July 13, 2004
Guest Lecture ESD.33
“Isoperformance”
Olivier de Weck
1
© Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox
Engineering Systems Division and Dept. of Aeronautics and Astronautics
Why not performance-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.
2
© Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox
Engineering Systems Division and Dept. of Aeronautics and Astronautics
Lecture Outline
• Motivation - why isoperformance ?
• 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
Target
Vector
T
x
Forward Perspective
What is J ?
Choose x
Backward Perspective
Choose J
3
What x satisfy this?
© Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox
Engineering Systems Division and Dept. of Aeronautics and Astronautics
J
Goal Seeking
max(J)
J
xi ,iso
T=Jreq
min(J)
xi , LB
4
*
max
x
*
min
x
xi ,UB xi
© Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox
Engineering Systems Division and Dept. of Aeronautics and Astronautics
Excel: Tools – Goal Seek
Excel - example
J=sin(x)/x
1.2
1
0.8
J
0.6
0.4
0.2
4
2
5.
6
6
3.
4.
2
8
1.
2.
4
2
0.
-2
-1
.2
-0
.4
-0.2
-5
.2
-4
.4
-3
.6
-2
.8
-6
0
-0.4
x
sin(x)/x - example
• single variable x
• no solution if T is
out of range
5
© Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox
Engineering Systems Division and Dept. of Aeronautics and Astronautics
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 J req
Find all x such that
dH
J req
Example
Target
Vector:
6
J req ( x)
ª msat º ª 1000kg º
« R » { «1.5Mbps »
« data » «
»
«¬ Csc »¼ «¬ 15M $ »¼
Target mass
Target data rate
Target Cost
© Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox
Engineering Systems Division and Dept. of Aeronautics and Astronautics
Goal Programming vs. Isoperformance
Decision Space
(Design Space)
x2
c2
S
x1
J3
x3
x2
Criterion Space
(Objective Space)
J2
J2
Z
T1
x4
J2 J1
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
7
© Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox
Engineering Systems Division and Dept. of Aeronautics and Astronautics
T2
J1
Isoperformance Analogy
Analogy: Sea Level Pressure [mbar]
Chart: 1600 Z, Tue 9 May 2000
Non-Uniqueness of Design if n > z
Performance: Buckling Load
Constants: l=15 [m], c=2.05 PE
Variable Parameters: E, I(r)
Requirement:
PE , REQ
cS 2 EI
l2
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
PE
c
1008
2r
1012
Bridge-Column
1012
L
1008
1008
1016
l
1012
L
1012
E,I
8
Isobars = Contours of Equal Pressure
Parameters = Longitude and Latitude
1012
H
L
1016
1012
1004
1008
1012
1016
1012
1008
Isoperformance Contours = Locus of
constant system performance
Parameters = e.g. Wheel Imbalance Us,
Support Beam Ixx, Control Bandwidth Zc
© Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox
Engineering Systems Division and Dept. of Aeronautics and Astronautics
Isoperformance Approaches
(a) deterministic I soperformance Approach
Deterministic
System
Model
Jz,req
Design A
Design C
Isoperformance
Algorithms
Jz,req
Design B
(b) stocha stic I soperformance Approach
90%
Design A
Design B
Jz,req
80%
Empirical
Isoperformance
System
Algorithms
Model
Empirical
Isoperformance
System
Model
Algorithms
Design Space
Ind
1
2
3
x
y
Jz
0.75 9.21 17.34
0.91 3.11 8.343
...... ......
......
50%
Statistical Data
Jz,req
9
P(Jz)
© Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox
Engineering Systems Division and Dept. of Aeronautics and Astronautics
Bivariate Exhaustive Search (2D)
“Simple” Start: Bivariate Isoperformance Problem
Performance J z ( x1 , x2 ) : z 1
Variables x j , j 1, 2 : n
Number of points along j-th axis:
2
nj
x2
x1
10
First Algorithm: Exhaustive Search
coupled with bilinear interpolation
ª x j ,UB x j , LB º
«
»
'
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
Contour Following (2D)
k-th isoperformance point:
Taylor series expansion
1 T
J z ( x ) ’J
'x 'x H xk 'x H.O.T.
