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Page 1: Goal Programming and IsoperformanceIsoperformancedspace.mit.edu/.../lecture-notes/l15_gpiso.pdfGoal Programming vs. IsoperformanceGoal Programming vs. Isoperformance Criterion Space

1

Goal Programming and 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

Page 2: Goal Programming and IsoperformanceIsoperformancedspace.mit.edu/.../lecture-notes/l15_gpiso.pdfGoal Programming vs. IsoperformanceGoal Programming vs. Isoperformance Criterion Space

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Why not performanceWhy 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

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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

fa

Jx

Target Vector

Many-To-One

Page 4: Goal Programming and IsoperformanceIsoperformancedspace.mit.edu/.../lecture-notes/l15_gpiso.pdfGoal Programming vs. IsoperformanceGoal Programming vs. Isoperformance Criterion Space

Goal Seeking Goal Seeking

max(J)

T=Jreq

J

x *

min(J)

*

,i iso x

xi LB x ,, xmin

xi UB i

max

4 © Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox Engineering Systems Division and Dept. of Aeronautics and Astronautics

Page 5: Goal Programming and IsoperformanceIsoperformancedspace.mit.edu/.../lecture-notes/l15_gpiso.pdfGoal Programming vs. IsoperformanceGoal Programming vs. Isoperformance Criterion Space

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Excel: ToolsExcel: 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.

41.

2 22.

8 3.

64.

45.

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

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Goal Seeking and Equality ConstraintsGoal 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 ( ) − JreqFind all x such that ≤ ε

Jreq

Target massExample m 1000kgsat

Target data ratexTarget J ( ) = Rdata ≡ 1.5Mbpsreq

Vector: Target Cost15M $Csc

© Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox Engineering Systems Division and Dept. of Aeronautics and Astronautics

Page 7: Goal Programming and IsoperformanceIsoperformancedspace.mit.edu/.../lecture-notes/l15_gpiso.pdfGoal Programming vs. IsoperformanceGoal Programming vs. Isoperformance Criterion Space

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Goal Programming vs. IsoperformanceGoal Programming vs. Isoperformance

Criterion SpaceDecision Space

(Objective Space)(Design Space) J2

is not in Z - don’t get a solution - find closest

x2

J1

S Zc2

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

Page 8: Goal Programming and IsoperformanceIsoperformancedspace.mit.edu/.../lecture-notes/l15_gpiso.pdfGoal Programming vs. IsoperformanceGoal Programming vs. Isoperformance Criterion Space

Isoperformance Analogy Isoperformance Analogy

Non-Uniqueness of Design if n > z Analogy: Sea Level Pressure [mbar] Chart: 1600 Z, Tue 9 May 2000

2Performance: Buckling Load c EIπ =P Isobars = Contours of Equal Pressure Constants: l=15 [m], c=2.05 E

l2 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

LL

LL LL

HH 1008

1008

1012

1008

1008

1008

1012

1016

1012

1012

1012

1016

1012

1004

1016

1012

1012

l2r

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 c8 © Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox

Engineering Systems Division and Dept. of Aeronautics and Astronautics

Page 9: Goal Programming and IsoperformanceIsoperformancedspace.mit.edu/.../lecture-notes/l15_gpiso.pdfGoal Programming vs. IsoperformanceGoal Programming vs. Isoperformance Criterion Space

Isoperformance and LP Isoperformance and LPT

• In LP the isoperformance surfaces are hyperplanes min c x

• Let cTx be performance objective and kTx a cost objective s t x ≤ ≤ x. . xLB 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

EfficientSolutionIso

perform

ance hyperp

lanex**

B (primal feasibility)

PerformanceOptimal SolutioncTx*+εεεε xiso9 © Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox

Engineering Systems Division and Dept. of Aeronautics and Astronautics

Page 10: Goal Programming and IsoperformanceIsoperformancedspace.mit.edu/.../lecture-notes/l15_gpiso.pdfGoal Programming vs. IsoperformanceGoal Programming vs. Isoperformance Criterion Space

IsoperformanceAlgorithms

EmpiricalSystemModel

10

Isoperformance ApproachesIsoperformance 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

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Nonlinear Problem Setting Nonlinear Problem Setting

“Science Target Observation Mode”White Noise Input

Appended LTI System Dynamics

dJ

Disturbances

Opto-Structural Plant

Control

(ACS, FSM)

(RWA, Cryo)

w

u y

z

ΣΣΣΣ ΣΣΣΣActuator

Noise Sensor

Noise

[Ad,Bd,Cd,Dd]

[Ap,Bp,Cp,Dp]

