James Steven Gibson,
Chi-Chao Chang,
and Brent L. Ellerbroek
J. S. Gibson (gibson@seas.ucla.edu) and C.-C. Chao (chichao@seas.ucla.edu) are with the Department of Mechanical and Aerospace Engineering, University of California, Los Angeles, California 90095-1597.
When this study was performed, B. L. Ellerbroek (bellerbroek@noao.edu) was with the Starfire Optical Range Directed Energy Directorate/Directed Energy Starfire, Kirtland Air Force Base, New Mexico 87117-5776;
he is now with the Gemini Observatory, Hilo, Hawaii 96720.
James Steven Gibson, Chi-Chao Chang, and Brent L. Ellerbroek, "Adaptive optics: wave-front correction by use of adaptive filtering and control," Appl. Opt. 39, 2525-2538 (2000)
A class of adaptive-optics problems is described in which phase
distortions caused by atmospheric turbulence are corrected by adaptive
wave-front reconstruction with a deformable mirror, i.e., the control
loop that drives the mirror adapts in real time to time-varying
atmospheric conditions, as opposed to the linear time-invariant control
loops used in conventional adaptive optics. The basic problem is
posed as an adaptive disturbance-rejection problem with many
channels. The solution given is an adaptive feedforward control
loop built around a multichannel adaptive lattice
filter. Simulation results are presented for a 1-m telescope with
both one-layer and two-layer atmospheric turbulence profiles. These
results demonstrate the significant improvement in imaging resolution
produced by the adaptive control loop compared with a classical linear
time-invariant control loop.
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Values of J(6001, 8000)
are normalized by J(6001, 8000) for no control. The
notation (VR) indicates that the actuator space was reparameterized
by the transformation V before the degrees of freedom were
reduced; the notation (RV) indicates that the degrees of freedom
were reduced before reparameterization by V. In all
cases, NL = 8 and E2 = E2Opt.
Table 4
Values of the Normalized Performance Index
J(6001, 8000) for LTI Lattice Filters (VR)
Type of Control
Number of Channels
γG = 0, γy = 0
γG = 0.05, γy = 0
γG = 0, γy = 0.05
γG = 0.05, γy = 0.05
Control Nv
Reference Disturbance Nr
No control
0
0
1.000
1.000
1.000
1.000
Lattice filters of order
NL = 2
69
69
0.096
0.111
0.131
0.142
NL = 3
69
69
0.075
0.093
0.115
0.128
NL = 4
69
69
0.070
0.089
0.111
0.124
NL = 8
69
69
0.066
0.086
0.107
0.120
NL = 8 and ê = z-1
e
69
69
0.062
0.063
0.101
0.101
Table 5
Values of
JE2(t1,
t2) and
J(t1,
t2) for E2 Chosen to
Optimize JE2(t1,
t2) over Different
Intervalsa
(t
1
, t
2
)
JE
2
(t
1
, t
2
)
JE
2
(1, 8000)
J(6001, 8000)
(1, 8000)
0.034
0.066
(3001, 3500)
0.025
0.060
0.078
(4001, 4500)
0.021
0.057
0.080
(7001, 7500)
0.022
0.059
0.076
The numbers of channels are
Nv = Nr = 69. The
lattice order is NL = 8. All cases
satisfy γG = γy = 0.
Tables (5)
Table 1
Definitions of Parameters for Vector Signals, Matrices,
and Filters
Parameter
Definition
Actuator space
ϕ
Uncorrected phase vector
η
Actuator-error vector
c
Actuator-command vector
∊
Error vector (not measured)
ψ
Uncorrected phase vector including the actuator error
Control space
v
Control-command vector
u
Feedforward control vector
e
Error vector (not measured)
ê
Tuning signal for adaptive feedforward control (estimate of e)
r
Constructed reference signal for adaptive feedforward control
Sensor space
y
Uncorrected WFS measurement, including any sensor noise
w
Uncorrected WFS measurement, including actuator error
Values of J(6001, 8000)
are normalized by J(6001, 8000) for no control. The
notation (VR) indicates that the actuator space was reparameterized
by the transformation V before the degrees of freedom were
reduced; the notation (RV) indicates that the degrees of freedom
were reduced before reparameterization by V. In all
cases, NL = 8 and E2 = E2Opt.
Table 4
Values of the Normalized Performance Index
J(6001, 8000) for LTI Lattice Filters (VR)
Type of Control
Number of Channels
γG = 0, γy = 0
γG = 0.05, γy = 0
γG = 0, γy = 0.05
γG = 0.05, γy = 0.05
Control Nv
Reference Disturbance Nr
No control
0
0
1.000
1.000
1.000
1.000
Lattice filters of order
NL = 2
69
69
0.096
0.111
0.131
0.142
NL = 3
69
69
0.075
0.093
0.115
0.128
NL = 4
69
69
0.070
0.089
0.111
0.124
NL = 8
69
69
0.066
0.086
0.107
0.120
NL = 8 and ê = z-1
e
69
69
0.062
0.063
0.101
0.101
Table 5
Values of
JE2(t1,
t2) and
J(t1,
t2) for E2 Chosen to
Optimize JE2(t1,
t2) over Different
Intervalsa
(t
1
, t
2
)
JE
2
(t
1
, t
2
)
JE
2
(1, 8000)
J(6001, 8000)
(1, 8000)
0.034
0.066
(3001, 3500)
0.025
0.060
0.078
(4001, 4500)
0.021
0.057
0.080
(7001, 7500)
0.022
0.059
0.076
The numbers of channels are
Nv = Nr = 69. The
lattice order is NL = 8. All cases
satisfy γG = γy = 0.