Abstract

We apply robust control techniques to an adaptive optics system including a dynamic model of the deformable mirror. The dynamic model of the mirror is a modification of the usual plate equation. We propose also a state–space approach to model the turbulent phase. A continuous time control of our model is suggested, taking into account the frequential behavior of the turbulent phase. An H controller is designed in an infinite-dimensional setting. Because of the multivariable nature of the control problem involved in adaptive optics systems, a significant improvement is obtained with respect to traditional single input–single output methods.

© 2008 Optical Society of America

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References

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  1. F. Rodier, Adaptive Optics in Astronomy (Cambridge University Press, 1999).
    [CrossRef]
  2. H.-F. Raynaud, C. Kulcsár, C. Petit, J.-M. Conan, and P. V. de Lesegno, “Optimal control, observers and integrators in adaptive optics,” Opt. Express 14, 7464-7476 (2006).
    [CrossRef] [PubMed]
  3. B. W. Frazier, R. K. Tyson, M. Smith, and J. Roche, “Theory and operation of a robust controller for a compact adaptive optics system,” Opt. Eng. 43, 2912-2920 (2004).
    [CrossRef]
  4. L. Baudouin, C. Prieur, and D. Arzelier, Robust control of a bimorph mirror for adaptive optics system, in 17th International Symposium on Mathematical Theory of Networks and Systems (MTNS, 2006).
  5. D. W. Miller and S. C.O. Grocott, “Robust control of the multiple mirror telescope adaptive secondary mirror,” Opt. Eng. 38, 1276-1287 (1999).
    [CrossRef]
  6. J.-M. Conan, G. Rousset, and P. Y Madec, “Wavefront temporal spectra in high resolution imaging through turbulence,” J. Opt. Soc. Am. A 12, 1559-1570 (1995).
    [CrossRef]
  7. M. Lenczner and C. Prieur, “Asymptotic model of an active mirror,” presented at the 13th International Federation of Automatic Control Workshop on Control Applications of Optimization, Cachan, France, 2006.
  8. B. Van Keulen, “H∞ control with measurement-feedback for linear infinite-dimensional systems,” J. Math. Syst. Estim. Control 3 (4), 373-411 (1993).
  9. R. Paschall and D. Anderson, “Linear quadratic Gausian control of a deformable mirror adaptive optics system with time-delayed measurements,” Appl. Opt. 32, 6347-6358(1993).
    [CrossRef] [PubMed]
  10. S. Skogestad and I. Postlethwaite, Multivariable Feedback Control--Analysis and Design (Wiley, 1996).
  11. J. F. Nye, Physical Properties of Crystals, Their Representation by Tensors and Matrices (Oxford University Press, 1985).
  12. R. Noll, “Zernike polynomials and atmospheric turbulence,” J. Opt. Soc. Am. 66, 207-211 (1976).
    [CrossRef]
  13. C. Hogge and R. Butts, “Frequency spectra for the geometric representation of wavefront distortions due to atmospheric turbulence,” IEEE Trans. Antennas Propag. 24 (2), 144-154(1976).
  14. A. Bensoussan and P. Bernhard, “On the standard problem of H-optimal control for infinite-dimensional systems,” in Identification and Control in Systems Governed by Partial Differential Equations (Society for Industrial and Applied Mathematics, 1993) pp. 117-140.
  15. R. Curtain, A. M. Peters, and B. Van Keulen, “H∞-control with state-feedback: the infinite-dimensional case,” J. Math. Syst. Estim. Control 3 (1), 1-39 (1993).
  16. J.-L. Lions and G. Duvaut, Les Inéquations en Mécanique et en Physique (Dunod, 1972).
  17. M. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences (Springer-Verlag, 1983), Vol. 44.
    [CrossRef]
  18. Z.-H. Luo, B.-Z. Guo, and O. Morgul, Stability and Stabilization of Infinite Dimensional Systems with Applications, Communications and Control Engineering (Springer-Verlag, 1999).
    [CrossRef]
  19. M. Amabili, A. Pasqualini, and G. Dalpiaz, “Natural frequencies and modes of free-edge circular plates vibrating in vacuum or in contact with liquid,” J. Sound Vibrat. 188, 685-699 (1995).
    [CrossRef]
  20. R. D. Blevins, Formulas for Natural Frequency and Mode Shape (Kriger, 1979).

