Post on 25-Mar-2020
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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
scaled
Pre-balance:• modal form• balance blocks
Pre-balance:• modal form• balance blocks
Compute Gramians:• controllability• observability
Compute Gramians:• controllability• observability
Perform SVDon Gramians
Perform SVDon Gramians
Removes.v.’s below threshold
Removes.v.’s below threshold
Computetransform.matrices
Computetransform.matrices
Indicates modification to regular algorithm
0 50 100 150 200 250 300 35010
-15
10-10
10-5
100
105
State Number i
Han
kel S
ingu
lar
Val
ue σ
iH
σiH of internally balanced system
1 12 21 T
c c b HT U U− −− = Σ Σ
SIM Model ReductionSIM Model Reduction
0 200 400 600 800 1000 1200 1400 1600 1800 200010-8
10-6
10-4
10-2
100
102
104
106Gramian of the Balanced Realization
State number
Han
kel S
ingu
lar V
alue
s
> removed states
#1#2
#3#4
#5#6
#7#8
Intr #1 Intr #2 Intr #3
Telescope #
∑∑
=
+=<∆
k
iHi
n
kiHi
z
z
i
i
1
1
21
σ
σσσ
Select number of states to retain based on % RMS difference between reduced system and full-order system
0.01% difference → 316 retained statesKeep more to match original T.F. more accurately
...i th VSHHi =σ
10-5
100
105
1010
Mag
nitu
de [n
m/N
]
Transfer Function of RWAFx1 to Star Opd #1 for SIM model v2.2
Original JPL Reduced MIT1063Reduced MIT 316
10-1 100 101 102 10310-2
100
102TF Normalized to JPL Original
Frequency [Hz]
RWA Testing & Modeling
• Reaction wheels are anticipated to be largest source of disturbances for Precision Space Structures
• Static, dynamic imbalances induce disturbance at freq. of wheel spin• Bearing, motor, dynamic lubricant disturbances induce vibration at higher
(and sub) harmonics of wheel speed• Experiment & empirical modeling• Analytical modeling
Ithaco “B” RW
Impedance Coupling Analysis
zpredictedzmeasuredHardmountedCoupled
Testbed IMOS Model
Σ+__ RWAΦzpΦ
HRWAzp GGΦ=Φ
Disturbance Analysis
ComparisonPerformed a disturbance analysisand modal parameter sensitivity
analysis on closed-loop SIMClassic Model to validate framework
Results Full Model Red Model# States 308 110RMS (PSD) 4.21 nm 4.21 nmRMS (Lyap) 4.3321 nm 4.1077 nmCPU (Lyap) 39.567 sec 1.552 sec
Nominal RMS error < 0.2 %Disturbance (left): Reaction Wheel AssemblyPerformance (right): Total OPD (int. #1)
SIM Classic Total OPD (int. #1) Power Spectral DensityDisturbance PSD's and cumulative RMS curves
0246
nm
100 101 10210-10
10-5
100
105
nm2 /H
z
Frequency [Hz]
Reduced Model (110 st.) Full Order Model (308 st.)
Requirement 4.4 nm
100 101 102 10310-15
10-10
10-5
100
Frequency [Hz]
Mag
nitu
de
RWA Fx [(N) 2/Hz]
RWA Fy [(N) 2/Hz]
RWA Fz [(N)2/Hz]
RWA Mx [(N-m)2/Hz]
RWA My [(N-m)2/Hz]
RWA Mz [(N-m) 2/Hz]
100 101 102 1030
0.5
1
1.5
SIM Broadband Disturbance AnalysisSIM Broadband Disturbance AnalysisQuestion: How does the choice of reaction wheel (disturbance
model) affect the broadband disturbance analysis results ?
0
5
10
Wheel Model Comparison for Star OPD #1 (CL optics)
nm
Requirement
10-1 100 101 102 103
10-10
10-5
100
nm2/Hz
PSD
Frequency (Hz)
HST wheelIthaco E Ithaco B
ITHACO E
ITHACO B
HST Wheel*
Note: Teldix Wheel information not released to MIT* actually represents Honeywell HR2020 wheel
SIM Disturbance Analysis ResultsSIM Disturbance Analysis ResultsDiscrete Discrete -- Star OPD #1Star OPD #1
10 20 30 40 50 60 700
20
40
60
80
100
120
140
160
180
σ zR
MS
[nm
]
Wheel Speed [RPS]
Star OPD #1
Open Loop OpticsClosed Loop Optics
Plot shows total performance RMS at each wheel speed.
Closed loop performances improve at low wheels speeds (< 30 RPS); not affected at higher wheel speeds.