2 k
J z ( x)
T
z xk
first order term
’J zT
t
k
p
k
Dk
nk
ª0 1º k
«1 0 » ˜ n
¬
¼
tk: tangential
step direction
k+1-th isoperformance point:
11
second order term
'x { 0
’J z
ƒ˜
’J z p k
’J z
ª W J z ,req T
tk H
«2
¬ 100
1/ 2
xk
tk H: Hessian
Dk : Step size
'x D k ˜ t k
x k 'x
1
º
»
¼
x k 1 J z ,req
Demo
xk
x k 1
x k 1
ª wJ z º
« wx »
« 1»
« wJ z »
« wx »
¬ 2¼
t
k
© Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox
Engineering Systems Division and Dept. of Aeronautics and Astronautics
nk
Progressive Spline
Approximation (III)
(b)
t=1
•
•
First find iso-points on boundary
Then progressive spline approximation
via segment-wise bisection
Makes use of MATLAB spline toolbox ,
e.g. function csape.m
•
piso
t 6 Pl t t=0
12
ª f1 t º
«
»
f
t
¬ 2 ¼
t  > 0,1@ 6 Pl t  > a, b @
(a)
Demo
ª xiso ,1 t º
«
»
x
t
¬ iso ,2 ¼
Use cubic
splines: k=4
f j ,l t k i
t ] l c , t  ] !]
> l l 1 @
¦ k i ! j ,l , k
i 1 k
© Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox
Engineering Systems Division and Dept. of Aeronautics and Astronautics
Bivariate Algorithm
Comparison
Metric
Exhaustive
Search (I)
Contour
Spline
Follow (II) Approx (III)
FLOPS
2,140,897
783,761
377,196
CPU time [sec]
1.15
0.55
0.33
Tolerance W
1.0%
1.0%
1.0%
Actual Error Jiso
0.057%
0.347%
0.087%
# of isopoints
35
45
7
x 10
Isoperformance Quality Metric
13
1/ 2
º
»
»
»
»¼
Performance RMS z m
biso
2
Conclusions:
(I) most general but expensive
(II) robust, but requires guesses
(III) most efficient, but requires
monotonic performance Jz
Quality of Isoperformance Solution Plot
8.2
“Normalized Error”
ª niso
J z xiso ,k J z ,req
100 « ¦
«r 1
J z ,req «
niso
«¬
-4
Results for SDOF Problem
Normalized Error : 0.34685 [%]
Allowable Error: 1 [%]
8.15
8.1
8.05
8
7.95
7.9
correction step
7.85
7.8
0
5
10
15
20
25
30
35
Isoperformance Solution Number
© Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox
Engineering Systems Division and Dept. of Aeronautics and Astronautics
40
45
Multivariable Branch-and-Bound
Exhaustive Search requires
np-nested loops -> NP-cost: e.g.
ª xUB , j xLB , j º
»
– « 'x
j 1«
»
j
np
N
Branch-and-Bound only retains points/branches
which meet the condition:
ª J z xi t J z ,req t J z x j º ‰ ª J z xi d J z ,req d J z x j º
¬
¼ ¬
¼
Expensive for small tolerance W
Need initial branches to be fine enough
14
© Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox
Engineering Systems Division and Dept. of Aeronautics and Astronautics
Tangential Front Following
U
U 6V T
ª¬u1 " unz º¼
zxz
0nz x ( n p nz ) º¼
6 = ª¬diag V 1 " V nz
wJ z º
ª wJ1
« wx
wx1 »
1
«
»
« # % # »
«
»
J
J
w
w
z »
« 1
«¬ wxn
wx »¼
’J zT
’J z
zxn
V
ª
º
«v1 " vz vz 1 " vn »
»
« nullspace
¬ column space
¼
'x D ˜ E1vz 1 ! E n z vn DVt E
V2
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 !
15
© Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox
Engineering Systems Division and Dept. of Aeronautics and Astronautics
V1
Vector Spline Approximation
Tangential front following is
more efficient than branch-and-bound
but can still be expensive for np large.
Centroid
Idea: Find a representative
subset off all isoperformance
points, which are different
from other.
