Abstract

We present a first experimental validation of vibration filtering with a Linear Quadratic Gaussian (LQG) control law in Adaptive Optics (AO). A quasi-pure mechanical vibration is generated on a classic AO bench and filtered by the control law, leading to an improvement of the Strehl Ratio and image stability. Vibration filtering may be applied to any AO system, but these results are of particular interest for eXtrem AO, and for instance for the SPHERE AO design, where high performance is required.

© 2008 Optical Society of America

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  1. G. Rousset, F. Lacombe, P. Puget, N. Hubin, E. Gendron, T. Fusco, R. Arsenault, J. Charton, P. Gigan, P. Kern, A.-M. Lagrange, P.-Y. Madec, D. Mouillet, D. Rabaud, P. Rabou, E. Stadler and G. Zins, "NAOS, the first AO system of the VLT: on sky performance," in Adaptive Optical System Technology II, Bellingham, Proc. SPIE 4839, 140-149, (2002).
    [CrossRef]
  2. Y. Cl’enet, M.E. Kasper, N. Ageorges, C. Lidman, T. Fusco, O. Marco, M. Hartung, D. Mouillet, B. Koehler, G. Rousset and N. Hubin, "NACO performance: status after 2 years of operation," in Advancements in Adaptive Optics, D. Bonaccini Calia, B. L. Ellerbroek and R. Ragazzoni eds., Proc. SPIE 5490, 107-117, (2004).
    [CrossRef]
  3. J. A. Stoesz, J.-P. V’eran, F. Rigaut, G. Herriot, L. Jolissaint and D. Frenette, "Evaluation of the on-sky performance of Altair," in Advancements in Adaptive Optics, D. Bonaccini Calia, B. L. Ellerbroek and R. Ragazzoni eds., Proc. SPIE 5490, 67-78, (2004).
    [CrossRef]
  4. C. Dessenne, P.-Y. Madec, G. Rousset, "Optimization of a predictive controller for closed-loop adaptive optics," Appl. Opt.,  37, 4623-4633, (1998).
    [CrossRef]
  5. C. Petit, J.-M. Conan, C. Kulcs’ar, H.-F. Raynaud, T. Fusco, J. Montri and D. Rabaud,"Optimal Control for Multi-conjugate Adaptive Optics," Comptes Rendus de l’Acad’emie des Science Physique 6, 1059-1069, (2005) (http://www.sciencedirect.com/science/journal/16310705).
  6. B. Le Roux, J.-M. Conan, C. Kulcs’ar, H.-F. Raynaud, L. Mugnier and T. Fusco,"Optimal control law for classical and multiconjugate adaptive optics," J. Opt. Soc. Am. A 21, 1261-1276, (2004).
    [CrossRef]
  7. C. Petit, F. Quiros-Pacheco, J.-M. Conan, C. Kulcs’ar, H.-F. Raynaud, T. Fusco, and G. Rousset, "Kalman Filter based control loop for Adaptive Optics," in Advancements in Adaptive Optics, D. Bonaccini Calia, B. L. Ellerbroek and R. Ragazzoni eds., Proc. SPIE 5490,1414-1425,(2004).
    [CrossRef]
  8. C. Kulcs’ar, H.-F. Raynaud, C. Petit,J.-M. Conan, and P. Viaris de Lesegno, "Optimal control, observers and integrators in adaptive optics," Opt. Express 14, 7464-7476, (2006)
    [CrossRef] [PubMed]
  9. F. Cottet, Traitement des signaux et acquisition de donn’ees (Dunod,2002).
  10. C. Mohtadi., "Bode’s integral theorem for discrete-time systems," Proceedings of the IEE 137, 57-66, (1990).
  11. T. Fusco, C. Petit, G. Rousset, J.-F. Sauvage, J.-M. Conan, A. Blanc and J.-L. Beuzit, "Optimization of the pre-compensation of non-common path aberrations for adaptive optics systems," in Adaptive Optics: Analysis and Methods/Computational Optical Sensing and Imaging/Information Photonics/Signal Recovery and Synthesis Topical Meetings on CD ROM, OSA electronic proceedings Technical Digest AWB 2, Optical Society of America, Charlotte, (2005).
  12. C. Dessenne, "Commande modale et pr’edictive en optique adaptative," Ph. D. thesis, Univ. Paris VII, (1998)
  13. C. Petit, "Etude de la commande optimale en Optique Adaptative et Optique Adaptative MultiConjugu’ee, validation num’erique et exp’erimentale," Ph. D. thesis, Univ. Paris XIII, (2006).
  14. R. Paschall and D. Anderson, "Linear Quadratic Gaussian control of a deformable mirror adaptive optics system with time-delayed measurements," Appl. Opt. 32, 6347-6358, (1993).
    [CrossRef] [PubMed]
  15. D. M. Wiberg, C. E. Max, and D. T. Gavel. "A special non-dynamic LQG controller: part I, application to adaptive optics," Proceedings of the 43th IEEE Conference on Decision and Control, 3:3333-3338, (2004).
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  17. T. Fusco, G. Rousset, J. -F. Sauvage, C. Petit, J. -L. Beuzit, K. Dohlen, D. Mouillet, J. Charton, M. Nicolle, M. Kasper, P. Baudoz, and P. Puget, "High-order adaptive optics requirements for direct detection of extrasolar planets: Application to the SPHERE instrument," Opt. Express 14, 7515-7534 (2006).
    [CrossRef] [PubMed]
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    [CrossRef]