[Ac,Bc,Cc,Dc]

[Azd, Bzd, Czd, Dzd]

Variables: xj J = RSS LOSz,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=Czd qzd

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Problem Statement Problem Statement

Given

x q + Bzd ( ) x r LTI System Dynamicsq = Azd ( ) x d + Bzr ( )j j j

z = Czd ( ) x d + D ( x r , where j = 1, 2,..., npx q + Dzd ( ) zr j )j j

And Performance Objectives T

T1/ 2

1/ 2

J = F ( ) , e.g. J =z z z t= E z z =T

1( )2 dt RMS,z z i 2

0

Find Solutions x such thatiso

J ( x ) ≡ J ∀ i = 1, 2,..., nzz i iso z req i, , ,

− ≥1 and x j LB ≤ x j ≤ x j ,UB ∀ j = 1, 2,..., nAssuming n z ,

J x( ) − J τ,z z reqSubject to a numerical tolerance τ : ≤ = ε

J 100,z req

© Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox Engineering Systems Division and Dept. of Aeronautics and Astronautics

Page 13: Goal Programming and IsoperformanceIsoperformancedspace.mit.edu/.../lecture-notes/l15_gpiso.pdfGoal Programming vs. IsoperformanceGoal Programming vs. Isoperformance Criterion Space

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Bivariate Exhaustive Search (2D) Bivariate Exhaustive Search (2D)

“Simple” Start: Bivariate Isoperformance Problem First Algorithm: Exhaustive Search

Performance J (x x2 ) : z = 1z 1,coupled with bilinear interpolation

Variables x j = 1, 2 :j , n = 2 Number of points along j-th axis:

x − xj ,UB j ,LB

x1

x2n = 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

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Contour Following (2D) Contour Following (2D)

k-th isoperformance point:

Taylor series expansion

1 TTJ x ( k( ) = J x ) + ∇J x∆ + ∆x H x∆ + H.O.T.kkz z z xx 2first order term second order term

T x∇J ∆ ≡ 0 τ Jz kpTα (

∂Jz

∂xJ

1∇ = z ∂Jz

∂x2

1/ 2 −1

= 2z req,

t H k tk ) k k x100H: Hessian0 −1−∇Jk z kt = ℜ⋅ =

1 0⋅ n

αk : Step size J1kx −

kx1kx +

knkt

∇Jkz p

,z reqkn ktk: tangential ∆ = α k ⋅ tx

step direction

k +1 kk+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

Page 15: Goal Programming and IsoperformanceIsoperformancedspace.mit.edu/.../lecture-notes/l15_gpiso.pdfGoal Programming vs. IsoperformanceGoal Programming vs. Isoperformance Criterion Space

Mas

s m

[kg]

Progressive Spline 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

piso 4

1

t = 0

e.g. function csape.m4.5

4

t tl ( ) =

x ,1 ( ) f1 ( )3.5 isot P t = 3

t tisox ,2 ( ) f2 ( )2.5

2

1.5

t ∈[0,1] P t( )∈[a,b]1 l

0.50 10 (a)(a) 20 30 40 50 60 70

Disturbance Corner ωd [rad/sec]

−Use cubic k ik (t −ζ )lsplines: k=4 f j l ( ) =t

i=1 (k i)!c j l k , 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

Page 16: Goal Programming and IsoperformanceIsoperformancedspace.mit.edu/.../lecture-notes/l15_gpiso.pdfGoal Programming vs. IsoperformanceGoal Programming vs. Isoperformance Criterion Space

Bivariate Algorithm Bivariate Algorithm

Comparison ComparisonMetric 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 γiso 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 Jz

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

Pe

rfo

rma

nce

RM

S z

m 8.15

8.1“Normalized Error” 8.05

1/ 2niso 2( , ,( J xiso k ) z req ) 8

− J 7.95z100ϒ =isor=1 7.9

J 7.85 ,z req niso

Page 17: Goal Programming and IsoperformanceIsoperformancedspace.mit.edu/.../lecture-notes/l15_gpiso.pdfGoal Programming vs. IsoperformanceGoal Programming vs. Isoperformance Criterion Space

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Multivariable BranchMultivariable Branch--andand--Bound Bound

Exhaustive Search requires np-nested loops -> NP-cost: e.g.

pn x − xUB, j LB, jN = ∏

j=1 ∆x j

Branch-and-Bound only retains points/branches which meet the condition:

J ( ) ≥ J , ≥ J ( ) ∪ J ( ) ≤ J , ≤ Jz ( )x x x xz i z req z j z i z req 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

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Tangential Front Following Tangential Front Following

