Abstract

The objective of an astronomical adaptive optics control system is to minimize the residual wave-front error remaining on the science-object wave fronts after being compensated for atmospheric turbulence and telescope aberrations. Minimizing the mean square wave-front residual maximizes the Strehl ratio and the encircled energy in pointlike images and maximizes the contrast and resolution of extended images. We prove the separation principle of optimal control for application to adaptive optics so as to minimize the mean square wave-front residual. This shows that the residual wave-front error attributable to the control system can be decomposed into three independent terms that can be treated separately in design. The first term depends on the geometry of the wave-front sensor(s), the second term depends on the geometry of the deformable mirror(s), and the third term is a stochastic term that depends on the signal-to-noise ratio. The geometric view comes from understanding that the underlying quantity of interest, the wave-front phase surface, is really an infinite-dimensional vector within a Hilbert space and that this vector space is projected into subspaces we can control and measure by the deformable mirrors and wave-front sensors, respectively. When the control and estimation algorithms are optimal, the residual wave front is in a subspace that is the union of subspaces orthogonal to both of these projections. The method is general in that it applies both to conventional (on-axis, ground-layer conjugate) adaptive optics architectures and to more complicated multi-guide-star- and multiconjugate-layer architectures envisaged for future giant telescopes. We illustrate the approach by using a simple example that has been worked out previously [J. Opt. Soc. Am. A 73, 1171 (1983) ] for a single-conjugate, static atmosphere case and follow up with a discussion of how it is extendable to general adaptive optics architectures.

© 2005 Optical Society of America

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References

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  1. M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1980), p. 469.
  2. P. D. Joseph, J. T. Tou, “On linear control theory,” Trans. Am. Inst. Electr. Eng. 80, 193–196 (1961).
  3. T. L. Gunckel, G. F. Franklin, “A general solution for linear sampled data control,” J. Basic Eng. 85-D, 197–201 (1963).
    [CrossRef]
  4. W. M. Wonham, “On the separation theorem of stochastic control,” SIAM J. Control 6, 312–326 (1968).
    [CrossRef]
  5. M. S. Sholar, D. M. Wiberg, “Canonical equations for boundary feedback control of stochastic distributed parameter systems,” Automatica 8, 287–298 (1972).
    [CrossRef]
  6. A. V. Balakrishnan, Applied Functional Analysis (Springer-Verlag, New York, 1976), Chap. 6.
  7. E. P. Wallner, “Optimal wave-front correction using slope measurements,” J. Opt. Soc. Am. 73, 1771–1776 (1983).
    [CrossRef]
  8. B. L. Ellerbroek, “First-order performance evaluation of adaptive-optics systems for atmospheric-turbulence compensation in extended-field-of-view astronomical telescopes,” J. Opt. Soc. Am. A 11, 783–805 (1994).
    [CrossRef]
  9. B. L. Ellerbroek, F. Rigaut, “Methods for correcting tilt anisoplanatism in laser-guide-star-based multiconjugate adaptive optics,” J. Opt. Soc. Am. A 18, 2539–2547 (2001).
    [CrossRef]
  10. W. J. Wild, “Predictive optimal estimators for adaptive-optics systems,” Opt. Lett. 21, 1433–1435 (1996).
    [CrossRef] [PubMed]
  11. T. Fusco, J.-M. Conan, G. Rousset, L. Mugnier, V. Michau, “Optimal wave-front reconstruction strategies for multiconjugate adaptive optics,” J. Opt. Soc. Am. A 18, 2527–2538 (2001).
    [CrossRef]
  12. B. Le Roux, J. M. Conan, C. Kulcsar, H. F. Reynaud, L. M. Mugnier, T. Fusco, “Optimal control law for multiconjugate adaptive optics,” in Adaptive Optical System Technologies II, P. L. Wizinowich and D. Bonaccini, eds., Proc. SPIE4839, 878–889 (2002).
  13. D. Gavel, D. Wiberg, “Toward Strehl-optimizing adaptive optics controllers,” in Adaptive Optical System Technologies II, P. L. Wizinowich and D. Bonaccini, eds., Proc. SPIE4839, 890–901 (2002).
  14. P. G. Hoel, S. C. Port, C. J. Stone, Introduction to Stochastic Processes (Houghton Mifflin, Boston, Mass., 1972), pp. 171–174.
  15. R. E. Kalman, “Contributions to the theory of optimal control,” Bol. Soc. Mat. Mex.102–119 (1960).
  16. D. L. Fried, “Least-square fitting a wave-front distortion estimate to an array of phase-difference measurements,” J. Opt. Soc. Am. 67, 370–374 (1977).
    [CrossRef]
  17. R. Courant, D. Hilbert, Methods of Mathematical Physics (Interscience, New York, 1953), Vol. 1.
  18. MATLAB Partial Differential Equation Toolbox, http://www.mathworks.com.
  19. G. J. Baker, personal communication, g.j.baker@lmco.com.

