Abstract

This paper develops optimal statistical estimation formulation for multidither adaptive optics control loops which potentially can enhance stable beam control performance in the presence of spurious signal-like noise. A multidither autofocus example is chosen to compare estimated assisted and conventional adaptive optical performance in the presence of speckle-generated noise.

© 1977 Optical Society of America

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References

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  1. J. D’Azzo and C. Houpis, Feedback Control System Analysis and Synthesis (McGraw-Hill, New York, 1966).
  2. A. H. Jazwinski, Stochastic Processes and Filtering Theory (Academic, New York, 1970).
  3. A. Gelb, Applied Optimal Estimation (M. I. T. Press, Cambridge, Mass., 1974).
  4. J. Meditch, Stochastic Linear Estimation Control (McGraw-Hill, New York, 1969).
  5. R. B. Asher and R. D. Neal, “Adaptive Estimation of Aberration Coefficients in Adaptive Optics,” in Proceedings of the 1976 IEEE Conference on Decision and Control.
  6. J. E. Pearson, W. P. Brown, S. A. Kokorowski, M. E. Pedinoff, and C. Yeh, “COAT Measurements and Analysis,” , November1975, Rome Air Development Center, Griffiss AFB, New York.
  7. L. I. Goldfischer, J. Opt. Soc. Am. 55, 247 (1965).
    [Crossref]
  8. R. B. Crane, J. Opt. Soc. Am. 60, 1658 (1970).
    [Crossref]
  9. Ya Z. Tsypkin, Adaption and Learning in Automatic Systems (Academic, New York, 1971).
  10. R. B. Asher, D. Andrisani, and P. Dorato, “Bibliography on Adaptive Control Systems,” Proc. IEEE 64, 1226–1240 (1976).
    [Crossref]
  11. A. Erteza, “A Method of Active Autofocusing Using an Apertured Gaussian Beam,” , September1974.
  12. R. B. Asher, L. Sher, and R. D. Neal, “Focal Length and Target Range Estimation for Focused, Apertured Gaussian Laser Beams,” (unpublished).

1976 (1)

R. B. Asher, D. Andrisani, and P. Dorato, “Bibliography on Adaptive Control Systems,” Proc. IEEE 64, 1226–1240 (1976).
[Crossref]

1970 (1)

1965 (1)

Andrisani, D.

R. B. Asher, D. Andrisani, and P. Dorato, “Bibliography on Adaptive Control Systems,” Proc. IEEE 64, 1226–1240 (1976).
[Crossref]

Asher, R. B.

R. B. Asher, D. Andrisani, and P. Dorato, “Bibliography on Adaptive Control Systems,” Proc. IEEE 64, 1226–1240 (1976).
[Crossref]

R. B. Asher and R. D. Neal, “Adaptive Estimation of Aberration Coefficients in Adaptive Optics,” in Proceedings of the 1976 IEEE Conference on Decision and Control.

R. B. Asher, L. Sher, and R. D. Neal, “Focal Length and Target Range Estimation for Focused, Apertured Gaussian Laser Beams,” (unpublished).

Brown, W. P.

J. E. Pearson, W. P. Brown, S. A. Kokorowski, M. E. Pedinoff, and C. Yeh, “COAT Measurements and Analysis,” , November1975, Rome Air Development Center, Griffiss AFB, New York.

Crane, R. B.

D’Azzo, J.

J. D’Azzo and C. Houpis, Feedback Control System Analysis and Synthesis (McGraw-Hill, New York, 1966).

Dorato, P.

R. B. Asher, D. Andrisani, and P. Dorato, “Bibliography on Adaptive Control Systems,” Proc. IEEE 64, 1226–1240 (1976).
[Crossref]

Erteza, A.

A. Erteza, “A Method of Active Autofocusing Using an Apertured Gaussian Beam,” , September1974.

Gelb, A.

A. Gelb, Applied Optimal Estimation (M. I. T. Press, Cambridge, Mass., 1974).

Goldfischer, L. I.

Houpis, C.

J. D’Azzo and C. Houpis, Feedback Control System Analysis and Synthesis (McGraw-Hill, New York, 1966).

Jazwinski, A. H.

A. H. Jazwinski, Stochastic Processes and Filtering Theory (Academic, New York, 1970).

Kokorowski, S. A.

J. E. Pearson, W. P. Brown, S. A. Kokorowski, M. E. Pedinoff, and C. Yeh, “COAT Measurements and Analysis,” , November1975, Rome Air Development Center, Griffiss AFB, New York.

