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# Infinite impulse response modal filtering in visible adaptive optics

G. Agapitoa  C. Arcidiaconob  F. Quirós-Pachecoa  A. Puglisia  S. Espositoa
aOsservatorio Astrofisico di Arcetri  Largo E. Fermi 5  Firenze  Italy;
bOsservatorio Astronomico di Bologna  Via Ranzani 1  Bologna  Italy

###### Abstract

Diffraction limited resolution adaptive optics (AO) correction in visible
wavelengths requires a high performance control. In this paper we investigate
infinite impulse response filters that optimize the wavefront correction: we
tested these algorithms through full numerical simulations of a single-
conjugate AO system comprising an adaptive secondary mirror with 1127
actuators and a pyramid wavefront sensor (WFS). The actual practicability of
the algorithms depends on both robustness and knowledge of the real system:
errors in the system model may even worsen the performance. In particular we
checked the robustness of the algorithms in different conditions, proving that
the proposed method can reject both disturbance and calibration errors.

###### keywords:

Adaptive optics, Pyramid wavefront sensor, telescope vibrations, adaptive
secondary mirror, modal control optimization

##  1 Introduction

Modern 8m telescopes are able to provide up to 90% Strehl Ratio (SR) on high
flux regime in H band (Esposito _et al._[1]), and reach a maximum resolution
of about 30mas (FLAO@LBT at the J band or the Hubble Space Telescope at the U
band or using lucky imaging technique on large telescope at R band Law _et
al._[2]). Since the maximum achievable resolution is fixed by diffraction
retrieving Point Spread Function (PSF), which Full Width at Half Maximum
(FWHM) is $\lambda/D$, where $\lambda$ is the imaged wavelength and $D$ the
telescope diameter. 8-m class telescope could improve the maximum achievable
resolution making images at shorter wavelenghts, such as V band, which could
provide a 12mas resolution. However behind these numbers is hidden a
limitation: an Adaptive Optics (AO) system for visible wavelengths has tighter
requirements compared to an near infra-red one. In fact shorter wavelengths
correspond to smaller Fried’s parameters, $r_{0}$, hence more degrees of
correction and low measurement noise values are required to obtain a good
correction.
Optimized control techniques such as a Modal Control Optimization[3],
observer-based and model-based[4, 5, 6, 7, 8, 9, 10] approaches can increase
the performance of an AO system in comparison to an integrator controller.
These methods need to work in realistic conditions, _i.e._ without an accurate
knowledge of the real system and with complex disturbances like calibration
errors and telescope vibrations. In particular, Modal Control Optimization has
only one degree of freedom per mode, that is the integrator gain, so it cannot
efficiently reject disturbance condition such as telescope vibrations, while
the other methods rely on the accuracy of the system model.
For these reasons, starting from the work of Dessenne _et al._[11], we have
chosen to design a method that is data driven, so it relies as little as
possible on the system model, and can have many degrees of freedoms as to
adapt itself to the disturbances.
The remainder of this paper is as follows. Sec.2 introduces the Infinite
Impulse Response (IIR) filter based control along with other controllers that
we will use in the performance evaluation as comparison. Sec.3 briefly
describes the AO system considered in our simulations. Finally in Sec.4 the
controllers are analyzed through numerical simulations from both performance
and robustness viewpoint.

##  2 AO Control Systems

Figure 1: Adaptive Optics Control loop.

The AO system control loop considered in this work is represented in Fig. 1.
The controller gets the modal measurements as input and gives the modal
commands as output. The most used controller in AO systems is the integrator,
so we have chosen it to make a comparison that will be presented in the next
sections.