2006 (1)

2004 (1)

B. W. Frazier, R. K. Tyson, M. Smith, and J. Roche, “Theory and operation of a robust controller for a compact adaptive optics system,” Opt. Eng. 43, 2912-2920 (2004).
[CrossRef]

1999 (1)

D. W. Miller and S. C.O. Grocott, “Robust control of the multiple mirror telescope adaptive secondary mirror,” Opt. Eng. 38, 1276-1287 (1999).
[CrossRef]

1995 (2)

M. Amabili, A. Pasqualini, and G. Dalpiaz, “Natural frequencies and modes of free-edge circular plates vibrating in vacuum or in contact with liquid,” J. Sound Vibrat. 188, 685-699 (1995).
[CrossRef]

J.-M. Conan, G. Rousset, and P. Y Madec, “Wavefront temporal spectra in high resolution imaging through turbulence,” J. Opt. Soc. Am. A 12, 1559-1570 (1995).
[CrossRef]

1993 (3)

R. Paschall and D. Anderson, “Linear quadratic Gausian control of a deformable mirror adaptive optics system with time-delayed measurements,” Appl. Opt. 32, 6347-6358(1993).
[CrossRef] [PubMed]

B. Van Keulen, “H∞ control with measurement-feedback for linear infinite-dimensional systems,” J. Math. Syst. Estim. Control 3 (4), 373-411 (1993).

R. Curtain, A. M. Peters, and B. Van Keulen, “H∞-control with state-feedback: the infinite-dimensional case,” J. Math. Syst. Estim. Control 3 (1), 1-39 (1993).

1976 (1)

Amabili, M.

M. Amabili, A. Pasqualini, and G. Dalpiaz, “Natural frequencies and modes of free-edge circular plates vibrating in vacuum or in contact with liquid,” J. Sound Vibrat. 188, 685-699 (1995).
[CrossRef]

Anderson, D.

Arzelier, D.

L. Baudouin, C. Prieur, and D. Arzelier, Robust control of a bimorph mirror for adaptive optics system, in 17th International Symposium on Mathematical Theory of Networks and Systems (MTNS, 2006).

Baudouin, L.

L. Baudouin, C. Prieur, and D. Arzelier, Robust control of a bimorph mirror for adaptive optics system, in 17th International Symposium on Mathematical Theory of Networks and Systems (MTNS, 2006).

Bensoussan, A.

A. Bensoussan and P. Bernhard, “On the standard problem of H-optimal control for infinite-dimensional systems,” in Identification and Control in Systems Governed by Partial Differential Equations (Society for Industrial and Applied Mathematics, 1993) pp. 117-140.

Bernhard, P.

A. Bensoussan and P. Bernhard, “On the standard problem of H-optimal control for infinite-dimensional systems,” in Identification and Control in Systems Governed by Partial Differential Equations (Society for Industrial and Applied Mathematics, 1993) pp. 117-140.

Blevins, R. D.

R. D. Blevins, Formulas for Natural Frequency and Mode Shape (Kriger, 1979).

Butts, R.

C. Hogge and R. Butts, “Frequency spectra for the geometric representation of wavefront distortions due to atmospheric turbulence,” IEEE Trans. Antennas Propag. 24 (2), 144-154(1976).

Conan, J.-M.

Curtain, R.

R. Curtain, A. M. Peters, and B. Van Keulen, “H∞-control with state-feedback: the infinite-dimensional case,” J. Math. Syst. Estim. Control 3 (1), 1-39 (1993).

Dalpiaz, G.

M. Amabili, A. Pasqualini, and G. Dalpiaz, “Natural frequencies and modes of free-edge circular plates vibrating in vacuum or in contact with liquid,” J. Sound Vibrat. 188, 685-699 (1995).
[CrossRef]

de Lesegno, P. V.

