1. Fundamentals of closed loop wave-front control
M. Le Louarn
ESO
Many thanks to
E. Fedrigo
for his help !

2. Introduction 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

3. Temporal 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

4. Atmospheric structure (Cn2)

5. Atmospheric structure (v(h))

6. AO Closed loop

7. AO and atmospheric turbulence
AO must be fast enough to follow turbulence evolution
Greenwood & Fried (1976), Greenwood (1977), Tyler 1994  BW requirements for AO
Greenwood
frequency
fG~10-30 Hz at 0.5 um
t0 » 3-6 ms at 0.5 um
Correlation
time

8. Errors in AO photon noise + read-out noise + aliasing + fitting +
temporal delay
gather enough photons to reduce measurement noise
read the detector
compute the DM commands
® atmosphere has evolved between measurement and command
® temporal delay error:
t : delay between the beginning of the measurement and
the actuation of the DM
t0 : atmospheric correlation time
Rousset 94, Parenti & Sasiela 94

9. 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

10. Interaction 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 :

11. Optimizing 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

12. A 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)

13. Optimized 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:

14. Requirements We need to: Identify delays in the system
 Model system’s transfer function
Measure the measurement noise in the WFS
Atmospheric noise

15. Major 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

16. MACAO Control Loop Model
Corrected
Wavefront
Digital System
Aberrated
Wavefront +
WFS
RTC
DAC
HVA DM -
WFS: integrator from t to t+T
RTC: computational delay + digital controller
DAC: zero-order holder
HVA: low-pass filter, all pass inside our bandwidth
DM: low-pass filter, all pass inside our bandwidth (first approximation)
System frequency: 350 Hz
T = 2.86ms
τ = 0.5 ms
HVA bandwidth: 3Khz
DM first resonance: 200 Hz
E. Fedrigo

17. AO 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))
t: Pure delay
Simplistic model (continuous, not all errors…), can of course be improved

18. Noise estimation It is possible to compute (Gendron & Léna 1994):
Residual error
on a mode
Bandwidth error
Noise error

19. Noise 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 freq Hcor(f,g): correction transfer function T(f): PSD of fluctuations due to turbulence B(f): PSD of noise propagated on mode b0: average level of B(f): Hn: transfer fn white noise input  noise output on mirror mode controls.

20. Measuring the noise variance
Gendron
&
Lena
1994

21. Optimized modal control
Gendron
&
Lena
1994
Correction BW not very
sensitive to b0 estimate

22. Optimized modal control
High order modes have less BW
than low order modes
Gendron
&
Lena
1994

23. Closed 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.

24. An example AO system: MACAO
Multi Application Curvature Adaptive Optics system
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)

25. A real system: MACAO E. Fedrigo Corrected Wavefront Telescope Control
Guide
Probe M2
1Hz
ATM+TEL
CWFS DM
Adaptive
Control
TTM
350Hz
Low Pass
5Hz
Low Pass
1Hz
E. Fedrigo

26. LGS Control E. Fedrigo Corrected Wavefront Telescope Control Airmass
Defocus
Corrected
Wavefront
Guide
Probe M2
Low Pass
Trombone
1Hz
ATM+TEL
CWFS DM
Adaptive
Control
TTM
500Hz
0.03Hz
Low Pass
5Hz
Low Pass
E. Fedrigo
1Hz

27. Plots: courtesy of Liviu Ivanescu, ESO
System Bandwidth
Measured Close-Loop Error Transfer Function
(0.45” seeing in V)
Plots: courtesy of Liviu Ivanescu, ESO

28. Predictive & 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

29. Dessenne, Madec, Rousset, 1999
Predictive control
Dessenne, Madec, Rousset, 1999

30. MCAO / 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 ?

31. The 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