A Multiobjective, Multidisciplinary Design Optimization
Methodology for the Conceptual Design of
Distributed Satellite Systems
Cyrus D. Jilla
Presented by Dan Hastings
SSPARC IAB Meeting
May 16, 2002
Chart: 1
MIT Space Systems Laboratory
Motivation
• Objective: To develop a methodology for mathematically modeling the
Distributed Satellite System (DSS) conceptual design problem as an
optimization problem to enable an efficient search for the best families of
solutions within the system trade space.
• Motivation:
– The trade-space for DSS’s is enormous - too large to enumerate, analyze,
and compare all possible system architectures.
– MDO techniques have been applied successfully across many fields to find
the solutions to complex problems.
– “… the conceptual space systems design process is very unstructured …
designers often pursue a single design concept, patching and repairing their
original idea rather than generating new alternatives.” - Proceedings of the
1998 IEEE Aerospace Conference
– A design that is globally optimized on the basis of a system metric(s) is
likely to vary drastically from a design that has been developed by optimizing
each subsystem.
– A methodology is needed that will enable a greater search of the trade
space and explore design options that might not otherwise be considered
during the Conceptual Design Phase (when lifecycle cost gets locked in).
Chart: 2
MIT Space Systems Laboratory
Case Studies
•
•
Name:
Mission:
•
•
•
Sector:
Sponsor:
TS Size:
Terrestrial Planet Finder
Terrestrial Planet Detection/
Characterization
Civil
NASA
640 Architectures
Image removed due to copyright considerations.
[Beichman et al, 1999]
Image removed due to copyright considerations.
[Martin, 2000]
•
•
Name:
Mission:
•
•
•
Sector:
Sponsor:
TS Size:
Chart: 3
•
•
Name:
Mission:
•
•
•
Sector:
Sponsor:
TS Size:
TechSat 21
Ground Moving Target
Indicator (GMTI)
Military
AFRL
732,160
Broadband Communication
High Data Rate
Communication Services
Commercial
Private Company
42,400 Architectures
Image removed due to copyright conisderations.
[Boeing, 2002]
MIT Space Systems Laboratory
Past Work – Literature Review
Aeronautics
Astronautics
1996-1998
University of Colorado
Mosher (Genetic Alg.)
Riddle (Dynamic Prog.)
Subsystems: 1980’s onward
Aerodynamics NASA Langley Research Center
Sobieski et al
Structures
Launch
Vehicles
1995
Constellation University of Colorado
Design
Matossian (MILP)
Spacecraft
Design
Distributed
Satellite
Systems
Chart: 4
Middle 1990’s
Stanford, The Aerospace Corporation,
German Academic Institutions
Kroo, Braun, et al
1999-2002
MIT – Jilla & Miller
- System of Systems
- Nonlinear
- Multiobjective
MIT Space Systems Laboratory
The MMDOSA Methodology
1
2
3
Create the
GINA Model
Perform Univariate
Studies
Take a Random
Sample of the
Trade Space
7
5
Converge on System
Architectures for
the Next Phase of
the Design Process
Interpret Results
(Sensitivity
Analysis)
Formulate and
Apply MDO
Algorithm(s)
4
Iterate
6
Chart: 5
MIT Space Systems Laboratory
Step 1 – Create the GINA Model
Top Trades Number of
Satellites
Capability
Per
Cluster
Metrics
Isolation
Number of
Clusters
N/A
N/A
Minimum
revisit time
Rate
N/A
Integrity
Availability
Cost per
Function
Constellation
Chart: 6
Antenna gain
gain
Antenna
patternfrom
from
pattern
cluster
cluster
projection
projection
N/A
Overall
system
reliability
Overall
system
reliability
Learning
curve
effects
Learning
curve
effects
Radar
Payload
Altitude
MDV and
maximum
cluster
baseline
Aperture
Diameter
Transmission
Power
N/A
N/A
Revisit time
and area
search rate
Footprint
coverage
area
Required
dwell time
Received
signal
strength
Mainlobe vs.
sidelobe gain
Impacts the
signal-tonoise ratio
N/A
N/A
Time required
to scan a
target area
Launch,
eclipse,
& drag
Antenna
mass & cost
Size of
spacecraft
S/C Bus
Deploy/Ops
System Analysis
MIT Space Systems Laboratory
Step 1 – Develop Simulation Software
Inputs (Design Vector)
# S/C per Cluster
MATLAB Models
….