Meditch, J.

J. Meditch, Stochastic Linear Estimation Control (McGraw-Hill, New York, 1969).

Neal, R. D.

R. B. Asher and R. D. Neal, “Adaptive Estimation of Aberration Coefficients in Adaptive Optics,” in Proceedings of the 1976 IEEE Conference on Decision and Control.

R. B. Asher, L. Sher, and R. D. Neal, “Focal Length and Target Range Estimation for Focused, Apertured Gaussian Laser Beams,” (unpublished).

Pearson, J. E.

J. E. Pearson, W. P. Brown, S. A. Kokorowski, M. E. Pedinoff, and C. Yeh, “COAT Measurements and Analysis,” , November1975, Rome Air Development Center, Griffiss AFB, New York.

Pedinoff, M. E.

J. E. Pearson, W. P. Brown, S. A. Kokorowski, M. E. Pedinoff, and C. Yeh, “COAT Measurements and Analysis,” , November1975, Rome Air Development Center, Griffiss AFB, New York.

Sher, L.

R. B. Asher, L. Sher, and R. D. Neal, “Focal Length and Target Range Estimation for Focused, Apertured Gaussian Laser Beams,” (unpublished).

Tsypkin, Ya Z.

Ya Z. Tsypkin, Adaption and Learning in Automatic Systems (Academic, New York, 1971).

Yeh, C.

J. E. Pearson, W. P. Brown, S. A. Kokorowski, M. E. Pedinoff, and C. Yeh, “COAT Measurements and Analysis,” , November1975, Rome Air Development Center, Griffiss AFB, New York.

J. Opt. Soc. Am. (2)

Proc. IEEE (1)

R. B. Asher, D. Andrisani, and P. Dorato, “Bibliography on Adaptive Control Systems,” Proc. IEEE 64, 1226–1240 (1976).
[Crossref]

Other (9)

A. Erteza, “A Method of Active Autofocusing Using an Apertured Gaussian Beam,” , September1974.

R. B. Asher, L. Sher, and R. D. Neal, “Focal Length and Target Range Estimation for Focused, Apertured Gaussian Laser Beams,” (unpublished).

Ya Z. Tsypkin, Adaption and Learning in Automatic Systems (Academic, New York, 1971).

J. D’Azzo and C. Houpis, Feedback Control System Analysis and Synthesis (McGraw-Hill, New York, 1966).

A. H. Jazwinski, Stochastic Processes and Filtering Theory (Academic, New York, 1970).

A. Gelb, Applied Optimal Estimation (M. I. T. Press, Cambridge, Mass., 1974).

J. Meditch, Stochastic Linear Estimation Control (McGraw-Hill, New York, 1969).

R. B. Asher and R. D. Neal, “Adaptive Estimation of Aberration Coefficients in Adaptive Optics,” in Proceedings of the 1976 IEEE Conference on Decision and Control.

J. E. Pearson, W. P. Brown, S. A. Kokorowski, M. E. Pedinoff, and C. Yeh, “COAT Measurements and Analysis,” , November1975, Rome Air Development Center, Griffiss AFB, New York.

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Figures (7)

FIG. 1
FIG. 1

Geometry of illuminated object with illumination function projected to the (u, v) plane, and the receiver plane (x, y) located h from the object.

FIG. 2
FIG. 2

Power spectral density taking into account target glint (specular regions), distributed diffuse regions, and cross-correlated adaptive subaperture factors. Only a point receiver is considered.

FIG. 3
FIG. 3

Autofocus Cassegrain geometry.

FIG. 4
FIG. 4

Target intensity—autofocus performance—fixed aspect. Nose-on missile scenario. Multidither with and without estimation cancellation of speckle noise spectrum.

FIG. 5
FIG. 5

Covariance factors for the dither domain spectrum, Ps; diffuse domain, PT; and glint domain, Pg.

FIG. 6
FIG. 6

Covariance for estimated intensity, I, and slope maximization, ∇I.

FIG. 7
FIG. 7

Control-calculated average mirror separation.