2001

1996

1994

1983

1977

1972

M. S. Sholar, D. M. Wiberg, “Canonical equations for boundary feedback control of stochastic distributed parameter systems,” Automatica 8, 287–298 (1972).
[CrossRef]

1968

W. M. Wonham, “On the separation theorem of stochastic control,” SIAM J. Control 6, 312–326 (1968).
[CrossRef]

1963

T. L. Gunckel, G. F. Franklin, “A general solution for linear sampled data control,” J. Basic Eng. 85-D, 197–201 (1963).
[CrossRef]

1961

P. D. Joseph, J. T. Tou, “On linear control theory,” Trans. Am. Inst. Electr. Eng. 80, 193–196 (1961).

1960

R. E. Kalman, “Contributions to the theory of optimal control,” Bol. Soc. Mat. Mex.102–119 (1960).

Baker, G. J.

G. J. Baker, personal communication, g.j.baker@lmco.com.

Balakrishnan, A. V.

A. V. Balakrishnan, Applied Functional Analysis (Springer-Verlag, New York, 1976), Chap. 6.

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1980), p. 469.

Conan, J. M.

B. Le Roux, J. M. Conan, C. Kulcsar, H. F. Reynaud, L. M. Mugnier, T. Fusco, “Optimal control law for multiconjugate adaptive optics,” in Adaptive Optical System Technologies II, P. L. Wizinowich and D. Bonaccini, eds., Proc. SPIE4839, 878–889 (2002).

Conan, J.-M.

Courant, R.

R. Courant, D. Hilbert, Methods of Mathematical Physics (Interscience, New York, 1953), Vol. 1.

Ellerbroek, B. L.

Franklin, G. F.

T. L. Gunckel, G. F. Franklin, “A general solution for linear sampled data control,” J. Basic Eng. 85-D, 197–201 (1963).
[CrossRef]

Fried, D. L.

Fusco, T.

T. Fusco, J.-M. Conan, G. Rousset, L. Mugnier, V. Michau, “Optimal wave-front reconstruction strategies for multiconjugate adaptive optics,” J. Opt. Soc. Am. A 18, 2527–2538 (2001).
[CrossRef]

B. Le Roux, J. M. Conan, C. Kulcsar, H. F. Reynaud, L. M. Mugnier, T. Fusco, “Optimal control law for multiconjugate adaptive optics,” in Adaptive Optical System Technologies II, P. L. Wizinowich and D. Bonaccini, eds., Proc. SPIE4839, 878–889 (2002).

Gavel, D.

D. Gavel, D. Wiberg, “Toward Strehl-optimizing adaptive optics controllers,” in Adaptive Optical System Technologies II, P. L. Wizinowich and D. Bonaccini, eds., Proc. SPIE4839, 890–901 (2002).

Gunckel, T. L.

