Abstract

We present a technique for controlling a ground-based deformable mirror adaptive optics telescope to compensate for optical wave-front phase distortion induced by a turbulent atmosphere. Specifically, a predictive linear quadratic Gaussian (LQG) controller is designed that generates commanded control voltages to the mirror actuators based on a set of time-delayed wave-front slope measurements from a Hartmann-type wave-front sensor.

© 1993 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. J. W. Hardy, J. E. Lefebvre, C. L. Koliopoulos, “Real-time atmospheric compensation,” J. Opt. Soc. Am. 67, 360–369 (1977).
    [CrossRef]
  2. D. J. Anderson, “LQG control of a deformable mirror adaptive optics system with time-delayed measurements,” M.S. thesis, AFIT/GE/ENG/91D-03 [School of Engineering, Air Force Institute of Technology (AU), Wright-Patterson Air Force Base, Ohio, 1991].
  3. M. A. Von Bokern, “Design of a linear quadratic Gaussian control law for an adaptive optics system,” M.S. thesis, AFIT/GE/ENG/90D-65 [School of Engineering, Air Force Institute of Technology (AU), Wright-Patterson Air Force Base, Ohio, 1990].
  4. C. B. Hogge, R. R. Butts, “Frequency spectra for the geometric representation of wavefront distortions due to atmospheric turbulence,” IEEE Trans. Antennas Propag. AP-24, 144–154 (1976).
    [CrossRef]
  5. R. J. Noll, “Zernike polynomials and atmospheric turbulence,” J. Opt. Soc. Am. 66, 207–211 (1976).
    [CrossRef]
  6. N. Roddier, “Atmospheric wavefront simulation using Zernike polynomials,” Opt. Eng. 29, 1174–1180 (1990).
    [CrossRef]
  7. R. Cubalchini, “Modal wave-front estimation from phase derivative measurements,” J. Opt. Soc. Am. 69, 972–977 (1979).
    [CrossRef]
  8. G. A. Tyler, D. L. Fried, J. F. Belsher, “Adaptive optics technology and atmospheric turbulence correction, Volume I: Technical descriptions,” Contract F2960182C0104 (The Optical Sciences Company, Placentia, Calif., 1988).
  9. J. Y. Wang, “Phase-compensated optical beam propagation through atmospheric turbulence,” Appl. Opt. 17, 2580–2590 (1978).
    [PubMed]
  10. J. Y. Wang, J. K. Markey, “Modal compensation of atmospheric turbulence phase distortion,” J. Opt. Soc. Am. 68, 78–87 (1978).
    [CrossRef]
  11. “97-Channel, low-voltage, electrodistortive mirror: operation manual,” Tech. Rep. (Litton Itek Optical Systems, Lexington, Mass., 1988).
  12. P. S. Maybeck, Stochastic Models, Estimation, and Control (Academic, New York, 1979), Vol. 1.
  13. S. F. Clifford, “The classical theory of wave propagation in a turbulent medium,” in Laser Beam Propagation in the Atmosphere, J. W. Strohbehn, ed., Vol. 25 of Topics in Applied Physics (Springer-Verlag, New York, 1978).
    [CrossRef]
  14. B. M. Welsh, C. S. Gardner, “Performance analysis of adaptive optics systems using laser guide stars and slope sensors,” Technical Report (University of Illinois, Champaign-Urbana, Ill., 1989).
  15. M. Athans, P. Falb, Optimal Control (McGraw-Hill, New York, 1966).
  16. H. Kwakernaak, R. Sivan, Linear Optimal Control Systems (Wiley, New York, 1972).
  17. P. S. Maybeck, Stochastic Models, Estimation, and Control (Academic, New York, 1982), Vol. 3.
  18. R. G. Miller, “Vector space approach for modeling a Hartmann wavefront sensor,” M.S. thesis, AFIT/GE/ENG/89D-33 [School of Engineering, Air Force Institute of Technology (AU), Wright-Patterson Air Force Base, Ohio, 1989].
  19. S. H. Musick, N. A. Carlson, “User’s manual for a multimode simulation for optimal filter evaluation (MSOFE), Contract F3361586C1047 (Integrity Systems, Incorporated, Winchester, Mass., 1990).
  20. D. L. Kleinman, S. Baron, W. H. Levison, “An optimal control model of human response part I: Theory and validation,” Automatica 6, 357–369 (1970).
    [CrossRef]
  21. D. L. Kleinman, “Optimal control of linear systems with time-delay and observation noise,” IEEE Trans. Autom. Control AC-14, 524–527 (1969).
    [CrossRef]