Disturbance contributions from higher harmonics at high wheel speeds are above the optical bandwidth (100 Hz).
Requirement
Average values represented by dashed lines.Requirement represented by black line.
Lyapunov Block Diagonal Solutions
A1
A2
A3
An
0
0…
X11 X12
X21 X22
……
…
[0] = +
X11 X12
X21 X22
……
… A1
A2
A3
An
0
0…
T
T
T
T
+
BB11
…
BB21
BB12 …
BB22
…
A1X11 + X11A1T + BB11 = 0 A1X12 + X12A2
T + BB12 = 0
A2X21 + X21A1T + BB21 = 0 ….
Computation Times of New Gramian Solver
• Decouple the full Lyapunov equation (solving for a nxnmatrix) into (n2+n)/2 separate Lyapunov equations, each solving for a mxm (block size) matrix.
• Trade-off between faster lyap.m computations for small block size and for…end loop losses.
• Improvement in computation time tested with SIM model in 2x2 block diagonal, 2nd order form.
• Dramatic improvement in time; equal accuracy. Note odd block sizes fail.
2.55*10-155.15m=94
Solution does not existm=47
2.55*10-153.75m=46
Solution does not existm=23
2.55*10-1533.29m=2
2.55*10-15169.14lyap.m on full n x n
Max. Resultant
Time [sec]Method
SIM Modal Sensitivity AnalysisSIM Modal Sensitivity Analysis
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
-0.04 -0.03 -0.02 -0.01 0 0.01 0.02 0.03 0.04-0.04
-0.03
-0.02
-0.01
0
0.01
0.02
0.03
Centroid Pos
2
0
-2
0.005 <= Qc <= 0.05Qc: Amplification Factor [-]
Dynamics and Controls Error Budgeting (2)
ERROR SOURCE 1
ERROR SOURCE 2ERROR SOURCE 3
PERFORMANCE 1
PERFORMANCE 2
-K-
ps-1
ps
ps
nm
nm
-K-
mean
XY Graph
White Noise
Ch X
WhiteNoise
RWA
WhiteNoiseCryo
WhiteNoiseCh Y
x' = Ax+Bu y = Cx+Du
SpacecraftStructuralDynamics
Scope
Mux
Mux
Mux
Mux sqrtMathFunction1
sqrt
MathFunction
t_losLOS Jitter
K
Kwfe
K
Klos
K KfsmK
Kacs
x' = Ax+Bu y = Cx+Du
HexapodIsolator
x' = Ax+Bu y = Cx+Du
FSM Controller
x' = Ax+Bu y = Cx+Du
FGSNoise
t_wfeDynamic WFE
Dot Product2
Dot Product
Demux
Demux
x' = Ax+Bu y = Cx+DuCryocoolerDisturbance
x' = Ax+Bu y = Cx+Du
AttitudeControl System
x' = Ax+Bu y = Cx+Du
Reaction Wheel Assembly Disturbance
Dynamics and Controls Block Diagram for NGST
486 states
Control Parameters(Homogeneous Dynamics)
(fixed)
Plant Parameters(Homogeneous Dynamics)
(fixed)
Disturbance Parameters(Inhomogeneous Dynamics)
(variable)
Dynamics and Controls Error Budgeting (3)
0.01 0.02 0.03 0.04 0.0551015202530
0.010.020.030.040.050.060.070.080.090.1
Cryo Attenuation Qc [-]
1st-Isopoint
Initial Guess
Static Imbalance Us [gcm]
Inte
grat
ion
Tim
e T i
nt[s
ec]
Parameter Bounding Box
-0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
Qc
Us
Tint
Normalized Performance Sensitivities
Dis
turb
ance
Par
amet
ers
p
WFE z1LOS Jitter z2
( )( / ) /nom z zp pσ σ⋅ ∂ ∂
0246x 10-3
[ase
c]
PSD and cumulative RMS (LOS Jitter)
10-2 10-1 100 101 102
10-10
10-5
[ase
c2/H
z]
Frequency (Hz)
Isoperformance Analysis Results
Isocontour for Performances: σz, WFE = 55 [nm] ; σz,LOS = 0.005 [asec]
Dynamics and Controls Error Budgeting (4)Example: NGST Error Budget (Excel)
LTI System , σz,req, p_bounds, p_nom z1: WFE RMS [nm] Req: 55.00 z2: LOS Jitter [asec] Req: 0.005
Error Source VAR % Allocation VAR% Capability VAR % Allocation VAR% CapabilityCryocooler 0.49 38.50 0.72 46.6467 0.40 0.003162 0.31 0.002823RWA 0.49 38.50 0.28 29.1543 0.40 0.003162 0.53 0.0037016GStar Noise 0.02 7.78 0.00 0.0000 0.20 0.002236 0.17 0.0020829Total 55.00 55.0081 0.005000 0.0051