U V T = ∇J T ∂J ∂JΣ 1 zzU = u1 unz

∂x ∂x1 1zxz

JΣ= diag (σ σ n ) 0 ( p −nz ) ∇ =1 n x n zzz

zxn ∂J1 ∂Jz

V = v vz z+1 v

V1V2

∂xn ∂x1 n

column space nullspace

∆ = ⋅(β v + β − v ) = αV βx α 1 z+1 n z 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

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Vector Spline Approximation Vector Spline Approximation

Tangential front following is more efficient than branch-and-bound but can still be expensive for np large. Vector Spline Approximation of Isoperformance Set

1020

3040

5060 1

23

45

Isoperformance

mass disturbance corner

Isoperformance

Centroid

A

B

Idea: Find a representative subset off all isoperformance points, which are different

600

from other. 500

contr

ol corn

er

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

Page 20: Goal Programming and IsoperformanceIsoperformancedspace.mit.edu/.../lecture-notes/l15_gpiso.pdfGoal Programming vs. IsoperformanceGoal Programming vs. Isoperformance Criterion Space

Multivariable Algorithm Multivariable Algorithm

Comparison ComparisonChallenges if np > 2

• Computational complexity as a function of [ nz nd np ns ]• Visualization of isoperformance set in np-space

Table: Multivariable Algorithm Comparison for SDOF (np=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 Yiso 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 + cexs p s

ns = # of Branch-and-Bound: log ( J ) → n ( n log 2 + log β ) + 3log n + cbab 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 np© Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox Engineering Systems Division and Dept. of Aeronautics and Astronautics

20

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Graphics: Radar PlotsGraphics: 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 ωd

Oscillator mass m

Optical control bw ω o

A B

Interested in low COM

pairs

Multi-Dimensional Comparison of Isoperformance Points

ω d62.8

rad/sec

m5 kg

ωo

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

Page 22: Goal Programming and IsoperformanceIsoperformancedspace.mit.edu/.../lecture-notes/l15_gpiso.pdfGoal Programming vs. IsoperformanceGoal Programming vs. Isoperformance Criterion Space

22 © Massachusetts Institute of Technology - Prof. de Weck and Prof. WillcoxEngineering Systems Division and Dept. of Aeronautics and Astronautics

Nexus Case StudyNexus Case Study

on-orbitconfiguration

OTA

0 1 2

meters

NGST Precursor Mission2.8 m diameter apertureMass: 752.5 kgCost: 105.88 M$ (FY00)Target Orbit: L2 Sun/EarthProjected Launch: 2004

Demonstrate the usefulnessof Isoperformance on a realistic

conceptual design model ofa high-performance spacecraft

InstrumentModule

Sunshield

Delta IIFairing

launchconfiguration

• Integrated Modeling• Nexus Block Diagram• Baseline Performance Assessment• Sensitivity Analysis• Isoperformance Analysis (2)• Multiobjective Optimization• Error Budgeting

DeployablePM petal

Purpose of this case study:

The following results are shown:

Details are contained in CH7

NexusSpacecraftConcept

Pro/E models© NASA GSFC

Page 23: Goal Programming and IsoperformanceIsoperformancedspace.mit.edu/.../lecture-notes/l15_gpiso.pdfGoal Programming vs. IsoperformanceGoal Programming vs. Isoperformance Criterion Space

23 © Massachusetts Institute of Technology - Prof. de Weck and Prof. WillcoxEngineering Systems Division and Dept. of Aeronautics and Astronautics

Nexus Integrated ModelNexus Integrated Model

XY

Z

8 m2 solar panel

RWA and hexisolator (79-83)

SM (202)

sunshield

2 fixed PMpetals

deployablePM 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_yPMCassegrain

Telescope:

PM (2.8 m)PM f/# 1.25SM (0.27 m)f/24 OTA

(149,169)(207)

,Ω Φ

Page 24: Goal Programming and IsoperformanceIsoperformancedspace.mit.edu/.../lecture-notes/l15_gpiso.pdfGoal Programming vs. IsoperformanceGoal Programming vs. Isoperformance Criterion Space

Out1

© Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox Engineering Systems Division and Dept. of Aeronautics and Astronautics

24

Nexus Block DiagramNexus Block Diagram

Number of performances: nz=2 Number of states ns= 320 Number of design parameters: np=25 Number of disturbance sources: nd=4

RWA Noise InputsWFE

NEXUS Plant Dynamics Outputsphysical Sensitivity RMMS

24 [nm][m]30 dofs

[N,Nm] K In1 Out1 [m,rad]

