Frequency based design of modal controllers for adaptive optics systems

Guido Agapito; Giorgio Battistelli; Daniele Mari; Daniela Selvi; Alberto Tesi; Pietro Tesi

doi:10.1364/OE.20.027108

Frequency based design of modal controllers for adaptive optics systems

Guido Agapito, Giorgio Battistelli, Daniele Mari, Daniela Selvi, Alberto Tesi, and Pietro Tesi

Optics Express
Vol. 20,
Issue 24,
pp. 27108-27122
(2012)
•https://doi.org/10.1364/OE.20.027108

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Guido Agapito, Giorgio Battistelli, Daniele Mari, Daniela Selvi, Alberto Tesi, and Pietro Tesi, "Frequency based design of modal controllers for adaptive optics systems," Opt. Express 20, 27108-27122 (2012)

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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 turbulence (010.1330)

About this Article
History
- Original Manuscript: August 2, 2012
- Revised Manuscript: October 12, 2012
- Manuscript Accepted: October 14, 2012
- Published: November 16, 2012

Accessible

Open Access

Abstract

This paper addresses the problem of reducing the effects of wavefront distortions in ground-based telescopes within a “Modal-Control” framework. The proposed approach allows the designer to optimize the Youla parameter of a given modal controller with respect to a relevant adaptive optics performance criterion defined on a “sampled” frequency domain. This feature makes it possible to use turbulence/vibration profiles of arbitrary complexity (even empirical power spectral densities from data), while keeping the controller order at a moderate value. Effectiveness of the proposed solution is also illustrated through an adaptive optics numerical simulator.

