Application of complex-valued FXLMS adaptive filter to Fourier basis control of adaptive optics

Abstract

In this paper, the Filtered-X Least Mean Square (FXLMS) adaptive filter with bias integration technique is applied to an adaptive optics system where the Discrete Fourier Transform is used to project the measured phase onto the Fourier basis for modal control. The control law is applied in the complex-valued coefficient space and the FXLMS algorithm is modified accordingly for the complex-valued control. Numerical analysis is conducted for a feedback loop of a single Fourier mode in the presence of a disturbance representing a frozen flow atmospheric turbulence. The performance is compared with a Kalman estimator based control law proposed in the literature called Predictive Fourier Control (PFC). The proposed method demonstrated a similar performance for a stationary disturbance and improved performance for a slowly drifting disturbance. Whereas the performance of the PFC is very sensitive to the accuracy of the identification of the disturbance, the proposed method does not require such an explicit identification and produces minimum error for the given disturbance.

Full text

Page 1

2011 American Control Conference
on O'Farrell Street, San Francisco, CA, USA
June 29 - July 01, 2011

Abstract— In this paper, the Filtered-X Least Mean Square
(FXLMS) adaptive filter with bias integration technique is
applied to an adaptive optics system where the Discrete Fourier
Transform is used to project the measured phase onto the
Fourier basis for modal control. The control law is applied in the
complex-valued coefficient space and the FXLMS algorithm is
modified accordingly for the complex-valued control. Numerical
analysis is conducted for a feedback loop of a single Fourier
mode in the presence of a disturbance representing a frozen flow
atmospheric turbulence. The performance is compared with a
Kalman estimator based control law proposed in the literature
called Predictive Fourier Control (PFC). The proposed method
demonstrated a similar performance for a stationary
disturbance and improved performance for a slowly drifting
disturbance. Whereas the performance of the PFC is very
sensitive to the accuracy of the identification of the disturbance,
the proposed method does not require such an explicit
identification and produces minimum error for the given
disturbance.

Index Terms— Adaptive Optics, Adaptive Filter, Filtered-X
LMS, Predictive Fourier Control.

I. INTRODUCTION
DAPTIVE Optics (AO) refers to optical control systems
used for advanced telescopes or laser propagation
systems where the phase aberration of the optical waves is
measured by wave-front sensors (WFSs) and corrected by
deformable mirrors (DMs). In recent years, control techniques
that are more advanced than a simple classical control have
been proposed [1-10]. Due to the increased size of the arrays
in WFSs and DMs, the computational cost of control schemes
has become important in large AO applications, and the
two-dimensional Discrete Fourier Transform has been
introduced as a computationally efficient way for
reconstructing the phase from the phase gradient measurement
by Shack-Hartmann WFSs [11], [12].
Poyneer et al. pointed out another benefit of Fourier
decomposition for a class of disturbance and proposed a
control scheme called Predictive Fourier Control (PFC) [9]. In
the Fourier coefficient space, an atmospheric turbulence that
obeys the frozen flow hypothesis exhibits distinctive peaks in
the temporal frequency spectrum corresponding to each
turbulence layer, and the PFC takes advantage of this
observation for efficient modeling of the disturbance in the
Fourier coefficient space. The identified disturbance is used to
construct a Kalman filter which performs one-step-ahead
prediction of the disturbance state to suppress the disturbance
[9], [13]. Because the Fourier Transform is spatially
orthogonal, the AO system is uncoupled in the Fourier domain
and computational load for the control law is significantly
reduced compared with full-state Multi-Input Multi-Output
(MIMO) control laws. The PFC is optimized for the current
temporal dynamics of the disturbance and it requires
repetition of disturbance identification and computation of the
controller parameters.
The purpose of this paper is to introduce an adaptive
controller for the DFT based modal AO control system
referred to here as Fourier basis AO control. Because of its
adaptive nature, the proposed Filtered-X Least Mean Square
(FXLMS) method can eliminate the need for the explicit
disturbance identification and controller update required by
the PFC. The development is based on the assumptions that
the controller is applied on the completely uncoupled Fourier
modes and that a class of disturbance exists which exhibits
distinctive peaks in the Fourier domain. While whether these
assumptions are appropriate or not is an important question in
practice, it is out of scope of this paper and is not addressed.
Given the uncoupling assumption,
Single-Output (SISO) loop for an arbitrary mode is used to
investigate the behavior of the control law. A Fourier basis
AO control system is described in the next section followed by
the description of the proposed control law. A brief
description of the PFC is also provided. Numerical simulation
results are presented in Section IV and Section V provides a
discussion of the results.
a Single-Input
II. FOURIER BASIS CONTROL OF ADAPTIVE OPTICS
Fig. 1 shows the block diagram of a Fourier basis AO
control system considered in this study.