WFE363 3 Performance 2 Performance 1

30 x' = Ax+Bu DemuxOut1 Mux Centroidy = Cx+Du K[N]

Sensitivity LOSWFE

Cryo Noise 3 [microns]3 2

2[Nm] Demux

[m]

-K- m2mic

AttitudeControl

2 x' = Ax+Bu Torques

[m] y = Cx+Du

Centroid3 rates

[rad/sec] Measured FSM Controller

Centroid3 FSM

x' = Ax+Bu 8 K 2Mux 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

Page 25: Goal Programming and IsoperformanceIsoperformancedspace.mit.edu/.../lecture-notes/l15_gpiso.pdfGoal Programming vs. IsoperformanceGoal Programming vs. Isoperformance Criterion Space

25

JJzz(p(poo))

Cumulative RSS for LOS Results Lyap/Freq Time 20

requirement

Jz,1 (RMMS WFE) 25.61 19.51 [nm]10

0

RS

S [ µ

m]

Initial Performance AssessmentInitial Performance Assessment

Jz,2 (RSS LOS) 15.51 14.97 [µm]

-1 0 1 210 10 10 10

Frequency [Hz] Centroid Jitter on Focal Plane [RSS LOS]

0

Freq Domain Time Domain

Critical Mode

4010

60

T=5 sec

14.97 µµµµm

1 pixel

Requirement: J z,2=5 µm

2P

SD

[µm

/Hz]

Cen

tro

id Y

[µm

]

20

-510

0-1 0 1 2

10 10 10 10

Frequency [Hz] -20

Cent x S

igna

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

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26

Nexus Sensitivity Nexus Sensitivity

Analysis AnalysisNorm Sensitivities: RMMS WFE Norm Sensitivities: RSS LOS Graphical Representation of Ru

K_yPM

De

sig

n P

ara

me

ters

analytical finite difference

Ru Jacobian evaluated at designUs Us

contr

ol

optics

pla

nt

dis

turb

ance

para

ms

para

ms

para

mete

rs

para

mete

rs

po, normalized for comparison.Ud Ud

fc fc

∂J ∂JQc Qcz ,1 z , 2

Tst Tst

∂R ∂RSrg Srg u u Sst Sst

J∇ = poTgs Tgs z

Jm_SM m_SM ∂J ∂J,z o

K_yPM ,1 , 2z z

K∂KK_rISO K_rISO ∂cf cfm_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 gcm2]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 /∂ p p * J / p Kcf - FSM controller gain [-]∂ ∂ o z,1,o z,1 o

/Jz,2,o

∂ z,2

© Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox Engineering Systems Division and Dept. of Aeronautics and Astronautics

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27

2D2D--Isoperformance AnalysisIsoperformance Analysis

Ud=mrd

[gcm2]

CADModel

K_rISO

[Nm/rad]

isolator strut

ω

r

m

md

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 [gcm2]

K_

rIS

O

RW

A iso

lato

r jo

int

stiff

ne

ss

[Nm

/ra

d]

Isoperformance contour for RSS LOS : Jz,req = 5 µm

10

20

20

20

60

60

60

120

120

160

po

Parameter Bounding Box

testspec

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

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28

Nexus MultivariableNexus Multivariable

IsoperformanceIsoperformance nnpp=10=10Pareto-Optimal Designs p*Qc Ud iso

Cumulative RSS for LOS

Design A

ISO

3850 [RPM 5000 [Nm/rad]

Us2.7 [gcm

90 [gcm2]0.025 [-]

Tgs 0.4 [sec]

Kzpet18E+08 [N/m]

Kcf 1E+06 [-

-5performance 10tsp Mgs uncertainty

0.005 [m] 20 [mag] 0 210 10

Frequency [Hz]Performance Cost and Risk Objectives

Jz,1 Jz,2 Jc,1 Jc,2 Jr,1

6

RM

S [

µm]

Best “mid-range” 4] compromise 2

00 2

10 10Design B Frequency [Hz]

K RurSmallest FSM]

control gain 0

10

2P

SD

[µm

/Hz]

Design C ]

Smallest

A: min(Jc1)

B: min(Jc2)

C: min(Jr1)

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

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29

NexusNexus Initial pInitial poo vs. Final Design p**vs. Final Design p**isoiso

Parameters Initial Final secondary

+X+Z

+Y

hub

SM

Deployable segment

SM Spider Support tsp

Spider wall thickness

Dsp

Kzpet

Improvements are achieved by a

well balanced mix of changes in the

Ru 3000 3845Us 1.8 1.45Ud 60 47.2Qc 0.005 0.014Tgs 0.040 0.196KrISO 3000 2546Kzpet 0.9E+8 8.9E+8tsp 0.003 0.003Mgs 15 18.6Kcf 2E+3 4.7E+5