Equations (59)

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I = I ( P ) .
P = P ¯ + ( c 1 sin ω 1 t c 2 sin ω 2 t c q sin ω q t ) .
I ( P ) = I ( P ¯ + Ω ) ,
Ω = ( c 1 sin ω 1 t c q sin ω q t ) .
I ( P ) = I ( P ¯ ) + I T P Ω + 1 2 Ω T 2 I P 2 Ω + H . O . T . ,
I P = ( I / P 1 I / P q ) ,
2 I P 2 = ( 2 I / P 1 2 ,     2 I / P 2 P 1 , ... , 2 I / P q P 1 2 I / P 1 P q , ....... , 2 I / P q 2 ) ,
I ( P ) = I ( P ¯ ) + I T P Ω .
P ¯ new k + 1 = P ¯ old k + k I P k ,
δ I = I T P ( P ¯ new - P ¯ old ) = k I T P I P .
I P < δ ,             δ > 0
y ( I ) = k ( z ) ( I ( P ¯ ) + I T P Ω + s 1 + s 2 + + s q + T + g ) + η ,
x ˙ ( t ) = f { x ( t ) } + G ( t ) u ( t ) ,
E { u ( t ) u ( τ ) T } = Q ( t ) δ ( t - τ ) ,
y ( t ) = h ( t ) T x ( t ) + v ( t ) ,
E { v ( t ) v ( τ ) } = r ( t ) δ ( t - τ ) .
x ˆ ( t ) = F ( t ) x ˆ + K ( t ) { y ( t ) - h T ( t ) x ˆ ( t ) } ,
F ( t ) = f x | x ˆ ,
K ( t ) = [ P ( t ) h ( t ) ] / r ( t ) ,
P ˙ ( t ) = F ( t ) P ( t ) + P ( t ) F ( t ) T + G ( t ) Q ( t ) G ( t ) T - [ P ( t ) h T ( t ) h ( t ) P ( t ) / r ( t ) ] .
S ( ω , Ω ) - d u d p ( u , v ) p ( u - λ h ω 2 π ,     v - λ h Ω 2 π ) .
S y y ( ω ) = H ( j ω ) 2 S w n ( ω ) ,
S g g ( ω ) = 2 β g σ g 2 / ( ω 2 + β g 2 ) ,
S T T ( ω ) = 2 β T σ T 2 / ( ω 2 + β g 2 ) .
S s i s i ( ω ) = σ i 2 / [ ( ω 2 - ω i 2 ) + β i 2 ω 2 ] ,
g ˙ = - β g g + ( 2 β g ) 1 / 2 σ g u g , T ˙ = - β T T + ( 2 β T ) 1 / 2 σ T u T , S ˙ i = x 2 i ,
X ˙ 2 i = - β i x 2 i - ω i 2 S i + σ i u i ,             i = 1 , , q
d I d t = I T P d P d t ,
k = ¯ k Δ t ;
d P d t = ¯ k I P k ,
d I d t = ¯ k I T P I P .
d d t ( I P ) = 2 I P 2 d P d t ,
d d t ( I P ) = ¯ k 2 I P 2 ( I P ) .
d d t ( I P ) = ¯ k 2 I P 2 ( I P ) + w ,
E { w ( t ) w T ( τ ) } = Q w ( t ) δ ( t - τ ) .
d d t ( I P ) = - v ,
v ˙ i = - β v i v i + ( 2 β v i ) 1 / 2 σ i u v i ,             i = 1 , , g
v i = v 0 i > 0 ,
x T = { I ( P ¯ ) , p I T , s 1 , , s q , x 2 1 , , x 2 q , v , T , g } T .
y ( t ) = k ( z ) { 1 , Ω T , 1 , 1 , , 1 , 0 , 0 , 1 , 1 } [ I ( P ¯ ) p I s x 2 v T g ] + η ,
[ I ˆ ˙ ( P ¯ ) p I ˆ ˙ s ˆ ˙ x ˆ ˙ 2 1 x ˆ ˙ 2 g v ˆ ˙ T ˆ ˙ g ˆ ˙ ] = [ k ˆ p I T p ˆ I - v ˆ x ˆ 2 - β 1 x ˆ 2 1 - ω 1 2 s ˆ 1 - β 2 x ˆ 2 2 - ω 2 2 s ˆ 2 - β q x ˆ 2 q - ω q 2 s ˆ q - β v v ˆ - β T T ˆ - β g g ] + K { y - k ( z ) [ 1 , Ω T , 1 , 1 , , 1 , 0 , 0 , 1 , 1 ] [ I ˆ ( P ¯ ) p ˆ I s ˆ x ˆ 2 v ˆ T ˆ g ˆ ] }