e

m
v
d
z
Wavefront
Sensor
Cntrl.
Law
Deformable
Mirror
y
u
s
y
G
Controller
u
F
c
e
c
0



Fig. 1. Block diagram of a Fourier basis Adaptive Optics control system.
Application of Complex-Valued FXLMS Adaptive
Filter to Fourier Basis Control of Adaptive Optics
Masaki Nagashima and Brij Agrawal
A
978-1-4577-0079-8/11/$26.00 ©2011 AACC2939

Page 2

The dynamics of the DM and the WFS are all assumed to be
negligible except for the pure time delay denoted by d, and the
relationship from the actuator command u to the sensor output
ys, which can either be phase or phase gradient, is represented
by a constant influence matrix M. The disturbance is denoted
by  , and e and vm are the error and measurement noise,
respectively. For an AO system with m sensor output signals
and n actuator input signals, M is an m × n matrix and the
sensor output and the actuator command are represented by
vectors in
 and
 , respectively.
Commonly, the sensor output is transformed to a different
vector space by a matrix F whose dimension is k × m to apply
a control law. The controller output is then converted to the
actual command for the actuators by a matrix G. If the rank of
F is k and it is smaller than that of M, the path from u to ys
which is FM can be diagonalized by the pseudo inverse of FM,
i.e.,

IFMFM

, where † denotes the pseudo inverse and I
is an identity matrix of dimension k × k. Thus, choosing 
as G will uncouple the path from uc to ec, and a SISO
controller can be applied individually on each component of
ec. For example, let
MF 
, then
whose rank is n < = m and the open loop path is
.)(IMMMMFMG


If the rank of M is r < n, then modal reduction techniques
such as the singular value decomposition can be applied. For
instance, suppose
UF


, where U is the transpose of the
first r column vectors of U which is obtained by the singular
value decomposition
M

orthogonality of U and V, it can be shown that
))((VUUVUU FMG

  
usually cannot observe piston mode whereas the DM may be
able to produce it. In this case, the rank of M cannot exceed
n-1 even for M with n < m, and the dimension of controller
vector space needs to be reduced to less than the number of the
actuators in the DM to achieve the uncoupling of the system.
In the Fourier basis control, F represents the transform of
the sensor output to the Fourier coefficients of the phase. If the
sensor output is the phase, the two dimensional Discrete
Fourier Transform can directly be applied and F becomes a
matrix equivalent of the transform. When the sensor output is
not the phase but the gradient of the phase as is usually the
case with the wavefront sensors available today, the
techniques to obtain the Fourier coefficients from the gradient
information presented in [12] can be used. Due to the limited
space, the details are omitted but interested readers are
referred to the literature. Also, practical issues such as
spillover effect of the DM or spatial aliasing of the aberration
are not considered here. A detailed discussion on the latter can
be found in [14]. In general, choosing a DM that has a
geometry and specifications suitable for this particular control
method is important for successful application. In the
following, it is assumed the above mentioned techniques are
applied and the uncoupling of the open loop path is achieved
to the sufficient extent.
One of the benefits of applying a controller in the Fourier
mn
†
†
FM
†
IMM(G

††
)
for M
†††
T
.
T
VU
 
Because of the
.
†
I

TTTT
Wavefront sensors
coefficient space is that a class of disturbance exhibits
distinctive temporal frequency peaks which can be
represented by a low order model with sufficiently high
accuracy [9]. This result, however, is obtained for square-grid
measurement points whereas hexagonal array geometry is
often found in AO systems. Thus, it is of interest to investigate
if the temporal characteristics of the frozen flow disturbance
in the Fourier coefficient space shown in [9] are preserved
even in the case of hexagonal geometry.
Suppose a very simple cosine phase function moving with
velocities
spatially sampled function is given as follows.



x v and
y v in x and y direction, respectively, whose
yyxx
tn
N
l
m
M
k
jtn
N
l
m
M
k
j
NdvlMdv

k
eetnm
'22
2
1
),,(
'2'2'2'2





























(1)

Here, M and N are the number of sample points and dx and dy
are their intervals in the x and y directions, respectively. For
simplicity, the aperture of the hexagonal array is assumed to
be rectangular. The Fourier Transform of this function has a
temporal frequency
(2exp[kj

In a hexagonal array, the sampling points of the alternate
rows are shifted in the x direction by one half of the sampling
interval. Let this spatial shift be x
with respect to the interval be

the hexagonal array can be expressed as follows using the shift
property of DFT.