[RPM] [gcm] [gcm2] [-] [sec] [Nm/rad] [N/m] [m] [Mag] [-]

Centroid Jitter on 50

40

30 Focal Plane

Cen

tro

id Y

0

-10

T=5 sec

µmµm

[RSS LOS]20

10Initial: 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

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30

Applicability to other Fields Applicability to other Fields

Parameters pj j=1,2 Example : Crack Growth Predictions in Metal Fatigue ao Initial Crack Length

sec π a

w

∆σ∆σ∆σ∆σ Stress Amplitude Paris daaσ πC K m= ∆ K∆ = ∆ ⋅ Law: dN

2a

w=6”

CenterCracked

Panel

stress ∆σ∆σ∆σ∆σ

Cra

ck

le

ng

th a

[in

ch

]

σ max

∆σ Isoperformance Curve: Requirement=CGL=Nc: 25000

0

0.1

0.2

0.3

0.4

0.5

Init

ial

Cra

ck L

en

gth

ao

[in

ch

]

Parameter Bounding Box

Performance

Jz = Nc =25000

cycles to failure

2.5

Critical Load Number of Cycles N

C=4e-9m=3.5

σmin R=0

0.5

1.5

2.5 Nc= 24647

3

2

1

00 0.5 1 1.5 2 4 8 10 12 14 16 18 20Load cycles N [-] x 10

Stress Amplitude ∆∆∆∆σσσσ [ksi]

© Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox Engineering Systems Division and Dept. of Aeronautics and Astronautics

22

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31

Isoperformance with Stochastic DataIsoperformance 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 M.B. Jones. Used with permission.

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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 M.B. Jones. Used with permission.

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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

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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 x1,I and x2,i

1.1 assumed model

E [J ] = a + a ( x ) + a ( x ) + a ( x − x1 )( x − x ) (1)i 0 1 1,i 2 2,i 12 1,i 2,i 2

1.2 model fitting

1 N Used Matlab ao = J fminunc.m for

General mean N j=1

j

optimal surface fit

Baseball: Obtain an expression for expected final standings (FS ) ofi

individual Team i as a function of RBIi and ERAi

E F ] = + a RBI ) + b ERA ) + c RBI − RBI )(ERA − ERA)[ i m ( i ( i ( i i

© Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox Engineering Systems Division and Dept. of Aeronautics and Astronautics

Courtesy of M.B. Jones. Used with permission.

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35

Fitted Model Fitted Model

Coefficients:

ao=0.7450

a1=0.0321

a2=-0.0869

a12= -0.0369

RMSE:Error

ErrorDistribution

σ = 0.0493e© Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox Engineering Systems Division and Dept. of Aeronautics and Astronautics

Courtesy of M.B. Jones. Used with permission.

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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 + zσε (2)i req

Error TermRequired (total variance)

performance

Φ ( ) levelz

Confidence level normal variable z (Lookup Table)Specify

2

Φ ( ) = 1z

− z

z e 2 dzz 2π −∞

© Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox Engineering Systems Division and Dept. of Aeronautics and Astronautics

Courtesy of M.B. Jones. Used with permission.

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37

Expected Performance Expected Performance

Baseball:

Performance criterion

- User specifies a final desired standing of FSi=0.550

Confidence Level

- User specifies a .80 confidence level that this is achieved

From normal Error term from dataSpec is met if for Team i: table lookup

E F ] = .550 + zσ = .550 + 0.84 0.0493) = 0.5914[ i r ( 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 M.B. Jones. Used with permission.

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38

Get Isoperformance CurveGet Isoperformance Curve

Step 3: Put equations (1) and (2) together

J + zσ = E [ i ] = a0 + a1 ( x1,i ) + a2 ( x ) + a ( x − x1 )( x − x2 )req r 2,i 12 1,i 2,i

(3), , a a12Four constant parameters: a a1 2 ,o

Two sample statistics: x1, x2

Two design variables: x1,i and x2,i

Then rearrange: x2,i = f x1,i ) ( .5914 − − bERA + cRBI (ERA − ERA)

RBIi = m i iBaseball:

a + c ERA − ERA)( i

Equation for isoperformance

© Massachusetts Institute of Technology - Prof. de Weck and Prof. WillcoxcurveEngineering Systems Division and Dept. of Aeronautics and Astronautics

Courtesy of M.B. Jones. Used with permission.

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39

Stochastic IsoperformanceStochastic 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

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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