Lèonard Prengère, Caroline Kulcsár, and Henri-François Raynaud, "Zonal-based high-performance control in adaptive optics systems with application to astronomy and satellite tracking," J. Opt. Soc. Am. A 37, 1083-1099 (2020)
This paper presents a model-based approach to adaptive optics (AO)
control based on a zonal (i.e., pixelized) representation of the
incoming atmospheric turbulence. Describing the turbulence on a zonal
basis enables the encapsulation of the standard frozen-flow assumption
into a control-oriented model. A multilayer zonal model is proposed
for single-conjugate AO (SCAO) systems. It includes an edge
compensation mechanism involving limited support, which results in a
sparser model structure. To further reduce the computational
complexity, new resultant zonal models localized in the telescope
pupil are proposed, with AR1 or AR2 structures, that match the spatial
and temporal cross-correlations of the incoming turbulence. The global
performance of the resulting linear quadratic Gaussian (LQG) regulator
is evaluated using end-to-end simulations and compared to several
existing controllers for two different configurations: a very large
telescope SCAO and low earth orbit satellite tracking. The results
show the high potential of the new approach and highlight possible
trade-offs between the performance and complexity.
Lucas Marquis, Henri-François Raynaud, Nicolas Galland, Jose Marco de la Rosa, Icíar Montilla, Óscar Tubío Araújo, Marcos Reyes García-Talavera, and Caroline Kulcsár J. Opt. Soc. Am. A 41(1) 111-126 (2024)
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Pseudo-Boiling Atmosphere Dynamical Parameters for End-to-End
Simulationsa
Blue corresponds to the first layer, green to the second
layer, and purple to the third layer.
Table 3.
Strehl Ratio at 1.654 µm of Zonal-Based LQG Regulator with a
Boiling AR1 Dynamical Model, Using Different Sampling of the
Zonal Basis
Linear sampling
(points/actuator pitch)
1
2
3
4
LQG-KF AR1 boiling
42.2%
50.9%
51.7%
52.1%
Table 4.
Zonal LQG Regulators According to Associated Modelsab
Regulator Names
Associated Models
(a)
LQG-DKF multilayer AR1
(b)
LQG-KF multilayer AR1 OS
(c)
LQG-KF multilayer AR1
(d)
LQG-KF + MAP multilayer
AR1
All turbulence models are of AR1 type in each layer,
oversampling is two linear points per actuator.
To ease reading of the table, here are a few short
descriptions: (a): spatially invariant
models based on convolution kernels (that is, the DKF case
based on the infinite pupil hypothesis) where the
coefficient $\alpha \lt
1$ insures
stability; (b): same oversampling for
phase and measurements in matrix form (finite pupil) and
without edge compensation; (c): any
sampling of the phase, but non-oversampled measurements,
in matrix form (finite pupil) without edge compensation;
and (d): any sampling of the phase, but
non-oversampled measurements, in matrix form (finite
pupil) with edge compensation using MAP estimation.
Table 5.
Strehl Ratio at 1.654 µm of Zonal-Based LQG Regulators with
Spatially Invariant or Localized Dynamical and Measurement
Modelsa
We evaluated the performance with one turbulent layer in
frozen flow with a wind of 10 m/s on x-direction.
Table 6.
Zonal Models with Localized Resultant AR1 and AR2 Dynamical
Models and Their Measurement Equationab
Regulator Names
Associated Models
Lazy SA-LQG (resultant
AR1)
LQG-KF + MAP (resultant
AR2)
The spatial sampling of the zonal basis used for the
turbulent phase model can be chosen independently of the
WFS spatial resolution.
The left column gives the labels used for the associated
LQG regulators.
Table 7.
Atmosphere Configurations Used in End-to-End Simulations, from
Pseudo-Boiling to Frozen Flow Behaviorab
On the graphical representations, blue corresponds to the
first layer, green to the second and purple to the third
layer.
Only one layer is considered in pure frozen flow
configuration.
Table 8.
Strehl Ratio at 1.654 µm of Zonal Based LQG Regulators,
Compared to a Standard Integral Action Regulator for Five
Different Atmosphere Behaviors for a VLT NAOS-Like Casea
The numerical accuracy of the Strehl ratio calculations is
under 0.05 point at the science camera wavelength.
Table 9.
Performance of the Lazy SA-LQG and LQG-KF + MAP Resultant AR2
Regulators at 1.654 µm without Model Error, and with Weak and
Strong Errors on Wind Profiles Priors Used for Model
Computationsabc
Regulator
Atmosphere
No Error
Weak Error Std Dev
Strong Error Std Dev
Lazy SA-LQG
(resultant AR1)
Pseudo-Boiling Mainly Frozen
Flow
52.5%55.3%
LQG-KF + MAP
(resultant AR2)
Pseudo-Boiling Mainly Frozen
Flow
53.8%55.9%
The value of the standard deviation is also indicated. The
numerical accuracy of the Strehl ratio is under 0.05
points at the science camera wavelength.
The two different atmosphere configurations can be found in
Table 7.
The linear sampling of the zonal basis is two points per
actuator pitch.
Table 10.
Atmosphere Parameters for the LEO Satellite Tracking End-to-End
Simulationsab
Layers
Fractional Energy (%)
Altitude (km)
Wind Speed (m/s)
Wind Direction (°)
1
0.45
0
10
60
2
0.1
2
19.60
0
3
0.125
5
49.02
0
4
0.125
7
68.63
0
5
0.15
10
98.04
0
6
0.05
12
117.64
0
The $C_n^2$ profile is adapted from
the six-layer atmosphere considered in [35].
Layers two to six have wind speeds that are the projection
of the satellite speed on their altitudes.