)(),,(
nm
])''t NdvlMdv
yyxx

.
 and its normalized value
.xm

Then, the DFT of
x d












1
2
1
2
),,(
N
n
N
l
j
M
m
M
k
j
hex
eetnmnctlk



(2)


1

1










)(1
)(
n
2
1
2
1
)(
2
2
) 1(
even
oddne
eeenc
m
M
k
m
M
k
j
njnj



(3)

The function c(n) alternately applies the shift in the x direction.
Substituting (3) into (2) yields





Here,
),,(tlk

denotes the Fourier transform of the phase on
the rectangular grid and

).,, ( '
2
1
),,(1
2
1
),,(
22
tlkeetlketlk
m
M
k
j
j
m
M
k
j
hex



















(4)








 
n

n
N
l
m
M
k
j
NM
m
nj
etnmetlk

2

2

11
),,(),,(
(5)

is
It can be seen from this result that the geometry change of the
),,(tlk

shifted in the l direction in the frequency domain.
2940

Page 3

measurement points introduces extra frequency components
along with the magnitude and phase changes, but the temporal
frequency
'(2 exp[ Ndvkj
xx

movement of the layer with velocities vx and vy is preserved in
Eq. (4) to be exploited for disturbance attenuation.
])'tMdvl
yy

caused by the
III. FOURIER COEFFICIENT SPACE CONTROLLER
When complete uncoupling of the system is achieved, the
MIMO system becomes a set of complex-valued SISO
systems in the Fourier coefficient space as shown in Fig. 2.

Controller




e

d
z
d
v
Disturbance
Model
m
v

0

y

Fig. 2. Diagram of the simulation model. All variables are complex-valued
and the disturbance and measurement noise represent those projected onto
the Fourier basis.

Here, the projections of the disturbance and reconstructed
phase onto the Fourier basis are denoted by  and y,
respectively. The vd is the white noise driving the disturbance
model and vm is the white measurement noise. White noise is
used here instead of other disturbance models in order to
evaluate the ideal performance of the PFC which is based on
this white noise assumption. Any delay in the actual physical
system is represented by an integer d.

A. Normalized FX LMS with Bias Integration
A block diagram of the normalized FXLMS adaptive filter
proposed in this paper, which augments an existing integral
control feedback loop, is shown in Fig. 3. The actuator and
sensor delays are represented by pure time delay d1 and d2.

Integral
Controller
Normalized
LMS




x
e
FIR
Sˆ
Sˆ



w
AF
u
u
2 d
z

m
v
0


y

1 d
z
x
B. I.
xb
sy

Fig. 3. Block diagram of normalized Filtered-X LMS adaptive filter
augmenting an existing integral control feedback loop. B. I. stands for bias
integration.

The control law of the adaptive filter loop is a Finite
Impulse Response (FIR) filter given by Eq. (6), whose input
signal and adaptive coefficients
expressed as vectors defined in Eq. (7) and Eq. (8).

)(kwi
called weights are
][][)()()()(
0
kkxkwikxkwku
T
bb
L
i
i AF
xw
 

(6)

The bias integration technique originally introduced by Yoon,
et al. [15], and modified by Corley, et al. [5], is applied here
for additional robustness, and the input and weight vectors are
augmented by the bias terms,


kxk,),(][


x

kwk,),(][
0


w

where xb is an arbitrary constant and wb (k) is the
corresponding weight. The input of the FIR filter, called
disturbance correlated signal, or reference signal, denoted by
x is the estimate of the disturbance to be canceled, which is
obtained as

T
T
)
b xLkx),(

(7)
bL
kwkw(),(
(8)
).(
][1
][
H
)()(][ˆ
S)()(
)(
)(
21
21
zu
zz
zzH

zezuzzezx
AF
dd
dd
AF



(9)

Here,
represents so-called secondary path dynamics, and
) 1(][

zzKzH
i
is the transfer function of the integral
controller with gain Ki. The filter output uAF is added to the
input of the integral controller and the filter adaptively
modifies the frequency spectrum of x such that the filter output
uAF cancels the disturbance observed in the error. The
Filtered-X algorithm takes into account the effect of the
secondary path dynamics from uAF(z) to ys(z) by filtering x[z]
with
],[ˆzS
i.e.,

[
][][ˆ
][
H


The normalized Filtered-X algorithm further processes this
signal
][kx
by the following normalization formula, where
 is a small constant to avoid division by zero.