T. L. Gunckel, G. F. Franklin, “A general solution for linear sampled data control,” J. Basic Eng. 85-D, 197–201 (1963).
[CrossRef]

Hilbert, D.

R. Courant, D. Hilbert, Methods of Mathematical Physics (Interscience, New York, 1953), Vol. 1.

Hoel, P. G.

P. G. Hoel, S. C. Port, C. J. Stone, Introduction to Stochastic Processes (Houghton Mifflin, Boston, Mass., 1972), pp. 171–174.

Joseph, P. D.

P. D. Joseph, J. T. Tou, “On linear control theory,” Trans. Am. Inst. Electr. Eng. 80, 193–196 (1961).

Kalman, R. E.

R. E. Kalman, “Contributions to the theory of optimal control,” Bol. Soc. Mat. Mex.102–119 (1960).

Kulcsar, C.

B. Le Roux, J. M. Conan, C. Kulcsar, H. F. Reynaud, L. M. Mugnier, T. Fusco, “Optimal control law for multiconjugate adaptive optics,” in Adaptive Optical System Technologies II, P. L. Wizinowich and D. Bonaccini, eds., Proc. SPIE4839, 878–889 (2002).

Le Roux, B.

B. Le Roux, J. M. Conan, C. Kulcsar, H. F. Reynaud, L. M. Mugnier, T. Fusco, “Optimal control law for multiconjugate adaptive optics,” in Adaptive Optical System Technologies II, P. L. Wizinowich and D. Bonaccini, eds., Proc. SPIE4839, 878–889 (2002).

Michau, V.

Mugnier, L.

Mugnier, L. M.

B. Le Roux, J. M. Conan, C. Kulcsar, H. F. Reynaud, L. M. Mugnier, T. Fusco, “Optimal control law for multiconjugate adaptive optics,” in Adaptive Optical System Technologies II, P. L. Wizinowich and D. Bonaccini, eds., Proc. SPIE4839, 878–889 (2002).

Port, S. C.

P. G. Hoel, S. C. Port, C. J. Stone, Introduction to Stochastic Processes (Houghton Mifflin, Boston, Mass., 1972), pp. 171–174.

Reynaud, H. F.

B. Le Roux, J. M. Conan, C. Kulcsar, H. F. Reynaud, L. M. Mugnier, T. Fusco, “Optimal control law for multiconjugate adaptive optics,” in Adaptive Optical System Technologies II, P. L. Wizinowich and D. Bonaccini, eds., Proc. SPIE4839, 878–889 (2002).

Rigaut, F.

Rousset, G.

Sholar, M. S.

M. S. Sholar, D. M. Wiberg, “Canonical equations for boundary feedback control of stochastic distributed parameter systems,” Automatica 8, 287–298 (1972).
[CrossRef]

Stone, C. J.

P. G. Hoel, S. C. Port, C. J. Stone, Introduction to Stochastic Processes (Houghton Mifflin, Boston, Mass., 1972), pp. 171–174.

Tou, J. T.

P. D. Joseph, J. T. Tou, “On linear control theory,” Trans. Am. Inst. Electr. Eng. 80, 193–196 (1961).

Wallner, E. P.

Wiberg, D.

D. Gavel, D. Wiberg, “Toward Strehl-optimizing adaptive optics controllers,” in Adaptive Optical System Technologies II, P. L. Wizinowich and D. Bonaccini, eds., Proc. SPIE4839, 890–901 (2002).

Wiberg, D. M.

M. S. Sholar, D. M. Wiberg, “Canonical equations for boundary feedback control of stochastic distributed parameter systems,” Automatica 8, 287–298 (1972).
[CrossRef]

Wild, W. J.

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1980), p. 469.

Wonham, W. M.