###  2.1 IIR filter based control

To determine the optimal control expressed as an IIR filter we realized an
algorithm that minimized the residual variance for each mode $i$ as a function
of the IIR filter parameters:

$[\hat{a}_{i},\hat{b}_{i}]=\min_{[a_{i},b_{i}]}J_{i}\,,$ (1)

where $a_{i}$ and $b_{i}$ are two vectors of the IIR filter denominator and
numerator coefficients:

$C_{i}(z)=\frac{\sum\limits_{j=1}^{N_{b}}b_{i}(j)z^{-j}}{\sum\limits_{l=1}^{N_{a}}a_{i}(l)z^{-l}}\,,$
(2)

where $N_{a}$ and $N_{b}$ are the order of respectively the denominator and
numerator. When the modes are orthonormal, minimizing the sum of the residual
modal variances is equivalent to minimizing the residual phase variance in the
direction of the Natural Guide Star (NGS), which is the goal of the control
system studied in this paper. The cost, that is the measured residual modal
variance of the i-_th_ mode $J_{i}$, is:

$J_{i}=\sum\limits_{f=\frac{T}{n}}^{\frac{T}{2}}\Phi_{i}^{meas}(\omega)=\sum\limits_{f=\frac{T}{n}}^{\frac{T}{2}}\|W(z)\|^{2}\Phi_{i}^{ol}(\omega)\,,$
(3)

where:

$W(z)=\frac{1}{1+\mathrm{C}(z)\mathrm{WFS}(z)\mathrm{DM}(z)}.$ (4)

$\omega$ is the discrete frequency, $n$ is the number of points of the
discrete signals, $T$ is the sampling period, $\Phi_{i}^{meas}(\omega)$ is the
Power Spectral Density (PSD) of the closed loop i-_th_ modal coefficient
determined from the WFS measurements, and $\Phi_{i}^{ol}(\omega)$ is the PSD
of the open loop i-_th_ modal coefficient. The open loop measurement $o_{i}$
would be the input signal of the AO system without the control feedback, and
we reconstruct it as the sum of the closed loop measurement $s_{i}$ and
command $c_{i}$. At iteration $k$, taking into account the delay $d$, it can
be written as:

$o_{i}(k)=s_{i}(k)+c_{i}(k-d)\,.$ (5)

It can be shown (see Dessenne _et al._[11]) that:

$s_{i}(k)=W(z)o_{i}(k)\,,$ (6)

so that the relationship shown in Eq.3 is demonstrated.
This algorithm must verify the stability of the loop during the minimization
of the cost $J_{i}$, and it must discard the controller that do not stabilize
the loop. In fact, an unstable controller could produce the minimum cost, but
it could not be implemented.
To take into account the modelling errors, like pupil shifts or approximations
of DM and WFS models, a stability constrain is not enough. Therefore the
controller must guarantee some robustness, so small variations of the model
parameters would not drive the controller to instability. Thus we impose
$\max\|{\mu}\left(\chi_{i}(z)\right)\|^{2}\leq\eta$, where $\mu(\cdot)$ gives
the roots of the polynomial, $0<\eta<1$, and $\chi_{i}(z)=1+C_{i}(z)P(z)$ is
the characteristic polynomial, where $P(z)$ is the AO system model:

$P(z)=\mathrm{WFS}(z)\mathrm{DM}(z)\,,$ (7)

we have simply assumed that $\mathrm{WFS}(z)=z^{-1}$ and
$\mathrm{DM}(z)=z^{-1}$, in order to model a simple AO loop with two frames
delay.
Note that $\eta$ gives a measurement of the stability margin since if $\eta$
is equal to 1 only stability is guaranteed, while smaller values of $\eta$
correspond to a greater distance from instability, _i.e._ a greater
robustness.
In this work we choose $\eta=0.8$, which corresponds to a gain margin of 0.2.
Hence the algorithm can be summarized as:

  * •

Acquire with a closed loop the mode measurement $s_{i}$ and commands $c_{i}$.

  * •

Determine $\Phi_{i}^{ol}(f)$.

  * •

Choose $N_{a}$ and $N_{b}$ and the starting IIR filter parameters $a_{i}(0)$,
$b_{i}(0)$.

  * •

Search the combination of IIR filter parameters $\hat{a}_{i}$, $\hat{b}_{i}$
which minimizes $J_{i}$ and satisfies
$\max\|{\mu}\left(\chi_{i}(z)\right)\|^{2}\leq\eta$.

