Abstract

Vibrations are detrimental to the performance of modern adaptive optics (AO) systems. In this paper, we describe new methods tested to mitigate the vibrations encountered in some of the instruments of the Gemini South telescope. By implementing a spectral analysis of the slope measurements from several wavefront sensors and an imager, we can determine the frequencies and magnitude of these vibrations. We found a persistent vibration at 55 Hz with others occurring occasionally at 14 and 100 Hz. Two types of AO controllers were designed and implemented, Kalman and H, in the multiconjugate AO tip–tilt loop. The first results show a similar performance for these advanced controllers and a clear improvement in vibration rejection and overall performance over the classical integrator scheme. It is shown that the reduction in the standard deviation of the residual slopes (as measured by wavefront sensors) is highly dependent on turbulence, wind speed, and vibration conditions, ranging—in terms of slopes RMS value—from an almost negligible reduction for high speed wind to a factor of 5 for a combination of low wind and strong vibrations.

© 2012 Optical Society of America

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    [CrossRef]
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  4. J. P. Verán and L. Poyneer, “Evaluation of the T/T conditions at Gemini South using NICI AO telemetry data,” in First AO4ELT Conference (EdP Sciences, 2009), p. 05002.
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  6. C. Petit, J.-M. Conan, C. Kulcsár, and H. F. Raynaud, “Linear quadratic Gaussian control for adaptive optics and multiconjugate adaptive optics: experimental and numerical analysis,” J. Opt. Soc. Am. A 26, 1307–1325 (2009).
    [CrossRef]
  7. C. Petit, J. M. Conan, T. Fusco, J. Montri, C. Kulcsár, H. F. Raynaud, and D. Rabaud, “First laboratory demonstration of closed-loop Kalman based optimal control for vibration filtering and simplified MCAO,” Proc. SPIE 6272, 62721T (2006).
    [CrossRef]
  8. B. Neichel, F. Rigaut, A. Guesalaga, I. Rodriguez, and D. Guzman, “Kalman and H-infinity controllers for GeMS,” in Imaging Systems Applications OSA Technical Digest (CD) (Optical Society of America, 2011), paper JWA32.
  9. G. Agapito, F. Quirós-Pacheco, P. Tesi, A. Riccardi, and S. Esposito, “Observer-based control techniques for the LBT adaptive optics under telescope vibrations,” Eur. J. Control 17, 316–326 (2011).
    [CrossRef]
  10. J. M. Conan, H. F. Raynaud, C. Kulcsár, and S. Meimon, “Are integral controllers adapted to the new era of ELT adaptive optics?” presented at Second AO4ELT Conference, Victoria, British Columbia, Canada, 25–30Sept.2011.
  11. E. Fedrigo, R. Muradore, and D. Zilio, “High performance adaptive optics system with fine tip/tilt control,” Contr. Eng. Pract. 17, 122–135 (2009).
    [CrossRef]
  12. B. Le Roux, J. M. Conan, C. Kulcsár, H. F. Raynaud, L. M. Mugnier, and T. Fusco, “Optimal control law for classical and multiconjugate adaptative optics,” J. Opt. Soc. Am. A 21, 1261–1276 (2004).
    [CrossRef]
  13. 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, 7464–7476 (2006).
    [CrossRef]
  14. J. C. Doyle, K. Glover, P. P. Khargonekar, and B. A. Francis, “State-space solutions to standard H2 and H∞ control problems,” IEEE Trans. Autom. Contr. 34, 831–847(1989).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  19. F. Rigaut and B. Neichel, “Gemini South MCAO on-sky results,” presented at Second AO4ELT Conference, Victoria, British Columbia, Canada, 25–30Sept.2011.
  20. J. M. Conan, G. Rousset, and P. Y. Madec, “Wave-front temporal spectra in high-resolution imaging through turbulence,” J. Opt. Soc. Am. A 12, 1559–1570 (1995).
    [CrossRef]
  21. E. Gendron and P. Lena, “Astronomical adaptive optics I. modal control optimization,” Astron. Astrophys. 291, 337–347 (1994).
  22. C. Correia, H. F. Raynaud, C. Kulcsár, and J. M. Conan, “On the optimal reconstruction and control of adaptive optical systems with mirror dynamics,” J. Opt. Soc. Am. A 27, 333–349 (2010).
    [CrossRef]
  23. S. Meimon, C. Petit, T. Fusco, and C. Kulcsár, “Tip–tilt disturbance model identification for Kalman-based control scheme: application to XAO and ELT systems,” J. Opt. Soc. Am. A 27, A122–A132 (2010).
    [CrossRef]
  24. C. Dessenne, P.-Y. Madec, and G. Rousset, “Modal prediction for closed-loop adaptive optics,” Opt. Lett. 22, 1535–1537 (1997).
    [CrossRef]
  25. C. Mohtadi, “Bode’s integral theorem for discrete-time systems,” IEE Proc. D Contr. Theory Appl. 137, 57–66(1990).
    [CrossRef]

2011

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

2010

J. Maly, E. Darren, and T. Pargett, “Vibration suppression for the Gemini planet image,” Proc. SPIE 7733, 77331F (2010).
[CrossRef]