Duvaut, G.

J.-L. Lions and G. Duvaut, Les Inéquations en Mécanique et en Physique (Dunod, 1972).

Frazier, B. W.

B. W. Frazier, R. K. Tyson, M. Smith, and J. Roche, “Theory and operation of a robust controller for a compact adaptive optics system,” Opt. Eng. 43, 2912-2920 (2004).
[CrossRef]

Grocott, S. C.O.

D. W. Miller and S. C.O. Grocott, “Robust control of the multiple mirror telescope adaptive secondary mirror,” Opt. Eng. 38, 1276-1287 (1999).
[CrossRef]

Guo, B.-Z.

Z.-H. Luo, B.-Z. Guo, and O. Morgul, Stability and Stabilization of Infinite Dimensional Systems with Applications, Communications and Control Engineering (Springer-Verlag, 1999).
[CrossRef]

Hogge, C.

C. Hogge and R. Butts, “Frequency spectra for the geometric representation of wavefront distortions due to atmospheric turbulence,” IEEE Trans. Antennas Propag. 24 (2), 144-154(1976).

Keulen, B. Van

R. Curtain, A. M. Peters, and B. Van Keulen, “H∞-control with state-feedback: the infinite-dimensional case,” J. Math. Syst. Estim. Control 3 (1), 1-39 (1993).

Kulcsár, C.

Lenczner, M.

M. Lenczner and C. Prieur, “Asymptotic model of an active mirror,” presented at the 13th International Federation of Automatic Control Workshop on Control Applications of Optimization, Cachan, France, 2006.

Lions, J.-L.

J.-L. Lions and G. Duvaut, Les Inéquations en Mécanique et en Physique (Dunod, 1972).

Luo, Z.-H.

Z.-H. Luo, B.-Z. Guo, and O. Morgul, Stability and Stabilization of Infinite Dimensional Systems with Applications, Communications and Control Engineering (Springer-Verlag, 1999).
[CrossRef]

Madec, P. Y

Miller, D. W.

D. W. Miller and S. C.O. Grocott, “Robust control of the multiple mirror telescope adaptive secondary mirror,” Opt. Eng. 38, 1276-1287 (1999).
[CrossRef]

Morgul, O.

Z.-H. Luo, B.-Z. Guo, and O. Morgul, Stability and Stabilization of Infinite Dimensional Systems with Applications, Communications and Control Engineering (Springer-Verlag, 1999).
[CrossRef]

Noll, R.

Nye, J. F.

J. F. Nye, Physical Properties of Crystals, Their Representation by Tensors and Matrices (Oxford University Press, 1985).

Paschall, R.

Pasqualini, A.

M. Amabili, A. Pasqualini, and G. Dalpiaz, “Natural frequencies and modes of free-edge circular plates vibrating in vacuum or in contact with liquid,” J. Sound Vibrat. 188, 685-699 (1995).
[CrossRef]

Pazy, M.

M. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences (Springer-Verlag, 1983), Vol. 44.
[CrossRef]

Peters, A. M.

R. Curtain, A. M. Peters, and B. Van Keulen, “H∞-control with state-feedback: the infinite-dimensional case,” J. Math. Syst. Estim. Control 3 (1), 1-39 (1993).

Petit, C.

Postlethwaite, I.

S. Skogestad and I. Postlethwaite, Multivariable Feedback Control--Analysis and Design (Wiley, 1996).

Prieur, C.

M. Lenczner and C. Prieur, “Asymptotic model of an active mirror,” presented at the 13th International Federation of Automatic Control Workshop on Control Applications of Optimization, Cachan, France, 2006.

L. Baudouin, C. Prieur, and D. Arzelier, Robust control of a bimorph mirror for adaptive optics system, in 17th International Symposium on Mathematical Theory of Networks and Systems (MTNS, 2006).

Raynaud, H.-F.

Roche, J.