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References

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Publication

F. Roddier, Adaptive Optics in Astronomy (Cambridge University Press, 1999).
[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]
C. Petit, J.-M. Conan, C. Kulcsár, H.-F. Raynaud, and T. Fusco, “First laboratory validation of vibration filtering with LQG control law for adaptive optics,” Opt. Express 16(1), 87–97 (2008).
[Crossref] [PubMed]
E. Fedrigo, R. Muradore, and D. Zillo, “High performance adaptive optics system with fine tip/tilt control,” Control Eng. Pract. 17(1), 122–135 (2009).
[Crossref]
L.A. Poyneer, B.A. Macintosh, and J.-P. Véran, “Fourier transform wavefront control with adaptive prediction of the atmosphere,” J. Opt. Soc. Am. A, 24(9), 2645–2660 (2007).
[Crossref]
K. Hinnen, M. Verhaegen, and N. Doelman, “A data-driven H2-optimal control approach for adaptive optics,” IEEE Trans. Control Syst. Technol, 16(3), 381–395 (2008).
[Crossref]
“Adaptive optics control for ground based telescopes,” in Special Issue of the European Journal of Control, Vol. 17, No. 3, J.-M. Conan, C. Kulcsár, and H.-F. Raynaud, eds. (Lavoisier, 2011).
G. Agapito, F. Quiros-Pacheco, P. Tesi, A. Riccardi, and S. Esposito, “Observer-based control techniques for the LBT adaptive optics under telescope vibrations,” Eur. J. Control 17(3), 316–326 (2011).
[Crossref]
G. Agapito, S. Baldi, G. Battistelli, D. Mari, E. Mosca, and A. Riccardi, “Automatic tuning of the internal position control of an adaptive secondary mirror,” Eur. J. Control 17(3), 273–289 (2011).
[Crossref]
G. Agapito, F. Quiros-Pacheco, P. Tesi, S. Esposito, and M. Xompero, “Optimal control techniques for the adaptive optics system of the LBT,” Proc. SPIE 7015, 70153G (2008).
[Crossref]
L. Close, V. Gasho, D. Kopon, J. Males, K.B. Follette, K. Brutlag, A. Uomoto, and T. Hare, “The Magellan telescope adaptive secondary AO system: a visible and mid-IR AO facility,” Proc. SPIE 7736, 773605 (2010).
[Crossref]
E. Marchetti, M. Le Louarn, C. Soenke, E. Fedrigo, P.-Y. Madec, and N. Hubin, “ERIS adaptive optics system design,” Proc. SPIE 8447, 84473M (2012).
S. Esposito, E. Pinna, F. Quirós-Pacheco, A. Puglisi, L. Carbonaro, M. Bonaglia, V. Biliotti, R. Briguglio, G. Agapito, C. Arcidiacono, L. Busoni, M. Xompero, A. Riccardi, L. Fini, and A. Bouchez, “Wavefront sensor design for the GMT natural guide star AO system,” Proc. SPIE 8447, 84471L (2012).
M. Born and E. Wolf, Principles of Optics (Cambridge University Press, Cambridge, U.K, 1980).
C. Dessenne, P.-Y. Madec, and G. Rousset, “Optimization of a predictive controller for closed-loop adaptive optics,” Appl. Optim., 37(21), 4623–4633 (1998).
[Crossref]
E. Gendron and P. Lena, “Modal control optimization,” Astron. Astrophys., 291, 337–347 (1994).
C. Petit, F. Quiros-Pacheco, J.-M. Conan, C. Kulcsár, H.-F. Raynaud, T. Fusco, and G. Rousset, “Kalman filter based control for adaptive optics,” Proc. SPIE 5490, 1414–1425 (2004).
[Crossref]
C. Correia, H.F. Raynaud, C. Kulcsar, and J.M. Conan, “Minimum-variance control for astronomical adaptive optics with resonant deformable mirrors,” Eur. J. Control 17(3), 222–236 (2011).
[Crossref]
D. P. Looze, M. Kasper, S. Hippler, O. Beker, and R. Weiss, “Optimal compensation and implementation for adaptive optics systems,” Exp. Astron., 15, 67–88 (2004).
[Crossref]
L. A. Poyneer and J.-P. Véran, “Optimal modal Fourier transform wavefront control,” J. Opt. Soc. Am. A, 22(2), 1515–1526 (2005).
[Crossref]
B. Le Roux, J.-M. Conan, C. Kulcsár, H.-F. Raynaud, L. Mugnier, and T. Fusco, “Optimal control law for classical and multiconjugate adaptive optics,” J. Opt. Soc. Am. A, 21(7), 1261–1276 (2004).
[Crossref]
A. Riccardi, G. Brusa, P. Salinari, D. Gallieni, R. Biasi, M. Andrighettoni, and H. M. Martin, “Adaptive secondary mirrors for the Large Binocular Telescope,” Proc. SPIE 4839, 721–732 (2003).
[Crossref]
J. Herrmann, “Phase variance and Strehl ratio in adaptive optics,” J. Opt. Soc. Am. A, 9, 2257–2258 (1992).
[Crossref]
R.A. de Callafon and C.E. Kinney, “Robust estimation and adaptive controller tuning for variance minimization in servo systems,” JSME J. Adv. Mech. Des. Syst. Manuf., 4(1), 130–142 (2010).
[Crossref]
J. Doyle, B. Francis, and A. Tannenbaum, Feedback Control Theory (Macmillan Publishing Company, 1992).
A. Karimi and G. Galdos, “Fixed-order H∞ controller design for nonparametric models by convex optimization,” Automatica 46, 1388–1394 (2010).
[Crossref]
G. Goodwin, S. Graebe, and M. Salgado, Control System Design (Prentice Hall, 2001).
R. Ragazzoni, “Pupil plane wavefront sensing with an oscillating prism,” J. Mod. Optic. 43, 289–293 (1996).
[Crossref]
C. Vérinaud, “On the nature of the measurements provided by a pyramid wave-front sensor,” Opt. Commun. 233, 27 – 38 (2004).
[Crossref]
B. L. McGlamery, “Computer simulation studies of compensation of turbulence degraded images,” Proc. SPIE, 74(9), 225–233 (1976).
[Crossref]

2012 (2)

E. Marchetti, M. Le Louarn, C. Soenke, E. Fedrigo, P.-Y. Madec, and N. Hubin, “ERIS adaptive optics system design,” Proc. SPIE 8447, 84473M (2012).

S. Esposito, E. Pinna, F. Quirós-Pacheco, A. Puglisi, L. Carbonaro, M. Bonaglia, V. Biliotti, R. Briguglio, G. Agapito, C. Arcidiacono, L. Busoni, M. Xompero, A. Riccardi, L. Fini, and A. Bouchez, “Wavefront sensor design for the GMT natural guide star AO system,” Proc. SPIE 8447, 84471L (2012).