][ˆzS
is the transfer function from uAF(z) to ys(z), which
].[
][1
]
)(
)(
21
21
z
zz
zzH
zzSz
dd
dd
xxx




(10)



x

 
][][
]
k
[
][
k
k
k
x
x
x
T
(11)

The weight vector is updated by the Least Mean Square
(LMS) algorithm, which is a steepest gradient descent method
to minimize the instantaneous square error at time k,
2)()(kekJ

[16]. The weight update formula for a
real-valued system is written using x''[k] as

[
2w

][][][
][
]
][] 1

[kkek
k
kJ
kkxwww
 




(12)

where  is the update gain and the normalization effect is
ignored. For a complex-valued system, the instantaneous error
2941

Page 4

is evaluated by
conjugate, and Eq. (12) is modified as follows [17].

] 1[kw

)()()(
*kekekJ

, where * denotes the complex
][][][kkek
*
xw
 

(13)

B. Predictive Fourier Controller
The Predictive Fourier Control is a Linear Time Invariant
(LTI) controller optimized for the identified disturbance and
measurement noise in the Fourier coefficient space. In the
following, only a brief description of the PFC is presented as a
reference. The details of the method can be found in [9].
First, the parameters of the following complex-valued
AutoRegressive (AR) disturbance model



k


k



L
k
k
L
k
z
zPzP
0
2
1
2
0
1
)()(


(14)

is identified from the temporal Power Spectrum Density
(PSD) of the disturbance. Each
Tf
kk

2} arg{

Hz, and this resonance frequency is
determined by computing the cross-correlation of a template
AR model and the disturbance PSD for all peaks whose
magnitudes exceed a certain threshold. The static dc
component is represented by k = 0, and L corresponds to the
number of layers in the disturbance moving in different
direction at various velocities. When two or more frequency
peaks are very close, it appears as a single peak and L becomes
less than the number of the layers.
Since this process only determines the phase of
magnitude of
k
 which represents the damping of the mode is
determined by an empirical formula
where fs is the sampling frequency. The magnitude parameter
2
k
 of each mode is obtained by fitting the corresponding AR
model to the disturbance PSD. The measurement noise power
2
m
 is estimated from the power of the disturbance at a very
high frequency range where no disturbance component is
expected to appear.
The obtained disturbance model is then converted to a state
space form to construct a Kalman estimator which performs
one-step-ahead prediction of the disturbance state whose
conjugate is used to cancel the disturbance. A steady state
optimal gain matrix for the Kalman estimator is obtained by
solving an Algebraic Riccati Equation (ARE) formulated with
the identified disturbance model and the measurement noise
power. Because this state space system is SISO, the resulting
controller can be expresses by the following transfer function.




k
qz1
)(zPk
has a resonance peak at
k
 , the
20//21
Skk
ff






L
k
k
z
p

zC
0
11
1
1
)(
(15)
IV. NUMERICAL SIMULATION
Numerical simulation of the system shown in Fig. 3 was
conducted with the delay d1 = d2=1 for a single feedback loop
representing a control loop for an arbitrary Fourier mode. This
is considered to be sufficient to demonstrate the qualitative
behavior of the proposed method for the given class of
disturbance. For the PFC, the controller replaces the integral
controller and adaptive filter is removed. A disturbance
representing a four-layer frozen flow atmospheric turbulence
with a static phase aberration was generated to imitate the data
shown in [9]. The parameters are shown in Table 1. The power
of the disturbance noise
are 10-8 and 10-6, respectively. The sample rate of the system is
1000Hz.

TABLE 1
PARAMETERS OF THE DISTURBANCE
d v and the measurement noise
m
v
Layers
0 1 2 3 4
k


0.999 0.996 0.998 0.994 0.997
kf (Hz)
0 -30 -10 15 50

A normalized FXLMS adaptive filter with bias integration
was designed for this simulation and the parameters are shown
in Table 2. The gain of the integral controller was determined
so that the control loop is stable with sufficient margin.

TABLE 2
ADAPTIVE FILTER CONTROLLER PARAMETERS
Number of
Weights


Integrator gain
Ki
bx
15 0.030 1.2 10-7 0.09 0.001

The disturbance model and the controller parameters for the
PFC were obtained as shown in Table 3. The hat ^ indicates
the value is an estimate. The measurement noise estimate
is 10-6.