Constellation
Radar
# Clusters
Key Outputs
….
Constellation Altitude
S/C Bus
Payload
Aperture Diameter
Lifecycle Cost
Availability
Probability of
Detection
Revisit Rate
Resolution & MDV
….
Launch &
Operations
Antenna Power
System
Analysis
100
90
80
70
60
50
40
30
20
10
0
1
Chart: 7
2
3
4
5
6
7
8
9
10
MIT Space Systems Laboratory
Step 2 – Perform Univariate Studies
•
•
Univariate Studies
)
B
$
( 13
Strengths
t 12
s
o 11
C
10
e
l
9
$1.6B
c
y
8
c
e 7
f
i
L 6
– Select a baseline . Holding
everything else constant, vary
an individual i over its entire
range of values and observe
how the system attributes vary.
– Provides the systems engineer
with an initial feel of the trade
space.
– Further assessment of model
fidelity.
•
Weaknesses
– Ignores couplings between
elements of the design vector.
– Focuses on only a local portion
of the global trade space.
Chart: 8
TechSat21 Trade Space Sample
Baseline Design
# Satellites/Cluster
Altitude
Ant. Diameter
Ant. Power
# Planes
# Clusters/Plane
$0.1B
$0.8B
$0.4B
$0.2B
Baseline
Design
5
4
3
$0.05B
0
0.2
0.4
0.6
0.8
Probability of Detection at 99% Availability
1
* Family Baseline Design Vector
# Satellites Per Cluster:
Aperture Diameter:
Radar Transmission Power:
Constellation Altitude:
# Clusters Per Orbital Plane:
# Orbital Planes:
8
2m
300 W
1000 km
6
6
MIT Space Systems Laboratory
Step 3 – Random Sample
•
Random Sampling
–
•
95% Confidence
Intervals
X
vx
95%CI
2()
=
n±
–
•
“Average” vs. Optimized Designs
Initial Optimization Bounds
–
•
Collecting and deriving data on a population (i.e. the complete set of architectures in the DSS
global trade space) after measuring only a small subset of the population in such a way that every
architecture in the trade space is equally likely to be selected.
LP Relaxation Analogy
Gather Data to be Used Downstream
to Tailor MDO Algorithms
–
Chart: 9
Simulated Annealing Δ Parameter
TPF Random Sample (n=48)
Parameter
Lifecycle Cost
($B)
Maximum
2.11
Minimum
0.74
Range
1.37
Mean
1.23
Standard Deviation
0.30
Performance Cost Per Image
(# Images)
($K)
3839
2429
390
474
3449
4955
1681
849
731
367
MIT Space Systems Laboratory
Step 4 – Apply MDO Algorithms:
Single Objective Optimization
•
Application of 4 MDO Techniques
–
–
–
–
•
•
Taguchi Methods
Simulated Annealing
Pseudo-Gradient Search
Univariate Search
Heuristic techniques found to be the
least susceptible to getting stuck in the
local optima present in the trade space of
DSS’s.
Simulated Annealing forms the core
algorithm within MMDOSA.