B. W. Frazier, R. K. Tyson, M. Smith, and J. Roche, “Theory and operation of a robust controller for a compact adaptive optics system,” Opt. Eng. 43, 2912-2920 (2004).
[CrossRef]

Rodier, F.

F. Rodier, Adaptive Optics in Astronomy (Cambridge University Press, 1999).
[CrossRef]

Rousset, G.

Skogestad, S.

S. Skogestad and I. Postlethwaite, Multivariable Feedback Control--Analysis and Design (Wiley, 1996).

Smith, M.

B. W. Frazier, R. K. Tyson, M. Smith, and J. Roche, “Theory and operation of a robust controller for a compact adaptive optics system,” Opt. Eng. 43, 2912-2920 (2004).
[CrossRef]

Tyson, R. K.

B. W. Frazier, R. K. Tyson, M. Smith, and J. Roche, “Theory and operation of a robust controller for a compact adaptive optics system,” Opt. Eng. 43, 2912-2920 (2004).
[CrossRef]

Van Keulen, B.

B. Van Keulen, “H∞ control with measurement-feedback for linear infinite-dimensional systems,” J. Math. Syst. Estim. Control 3 (4), 373-411 (1993).

Appl. Opt. (1)

IEEE Trans. Antennas Propag. (1)

C. Hogge and R. Butts, “Frequency spectra for the geometric representation of wavefront distortions due to atmospheric turbulence,” IEEE Trans. Antennas Propag. 24 (2), 144-154(1976).

J. Math. Syst. Estim. Control (2)

B. Van Keulen, “H∞ control with measurement-feedback for linear infinite-dimensional systems,” J. Math. Syst. Estim. Control 3 (4), 373-411 (1993).

R. Curtain, A. M. Peters, and B. Van Keulen, “H∞-control with state-feedback: the infinite-dimensional case,” J. Math. Syst. Estim. Control 3 (1), 1-39 (1993).

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (1)

J. Sound Vibrat. (1)

M. Amabili, A. Pasqualini, and G. Dalpiaz, “Natural frequencies and modes of free-edge circular plates vibrating in vacuum or in contact with liquid,” J. Sound Vibrat. 188, 685-699 (1995).
[CrossRef]

Opt. Eng. (2)

B. W. Frazier, R. K. Tyson, M. Smith, and J. Roche, “Theory and operation of a robust controller for a compact adaptive optics system,” Opt. Eng. 43, 2912-2920 (2004).
[CrossRef]

D. W. Miller and S. C.O. Grocott, “Robust control of the multiple mirror telescope adaptive secondary mirror,” Opt. Eng. 38, 1276-1287 (1999).
[CrossRef]

Opt. Express (1)

Other (10)

F. Rodier, Adaptive Optics in Astronomy (Cambridge University Press, 1999).
[CrossRef]

M. Lenczner and C. Prieur, “Asymptotic model of an active mirror,” presented at the 13th International Federation of Automatic Control Workshop on Control Applications of Optimization, Cachan, France, 2006.

L. Baudouin, C. Prieur, and D. Arzelier, Robust control of a bimorph mirror for adaptive optics system, in 17th International Symposium on Mathematical Theory of Networks and Systems (MTNS, 2006).

J.-L. Lions and G. Duvaut, Les Inéquations en Mécanique et en Physique (Dunod, 1972).

M. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences (Springer-Verlag, 1983), Vol. 44.
[CrossRef]

Z.-H. Luo, B.-Z. Guo, and O. Morgul, Stability and Stabilization of Infinite Dimensional Systems with Applications, Communications and Control Engineering (Springer-Verlag, 1999).
[CrossRef]

R. D. Blevins, Formulas for Natural Frequency and Mode Shape (Kriger, 1979).

S. Skogestad and I. Postlethwaite, Multivariable Feedback Control--Analysis and Design (Wiley, 1996).

J. F. Nye, Physical Properties of Crystals, Their Representation by Tensors and Matrices (Oxford University Press, 1985).