B. Neichel, F. Rigaut, M. Bec, M. Boccas, F. Daruich, C. D’Orgeville, V. Fesquet, R. Galvez, A. Garcia-Rissmann, G. Gausachs, M. Lombini, G. Perez, G. Trancho, V. Upadhya, and T. Vucina, “The Gemini MCAO systems GeMS: nearing the end of a lab-story,” Proc. SPIE 7736, 773606 (2010).
[CrossRef]

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

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

2009

2006

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

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

2004

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

Y. Clenet, M. Kasper, N. Ageorges, C. Lidman, T. Fusco, G. Rousset, O. Marco, M. Hartung, D. Mouillet, B. Koehler, and N. Hubin, “NACO performance: status after 2 years of operation,” Proc. SPIE 5490, 107–117 (2004).
[CrossRef]

2001

1997

1995

1994

E. Gendron and P. Lena, “Astronomical adaptive optics I. modal control optimization,” Astron. Astrophys. 291, 337–347 (1994).

1992

B. Bamieh and J. B. Pearson, “A general framework for linear periodic systems with application to H∞ sampled-data control,” IEEE Trans. Autom. Contr. 37, 418–435(1992).
[CrossRef]

1990

C. Mohtadi, “Bode’s integral theorem for discrete-time systems,” IEE Proc. D Contr. Theory Appl. 137, 57–66(1990).
[CrossRef]

1989

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

M. G. Safonov, D. J. N. Limebeer, and R. Y. Chiang, “Simplifying the H∞ theory via loop shifting, matrix pencil and descriptor concepts,” Int. J. Contr. 50, 2467–2488 (1989).
[CrossRef]

Agapito, G.

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

Ageorges, N.

Y. Clenet, M. Kasper, N. Ageorges, C. Lidman, T. Fusco, G. Rousset, O. Marco, M. Hartung, D. Mouillet, B. Koehler, and N. Hubin, “NACO performance: status after 2 years of operation,” Proc. SPIE 5490, 107–117 (2004).
[CrossRef]

Y. Clenet, M. Kasper, N. Ageorges, C. Lidman, T. Fusco, O. P. Marco, M. Hartung, D. Mouillet, B. Koehler, G. Rousset, and N. Hubin, “NAOS performances: impact of the telescope vibrations and possible origins,” in Semaine de l’Astrophysique Francaise (EdP Sciences, 2004), p. 179.

Bamieh, B.

B. Bamieh and J. B. Pearson, “A general framework for linear periodic systems with application to H∞ sampled-data control,” IEEE Trans. Autom. Contr. 37, 418–435(1992).
[CrossRef]

Bec, M.

B. Neichel, F. Rigaut, M. Bec, M. Boccas, F. Daruich, C. D’Orgeville, V. Fesquet, R. Galvez, A. Garcia-Rissmann, G. Gausachs, M. Lombini, G. Perez, G. Trancho, V. Upadhya, and T. Vucina, “The Gemini MCAO systems GeMS: nearing the end of a lab-story,” Proc. SPIE 7736, 773606 (2010).
[CrossRef]

Boccas, M.

B. Neichel, F. Rigaut, M. Bec, M. Boccas, F. Daruich, C. D’Orgeville, V. Fesquet, R. Galvez, A. Garcia-Rissmann, G. Gausachs, M. Lombini, G. Perez, G. Trancho, V. Upadhya, and T. Vucina, “The Gemini MCAO systems GeMS: nearing the end of a lab-story,” Proc. SPIE 7736, 773606 (2010).
[CrossRef]

Chiang, R. Y.

M. G. Safonov, D. J. N. Limebeer, and R. Y. Chiang, “Simplifying the H∞ theory via loop shifting, matrix pencil and descriptor concepts,” Int. J. Contr. 50, 2467–2488 (1989).
[CrossRef]

Christou, J.

J. Christou, C. Trujillo, B. Neichel, F. Rigaut, B. Walls, D. Coulson, J. White, A. Stephens, and M. Sheehan, “The effect of the instrument environment on the Altair AO system,” presented at Second AO4ELT Conference, Victoria, British Columbia, Canada, 25–30Sept.2011.

Clenet, Y.

Y. Clenet, M. Kasper, N. Ageorges, C. Lidman, T. Fusco, G. Rousset, O. Marco, M. Hartung, D. Mouillet, B. Koehler, and N. Hubin, “NACO performance: status after 2 years of operation,” Proc. SPIE 5490, 107–117 (2004).
[CrossRef]

Y. Clenet, M. Kasper, N. Ageorges, C. Lidman, T. Fusco, O. P. Marco, M. Hartung, D. Mouillet, B. Koehler, G. Rousset, and N. Hubin, “NAOS performances: impact of the telescope vibrations and possible origins,” in Semaine de l’Astrophysique Francaise (EdP Sciences, 2004), p. 179.

Conan, J. M.

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

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

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

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

J. M. Conan, H. F. Raynaud, C. Kulcsár, and S. Meimon, “Are integral controllers adapted to the new era of ELT adaptive optics?” presented at Second AO4ELT Conference, Victoria, British Columbia, Canada, 25–30Sept.2011.