Potential
Neighbors
Simulated
Annealing
Gradient
Gradient
MDO Technique Performance
Trial
Simulated
Annealing
($K/Image)
Univariate
Algorithm
($K/Image)
493.8
470.0
469.6
505.1
470.8
470.8
493.8
470.0
498.2
496.4
PseudoGradient
Search
($K/Image)
470.8
470.0
469.6
520.9
469.6
469.6
532.6
469.6
470.8
483.0
1
2
3
4
5
6
7
8
9
10
MSE
($K/Image)2
397.1
678.3
1027.8
469.6
469.6
513.8
469.6
526.0
513.8
470.8
525.9
469.6
473.9
21MPICPI
iC
n
=
n
(*)
SE=
Local Optimum
Global Optimum
Chart: 10
MIT Space Systems Laboratory
n
o
i
l
l
i
m
$
(
n
o
i
l
l
i
m
$
(
Step 4 – Apply MDO Algorithms:
Single Objective Optimization
TPF System Trade Space
t
s 2400
o 2200
C
2000
e
l 1800
c
y 1600
c
e 1400
f 1200
i
L 1000
$2M/Image
t 1400
s
o
C 1350
$1M/Image
$0.5M/Image
Zoom in of the TPF System Trade Space
$0.55M/Image
$0.5M/Image
e
l 1300
c
y
c
1250
e
f
i
L 1200
$0.45M/Image
$0.4M/Image
$0.25M/Image
800
0
500
1000 1500 2000 2500 3000 3500 4000
Performance (total # of images)
1150
2300
2400
2500 2600 2700 2800 2900
Performance (total # of images)
3000
“Optimal” Solution
Orbit
Interf.
# Apert.
Apert.
Chart: 11
4 AU
SCI-2D
8
4m
Number of Images 2759
Total Cost
$1.3 Billion
Cost Per Image $469,000
MIT Space Systems Laboratory
Step 4 – Apply MDO Algorithms:
Multiobjective Optimization
•
•
•
Motivation: True systems methods handles trades, not just a single metric. In
real-world systems engineering problems, one has to balance multiple
requirements while simultaneously trying to achieve multiple goals.
)
s
n
o
i
l
l
i
b
$
(
Differences between single objective and multiobjective problems.
• Single objective problems have only one true solution.
• Multiobjective problems can have more than one solution.
Terminology:
– Dominated Solutions
– Non-Dominated Solutions
– Pareto Optimal Set
O j ( xi ) < O j ( y ) for all i and j
Multi-Objective Illustration
2
(1600,$1.8B)
t
s
o
C 1.5
e
l
c
1
y
c
e
f
i
L 0.5
Design 4
CPI=$1.13M
(2000,$1.5B)
Design 3
(1000,$1.3B)
CPI=$0.75M
Design 5
CPI=$1.30M
(1400,$0.8B)
Design 2
CPI=$0.57M
(500,$0.5B)
Design 1
CPI=$1.00M
0
0
500
1000
1500
2000
Performance (total # images)
Chart: 12
MIT Space Systems Laboratory
Step 4 – Apply MDO Algorithms:
Multiobjective Multiple Solution Algorithm
)
s
n
o
i
l
l
i
m 1600
$
(
1500
t
s 1400
o
C 1300
Goal:
To find multiple architectures in
the Pareto optimal set.
IF En ( Ãk ) < En ( Ãi ) for ALL n = 1,2 ,K ,N
Ãi ∉ P
k = 1,2, K , i - 1
END
IF Ãi +1 ∈ P OR χ
Ãb = Ãi +1
Chart: 13
$1M/Image
$2M/Image
$0.5M/Image
$0.25M/Image
0
500
1500
2000
1000
Performance (total # images
2500
3000
Observations:
•
ELSE
Ãb = Ãi
END
Current Solution Path
Pareto-Optimal Set
800
700
Δ
−
<e T
Multi-Objective SA Exploration of the TPF Trade Space
e
l 1200
c
y 1100
c
e
f 1000
i
900
L
New Decision Logic
ELSE
Ãi ∈ P
Pareto Front
•
Along this boundary, the systems
engineer cannot improve the
performance of the design without also
increasing lifecycle cost.
This boundary quantitatively captures
the trades between the DSS design
criteria.
