no slide titleresearch motivation - problem statement traditionally: define system parameters pj =...

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


HexapodIsolator


FSM Controller


FGSNoise

t_wfeDynamic WFE

Dot Product2

Dot Product

Demux

Demux

x' = Ax+Bu y = Cx+DuCryocoolerDisturbance


AttitudeControl System


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


SIM (Open Loop)


RWA Noise

opd_time

OPD Science Int.


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

no slide titleresearch motivation - problem statement traditionally: define system parameters pj =...

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