Mitigation of vibrations in adaptive optics by minimization of closed-loop residuals

Andres Guesalaga; Benoit Neichel; Jared O’Neal; Dani Guzman

doi:10.1364/OE.21.010676

Mitigation of vibrations in adaptive optics by minimization of closed-loop residuals

Andres Guesalaga, Benoit Neichel, Jared O’Neal, and Dani Guzman

Optics Express
Vol. 21,
Issue 9,
pp. 10676-10696
(2013)
•https://doi.org/10.1364/OE.21.010676

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Andres Guesalaga, Benoit Neichel, Jared O’Neal, and Dani Guzman, "Mitigation of vibrations in adaptive optics by minimization of closed-loop residuals," Opt. Express 21, 10676-10696 (2013)

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Related Topics
Table of Contents Category
- Adaptive Optics
Optics & Photonics Topics
?

The topics in this list come from the OSA Optics and Photonics Topics applied to this article.
Previously assigned OCIS codes
- Active or adaptive optics (010.1080)
- Atmospheric correction (010.1285)
- Astronomical optics (350.1260)

About this Article
History
- Original Manuscript: January 25, 2013
- Revised Manuscript: April 15, 2013
- Manuscript Accepted: April 20, 2013
- Published: April 24, 2013

Accessible

Open Access

Abstract

We describe a new technique to reduce tip and tilt vibrations via the design of adaptive optics controllers in a frequency framework. The method synthesizes controllers by minimizing an H₂ norm of the tip and tilt residuals. In this approach, open loop slopes (pseudo-open-loop) are reconstructed from on-sky data and input into off-line simulations of the adaptive optics system. The proposed procedure executes a sequence of off-line closed-loop runs with increasing controller complexity and searches for the controller that minimizes the variance of residuals. Although the method avoids any identification of the vibration and turbulence models during the controller synthesis, the actual models are indirectly constructed as a by-product of the H₂ norm minimization. The technique has been implemented on and tested with two operational instruments, namely Paranal’s NACO and Gemini-South’s GeMS, showing an effective rejection of the main vibrations in the loop and also improving the overall performance of the system over varying turbulence conditions. It is shown that a superior performance is obtained when compared to the standard integrator controller.