1990 (1)

N. Roddier, “Atmospheric wavefront simulation using Zernike polynomials,” Opt. Eng. 29, 1174–1180 (1990).
[CrossRef]

1979 (1)

1978 (2)

1977 (1)

1976 (2)

C. B. Hogge, R. R. Butts, “Frequency spectra for the geometric representation of wavefront distortions due to atmospheric turbulence,” IEEE Trans. Antennas Propag. AP-24, 144–154 (1976).
[CrossRef]

R. J. Noll, “Zernike polynomials and atmospheric turbulence,” J. Opt. Soc. Am. 66, 207–211 (1976).
[CrossRef]

1970 (1)

D. L. Kleinman, S. Baron, W. H. Levison, “An optimal control model of human response part I: Theory and validation,” Automatica 6, 357–369 (1970).
[CrossRef]

1969 (1)

D. L. Kleinman, “Optimal control of linear systems with time-delay and observation noise,” IEEE Trans. Autom. Control AC-14, 524–527 (1969).
[CrossRef]

Anderson, D. J.

D. J. Anderson, “LQG control of a deformable mirror adaptive optics system with time-delayed measurements,” M.S. thesis, AFIT/GE/ENG/91D-03 [School of Engineering, Air Force Institute of Technology (AU), Wright-Patterson Air Force Base, Ohio, 1991].

Athans, M.

M. Athans, P. Falb, Optimal Control (McGraw-Hill, New York, 1966).

Baron, S.

D. L. Kleinman, S. Baron, W. H. Levison, “An optimal control model of human response part I: Theory and validation,” Automatica 6, 357–369 (1970).
[CrossRef]

Belsher, J. F.

G. A. Tyler, D. L. Fried, J. F. Belsher, “Adaptive optics technology and atmospheric turbulence correction, Volume I: Technical descriptions,” Contract F2960182C0104 (The Optical Sciences Company, Placentia, Calif., 1988).

Butts, R. R.

C. B. Hogge, R. R. Butts, “Frequency spectra for the geometric representation of wavefront distortions due to atmospheric turbulence,” IEEE Trans. Antennas Propag. AP-24, 144–154 (1976).
[CrossRef]

Carlson, N. A.

S. H. Musick, N. A. Carlson, “User’s manual for a multimode simulation for optimal filter evaluation (MSOFE), Contract F3361586C1047 (Integrity Systems, Incorporated, Winchester, Mass., 1990).

Clifford, S. F.

S. F. Clifford, “The classical theory of wave propagation in a turbulent medium,” in Laser Beam Propagation in the Atmosphere, J. W. Strohbehn, ed., Vol. 25 of Topics in Applied Physics (Springer-Verlag, New York, 1978).
[CrossRef]

Cubalchini, R.

Falb, P.

M. Athans, P. Falb, Optimal Control (McGraw-Hill, New York, 1966).

Fried, D. L.

G. A. Tyler, D. L. Fried, J. F. Belsher, “Adaptive optics technology and atmospheric turbulence correction, Volume I: Technical descriptions,” Contract F2960182C0104 (The Optical Sciences Company, Placentia, Calif., 1988).

Gardner, C. S.

B. M. Welsh, C. S. Gardner, “Performance analysis of adaptive optics systems using laser guide stars and slope sensors,” Technical Report (University of Illinois, Champaign-Urbana, Ill., 1989).

Hardy, J. W.

Hogge, C. B.

C. B. Hogge, R. R. Butts, “Frequency spectra for the geometric representation of wavefront distortions due to atmospheric turbulence,” IEEE Trans. Antennas Propag. AP-24, 144–154 (1976).
[CrossRef]

Kleinman, D. L.