In this work, the minimization of $J_{i}$ is based on the downhill simplex
method of Nelder & Mead [12], and we opted for $N_{a}=N_{b}=3$ in case of
turbulence and 2 coefficients more for each vibration. So, in the next
simulations, Tip and Tilt have $N_{a}=N_{b}=7$ while the other modes have
$N_{a}=N_{b}=3$, because, as presented in Sec.3, Tip and Tilt are affected by
two vibrations.

###  2.2 Modal Gain Integrator

The Transfer Function of the modal integrator is:

$C(z)=\frac{g_{i}}{1-z^{-1}}$ (8)

where $g_{i}$ is the integrator gain of the _i_ -th mode, and $z$ is the
variable of the Z-transform.
In this work, we refer as integrator the particular case in which all the
gains are equal, and as Optimized Modal Gain Integrator[3] (OMGI) the
particular case in which each modal gain is optimized. For both cases we
optimize the gain: in the integrator case we optimized the global gain by a
trial and error procedure _i.e._ we ran many simulations with different gains
and then we chose the one that produced the best results in terms of Strehl
Ratio (SR); instead, in case of OMGI, we determine the gain following the same
algorithm presented in the previous section, but instead of minimizing the
cost $J_{i}$ as a function of the IIR filter parameters, we minimized it as
function of the integrator gain $g_{i}$:

$\hat{g}_{i}=\min_{g_{i}}J_{i}\,,$ (9)

since the integrator controller is an IIR filter with $N_{a}=2$ $N_{b}=1$ and
a single variable parameter, the gain $g$.
An example of IIR and OMGI Rejection Transfer Functions (RTF) $W$ is shown in
Fig.2. Note that the OMGI give better rejection at the lowest frequencies, but
the bandwidth rapidly decreases with the mode order. The integrator controller
bandwidth is in direct proportion to the peak at high frequencies, and this
peak increase the residual phase due to the measurement noise. Hence, the high
order modes, that are more affected by noise, have a lower bandwidth. Instead
the IIR filter gives a bandwidth greater than $20\,\mathrm{Hz}$ for all the
modes with low peaks at high frequencies.

Figure 2: RTFs for various modes with the IIR filter based control with
$N_{a}=N_{b}=3$ (left), and the OMGI (right). This RTFs are determined for the
case magnitude 12.5 without vibrations nor pupil shifts. All the other
parameters may be found in Tab.1.

##  3 AO system and simulator

The AO system analyzed in this work is based on a pyramid wavefront sensor
(WFS) with tilt modulation (Ragazzoni[14]) and on an adaptive secondary mirror
(ASM) (Salinari _et al._ [15]). We chose to work with an 8m class telescope,
an ASM with 1127 actuators, and a WFS with 40 $\times$ 40 sub-apertures. As
detector of the WFS we considered an ANDOR iXon X3 897 camera[16] with Read
Out Noise (RON), Excess Noise, and Clock Induced Charge (CIC) (Daigle _et
al._[17]) values shown in Tab.1. All simulations reported hereafter rely on
the End-to-End numerical simulator used to perform the performance analysis
and optimization of the First Light Adaptive Optics (FLAO) system of the LBT
telescope [18].
The atmospheric turbulence was simulated with a set of phase screens with Von-
Karman statistics that were displaced in front of the telescope pupil to
emulate the time-evolving turbulence according to the Taylor hypothesis.
Pupil shifts due to the sensor optical misalignments are a common calibration
error. We will evaluate in our simulations the robustness of the controller to
this kind of errors. The pupil shifts considered in the simulations have an
amplitude of $0.1\,\mathrm{pixel}$ in both _x_ and _y_ directions.
We add in the simulations, as external disturbance, telescope vibrations.
Vibrations may arise from many different situations, e.g. telescope
orientation, telescope tracking errors, and wind shaking. In particular, since
vibrations cause displacements of the image they have a major impact on so-
called tip/tilt modes. Hence, we chose to introduce Tip/Tilt vibrations with
central frequencies of $13\,\mathrm{Hz}$ and $22\,\mathrm{Hz}$, an RMS of
$20\,\mathrm{mas}$ each, and a damping ratio of $0.01$.