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References

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C. Kulcsár, G. Sivo, H.-F. Raynaud, B. Neichel, F. Rigaut, J. Christou, A. Guesalaga, C. Correia, J.-P. Véran, E. Gendron, F. Vidal, G. Rousset, T. Morris, S. Esposito, F. Quiros-Pacheco, G. Agapito, E. Fedrigo, L. Pettazzi, R. Clare, R. Muradore, O. Guyon, F. Martinache, S. Meimon, and J. M. Conan, “Vibrations in AO control: a short analysis of on-sky data around the world,” Proc. SPIE 8447, Adaptive Optics Systems III, 84471C (2012).
J. Maly, E. Darren, and T. Pargett, “Vibration suppression for the Gemini planet image,” Proc. SPIE 7733, Ground-based and Airborne Telescopes III,844711 (2010).
C. Petit, J.-M. Conan, C. Kulcsár, H.-F. Raynaud, T. Fusco, J. Montri, and D. Rabaud, “Linear quadratic Gaussian control for adaptive optics and multiconjugate adaptive optics: experimental and numerical analysis,” J. Opt. Soc. Am. A 26(6), 1307–1325 (2009).
[Crossref] [PubMed]
C. Petit, J. M. Conan, T. Fusco, J. Montri, C. Kulcsár, H. F. Raynaud, and D. Rabaud, “First laboratory demonstration of closed-loop Kalman based optimal control for vibration filtering and simplified MCAO,” Proc. SPIE 6272, Advances in Adaptive Optics II, 62721T (2006).
[Crossref]
B. Neichel, F. Rigaut, A. Guesalaga, I. Rodriguez, and D. Guzman, “Kalman and H-infinity controllers for GeMS,” OSA Conf. on Adaptive Optics (2011).
G. Agapito, F. Quirós-Pacheco, P. Tesi, A. Riccardi, and S. Esposito, “Observer-based control techniques for the LBT adaptive optics under telescope vibrations,” Euro. J. of Control 17(3), 316–326 (2011).
[Crossref]
J. M. Conan, H. F. Raynaud, C. Kulcsár, and S. Meimon, “Are integral controllers adapted to the new era of ELT adaptive optics?” Proc. of the 2nd AO4ELT Conference (2011).
E. Fedrigo, R. Muradore, and D. Zilio, “High performance adaptive optics system with fine tip/tilt control,” Control Eng. Pract. 17(1), 122–135 (2009).
[Crossref]
B. Le Roux, J. M. Conan, C. Kulcsár, H. F. Raynaud, L. M. Mugnier, and T. Fusco, “Optimal control law for classical and multiconjugate adaptative optics,” J. Opt. Soc. Am. A 21(7), 1261–1276 (2004).
[Crossref]
C. Kulcsár, H. F. Raynaud, C. Petit, J. M. Conan, and P. Viaris de Lesegno, “Optimal control, observers and integrators in adaptive optics,” Opt. Express 14(17), 7464–7476 (2006).
[Crossref] [PubMed]
A. Guesalaga, B. Neichel, F. Rigaut, J. Osborn, and D. Guzman, “Comparison of vibration mitigation controllers for adaptive optics systems,” Appl. Opt. 51(19), 4520–4535 (2012).
[Crossref] [PubMed]
C. Correia, H. F. Raynaud, C. Kulcsár, and J. M. Conan, “On the optimal reconstruction and control of adaptive optical systems with mirror dynamics,” J. Opt. Soc. Am. A 27(2), 333–349 (2010).
[Crossref] [PubMed]
C. Correia, J. P. Véran, and G. Herriot, “Advanced vibration suppression algorithms in adaptive optics systems,” J. Opt. Soc. Am. A 29(3), 185–194 (2012).
[Crossref] [PubMed]
J. W. Hardy, Adaptive Optics for Astronomical Telescopes (Oxford University, 1998).
J. C. Doyle, K. Glover, P. P. Khargonekar, and B. A. Francis, “State-space solutions to standard H2 and H∞ control problems,” IEEE Trans. Automat. Contr. 34(8), 831–847 (1989).
[Crossref]
A. Guesalaga, B. Neichel, F. Rigaut, J. Osborn, and D. Guzman, “Design of frequency-based controllers for vibration mitigation at the Gemini-South telescope,” Proc. SPIE. 8447, Adaptive Optics Systems III, 844711 (2012).
B. L. Ellerbroek and F. Rigaut, “Methods for correcting tilt anisoplanatism in laser-guide-star-based multiconjugate adaptive optics,” J. Opt. Soc. Am. A 18(10), 2539–2547 (2001).
[Crossref] [PubMed]
B. Neichel, F. Rigaut, A. Serio, G. Arriagada, M. Boccas, C. d'Orgeville, V. Fesquet, C. Trujillo, W. N. Rambold, R. L. Galvez, G. Gausachs, T. B. Vucina, V. Montes, C. Urrutia, C. Moreno, S. J. Diggs, C. Araya, J. Lührs, G. Trancho, M. Bec, C. Marchant, F. Collao, E. R. Carrasco, M. L. Edwards, P. Pessev, A. Lopez, and H. Diaz, “Science readiness of the Gemini MCAO System: GeMS,” Proc. SPIE 8447, Adaptive Optics Systems III, 84474Q (2012).
G. Rousset, F. Lacombe, P. Puget, N. N. Hubin, E. Gendron, T. Fusco, R. Arsenault, J. Charton, P. Feautrier, P. Gigan, P. Y. 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,” Proc. SPIE. 4839, Adaptive Optical System Technologies II,140 (2003).
S. Meimon, C. Petit, T. Fusco, and C. Kulcsár, “Tip-tilt disturbance model identification for Kalman-based control scheme: application to XAO and ELT systems,” J. Opt. Soc. Am. A 27(11), A122–A132 (2010).
[Crossref] [PubMed]
J. M. Conan, G. Rousset, and P. Y. Madec, “Wave-front temporal spectra in high-resolution imaging through turbulence,” J. Opt. Soc. Am. A 12(7), 1559–1570 (1995).
[Crossref]
E. Gendron and P. Lena, “Astronomical adaptive optics I. modal control optimization,” Astron. Astrophys. 291, 337 (1994).
J. A. Nelder and R. Mead, “A Simplex Method for Function Minimization,” Comput. J. 7(4), 308–313 (1965).
[Crossref]