Table 11.
Strehl Ratio at 800 nm for Several LQG Regulators with Zonal or
Modal Basis Dynamical Models Compared to a Standard Integral
Action Regulatora
The LEO tracking atmosphere behavior is described in
Table 10.
Table 12.
Atmosphere Parameter for Zonal LQG Regulators with a Multilayer
Reconstruction with a Reduced Number of Layersa
Controller
Fractional Energy (%)
Wind Speed (m/s)
Wind Direction (°)
LQG-KF + MAP 2-layer AR1
[0.45, 0.55]
[10, 72.54]
[60, 0]
LQG-KF + MAP 4-layer AR1
[0.45, 0.1, 0.25, 0.2]
[10, 19.60, 59.37,
103.18]
[60, 0, 0, 0]
The wind speeds and the $C_n^2$ profiles are computed
with the atmosphere parameters from Table 10.
Tables (12)
Table 1.
VLT-NAOS-Like AO Parameters
Diameter
8 m
Central occultation
1 m
DM cartesian grid (Fried
geometry)
actuators with 185
valid
Gaussian influence function
Coupling factor 0.3
Shack–Hartmann (squared
subaperture grid)
sub-apertures with
152 valid
AO loop frequency
500 Hz
0.55 µm
1.654 µm
Table 2.
Pseudo-Boiling Atmosphere Dynamical Parameters for End-to-End
Simulationsa
Blue corresponds to the first layer, green to the second
layer, and purple to the third layer.
Table 3.
Strehl Ratio at 1.654 µm of Zonal-Based LQG Regulator with a
Boiling AR1 Dynamical Model, Using Different Sampling of the
Zonal Basis
Linear sampling
(points/actuator pitch)
1
2
3
4
LQG-KF AR1 boiling
42.2%
50.9%
51.7%
52.1%
Table 4.
Zonal LQG Regulators According to Associated Modelsab
Regulator Names
Associated Models
(a)
LQG-DKF multilayer AR1
(b)
LQG-KF multilayer AR1 OS
(c)
LQG-KF multilayer AR1
(d)
LQG-KF + MAP multilayer
AR1
All turbulence models are of AR1 type in each layer,
oversampling is two linear points per actuator.
To ease reading of the table, here are a few short
descriptions: (a): spatially invariant
models based on convolution kernels (that is, the DKF case
based on the infinite pupil hypothesis) where the
coefficient $\alpha \lt
1$ insures
stability; (b): same oversampling for
phase and measurements in matrix form (finite pupil) and
without edge compensation; (c): any
sampling of the phase, but non-oversampled measurements,
in matrix form (finite pupil) without edge compensation;
and (d): any sampling of the phase, but
non-oversampled measurements, in matrix form (finite
pupil) with edge compensation using MAP estimation.
Table 5.
Strehl Ratio at 1.654 µm of Zonal-Based LQG Regulators with
Spatially Invariant or Localized Dynamical and Measurement
Modelsa
We evaluated the performance with one turbulent layer in
frozen flow with a wind of 10 m/s on x-direction.
Table 6.
Zonal Models with Localized Resultant AR1 and AR2 Dynamical
Models and Their Measurement Equationab
Regulator Names
Associated Models
Lazy SA-LQG (resultant
AR1)
LQG-KF + MAP (resultant
AR2)
The spatial sampling of the zonal basis used for the
turbulent phase model can be chosen independently of the
WFS spatial resolution.
The left column gives the labels used for the associated
LQG regulators.
Table 7.
Atmosphere Configurations Used in End-to-End Simulations, from
Pseudo-Boiling to Frozen Flow Behaviorab
On the graphical representations, blue corresponds to the
first layer, green to the second and purple to the third
layer.
Only one layer is considered in pure frozen flow
configuration.
Table 8.
Strehl Ratio at 1.654 µm of Zonal Based LQG Regulators,
Compared to a Standard Integral Action Regulator for Five
Different Atmosphere Behaviors for a VLT NAOS-Like Casea
The numerical accuracy of the Strehl ratio calculations is
under 0.05 point at the science camera wavelength.
Table 9.
Performance of the Lazy SA-LQG and LQG-KF + MAP Resultant AR2
Regulators at 1.654 µm without Model Error, and with Weak and
Strong Errors on Wind Profiles Priors Used for Model
Computationsabc
Regulator
Atmosphere
No Error
Weak Error Std Dev
Strong Error Std Dev
Lazy SA-LQG
(resultant AR1)
Pseudo-Boiling Mainly Frozen
Flow
52.5%55.3%
LQG-KF + MAP
(resultant AR2)
Pseudo-Boiling Mainly Frozen
Flow
53.8%55.9%
The value of the standard deviation is also indicated. The
numerical accuracy of the Strehl ratio is under 0.05
points at the science camera wavelength.
The two different atmosphere configurations can be found in
Table 7.
The linear sampling of the zonal basis is two points per
actuator pitch.
Table 10.
Atmosphere Parameters for the LEO Satellite Tracking End-to-End
Simulationsab
Layers
Fractional Energy (%)
Altitude (km)
Wind Speed (m/s)
Wind Direction (°)
1
0.45
0
10
60
2
0.1
2
19.60
0
3
0.125
5
49.02
0
4
0.125
7
68.63
0
5
0.15
10
98.04
0
6
0.05
12
117.64
0
The $C_n^2$ profile is adapted from
the six-layer atmosphere considered in [35].
Layers two to six have wind speeds that are the projection
of the satellite speed on their altitudes.
Table 11.
Strehl Ratio at 800 nm for Several LQG Regulators with Zonal or
Modal Basis Dynamical Models Compared to a Standard Integral
Action Regulatora
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