W. M. Wonham, “On the separation theorem of stochastic control,” SIAM J. Control 6, 312–326 (1968).
[CrossRef]

Automatica

M. S. Sholar, D. M. Wiberg, “Canonical equations for boundary feedback control of stochastic distributed parameter systems,” Automatica 8, 287–298 (1972).
[CrossRef]

Bol. Soc. Mat. Mex.

R. E. Kalman, “Contributions to the theory of optimal control,” Bol. Soc. Mat. Mex.102–119 (1960).

J. Basic Eng.

T. L. Gunckel, G. F. Franklin, “A general solution for linear sampled data control,” J. Basic Eng. 85-D, 197–201 (1963).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Opt. Lett.

SIAM J. Control

W. M. Wonham, “On the separation theorem of stochastic control,” SIAM J. Control 6, 312–326 (1968).
[CrossRef]

Trans. Am. Inst. Electr. Eng.

P. D. Joseph, J. T. Tou, “On linear control theory,” Trans. Am. Inst. Electr. Eng. 80, 193–196 (1961).

Other

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1980), p. 469.

B. Le Roux, J. M. Conan, C. Kulcsar, H. F. Reynaud, L. M. Mugnier, T. Fusco, “Optimal control law for multiconjugate adaptive optics,” in Adaptive Optical System Technologies II, P. L. Wizinowich and D. Bonaccini, eds., Proc. SPIE4839, 878–889 (2002).

D. Gavel, D. Wiberg, “Toward Strehl-optimizing adaptive optics controllers,” in Adaptive Optical System Technologies II, P. L. Wizinowich and D. Bonaccini, eds., Proc. SPIE4839, 890–901 (2002).

P. G. Hoel, S. C. Port, C. J. Stone, Introduction to Stochastic Processes (Houghton Mifflin, Boston, Mass., 1972), pp. 171–174.

R. Courant, D. Hilbert, Methods of Mathematical Physics (Interscience, New York, 1953), Vol. 1.

MATLAB Partial Differential Equation Toolbox, http://www.mathworks.com.

G. J. Baker, personal communication, g.j.baker@lmco.com.

A. V. Balakrishnan, Applied Functional Analysis (Springer-Verlag, New York, 1976), Chap. 6.

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Figures (3)

Fig. 1
Fig. 1

(a) Feedforward AO control system. (b) Feedback AO control system.

Fig. 2
Fig. 2

Gaussian random variable ϕ ( x ) and its conditional mean ϕ ̂ ( x ) , illustrating ϕ ̂ ( x ) orthogonal to the error ϕ ̃ ( x ) = ϕ ( x ) ϕ ̂ ( x ) . The wave front ϕ ( x ) is in an infinite-dimensional Hilbert space, and the best linear estimate ϕ ̂ ( x ) is a linear combination of N s measurements in a finite-dimensional space spanned by q ( x ) .

Fig. 3
Fig. 3

Spatial influence function r n ( x ) of the center actuator of 61 actuators in the hexagonal array (similar to the Lick Observatory’s AO system). Note that r n ( 0 , 0 ) = 1 and r n ( x n ) = 0 at all other actuator center points x n . The function r n ( x ) obeys the thin-plate equation. DM, deformable mirror.

Equations (88)