B
600
500
control corner
“Frame-but-not-panels”
analogy in construction
Vector Spline Approximation of Isoperformance Set
400
300
200
100
0
Algorithm:
50
1. Find Boundary (Edge) Points
2. Approximate Boundary curves
3. Find Centroid point
4. Approximate Internal curves
16
A
60
1
2
40
30
3
20
10
disturbance corner
Isoperformance
Boundary Curves
4
5
mass
Isoperformance
Boundary Points
© Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox
Engineering Systems Division and Dept. of Aeronautics and Astronautics
Multivariable Algorithm
Comparison
Challenges if np > 2
Problem Size:
• 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)
Metric
MFLOPS
Exhaustive
Search
6,163.72
Branch-andBound
891.35
Tang Front
Following
106.04
V- Spline
Approx
1.49
CPU [sec]
5078.19
498.56
69.59
4.45
Error Yiso
0.87 %
2.43%
0.22%
0.42%
2073
7421
4999
20
z = # of
performances
d = # of
disturbances
n = # of
variables
# of points
From Complexity Theory: Asymptotic Cost
Exhaustive Search:
ns = # of
states
Branch-and-Bound:
[FLOPS]
log J exs o n p log D 3log ns c
log J bab o ng n p log 2 log E 3log ns c
Tang Front Follow: log J tff o n p nz log J log 1 nz 3log ns c
V-Spline Approx:
log J vsa o n p log 2 3log ns log(nz +1)+c
Conclusion: Isoperformance problem is non-polynomial in np
17
© Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox
Engineering Systems Division and Dept. of Aeronautics and Astronautics
Graphics: Radar Plots
Disturbance corner Zd 6.2832
21.3705
Oscillator mass m 5.0000
0.5000
Optical control bw Zo 186.5751 628.3185
For np >3
B
A
Multi-Dimensional Comparison
of Isoperformance Points
m
5 kg
Interested
in low COM
pairs
A
B
Cross Orthogonality Matrix
COM (i, j )
18
Zd
piso ,i ˜ piso , j
piso ,i ˜ piso , j
Zo
628.3 rad/sec
© Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox
Engineering Systems Division and Dept. of Aeronautics and Astronautics
62.8
rad/sec
Nexus Case Study
on-orbit
configuration
Purpose of this case study:
Pro/E models
© NASA GSFC
Nexus
Spacecraft
Concept
Demonstrate the usefulness
of Isoperformance on a realistic
conceptual design model of
a high-performance spacecraft
OTA
launch
configuration
The following results are shown:
Sunshield
• Integrated Modeling
• Nexus Block Diagram
• Baseline Performance Assessment
• Sensitivity Analysis
• Isoperformance Analysis (2)
Deployable
• Multiobjective Optimization PM petal
• Error Budgeting
Details are contained in CH7
19
Delta II
Fairing
Instrument
Module
0
1
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
© Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox
Engineering Systems Division and Dept. of Aeronautics and Astronautics
2
Nexus Integrated Model
Legend:
Design Parameters
(I/O Nodes)
Spacecraft bus
(84) m_bus
sunshield
I_ss
Instrument
(207)
Cassegrain
Telescope:
8 m2 solar panel
RWA and hex
isolator (79-83)
K_rISO
2 fixed PM
petals (149,169)
K_yPM
Z
X
PM (2.8 m)
PM f/# 1.25
SM (0.27 m)
f/24 OTA
Y
deployable
PM petal (129) K_zpet
Structural Model (FEM)
(Nastran, IMOS)
20
SM spider
t_sp
SM (202)
m_SM
:, )
© Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox
Engineering Systems Division and Dept. of Aeronautics and Astronautics
Nexus Block Diagram
Number of performances: nz=2
Number of design parameters: np=25
Number of states ns= 320
Number of disturbance sources: nd=4
Inputs
RWA Noise
NEXUS Plant Dynamics
WFE
physical Sensitivity
[m]
30 dofs
K
[m,rad]
Outputs
24
Out1
[N,Nm]
3
Out1
Mux
30
x' = Ax+Bu
y = Cx+Du
[N]
36
3
Demux
3
3
[Nm]
[nm]
In1 Out1
WFE
Performance 1
Performance 2
Centroid
Sensitivity
K
Cryo Noise
RMMS
LOS
WFE
[microns]
2
2
Demux
[m]
-K- m2mic
Attitude
Control
Torques
2
3
rates
x' = Ax+Bu
y = Cx+Du
8
3
Mux
Out1
Angles
2
[rad]
21
K
[rad]
ACS Controller
[m]
Centroid
[rad/sec]
FSM
Coupling
x' = Ax+Bu
y = Cx+Du
FSM Controller
Measured
Centroid
2
Out1
[m]
GS Noise