Telescope
---
Effective diameter ($D$) | $8\,\mathrm{m}$
Central obstruction | $0.138\,\mathrm{D}$
Pyramid WFS
Sensing wavelength ($\lambda$) | $0.75\,\mathrm{\mu m}$
Number of sub-apertures | $40\times 40$
Tilt modulation radius | $2.0\frac{\lambda}{D}$
Total average transmission | $0.41$
RON (electrons per pixel) | $0.06\,\mathrm{e^{-}RMS}$
Clock Induced Charge | $0.005\,\mathrm{e^{-}/pixel/frame}$
Excess Noise | $\sqrt{2}$
Guide star
Mag. zero point (Johnson [13]) | $1.76\,10^{-8}\,\mathrm{J/s/m^{2}/\mu m}$

ASM
---
Modes (Karhunen-Loève) | $1127$
Turbulence
Outer scale ($L_{0}$) | $25\,\mathrm{m}$
Mean wind speed | $10.6\,\mathrm{m/s}$
Telescope Vibration
Frequencies | $13,22\,\mathrm{Hz}$
Standard deviation | $20\,\mathrm{mas}$
Damping ratio | $0.01$
Loop paramenters
Sampling frequency | $1000\,\mathrm{Hz}$
Delay | 2 frames

Table 1: Summary of the simulation parameters.

##  4 Simulations

Firstly we will check the performance of the integrator with turbulence and
without pupil shifts nor vibrations. We chose to run all the simulations in
between magnitude 8.5 and 12.5. All the simulation parameters are listed in
Tab.1. The gain is determined with the method described in Sec.2.2. In Fig.3
and Tab.2 are shown the results. As we expect the performance of the system
rapidly decreases at the shortest wavelengths. Moreover the decrease is
greater in the visible wavelengths, in fact, while at K band from magnitude
$8.5$ to $12.5$ less than $8\%$ of SR is lost, at V band the difference is
about $40\%$.

Figure 3: Summary of simulation results for the integrator controller in case of turbulence without pupil shifts nor vibrations. Integrator gain is determined as described in Sec.2.2. Seeing 0.8′′, see Table 1 for the other parameters. band | V | R | H | K
---|---|---|---|---
magnitude |  Strehl Ratio $\%$
8.5 | 54.8 | 73.4 | 94.5 | 96.8
9.5 | 50.1 | 70.1 | 93.7 | 96.4
10.5 | 42.4 | 64.3 | 92.2 | 95.5
11.5 | 30.7 | 54.4 | 89.5 | 93.9
12.5 | 14.9 | 37.1 | 83.3 | 90.1
Table 2: Summary of simulation results for the integrator controller.
Integrator gain is determined as described in Sec.2.2. Seeing 0.8′′, see Table
1 for the other parameters.

In the next simulations we considered only these two magnitudes, $8.5$ and
$12.5$, to make a comparison between all the controllers. We will compare the
controller in 4 difference observation conditions: the first one is with
turbulence and without pupil shifts nor vibrations, then second and third case
with turbulence and with pupil shifts or vibrations, respectively, and the
last one with all the disturbances included.
The controllers parameters are optimized for each condition. The gain found
for the integrator and the minimum, maximum and mean gain of the OMGI are
shown in Tab.3. Regarding the IIR filter Fig.4 shown, as an example, two RTFs
for the tip/tilt modes in two different cases: on the left, with only
turbulence and, on the right, with turbulence, pupil shifts, and vibrations.
Note that in the first case we design a filter with $N_{a}=N_{b}=3$, while
with all the disturbances $N_{a}=N_{b}=7$.

Controller | integrator | OMGI
---|---|---
magnitude | g | g min | g max | g mean
Turbulence
8.5 | 0.6 | 0.23 | 0.88 | 0.50
12.5 | 0.3 | 0.01 | 0.85 | 0.20
Turbulence + Pupil shifts
8.5 | 0.6 | 0.08 | 0.92 | 0.33
12.5 | 0.3 | 0.03 | 0.85 | 0.23
Turbulence + Vibrations
8.5 | 0.8 | 0.23 | 0.92 | 0.50
12.5 | 0.4 | 0.01 | 0.93 | 0.22
Turbulence + Pupil shifts + Vibration
8.5 | 0.7 | 0.08 | 0.92 | 0.33
12.5 | 0.4 | 0.01 | 0.92 | 0.23
Table 3: Summary of simulation optimal gain. Comparison between Integrator and
OMGI. Seeing 0.8′′.