Conan, J.-M.

Correia, C.

Coulson, D.

J. Christou, C. Trujillo, B. Neichel, F. Rigaut, B. Walls, D. Coulson, J. White, A. Stephens, and M. Sheehan, “The effect of the instrument environment on the Altair AO system,” presented at Second AO4ELT Conference, Victoria, British Columbia, Canada, 25–30Sept.2011.

D’Orgeville, C.

B. Neichel, F. Rigaut, M. Bec, M. Boccas, F. Daruich, C. D’Orgeville, V. Fesquet, R. Galvez, A. Garcia-Rissmann, G. Gausachs, M. Lombini, G. Perez, G. Trancho, V. Upadhya, and T. Vucina, “The Gemini MCAO systems GeMS: nearing the end of a lab-story,” Proc. SPIE 7736, 773606 (2010).
[CrossRef]

Darren, E.

J. Maly, E. Darren, and T. Pargett, “Vibration suppression for the Gemini planet image,” Proc. SPIE 7733, 77331F (2010).
[CrossRef]

Daruich, F.

B. Neichel, F. Rigaut, M. Bec, M. Boccas, F. Daruich, C. D’Orgeville, V. Fesquet, R. Galvez, A. Garcia-Rissmann, G. Gausachs, M. Lombini, G. Perez, G. Trancho, V. Upadhya, and T. Vucina, “The Gemini MCAO systems GeMS: nearing the end of a lab-story,” Proc. SPIE 7736, 773606 (2010).
[CrossRef]

de Lesegno, P. Viaris

Dessenne, C.

Doyle, J. C.

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

Ellerbroek, B. L.

Esposito, S.

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

Fedrigo, E.

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

Fesquet, V.

B. Neichel, F. Rigaut, M. Bec, M. Boccas, F. Daruich, C. D’Orgeville, V. Fesquet, R. Galvez, A. Garcia-Rissmann, G. Gausachs, M. Lombini, G. Perez, G. Trancho, V. Upadhya, and T. Vucina, “The Gemini MCAO systems GeMS: nearing the end of a lab-story,” Proc. SPIE 7736, 773606 (2010).
[CrossRef]

Francis, B. A.

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

Fusco, T.

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

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

Y. Clenet, M. Kasper, N. Ageorges, C. Lidman, T. Fusco, G. Rousset, O. Marco, M. Hartung, D. Mouillet, B. Koehler, and N. Hubin, “NACO performance: status after 2 years of operation,” Proc. SPIE 5490, 107–117 (2004).
[CrossRef]

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

Y. Clenet, M. Kasper, N. Ageorges, C. Lidman, T. Fusco, O. P. Marco, M. Hartung, D. Mouillet, B. Koehler, G. Rousset, and N. Hubin, “NAOS performances: impact of the telescope vibrations and possible origins,” in Semaine de l’Astrophysique Francaise (EdP Sciences, 2004), p. 179.

Galvez, R.

B. Neichel, F. Rigaut, M. Bec, M. Boccas, F. Daruich, C. D’Orgeville, V. Fesquet, R. Galvez, A. Garcia-Rissmann, G. Gausachs, M. Lombini, G. Perez, G. Trancho, V. Upadhya, and T. Vucina, “The Gemini MCAO systems GeMS: nearing the end of a lab-story,” Proc. SPIE 7736, 773606 (2010).
[CrossRef]

Garcia-Rissmann, A.

B. Neichel, F. Rigaut, M. Bec, M. Boccas, F. Daruich, C. D’Orgeville, V. Fesquet, R. Galvez, A. Garcia-Rissmann, G. Gausachs, M. Lombini, G. Perez, G. Trancho, V. Upadhya, and T. Vucina, “The Gemini MCAO systems GeMS: nearing the end of a lab-story,” Proc. SPIE 7736, 773606 (2010).
[CrossRef]

Gausachs, G.

B. Neichel, F. Rigaut, M. Bec, M. Boccas, F. Daruich, C. D’Orgeville, V. Fesquet, R. Galvez, A. Garcia-Rissmann, G. Gausachs, M. Lombini, G. Perez, G. Trancho, V. Upadhya, and T. Vucina, “The Gemini MCAO systems GeMS: nearing the end of a lab-story,” Proc. SPIE 7736, 773606 (2010).
[CrossRef]

Gendron, E.

E. Gendron and P. Lena, “Astronomical adaptive optics I. modal control optimization,” Astron. Astrophys. 291, 337–347 (1994).

Glover, K.

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

Guesalaga, A.

B. Neichel, F. Rigaut, A. Guesalaga, I. Rodriguez, and D. Guzman, “Kalman and H-infinity controllers for GeMS,” in Imaging Systems Applications OSA Technical Digest (CD) (Optical Society of America, 2011), paper JWA32.

Guzman, D.

B. Neichel, F. Rigaut, A. Guesalaga, I. Rodriguez, and D. Guzman, “Kalman and H-infinity controllers for GeMS,” in Imaging Systems Applications OSA Technical Digest (CD) (Optical Society of America, 2011), paper JWA32.

Hartung, M.