A. Bensoussan and P. Bernhard, “On the standard problem of H-optimal control for infinite-dimensional systems,” in Identification and Control in Systems Governed by Partial Differential Equations (Society for Industrial and Applied Mathematics, 1993) pp. 117-140.

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Figures (5)

Fig. 1
Fig. 1

An adaptive optics system. The control loop consists of a SH sensor analyzing the incoming wavefront and a layer of piezoelectric sensors giving the precise position of the bimorph mirror, both of them allowing the calculation of the appropriate command of the DM in order to recover the genuine wavefront.

Fig. 2
Fig. 2

Shaping filter generating ϕ. The turbulent phase ϕ is modeled through a linear shaping filter of transfer function H from the noise w.

Fig. 3
Fig. 3

Closed-loop system. P is the system, w is the disturbance, and K is the controller that calculates the control u from the measured output y in order to control the output z.

Fig. 4
Fig. 4

Standard model for an adaptive optics system control loop.

Fig. 5
Fig. 5

Time evolution of ϕ tur L 2 ( Ω ) (solid line) and ϕ res L 2 ( Ω ) (dashed line).

Tables (4)

Tables Icon

Table 1 First 15 Zernike Functions

Tables Icon

Table 2 Atmospheric Phase Distortion State–Space Model with Average Wind Speed V = 9 m / s 1 , D r 0 = 8 , and Wavelength λ = 550 nm

Tables Icon

Table 3 Coefficients of the Eigenvectors L k j and M k j

Tables Icon

Table 4 Physical Parameters for the Numerical Simulations

Equations (38)