2012 (5)

C. Kulcsár, G. Sivo, H.-F. Raynaud, B. Neichel, F. Rigaut, J. Christou, A. Guesalaga, C. Correia, J.-P. Véran, E. Gendron, F. Vidal, G. Rousset, T. Morris, S. Esposito, F. Quiros-Pacheco, G. Agapito, E. Fedrigo, L. Pettazzi, R. Clare, R. Muradore, O. Guyon, F. Martinache, S. Meimon, and J. M. Conan, “Vibrations in AO control: a short analysis of on-sky data around the world,” Proc. SPIE 8447, Adaptive Optics Systems III, 84471C (2012).

A. Guesalaga, B. Neichel, F. Rigaut, J. Osborn, and D. Guzman, “Comparison of vibration mitigation controllers for adaptive optics systems,” Appl. Opt. 51(19), 4520–4535 (2012).
[Crossref] [PubMed]

C. Correia, J. P. Véran, and G. Herriot, “Advanced vibration suppression algorithms in adaptive optics systems,” J. Opt. Soc. Am. A 29(3), 185–194 (2012).
[Crossref] [PubMed]

A. Guesalaga, B. Neichel, F. Rigaut, J. Osborn, and D. Guzman, “Design of frequency-based controllers for vibration mitigation at the Gemini-South telescope,” Proc. SPIE. 8447, Adaptive Optics Systems III, 844711 (2012).

B. Neichel, F. Rigaut, A. Serio, G. Arriagada, M. Boccas, C. d'Orgeville, V. Fesquet, C. Trujillo, W. N. Rambold, R. L. Galvez, G. Gausachs, T. B. Vucina, V. Montes, C. Urrutia, C. Moreno, S. J. Diggs, C. Araya, J. Lührs, G. Trancho, M. Bec, C. Marchant, F. Collao, E. R. Carrasco, M. L. Edwards, P. Pessev, A. Lopez, and H. Diaz, “Science readiness of the Gemini MCAO System: GeMS,” Proc. SPIE 8447, Adaptive Optics Systems III, 84474Q (2012).

2011 (1)

G. Agapito, F. Quirós-Pacheco, P. Tesi, A. Riccardi, and S. Esposito, “Observer-based control techniques for the LBT adaptive optics under telescope vibrations,” Euro. J. of Control 17(3), 316–326 (2011).
[Crossref]

2010 (3)

C. Correia, H. F. Raynaud, C. Kulcsár, and J. M. Conan, “On the optimal reconstruction and control of adaptive optical systems with mirror dynamics,” J. Opt. Soc. Am. A 27(2), 333–349 (2010).
[Crossref] [PubMed]

J. Maly, E. Darren, and T. Pargett, “Vibration suppression for the Gemini planet image,” Proc. SPIE 7733, Ground-based and Airborne Telescopes III,844711 (2010).

S. Meimon, C. Petit, T. Fusco, and C. Kulcsár, “Tip-tilt disturbance model identification for Kalman-based control scheme: application to XAO and ELT systems,” J. Opt. Soc. Am. A 27(11), A122–A132 (2010).
[Crossref] [PubMed]

2009 (2)

C. Petit, J.-M. Conan, C. Kulcsár, H.-F. Raynaud, T. Fusco, J. Montri, and D. Rabaud, “Linear quadratic Gaussian control for adaptive optics and multiconjugate adaptive optics: experimental and numerical analysis,” J. Opt. Soc. Am. A 26(6), 1307–1325 (2009).
[Crossref] [PubMed]