ACS Noise
K
desaturation signal
gimbal
angles
FSM Plant
© Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox
Engineering Systems Division and Dept. of Aeronautics and Astronautics
2
[rad]
Initial Performance Assessment
Jz(po)
RSS [ Pm]
Results
Lyap/Freq
Jz,1 (RMMS WFE)
Jz,2 (RSS LOS)
25.61
15.51
Cumulative RSS for LOS
20
10
requirement
0
-1
10
0
1
10
2
10
10
Frequency [Hz]
23.1 Hz
Time
[nm]
[Pm]
19.51
14.97
Centroid Jitter on Focal Plane [RSS LOS]
Critical Mode
40
0
10
Freq Domain
Time Domain
-5
10
-1
0
10
1
10
2
10
10
Cent x Signal [Pm]
Frequency [Hz]
22
1 pixel
T=5 sec
Centroid Y [Pm]
PSD [ Pm2/Hz]
60
20
0
14.97 Pm
-20
50
-40
0
-50
5
Requirement: Jz,2=5 Pm
6
7
8
9
10
-60
-60
-40
-20
0
20
Centroid X [Pm]
Time [sec]
© Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox
Engineering Systems Division and Dept. of Aeronautics and Astronautics
40
60
Nexus Sensitivity
Analysis
Srg
Sst
Tgs
m_SM
K_yPM
K_rISO
Srg
Sst
Tgs
m_SM
K_yPM
K_rISO
m_bus
K_zpet
t_sp
I_ss
I_propt
zeta
lambda
m_bus
K_zpet
t_sp
I_ss
I_propt
zeta
lambda
Ro
QE
Mgs
fca
Kc
Kcf
Ro
QE
Mgs
fca
Kc
Kcf
analytical
finite difference
-0.5
0
po
23
0.5
1
1.5
/Jz,1,o *w Jz,1 /w p
Graphical Representation of
Jacobian evaluated at design
po, normalized for comparison.
’J z
plant
parameters
Ru
Us
Ud
fc
Qc
Tst
Design Parameters
Ru
Us
Ud
fc
Qc
Tst
disturbance
parameters
Norm Sensitivities: RSS LOS
-0.5
0
po
0.5
1
1.5
/Jz,2,o *w Jz,2 /w p
control optics
params params
Norm Sensitivities: RMMS WFE
ª wJ z ,1
«
wRu
«
po «
"
J z ,o «
« wJ z ,1
« wK
¬ cf
wJ z ,2 º
»
wRu »
" »
»
wJ z ,2 »
wK cf »¼
RMMS WFE most sensitive to:
Ru - upper op wheel speed [RPM]
Sst - star track noise 1V [asec]
K_rISO - isolator joint stiffness [Nm/rad]
K_zpet - deploy petal stiffness [N/m]
RSS LOS most sensitive to:
Ud - dynamic wheel imbalance [gcm2]
K_rISO - isolator joint stiffness [Nm/rad]
zeta - proportional damping ratio [-]
Mgs - guide star magnitude [mag]
Kcf - FSM controller gain [-]
© Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox
Engineering Systems Division and Dept. of Aeronautics and Astronautics
2D-Isoperformance Analysis
joint
isolator
strut
Isoperformance contour for RSS LOS : Jz,req = 5 Pm
Parameter Bounding Box
10000
K_rISO RWA isolator joint stiffness [Nm/rad]
K_rISO
[Nm/rad]
CAD
Model
Ud=mrd
[gcm2]
Z
E-wheel
m
m
d
9000
60
20
8000
120
160
7000
60
120
6000
60
5000
20
Initial
design
4000
HST
3000
20
p
o
5
2000
10
test
1000
spec
5
0
0
10
Ud
20
30
40
50
60
dynamic wheel imbalance
70
80
[gcm2]
r
24
© Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox
Engineering Systems Division and Dept. of Aeronautics and Astronautics
90
Nexus Multivariable
Qc
0.025 [-]
Isoperformance
n
=10
p
Pareto-Optimal Designs p
Ud
*
iso
2
90 [gcm ]
Cumulative RSS for LOS
Design A
2.7 [gcm]
6
RMS [Pm]
Us
Tgs
0.4 [sec]
Best “mid-range”
compromise
4
2
0
Design B
K ISO
r
Ru
3850 [RPM]
5000 [Nm/rad]
0
2
10
10
Frequency [Hz]
Smallest FSM
control gain
K pet
Design C
Kcf
z
18E+08 [N/m]
1E+06 [-]
t p
s
0.005 [m]
Smallest
performance
uncertainty
Mgs
20 [mag]
Jz,1
25
Design A
Design B
Design C
-5
10
0
2
10
Performance
A: min(Jc1)
B: min(Jc2)
C: min(Jr1)
PSD [ Pm2/Hz]
0
10
20.0000
20.0012
20.0001
Jz,2
5.2013
5.0253
4.8559
Cost and Risk Objectives
Jc,1
Jc,2
0.6324
0.8960
1.5627
0.4668
0.0017
1.0000
10
Frequency [Hz]
Jr,1
+/- 14.3218 %
+/- 8.7883 %
+/- 5.3067 %
© Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox
Engineering Systems Division and Dept. of Aeronautics and Astronautics
Nexus Initial po vs. Final Design p**iso
+Y
secondary
hub
+Z
SM
+X
SM Spider Support
Spider wall
thickness
tsp
Dsp
Deployable
segment
Parameters
Ru
Us
Ud
Qc
Tgs
KrISO
Kzpet
tsp
Mgs
Kcf
Improvements are achieved by a
well balanced mix of changes in the
disturbance parameters, structural
redesign and increase in control gain
of the FSM fine pointing loop.