Find Error SourceContributions
EvaluateError Contribution
Sphere
Cap
abili
ty=
Clo
sest
Fea
sibl
e Er
ror B
udge
t
Parameters: Qc=0.029, Us=14.09, Tint=0.0407
0.20.4
0.60.8
00.2
0.40.6
0.8
0
0.2
0.4
0.6
0.8
CRYO
Error Contribution
Sphere
RWA
GS
Noi
se
WFE BudgetLOS BudgetCapability
var_contr
1.0
Allo
cate
d B
udge
t
Isoperformance Engine
Isoperformancesolution set
(Bivariate) Isoperformance Fundamentals
1/ 22
,1
,
100
ison
z k zk
isoz req ison
σ σ
σ=
−
ϒ = ⋅
∑
Quality of isoperformance solution
Taylor series expansion :
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
Vectorfunction: ( ) 2, k z kp p whereσa a
1,
2,
kk
k
pp
p
=
H: Hessian0k
Tz p
pσ∇ ∆ ≡
tk: Tangent vectoris in the nullspace
αk : Step sizetκ : Step direction
k
T Tk k k z p
U S V σ= ∇
k kp tα∆ = ⋅[ ]k k kV n t=
( )1/ 2
1,2100 k
iso z req Tk k kp
t H tε σ
α−
=
1k kp p p+ = + ∆nk: Normal Vector
Pk+1 : k+1th isopoint, where k=1,…,niso
k-thisopoint:
Isoperformance for SIM (1)
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
become more stringent.
Version 2.0
Parameters Bounding Box
Taylor Exp:
Isoperformance Results SIM V2.2Isoperformance Results SIM V2.2
Solution 1Solution 10.2 0.4 0.6 0.8 1 1.2
80
100
120
140
160
180
200
220
Initial Point
RWA Disturbance Gain Krwa [-]
Optical Control Bandwidth f o [Hz]
SIM Version 2.2 Star OPD#1 - Isoperformance Chart
Parameter Bounding Box
3 nm 6nm
Results suggest that Star OPD #1performance benefits more from
reduction in wheel disturbance thanfrom increasing optical control BW
0
5
10
nm
PSD and cumulative RMS (Star OPD #1)
10-1 100 101 10210-8
10-6
10-4
10-2
100
102
nm2/Hz
Frequency (Hz)
Solution 1: Krw a=1.0 fo=180 Hz Solution 2: Krw a=0.725 fo=100 Hz
2% tolerance
8.26nm
Solution 2Solution 2Higher control
bandwidth (180 Hz)and current wheels
Reduced wheeldisturbance (0.75)and 100 Hz optical
controlSolution 1Solution 1
Model Sensor/Actuator Topology
Dual development for sensors and actuators
Weight modalobservabilityby Modal costs
Weight modalobservabilityby Modal costs
Compute MS modalcost from disturbance to each sensor
Compute MS modalcost from disturbance to each sensor
Determine modalobservability foreach sensor
Determine modalobservability foreach sensor
Compute MS modalcost from actuator toperformance
Compute MS m
[ ] [ ][ ]Tnnz
Tzjz
Tzjj
Tjuju
Tjj
CCXCCXJ
BBAXAX
L11
,, 0
=
=++For the i-th state and j-th actuator:
Dot product to quantify alignment
odalcost from actuator toperformance
Determine modalcontrollability foreach actuator
Determine modalcontrollability foreach actuator
Weight modalcontrollabilityby Modal costs
Weight modalcontrollabilityby Modal costs
Scalesystem
juH
iji
Tii
Ti
Bqf
qAq
,, =
= λ
xCyxC
uBwBAxx
y
z
uw
==
++=
For the i-th state and j-th actuator:
Scalesystem
Modelreduction
Modelreduction
Improve numerical robustness of balanced reduction
Scales with sensor and actuator resolutions
Combine weightedmodal observabilitywith weightmodal controllability
Combine weightedmodal observabilitywith weightmodal controllability
zReduced open-loop model
State by state multiply of Jj and fj