E. Fedrigo, R. Muradore, and D. Zilio, “High performance adaptive optics system with fine tip/tilt control,” Control Eng. Pract. 17(1), 122–135 (2009).
[Crossref]

2006 (2)

C. Petit, J. M. Conan, T. Fusco, J. Montri, C. Kulcsár, H. F. Raynaud, and D. Rabaud, “First laboratory demonstration of closed-loop Kalman based optimal control for vibration filtering and simplified MCAO,” Proc. SPIE 6272, Advances in Adaptive Optics II, 62721T (2006).
[Crossref]

C. Kulcsár, H. F. Raynaud, C. Petit, J. M. Conan, and P. Viaris de Lesegno, “Optimal control, observers and integrators in adaptive optics,” Opt. Express 14(17), 7464–7476 (2006).
[Crossref] [PubMed]

2004 (1)

B. Le Roux, J. M. Conan, C. Kulcsár, H. F. Raynaud, L. M. Mugnier, and T. Fusco, “Optimal control law for classical and multiconjugate adaptative optics,” J. Opt. Soc. Am. A 21(7), 1261–1276 (2004).
[Crossref]

2003 (1)

G. Rousset, F. Lacombe, P. Puget, N. N. Hubin, E. Gendron, T. Fusco, R. Arsenault, J. Charton, P. Feautrier, P. Gigan, P. Y. 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,” Proc. SPIE. 4839, Adaptive Optical System Technologies II,140 (2003).

2001 (1)

B. L. Ellerbroek and F. Rigaut, “Methods for correcting tilt anisoplanatism in laser-guide-star-based multiconjugate adaptive optics,” J. Opt. Soc. Am. A 18(10), 2539–2547 (2001).
[Crossref] [PubMed]

1995 (1)

J. M. Conan, G. Rousset, and P. Y. Madec, “Wave-front temporal spectra in high-resolution imaging through turbulence,” J. Opt. Soc. Am. A 12(7), 1559–1570 (1995).
[Crossref]

1994 (1)

E. Gendron and P. Lena, “Astronomical adaptive optics I. modal control optimization,” Astron. Astrophys. 291, 337 (1994).

1989 (1)

J. C. Doyle, K. Glover, P. P. Khargonekar, and B. A. Francis, “State-space solutions to standard H2 and H∞ control problems,” IEEE Trans. Automat. Contr. 34(8), 831–847 (1989).
[Crossref]

1965 (1)

J. A. Nelder and R. Mead, “A Simplex Method for Function Minimization,” Comput. J. 7(4), 308–313 (1965).
[Crossref]

Agapito, G.

Araya, C.

Arriagada, G.

Arsenault, R.

Bec, M.

Boccas, M.

Carrasco, E. R.

Charton, J.

Christou, J.

Clare, R.

Collao, F.

Conan, J. M.

J. M. Conan, G. Rousset, and P. Y. Madec, “Wave-front temporal spectra in high-resolution imaging through turbulence,” J. Opt. Soc. Am. A 12(7), 1559–1570 (1995).
[Crossref]

Conan, J.-M.

Correia, C.

C. Correia, J. P. Véran, and G. Herriot, “Advanced vibration suppression algorithms in adaptive optics systems,” J. Opt. Soc. Am. A 29(3), 185–194 (2012).
[Crossref] [PubMed]

Darren, E.

J. Maly, E. Darren, and T. Pargett, “Vibration suppression for the Gemini planet image,” Proc. SPIE 7733, Ground-based and Airborne Telescopes III,844711 (2010).

Diaz, H.

Diggs, S. J.

d'Orgeville, C.

Doyle, J. C.

J. C. Doyle, K. Glover, P. P. Khargonekar, and B. A. Francis, “State-space solutions to standard H2 and H∞ control problems,” IEEE Trans. Automat. Contr. 34(8), 831–847 (1989).
[Crossref]

Edwards, M. L.

Ellerbroek, B. L.

Esposito, S.

Feautrier, P.

Fedrigo, E.

E. Fedrigo, R. Muradore, and D. Zilio, “High performance adaptive optics system with fine tip/tilt control,” Control Eng. Pract. 17(1), 122–135 (2009).
[Crossref]

Fesquet, V.