26
Centroid Y
Kzpet
50
40
30
20
10
0
-10
-20
-30
-40
-50
Initial
3000
1.8
60
0.005
0.040
3000
0.9E+8
0.003
15
2E+3
Final
3845
1.45
47.2
0.014
0.196
2546
8.9E+8
0.003
18.6
4.7E+5
T=5 sec
Pm
[RPM]
[gcm]
[gcm2]
[-]
[sec]
[Nm/rad]
[N/m]
[m]
[Mag]
[-]
Centroid Jitter on
Focal Plane
[RSS LOS]
Initial: 14.97
Final: 5.155
-50
0
50 Centroid X
© Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox
Engineering Systems Division and Dept. of Aeronautics and Astronautics
Isoperformance with Stochastic Data
Example: Baseball season has started
What determines success of a team ?
Pitching
Batting
ERA
“Earned Run Average”
RBI
“Runs Batted In”
How is success of team measured ?
27
FS= Wins/Decisions
© Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox
Engineering Systems Division and Dept. of Aeronautics and Astronautics
Raw Data
Team results for 2000, 2001 seasons: RBI,ERA,FS
28
© Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox
Engineering Systems Division and Dept. of Aeronautics and Astronautics
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
29
© Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox
Engineering Systems Division and Dept. of Aeronautics and Astronautics
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 i @ a0 a1 x1,i a2 x2,i a12 x1,i x1
x
2, i
x2
(1)
1.2 model fitting
General mean
ao
1
N
Used Matlab
fminunc.m for
optimal surface fit
N
¦J
j
j 1
Baseball: Obtain an expression for expected final standings (FSi) of
individual Team i as a function of RBIi and ERAi
E > FSi @ m a RBI i b ERAi c RBI i RBI
30
ERA ERA
i
© Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox
Engineering Systems Division and Dept. of Aeronautics and Astronautics
Fitted Model
Coefficients:
ao=0.7450
a1=0.0321
a2=-0.0869
a12= -0.0369
RMSE:
Error
Ve= 0.0493
31
Error
Distribution
© Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox
Engineering Systems Division and Dept. of Aeronautics and Astronautics
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 > Ji @
)z
J req zV H
Error Term
(total variance)
Required
performance
level
Confidence level
normal variable z
(Lookup Table)
Specify
z
32
)z
(2)
1
2S
z
³e
z2
2
dz
f
© Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox
Engineering Systems Division and Dept. of Aeronautics and Astronautics
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
Spec is met if for Team i:
E > FSi @ .550 zV r
From normal
table lookup
Error term
from data
.550 0.84 0.0493 0.5914
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
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© Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox
Engineering Systems Division and Dept. of Aeronautics and Astronautics
Get Isoperformance Curve
Step 3:
J req zV r
Put equations (1) and (2) together
E > J i @ a0 a1 x1,i a2 x2,i a12 x1,i x1
x
Two sample statistics: x1 , x 2
Then rearrange:
Baseball:
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RBI i
Equation
for isoperformance
curve
x2,i
x1,i and x2,i
f x1,i .5914 m bERAi cRBI ERAi ERA
a c ERAi ERA
x2
(3)
Four constant parameters: ao , a1 , a2 , a12
Two design variables:
2, i
© Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox
Engineering Systems Division and Dept. of Aeronautics and Astronautics
Stochastic Isoperformance
This is our desired tradeoff curve
35
© Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox
Engineering Systems Division and Dept. of Aeronautics and Astronautics
Summary
• Isoperformance fixes a target level of
“expected” performance and finds a set of
points (contours) that meet that requirement
• Model can be physics-based or empirical
• Helps to achieve a “balanced” system
design, rather than an “optimal design”.
36
© Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox
Engineering Systems Division and Dept. of Aeronautics and Astronautics