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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
2
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
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
fa
Jx
Target Vector
Many-To-One
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
5
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
6
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
7
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
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
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
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
11
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
12
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
13
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
14
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
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
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
17
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
18
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
19
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
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
21
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
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
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
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
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
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
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
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
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
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
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.
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.
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 M.B. Jones. 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 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.
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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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.
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.
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.
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
Courtesy of M.B. Jones. Used with permission.
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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
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