SIM: Sensor/Actuator Effectiveness Matrix• Two interfer-
ometers: guide 1 and guide 2
• Actuators:– Voice coil– Piezo (PZT) with
mirror– Tip fast-steering
mirror (FSM)– Tilt FSM
• Sensors:– Total differential
pathlength (DPL)– Internal DPL– x-direction
wavefront tilt– y-direction
wavefront tilt
Voicecoil
PZTmirror
Voicecoil
PZTmirror
Tip Tilt Tip Tilt
TotalDPL
3.8 6.7 0.9 3.2 4.2 -1.8 0.3 -3.1
Int. DPL 4.9 6.4 2.1 2.9 4.5 -1.4 0.4 -2.1
TotalDPL
0.9 3.2 3.6 6.7 -0.1 -1.5 3.8 -0.2
Int. DPL 2.1 3.4 4.8 6.9 0.1 -1.7 4.4 0.5
WavefrontX tilt
-13.4 -3.6 -15.0 -7.1 6.3 -0.6 -0.3 -3.4
WavefrontY tilt
-14.7 -23.3 -15.5 -22.7 0.5 3.1 -2.2 -3.6
WavefrontX tilt
-12.9 -7.5 -14.0 -4.0 0.5 -1.5 5.8 1.4
WavefrontY tilt
-15.6 -23.9 -16.4 -23.4 -1.4 -2.9 1.2 3.2-3.4
Guide channel 1
Guide channel 1
Phasing control block
Fine-pointing control block
Cross-couplingblock
Control Tuning Framework
Formal Tuning Problem:
Minimize Performance subject to (1) Stab. Robustness requirement
(2) Limited deviation from baseline controller(3) Control channel gain limitations
Simplified Tuning Problem:
Minimize Augmented Cost:JA= Performance
+ Stab. Robustness metric+ Penalty for baseline dev.+ Penalty for control gain
•Gradients of each cost term are computable analytically•Each cost term defined for (1) design model and (2) measured data
Control Tuning: Nonlinear Program
Compute param.Update (Hessian)-1
• Extension of MACE control design strategy
• BFGS nonlinear program• Closed-loop stability-
preserving iterations• With each iteration a
stabilizing tuned controller is designed
Cost expressions for either design model tuning or measured data tuning
Augmented cost includes explicit cost term for stability non-robustness, and for controller deviation
Parameterize controller • allow general control
structure (e.g. PID)• allow state addition
Problem set-up:Form cost JA
Compute cost and gradient
Compute param.update controller
Decreasestepsize
Linesearchalgorithm
No
Update costand gradient
If gradient smallor exceed # iterations
exitComputecontroller
Yes
If stepsizetoo small
No
If closed-loop unstable Yes
Automate graphical evaluation of stabilityThesis
contributionNo, next iteration Yes
Controller Tuning: Design Knobs
100
101
102
0
0.5
1
1.5
Ws
Max S s.v.
100
101
102
0
5
10
15
Ts
f (Hz)
100
101
102
0
0.5
1
1.5
Wcr
Nyquist Dev
f (Hz)
100
101
102
0
0.5
1
1.5
2
2.5
U1
Y1
100
101
102
0
0.5
1
1.5
2
2.5Y2
100
101
102
0
0.5
1
1.5
2
2.5Y3
100
101
102
0
0.5
1
1.5
2
2.5
U2
f (Hz)10
010
110
20
0.5
1
1.5
2
2.5
f (Hz)10
010
110
20
0.5
1
1.5
2
2.5
f (Hz)
)()()()()( pMpdpSpJpJ RA +++=
Performance
Channel 1: X 100Channel 2: X 1
Stability non-robustness penalty
Control channel magnitude gain
Controller deviationαd=100
Augmented Cost:Performance• Weight channel 1
to be more important than 2
Stability• Penalize Sens.
s.v.>10dB bumps for f>10 Hz
• Roll-off with penalty on s.v.>5dB bumps for f>70 Hz
• Penalize a close pass of critical point for f~75 Hz
Channel Gain• Penalize Y2 to U1• Penalize low freq.
use of U2
SIM Control Tuning: Family
OL BC S1 S2 S30
10
20
30
40
50
60
RMS phasing
T D
PL
G1
(nm
)
OL BC S1 S2 S30
0.05
0.1
0.15
0.2
0.25
0.3
0.35RMS pointing
DW
FT
x G
1 (a
sec)
101
102
103
104
0
2
4
6
8
10
12
14
f (Hz)
Max Sensitivity S.V.
