m Optical control bw ω o A B Interested in low COM pairs Multi-Dimensional Comparison of Isoperformance Points ω d 62.8 rad/sec m 5 kg ω o 628.3 rad/sec A B ? Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox Engineering Systems Division and Dept. of Aeronautics and Astronautics 22 ? Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox Engineering Systems Division and Dept. of Aeronautics and Astronautics Nexus Case Study on-orbit configuration OTA 012 meters NGST Precursor Mission 2.8 m diameter aperture Mass: 752.5 kg Cost: 105.88 M$ (FY00) Target Orbit: L2 Sun/Earth Projected Launch: 2004 Demonstrate the usefulness of Isoperformance on a realistic conceptual design model of a high-performance spacecraft Instrument Module Sunshield Delta II Fairing launch configuration Integrated Modeling Nexus Block Diagram Baseline Performance Assessment Sensitivity Analysis Isoperformance Analysis (2) Multiobjective Optimization Error Budgeting Deployable PM petal Purpose of this case study: The following results are shown: Details are contained in CH7 Nexus Spacecraft Concept Pro/E models ? NASA GSFC 23 ? Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox Engineering Systems Division and Dept. of Aeronautics and Astronautics Nexus Integrated Model X Y Z 8 m 2 solar panel RWA and hex isolator (79-83) SM (202) sunshield 2 fixed PM petals deployable PM petal (129) SM spider (I/O Nodes) Design Parameters Instrument Spacecraft bus (84) t_sp I_ss Structural Model (FEM) (Nastran, IMOS) Legend: m_SM K_zpet m_bus K_rISO K_yPM Cassegrain Telescope: PM (2.8 m) PM f/# 1.25 SM (0.27 m) f/24 OTA (149,169) (207) ,?Φ Out1 ? Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox Engineering Systems Division and Dept. of Aeronautics and Astronautics 24 Nexus Block Diagram Number of performances: n z =2 Number of states n s = 320 Number of design parameters: n p =25 Number of disturbance sources: n d =4 RWA Noise Inputs WFE NEXUS Plant Dynamics Outputs physical Sensitivity RMMS 24 [nm][m] 30 dofs [N,Nm] K In1 Out1 [m,rad] WFE 36 3 3 Performance 2 Performance 1 30 x' = Ax+Bu DemuxOut1 Mux Centroid y = Cx+Du K [N] Sensitivity LOS WFE Cryo Noise 3 [microns] 3 2 2 [Nm] Demux [m] -K- m2mic Attitude Control 2 x' = Ax+Bu Torques [m] y = Cx+Du Centroid 3 rates [rad/sec] Measured FSM Controller Centroid 3 FSM x' = Ax+Bu 8 K 2 Mux Coupling Out1 y = Cx+Du [rad] [m] Out1 GS NoiseACS Controller Angles ACS Noise 2 K [rad][rad] desaturation signal gimbal angles FSM Plant 2 25 J z (p o ) Cumulative RSS for LOS Results Lyap/Freq Time 20 requirement J z,1 (RMMS WFE) 25.61 19.51 [nm] 10 0 RSS [ μ m] Initial Performance Assessment Initial Performance Assessment J z,2 (RSS LOS) 15.51 14.97 [μm] -1 0 1 2 10 10 10 10 Frequency [Hz] Centroid Jitter on Focal Plane [RSS LOS] 0 Freq Domain Time Domain Critical Mode 40 10 60 T=5 sec 14.97 μm 1 pixel Requirement: J z,2 =5 μm 2 PSD [ μ m /Hz ] Cen t ro i d Y [ μ m] 20 -5 10 0 -1 0 1 2 10 10 10 10 Frequency [Hz] -20 Cent x Signa l [ μ m] 50 -40 0 -60 -50 -60 -40 -20 0 20 40 60 5 6 7 8 9 10 Time [sec] Centroid X [μm] ? Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox Engineering Systems Division and Dept. of Aeronautics and Astronautics 26 Nexus Sensitivity Analysis Analysis Norm Sensitivities: RMMS WFE Norm Sensitivities: RSS LOS Graphical Representation of Ru K_yPM De sig n Pa rame t e rs analytical finite difference Ru Jacobian evaluated at design Us Us co ntrol op t i cs plan t d i st ur b anc e par am s par am s par ame t e r s par ame t e r s p o , normalized for comparison. Ud Ud fc fc a ?J ?J o Qc Qc z,1 z, 2 Tst Tst ? ?R ?R ? Srg Srg ?? uu Sst Sst J?= p o ? "" ? Tgs Tgs z J ?? m_SM m_SM ?J ?J ,zo K_yPM ,1 , 2zz K ?? ?K ?? K_rISO K_rISO ? cf cf m_bus m_bus K_zpet K_zpet RMMS WFE most sensitive to: t_sp t_sp Ru - upper op wheel speed [RPM] I_ss I_ss Sst - star track noise 1σ [asec] I_propt I_propt K_rISO - isolator joint stiffness [Nm/rad]zeta zeta K_zpet - deploy petal stiffness [N/m] lambda lambda Ro Ro RSS LOS most sensitive to:QE QE Mgs Mgs gcm 2 ]Ud - dynamic wheel imbalance [ fca fca K_rISO - isolator joint stiffness [Nm/rad] Kc Kc zeta - proportional damping ratio [-] Kcf Kcf Mgs - guide star magnitude [mag] -0.5 0 0.5 1 1.5 -0.5 0 0.5 1 1.5 p /J * J /? pp*J/p Kcf - FSM controller gain [-] ? ? o z,1,o z,1 o /J z,2,o ? z,2 ? Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox Engineering Systems Division and Dept. of Aeronautics and Astronautics 27 2D-Isoperformance Analysis Ud=mrd [gcm 2 ] CAD Model K_rISO [Nm/rad] isolator strut ω r m m d joint 0 10 20 30 40 50 60 70 80 90 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 Ud dynamic wheel imbalance [gcm 2 ] K_rISO RW A is ol at or j o i n t stif f n ess [Nm/ rad ] Isoperformance contour for RSS LOS : Jz,req = 5 μm 10 2 0 2 0 20 60 6 0 60 12 0 12 0 16 0 p o Parameter Bounding Box test spec HST Initial design 5 5 E-wheel ? Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox Engineering Systems Division and Dept. of Aeronautics and Astronautics 28 Nexus Multivariable Nexus Multivariable Isoperformance n p =10 Pareto-Optimal Designs p * Qc Ud iso Cumulative RSS for LOS Design A ISO 3850 [RPM 5000 [Nm/rad] Us 2.7 [gcm 90 [gcm 2 ]0.025 [-] Tgs 0.4 [sec] K z pet 18E+08 [N/m] Kcf 1E+06 [- -5 performance 10 t s p Mgs uncertainty 0.005 [m] 20 [mag] 02 10 10 Frequency [Hz] Performance Cost and Risk Objectives Jz,1 Jz,2 Jc,1 Jc,2 Jr,1 6 RMS [ μ m] Best “mid-range” 4 ] compromise 2 0 02 10 10 Design B Frequency [Hz] K Ru r Smallest FSM ] control gain 0 10 2 PSD [ μ m /Hz ] Design C ] Smallest A: min(J c1 ) B: min(J c2 ) C: min(J r1 ) Design A 20.0000 5.2013 0.6324 0.4668 +/- 14.3218 % Design B 20.0012 5.0253 0.8960 0.0017 +/- 8.7883 % Design C 20.0001 4.8559 1.5627 1.0000 +/- 5.3067 % ? Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox Engineering Systems Division and Dept. of Aeronautics and Astronautics 29 Nexus Initial p o vs. Final Design p** iso Parameters Initial Final secondary +X +Z +Y hub SM Deployable segment SM Spider Support t sp Spider wall thickness D sp K zpet Improvements are achieved by a well balanced mix of changes in the Ru 3000 3845 Us 1.8 1.45 Ud 60 47.2 Qc 0.005 0.014 Tgs 0.040 0.196 KrISO 3000 2546 Kzpet 0.9E+8 8.9E+8 tsp 0.003 0.003 Mgs 15 18.6 Kcf 2E+3 4.7E+5 [RPM] [gcm] [gcm 2 ] [-] [sec] [Nm/rad] [N/m] [m] [Mag] [-] Centroid Jitter on 50 40 30 Focal Plane C e ntr oi d Y 0 -10 T=5 sec μm [RSS LOS] 20 10 Initial: 14.97 Final: 5.155 -20 disturbance parameters, structural -30 redesign and increase in control gain of the FSM fine pointing loop. -40 -50 -50 0 50 Centroid X ? Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox Engineering Systems Division and Dept. of Aeronautics and Astronautics 30 Applicability to other Fields Applicability to other Fields Parameters p j j=1,2 Example : Crack Growth Predictions in Metal Fatigue a o Initial Crack Length sec · ? § ¨ ? π a w 1 ?σ Stress Amplitude Paris da aσπ CK m =? K?= ? ? Law: dN 2a w=6” Center Cracked Panel stress ?σ Cra c k l e ngth a [i nch] σ max ?σ Isoperformance Curve: Requirement=CGL=Nc: 25000 0 0.1 0.2 0.3 0.4 0.5 Initial Cra ck L e ngth a o [inch] Parameter Bounding Box Performance Jz = Nc =25000 cycles to failure 2.5 Critical Load Number of Cycles N C=4e-9 m=3.5 σ min R=0 0.5 1.5 2.5 Nc= 24647 3 2 1 0 0 0.5 1 1.5 2 4 8 10 12 14 16 18 20Load cycles N [-] x 10 Stress Amplitude ?σ [ksi] ? Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox Engineering Systems Division and Dept. of Aeronautics and Astronautics 22 31 Isoperformance with Stochastic Data Example: Baseball season has started What determines success of a team ? Pitching Batting ERA RBI “Earned Run Average” “Runs Batted In” How is success of team measured ? Source: Prof. M.B. Jones, Penn State FS= Wins/Decisions ? Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox Engineering Systems Division and Dept. of Aeronautics and Astronautics Courtesy of Prof. M.B. Jones, Penn State. Used with permission. 32 Raw Data Raw Data Team results for 2000, 2001 seasons: RBI,ERA,FS ? Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox Engineering Systems Division and Dept. of Aeronautics and Astronautics Courtesy of Prof. M.B. Jones, Penn State. Used with permission. 33 Stochastic Isoperformance (I) Stochastic Isoperformance (I) Step-by-step process for obtaining (bivariate) isoperformance curves given statistical data: Starting point, need: - Model - derived from empirical data set - (Performance) Criterion - Desired Confidence Level ? Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox Engineering Systems Division and Dept. of Aeronautics and Astronautics Courtesy of Prof. M.B. Jones, Penn State. Used with permission. 34 Model Model Step 1: Obtain an expression from model for expected performance of a “system” for individual design i as a function of design variables x 1,I and x 2,i 1.1 assumed model E [ J ] = a + a ( x ) + a ( x ) + a ( x ? x 1 )( x ? x ) (1) i 011,i 22,i 12 1,i 2,i 2 1.2 model fitting 1 N Used Matlab a o = | J fminunc.m for General mean N j=1 j optimal surface fit Baseball: Obtain an expression for expected final standings (FS ) of i individual Team i as a function of RBI i and ERA i E F ] =+ a RBI ) + b ERA ) + c RBI ? RBI )( ERA ? ERA ) [ i m ( i ( i ( ii ? Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox Engineering Systems Division and Dept. of Aeronautics and Astronautics Courtesy of Prof. M.B. Jones, Penn State. Used with permission. 35 Fitted Model Fitted Model Coefficients: ao=0.7450 a1=0.0321 a2=-0.0869 a12= -0.0369 RMSE: Error Error Distribution σ = 0.0493 e ? Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox Engineering Systems Division and Dept. of Aeronautics and Astronautics Courtesy of Prof. M.B. Jones, Penn State. Used with permission. 36 Expected Performance Expected Performance Step 2: Determine expected level of performance for design i such that the probability of adequate performance is equal to specified confidence level E [ J ] = J + zσ ε (2) i req Error Term Required (total variance) performance Φ () level z Confidence level normal variable z (Lookup Table) Specify 2 Φ () = 1 z ? z ze 2 dz 3 z 2π ?∞ ? Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox Engineering Systems Division and Dept. of Aeronautics and Astronautics Courtesy of Prof. M.B. Jones, Penn State. Used with permission. 37 Expected Performance Expected Performance Baseball: Performance criterion - User specifies a final desired standing of FS i =0.550 Confidence Level - User specifies a .80 confidence level that this is achieved From normal Error term from data Spec is met if for Team i: table lookup EF ] = .550 + zσ = .550 + 0.84 0.0493 ) = 0.5914 [ ir ( If the final standing of team i is to equal or exceed .550 with a probability of .80, then the expected final standing for Team i must equal 0.5914 ? Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox Engineering Systems Division and Dept. of Aeronautics and Astronautics Courtesy of Prof. M.B. Jones, Penn State. Used with permission. 38 Get Isoperformance Curve Step 3: Put equations (1) and (2) together J + zσ = E [ i ] = a 0 + a 1 ( x 1,i ) + a 2 ( x ) + a ( x ? x 1 )( x ? x 2 ) req r 2,i 12 1,i 2,i (3) ,,aa 12 Four constant parameters: aa 1 2 , o Two sample statistics: x 1 , x 2 Two design variables: x 1,i and x 2,i Then rearrange: x 2,i = fx 1,i ) ( .5914 ??bERA + cRBI ( ERA ? ERA ) RBI i = m ii Baseball: a + c ERA ? ERA )( i Equation for isoperformance ? Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox curve Engineering Systems Division and Dept. of Aeronautics and Astronautics Courtesy of Prof. M.B. Jones, Penn State. Used with permission. 39 Stochastic Isoperformance This is our desired tradeoff curve ? Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox Engineering Systems Division and Dept. of Aeronautics and Astronautics Courtesy of Prof. M.B. Jones, Penn State. Used with permission. 40 Lecture Summary Lecture Summary Traditional process goes from design spacex ? objective space J (forward process) Many systems are designed to meet “targets” - Performance, Cost, Stability Margins, Mass … - Methodological Options - Formulate optimization problem with equality constraints given by targets - Goal Programming minimizes the “distance” between a desired “target” state and the achievable design - Isoperformance finds a set of (non-unique) performance invariant solutions - Isoperformance works backwards from objective space J ? design space x (reverse process) - Deterministically - Stochastically ? Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox Engineering Systems Division and Dept. of Aeronautics and Astronautics