2011 (3)

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

G. Agapito, S. Baldi, G. Battistelli, D. Mari, E. Mosca, and A. Riccardi, “Automatic tuning of the internal position control of an adaptive secondary mirror,” Eur. J. Control 17(3), 273–289 (2011).
[Crossref]

C. Correia, H.F. Raynaud, C. Kulcsar, and J.M. Conan, “Minimum-variance control for astronomical adaptive optics with resonant deformable mirrors,” Eur. J. Control 17(3), 222–236 (2011).
[Crossref]

2010 (3)

R.A. de Callafon and C.E. Kinney, “Robust estimation and adaptive controller tuning for variance minimization in servo systems,” JSME J. Adv. Mech. Des. Syst. Manuf., 4(1), 130–142 (2010).
[Crossref]

A. Karimi and G. Galdos, “Fixed-order H∞ controller design for nonparametric models by convex optimization,” Automatica 46, 1388–1394 (2010).
[Crossref]

L. Close, V. Gasho, D. Kopon, J. Males, K.B. Follette, K. Brutlag, A. Uomoto, and T. Hare, “The Magellan telescope adaptive secondary AO system: a visible and mid-IR AO facility,” Proc. SPIE 7736, 773605 (2010).
[Crossref]

2009 (1)

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

2008 (3)

C. Petit, J.-M. Conan, C. Kulcsár, H.-F. Raynaud, and T. Fusco, “First laboratory validation of vibration filtering with LQG control law for adaptive optics,” Opt. Express 16(1), 87–97 (2008).
[Crossref] [PubMed]

K. Hinnen, M. Verhaegen, and N. Doelman, “A data-driven H2-optimal control approach for adaptive optics,” IEEE Trans. Control Syst. Technol, 16(3), 381–395 (2008).
[Crossref]

G. Agapito, F. Quiros-Pacheco, P. Tesi, S. Esposito, and M. Xompero, “Optimal control techniques for the adaptive optics system of the LBT,” Proc. SPIE 7015, 70153G (2008).
[Crossref]

2007 (1)

L.A. Poyneer, B.A. Macintosh, and J.-P. Véran, “Fourier transform wavefront control with adaptive prediction of the atmosphere,” J. Opt. Soc. Am. A, 24(9), 2645–2660 (2007).
[Crossref]

2006 (1)

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]

2005 (1)

L. A. Poyneer and J.-P. Véran, “Optimal modal Fourier transform wavefront control,” J. Opt. Soc. Am. A, 22(2), 1515–1526 (2005).
[Crossref]

2004 (4)

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

D. P. Looze, M. Kasper, S. Hippler, O. Beker, and R. Weiss, “Optimal compensation and implementation for adaptive optics systems,” Exp. Astron., 15, 67–88 (2004).
[Crossref]

C. Vérinaud, “On the nature of the measurements provided by a pyramid wave-front sensor,” Opt. Commun. 233, 27 – 38 (2004).
[Crossref]

C. Petit, F. Quiros-Pacheco, J.-M. Conan, C. Kulcsár, H.-F. Raynaud, T. Fusco, and G. Rousset, “Kalman filter based control for adaptive optics,” Proc. SPIE 5490, 1414–1425 (2004).
[Crossref]

2003 (1)

A. Riccardi, G. Brusa, P. Salinari, D. Gallieni, R. Biasi, M. Andrighettoni, and H. M. Martin, “Adaptive secondary mirrors for the Large Binocular Telescope,” Proc. SPIE 4839, 721–732 (2003).
[Crossref]

1998 (1)

C. Dessenne, P.-Y. Madec, and G. Rousset, “Optimization of a predictive controller for closed-loop adaptive optics,” Appl. Optim., 37(21), 4623–4633 (1998).
[Crossref]

1996 (1)

R. Ragazzoni, “Pupil plane wavefront sensing with an oscillating prism,” J. Mod. Optic. 43, 289–293 (1996).
[Crossref]

1994 (1)

E. Gendron and P. Lena, “Modal control optimization,” Astron. Astrophys., 291, 337–347 (1994).

1992 (1)

J. Herrmann, “Phase variance and Strehl ratio in adaptive optics,” J. Opt. Soc. Am. A, 9, 2257–2258 (1992).
[Crossref]

1976 (1)

B. L. McGlamery, “Computer simulation studies of compensation of turbulence degraded images,” Proc. SPIE, 74(9), 225–233 (1976).
[Crossref]

Agapito, G.