2007 (1)

2006 (3)

2004 (4)

Y. Cl’enet, M.E. Kasper, N. Ageorges, C. Lidman, T. Fusco, O. Marco, M. Hartung, D. Mouillet, B. Koehler, G. Rousset and N. Hubin, "NACO performance: status after 2 years of operation," in Advancements in Adaptive Optics, D. Bonaccini Calia, B. L. Ellerbroek and R. Ragazzoni eds., Proc. SPIE 5490, 107-117, (2004).
[CrossRef]

J. A. Stoesz, J.-P. V’eran, F. Rigaut, G. Herriot, L. Jolissaint and D. Frenette, "Evaluation of the on-sky performance of Altair," in Advancements in Adaptive Optics, D. Bonaccini Calia, B. L. Ellerbroek and R. Ragazzoni eds., Proc. SPIE 5490, 67-78, (2004).
[CrossRef]

B. Le Roux, J.-M. Conan, C. Kulcs’ar, H.-F. Raynaud, L. Mugnier and T. Fusco,"Optimal control law for classical and multiconjugate adaptive optics," J. Opt. Soc. Am. A 21, 1261-1276, (2004).
[CrossRef]

C. Petit, F. Quiros-Pacheco, J.-M. Conan, C. Kulcs’ar, H.-F. Raynaud, T. Fusco, and G. Rousset, "Kalman Filter based control loop for Adaptive Optics," in Advancements in Adaptive Optics, D. Bonaccini Calia, B. L. Ellerbroek and R. Ragazzoni eds., Proc. SPIE 5490,1414-1425,(2004).
[CrossRef]

2002 (1)

G. Rousset, F. Lacombe, P. Puget, N. Hubin, E. Gendron, T. Fusco, R. Arsenault, J. Charton, P. Gigan, P. Kern, A.-M. Lagrange, P.-Y. Madec, D. Mouillet, D. Rabaud, P. Rabou, E. Stadler and G. Zins, "NAOS, the first AO system of the VLT: on sky performance," in Adaptive Optical System Technology II, Bellingham, Proc. SPIE 4839, 140-149, (2002).
[CrossRef]

1998 (1)

1993 (1)

1990 (1)

C. Mohtadi., "Bode’s integral theorem for discrete-time systems," Proceedings of the IEE 137, 57-66, (1990).

Appl. Opt. (2)

J. Opt. Soc. Am. A (3)

Opt. Express (2)

Proc. SPIE (4)

C. Petit, F. Quiros-Pacheco, J.-M. Conan, C. Kulcs’ar, H.-F. Raynaud, T. Fusco, and G. Rousset, "Kalman Filter based control loop for Adaptive Optics," in Advancements in Adaptive Optics, D. Bonaccini Calia, B. L. Ellerbroek and R. Ragazzoni eds., Proc. SPIE 5490,1414-1425,(2004).
[CrossRef]

G. Rousset, F. Lacombe, P. Puget, N. Hubin, E. Gendron, T. Fusco, R. Arsenault, J. Charton, P. Gigan, P. Kern, A.-M. Lagrange, P.-Y. Madec, D. Mouillet, D. Rabaud, P. Rabou, E. Stadler and G. Zins, "NAOS, the first AO system of the VLT: on sky performance," in Adaptive Optical System Technology II, Bellingham, Proc. SPIE 4839, 140-149, (2002).
[CrossRef]

Y. Cl’enet, M.E. Kasper, N. Ageorges, C. Lidman, T. Fusco, O. Marco, M. Hartung, D. Mouillet, B. Koehler, G. Rousset and N. Hubin, "NACO performance: status after 2 years of operation," in Advancements in Adaptive Optics, D. Bonaccini Calia, B. L. Ellerbroek and R. Ragazzoni eds., Proc. SPIE 5490, 107-117, (2004).
[CrossRef]

J. A. Stoesz, J.-P. V’eran, F. Rigaut, G. Herriot, L. Jolissaint and D. Frenette, "Evaluation of the on-sky performance of Altair," in Advancements in Adaptive Optics, D. Bonaccini Calia, B. L. Ellerbroek and R. Ragazzoni eds., Proc. SPIE 5490, 67-78, (2004).
[CrossRef]

Proceedings of the IEE (1)

C. Mohtadi., "Bode’s integral theorem for discrete-time systems," Proceedings of the IEE 137, 57-66, (1990).

Other (6)

T. Fusco, C. Petit, G. Rousset, J.-F. Sauvage, J.-M. Conan, A. Blanc and J.-L. Beuzit, "Optimization of the pre-compensation of non-common path aberrations for adaptive optics systems," in Adaptive Optics: Analysis and Methods/Computational Optical Sensing and Imaging/Information Photonics/Signal Recovery and Synthesis Topical Meetings on CD ROM, OSA electronic proceedings Technical Digest AWB 2, Optical Society of America, Charlotte, (2005).