MIT Space Systems Laboratory
Step 4 – Apply MDO Algorithms:
Approximating the Global Pareto Boundary
TPF System Trade Space Pareto-Optimal Front Estimation
2200
2200
Dominated Solutions
Non-Dominated Solutions
2000
Dominated Solutions
Non-Dominated Solutions
Local Pareto Front
Global Pareto Front
2000
1800
Lifecycle Cost ($M)
1800
1600
1400
1200
1600
1400
1200
1000
1000
800
800
0
500
1000
1500
2000
2500
3000
3500
4000
Single Objective SA
0
# Iterations vs. RMSE of Pareto Boundary Curve Fit
1000 1500 2000 2500 3000
Performance (total # of images)
3500
4000
TPF System Trade Space Pareto-Optimal Front Estimation
450
2200
Single Objective SA
Multi-Objective Single Point SA
Multi-Objective Multiple Point SA
400
Dominated Solutions
Non-Dominated Solutions
Local Pareto Front
Global Pareto Front
2000
350
1800
Lifecycle Cost ($M)
Mean Error ($M) - 100 Trials
500
300
250
200
150
100
1600
1400
1200
Multiobjective
Multiple Point SA
1000
50
800
0
Chart: 14
0
50
100
150
200
250
# Iterations in Algorithm
300
350
400
0
500
1000 1500 2000 2500 3000
Performance (total # of images)
3500
4000
MIT Space Systems Laboratory
Step 4 – Apply MDO Algorithms:
Multiobjective Optimization
•
•
•
In a two-dimensional trade space, the Pareto Optimal set represents the
boundary of the most design efficient solutions.
The same principles of Pareto Optimality hold for a trade space with any
number n dimensions (ie. any number of decision criteria).
3 Criteria Example for Space-Based Radar
– Minimize(Lifecycle Cost) AND
– Minimize(Maximum Revisit Time) AND
– Maximize(Target Probability of Detection)
Chart: 15
MIT Space Systems Laboratory
Step 5 – Interpret Results:
Sensitivity Analysis
•
Finite Differencing
– Computationally expensive
– Local Sensitivity
•
Key Question: Which variables in
the design vector have the most
significant effect on the metrics of
interest for system?
•
Analysis of Variance (ANOVA)
– A statistical technique used to
detect differences in the average
performance of groups of items
tested.
•
ANOVA provides the systems
engineer with a tool to identify key
models and guide technology
investment strategies during the
Conceptual Design Phase of a
program.
Chart: 16
e
c
n
1
e
u 0.9
l
f 0.8
n
I 0.7
e
v
i
t
a
l
e
R
Relative Sensitivity of CPI to Design Parameters
Aperture
Diameter
0.6
0.5
0.4
0.3
0.2
0.1
Orbit
0
1
Number
Aperture
Apertures
Geometry/
Connectivity
2
3
Design Vector Parameter
N
SST = (yi T )
2
i=1
n
(
SS = n i y i T
i=1
4
SS RI = 100
SS
T
2
)
MIT Space Systems Laboratory
Step 6 – Iterate
• Model Fidelity
• Simulated Annealing
Algorithm Parameters
– Cooling Schedule
– DOF
• “Warm Starting”
• Run additional trials
Warm Start Example
State
Trial 1 Initial Architecture
Trial 1 Final Architecture
Trial 2 Warm Start Initial Architecture
Trial 2 Warm Start Final Architecture
Chart: 17
Orbit (AU)
Number
Apertures
1
4.5
4.5
4.5
4
8
8
8
Collector
Aperture
Connectivity/ Diameter (m)
Geometry
SCI-1D
1
SCI-2D
3
SCI-2D
3
SCI-2D
4
Cost Per
Image ($K)
2475
526
526
471
MIT Space Systems Laboratory
Step 7 – Recommended Architectures
•
•
Single Objective DSS Conceptual Design Problem
– The best family(s) of architectures with respect to the metric of interest
found by the single objective simulated annealing algorithm.
Multiobjective DSS Conceptual Design Problem
– The Pareto optimal set with respect to the selected decision criteria as
found by the multiobjective, multiple solution simulated annealing algorithm.