G. Agapito, F. Quiros-Pacheco, P. Tesi, S. Esposito, and M. Xompero, “Optimal control techniques for the adaptive optics system of the LBT,” Proc. SPIE 7015, 70153G (2008).
[Crossref]

Andrighettoni, M.

Arcidiacono, C.

Baldi, S.

Battistelli, G.

Beker, O.

D. P. Looze, M. Kasper, S. Hippler, O. Beker, and R. Weiss, “Optimal compensation and implementation for adaptive optics systems,” Exp. Astron., 15, 67–88 (2004).
[Crossref]

Biasi, R.

Biliotti, V.

Bonaglia, M.

Born, M.

M. Born and E. Wolf, Principles of Optics (Cambridge University Press, Cambridge, U.K, 1980).

Bouchez, A.

Briguglio, R.

Brusa, G.

Brutlag, K.

Busoni, L.

Carbonaro, L.

Close, L.

Conan, J.M.

Conan, J.-M.

C. Petit, F. Quiros-Pacheco, J.-M. Conan, C. Kulcsár, H.-F. Raynaud, T. Fusco, and G. Rousset, “Kalman filter based control for adaptive optics,” Proc. SPIE 5490, 1414–1425 (2004).
[Crossref]

Correia, C.

de Callafon, R.A.

Dessenne, C.

C. Dessenne, P.-Y. Madec, and G. Rousset, “Optimization of a predictive controller for closed-loop adaptive optics,” Appl. Optim., 37(21), 4623–4633 (1998).
[Crossref]

Doelman, N.

K. Hinnen, M. Verhaegen, and N. Doelman, “A data-driven H2-optimal control approach for adaptive optics,” IEEE Trans. Control Syst. Technol, 16(3), 381–395 (2008).
[Crossref]

Doyle, J.

J. Doyle, B. Francis, and A. Tannenbaum, Feedback Control Theory (Macmillan Publishing Company, 1992).

Esposito, S.

G. Agapito, F. Quiros-Pacheco, P. Tesi, S. Esposito, and M. Xompero, “Optimal control techniques for the adaptive optics system of the LBT,” Proc. SPIE 7015, 70153G (2008).
[Crossref]

Fedrigo, E.

E. Marchetti, M. Le Louarn, C. Soenke, E. Fedrigo, P.-Y. Madec, and N. Hubin, “ERIS adaptive optics system design,” Proc. SPIE 8447, 84473M (2012).

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

Fini, L.

Follette, K.B.

Francis, B.

J. Doyle, B. Francis, and A. Tannenbaum, Feedback Control Theory (Macmillan Publishing Company, 1992).

Fusco, T.

C. Petit, F. Quiros-Pacheco, J.-M. Conan, C. Kulcsár, H.-F. Raynaud, T. Fusco, and G. Rousset, “Kalman filter based control for adaptive optics,” Proc. SPIE 5490, 1414–1425 (2004).
[Crossref]

Galdos, G.

A. Karimi and G. Galdos, “Fixed-order H∞ controller design for nonparametric models by convex optimization,” Automatica 46, 1388–1394 (2010).
[Crossref]

Gallieni, D.

Gasho, V.

Gendron, E.

E. Gendron and P. Lena, “Modal control optimization,” Astron. Astrophys., 291, 337–347 (1994).

Goodwin, G.

G. Goodwin, S. Graebe, and M. Salgado, Control System Design (Prentice Hall, 2001).

Graebe, S.

G. Goodwin, S. Graebe, and M. Salgado, Control System Design (Prentice Hall, 2001).

Hare, T.

Herrmann, J.

J. Herrmann, “Phase variance and Strehl ratio in adaptive optics,” J. Opt. Soc. Am. A, 9, 2257–2258 (1992).
[Crossref]

Hinnen, K.

K. Hinnen, M. Verhaegen, and N. Doelman, “A data-driven H2-optimal control approach for adaptive optics,” IEEE Trans. Control Syst. Technol, 16(3), 381–395 (2008).
[Crossref]

Hippler, S.

