Towards Strehl-Optimal Adaptive
Optics Control
Donald Gavel, Donald Wiberg,
Center for Adaptive Optics, U.C. Santa Cruz
Marcos Van Dam,
Lawrence Livermore National Laboaratory
The goal of adaptive optics is to
Maximize Strehl
• Piston-removed atmospheric phase:
 x    x     x  WA  x dx 
• Phase correction by DM: n
a
 a  x    ai ri  x 
i 1
 WA  xdx  1
vector of actuator commands
ai  ai s
vector of wavefront sensor readings
actuator response functions
• Max Strehl  minimize residual wavefront variance (Marechal’s
aproximation)
 x    x    a  x 
 2    2  x  WA  x dx
Strehl  e
aperture averaged residual
 2
IPAM Workshop on Estimation and Control Problems in Adaptive Optics, Jan., 2004
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Strehl-optimizing adaptive optics
Define the cost function, J = mean square wavefront residual:
J   2    x  WA x dx
2
Wavefront estimation and control problems are separable (proven on subsequent pages):
J  J E  JC
where
• JE is the estimation part:




2
J E    x   ˆx  WA x dx
• JC is the control part:
JC  
and ˆx  
2
ˆ
 x   a x  WA x dx
 x  | s is the conditional mean of the wavefront
IPAM Workshop on Estimation and Control Problems in Adaptive Optics, Jan., 2004
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The Conditional Mean
The conditional probability distribution is defined via Bayes theorem:
P|S  | s  
P , S  , s 
PS s 
The conditional mean is the expected value over the conditional distribution:
P , S  , s 
ˆ
 x    x  | s   P|S  | s d   
d
PS s 
IPAM Workshop on Estimation and Control Problems in Adaptive Optics, Jan., 2004
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Properties of the conditional mean
1. The conditional mean is unbiased:
~
      ˆ P , S  , s dds   PS s  P , S  | s dds   PS s  ˆP|S  | s dds


  PS s ˆds   PS s ˆ  P|S  | s dds   PS s ˆds   PS s ˆds  0
2. The error in the conditional mean is uncorrelated to the data it is conditioned on:
P  , s 
~
 x s    ˆ s    sP , S  , s dds   s    , S
d PS s ds  0


PS s 
3. The error in the conditional mean is uncorrelated to the conditional mean:


 x ˆx     ˆ ˆ       P|S   | s d P , S  , s dds   PS s  P|S  | s d   P|S   | s d ds
~
   P , S  , s d   P|S   | s d ds   PS s  P|S  | s d   P|S   | s d ds
  PS s  P , S  | s d   P|S   | s d ds   PS s  P|S  | s d   P|S   | s d ds  0
4. The error in the conditional mean is uncorrelated to the actuator commands:
na


na
~
 x  a x    ri x   ai s    ri  x  ai s  ˆ   P|S  | s dds
~
i 1
i 1

na

  ri  x  ai s    P|S   | s d    P|S  | s dds
i 1
na


  ri  x  ai s    P|S   | s d    P|S  | s d ds  0
i 1
IPAM Workshop on Estimation and Control Problems in Adaptive Optics, Jan., 2004
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Proof that J = JE+JC (the estimation
and control problems are separable)
J    x   ˆ x    x   ˆ x  W  x dx
    x   ˆ x   2  x   ˆ x  x   ˆ x     x   ˆ x 

2
a
A
2
2


a
a
W  x dx
 A

~
~
 J E  2   x a x     x ˆ x  WA x dx  J C
 J E  JC
0
0
IPAM Workshop on Estimation and Control Problems in Adaptive Optics, Jan., 2004
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1) The conditional mean wavefront is the
optimal estimate (minimizes JE)
Proof:
We show that any other wavefront estimate results in larger JE
Let
E s   ˆs   s 
  E s 2

~
 2
~
 2
~
 2
~
 2


2
2
ˆ
J E        E 
Therefore,  E

2
~
   
s   ˆs 
~
 2     2
~
 2    , s s P( , s )dds   2
~
 2 s PS s     , s P( | s )d ds   2