(dB
)
BCS1S2S3
• Tuning allows improvement in RMS phasing and pointing:
Tune fine-pointingblocks
Tunephasing blocks
Tune all controlblocks
Final tunedcontroller
(BC) (S1) (S2) (S3)Open loopsystem
Baselinecontroller
40 state JPL-designedGuide 1 and 2 decoupledphasing and pointing decoupled
control phasing(nm)
pointing(asec)
Baseline 3.8 0.0047Tuned 1.2 0.0014
• Approach phasing nulling requirement
• Slight stability robustness penalty
Origins Testbed
• Offers the combination of large angle slewing with fine phasing and pointing control in the presence of realistic disturbances
• Traceable to precision space telescopes, e.g. SIM, NGST
• Investigate methods for modeling and control of flexible structures– global MIMO control– vibration isolation and suppression schemes to
meet future space-based telescope reqs.– dynamic system scaling issues– automation issues for complex optical systems
Slew Optical capture
ObservationSelect target
Integral com- mand tracking
Hold at position Fine phasing and pointing Dump wheel momentum
Settle structure Acquire target
Origins Testbed: Transfer Function Matrix
10−4
10−3
10−2
10−1
|Enc
| (de
g/V
)
RWA VC PZT FSM
10−2
10−1
100
101
|RG
A| (
deg/
s/V
)
10−1
100
101
102
|DP
L| (
µm/V
)
100
101
102
10−3
10−2
10−1
|QC
| (de
g/V
)
Freq (Hz)10
010
110
210
−3
10−2
10−1
Freq (Hz)10
010
110
210
−3
10−2
10−1
Freq (Hz)10
010
110
210
−3
10−2
10−1
Freq (Hz)
RWA VC PZT FSM
ENC 20.3 5.0 10.2 6.3RGA 19.1 8.8 12.7 11.4DPL 12.2 22.8 26.3 14.5QC 17.4 8.9 13.7 20.6
Sensors:• ENC - encoder• RGA - rate-gyro assembly• DPL - interferometer• QC - quad cell photodiode
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.
Conclusions
• Despite differing scientific objectives, future space-based telescopes are dynamically similar.
• 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
( )
( ) ( ) ( ) ( )
+++++∑ ∑
++∑=Θ
−
= +=
=
ililililllliiN
i
N
ili
kkN
kk
PyxGG
G
δλπδθγδθγβδθθδθθ
δθθ
2coscoscos1
1 1
1
2
10-2 10010-9
10-8
10-7
10-6
RMS Aperture Shear (m)
Nul
l Dep
th
10-2 10010-9
10-8
10-7
10-6
RMS OPD (nm)1.5m 5nm
Example: 4 aperture linear array( ) ( ) liililil yx φφθγθγβ −++= coscos
liil yyy −=,liil xxx −=
• Assume zero-mean, white Gaussian perturbations:
( )pathlength aldifferenti
motion aperture,angle tilt aperture
==
=
il
ilil
k
Pyx
δδδ
δθ
Derive Performance Penalty Matrix
• Utilize the transmissivity function to generate performance requirements
∫Θ=Θ−∞→
T
TTdt
T
21lim
( ) ( ) ( ) ( )
( ) ( ) ( )4444444444444 34444444444444 21
4444444 34444444 21
onperturbati
222221
1 1
nominal
1
1 11
2
2212
21cos2
cos2
22
++
∑ ∑−
+∑ ∑+∑=Θ
∆∆
−
= +=
−
= +==
ililililPP
N
i
N
ililllii
N
i
N
ilillliik
N
kk
GG
GGG
σλπγσγσ
λπβθθ
βθθθ
[ ]zQzEJ zzT
t ∞→= lim [ ]TNNPPPz 1131211312 ... , ... ∆∆∆=
• Define system cost from perturbation terms and write it in a matrix form
=zzQ Cost matrix depending on the coefficients of the perturbation terms
Optical Sensitivities
• Map physical coordinates to optical sensitivities– Optical Path Difference (OPD): irefri OPLOPLOPD −=
)()( ioiireforefrefrefiri ududududzzOPD •−•−•−•+−=
– Aperture Shear (AS): irefri xxAS −=
x
y
Aref xref
yref
Ai xi
yi
xo
yo
uref ui
di do
dref
zi
zref
Hub
OPL
i
OPLref
ykkAT δθ=
θk
θk
ζzT ζζzTz =• Define transformation matrix :vectoreperformanc=z s/ceach ofattitudeandposition =ζ
– Aperture Tilt (AT):
* Figure courtesy of Olivier de Weck
Control Effort Analysis• Assess control effort for different baseline configurations
– Perform small perturbation analysis on the system to generate a set of linear equations which describe system dynamics
wBuBA wu ++= ζζ&
ζζzTz =– Solve the standard Linear Quadratic Regulator (LQR) problem
ζKu −=
dt )(0∫ +=∞
uRuTQTJ uuT
zzzTz
T ζζ ζζ
[ ] TTu KKuuE ζζΣ==Σ
– Compute the closed-loop control covariance matrix
=Σζζ Closed-loop steady-state covariance matrix
0=+Σ+Σ Tww
Tclcl BBAA ζζζζ