C. Dessenne, "Commande modale et pr’edictive en optique adaptative," Ph. D. thesis, Univ. Paris VII, (1998)

C. Petit, "Etude de la commande optimale en Optique Adaptative et Optique Adaptative MultiConjugu’ee, validation num’erique et exp’erimentale," Ph. D. thesis, Univ. Paris XIII, (2006).

D. M. Wiberg, C. E. Max, and D. T. Gavel. "A special non-dynamic LQG controller: part I, application to adaptive optics," Proceedings of the 43th IEEE Conference on Decision and Control, 3:3333-3338, (2004).

C. Petit, J.-M. Conan, C. Kulcs’ar, H.-F. Raynaud, T. Fusco, J. Montri and D. Rabaud,"Optimal Control for Multi-conjugate Adaptive Optics," Comptes Rendus de l’Acad’emie des Science Physique 6, 1059-1069, (2005) (http://www.sciencedirect.com/science/journal/16310705).

F. Cottet, Traitement des signaux et acquisition de donn’ees (Dunod,2002).

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Figures (4)

Fig. 1.
Fig. 1.

Numerical simulation of three vibrations filtering (pure at 6,15 Hz, large at 8 Hz). Transfer |Tx (ν)| of the x average slope is plotted for standard and vibration filtering LQG control law.

Fig. 2.
Fig. 2.

Picture of the experimental set-up for vibration filtering validation. The AO bench is on the right, under its baffling. The seismic vibration generator is in the middle of the picture, placed on its stage and against the AO bench (cylindrical black component).

Fig. 3.
Fig. 3.

Experimental CTPSD of the x average slope for a LQG control law, without vibration (dotted line), with vibration and no specific filtering (dashed), and with vibration filtering (solid). Experimental error bound is lower than ±2.5 10-4(λ/D)2.

Fig. 4.
Fig. 4.

Comparison of the experimental x average slope transfer. Top is obtained with integrator (dotted) and compared to theory (solid). Bottom is obtained with vibration filtering LQG control (dotted) and compared to numerical simulation (solid).

Tables (1)

Tables Icon

Table 1. Performance in terms of Strehl Ratio (SR) for the various control laws, in presence or not of vibration. SR are given with a ±0.5% error bar. Error on SR evaluation is statistically estimated on a large number of measurements and error bar corresponds to ± the root mean square.

Equations (26)

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ϕ n 1 cor = Nu n 2 ,
y n = D ( ϕ n 1 ϕ n 1 cor ) + w n ,
ε ( u n ) = ϕ n + 1 N u n 2 turb , noise .
x n + 1 = A x n + B u n + ν n ,
y n = C x n + w n ,
x n t = ( ϕ n t , ϕ n 1 t , u n 1 t , u n 2 t ) ,
ϕ n + 1 = A tur ϕ n + ν n ,
x n + 1 = ( A tur 0 0 0 Id 0 0 0 0 0 0 0 0 0 Id 0 ) x n + ( 0 0 Id 0 ) u n + ( Id 0 0 0 ) ν n ,
y n = D ( 0 Id 0 N ) x n + w n .
x ̂ n + 1 n = A x ̂ n n 1 + B u n + L n ( y n C x ̂ n n 1 ) .
L n = A n n 1 C t ( C n n 1 C t + w ) 1 ,
n + 1 n = A n n 1 A t + v A n n 1 C t ( C n n 1 C t + w ) 1 C n n 1 A t .
u n = K x ̂ n + 1 n ,
ϕ n glob = ϕ n + ϕ n vib .
ϕ ̈ + 2 K ω 0 ϕ ̇ + ω 0 2 ϕ = C ω 0 2 ξ ,
ϕ ˜ ( p ) = H ( p ) ξ ˜ ( p ) ,
H ( p ) = C ω 0 2 p 2 + 2 K ω 0 p + ω 0 2 .
ϕ n = ϕ ( n T ) , ξ n = ξ ( n T ) .
ϕ ˜ ( z ) = H ( z ) ξ ˜ ( z ) .
z = e pT .
H ( z ) = C ω 0 2 T 1 1 2 e K ω 0 T cos ( ω 0 T 1 K 2 ) z 1 + e 2 K ω 0 T z 2 .
ϕ n vib = a 1 ϕ n 1 vib + a 2 ϕ n 2 vib + ξ n ,
a 1 = 2 e K ω 0 T cos ( ω 0 T 1 K 2 ) , a 2 = e 2 K ω 0 T .
x n t = ( ϕ n vib t , ϕ n 1 vib t , ϕ n t , ϕ n 1 t , u n 1 t , u n 2 t ) .
y n = D ( ϕ n 1 glob ϕ n 1 cor ) + w n .
T x , y ( ν ) = PSD cl x , y ( ν ) PSD ol x , y ( ν ) .

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