Y. Clenet, M. Kasper, N. Ageorges, C. Lidman, T. Fusco, G. Rousset, O. Marco, M. Hartung, D. Mouillet, B. Koehler, and N. Hubin, “NACO performance: status after 2 years of operation,” Proc. SPIE 5490, 107–117 (2004).
[CrossRef]

Y. Clenet, M. Kasper, N. Ageorges, C. Lidman, T. Fusco, O. P. Marco, M. Hartung, D. Mouillet, B. Koehler, G. Rousset, and N. Hubin, “NAOS performances: impact of the telescope vibrations and possible origins,” in Semaine de l’Astrophysique Francaise (EdP Sciences, 2004), p. 179.

Hubin, N.

Y. Clenet, M. Kasper, N. Ageorges, C. Lidman, T. Fusco, G. Rousset, O. Marco, M. Hartung, D. Mouillet, B. Koehler, and N. Hubin, “NACO performance: status after 2 years of operation,” Proc. SPIE 5490, 107–117 (2004).
[CrossRef]

Y. Clenet, M. Kasper, N. Ageorges, C. Lidman, T. Fusco, O. P. Marco, M. Hartung, D. Mouillet, B. Koehler, G. Rousset, and N. Hubin, “NAOS performances: impact of the telescope vibrations and possible origins,” in Semaine de l’Astrophysique Francaise (EdP Sciences, 2004), p. 179.

Kasper, M.

Y. Clenet, M. Kasper, N. Ageorges, C. Lidman, T. Fusco, G. Rousset, O. Marco, M. Hartung, D. Mouillet, B. Koehler, and N. Hubin, “NACO performance: status after 2 years of operation,” Proc. SPIE 5490, 107–117 (2004).
[CrossRef]

Y. Clenet, M. Kasper, N. Ageorges, C. Lidman, T. Fusco, O. P. Marco, M. Hartung, D. Mouillet, B. Koehler, G. Rousset, and N. Hubin, “NAOS performances: impact of the telescope vibrations and possible origins,” in Semaine de l’Astrophysique Francaise (EdP Sciences, 2004), p. 179.

Khargonekar, P. P.

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

Koehler, B.

Y. Clenet, M. Kasper, N. Ageorges, C. Lidman, T. Fusco, G. Rousset, O. Marco, M. Hartung, D. Mouillet, B. Koehler, and N. Hubin, “NACO performance: status after 2 years of operation,” Proc. SPIE 5490, 107–117 (2004).
[CrossRef]

Y. Clenet, M. Kasper, N. Ageorges, C. Lidman, T. Fusco, O. P. Marco, M. Hartung, D. Mouillet, B. Koehler, G. Rousset, and N. Hubin, “NAOS performances: impact of the telescope vibrations and possible origins,” in Semaine de l’Astrophysique Francaise (EdP Sciences, 2004), p. 179.

Kulcsár, C.

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

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

C. Petit, J.-M. Conan, C. Kulcsár, and H. F. Raynaud, “Linear quadratic Gaussian control for adaptive optics and multiconjugate adaptive optics: experimental and numerical analysis,” J. Opt. Soc. Am. A 26, 1307–1325 (2009).
[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, 7464–7476 (2006).
[CrossRef]

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

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

J. M. Conan, H. F. Raynaud, C. Kulcsár, and S. Meimon, “Are integral controllers adapted to the new era of ELT adaptive optics?” presented at Second AO4ELT Conference, Victoria, British Columbia, Canada, 25–30Sept.2011.

Le Roux, B.

Lena, P.

E. Gendron and P. Lena, “Astronomical adaptive optics I. modal control optimization,” Astron. Astrophys. 291, 337–347 (1994).

Lidman, C.

Y. Clenet, M. Kasper, N. Ageorges, C. Lidman, T. Fusco, G. Rousset, O. Marco, M. Hartung, D. Mouillet, B. Koehler, and N. Hubin, “NACO performance: status after 2 years of operation,” Proc. SPIE 5490, 107–117 (2004).
[CrossRef]

Y. Clenet, M. Kasper, N. Ageorges, C. Lidman, T. Fusco, O. P. Marco, M. Hartung, D. Mouillet, B. Koehler, G. Rousset, and N. Hubin, “NAOS performances: impact of the telescope vibrations and possible origins,” in Semaine de l’Astrophysique Francaise (EdP Sciences, 2004), p. 179.

Limebeer, D. J. N.

M. G. Safonov, D. J. N. Limebeer, and R. Y. Chiang, “Simplifying the H∞ theory via loop shifting, matrix pencil and descriptor concepts,” Int. J. Contr. 50, 2467–2488 (1989).
[CrossRef]

Lombini, M.

B. Neichel, F. Rigaut, M. Bec, M. Boccas, F. Daruich, C. D’Orgeville, V. Fesquet, R. Galvez, A. Garcia-Rissmann, G. Gausachs, M. Lombini, G. Perez, G. Trancho, V. Upadhya, and T. Vucina, “The Gemini MCAO systems GeMS: nearing the end of a lab-story,” Proc. SPIE 7736, 773606 (2010).
[CrossRef]

Madec, P. Y.

Madec, P.-Y.

Maly, J.