Figure 4: Rejection Transfer Function (RTF) for the Tip mode with an IIR
filter based control with $N_{a}=N_{b}=3$ (left), and with $N_{a}=N_{b}=7$
(right). Note that the order 7 is used to reject both vibrations at
$13\,\mathrm{}$ and $22\,\mathrm{Hz}$ and atmospheric turbulence. These RTFs
are determined for the case magnitude 8.5. All the other parameters may be
found in Tab.1

All the results of the simulations are summarized in Tab.4 in term of SR for
all the considered observing conditions. Fig.5 shows the SR at different
wavelengths at magnitude 8.5, Fig.6 shows the corresponding results at
magnitude 12.5.

Figure 5: Summary of simulation results for turbulence, pupil shifts and
vibrations at magnitude 8.5. Comparison between Integrator, IIR filter based
control, and autogain. Seeing 0.8′′.

Figure 6: Summary of simulation results for turbulence, pupil shifts and vibrations at magnitude 12.5. Comparison between Integrator, IIR filter based control, and autogain. Seeing 0.8′′. band | V | R | H | K | V | R | H | K | V | R | H | K
---|---|---|---|---|---|---|---|---|---|---|---|---
Controller | integrator | IIR filter | OMGI
magnitude |  Strehl Ratio $\%$
Turbulence
8.5 | 54.8 | 73.4 | 94.5 | 96.8 | 59.7 | 76.7 | 95.2 | 97.2 | 56.2 | 74.4 | 94.7 | 97.0
12.5 | 14.9 | 37.1 | 83.3 | 90.1 | 34.1 | 57.5 | 90.4 | 94.4 | 27.2 | 51.1 | 88.4 | 93.3
Turbulence + Pupil shifts
8.5 | 53.3 | 72.4 | 94.2 | 96.7 | 56.3 | 74.5 | 94.7 | 97.0 | 52.1 | 71.5 | 94.0 | 96.5
12.5 | 9.3 | 28.6 | 79.1 | 87.5 | 32.5 | 56.1 | 90.0 | 94.2 | 24.5 | 48.3 | 87.5 | 92.7
Turbulence + Vibrations
8.5 | 23.0 | 40.8 | 81.6 | 88.9 | 56.8 | 74.7 | 94.8 | 97.0 | 32.2 | 52.9 | 88.0 | 93.0
12.5 | 2.2 | 8.4 | 45.3 | 60.5 | 38.5 | 61.2 | 91.4 | 95.0 | 12.9 | 31.0 | 78.5 | 87.0
Turbulence + Pupil shifts + Vibrations
8.5 | 22.1 | 39.7 | 81.1 | 88.6 | 52.1 | 71.5 | 94.0 | 96.5 | 29.2 | 50.1 | 87.1 | 92.4
12.5 | 1.9 | 7.6 | 43.1 | 58.3 | 27.1 | 50.9 | 88.3 | 93.2 | 11.1 | 28.4 | 76.9 | 86.0
Table 4: Summary of simulation results. Comparison between Integrator, IIR
filter based control, and OMGI. Seeing 0.8′′.

The integrator and the OMGI show similar results in case of turbulence and
pupil shifts, while the vibrations affect more the integrator. These results
is expected, because the OMGI has more degrees of freedoms, one for each mode,
with respect to the single degree of the integrator. In fact it can be shown
in Tab.3 that the OMGI maximum gain value is greater in case of vibrations, so
it has a greater bandwidth and can better reject the Tip/Tilt disturbances.
Note that in case of pupil shifts the gains are lower to guarantee greater
robustness. At the faintest magnitude the differences between the controller
are greater, because, as we expected, the IIR filters reject better the noise
without decreasing the bandwidth (see Sec.2.2 and Fig.2).
The IIR filter based control produces similar performance in all the tested
conditions: it can efficiently reject both calibration errors and
disturbances. In the worst case – turbulence, pupil shifts and vibrations –
the IIR filter at V band gives the same SR of the OMGI and about two times the
one of the integrator both in the condition without pupil shifts nor
vibrations.