Francis, B. A.

J. C. Doyle, K. Glover, P. P. Khargonekar, and B. A. Francis, “State-space solutions to standard H2 and H∞ control problems,” IEEE Trans. Automat. Contr. 34(8), 831–847 (1989).
[Crossref]

Fusco, T.

Galvez, R. L.

Gausachs, G.

Gendron, E.

E. Gendron and P. Lena, “Astronomical adaptive optics I. modal control optimization,” Astron. Astrophys. 291, 337 (1994).

Gigan, P.

Glover, K.

J. C. Doyle, K. Glover, P. P. Khargonekar, and B. A. Francis, “State-space solutions to standard H2 and H∞ control problems,” IEEE Trans. Automat. Contr. 34(8), 831–847 (1989).
[Crossref]

Guesalaga, A.

B. Neichel, F. Rigaut, A. Guesalaga, I. Rodriguez, and D. Guzman, “Kalman and H-infinity controllers for GeMS,” OSA Conf. on Adaptive Optics (2011).

Guyon, O.

Guzman, D.

B. Neichel, F. Rigaut, A. Guesalaga, I. Rodriguez, and D. Guzman, “Kalman and H-infinity controllers for GeMS,” OSA Conf. on Adaptive Optics (2011).

Herriot, G.

C. Correia, J. P. Véran, and G. Herriot, “Advanced vibration suppression algorithms in adaptive optics systems,” J. Opt. Soc. Am. A 29(3), 185–194 (2012).
[Crossref] [PubMed]

Hubin, N. N.

Kern, P. Y.

Khargonekar, P. P.

J. C. Doyle, K. Glover, P. P. Khargonekar, and B. A. Francis, “State-space solutions to standard H2 and H∞ control problems,” IEEE Trans. Automat. Contr. 34(8), 831–847 (1989).
[Crossref]

Kulcsár, C.

Lacombe, F.

Lagrange, A.-M.

Le Roux, B.

Lena, P.

E. Gendron and P. Lena, “Astronomical adaptive optics I. modal control optimization,” Astron. Astrophys. 291, 337 (1994).

Lopez, A.

Lührs, J.

Madec, P. Y.

J. M. Conan, G. Rousset, and P. Y. Madec, “Wave-front temporal spectra in high-resolution imaging through turbulence,” J. Opt. Soc. Am. A 12(7), 1559–1570 (1995).
[Crossref]

Maly, J.

J. Maly, E. Darren, and T. Pargett, “Vibration suppression for the Gemini planet image,” Proc. SPIE 7733, Ground-based and Airborne Telescopes III,844711 (2010).

Marchant, C.

Martinache, F.

Mead, R.

J. A. Nelder and R. Mead, “A Simplex Method for Function Minimization,” Comput. J. 7(4), 308–313 (1965).
[Crossref]

Meimon, S.

Montes, V.

Montri, J.

Moreno, C.

Morris, T.

Mouillet, D.

Mugnier, L. M.

Muradore, R.

E. Fedrigo, R. Muradore, and D. Zilio, “High performance adaptive optics system with fine tip/tilt control,” Control Eng. Pract. 17(1), 122–135 (2009).
[Crossref]

Neichel, B.

B. Neichel, F. Rigaut, A. Guesalaga, I. Rodriguez, and D. Guzman, “Kalman and H-infinity controllers for GeMS,” OSA Conf. on Adaptive Optics (2011).

Nelder, J. A.

J. A. Nelder and R. Mead, “A Simplex Method for Function Minimization,” Comput. J. 7(4), 308–313 (1965).
[Crossref]

Osborn, J.

Pargett, T.

J. Maly, E. Darren, and T. Pargett, “Vibration suppression for the Gemini planet image,” Proc. SPIE 7733, Ground-based and Airborne Telescopes III,844711 (2010).

Pessev, P.

Petit, C.

Pettazzi, L.

Puget, P.

Quiros-Pacheco, F.

Quirós-Pacheco, F.