J. Maly, E. Darren, and T. Pargett, “Vibration suppression for the Gemini planet image,” Proc. SPIE 7733, 77331F (2010).
[CrossRef]

Marco, O.

Y. Clenet, M. Kasper, N. Ageorges, C. Lidman, T. Fusco, G. Rousset, O. Marco, M. Hartung, D. Mouillet, B. Koehler, and N. Hubin, “NACO performance: status after 2 years of operation,” Proc. SPIE 5490, 107–117 (2004).
[CrossRef]

Marco, O. P.

Y. Clenet, M. Kasper, N. Ageorges, C. Lidman, T. Fusco, O. P. Marco, M. Hartung, D. Mouillet, B. Koehler, G. Rousset, and N. Hubin, “NAOS performances: impact of the telescope vibrations and possible origins,” in Semaine de l’Astrophysique Francaise (EdP Sciences, 2004), p. 179.

Meimon, S.

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

J. M. Conan, H. F. Raynaud, C. Kulcsár, and S. Meimon, “Are integral controllers adapted to the new era of ELT adaptive optics?” presented at Second AO4ELT Conference, Victoria, British Columbia, Canada, 25–30Sept.2011.

Mohtadi, C.

C. Mohtadi, “Bode’s integral theorem for discrete-time systems,” IEE Proc. D Contr. Theory Appl. 137, 57–66(1990).
[CrossRef]

Montri, J.

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

Mouillet, D.

Y. Clenet, M. Kasper, N. Ageorges, C. Lidman, T. Fusco, G. Rousset, O. Marco, M. Hartung, D. Mouillet, B. Koehler, and N. Hubin, “NACO performance: status after 2 years of operation,” Proc. SPIE 5490, 107–117 (2004).
[CrossRef]

Y. Clenet, M. Kasper, N. Ageorges, C. Lidman, T. Fusco, O. P. Marco, M. Hartung, D. Mouillet, B. Koehler, G. Rousset, and N. Hubin, “NAOS performances: impact of the telescope vibrations and possible origins,” in Semaine de l’Astrophysique Francaise (EdP Sciences, 2004), p. 179.

Mugnier, L. M.

Muradore, R.

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

Neichel, B.

B. Neichel, F. Rigaut, M. Bec, M. Boccas, F. Daruich, C. D’Orgeville, V. Fesquet, R. Galvez, A. Garcia-Rissmann, G. Gausachs, M. Lombini, G. Perez, G. Trancho, V. Upadhya, and T. Vucina, “The Gemini MCAO systems GeMS: nearing the end of a lab-story,” Proc. SPIE 7736, 773606 (2010).
[CrossRef]

J. Christou, C. Trujillo, B. Neichel, F. Rigaut, B. Walls, D. Coulson, J. White, A. Stephens, and M. Sheehan, “The effect of the instrument environment on the Altair AO system,” presented at Second AO4ELT Conference, Victoria, British Columbia, Canada, 25–30Sept.2011.

F. Rigaut and B. Neichel, “Gemini South MCAO on-sky results,” presented at Second AO4ELT Conference, Victoria, British Columbia, Canada, 25–30Sept.2011.

B. Neichel, F. Rigaut, A. Guesalaga, I. Rodriguez, and D. Guzman, “Kalman and H-infinity controllers for GeMS,” in Imaging Systems Applications OSA Technical Digest (CD) (Optical Society of America, 2011), paper JWA32.

Pargett, T.

J. Maly, E. Darren, and T. Pargett, “Vibration suppression for the Gemini planet image,” Proc. SPIE 7733, 77331F (2010).
[CrossRef]

Pearson, J. B.

B. Bamieh and J. B. Pearson, “A general framework for linear periodic systems with application to H∞ sampled-data control,” IEEE Trans. Autom. Contr. 37, 418–435(1992).
[CrossRef]

Perez, G.

B. Neichel, F. Rigaut, M. Bec, M. Boccas, F. Daruich, C. D’Orgeville, V. Fesquet, R. Galvez, A. Garcia-Rissmann, G. Gausachs, M. Lombini, G. Perez, G. Trancho, V. Upadhya, and T. Vucina, “The Gemini MCAO systems GeMS: nearing the end of a lab-story,” Proc. SPIE 7736, 773606 (2010).
[CrossRef]

Petit, C.

Poyneer, L.

J. P. Verán and L. Poyneer, “Evaluation of the T/T conditions at Gemini South using NICI AO telemetry data,” in First AO4ELT Conference (EdP Sciences, 2009), p. 05002.

Quirós-Pacheco, F.

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

Rabaud, D.

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

Raynaud, H. F.

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

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

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

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

J. M. Conan, H. F. Raynaud, C. Kulcsár, and S. Meimon, “Are integral controllers adapted to the new era of ELT adaptive optics?” presented at Second AO4ELT Conference, Victoria, British Columbia, Canada, 25–30Sept.2011.

Raynaud, H.-F.

Riccardi, A.

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

Rigaut, F.