D. P. Looze, M. Kasper, S. Hippler, O. Beker, and R. Weiss, “Optimal compensation and implementation for adaptive optics systems,” Exp. Astron., 15, 67–88 (2004).
[Crossref]

Hubin, N.

E. Marchetti, M. Le Louarn, C. Soenke, E. Fedrigo, P.-Y. Madec, and N. Hubin, “ERIS adaptive optics system design,” Proc. SPIE 8447, 84473M (2012).

Karimi, A.

A. Karimi and G. Galdos, “Fixed-order H∞ controller design for nonparametric models by convex optimization,” Automatica 46, 1388–1394 (2010).
[Crossref]

Kasper, M.

D. P. Looze, M. Kasper, S. Hippler, O. Beker, and R. Weiss, “Optimal compensation and implementation for adaptive optics systems,” Exp. Astron., 15, 67–88 (2004).
[Crossref]

Kinney, C.E.

Kopon, D.

Kulcsar, C.

Kulcsár, C.

C. Petit, F. Quiros-Pacheco, J.-M. Conan, C. Kulcsár, H.-F. Raynaud, T. Fusco, and G. Rousset, “Kalman filter based control for adaptive optics,” Proc. SPIE 5490, 1414–1425 (2004).
[Crossref]

Le Louarn, M.

E. Marchetti, M. Le Louarn, C. Soenke, E. Fedrigo, P.-Y. Madec, and N. Hubin, “ERIS adaptive optics system design,” Proc. SPIE 8447, 84473M (2012).

Le Roux, B.

Lena, P.

E. Gendron and P. Lena, “Modal control optimization,” Astron. Astrophys., 291, 337–347 (1994).

Looze, D. P.

D. P. Looze, M. Kasper, S. Hippler, O. Beker, and R. Weiss, “Optimal compensation and implementation for adaptive optics systems,” Exp. Astron., 15, 67–88 (2004).
[Crossref]

Macintosh, B.A.

L.A. Poyneer, B.A. Macintosh, and J.-P. Véran, “Fourier transform wavefront control with adaptive prediction of the atmosphere,” J. Opt. Soc. Am. A, 24(9), 2645–2660 (2007).
[Crossref]

Madec, P.-Y.

E. Marchetti, M. Le Louarn, C. Soenke, E. Fedrigo, P.-Y. Madec, and N. Hubin, “ERIS adaptive optics system design,” Proc. SPIE 8447, 84473M (2012).

C. Dessenne, P.-Y. Madec, and G. Rousset, “Optimization of a predictive controller for closed-loop adaptive optics,” Appl. Optim., 37(21), 4623–4633 (1998).
[Crossref]

Males, J.

Marchetti, E.

E. Marchetti, M. Le Louarn, C. Soenke, E. Fedrigo, P.-Y. Madec, and N. Hubin, “ERIS adaptive optics system design,” Proc. SPIE 8447, 84473M (2012).

Mari, D.

Martin, H. M.

McGlamery, B. L.

B. L. McGlamery, “Computer simulation studies of compensation of turbulence degraded images,” Proc. SPIE, 74(9), 225–233 (1976).
[Crossref]

Mosca, E.

Mugnier, L.

Muradore, R.

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

Petit, C.

C. Petit, F. Quiros-Pacheco, J.-M. Conan, C. Kulcsár, H.-F. Raynaud, T. Fusco, and G. Rousset, “Kalman filter based control for adaptive optics,” Proc. SPIE 5490, 1414–1425 (2004).
[Crossref]

Pinna, E.

Poyneer, L. A.

L. A. Poyneer and J.-P. Véran, “Optimal modal Fourier transform wavefront control,” J. Opt. Soc. Am. A, 22(2), 1515–1526 (2005).
[Crossref]

Poyneer, L.A.

L.A. Poyneer, B.A. Macintosh, and J.-P. Véran, “Fourier transform wavefront control with adaptive prediction of the atmosphere,” J. Opt. Soc. Am. A, 24(9), 2645–2660 (2007).
[Crossref]

Puglisi, A.

Quiros-Pacheco, F.