D. L. Kleinman, S. Baron, W. H. Levison, “An optimal control model of human response part I: Theory and validation,” Automatica 6, 357–369 (1970).
[CrossRef]

D. L. Kleinman, “Optimal control of linear systems with time-delay and observation noise,” IEEE Trans. Autom. Control AC-14, 524–527 (1969).
[CrossRef]

Koliopoulos, C. L.

Kwakernaak, H.

H. Kwakernaak, R. Sivan, Linear Optimal Control Systems (Wiley, New York, 1972).

Lefebvre, J. E.

Levison, W. H.

D. L. Kleinman, S. Baron, W. H. Levison, “An optimal control model of human response part I: Theory and validation,” Automatica 6, 357–369 (1970).
[CrossRef]

Markey, J. K.

Maybeck, P. S.

P. S. Maybeck, Stochastic Models, Estimation, and Control (Academic, New York, 1979), Vol. 1.

P. S. Maybeck, Stochastic Models, Estimation, and Control (Academic, New York, 1982), Vol. 3.

Miller, R. G.

R. G. Miller, “Vector space approach for modeling a Hartmann wavefront sensor,” M.S. thesis, AFIT/GE/ENG/89D-33 [School of Engineering, Air Force Institute of Technology (AU), Wright-Patterson Air Force Base, Ohio, 1989].

Musick, S. H.

S. H. Musick, N. A. Carlson, “User’s manual for a multimode simulation for optimal filter evaluation (MSOFE), Contract F3361586C1047 (Integrity Systems, Incorporated, Winchester, Mass., 1990).

Noll, R. J.

Roddier, N.

N. Roddier, “Atmospheric wavefront simulation using Zernike polynomials,” Opt. Eng. 29, 1174–1180 (1990).
[CrossRef]

Sivan, R.

H. Kwakernaak, R. Sivan, Linear Optimal Control Systems (Wiley, New York, 1972).

Tyler, G. A.

G. A. Tyler, D. L. Fried, J. F. Belsher, “Adaptive optics technology and atmospheric turbulence correction, Volume I: Technical descriptions,” Contract F2960182C0104 (The Optical Sciences Company, Placentia, Calif., 1988).

Von Bokern, M. A.

M. A. Von Bokern, “Design of a linear quadratic Gaussian control law for an adaptive optics system,” M.S. thesis, AFIT/GE/ENG/90D-65 [School of Engineering, Air Force Institute of Technology (AU), Wright-Patterson Air Force Base, Ohio, 1990].

Wang, J. Y.

Welsh, B. M.

B. M. Welsh, C. S. Gardner, “Performance analysis of adaptive optics systems using laser guide stars and slope sensors,” Technical Report (University of Illinois, Champaign-Urbana, Ill., 1989).

Appl. Opt. (1)

Automatica (1)

D. L. Kleinman, S. Baron, W. H. Levison, “An optimal control model of human response part I: Theory and validation,” Automatica 6, 357–369 (1970).
[CrossRef]

IEEE Trans. Antennas Propag. (1)

C. B. Hogge, R. R. Butts, “Frequency spectra for the geometric representation of wavefront distortions due to atmospheric turbulence,” IEEE Trans. Antennas Propag. AP-24, 144–154 (1976).
[CrossRef]

IEEE Trans. Autom. Control (1)

D. L. Kleinman, “Optimal control of linear systems with time-delay and observation noise,” IEEE Trans. Autom. Control AC-14, 524–527 (1969).
[CrossRef]

J. Opt. Soc. Am. (4)

Opt. Eng. (1)

N. Roddier, “Atmospheric wavefront simulation using Zernike polynomials,” Opt. Eng. 29, 1174–1180 (1990).
[CrossRef]

Other (12)

D. J. Anderson, “LQG control of a deformable mirror adaptive optics system with time-delayed measurements,” M.S. thesis, AFIT/GE/ENG/91D-03 [School of Engineering, Air Force Institute of Technology (AU), Wright-Patterson Air Force Base, Ohio, 1991].