B. Neichel, F. Rigaut, M. Bec, M. Boccas, F. Daruich, C. D’Orgeville, V. Fesquet, R. Galvez, A. Garcia-Rissmann, G. Gausachs, M. Lombini, G. Perez, G. Trancho, V. Upadhya, and T. Vucina, “The Gemini MCAO systems GeMS: nearing the end of a lab-story,” Proc. SPIE 7736, 773606 (2010).
[CrossRef]

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

B. Neichel, F. Rigaut, A. Guesalaga, I. Rodriguez, and D. Guzman, “Kalman and H-infinity controllers for GeMS,” in Imaging Systems Applications OSA Technical Digest (CD) (Optical Society of America, 2011), paper JWA32.

F. Rigaut and B. Neichel, “Gemini South MCAO on-sky results,” presented at Second AO4ELT Conference, Victoria, British Columbia, Canada, 25–30Sept.2011.

J. Christou, C. Trujillo, B. Neichel, F. Rigaut, B. Walls, D. Coulson, J. White, A. Stephens, and M. Sheehan, “The effect of the instrument environment on the Altair AO system,” presented at Second AO4ELT Conference, Victoria, British Columbia, Canada, 25–30Sept.2011.

Rodriguez, I.

B. Neichel, F. Rigaut, A. Guesalaga, I. Rodriguez, and D. Guzman, “Kalman and H-infinity controllers for GeMS,” in Imaging Systems Applications OSA Technical Digest (CD) (Optical Society of America, 2011), paper JWA32.

Rousset, G.

Y. Clenet, M. Kasper, N. Ageorges, C. Lidman, T. Fusco, G. Rousset, O. Marco, M. Hartung, D. Mouillet, B. Koehler, and N. Hubin, “NACO performance: status after 2 years of operation,” Proc. SPIE 5490, 107–117 (2004).
[CrossRef]

C. Dessenne, P.-Y. Madec, and G. Rousset, “Modal prediction for closed-loop adaptive optics,” Opt. Lett. 22, 1535–1537 (1997).
[CrossRef]

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

Y. Clenet, M. Kasper, N. Ageorges, C. Lidman, T. Fusco, O. P. Marco, M. Hartung, D. Mouillet, B. Koehler, G. Rousset, and N. Hubin, “NAOS performances: impact of the telescope vibrations and possible origins,” in Semaine de l’Astrophysique Francaise (EdP Sciences, 2004), p. 179.

Safonov, M. G.

M. G. Safonov, D. J. N. Limebeer, and R. Y. Chiang, “Simplifying the H∞ theory via loop shifting, matrix pencil and descriptor concepts,” Int. J. Contr. 50, 2467–2488 (1989).
[CrossRef]

Sheehan, M.

J. Christou, C. Trujillo, B. Neichel, F. Rigaut, B. Walls, D. Coulson, J. White, A. Stephens, and M. Sheehan, “The effect of the instrument environment on the Altair AO system,” presented at Second AO4ELT Conference, Victoria, British Columbia, Canada, 25–30Sept.2011.

Stephens, A.

J. Christou, C. Trujillo, B. Neichel, F. Rigaut, B. Walls, D. Coulson, J. White, A. Stephens, and M. Sheehan, “The effect of the instrument environment on the Altair AO system,” presented at Second AO4ELT Conference, Victoria, British Columbia, Canada, 25–30Sept.2011.

Tesi, P.

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

Trancho, G.

B. Neichel, F. Rigaut, M. Bec, M. Boccas, F. Daruich, C. D’Orgeville, V. Fesquet, R. Galvez, A. Garcia-Rissmann, G. Gausachs, M. Lombini, G. Perez, G. Trancho, V. Upadhya, and T. Vucina, “The Gemini MCAO systems GeMS: nearing the end of a lab-story,” Proc. SPIE 7736, 773606 (2010).
[CrossRef]

Trujillo, C.

J. Christou, C. Trujillo, B. Neichel, F. Rigaut, B. Walls, D. Coulson, J. White, A. Stephens, and M. Sheehan, “The effect of the instrument environment on the Altair AO system,” presented at Second AO4ELT Conference, Victoria, British Columbia, Canada, 25–30Sept.2011.

Upadhya, V.

B. Neichel, F. Rigaut, M. Bec, M. Boccas, F. Daruich, C. D’Orgeville, V. Fesquet, R. Galvez, A. Garcia-Rissmann, G. Gausachs, M. Lombini, G. Perez, G. Trancho, V. Upadhya, and T. Vucina, “The Gemini MCAO systems GeMS: nearing the end of a lab-story,” Proc. SPIE 7736, 773606 (2010).
[CrossRef]

Verán, J. P.

J. P. Verán and L. Poyneer, “Evaluation of the T/T conditions at Gemini South using NICI AO telemetry data,” in First AO4ELT Conference (EdP Sciences, 2009), p. 05002.

Vucina, T.

B. Neichel, F. Rigaut, M. Bec, M. Boccas, F. Daruich, C. D’Orgeville, V. Fesquet, R. Galvez, A. Garcia-Rissmann, G. Gausachs, M. Lombini, G. Perez, G. Trancho, V. Upadhya, and T. Vucina, “The Gemini MCAO systems GeMS: nearing the end of a lab-story,” Proc. SPIE 7736, 773606 (2010).
[CrossRef]

Walls, B.