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d x W A ( x ) = 1 .
ϕ ( x ) = ψ ( x ) d x W A ( x ) ψ ( x ) .
s = d x w ( x ) ϕ ( x ) + v ,
v = v ϕ ( x ) = 0 .
c = l ( s ) ,
ϕ c ( x ) = r T ( x ) c ,
ε ( x ) = ϕ c ( x ) ϕ ( x ) .
s = d x w ( x ) ϕ ( x ) d x w ( x ) r T ( x ) K FB s + v .
P = d x w ( x ) r T ( x ) .
s = s P K FB s .
K FB = ( I K P ) 1 K = K ( I P K ) 1 ,
J = d x W A ( x ) ε ( x ) 2 ,
ϕ ̂ ( x ) = ϕ ( x ) s ,
ϕ ̃ ( x ) = ϕ ( x ) ϕ ̂ ( x ) .
ε 2 = ( ϕ ϕ c ) 2 = [ ϕ r T l ( s ) ] 2
ε 2 = [ ( ϕ ϕ ̂ ) + ( ϕ ̂ ϕ c ) ] 2 = ϕ ̃ 2 + 2 ϕ ̃ [ ϕ ̂ r T l ( s ) ] + [ ϕ ̂ r T l ( s ) ] 2 .
2 ϕ ̃ ( ϕ ̂ r T l ) = 2 P S , Φ ( s , ϕ ) ϕ ̃ [ ϕ ̂ r T l ( s ) ] d s d ϕ = 2 P S ( s ) { P Φ S ( ϕ s ) [ ϕ P Φ S ( ϕ s ) ϕ d ϕ ] d ϕ } [ P Φ S ( ϕ s ) ϕ d ϕ r T l ( s ) ] d s .
P Φ S ( ϕ s ) [ ϕ P Φ S ( ϕ s ) ϕ d ϕ ] d ϕ
= ϕ ̃ s = ϕ ( x ) s ϕ ̂ ( x ) = 0 .
ϕ ̃ f ( s ) = P Φ S ( ϕ s ) ϕ ̃ f ( s ) d ϕ d s = P S ( s ) f ( s ) [ P Φ S ( ϕ s ) ϕ ̃ d ϕ ] d s = 0 .
[ ϕ ϕ E ( s ) ] 2 = ( ϕ ̃ δ ϕ ) 2 = ϕ ̃ 2 2 ϕ ̃ δ ϕ + δ ϕ 2 = ϕ ̃ 2 + δ ϕ 2 ,
J = J E + J C ,
J E = d x W A ( x ) ϕ ̃ 2 ( x ) ,
J C = d x W A ( x ) [ ϕ ̂ ( x ) r T ( x ) l ( s ) ] 2 s .
J C = d x W A ( x ) [ ϕ ( x ) r T ( x ) c ] 2 .
c opt = [ d x W A ( x ) r ( x ) r T ( x ) ] 1 d x W A ( x ) r ( x ) ϕ ( x ) ,
ϕ con ( x ) = r T ( x ) c .
d x W A ( x ) r ( x ) ϕ unc ( x ) = 0 ,
d x W A ( x ) ϕ con ( x ) ϕ unc ( x ) = c T d x W A ( x ) r ( x ) ϕ unc ( x ) = 0 .
ϕ ( x ) = ϕ con ( x ) + ϕ unc ( x )
J C opt = d x W A ( x ) ϕ unc ( x ) 2 .
[ ϕ ( x ) ϕ ̂ ( x ) ] b T ( x ) s = 0 ,
ϕ ̂ ( x ) = p T ( x ) s ,
0 = [ ϕ ( x ) p T ( x ) s ] b T ( x ) [ d x w ( x ) ϕ ( x ) + v ] .
0 = b T ( x ) [ q ( x ) S p ( x ) ] ,
q ( x ) = d x w ( x ) ϕ ( x ) ϕ ( x ) ,
S = s s T = d x d x w ( x ) w T ( x ) ϕ ( x ) ϕ ( x ) + v v T .
ϕ ̂ ( x ) = q T ( x ) S 1 s .
0 = d x W A ( x ) q ( x ) = d x d x W A ( x ) w ( x ) ϕ ( x ) ϕ ( x ) ,
0 = d x w ( x ) ϕ uno ( x ) .
ϕ obs ( x ) = q T ( x ) S 0 1 d x w ( x ) ϕ ( x ) .
c opt = R 1 A S 1 s ,
R = d x W A ( x ) r ( x ) r T ( x ) ,