M. A. Von Bokern, “Design of a linear quadratic Gaussian control law for an adaptive optics system,” M.S. thesis, AFIT/GE/ENG/90D-65 [School of Engineering, Air Force Institute of Technology (AU), Wright-Patterson Air Force Base, Ohio, 1990].

G. A. Tyler, D. L. Fried, J. F. Belsher, “Adaptive optics technology and atmospheric turbulence correction, Volume I: Technical descriptions,” Contract F2960182C0104 (The Optical Sciences Company, Placentia, Calif., 1988).

“97-Channel, low-voltage, electrodistortive mirror: operation manual,” Tech. Rep. (Litton Itek Optical Systems, Lexington, Mass., 1988).

P. S. Maybeck, Stochastic Models, Estimation, and Control (Academic, New York, 1979), Vol. 1.

S. F. Clifford, “The classical theory of wave propagation in a turbulent medium,” in Laser Beam Propagation in the Atmosphere, J. W. Strohbehn, ed., Vol. 25 of Topics in Applied Physics (Springer-Verlag, New York, 1978).
[CrossRef]

B. M. Welsh, C. S. Gardner, “Performance analysis of adaptive optics systems using laser guide stars and slope sensors,” Technical Report (University of Illinois, Champaign-Urbana, Ill., 1989).

M. Athans, P. Falb, Optimal Control (McGraw-Hill, New York, 1966).

H. Kwakernaak, R. Sivan, Linear Optimal Control Systems (Wiley, New York, 1972).

P. S. Maybeck, Stochastic Models, Estimation, and Control (Academic, New York, 1982), Vol. 3.

R. G. Miller, “Vector space approach for modeling a Hartmann wavefront sensor,” M.S. thesis, AFIT/GE/ENG/89D-33 [School of Engineering, Air Force Institute of Technology (AU), Wright-Patterson Air Force Base, Ohio, 1989].

S. H. Musick, N. A. Carlson, “User’s manual for a multimode simulation for optimal filter evaluation (MSOFE), Contract F3361586C1047 (Integrity Systems, Incorporated, Winchester, Mass., 1990).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (13)

Fig. 1
Fig. 1

Adaptive optics system functional diagram.

Fig. 2
Fig. 2

Active mirror actuator locations.

Fig. 3
Fig. 3

Y-tilt truth model and filter states.

Fig. 4
Fig. 4

Y-tilt filter estimation error (mean ±1σ over 10 Monte Carlo runs).

Fig. 5
Fig. 5

Incident and reflected rms phase distortions (mean ±1σ over 10 Monte Carlo runs).

Fig. 6
Fig. 6

Y-tilt truth and filter states with measurement delay.

Fig. 7
Fig. 7

Y-tilt estimation error with measurement delay (mean ±1σ over 10 runs).

Fig. 8
Fig. 8

Incident and reflected rms phase distortions with measurement delay (mean).

Fig. 9
Fig. 9

Y-tilt truth and filter states with predictor compensation.

Fig. 10
Fig. 10

Y-tilt estimation error with predictor compensation (mean ± 1σ over 10 runs).

Fig. 11
Fig. 11

Incident and reflected rms phase distortions with predictor compensation (mean ±1σ).

Fig. 12
Fig. 12

Suboptimal filter versus filter-predictor estimation performance.

Fig. 13
Fig. 13

Rms distortion with predictor compensation and 3-ms sample rate (mean ± 1σ).

Tables (2)

Tables Icon

Table 1 First 15 Zernike Functions

Tables Icon

Table 2 Atmospheric Phase Distortion State-Space Model

Equations (41)

Equations on this page are rendered with MathJax. Learn more.