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

ϕ res = - 4 π λ e + ϕ tur .
y SH = - 4 π λ e + ϕ tur + c w SH ,
ρ t t e + Q 1 Δ 2 e + Q 2 e = d ˜ 31 Δ u + ρ b w mod
y pe = e ˜ 31 Δ e + d w pe .
Δ v = v r 2 + 1 r v r + 1 r 2 v θ 2 ;
2 e r 2 + ν ( 1 r e r + 1 r 2 2 e θ 2 ) | r = a = 0 , r ( Δ e ) + 1 r ( 1 - ν ) r ( 1 r e θ ) | r = a = 0.
ϕ tur ( r , θ , t ) i = 4 N Z ϕ i ( t ) Z i ( r , θ ) ,
ϕ = F ϕ + G w ,
f c i 0.3 ( n i + 1 ) V D ,
H i ( P ) = ϕ i ( p ) ω i ( p ) = 1 1 + τ i p with     τ i = 1 2 π f c i .
G G = - [ F P ϕ ( ) + P ϕ ( ) F ] .
P ϕ ( ) = cov ( ϕ i , ϕ j ) = E ( ϕ i ϕ j ) = 7.19 × 10 - 3 × ( - 1 ) ( n i + n j - m i - m j ) / 2 ( D r 0 ) 5 3 × ( n i + 1 ) ( n j + 1 ) π 8 3 × Γ ( 14 3 ) Γ ( n i + n j - 5 3 2 ) Γ ( n i - n j + 17 3 2 ) Γ ( n i - n j + 17 3 2 ) Γ ( n i + n j + 23 3 2 ) ,
{ x = A x + B 1 w + B 2 u z = C 1 x + D 12 u y = C 2 x + D 21 w ,
{ p = M p + N y u = L p + R y ,
{ t t e + Q 1 Δ 2 e + Q 2 e = d ˜ 31 Δ u + b ρ w mod t ϕ tur = F ϕ tur + G w tur ϕ res = ϕ tur - 4 π λ e y pe = e ˜ 31 Δ e + d w pe y SH = ϕ tur - 4 π λ e + c w SH .
ϕ tur i = 4 N Z ϕ i Z i and w tur = i = 4 N Z w i Z i ,
F ( φ ) = i = 4 N Z F i i φ , Z i L 2 ( Ω ) Z i , G ( φ ) = i = 4 N Z j = 4 N Z G i j φ , Z j L 2 ( Ω ) Z i ,
t ϕ tur = F ϕ tur + G w tur ,
A = ( 0 I 0 - Q 1 ρ Δ 2 - Q 2 ρ I 0 0 0 0 F ) , B 1 = ( 0 0 0 0 b 0 0 0 0 0 G 0 ) , B 2 = ( 0 d ˜ 31 ρ Δ 0 ) , C 1 = ( - 4 π λ I 0 I 0 0 0 ) , D 12 = ( 0 I ) , C 2 = ( e ˜ 31 Δ 0 0 - 4 π λ I 0 I ) , D 21 = ( 0 0 0 d 0 c 0 0 ) .
X = H b c 2 ( Ω ) × L 2 ( Ω ) × L 2 ( Ω ) = { e H 2 ( Ω ) , e satisfying   ( 5 ) } × [ L 2 ( Ω ) ] 2 ,
H 0 1 ( Ω ) = { f L 2 ( Ω ) / i = 1 , 2 , i f L 2 ( Ω ) , f | Ω = 0 } ,
H 2 ( Ω ) = { f L 2 ( Ω ) / i , j = 1 , 2 , i f , i j f L 2 ( Ω ) } .
( A * P + P A + P ( γ - 2 B 1 B 1 * - B 2 B 2 * ) P + C 1 * C 1 ) x = 0 ,
( A Q + Q A * + Q ( γ - 2 C 1 * C 1 - C 2 * C 2 ) Q + B 1 B 1 * ) x = 0 ,
M = A + ( γ - 2 B 1 B 1 * - B 2 B 2 * ) P Q ( I - γ - 2 P Q ) - 1 C 2 * C 2 , N = - Q ( I - γ - 2 P Q ) - 1 C 2 * , L = B 2 * P , R = 0 ,
ϕ res L 2 ( Ω ) + u L 2 ( Ω ) γ w [ L 2 ( Ω ) ] 4 .
A 1 D ( A 1 ) X ( e 0 e 1 e 2 ) ( 0 I 0 - Δ 2 0 0 0 0 0 ) ( e 0 e 1 e 2 ) = ( e 1 - Δ 2 e 0 0 ) ,
D ( A 1 ) = { e 0 H 4 ( Ω ) , e 0 st ( 5 ) } × H 2 ( Ω ) × L 2 ( Ω ) ,
A 1 x , x X 0
u , v H b c 2 ( Ω ) = Ω Δ u Δ v ( 1 ν ) ( 2 u x 1 2 2 v x 2 2 + 2 u x 2 2 2 v x 1 2 ) + 2 ( 1 ν ) ( 2 u x 1 x 2 2 v x 1 x 2 ) d Ω .
{ x N = A N x N + B 1 N w N + B 2 N u N z N = C 1 N x N + D 12 N u N y N = C 2 N x N + D 21 N w N ,
- Q 1 ρ Δ 2 - Q 2 ρ I
L k j ( r , θ ) = a k j [ J k ( λ k j r a ) + c k j I k ( λ k j r a ) ] cos ( k θ ) , M k j ( r , θ ) = a k j [ J k ( λ k j r a ) + c k j I k ( λ k j r a ) ] sin ( k θ ) ,
{ L k j , M k j , ( k , j ) N 2 }
    x X , x = n N , i 1 α i B i ( r , θ ) ,
ϕ res , i = - 4 π λ e i + j = 1 N Z - 2 Q i j B j + 2 .
A N = [ 0 1 N B 0 ω i 2 1 N B 0 0 0 0 F ] , B 1 N = [ 0 0 0 0 b 0 0 0 0 0 G 0 ] , B 2 N = [ 0 block i j ( d ˜ 31 ρ Δ B i , B j ) 0 ] , C 1 N = [ 4 π λ 1 N B 0 Q 0 0 0 ] , D 12 N = [ 0 1 N B ] , C 2 N = [ block i j ( e ˜ 31 Δ B i , B j ) 0 0 4 π λ 1 N B 0 Q ] , D 21 N = [ 0 0 0 d 0 c 0 0 ] ,
ϕ tur L 2 ( Ω ) = i = 4 N z ϕ i ( t ) 2 .

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