Technological Challenges HST - 1990 SIM-2006 Faint Star Interferometer Precision Astrometry Lightweight 8m-Optics IR Deep Field Observations Space-Based Observatory Multipurpose UV/Visual/IR Imaging and Spectroscopy The next generation of space based observatories is expected to provide significant improvements in angular resolution, spectral resolution and sensitivity. Science Requirements Engineering Requirements D&C System Requirements TPF-2011 5 year wide-angle astro- metric accuracy of 4 µasec to limit 20th Magnitude stars Fringe Visibility > 0.8 for astrometry Science Interferometer OPD < 10 nm RMS Sample Requirements Flowdown for SIM Nulling Interferometer Planet Detection NGST-2009 NEXUS-2004 Deployable Cold Optics NGST Precursor Mission Achieve requirements in a cost-effective manner with predictable risk level.
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No Slide TitleResearch Motivation - Problem Statement Traditionally: Define System Parameters pj = po Predict H 2 performances σz,i Isoperformance: Find Loci of Solutions pLB < pj
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Technological ChallengesHST - 1990
SIM-2006
Faint Star InterferometerPrecision Astrometry
Lightweight 8m-OpticsIR Deep Field Observations
Space-Based ObservatoryMultipurpose UV/Visual/IR Imaging and Spectroscopy
The next generation of space based observatories is expected to provide significant improvements in
angular resolution, spectral resolution and sensitivity.Science
RequirementsEngineering
RequirementsD&C SystemRequirements
TPF-2011
5 year wide-angle astro-metric accuracy of 4 µasec
to limit 20th Magnitude stars
Fringe Visibility> 0.8 for astrometry
Science InterferometerOPD < 10 nm RMS
Sample Requirements Flowdown for SIM
Nulling InterferometerPlanet Detection
NGST-2009NEXUS-2004
Deployable Cold OpticsNGST Precursor Mission
Achieve requirements in a cost-effective manner with predictable risk level.
Research Motivation - Problem Statement
Traditionally: Define System Parameters pj = po Predict H2 performances σz,iIsoperformance: Find Loci of Solutions pLB < pj < pUB Constrain performances σz,i = σz,req
Disturbances
Opto-Structural Plant
White Noise Input
Control
Performances
Phasing (WFE)
Pointing (LOS)
σz,2 = RSS LOS
Appended LTI System Dynamics
(ACS, FSM, ODL)
(RWA, Cryo)
d
w
u y
z
Σ ΣActuator Noise Sensor
Noise
σz,1=RMS WFE
[Ad,Bd,Cd,Dd]
[Ap,Bp,Cp,Dp]
[Ac,Bc,Cc,Dc]
disturbancestates
controllerstates
qdqpqc
[Azd, Bzd, Czd, Dzd]
z=Czd qzd
ProblemStatement:
# of parameters: j=1…np# of performances: i=1…nz
np>nz
Parameters: pj
Video Clip
(“sit and stare”)Science Target Observation ModeOverallState Vector
qzd=
plantstates
Dynamics-Optics-Controls-Structures Framework
Design Structure:IMOSStructure:IMOS
DisturbanceSources
DisturbanceSources
Baseline Control
Baseline Control
ModelAssemblyModel
Assembly
ModelUpdatingModel
Updating
DisturbanceAnalysis
DisturbanceAnalysis
UncertaintyAnalysis
UncertaintyAnalysis
ControlForgeControlForge
Data
SystemControlStrategy
Modeling Model Prep Analysis Design
CampbellBourgault
GutierrezMasterson
Jacques
Optics:MACOSOptics:MACOS
IsoperformanceIsoperformance
SubsystemRequirements
ErrorBudgets
JPL
ZhouHowHall
Miller
Balmes
Blaurock
SensitivitySensitivity
Σ Margins
MastersCrawley
Haftka
Gutierrez
System Requirements
Feron
Feron
van Schoor
Crawley
Moore Skelton
DYNAMODDYNAMOD
UncertaintyDatabase
UncertaintyDatabase
Model Reduction &Conditioning
Model Reduction &Conditioning
Sensor &Actuator
Topologies
Sensor &Actuator
Topologies
Mallory
ControlTuning
ControlTuning
OptimizationOptimization
Blaurock
Hasselman
Current Evaluation Framework
zd zd