0
  2
for any

  0
minimizes JE
IPAM Workshop on Estimation and Control Problems in Adaptive Optics, Jan., 2004
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Calculating the conditional mean wavefront
given wavefront sensor measurements
wavefront sensor operator: (average-gradient
operator in the Hartmann slope sensor case)
The measurement equation
Measurement noise
si    x Wi x dx  vi
S
For Gaussian distributed  and n, it is straightforward to show (see next page) that
the conditional mean of  must be a linear function of s:
ˆs  
ns
 k xs
i 1
i
i
 Ks
Cross-correlate both sides with s and solve for K
ˆsT  K ssT
K  ˆsT ssT
so
where
(known as the “normal” equation)
1
 s
T
T
ss
1
since
~
s  0
ˆ x   p T  x S 1s
pi x    x si     x   x Wi s  xdx
Sij  si s j   Wi s  x W js  x   x   x dxdx  vi v j
IPAM Workshop on Estimation and Control Problems in Adaptive Optics, Jan., 2004
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Aside: Proof that the conditional mean
is a linear function of measurements
if the wavefront and measurement noise are Gaussian
si    x Wi S x dx  vi
s  H
 s
P s |  P  d


S |
PS s 

1
 1

T
    exp  s  H  v1 s  H    T 1 d
2
 2

1
 1

T
 arg max  s  H  v1 s  H    T 1 
  2
2


 H T v1H  1
 Ks

1
Hv1s
Measurement equation
Measurement is a linear function
of wavefront
Bayesian conditional mean
Gaussian distribution
= maximum log-Likelihood of
a-posteriori distribution
= a linear (least squares)
solution
IPAM Workshop on Estimation and Control Problems in Adaptive Optics, Jan., 2004
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2) The best-fit of the DM response
functions to the conditional mean
wavefront minimizes JC