P = [ P I P 12 P 13 P 14 P 15 P 16 P 17 P 12 P I P 23 P 24 P 25 P 26 P 27 P 13 P 23 P s P 34 P 35 P 36 P 37 P 14 P 24 P 34 P x 2 P 45 P 46 P 47 P 15 P 25 P 35 P 45 P v P 56 P 57 P 16 P 26 P 36 P 46 P 56 P T P 67 P 17 P 27 P 37 P 47 P 57 P 67 P g ] ,
h = k ( z ) [ 1 , Ω T , 1 , 1 , , 1 , 0 , 0 , 1 , 1 ] .
f x = { 0 , 2 ¯ k 1 ˆ I , , 2 ¯ k q ˆ I , 0 , . 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 0 , 0 , , 0 , 0 , , 0 , 0 , 0 , - 1 , 0 , 0 , 0 , 0 0 , 0 , , 0 , 0 , , 0 , 0 , 0 , 0 , - 1 , 0 , 0 , 0 0 , 0 , , 0 , 0 , , 0 , 0 , 0 , 0 , 0 , - 1 , 0 , 0 0 , 0 , , 0 , 0 , , 0 , 1 , 0 , 0 , 0 , 0 , 0 , 0 , 0 0 , 0 , , 0 , 0 , , 0 , 0 , 1 , 0 , 0 , 0 , 0 , 0 , 0 0 , 0 , , 0 , 0 , , 0 , 0 , 0 , 1 , 0 , 0 , 0 , 0 0 , 0 , , 0 , - ω 1 2 , 0 , , 0 , - β 1 , 0 , 0 , 0 , 0 , 0 , 0 0 , 0 , , 0 , 0 , - ω q 2 , 0 , - β q , 0 , 0 , 0 , 0 0 , 0 , , 0 , 0 , , 0 , - β v 1 , 0 , 0 , 0 , 0 0 , 0 , , 0 , 0 , , 0 , 0 , 0 , - β v q , 0 , 0 0 , 0 , , 0 , 0 , , 0 , 0 , .. 0 , - β T , 0 0 , 0 , , 0 , 0 , , 0 , 0 , .. 0 , 0 , - β g } ,
G ( t )             { 0 , 0 , 0 , 0 , 0 , 0 , 0 0 , 0 , 0 , 0 , 0 , 0 , 0 σ 1 , 0 , 0 , 0 , 0 , 0 , 0 0 , 0 , σ g , 0 , 0 , 0 , 0 0 , 0 , 0 , ( 2 β v 1 ) 1 / 2 σ v 1 , 0 , 0 , 0 , 0 0 , 0 , 0 , 0 , ( 2 β v q ) 1 / 2 σ v q , 0 , 0 0 , 0 , 0 , 0 , 0 , ( 2 β T ) 1 / 2 σ T , 0 0 , 0 , 0 , 0 , 0 , 0 , ( 2 β g ) 1 / 2 σ g } ,
J ( P ) = max P E { I ( p ) } ,
E { p I } = 0 ,
P k + 1 = P k + k p I ,
inf [ ( P - P 0 ) T E { I ( P ) P } ] > 0 ,             > 0 < P - P 0 < - 1
E { p ˆ I 2 } h ( 1 + P - P 0 2 ) ,             h > 0
k 0 , k k =
k k 2 <
Prob { lim k P k = P 0 } = 1.
y = k ( z ) { I ( d ¯ ) + I d sin ω d t + s d + T + g } + η ,
I d = I f f d ,
θ ˙ δ Doppler λ / l T N
l T N f N [ 2.44 λ / 2 b ] ,
d N = r s 2 [ 1 - 2 f N [ ( 1 / r p ) - ( 1 / r s ) ] 2 ( f N / r p ) - 1 ] .
[ I ˆ ˙ ( r ) p I ˆ ˙ s ˆ ˙ x ˆ ˙ 2 V ˆ ˙ T ˆ ˙ g ˆ ˙ ] = [ k p I ˆ T p I ˆ - v ˆ x ˆ 2 - β d x ˆ 2 - ω d 2 s ˆ - β v V ˆ - β T T ˆ - β g g ˆ ] + K { y - K ( z ) [ 1 , Ω T , 1 , 0 , 0 , 1 , 1 ] [ I ˆ ( p ) p I ˆ s ˆ x ˆ 2 V ˆ T ˆ g ˆ ] } .