zd
q A q B dz C q= +=
&
Using Lyapunov Approach:
1 0T Tzd q q zd zd zdA A B BΣ + Σ + =2Steady-State Lyapunov Equation
, , 0T Ti zd zd i zd i zd iL A A L C C+ + =4 Lagrange Multiplier Matrix Equation
( ) ( )2, ,i
T TTzd i zd i zd zdz zd zd
q i q qj j j j j
C C B BA Atr tr Lp p p p pσ ∂ ∂∂ ∂ ∂ = Σ + Σ + Σ + ∂ ∂ ∂ ∂ ∂
5
Governing Sensitivity Equation (GSE)
( )1
2, ,i
Tz zd q zdi iC Cσ = Σ3
RMS 212
i i
i
z z
j z jp pσ σ
σ∂ ∂
= ⋅∂ ∂
6Sensitivity
Model Assembly and conditioning
Original Form of appended closed-loop state transition matrix
High orderFEMmodel
Balance &reduce stabledynamics
AppendRWA dist.dynamics
Close atti-tude controlloops
Reduced orderand conditionedmodel, 138 states
Star tracker and rate-gyro to reaction wheels.0.1 Hz bandwidth
Reduce SIM modelwith numericallyrobust balancing
Model RWA dist. witha low-order state-spacepre-whitening filter
0 0d
w p u czdd
c yw c y c c yu c
AB A B CA
CB D B C A B D C
= +
Create appendeddynamic LTIsystem:
SIM Classic
308 States (Full Order)
110 States (Balanced Reduced)
Numerically Robust Balancing Algorithm
100
101
102
−60
−40
−20
0
20
40
Mag
nitu
de (
dB)
100
101
102
−200
−100
0
100
200
Pha
se (
degr
ees)
f (Hz)
• Conventional model truncation occurs after balancing• Modified Algorithm: truncation occurs during balancing, controlled by threshold• Pre-balancing ensures that 2x2 blocks corresponding to each mode are acceptably
Find sensitivity (partial derivatives) of theperformances with respect to modal
or physical system parameters.Governing Sensitivity Equation (GSE)Governing Sensitivity Equation (GSE)
212
i i
i
z z
j z jp pσ σ
σ∂ ∂
= ⋅∂ ∂
( ) ( )2, ,i
T TTzd i zd i zd zdz zd zd
q i q qj j j j j
C C B BA Atr tr Lp p p p pσ ∂ ∂∂ ∂ ∂ = Σ + Σ + Σ + ∂ ∂ ∂ ∂ ∂
-40 -30 -20 -10 0 10 20 30 40
172.7
174.6
176.7
178.4181.4
183.6
184185.2
185.5
186.7
187.1187.6
187.8
188.1
188.4
Normalized Sensitivities of Star Opd #2 RMS value with respect to modal parameters
Open-loop modal frequency (Hz)
pnom/σz,nom*∂ σz /∂ p
p = ωp = ζ p = m
0
20
40
60Cumulative RMS (Star Opd #2)
m
100 101 10210-15
10-10
10-5
100
105
m2 /H
z
PSD
Frequency (Hz)
Physical Parameter Sensitivities
A, J, Iz,Iy,E,G,ρ: siderostat boom propertiesa1-a7: α scale factors for siderostat CELAS’sb1-b7: β scale factors for siderostat CONM’s
Physical Insight:G and J are the most important physical parameters for the siderostat boom.Also a3/b3 indicate that localsiderostat modes affect performance.
Physical Parameter Sensitivities can be obtained in 2 ways:
1 1
ˆ
ˆ
m onNj ijzd zd zd
oj ij ij
mB B Bp m p p
φφ= =
∂ ∂∂ ∂ ∂= ⋅ + ⋅ ∂ ∂ ∂ ∂ ∂ ∑ ∑
1
Njzd zd
j j
A Ap p
ωω=
∂∂ ∂= ⋅ ∂ ∂ ∂ ∑
1 1
m onN
ijzd zdo
j i ij
C Cp p
φφ= =
∂∂ ∂= ⋅ ∂ ∂ ∂ ∑∑
Matrix Derivativesfor a “StructuralPlant-only” System.
(1) Modal method (via chain rule):Sum only over important DOF’s and modes
that are kept in the reduced model. Only sum overopen loop modesthat are kept.
Only sum over“important” deg. of freedom
OR
(2) Direct method (in physical space)
Dynamics and Controls Error Budgeting (1)(1) Why is error budgeting important ?
(2) How is it done today?
(3) How can isoperformancehelp error budgeting ?
Goal: Balance anticipated error sources, which are given by physical process limits and imperfections of hardware in a predictable and physically realizable manner. Example: balancing of sensor vs. process noise.
NGST Example : Assume 3 Main Error Sources
Establishes feasibility of dynamic systemperformance given noise source assumptions.
Ad-Hoc error budgeting, RSS error tree,limited physical understanding of interactions.
Leverages sensitivity analysis and integrated modeling. Creates link to physical parameters.