G. Agapito, F. Quiros-Pacheco, P. Tesi, S. Esposito, and M. Xompero, “Optimal control techniques for the adaptive optics system of the LBT,” Proc. SPIE 7015, 70153G (2008).
[Crossref]

C. Petit, F. Quiros-Pacheco, J.-M. Conan, C. Kulcsár, H.-F. Raynaud, T. Fusco, and G. Rousset, “Kalman filter based control for adaptive optics,” Proc. SPIE 5490, 1414–1425 (2004).
[Crossref]

Quirós-Pacheco, F.

Ragazzoni, R.

R. Ragazzoni, “Pupil plane wavefront sensing with an oscillating prism,” J. Mod. Optic. 43, 289–293 (1996).
[Crossref]

Raynaud, H.F.

Raynaud, H.-F.

C. Petit, F. Quiros-Pacheco, J.-M. Conan, C. Kulcsár, H.-F. Raynaud, T. Fusco, and G. Rousset, “Kalman filter based control for adaptive optics,” Proc. SPIE 5490, 1414–1425 (2004).
[Crossref]

Riccardi, A.

Roddier, F.

F. Roddier, Adaptive Optics in Astronomy (Cambridge University Press, 1999).
[Crossref]

Rousset, G.

C. Petit, F. Quiros-Pacheco, J.-M. Conan, C. Kulcsár, H.-F. Raynaud, T. Fusco, and G. Rousset, “Kalman filter based control for adaptive optics,” Proc. SPIE 5490, 1414–1425 (2004).
[Crossref]

C. Dessenne, P.-Y. Madec, and G. Rousset, “Optimization of a predictive controller for closed-loop adaptive optics,” Appl. Optim., 37(21), 4623–4633 (1998).
[Crossref]

Salgado, M.

G. Goodwin, S. Graebe, and M. Salgado, Control System Design (Prentice Hall, 2001).

Salinari, P.

Soenke, C.

E. Marchetti, M. Le Louarn, C. Soenke, E. Fedrigo, P.-Y. Madec, and N. Hubin, “ERIS adaptive optics system design,” Proc. SPIE 8447, 84473M (2012).

Tannenbaum, A.

J. Doyle, B. Francis, and A. Tannenbaum, Feedback Control Theory (Macmillan Publishing Company, 1992).

Tesi, P.

G. Agapito, F. Quiros-Pacheco, P. Tesi, S. Esposito, and M. Xompero, “Optimal control techniques for the adaptive optics system of the LBT,” Proc. SPIE 7015, 70153G (2008).
[Crossref]

Uomoto, A.

Véran, J.-P.

L.A. Poyneer, B.A. Macintosh, and J.-P. Véran, “Fourier transform wavefront control with adaptive prediction of the atmosphere,” J. Opt. Soc. Am. A, 24(9), 2645–2660 (2007).
[Crossref]

L. A. Poyneer and J.-P. Véran, “Optimal modal Fourier transform wavefront control,” J. Opt. Soc. Am. A, 22(2), 1515–1526 (2005).
[Crossref]

Verhaegen, M.

K. Hinnen, M. Verhaegen, and N. Doelman, “A data-driven H2-optimal control approach for adaptive optics,” IEEE Trans. Control Syst. Technol, 16(3), 381–395 (2008).
[Crossref]

Vérinaud, C.

C. Vérinaud, “On the nature of the measurements provided by a pyramid wave-front sensor,” Opt. Commun. 233, 27 – 38 (2004).
[Crossref]

Viaris de Lesegno, P.

Weiss, R.

D. P. Looze, M. Kasper, S. Hippler, O. Beker, and R. Weiss, “Optimal compensation and implementation for adaptive optics systems,” Exp. Astron., 15, 67–88 (2004).
[Crossref]

Wolf, E.

M. Born and E. Wolf, Principles of Optics (Cambridge University Press, Cambridge, U.K, 1980).

Xompero, M.

G. Agapito, F. Quiros-Pacheco, P. Tesi, S. Esposito, and M. Xompero, “Optimal control techniques for the adaptive optics system of the LBT,” Proc. SPIE 7015, 70153G (2008).
[Crossref]

Zillo, D.