– Budget Capping, Multiattribute Utility Theory, Uncertainty Analysis,
Flexibility, and Policy Implications
Chart: 18
MIT Space Systems Laboratory
Terrestrial Planet Finder Case Study – Results (1)
MMDOSA
Minimum CPF
(% Improvement)
Pareto Optimal
Equivalent
Performance
(% Improvement)
Pareto Optimal
Equivalent LCC
(% Improvement)
JPL
49%
28%
73%
Ball Aerospace
47%
10%
52%
Lockheed Martin
53%
0%
31%
TRW SCI
34%
7%
20%
TRW SSI
55%
37%
119%
Architecture
Chart: 19
MIT Space Systems Laboratory
Terrestrial Planet Finder Case Study – Results (2)
Mission Cost
& Performance
Low
Medium
Family
4 ap.
SCI-1D
2 m Diam.
Family
6 ap.
SCI-2D
4 m Diam.
Family
6 ap.
SSI-2D
4 m Diam.
High
Chart: 20
Family
10 ap.
SSI-1D
4 m Diam.
# “Images”
LCC ($B)
Orbit (AU)
# Apert.’s
Architecture
502
577
651
1005
1114
1171
1195
1292
1317
1424
1426
1464
1631
1684
1687
1828
1881
1978
2035
2132
2285
2328
2398
2433
2472
2482
2487
2634
2700
2739
2759
2772
2779
2783
2788
2844
2872
2988
3177
3289
3360
3395
3551
3690
0.743
0.762
0.767
0.768
0.788
0.790
0.807
0.811
0.830
0.836
0.838
0.867
0.877
0.881
0.932
0.936
0.980
0.982
1.086
1.112
1.120
1.190
1197
1.212
1.221
1.227
1.232
1.273
1.280
1.288
1.296
1.305
1.312
1.317
1.569
1.609
1.655
1.691
1.698
1.739
1790
1.850
1.868
1.919
1.5
2.0
2.5
1.5
2.0
2.5
1.5
1.5
1.5
2.0
1.5
2.5
1.5
1.5
2.0
2.0
1.5
1.5
2.0
1.5
1.5
2.5
3.0
4.0
4.5
5.0
5.5
2.5
3.0
3.5
4.0
4.5
5.0
5.5
3.0
3.5
4.0
2.0
2.5
3.0
3.5
4.0
2.5
3.0
4
4
4
4
4
4
6
6
8
4
8
6
6
6
6
6
8
6
8
8
8
6
6
6
6
6
6
8
8
8
8
8
8
8
6
6
6
8
8
8
8
8
10
10
SCI-1D
SCI-1D
SCI-1D
SCI-1D
SCI-1D
SCI-1D
SCI-1D
SCI-2D
SCI-1D
SCI-1D
SCI-2D
SCI-2D
SCI-1D
SCI-2D
SCI-1D
SCI-2D
SCI-2D
SCI-1D
SCI-2D
SCI-1D
SCI-2D
SCI-2D
SCI-2D
SCI-2D
SCI-2D
SCI-2D
SCI-2D
SCI-2D
SCI-2D
SCI-2D
SCI-2D
SCI-2D
SCI-2D
SCI-2D
SSI-2D
SSI-2D
SSI-2D
SSI-1D
SSI-1D
SSI-1D
SSI-1D
SSI-1D
SSI-1D
SSI-1D
Apert.
Diam. (m)
1
1
1
2
2
2
2
2
2
3
2
2
3
3
3
3
3
4
3
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
Family
4 ap.
SCI-1D
1 m Diam.
Intersection
of Multiple
Families
Family
8 ap.
SCI-2D
4 m Diam.Transition from
SCI to SSI
Family
Designs
8 ap.
SSI-1D
4 m Diam.
