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.

© 2012 OSA

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    [Crossref] [PubMed]
  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]
  4. 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).
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    [Crossref]
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    [Crossref]
  9. 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]
  10. 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]
  11. 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]
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  13. 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).
  14. M. Born and E. Wolf, Principles of Optics (Cambridge University Press, Cambridge, U.K, 1980).
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    [Crossref]
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  17. 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]
  18. 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]
  19. 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]
  20. L. A. Poyneer and J.-P. Véran, “Optimal modal Fourier transform wavefront control,” J. Opt. Soc. Am. A, 22(2), 1515–1526 (2005).
    [Crossref]
  21. 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]
  22. 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]
  23. J. Herrmann, “Phase variance and Strehl ratio in adaptive optics,” J. Opt. Soc. Am. A, 9, 2257–2258 (1992).
    [Crossref]
  24. 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]
  25. J. Doyle, B. Francis, and A. Tannenbaum, Feedback Control Theory (Macmillan Publishing Company, 1992).
  26. A. Karimi and G. Galdos, “Fixed-order H∞ controller design for nonparametric models by convex optimization,” Automatica 46, 1388–1394 (2010).
    [Crossref]
  27. G. Goodwin, S. Graebe, and M. Salgado, Control System Design (Prentice Hall, 2001).
  28. R. Ragazzoni, “Pupil plane wavefront sensing with an oscillating prism,” J. Mod. Optic. 43, 289–293 (1996).
    [Crossref]
  29. C. Vérinaud, “On the nature of the measurements provided by a pyramid wave-front sensor,” Opt. Commun. 233, 27 – 38 (2004).
    [Crossref]
  30. 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)

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.

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).

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]

Andrighettoni, M.

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]

Arcidiacono, C.

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).

Baldi, S.

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]

Battistelli, G.

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]

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.

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]

Biliotti, V.

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).

Bonaglia, M.

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).

Born, M.

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

Bouchez, A.

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).

Briguglio, R.

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).

Brusa, G.

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]

Brutlag, K.

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]

Busoni, L.

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).

Carbonaro, L.

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).

Close, L.

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]

Conan, J.M.

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]

Conan, J.-M.

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]

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]

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]

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.

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]

de Callafon, R.A.

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]

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.

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).

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, 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.

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).

Follette, K.B.

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]

Francis, B.

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

Fusco, T.

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]

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]

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.

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]

Gasho, V.

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]

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.

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]

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.

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]

Kopon, D.

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]

Kulcsar, C.

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]

Kulcsár, C.

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]

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]

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]

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.

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]

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.

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]

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.

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]

Martin, H. M.

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]

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.

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]

Mugnier, L.

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]

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.

Pinna, E.

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).

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.

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).

Quiros-Pacheco, F.

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, 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.

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).

Ragazzoni, R.

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

Raynaud, H.F.

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]

Raynaud, H.-F.

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]

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]

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]

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.

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).

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]

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]

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.

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]

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, 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, 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.

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]

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.

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).

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)

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]

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)

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]

J. Mod. Optic. (1)

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

J. Opt. Soc. Am. A, (4)

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

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]

JSME J. Adv. Mech. Des. Syst. Manuf. (1)

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]

Opt. Commun. (1)

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

Opt. Express (2)

Proc. SPIE (7)

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).

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]

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]

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

Other (5)

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

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

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

“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).

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

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Figures (7)

Fig. 1
Fig. 1

AO control scheme.

Fig. 2
Fig. 2

Modal control scheme.

Fig. 3
Fig. 3

Turbulence and vibration PSD related to tip.

Fig. 4
Fig. 4

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

Fig. 5
Fig. 5

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

Fig. 6
Fig. 6

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

Fig. 7
Fig. 7

Bode diagram of the sensitivity function related to the controllers.

Tables (2)

Tables Icon

Table 1 Simulation parameters and conditions.

Tables Icon

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

Equations (36)

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

y ( k ) = 𝒟 ϕ res ( k 1 ) + w ( k ) ,
ϕ cor ( k ) = 𝒩 u ( k 1 ) ,
U ( z ) = 𝒦 M ( z ) Y ( z ) ,
Φ res ( z ) = Φ tot ( z ) 𝒩 z 1 U ( z ) = Φ tot ( z ) 𝒩 z 1 𝒦 M ( z ) Y ( z )
Δ ( z ) Φ res ( z ) = Φ tot ( z ) Ξ ( z ) W ( z )
f ( ρ ) = lim h 1 h + 1 k = 0 h | ϕ t res ( k ) | 2 .
f ( ρ ) = 1 2 π π π ( | S ( e j ω , ρ ) | 2 ϒ ϕ ( ω ) + | T ( e j ω , ρ ) | 2 ϒ w ( ω ) ) d ω ,
f ( ρ ) = 1 N f = 1 N ( | S ( e j ω f , ρ ) | 2 ϒ ^ ϕ ( ω f ) + | T ( e j ω f , ρ ) | 2 ϒ ^ w ( ω f ) ) ,
𝒞 ( P ) = { C ( z ) = Q ( z ) 1 P ( z ) Q ( z ) , Q ( z ) 𝒮 } ,
C ( z , ρ ) = 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 ) = z i 1 d ( z ) , i = 1 , 2 , , n ,
f ( ρ ) = 1 N f = 1 N { | ( 1 P ( e j ω f ) ψ T ( e j ω f ) ρ ) | 2 ϒ ^ ϕ ( ω f ) + | M ( e j ω f ) ψ T ( e j ω f ) ρ | 2 ϒ ^ w ( ω f ) } .
min ρ f ( ρ ) ,
C ( z , ρ ) 𝒮 .
Re { ( 1 P ( e j ω ) Q ( e j ω , ρ ^ ) ) ( 1 P ( e j ω ) Q ( e j ω , ρ ) ) } > 0 , ω ,
A ( ρ ^ ) ρ + b ( ρ ^ ) < 0
A ( ρ ^ ) = [ a 11 ( ρ ^ ) a 1 n ( ρ ^ ) a N 1 ( ρ ^ ) a N n ( ρ ^ ) ]
a f i ( ρ ^ ) = Re { ( 1 P ( e j ω f ) Q ( e j ω f , ρ ^ ) ) P ( e j ω f ) ψ i ( e j ω f ) } ,
b ( ρ ^ ) = [ b 1 ( ρ ^ ) b N ( ρ ^ ) ] T
b f ( ρ ^ ) = Re { 1 p ( e j ω f ) Q ( e j ω f , ρ ^ ) } .
Q ^ ( z ) = C ^ ( z ) 1 + P ( z ) C ^ ( z ) = n ^ ( z ) d ^ ( z )
min ρ f ( ρ )
A ( ρ ( i ) ) ρ + b ( ρ ( i ) ) < 0 ;
S R = PSF res ( o _ ) PSF Airy ( o _ ) ,
C ^ ( z ) = g z z 0.95 ,
Q ^ ( z ) = g z 2 z 2 0.95 z + g = n ^ ( z ) d ^ ( z ) .
ψ i ( z ) = z i 1 d ( z ) = z i 1 z γ + 2 0.95 z γ + 1 + g z γ , i = 1 , 2 , , γ + 3
ρ ( 0 ) = ρ ^ = [ 0 0 γ + 2 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 ) ,

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