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. Express16(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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  5. 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).
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  6. 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).
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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. Control17(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. SPIE7015, 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. SPIE7736, 773605 (2010).
    [CrossRef]
  12. E. Marchetti, M. Le Louarn, C. Soenke, E. Fedrigo, P.-Y. Madec, and N. Hubin, “ERIS adaptive optics system design,” Proc. SPIE8447, 84473M (2012).
  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. SPIE8447, 84471L (2012).
  14. M. Born and E. Wolf, Principles of Optics (Cambridge University Press, Cambridge, U.K, 1980).
  15. 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).
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    [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. Control17(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. SPIE4839, 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,” Automatica46, 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. SPIE8447, 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. SPIE8447, 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. Control17(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. Control17(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. Control17(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,” Automatica46, 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. SPIE7736, 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. Express16(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. SPIE7015, 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. SPIE5490, 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. SPIE4839, 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. SPIE8447, 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. Control17(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. Control17(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. SPIE7015, 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. SPIE4839, 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. SPIE8447, 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. Control17(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. Control17(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. SPIE4839, 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. SPIE8447, 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. SPIE8447, 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. SPIE8447, 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. SPIE8447, 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. SPIE4839, 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. SPIE7736, 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. SPIE8447, 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. SPIE8447, 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. SPIE7736, 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. Control17(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. Express16(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. Express14(17), 7464–7476 (2006).
[CrossRef] [PubMed]

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. SPIE5490, 1414–1425 (2004).
[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]

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. Control17(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. SPIE8447, 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. Control17(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. SPIE7015, 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. SPIE8447, 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. SPIE8447, 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. SPIE7736, 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. Express16(1), 87–97 (2008).
[CrossRef] [PubMed]

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. SPIE5490, 1414–1425 (2004).
[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]

Galdos, G.

A. Karimi and G. Galdos, “Fixed-order H∞ controller design for nonparametric models by convex optimization,” Automatica46, 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. SPIE4839, 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. SPIE7736, 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. SPIE7736, 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. SPIE8447, 84473M (2012).

Karimi, A.

A. Karimi and G. Galdos, “Fixed-order H∞ controller design for nonparametric models by convex optimization,” Automatica46, 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. SPIE7736, 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. Control17(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. Express16(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. Express14(17), 7464–7476 (2006).
[CrossRef] [PubMed]

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. SPIE5490, 1414–1425 (2004).
[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]

Le Louarn, M.

E. Marchetti, M. Le Louarn, C. Soenke, E. Fedrigo, P.-Y. Madec, and N. Hubin, “ERIS adaptive optics system design,” Proc. SPIE8447, 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. SPIE8447, 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. SPIE7736, 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. SPIE8447, 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. Control17(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. SPIE4839, 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. Control17(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. SPIE8447, 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. SPIE8447, 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. Control17(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. SPIE7015, 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. SPIE5490, 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. SPIE8447, 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. Control17(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. Express16(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. Express14(17), 7464–7476 (2006).
[CrossRef] [PubMed]

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. SPIE5490, 1414–1425 (2004).
[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]

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. SPIE8447, 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. Control17(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. Control17(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. SPIE4839, 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. SPIE5490, 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. SPIE4839, 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. SPIE8447, 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. Control17(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. SPIE7015, 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. SPIE7736, 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. SPIE8447, 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. SPIE7015, 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,” Automatica46, 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. Control17(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. Control17(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. Control17(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. SPIE7015, 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. SPIE7736, 773605 (2010).
[CrossRef]

E. Marchetti, M. Le Louarn, C. Soenke, E. Fedrigo, P.-Y. Madec, and N. Hubin, “ERIS adaptive optics system design,” Proc. SPIE8447, 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. SPIE8447, 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. SPIE4839, 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. SPIE5490, 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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