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

Appl. Optim. (1)

C. Dessenne, P.-Y. Madec, and G. Rousset, “Optimization of a predictive controller for closed-loop adaptive optics,” Appl. Optim., 37(21), 4623–4633 (1998).
[Crossref]

Astron. Astrophys. (1)

E. Gendron and P. Lena, “Modal control optimization,” Astron. Astrophys., 291, 337–347 (1994).

Automatica (1)

A. Karimi and G. Galdos, “Fixed-order H∞ controller design for nonparametric models by convex optimization,” Automatica 46, 1388–1394 (2010).
[Crossref]

Control Eng. Pract. (1)

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

Eur. J. Control (3)

Exp. Astron. (1)

D. P. Looze, M. Kasper, S. Hippler, O. Beker, and R. Weiss, “Optimal compensation and implementation for adaptive optics systems,” Exp. Astron., 15, 67–88 (2004).
[Crossref]

IEEE Trans. Control Syst. Technol (1)

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Fig. 1

AO control scheme.

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Fig. 2

Modal control scheme.

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Fig. 3

Turbulence and vibration PSD related to tip.

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Fig. 4

Residual phase variance corresponding to controllers synthesized with increasing values of γ.

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Fig. 5

PSF profiles resulting from the use of the controllers in the control loop.

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Fig. 6

PSDs of the residual phase (related to tip/tilt).

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Fig. 7

Bode diagram of the sensitivity function related to the controllers.

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Tables (2)

Table 1 Simulation parameters and conditions.

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Table 2 Performance exhibited by the controllers considered in the simulation tests.

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Equations (36)

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y (k) = 𝒟 ϕ^{res} (k - 1) + w (k),

ϕ^{cor} (k) = 𝒩 u (k - 1),

U (z) = ℳ 𝒦_{M} (z) ℛ Y (z),

Φ^{res} (z) = Φ^{tot} (z) - 𝒩 z^{- 1} U (z) = Φ^{tot} (z) - 𝒩 z^{- 1} ℳ 𝒦_{M} (z) ℛ Y (z)

Δ (z) Φ^{res} (z) = Φ^{tot} (z) - Ξ (z) W (z)

f (ρ) = lim_{h \to \infty} \frac{1}{h + 1} \sum_{k = 0}^{h} {| ϕ_{t}^{res} (k) |}^{2} .

f (ρ) = \frac{1}{2 π} \int_{- π}^{π} ({| S (e^{j ω}, ρ) |}^{2} ϒ_{ϕ} (ω) + {| T (e^{j ω}, ρ) |}^{2} ϒ_{w} (ω)) d ω,

f (ρ) = \frac{1}{N} \sum_{f = 1}^{N} ({| S (e^{{j ω}_{f}}, ρ) |}^{2} {\hat{ϒ}}_{ϕ} (ω_{f}) + {| T (e^{j ω_{f}}, ρ) |}^{2} {\hat{ϒ}}_{w} (ω_{f})),

𝒞 (P) = {C (z) = \frac{Q (z)}{1 - P (z) Q (z)}, Q (z) \in 𝒮},

C (z, ρ) = \frac{Q (z, ρ)}{1 - P (z) Q (z, ρ)} .

S (z, ρ) = 1 - P (z) Q (z, ρ),

T (z, ρ) = - M (z) Q (z, ρ) .

Q (z, ρ) = ψ^{T} (z) ρ,

S (z, ρ) = 1 - P (z) ψ^{T} (z) ρ,

T (z, ρ) = - M (z) ψ^{T} (z) ρ .

ψ_{i} (z) = \frac{z^{i - 1}}{d (z)}, i = 1, 2, \dots, n,

f (ρ) = \frac{1}{N} \sum_{f = 1}^{N} {{| (1 - P (e^{j ω_{f}}) ψ^{T} (e^{j ω_{f}}) ρ) |}^{2} {\hat{ϒ}}_{ϕ} (ω_{f}) + {| M (e^{j ω_{f}}) ψ^{T} (e^{j ω_{f}}) ρ |}^{2} {\hat{ϒ}}_{w} (ω_{f})} .

min_{ρ} f (ρ),

C (z, ρ) \in 𝒮 .