##  5 Processing power estimate

In this section we compare the processing power required for the IIR filter
based control described above, compared to the integrator case. Referencing
the scheme in Fig.1, we can split the required processing power into three
major steps:

  1. 1.

Reconstruction matrix

  2. 2.

Controller

  3. 3.

Modes to command projection matrix

Steps 1 and 3 are common to the different control strategies, while step 2 is
where the techniques differ.
For the estimate, we suppose the following system parameters (partially
following Tab.1):

  * •

Number of controlled modes: 1127 (including tip-tilt)

  * •

Number of DM actuators: 1127

  * •

Number of slopes measured by the WFS: 2480

  * •

Loop frequency: 1000 Hz

  * •

System delay: 2 frames

Step 1 is a straightforward vector/matrix multiplication. The required
processing power is $N_{\mathrm{modes}}\times N_{\mathrm{slopes}}$ multiply-
accumulate (MAC, a basic operation employed by most Digital Signal Processors
- DSPs where two operands are multiplied and summed to an accumulator). In our
example it is about 2.8 Mega-MAC (MMAC). Step 3 is also a vector/matrix
multiplication, this time of dimension $N_{\mathrm{modes}}\times
N_{\mathrm{commands}}$. In our example this amounts to about 1.3 MMAC.
Assuming a total delay of 2 frames at 1 KHz, and further assuming that this
computation cannot take more than 1/10th of the available time in order to be
considered negligible, the processing power requirements for steps 1 and 3
combined is 20 GMAC/sec. This is comparable with existing systems, like the
LBT one[19] where the total processing power, distributed over 168 DSPs, is
about 25 GMAC/sec (only a fraction of this processing power is used on the LBT
system, since the reconstrucion matrices have smaller dimensions and the DSPs
also have more tasks to attend to). Both steps 1 and 3 can be efficiently
parallelized over such a large number of DSPs, since each row of the matrix
multiplication can be processed independently of the others.
For step 2, the integrator case needs to sum the measurement vector from the
previous frame to the current one. Additionally, a gain value must be
multiplied to each measurement value. This requires about 1 Kilo-MAC (KMAC),
which is completely negligible (about 1/4000th) when compared to the previous
numbers. The only effect (and a rather large one) is to prevent full
pipelining of the two matrix multiplications.
For the IIR filters, we take into account a filter whith order 7 for the tip-
tilt modes, and 3 for the other modes (as in Sec.2.1). With $N_{a}=7$ and
$N_{b}=7$, this requires only a few dozens MACs for the tip-tilt IIR filters,
and about 6 KMAC for the other modes. While this number is 6 times bigger than
the integrator, it is again negligible in the context of the processing power
required by steps 1 and 3. Furthermore, no additional requirements are posed
on the other processing steps, except for the full separation of the two
matrix multiplications, which was already required for the integrator with
modal gain.
We can therefore conclude that the IIR filtering strategy outlined in this
article can readily be implemented on existing or future AO systems with
little to no penalty in terms of loop delay.

##  6 Conclusions

In this paper we have shown that IIR filter based control is a control
solution that can deliver good performance even in difficult conditions. It is
a possible choice for AO systems requiring high performance and robustness in
case of calibration errors and telescope vibrations.
Moreover, thanks to the data driven approach, the presented method need only a
basic knowledge of the system, and the computational burden is focused in the
parameters optimization that can be made off-line. In fact, the computational
power requested to the Real Time Computer of the AO system will be negligible
in comparison to the integrator as we proved in Sec.5.
As further work, more investigation on the robustness, on the coefficient
$\eta$ and on the impact of IIR filter order $N_{a}$, $N_{b}$ on the
performance must be done.

###### Acknowledgements.

This study was supported by the TECNO INAF 2009 grant from the italian
Istituto Nazionale di Astrofisica (INAF).

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