Rabaud, D.

Rabou, P.

Rambold, W. N.

Raynaud, H. F.

Raynaud, H.-F.

Riccardi, A.

Rigaut, F.

B. Neichel, F. Rigaut, A. Guesalaga, I. Rodriguez, and D. Guzman, “Kalman and H-infinity controllers for GeMS,” OSA Conf. on Adaptive Optics (2011).

Rodriguez, I.

B. Neichel, F. Rigaut, A. Guesalaga, I. Rodriguez, and D. Guzman, “Kalman and H-infinity controllers for GeMS,” OSA Conf. on Adaptive Optics (2011).

Rousset, G.

J. M. Conan, G. Rousset, and P. Y. Madec, “Wave-front temporal spectra in high-resolution imaging through turbulence,” J. Opt. Soc. Am. A 12(7), 1559–1570 (1995).
[Crossref]

Serio, A.

Sivo, G.

Stadler, E.

Tesi, P.

Trancho, G.

Trujillo, C.

Urrutia, C.

Véran, J. P.

C. Correia, J. P. Véran, and G. Herriot, “Advanced vibration suppression algorithms in adaptive optics systems,” J. Opt. Soc. Am. A 29(3), 185–194 (2012).
[Crossref] [PubMed]

Véran, J.-P.

Viaris de Lesegno, P.

Vidal, F.

Vucina, T. B.

Zilio, D.

E. Fedrigo, R. Muradore, and D. Zilio, “High performance adaptive optics system with fine tip/tilt control,” Control Eng. Pract. 17(1), 122–135 (2009).
[Crossref]

Zins, G.

Adaptive Optics Systems (1)

Advances in Adaptive Optics II (1)

Appl. Opt. (1)

Astron. Astrophys. (1)

E. Gendron and P. Lena, “Astronomical adaptive optics I. modal control optimization,” Astron. Astrophys. 291, 337 (1994).

Comput. J. (1)

J. A. Nelder and R. Mead, “A Simplex Method for Function Minimization,” Comput. J. 7(4), 308–313 (1965).
[Crossref]

Control Eng. Pract. (1)

E. Fedrigo, R. Muradore, and D. Zilio, “High performance adaptive optics system with fine tip/tilt control,” Control Eng. Pract. 17(1), 122–135 (2009).
[Crossref]

Euro. J. of Control (1)

Ground-based and Airborne Telescopes III (1)

J. Maly, E. Darren, and T. Pargett, “Vibration suppression for the Gemini planet image,” Proc. SPIE 7733, Ground-based and Airborne Telescopes III,844711 (2010).

IEEE Trans. Automat. Contr. (1)

J. C. Doyle, K. Glover, P. P. Khargonekar, and B. A. Francis, “State-space solutions to standard H2 and H∞ control problems,” IEEE Trans. Automat. Contr. 34(8), 831–847 (1989).
[Crossref]

J. Opt. Soc. Am. A (7)

C. Correia, J. P. Véran, and G. Herriot, “Advanced vibration suppression algorithms in adaptive optics systems,” J. Opt. Soc. Am. A 29(3), 185–194 (2012).
[Crossref] [PubMed]

J. M. Conan, G. Rousset, and P. Y. Madec, “Wave-front temporal spectra in high-resolution imaging through turbulence,” J. Opt. Soc. Am. A 12(7), 1559–1570 (1995).
[Crossref]

Opt. Express (1)

Proc. SPIE (3)

Other (3)

J. W. Hardy, Adaptive Optics for Astronomical Telescopes (Oxford University, 1998).

J. M. Conan, H. F. Raynaud, C. Kulcsár, and S. Meimon, “Are integral controllers adapted to the new era of ELT adaptive optics?” Proc. of the 2nd AO4ELT Conference (2011).

B. Neichel, F. Rigaut, A. Guesalaga, I. Rodriguez, and D. Guzman, “Kalman and H-infinity controllers for GeMS,” OSA Conf. on Adaptive Optics (2011).

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Figures (17)

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Fig. 1 Standard configuration for H_∞/H₂ synthesis [15].