Re {(1 - P (e^{- j ω}) Q (e^{- j ω}, \hat{ρ})) (1 - P (e^{j ω}) Q (e^{j ω}, ρ))} > 0, \forall ω,

A (\hat{ρ}) ρ + b (\hat{ρ}) < 0

A (\hat{ρ}) = [\begin{matrix} a_{11} (\hat{ρ}) & \dots & a_{1 n} (\hat{ρ}) \\ ⋮ & ⋱ & ⋮ \\ a_{N 1} (\hat{ρ}) & \dots & a_{N n} (\hat{ρ}) \end{matrix}]

a_{f i} (\hat{ρ}) = Re {(1 - P (e^{- j ω_{f}}) Q (e^{- j ω_{f}}, \hat{ρ})) P (e^{j ω_{f}}) ψ_{i} (e^{j ω_{f}})},

b (\hat{ρ}) = {[b_{1} (\hat{ρ}) \dots b_{N} (\hat{ρ})]}^{T}

b_{f} (\hat{ρ}) = - Re {1 - p (e^{- j ω_{f}}) Q (e^{- j ω_{f}}, \hat{ρ})} .

\hat{Q} (z) = \frac{\hat{C} (z)}{1 + P (z) \hat{C} (z)} = \frac{\hat{n} (z)}{\hat{d} (z)}

min_{ρ} f (ρ)

A (ρ^{(i)}) ρ + b (ρ^{(i)}) < 0;

S R = \frac{{PSF}_{res} (\underline{o})}{{PSF}_{Airy} (\underline{o})},

\hat{C} (z) = \frac{g z}{z - 0.95},

\hat{Q} (z) = \frac{g z^{2}}{z^{2} - 0.95 z + g} = \frac{\hat{n} (z)}{\hat{d} (z)} .

ψ_{i} (z) = \frac{z^{i - 1}}{d (z)} = \frac{z^{i - 1}}{z^{γ + 2} - 0.95 z^{γ + 1} + g z^{γ}}, i = 1, 2, \dots, γ + 3

ρ^{(0)} = \hat{ρ} = [\underset{γ + 2}{\underset{︸}{0 \dots 0}} g] .

ξ (k) = 1.9774 ξ (k - 1) - 0.9776 ξ (k - 2) + ν (k) .

ζ_{1} (k) = 1.9917 ζ_{1} (k - 1) - 0.9984 ζ_{1} (k - 2) + 1.8431 μ_{1} (k - 1) - 0.8493 μ_{1} (k - 2),

ζ_{2} (k) = 1.9782 ζ_{2} (k - 1) - 0.9972 ζ_{2} (k - 2) + 1.7418 μ_{2} (k - 1) - 0.7585 μ_{2} (k - 2),

Telescope	diameter = 8.222 m (central obstruction 11%)
Pyramid WFS	sensing wavelength = 0.750 μm
	tilt modulation radius = 2.0 $\frac{λ}{D}$
	number of subapertures = 30 × 30
	number of photons per integration time
	per subaperture = 213
	number of electrons per pixel of
	readout noise = 11.5 e⁻RMS
ASM	672 Karhunen-Loève (KL) modes projected onto the ASM influence functions
turbulence	seeing = 0.8″ (at 0.5 μm)
	outer scale L₀ = 40 m
	mean wind speed = 16.5 m/s
telescope structural vibrations	frequencies = 13, 22 Hz
	standard deviation = 20 mas
	damping ratio = 0.01
loop parameters	sampling frequency = 1000 Hz
loop parameters	2 frames total delay
Strehl Ratio calculated at	1.65 μm

Telescope	diameter = 8.222 m (central obstruction 11%)
Pyramid WFS	sensing wavelength = 0.750 μm
	tilt modulation radius = 2.0 $\frac{λ}{D}$
	number of subapertures = 30 × 30
	number of photons per integration time
	per subaperture = 213
	number of electrons per pixel of
	readout noise = 11.5 e⁻RMS
ASM	672 Karhunen-Loève (KL) modes projected onto the ASM influence functions
turbulence	seeing = 0.8″ (at 0.5 μm)
	outer scale L₀ = 40 m
	mean wind speed = 16.5 m/s
telescope structural vibrations	frequencies = 13, 22 Hz
	standard deviation = 20 mas
	damping ratio = 0.01
loop parameters	sampling frequency = 1000 Hz
loop parameters	2 frames total delay
Strehl Ratio calculated at	1.65 μm