Error Source 2: RWA
Us: Static Imbalance [gcm]0 0.5 1
Axial Force [N]
[sec] 0 1 2 3 4 5 6 7 8-5
-4
-3
-2
-1
0
1
2
3
4
5x 10
-3
Time [sec]
x
RWA Force Fx
1.0 <= Us <= 30.0
Error Source 1: CRYO Error Source 3: GS Noise
Tint: Integration Time [sec]0.020 <= Tint <= 0.100
SIM - Wheel Imbalance versus Corner Frequency Isoperformance Study
Scope
x' = Ax+Bu y = Cx+Du
SIM (Open Loop)
x' = Ax+Bu y = Cx+Du
RWA Noise
opd_time
OPD Science Int.
x' = Ax+Bu y = Cx+Du
IsolatorBand-Lim itedWhite Noise
100
101
102
-140
-120
-100
-80
-60
-40
Mag
nitu
de [d
B]
Disturbance Filter
100
101
102
-30
-20
-10
0
10
20
Mag
nitu
de [d
B]
Isolator Transmission
100
101
102
-40
-20
0
20
40
60
80
Mag
nitu
de [d
B]
P lant Dynamics
2 4 6 8 10 12 14 16 18 2010-2
10-1
100
First Iso-Point
Isolator Frequency f iso [Hz]
Stat
ic W
heel
Imba
lanc
e U
s[g
cm]
Iso-Performance Curve for SIM : σOPD#1= 20 nm
0z zz s iso
s isoU f
U fσ σσ ∂ ∂
∆ = ∆ + ∆ =∂ ∂
Isoperformance analysis for static wheel imbalance Us [gcm] versus isolator corner frequency fiso [Hz] at the RMS OPD #1 = 20 nm level for SIM Classic (version 1.0)
Observation:“Dip” in isoperformance contour
corresponds to region, where isolator amplifies the disturbance.
Frequency [Hz] Frequency [Hz] Frequency [Hz]
Isoperformance Analysis for SIM (2)
500 1000 1500 2000 2500 3000 3500 40000
200
400
600
800
1000
1200
1400
Bias Wheel Speed Ro [RPM]
Opt
ical
Con
trol
Ban
dwid
th ω
o[r
ad/s
ec]
SIM Classic Isoperformance : Star OPD #1 - dR=500 RPM
3 nm
10 nm
20 nmRWA Noise Plant Optical Control
Ro ωow
wR
WA
y pla
nt
z
Treat RMS performances σz,i of a dynamic opto-mechanical system as a constraint while trading key
disturbance, plant and control parameters pjwith each other
first order term second order term
1( ) ( ) HOT2 kk
T Tz z k z pp
p p p p H pσ σ σ= +∇ ∆ + ∆ ∆ +14243 1442443
Then Solve: 0k
Tz p
pσ∇ ∆ ≡
Conclusion: As Bias Wheel Speed Ro increases control requirements
Actuators:• RWA - reaction wheel• VC - mirror on voice coil• PZT - mirror on piezo stack• FSM - fast steering mirror
Sensor / Actuator Index Matrix
MACE Flight Validation
Middeck Active Control Experiment (MACE)1995: STS-672000:Currently an active experiment on ISSAssessed effectiveness and predictability of advanced modeling and control algorithms on precision attitude and instrument pointing of a small satellite.
MACE Test Article
Demonstrated gravity effects can be accounted for during control design. For weakly nonlinear systems the accurate fit of measurement models can be deceptive.
• The MIT SSL has developed a framework for the Dynamics, Optics, Structures and Control (DOCS) for these telescopes.
• Flexible tools are developed and demonstrated in each of four critical areas– Modeling: physics-based FEM, model integration– Model Preparation: model reduction and conditioning, system ID– Analysis: disturbance, performance, sensitivity, and
sensor/actuator coupling.– Design: isoperformance trades, control tuning
• Tools are validated on large-order models and on experimental test articles
Motivation
• Translate interferometer performance (null depth, sensitivity) to requirements on physical and optical motions– Aperture motion stability (AS)– Optical path difference (OPD)– Differential wavefront tilt (DWFT)
• Utilize the transmissivity function to characterize physical and optical effects on null depth
( ) ( )2
1expcos2exp∑
=
−=Θ
N
kkk
kk jrLjD φθθ
λπ
max||
DepthNullΘΘ
= oγ
• Derive control requirements from the perturbedtransmissivity function
Development of Stability Requirements I
• Describe transmissivity function of a two-dimensional aperture array as
( ) ( ) ( )2
1sin2cos2exp∑
++=Θ
=
N
kkkkkk yrxrjG φθ
λπθ
λπθ
( )
( ) ( )( )λπγθγθγ
φ
θ
r2 s,coordinateangular imagesin,cos
aperture k ofshift phase
aperture k oflocation D-2,x
aperture k of angletilt
aperture k ofdiameter
th
thk
thk
th
==
=
=
=
=
k
k
k
y
D
( ) ( )( )
( )λθπλθπ
θλ
πθ rD
rDJrD
kk
kk
kk
k −
−
−+= sin
sin1)cos1(
2Gk
),( 33 yx
Planet
x
y
),( 11 yx),( 22 yx
r
Starθ
Circular Aperture
Development of Stability Requirements II
• Add small perturbations to transmissivity function