2
J C   ˆx   a x  WA x dx
2
JC  
0
ˆ






x

a
r
x

i i

 WA x dx
i


J C

a j
ˆ






x

a
r
x

i i

 rj x  WA x dx
i


  ˆx rj x WA x dx   ai  ri x rj x WA x dx
i
 r  aT R
 a  R 1r
where

r   rj x ˆx WA x dx

and

R   ri ( x)rj ( x)WA ( x)dx

IPAM Workshop on Estimation and Control Problems in Adaptive Optics, Jan., 2004
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Comparing to Wallner’s1 solution
Combining the optimal estimator (1) and optimal controller (2) solutions gives
Wallner’s “optimal correction” result:
a  R 1 AS 1s
where
A   ri  x  p j  x WA  x dx
• The two methods give the same result, a set of Strehl-optimizing actuator
commands
• The conditional mean approach separates the problem into two independent
problems:
1) statistically optimal estimation of the wavefront given noisy data
2) deterministic optimal control of the wavefront to its optimal estimate given the
deformable mirror’s actuator influence functions
• We exploit the separation principle to derive a Strehl-optimizing closed-loop
controller
1E.
P. Wallner, Optimal wave-front correction using slope measurements, JOSA, 73, 1983.
IPAM Workshop on Estimation and Control Problems in Adaptive Optics, Jan., 2004
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The covariance statistics of (x)
(piston-removed phase over an aperture A)
 x    x      WA  d
 x  x   x  x    x  xWA xdx
    x  xWA xdx
    x  xWA xWA xdxdx
 x  x   D x  x  g x   g x  a
1
2
where
2
2
2
D x  x   x    x   x    x  2 x  x
g x  
1
D  x  xWA  xdx
2
1
a   WA  x D  x  xg  xWA  xdxdx
2
IPAM Workshop on Estimation and Control Problems in Adaptive Optics, Jan., 2004
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The g(x) function and a are “generic”
under Kolmogorov statistics
  x 2
g  x  6.88D r0 
53
6.88D r0 
53
• D(x) = 6.88(|x|/r0)5/3
• Circular aperture, diameter D
• Factor out parameters 6.88(D/r0)5/3 and integrals are computable numerically
x D
x D
a  0.149831  6.88 D r0 
53
IPAM Workshop on Estimation and Control Problems in Adaptive Optics, Jan., 2004
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Towards a Strehl-optimizing control
law for adaptive optics
Remember our goal is to maximize Strehl = minimize wavefront
variance in an adaptive optics system
• But adaptive optic systems measure and control the wavefront in
closed loop at sample times that are short compared to the wavefront
correlation time.
• So the optimum controller uses the conditional mean, conditioned on
all the previous data:
  x    x  s t , s t 1 , s t  2 ,
IPAM Workshop on Estimation and Control Problems in Adaptive Optics, Jan., 2004
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We need to progress the conditional mean
through time (the Kalman filter2 concept)
 t1  x    t 1  x  s t 1 , s t  2 ,
 t  x    t  x  s t 1 , s t  2 ,
 t  x    t  x  s t , s t 1 , s t  2 ,   t  x  s t ,  t  x 
1. Take a conditional mean at time t-1 and progress it forward to time t
2. Take data at time t
3. Instantaneously update the conditional mean, incorporating the new data
4. Progress forward to time step t+1
5. etc.
2Kalman,
R.E., A New Approach to Linear Filtering and Prediction Problems, J. Basic Eng., Trans.
ASME, 82,1, 1960.
IPAM Workshop on Estimation and Control Problems in Adaptive Optics, Jan., 2004
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Kalman filtering
new data
new data
st
 t1 x 
Time
progress
 t  x 
st 1
 t  x 
Update
Time
progress
 t1  x 
 t1  x 
Update
...
...
IPAM Workshop on Estimation and Control Problems in Adaptive Optics, Jan., 2004
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Problems with calculating and
progressing the conditional mean of an
atmospheric wavefront through time
• The wavefront is defined on a Hilbert Space (continuous domain) at an
infinite number of points, x A (A = the aperture).
• The progression of wavefronts with time is not a well-defined process
(Taylor’s frozen flow hypothesis, etc.)
• In addition to the estimate, the estimate’s error covariance must be updated
at each time step. In the Hilbert Space, these are covariance bi-functions:
ct (x,x’)=<t(x),t (x’)>, x A, x’ A.
IPAM Workshop on Estimation and Control Problems in Adaptive Optics, Jan., 2004
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Justifying the extra effort of the
optimal estimator/optimal controller
• If is interesting to compare “best possible” solutions to what we are
getting now, with “non-optimal” controllers
• Determine if there is room for much improvement.
• Gain insights into the sensitivity of optimal solutions to modeling
assumptions (e.g. knowledge of the wind, Cn2 profile, etc.)
• Preliminary analysis of tomographic (MCAO) reconstructors suggest
that Weiner (statistically optimal) filtering may be necessary to keep the
noise propagation manageable
IPAM Workshop on Estimation and Control Problems in Adaptive Optics, Jan., 2004
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Updating a conditional mean given
new data
Say we are given a conditional mean wavefront given previous wavefront measurements
ˆ    s t 1
s t  H  vt
And a measurement at time t
The residual
~
et  st  H  st 1  H   st 1   vt  H   vt
T
is uncorrelated to previous measurements, e t s t 1  0
Applying the normal equation on the two pieces of data et and st-1:

ˆ    s t , s t 1   et , s t 1  sT
where
t 1
eT
t

 s t 1sT
t 1

T
 et s t 1
~
  et  K t et  eTt et eTt
0
1
1
s t 1e t  s t 1 
~



s


et
0
t

1


T
e
et e t   t 
T
et
Summarizing:

ˆ  ˆ   K st  Hˆ

IPAM Workshop on Estimation and Control Problems in Adaptive Optics, Jan., 2004
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…written in Wallner’s notation
• Estimate-update, given new data st:
1
 t  x    t  x   p T
t  x S t s t  s t 
s t  s t s t 1 , s t  2 ,   W s  x  t  x dx
~
~
pt x    x et   t  x t  x W s xdx
et  st  sˆ t
Hartmann sensor applied to
the wavefront estimate
Correlation of wavefront to
measurement
T
~
~
St  et etT   W s x W s x t  x t  x dxdx  vv T
Correlation of measurement
to itself
• Covariance-update:
~ ~
~
~
1
t  x t  x   t  x t  x    p T
x
S


t
t p t  x 
where the estimate error is defined:
~
t  x    t  x    t  x 
IPAM Workshop on Estimation and Control Problems in Adaptive Optics, Jan., 2004
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How it works in closed loop
t x
+
 at  x 
  x 

t
 W  x  x dx

s
Wavefront
sensor
Best fit
to DM
1
t 
pT
t  x S t s t  s
Estimator
 t  x 
+

 t1  x 
Predictor
 t  x 
IPAM Workshop on Estimation and Control Problems in Adaptive Optics, Jan., 2004
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Closed-loop measurements need a
correction term
^ as s - s, the
…since what the wavefront sensor sees is not exactly the same
wavefront measurement prediction error
s
s
W
x

x
dx

W
       x   x    a  x dx

 