J. Christou, C. Trujillo, B. Neichel, F. Rigaut, B. Walls, D. Coulson, J. White, A. Stephens, and M. Sheehan, “The effect of the instrument environment on the Altair AO system,” presented at Second AO4ELT Conference, Victoria, British Columbia, Canada, 25–30Sept.2011.

White, J.

J. Christou, C. Trujillo, B. Neichel, F. Rigaut, B. Walls, D. Coulson, J. White, A. Stephens, and M. Sheehan, “The effect of the instrument environment on the Altair AO system,” presented at Second AO4ELT Conference, Victoria, British Columbia, Canada, 25–30Sept.2011.

Zilio, D.

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

Astron. Astrophys.

E. Gendron and P. Lena, “Astronomical adaptive optics I. modal control optimization,” Astron. Astrophys. 291, 337–347 (1994).

Contr. Eng. Pract.

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

Eur. J. Control

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

IEE Proc. D Contr. Theory Appl.

C. Mohtadi, “Bode’s integral theorem for discrete-time systems,” IEE Proc. D Contr. Theory Appl. 137, 57–66(1990).
[CrossRef]

IEEE Trans. Autom. Contr.

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

B. Bamieh and J. B. Pearson, “A general framework for linear periodic systems with application to H∞ sampled-data control,” IEEE Trans. Autom. Contr. 37, 418–435(1992).
[CrossRef]

Int. J. Contr.

M. G. Safonov, D. J. N. Limebeer, and R. Y. Chiang, “Simplifying the H∞ theory via loop shifting, matrix pencil and descriptor concepts,” Int. J. Contr. 50, 2467–2488 (1989).
[CrossRef]

J. Opt. Soc. Am. A

Opt. Express

Opt. Lett.

Proc. SPIE

B. Neichel, F. Rigaut, M. Bec, M. Boccas, F. Daruich, C. D’Orgeville, V. Fesquet, R. Galvez, A. Garcia-Rissmann, G. Gausachs, M. Lombini, G. Perez, G. Trancho, V. Upadhya, and T. Vucina, “The Gemini MCAO systems GeMS: nearing the end of a lab-story,” Proc. SPIE 7736, 773606 (2010).
[CrossRef]

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

J. Maly, E. Darren, and T. Pargett, “Vibration suppression for the Gemini planet image,” Proc. SPIE 7733, 77331F (2010).
[CrossRef]

Y. Clenet, M. Kasper, N. Ageorges, C. Lidman, T. Fusco, G. Rousset, O. Marco, M. Hartung, D. Mouillet, B. Koehler, and N. Hubin, “NACO performance: status after 2 years of operation,” Proc. SPIE 5490, 107–117 (2004).
[CrossRef]

Other

J. P. Verán and L. Poyneer, “Evaluation of the T/T conditions at Gemini South using NICI AO telemetry data,” in First AO4ELT Conference (EdP Sciences, 2009), p. 05002.

J. Christou, C. Trujillo, B. Neichel, F. Rigaut, B. Walls, D. Coulson, J. White, A. Stephens, and M. Sheehan, “The effect of the instrument environment on the Altair AO system,” presented at Second AO4ELT Conference, Victoria, British Columbia, Canada, 25–30Sept.2011.

B. Neichel, F. Rigaut, A. Guesalaga, I. Rodriguez, and D. Guzman, “Kalman and H-infinity controllers for GeMS,” in Imaging Systems Applications OSA Technical Digest (CD) (Optical Society of America, 2011), paper JWA32.

J. M. Conan, H. F. Raynaud, C. Kulcsár, and S. Meimon, “Are integral controllers adapted to the new era of ELT adaptive optics?” presented at Second AO4ELT Conference, Victoria, British Columbia, Canada, 25–30Sept.2011.

F. Rigaut and B. Neichel, “Gemini South MCAO on-sky results,” presented at Second AO4ELT Conference, Victoria, British Columbia, Canada, 25–30Sept.2011.

Y. Clenet, M. Kasper, N. Ageorges, C. Lidman, T. Fusco, O. P. Marco, M. Hartung, D. Mouillet, B. Koehler, G. Rousset, and N. Hubin, “NAOS performances: impact of the telescope vibrations and possible origins,” in Semaine de l’Astrophysique Francaise (EdP Sciences, 2004), p. 179.

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Figures (20)

Fig. 1.
Fig. 1.

GeMS.

Fig. 2.
Fig. 2.

Data from calibration sources. Left panel: PSDs for NGSWFS (tip or X is coarse line and Tilt or Y is fine line). Right panel: LGSWFS (fine line) and GSAOI-tip/tilt (coarse line). The plots on the right panel are in a PSD×frequency representation to emphasize the distribution of energy.

Fig. 3.
Fig. 3.

On-sky PSDs for the NGSWFS (tilt). Left: data acquired in April 2011; right: data acquired in March 2011.

Fig. 4.
Fig. 4.

On-sky PSDs for the LGSWFS. Data acquired in March 2011.

Fig. 5.
Fig. 5.

OL data simulated on the bench as seen by the LGSWFS. Left panel: slow seeing and 55 Hz vibration. Right panel: fast seeing and 55 Hz vibration (continuous line), slow seeing and 14 Hz vibration (dashed line), fast seeing and a 100 Hz vibration (dotted line).