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Fig. 2 Left: servo-control problem. Right: the rearrangement of the servo loop into the augmented configuration used to synthesize the H₂ controller. Functions W_e(s) and W_y(s) have been introduced to penalize control errors and weigh the noise, respectively.

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Fig. 3 On-sky PSDs for the measured tilt from NGSWFS POL slopes. Left: data from March 2011, Right: data acquired in April 2011. The noise level is estimated from the rms values of the residuals above 200 Hz.

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Fig. 4 Step 2: Minimum variance for optimal integrator gain for the slow and fast turbulences.

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Fig. 5 Step 3: residual PSD for optimum first order turbulence model. PSDs of integrator residuals are also shown.

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Fig. 6 Step 3: Weighting functions W_e and W_y that result from the optimal C₀ and C₁ values.

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Fig. 7 Frequency response of notch filters for ω_o = 2π∙50 rad/s.

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Fig. 8 Step 4: scanning of the notch filter in search for vibrations. The minimum of the variance (bottom plot) corresponds to the strong vibration shown in the top panel.

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Fig. 9 Step 4: PSDs of disturbances and noise for minimum variance.

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Fig. 10 Step 5: residual PSD for the fifth order W_e. PSDs of residuals generated by the integrators are also shown.

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Fig. 11 Step 5: PSD of tilt slopes and associated model for minimum variance.

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Fig. 12 Step 5: Sensitivity functions for integral and final H₂ controllers.

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Fig. 13 Progressive reduction of the variance of residuals for the sequence of design steps.

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Fig. 14 Left: open loop PSDs of residuals and associated model (coarse line); Right: PSDs of NAOS residuals using the default integrator and H₂ controllers. Data from the night of November 2nd, 2011.

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Fig. 15 Reduction in tip-tilt residual variance for increasing complexity of controller.

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Fig. 16 Result from optical bench experiments using artificially generated turbulence and vibrations. Left: PSDs of POL slopes and the associated tilt residuals for the H₂ and integrator controllers. Right: sensitivity functions for integral and H₂ controllers.

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Fig. 17 Results from on-line implementation for on-sky run (May 10th, 2012). Left: PSDs of POL tilt and residuals for the H₂ and integrator controllers. The variance for the integrator and H₂ controllers are 9.7e-4 arcsec² and 7.5e-4 arcsec², respectively. Right: sensitivity functions for integral and H₂ controllers.

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Tables (2)

Table 1 Reduction in residuals for the classical integrator and H₂ controllers

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Table 2 Processing time (per axis) required for different size of disturbance models and resulting controller size

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Equations (25)

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C (z) = \frac{K_{i}}{1 - a z^{- 1}},

[\begin{matrix} z \\ e \end{matrix}] = P (s) [\begin{matrix} w \\ u \end{matrix}] = [\begin{matrix} P_{11} (s) & P_{12} (s) \\ P_{21} (s) & P_{22} (s) \end{matrix}] [\begin{matrix} w \\ u \end{matrix}],

P (s) = [\begin{matrix} P_{11} (s) & P_{12} (s) \\ P_{21} (s) & P_{22} (s) \end{matrix}] = [\begin{matrix} W_{e} (s) & - W_{e} (s) M (s) \\ 0 & W_{y} (s) M (s) \\ 1 & - M (s) \end{matrix}],

P (s) : [\begin{matrix} A & B_{1} & B_{2} \\ C_{1} & D_{11} & D_{12} \\ C_{2} & D_{21} & D_{22} \end{matrix}],

P_{11} (s) = [\begin{matrix} W_{e} (s) \\ 0 \end{matrix}] = C_{1} {(s I - A)}^{- 1} B_{1} + D_{11},

P_{12} (s) = [\begin{matrix} - W_{e} (s) M (s) \\ W_{y} (s) M (s) \end{matrix}] = C_{1} {(s I - A)}^{- 1} B_{2} + D_{12},

P_{21} (s) = 1 = D_{21},

P_{22} (s) = - M (s) = C_{2} {(s I - A)}^{- 1} B_{2} + D_{22} .

z = F_{l} (P, G) w,

F_{l} (P, G) = P_{11} + P_{12} G {(I - P_{22} G)}^{- 1} P_{21} .

