Fundamentals of closed loop wave-front controlFundamentals of closed loop wave-front control

M. Le Louarn

ESO

Many thanks to

E. Fedrigo

for his help !

IntroductionIntroduction

Atmospheric turbulence: structure & temporal evolution

AO control and why a closed loop ? Optimal modal control & AO temporal model A real world example: MACAO Advanced concepts: Predictive control,

MCAO et al. Future directions

Temporal evolution of atmospheric turbulenceTemporal evolution of atmospheric turbulence

Model phase perturbations as thin turbulence layers

Temporal evolution: only shift of screens, i.e. assume frozen flow (“Taylor hypothesis”)

There is experimental evidence for this: Gendron & Léna, 1996 Schöck & Spillar, 2000

Idea: predict WFS measurements, if wind speed & direction are known/measured

But, when there are several layer, things get complicated

Atmospheric structure (CAtmospheric structure (Cnn22))

Atmospheric structure (v(h))Atmospheric structure (v(h))

AO Closed loopAO Closed loop

AO and atmospheric turbulenceAO and atmospheric turbulence

AO must be fast enough to follow turbulence evolution

Greenwood & Fried (1976), Greenwood (1977), Tyler 1994 BW requirements for AO

Correlation

time

53

0

220 )()()sec(91.2)( 3

5

dhhChvzk n

fG~10-30 Hz at 0.5 um0 3-6 ms at 0.5 um

Greenwood

frequency

53

0

256 35

10240)

(h)dh(h)Cvλ.(λf n

/G

Errors in AOErrors in AO

gather enough photons to reduce measurement noiseread the detectorcompute the DM commands atmosphere has evolved between measurement and command temporal delay error:

3/5

0

2 )(

: delay between the beginning of the measurement and the actuation of the DM0 : atmospheric correlation time

Rousset 94, Parenti & Sasiela 94

photon noise

+ read-out noise

+ aliasing

+ fitting

+ temporal delay

Why closed loop ?Why closed loop ? There is a good reconstruction algo (=get the right

answer in 1 iteration). Hardware issues:

DM hysteresis (i.e. don’t know accurately what shape DM has)

WFS dynamic range: much reduced if need only to measure residuals.E.g. on SH, Open loop requires more pixels more noise

WFS linearity range : for some WFSs, this is critical: SH with quad-cell, Curvature WFS…

Closed loop hides calibration / non-linearity problems

Closed loop statistics harder to model PSF reconstruction gets harder Loop optimization harder

Interaction matrix in (MC)AOInteraction matrix in (MC)AO

Move one actuator on the deformable mirror response DM influence function

Propagate this DM shape to the conjugation height of the WFS (usually ground)

measure of the response current WFS ( b ) store b in the interaction matrix (M)

as many rows as measurements and columns as actuators

Invert that matrix (+ filter some modes) command matrix: M+ (LS estimate)

command vector c of the DM :

bMMMbMc tt

1)(

Optimizing control matrixOptimizing control matrix

Problem: previous method doesn’t know anything about: Atmospheric turbulence power spectrum

A priori knowledge MAP Minumum variance methods (e.g. Ellerbroek 1994) […]

Guide star magnitude Temporal evolution (turbulent speed, system

bandwidth) See talks by Marcos van Dam & J.M. Conan

A word on modal controlA word on modal control

The previously generated command matrix controls “mirror modes”

Some filtering is usually required (there are ill conditioned modes)

Strategy: filter out “unlikely” modes Project system control space on some orthogonal

polynomials, like: Zernike polynomials KL-polynomials […]

Use atmospheric knowledge to guess which of those modes are not likely to appear in the atmosphere (see talk by J.-M. Conan)

Optimized modal controlOptimized modal control

Must evaluate S/N of measurements and include it in command matrix

Gendron & Léna (1994, 1995) Ellerbroek et al., 1994 Idea: low order modes should have better S/N because they

have lower spatial frequencies E.g. Tip-tilt has a lot of signal (measured over a large pupil) High orders need more integration time to get enough signal.

Compute, for each corrected mode, the optimal bandwidth: allows in effect to change integration time

Need to estimate Noise variance (at first in open loop) PSD of mode fluctuations Sys transfer fn

Include these gains in the command matrix:

toptWUVGM

RequirementsRequirements

We need to: Identify delays in the system Model system’s transfer function Measure the measurement noise in the

WFS Atmospheric noise

Major AO delay sourcesMajor AO delay sources

Integration time: need to get photons CCD read-out time ~ integration time WFS measurement processing:

Flat-fielding Thresholding CCD de-scrambling

Matrix multiplication Actuation (time between sending

command and new DM shape) NOTE: some of these operation can be (and

are) pipelined to increase performance

~4ms

~1ms

MACAO Control Loop ModelMACAO Control Loop Model

WFS DACRTC HVA DM+-

Digital System

WFS: integrator from t to t+TRTC: computational delay + digital controllerDAC: zero-order holderHVA: low-pass filter, all pass inside our bandwidthDM: low-pass filter, all pass inside our bandwidth (first approximation)

System frequency: 350 HzT = 2.86msτ = 0.5 msHVA bandwidth: 3KhzDM first resonance: 200 Hz

CorrectedWavefront

Abe

rrat

edW

avef

ront

E. Fedrigo

AO open loop transfer functionAO open loop transfer function

H(s): Open loop transfert function C(s): Compensator’s continuous transfer

function (usually ~integral…)