MIT Space Systems Laboratory
TechSat 21 Case Study – Results
CPF
%
Improvement
Lifecycle Cost
%
Improvement
Probability of
Detection %
Improvement
Max. Revisit
Time %
Improvement
CDC Trade Space
49%
51%
-4%
0%
Global Trade Space - SO
65%
66%
-3%
13%
Global Trade Space - MO
54%
57%
-5%
67%
Architecture
Chart: 21
MIT Space Systems Laboratory
Broadband Communication Case Study – Results (1)
MMDOSA
Minimum CPF
(% Improvement)
Pareto Optimal
Equivalent
Performance
(% Improvement)
Pareto Optimal
Equivalent LCC
(% Improvement)
Boeing
97%
85%
1392%
HughesNet
59%
22%
1%
Architecture
Chart: 22
MIT Space Systems Laboratory
Broadband Communication Case Study – Results (1)
Mission Cost
& Performance
Low
Single
GEO
Family
Small LEO’s
Low Inclination
Low Power
Med.
Family
Big LEO’s
High Inclination
High Power
High
Chart: 23
#
Subscriber
Years (M)
LCC ($B)
Const.
Altitude
Const.
Inclination
(°)
# Satellites
Per Plane
# Orbital
Planes
0.00120
0.00573
0.0140
0.0237
0.0553
0.0676
0.0794
0.095
0.157
1.32
2.15
6.37
10.4
14.6
14.8
15.5
24.3
24.8
29.0
29.4
29.4
29.5
0.81
0.83
0.93
0.96
1.12
1.18
1.28
1.44
1.56
1.58
1.70
1.78
2.71
2.79
2.95
3.09
3.62
5.22
7.53
8.98
10.20
20.43
GEO
MEO
MEO
MEO
MEO
MEO
MEO
MEO
MEO
LEO
LEO
LEO
LEO
LEO
LEO
LEO
LEO
LEO
LEO
LEO
LEO
LEO
0
0
0
0
0
0
15
0
30
0
0
30
45
45
30
30
60
45
75
75
60
75
1
3
7
6
7
5
5
8
5
7
7
7
3
5
5
5
6
7
7
7
5
7
1
1
1
1
1
1
2
1
2
1
1
4
10
8
6
6
10
10
8
8
10
8
Antenna
Payload
Transmission Transmission
Area (m2)
Power (kW)
1
1.0
1
1
1
1
1
1
1
1
2
2
1
1
1
1
1
1
1
3
4
5
10
0.5
0.5
1.0
2.0
3.5
2.0
3.0
4.0
0.5
1.5
1.0
2.0
2.0
4.0
4.5
3.5
3.0
4.0
4.0
3.0
3.5
Transition from
GEO to MEO
Architectures
Family
Small MEO’s
Low Inclination
Low Power
Transition from
MEO to LEO
Architectures
Family
Big LEO’s
High Inclination
Low Power
MIT Space Systems Laboratory
Thesis Contributions (1)
• Developed a methodology for mathematically formulating
DSS conceptual design problems as nonlinear
multiobjective optimization problems.
• Determined that a heuristic simulated annealing technique
finds the best system architectures with greater
consistency than Taguchi, gradient, and univariate
techniques when searching a nonconvex DSS trade space.
• Created two new multiobjective variants of the core
simulated annealing (SA) algorithm – the multiobjective
single solution SA algorithm and the multiobjective multiple
solution SA algorithm.
• For each of the three case study missions, identified
specific architectures that provide higher levels of
performance for lower lifecycle costs than prior proposed
designs.
Chart: 24
MIT Space Systems Laboratory
Thesis Contributions (2)
• Created a method for computing the simulated annealing
-parameter, originally developed and intended for single
objective optimization problems, for multiobjective
optimization problems.
• Gathered empirical evidence that the 2-DOF variant of the
simulated annealing algorithm is the most effective at both
single objective and multiobjective searches of a DSS trade
space.
• Developed an integer programming approach to model
and solve the DSS launch vehicle selection problem as an
optimization problem.
Chart: 25
MIT Space Systems Laboratory