  W s  x   x     x    a  x     x  dx


 s  s   W s  x   a  x     x  dx
 

Measurement prediction error
 Measurement prediction error
=
DM Fitting error
Hartmann sensor residual
(measured data)
+
DM Fitting error
(can be computed from the
wavefront estimate and
knowledge of the DM)
IPAM Workshop on Estimation and Control Problems in Adaptive Optics, Jan., 2004
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Time-progressing the conditional mean
 t1 x 
Given
how do we determine
 t  x 
?
Example 1:
On a finite aperture, the phase screen is unchanging and frozen in place
 t  x    t1  x 
~ ~
~
~
t  x t  x    t1  x  t1  x  
Consequences:
• Estimates corrections accrue (the integrator “has a pole at zero”)
• If the noise covariance <vvT> is non-zero, then the updates cause the
estimate error covariance to decrease monotonically with t.
IPAM Workshop on Estimation and Control Problems in Adaptive Optics, Jan., 2004
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Time-progressing the conditional mean
Example 2:
The aperture A is infinite, and the phase screen is frozen flow, with wind velocity w
 t  x    t1  x  w 
~ ~
~
~
t  x  t  x    t1  x  w t1  x   w
Consequence:
• An infinite plane of phase estimates must be updated at each measurement
IPAM Workshop on Estimation and Control Problems in Adaptive Optics, Jan., 2004
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Time-progressing the conditional mean
Example 3:
The aperture A is finite, and the phase screen is frozen flow, with wind velocity w

w
ˆt x  A  t x  A s t 1 , s t ,
 t x  A ˆt1 x  A

A
more on this
approximation
later
ˆ   xdx



F
x
,
x

t 1

A
A’
as we might expect
F x, x   x  x  w
for x in the overlap
region, AA’
The problem is to determine the progression operator, F(x,x’), for x in
the newly blown in region, A  A  A’ )
IPAM Workshop on Estimation and Control Problems in Adaptive Optics, Jan., 2004
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“Near Markov” approximation
t x   f t 1 x   wx 
The property
where w is random noise uncorrelated to t-1(x), is known as a Markov property.
We see that if  obeyed a Markov property
t x  st 1 , st 2 ,  t x  t 1 x  st 1 , st 2 ,  t x  ˆt1 x 
that is, the conditional mean on a finite sized aperture retains all of the relevant statistical
information from the growing history of prior measurements.
Phase  over the aperture however is not Markov, since some information in the “tail”
portion, A’’ - (A’’  A’ ), which correlated to st-1, is dropped off and ignored. The fractal nature
of Kolmogorov statistics does not allow us to write a Markov difference equation governing 
on a finite aperture.

w
We will nevertheless proceed assuming the Markov
property since the effect of neglecting  in A’’ - (A’’ A’ )
to estimates of  in A - (A  A’ ) is very small
A
A’
A’’
IPAM Workshop on Estimation and Control Problems in Adaptive Optics, Jan., 2004
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Validity of approximating wind-blown
Kolmogorov turbulence as near-Markov
using the
estimate at
this point
To predict
this point
what is the effect
of neglecting this
point?
A A , A  A A  A eA
eA  A  A A
var  A  A
contribution of
point in A’
var  A eA
A
A’
A’’
contribution of
neglected point
in A’’
wind
Information contained in points neglected by the nearMarkov approximation is negligible
IPAM Workshop on Estimation and Control Problems in Adaptive Optics, Jan., 2004
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The progression operator from A’ to A
We write the conditional mean of the wavefront in A, conditioned on knowing it in A’
 x    x   