T: Integration time (= sampling period (+ read-out))

: Pure delay Simplistic model (continuous, not all errors…),

can of course be improved

)(]exp[]exp[1

)(2

sCsTs

TssH

s

KsC )(

Noise estimationNoise estimation

It is possible to compute (Gendron & Léna 1994):

dfgfHbdfbfBfTgfHgncor ),())()((),()( 000

Bandwidth error Noise errorResidual erroron a mode

Noise estimationNoise estimation

It is possible to compute (Gendron & Léna, 1994):

where :0(g): residual phase error on a mode (for a mode)g: modal gain for mode ( BW), Fe: sampling freqHcor(f,g): correction transfer functionT(f): PSD of fluctuations due to turbulenceB(f): PSD of noise propagated on modeb0: average level of B(f):

Hn: transfer fn white noise input noise output on mirror mode controls.

dfgfHbdfbfBfTgfHgncor ),())()((),()( 000

2/

0

)(2

0eF

e

dffBF

b

Measuring the noise varianceMeasuring the noise variance

Gendron &

Lena 1994

Optimized modal controlOptimized modal control

Correction BW not very sensitive to b0 estimate

Gendron &

Lena 1994

Optimized modal controlOptimized modal control

High order modes have less BW

than low order modes

Gendron &

Lena 1994

Closed loop optimizationClosed loop optimization Problems:

noise estimation+transfer function model need (too) good accuracy

Turbulence is evolving rapidly must adapt gains Non linearity problems in WFS possible (e.g. curvature)

Rigaut, 1993: Use closed-loop data as well (PUEO)

Dessenne et al 1998: Minimization of residuals of WFS error Reconstruction of open loop data from CL

measurements Algorithm must be quick to follow turbulence evolution (few

minutes) Iterative process: initial gain “guess” (from simulation) improved with

closed-loop data, by minimizing WFE estimate.

An example AO system: MACAOAn example AO system: MACAO

60 elements Curvature System (vibrating membrane, radial geometry micro-lenses, Bimorph Deformable Mirror and Tip-Tilt mount)

2.1 kHz sampling, controlled 350 Hz, expected bandwidth ~50 Hz

Real Time Software running a PowerPC 400 MHz Real Time Computer

WaveFrontSensor detector: APD coupled with optical fibers no significant RON, or read-out time

Modular approach (4 VLTI units+SINFONI/CRIRES+spares) Strap quadrant detector tip-tilt sensor+ TCCD (VLTI)

Multi Application Curvature Adaptive Optics systemMulti Application Curvature Adaptive Optics system

A real system: MACAOA real system: MACAO

ATM+TEL

GuideProbe

TelescopeControl

M2

TTM

DM

CWFS

AdaptiveControl

Low Pass

Low Pass

CorrectedWavefront

350Hz

5Hz

1Hz

1Hz

E. Fedrigo

LGS ControlLGS Control

ATM+TEL

GuideProbe

TelescopeControl

M2

TTM

DM

Trombone

LGSDefocus

CWFS

AdaptiveControl

Low Pass

Low Pass

Low Pass

CorrectedWavefront

Airmass

500Hz

5Hz

1Hz

0.03Hz

1Hz

E. Fedrigo

100

101

102

-50

-40

-30

-20

-10

0

10M A C A O 1 inner ring c los ed-loop e rror t rans fe r func t ion fo r g= 70% , tu rb= 10 ,25,50%

frequenc y (H z )

ga

in (

dB

)

1 00

101

102

-50

-40

-30

-20

-10

0

10M A C A O 1 inner ring c los ed-loop e rror t rans fe r func t ion fo r g= 70% , tu rb= 10 ,25,50%

frequenc y (H z )

ga

in (

dB

)

System BandwidthSystem Bandwidth

Plots: courtesy of Liviu Ivanescu, ESOPlots: courtesy of Liviu Ivanescu, ESO(0.45” seeing in V)

Measured Close-Loop Error Transfer FunctionMeasured Close-Loop Error Transfer Function

Predictive & more elaborate controlPredictive & more elaborate control

How to improve control BW ? Turbulence is predictable

Schwarz, Baum & Ribak, 1994 Aitken & McGaughey, 1996

Several approaches (at least): Madec et al., 1991, Smith compensator : takes lag into account Paschall & Anderson, 1993 : Kalman filtering : see talk by Don Gavel Wild, 1996 : cross covariance matrices Dessenne, Madec, Rousset 1997: predictive control law

There is a performance increase in BW limited system Layers are not separated, so “hard job” Unfortunately, gain seems small for current single GS AO systems.

Static aberrations need to be taken into account: residual mode error 0 Telescope vibrations need special attention

Predictive controlPredictive control

Dessenne, Madec, Rousset, 1999

MCAO / GLAOMCAO / GLAO

General case of AO, Several WFSs, several DMs

Observation / optimization direction

Measurement Direction 1

Measurement Direction 2

Challenge:find clever ways to work in closed loop !Are the sensors always seeing the correction ?

The futureThe future

MCAO Allow better temporal predictability if layers are separated Control of several DMs, WFSs, possibly in “open loop”

ELTs / ExAO: fast reconstructors, because MVM grows quickly w/ number of actuators FFT-based Sparse matrix […] predictive methods to the rescue !?

Segmentation Algorithms investigated to use AO WFS info to co-phase

segmented telescopes Kalman filtering

Optimize correction in closed loop, reliable noise estimates, complex systems

Applicable to MCAO