A = d x W A ( x ) r ( x ) q T ( x ) ,
J c = s T S 1 [ Q A T R 1 A ] S 1 s ,
Q = d x W A ( x ) q ( x ) q T ( x ) .
r opt ( x ) = q ( x ) .
J c = tr { [ Q A T F T ( F R F T ) 1 F A ] S 1 } ,
ϕ ̃ 2 ( x ) = ϕ 2 ( x ) 2 q T ( x ) S 1 ϕ ( x ) s + q T ( x ) S 1 q ( x ) .
ϕ ( x ) s = q ( x ) .
J E = d x W A ( x ) ϕ ̃ 2 ( x ) = J 0 tr Q S 1 ,
J 0 = d x W A ( x ) ϕ 2 ( x ) .
J opt = J 0 tr A T R 1 A S 1 ,
J opt = J 0 tr Q S 1 ,
[ h , k ] f = d x h T ( x ) k ( x ) f ( x ) .
h f 2 = [ h , h ] f .
tr R = r W A 2 , tr A = [ r , q ] W A ,
tr Q = q W A 2 , tr S = s W A 2 ,
f = d x w ( x ) ϕ ( x ) ,
q ( x ) = f ϕ ( x ) ,
S 0 = f f T ,
ϕ obs ( x ) = q T ( x ) S 0 1 f ,
ϕ ̂ ( x ) = q T ( x ) S 1 ( f + v ) ,
ϕ uno ( x ) = ϕ ( x ) ϕ obs ( x )
S 0 1 S 1 = S 1 v v T S 0 1 = S 0 1 v v T S 1 .
J uno = d x W A ( x ) ϕ uno ( x ) 2 .
J uno = J 0 tr Q S 0 1 .
J E = J uno + tr Q S 0 1 tr Q S 1 .
ϕ unc obs ( x ) = ϕ obs ( x ) r T ( x ) R 1 d x W A ( x ) r ( x ) ϕ obs ( x ) .
ϕ unc obs ( x ) = [ q T ( x ) r T ( x ) R 1 A ] S 0 1 f .
J unc obs = tr Q S 0 1 tr A T R 1 A S 0 1 .
J V = tr A T R 1 A ( S 0 1 S 1 ) = tr A T R 1 A S 0 1 v v T ( S 0 + v v T ) 1
J opt = J uno + J unc obs + J V .
c = l ( s ) = K s .
J k = d x W A ( x ) [ ϕ ( x ) r T ( x ) K s ] 2 .
J k = J 0 2 tr K A T + tr K S K T R ,
J K J opt = tr R ( K R 1 A S 1 ) S ( K R 1 A S 1 ) T 0 .
J NS J opt = tr M 1 ( I + M ) 1 A S 0 1 ,
s n = ϕ ( x n + h ) ϕ ( x n h )
q n ( x ) = ϕ ( x ) ϕ ( x n + h ) ϕ ( x ) ϕ ( x n h ) .
H H d x q n ( x ) = ϕ ( x n + h ) H H d x ϕ ( x ) ϕ ( x n h ) H H d x ϕ ( x ) = 0 .
ϕ ( x ) ϕ ( x ) = D ( x x ) 2 + g ( x ) + g ( x ) a 0 ,
g ( x ) = 1 2 d x W A ( x ) D ( x x ) ,
a 0 = 1 2 d x W A ( x ) g ( x ) ,
D ( x x ) = [ ψ ( x ) ψ ( x ) ] 2 = 6.8839 x x r 0 5 3 .
q n ( x ) = 3.442 r 0 5 3 ( x x n + h 5 3 x x n h 5 3 γ n ) ,
4 r ( x , y ) x 4 + 2 4 r ( x , y ) x 2 y 2 + 4 r ( x , y ) y 4 = 0 .
[ ϕ obs , ϕ uno ] = [ ϕ obs , ϕ ] [ ϕ obs , ϕ obs ] = [ q T S 0 1 f , ϕ ] [ q T S 0 1 f , f T S 0 1 q ] = 0 ,

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