w
A’
x  A, x   A 
A
G(x,x’’) solves
A
 G  x , x   x  dx 
  x   x   
 G  x , x    x   x   dx 
(a normal equation)
A
We can then say that
 x    G  x , x   x  dx   q  x 
A
Note: q(x) = 0 and
G(x,x’) = (x-x’-w) for x
in the overlap A A’
where q(x) is the error in the conditional mean (x) - <(x)|(x’)>. q(x) is uncorrelated
to the “data” ((x’))
q x  x   0
Also true in the overlap
since q(x) = 0 there
q x  s t 1  0
and, consequently
since the measurement at t-1 depends only on (x’) and random measurement noise.
Then
 t  x    G  x , x   t1 x  dx 
i.e.
F x, x  Gx, x
A
IPAM Workshop on Estimation and Control Problems in Adaptive Optics, Jan., 2004
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In summary: The time-progression
of the conditional mean is

w
ˆt  x    F  x, xˆt1  xdx
A
A
A’
where F(x,x’) solves
  x   x   F  x, x   x  x dx
A
x  A, x  A
• If we assume the wavefront phase covariance function is constant or slowly
varying with time, then the Green’s function F(x,x’) need only be computed
infrequently (e.g. in slowly varying seeing conditions)
• To solve this equation, we now need the cross-covariance statistics of the phase,
piston-removed on two different apertures.
IPAM Workshop on Estimation and Control Problems in Adaptive Optics, Jan., 2004
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Cross-covariance of Kolmogorov phase,
piston-removed on two different apertures
  x  A  x  A 

A
A’
1
D  x  x   g  x  c   g  x  c   a  c  c 
2
x  A, x  A
Where c and c’ are the centers of the respective apertures, and
and
g x  
1
D  x  xWA  xdx

2
ax  
1
g x  x W A x dx

2
as before
also a “generic” function
IPAM Workshop on Estimation and Control Problems in Adaptive Optics, Jan., 2004
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The error covariance must also progress,
since it is used in the update formulas
using  t  x    t  x    t  x  the error in the conditional mean is
~
t   x    F  x, xt 1  xdx  qt  x 
~
~
A
and the error covariance is
t  x t  x 
~
~
~
~





   





F
x
,
x

x

t 1
t 1  x  F  x , x dx dx  Q  x, x 
A A
x, x  A and x, x  A
where
Q x, x  qx q x 
 x  x    F x, x  x x F x, xdxdx
A A
Q is defined simply to preserve the Kolmogorov turbulence strength on the
subsequent aperture
IPAM Workshop on Estimation and Control Problems in Adaptive Optics, Jan., 2004
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Simulations
•
Nominal parameters
– D = 3m, d = 43cm (D/d = 7)
– r0(l=0.5m) = 10cm ( r0(l=2m)  d )
– w = 11m/s  1 ms (w = D/300)
– Noise = 0.1 arcsec rms
•
Simulations
–
Wallner’s equations strictly applied, even though the wind is blowing
–
Strehl-optimal controller
–
Optimal controller with update matrix, K, set at converged value (allows precomputing error covariances)
–
Sensitivity to assumed r0
–
Sensitivity to assumed wind speed
–
Sensitivity to assumed wind direction
IPAM Workshop on Estimation and Control Problems in Adaptive Optics, Jan., 2004
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Noise performance after convergence
Single-step
(Wallner)
Strehl-optimal
IPAM Workshop on Estimation and Control Problems in Adaptive Optics, Jan., 2004
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Convergence time history
K matrix fixed at converged value
K matrix optimal at each time step
IPAM Workshop on Estimation and Control Problems in Adaptive Optics, Jan., 2004
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Sensitivity to r0
IPAM Workshop on Estimation and Control Problems in Adaptive Optics, Jan., 2004
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Sensitivity to wind speed and direction
IPAM Workshop on Estimation and Control Problems in Adaptive Optics, Jan., 2004
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Conclusions
• Kalman filtering techniques can be applied to better optimize the
closed-loop Strehl of adaptive optics wavefront controllers
• A-priori knowledge of r0 and wind velocity is required
• Simulations show
– Considerable improvement in performance over a single step
optimized control law (Wallner)
– Insensitivity to the exact knowledge of the seeing parameters over
reasonably practical variations in these parameters
IPAM Workshop on Estimation and Control Problems in Adaptive Optics, Jan., 2004
37