\min_{G} {‖ F_{l} (P, G) ‖}_{2} = \min_{G} {‖ \begin{matrix} W_{e} \cdot S F \\ W_{y} \cdot N S F \end{matrix} ‖}_{2},

{‖ F_{l} (P, G) ‖}_{2} = {(\frac{1}{2 π} \int_{- \infty}^{\infty} trace[F_{l} (P, G) {(j ω)}^{*} \times F_{l} (P, G) (j ω)] d ω)}^{1 / 2},

S F (s) = \frac{e (s)}{w (s)} = \frac{1}{1 + M (s) G (s)},

N T F (s) = \frac{y (s)}{w (s)} = \frac{M (s) G (s)}{1 + M (s) G (s)} .

A_{G} = A - B_{2} B_{2}^{'} X_{2} - Y_{2} C_{2}^{'} C_{2}, B_{G} = B_{2}^{'} X_{2}, C_{G} = Y_{2} C_{2}^{'}, D_{G} = 0,

X_{2} = Ric [\begin{matrix} A & - B_{2} B_{2}^{'} \\ B_{1} B_{1}^{'} & - A \end{matrix}] and Y_{2} = Ric [\begin{matrix} A' & - C_{2} C_{2}^{'} \\ - B_{1} B_{1}^{'} & - A \end{matrix}] .

G (s) \equiv C_{G} {(s I - A_{G})}^{- 1} B_{G} + D_{G} .

S_{k}^{t u r} = S_{k}^{r e s} + i M a t \cdot u_{k - 1}^{} .

w_{k}^{} = i M a t^{- 1} \cdot S_{k}^{t u r} .

\underset{G (s)}{M i n} σ_{e}^{2} = \frac{1}{N - 1} \sum_{k = 0}^{N - 1} {| e_{k} |}^{2},

〈 e^{2} 〉 = {\int_{ω} ‖ \frac{1}{1 + M (j ω) G (j ω)} ‖}^{2} {‖ w (j ω) ‖}^{2} d ω .

M (s) = \frac{e^{- 2 Δ s}}{0.0025 s + 1},

W_{e} (s) = \frac{C_{0}}{s + C_{1}} .

W_{e} (s) = \frac{C_{0}}{s + C_{1}} \frac{s^{2} + 2 η_{1} ω_{o} s + ω_{o}^{2}}{s^{2} + 2 η_{2} ω_{o} s + ω_{o}^{2}} .

\frac{η_{1} (j ω)}{η_{2} (j ω)} = A \sqrt{1 + \frac{{‖ W_{y} (j ω) ‖}^{2}}{{‖ W_{e} (j ω) ‖}^{2}}},

Controller	Integrator arcsec²/ mas RMS	H₂ arcsec²/ mas RMS
Average	5.6e-3 / 75	2.1e-3 / 45
Min	4.1e-3 / 64	1.2e-3 / 35
Max	6.2e-3 / 79	3.6e-3 / 60

Type of W_e	Order of W_e(s)	Order of G(s)	Computing time (mins)
First order (Eq. (23)	1	3	0.3
1st order + 1 notch (Eq. (24)	3	5	0.9
1st order + 2 notches	5	7	2.0
1st order + 4 notches	9	11	7.5
1st order + 7 notches	15	17	28.6

Controller	Integrator arcsec²/ mas RMS	H₂ arcsec²/ mas RMS
Average	5.6e-3 / 75	2.1e-3 / 45
Min	4.1e-3 / 64	1.2e-3 / 35
Max	6.2e-3 / 79	3.6e-3 / 60

Type of W_e	Order of W_e(s)	Order of G(s)	Computing time (mins)
First order (Eq. (23)	1	3	0.3
1st order + 1 notch (Eq. (24)	3	5	0.9
1st order + 2 notches	5	7	2.0
1st order + 4 notches	9	11	7.5
1st order + 7 notches	15	17	28.6