Fig. 6.
Fig. 6.

Closed-loop model of the tip–tilt AO loop.

Fig. 7.
Fig. 7.

AO closed-loop block diagram.

Fig. 8.
Fig. 8.

Standard controller-plant configuration for H/H2 synthesis.

Fig. 9.
Fig. 9.

Servo-control problem.

Fig. 10.
Fig. 10.

Augmented representation used to synthesize the H controller, G(s).

Fig. 11.
Fig. 11.

Fitting of We to open-loop NGSWFS data and definition of Wu according to manufacturer’s specs (We plot has been vertically displaced for clarity). Data are taken from observations on Feb. 11, 2011.

Fig. 12.
Fig. 12.

SF obtained through H synthesis (left) and PSD of residuals obtained with the H controller (right).

Fig. 13.
Fig. 13.

Effect of vibration rejection due to Bode’s theorem.

Fig. 14.
Fig. 14.

SF for the three controllers in the case of a notch at 14 Hz (top-left), 55 Hz (top-right), and 100 Hz (bottom).

Fig. 15.
Fig. 15.

Case I, 14 Hz vibration (top-left), 55 Hz (top-right), and 100 Hz (bottom-left): SF for OL, integrator, Kalman, and H controllers. The bottom-right shows a zoom around the 100 Hz, where Kalman starts underperforming due to the absence of the TTM dynamics in the design stage. A similar pattern is also observed for the 55 Hz vibration.

Fig. 16.
Fig. 16.

Case II: PSD plots for OL, integrator, Kalman, and H controllers. Turbulence with 14 Hz vibration (left) and 55 Hz vibration (right). Third and 11th harmonics are observed due to TTM saturation.

Fig. 17.
Fig. 17.

Case III: PSD plots for vibrations at 14 Hz (left) and 55 Hz (right) for OL, integrator, Kalman and H controllers.

Fig. 18.
Fig. 18.

Examples of H controllers for higher-order disturbances. Left: weighting functions modeling the disturbances; Right: resulting SFs in each case.

Fig. 19.
Fig. 19.

Effect of frequency mismatch between real and modeled vibration frequencies. The continuous line corresponds to a correctly identified frequency. Departure from the correct frequency can degrade the performance.

Fig. 20.
Fig. 20.

Kalman SF under TTM bandwidth uncertainty.

Tables (3)

Tables Icon

Table 1. Parameters for We and Wu

Tables Icon

Table 2. RMS of Residual Slopes (Normalized to the Integrator Residual Variance)a

Tables Icon

Table 3. Summary of Controller Characteristics

Equations (31)

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

yn=Dφn1res+wn,
G(z)=Ki1az1,
min|εn+1|2φ,noise=min|φn+1turφn+1cor|2φ,noise=min|φn+1turNun|2φ,noise.
φn=φntur+φnvib,1+φnvib,2++φnvib,i.
φn+1vib,i=a1vib,iφnvib,i+a2vib,iφn1vib,i+vnvib,i,
φn+1tur=a1turφntur+a2turφn1tur+vntur,
xn=[φnturφn1turφnvibφn1vib].
xn+1=Axn+vnyn=CxnDNun2+wn,
[φn+1turφnturφn+1vibφnvib]xn+1=[a1tura2tur00100000a1viba2vib0010]A[φnturφn1turφnvibφn1vib]xn+[vntur0vnvib0]vn,
yn=[0D0D]CxnDNun2+wn.
x^n/n=Ax^n/n1+L(ynCx^n/n1+DNun2),x^n+1/n=Ax^n/n,un=Kx^n+1/n,
G(z)=K[Id(IdLC)Az1+LDKz2]1L.
[ze]=P(s)[ru]=[P11(s)P12(s)P21(s)P22(s)][ru],
u=G(s)e.
z=Fl(P,G)r.
Fl(P,G)=P11+P12G(IP22G)1P21.
Fl(P,G)=supωσ¯(Fl(P,G)(jω)),
P[A,B1,B2,C1,C2,D11,D12,D21,D22] :[AB1B2C1D11D12C2D21D22],
AK=A+γ2B1B1XB2B2X(Iγ2XY)1YC2C2,BK=B2X,CK=(Iγ2XY)1YC2,DK=0,
X=Ric[Aγ2B1B1B2B2B1B1A],
Y=Ric[Aγ2C1C1C2C2B1B1A].
SF(s)=e(s)r(s)=11+M(s)G(s),
CSF(s)=u(s)r(s)=G(s)1+M(s)G(s),
NTF(s)=e(s)w(s)=M(s)G(s)1+M(s)G(s).
P(s)=[P11(s)P12(s)P21(s)P22(s)]=[We(s)We(s)M(s)0Wu(s)1M(s)].
minGFl(P,G)=minGWe·SFWu·CSF.
We(s)=C0s+C1·s2+2η1ωos+ωo2s2+2η2ωos+ωo2,
M(s)=e2Δs0.0025s+1,
Wu(S)=C2·SC3S+1,
|SF|21/We.
ffs/2ln|SF(f)|df=β,

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