ϕ ( r , Ө , t ) = i = 0 N c i ( t ) Z i ( r , Ө ) ,
1 π R 2 0 2 π d Ө 0 R Z i ( r , Ө ) Z j ( r , Ө ) r d r = δ i j ,
δ i j = { 1 , i = j 0 , i j .
ϕ ˜ = ( c 1 2 + c 2 2 + c 3 2 + + c N 2 ) 1 / 2 .
c i ( t ) = d Ө W ( r , Ө ) ϕ ( r , Ө , t ) Z i ( r , Ө ) r d r d Ө W ( r , Ө ) r d r ,
W ( r , Ө ) = { 1 , r R 0 , r > R .
g ˙ ( t ) = 1 0 . 00045 g ( t ) + 2150 u ( t ) ,
x ˙ ( t ) = F x ( t ) + B u ( t ) + G w ( t ) ,
E { w ( t ) w T ( t + τ ) } = Q δ ( t τ ) ,
x ˙ a ( t ) = F a x a ( t ) + G a w a ( t ) ,
x ˙ m ( t ) = F m x m ( t ) + B m u ( t ) ,
B m = diag [ 1 0 . 00045 ] M ,
[ x ˙ a ( t ) x ˙ m ( t ) ] = [ F a 0 0 F m ] [ x a ( t ) x m ( t ) ] + [ 0 B m ] u ( t ) + [ I 0 ] w a ( t ) .
a ( t i ) = x a ( t i ) + x m ( t i ) .
z ( t i ) = H a ( t i ) + ν ( t i ) ,
E { ν ( t i ) ν T ( t j ) } = R δ i j ,
J = E { i = 0 N 1 2 [ x T ( t i ) H x ( t i + u T ( t i ) U u ( t i ) ] } ,
H = [ C C C C ] ,
u * [ x ( t i ) , t i ] = G c * ( t i ) x ( t i ) ,
G c * ( t i ) = [ U + B d T K c B d ] 1 B d T K c Φ ( t i + 1 , t i ) ,
K c = H + Φ T ( t i + 1 , t i ) K c Φ ( t i + 1 , t i ) Φ T ( t i + 1 , t i ) × K c B d [ U + B d T K c B d ] 1 B d T K c Φ ( t i + 1 , t i ) ,
u * [ x ˆ ( t i + ) , t i ] = G c * ( t i ) x ˆ ( t i + ) .
x ˆ ( t i ) = E { x ( t i ) | Z ( t i 1 ) } , P ( t i ) = E { [ x ( t i ) x ˆ ( t i ) ] [ x ( t i ) x ˆ ( t i ) ] T | Z ( t i 1 ) } ,
x ˆ ( t i ) = Φ ( t i , t i 1 ) x ˆ ( t i 1 ) x ˆ ( t i 1 + ) + B d u ( t i 1 ) ,
P ( t i ) = Φ ( t i , t i 1 ) P ( t i 1 ) Φ T ( t i , t i 1 ) + G d Q d G d T ,
K = P H T [ H P H T + R ] 1 ,
x ˆ ( t i + ) = x ˆ ( t i ) + K [ z ( t i ) H x ˆ ( t i ) ] ,
P + = P K H P ,
x ( t i + 1 ) = Φ ( t i + 1 , t i ) x ( t i ) + B d u ( t i ) + w d ( t i ) .
z ( t i ) = H x ( t i 1 ) + ν ( t i ) ,
x p ( t i ) = x ( t i 1 ) .
z ( t i ) = H x p ( t i ) + ν ( t i ) ,
[ x ( t i ) x p ( t i ) ] = [ Φ 0 I 0 ] [ x ( t i 1 ) x p ( t i 1 ) ] + [ B d 0 ] u ( t i 1 ) + [ I 0 ] w d ( t i 1 ) .
x ˆ a p ( t i ) = [ x ˆ ( t i ) x ˆ p ( t i ) ] ,
μ e ( t ) 1 10 k = 1 10 e k ( t ) ,
σ e 2 ( t ) 1 10 1 k = 1 10 [ e k 2 ( t ) μ e 2 ( t ) ] ,
ϕ ˜ inc ( t ) = [ i = 1 14 x i 2 ( t ) ] 1 / 2 .
ϕ cor ( r , Ө ) = i = 4 14 [ c i ( t ) + c i + 14 ( t ) ] Z i ( r , Ө ) .
ϕ ˜ cor ( t ) = { i = 1 14 [ x i ( t ) + x i + 14 ( t ) ] 2 } 1 / 2 .
K a p = [ Φ K K ] .
u ( t i ) = G c * x ˆ ( t i ) .

Metrics