TABLE 3
PREDICTIVE FOURIER CONTROLLER PARAMETERS
2ˆm

Layers
0 1 2 3 4
k
 ˆ

0.999 0.997 0.999 0.9985 0.9950
kfˆ(Hz)

(10-8)
0 -30 -10 15 50
2ˆk
0.379
0.172 0.100 0.044 2.801
k p
0.0516
+0.0062i
0.0830
+0.1119i
0.0225
+0.0128i
0.0157
+0.0043i
0.0327
-0.0004i
q = 0.2557 - 0.0387i

Fig. 4 shows the disturbance PSD and the identified AR
model with the estimated measurement noise. Unlike the
frequency spectrum of a real-valued signal, the frequency
2942

Page 5

spectrum of a complex-valued signal is not symmetric with
respect to the imaginary axis and both positive and negative
frequencies are shown in the plot.

-200 -150 -100 -500 50100 150200
10
-6
10
-5
10
-4
10
-3
10
-2
Hz
Power


Open Loop PSD
AR Model

Fig. 4. AR model of the open loop disturbance PSD (dot line) and the actual
disturbance PSD (solid line).

Fig. 5 shows the frequency responses of the PFC transfer
functions. The PFC controller obtained using the nominal
disturbance parameters in Table 1 is also shown for
comparison. The sensitivity function has a slight increase
around 70 Hz compared with that of the nominal PFC, which
is caused by the identification error of the pole magnitudes.

-200-1000100200
-30
-20
-10
0
10
20
30
40
Hz
dB


(a)
PFC
PFC Nominal Params.

-200-1000100 200
-40
-30
-20
-10
0
10
Hz
dB


(b)
PFC
PFC Nominal Params.

Fig. 5. PFC Frequency response. (a) PFC controller, (b) PFC feedback loop
sensitivity function.

The duration of the simulation was 15 seconds. The integral
controller of the proposed method was turned on at 2.0 s and
the adaptive filter was turned on at 2.1 s. For the PFC, the
controller was turned on at 2.0 s. Fig. 6 shows the frequency
spectra and the transient responses of the output y by the
proposed method and the PFC. The steady state mean square
error is computed from the output between 14 s and 15 s. The
proposed method produces a transient about 27 times larger
than that of the PFC due to the high gain of the integral
controller, but it converges about 9 times faster and the steady
state mean square error is 35 % lower than that of the PFC.
The steady state error of the PFC is higher due to the increase
of the frequency component around 70 Hz as shown in (a) of
Fig. 6, which is caused by the bump of the sensitivity function
shown in (b) of Fig. 5.
Fig. 7 shows the result by the PFC with the nominal
disturbance parameters. The error increase around 70 Hz is
now disappeared and the controller achieves a steady state
error lower than that of the proposed method in Fig. 6. The
proposed method, however, also reduced the error when the
number of weights was increased to 30. The steady state mean
square errors of both methods approach the measurement
noise level
.106

m



2

-200-150 -100-50050100 150200
10
-6
10
-4
10
-2
Hz
Power


(a)
No Cntl., MS = 12.2e-6
PFC, MS = 2.57e-6
AF, MS = 1.67e-6
02468101214
10
-4
10
-2
sec
|error|2


(b)
PFC
AF

Fig. 6. Disturbance attenuation by PFC (identified disturbance parameters)
and proposed method (15 weights): (a) Power Spectral Density of the output,
(b) Transients of the squared output.

-200-150-100-50050100150200
10
-6
10
-4
10
-2
Hz
Power


(a)
No Cntl., MS = 12.2e-6
PFC, MS = 1.15e-6
AF, MS = 1.13e-6
02468101214
10
-4
10
-2
sec
|error|2


(b)
PFC
AF

Fig. 7. Disturbance attenuation by PFC (nominal disturbance parameters)
and proposed method (30 weights): (a) Power Spectral Density of the output,
(b) Transients of the squared output.

Fig. 8 shows the effect of the number of weights on the
steady state mean square error of the proposed controller. The
error decreases as the number of weights increases until it
reaches the lower limit at about 30 weights.

10203040
Number of Weights
5060708090100
10
-6
10
-5
Power


No Control
PFC Nominal
PFC
Adative Filter

Fig. 8. Effect of the number of weights on steady state error.

The results so far addressed a disturbance with a stationary
frequency spectrum. The performances of the proposed
method with 30 weights and the PFC with the nominal
disturbance parameters were investigated for a disturbance
with drifting frequency peaks. The rate of change introduced
was 0.3 Hz/s in randomly selected positive or negative
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