Abstract

Efficient and optimal prediction of frozen flow turbulence using the complete observation history of the wavefront sensor is an important issue in adaptive optics for large ground-based telescopes. At least for the sake of error budgeting and algorithm performance, the evaluation of an accurate estimate of the optimal performance of a particular adaptive optics configuration is important. However, due to the large number of grid points, high sampling rates, and the non-rationality of the turbulence power spectral density, the computational complexity of the optimal predictor is huge. This paper shows how a structure in the frozen flow propagation can be exploited to obtain a state-space innovation model with a particular sparsity structure. This sparsity structure enables one to efficiently compute a structured Kalman filter. By simulation it is shown that the performance can be improved and the computational complexity can be reduced in comparison with auto-regressive predictors of low order.

© 2010 Optical Society of America

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  1. M. Morari and E. Zafiriou, Robust Process Control (Prentice Hall, 1989).
  2. C. Vogel, “Sparse matrix methods for wavefront reconstruction revisited,” Proc. SPIE 5490, 1327–1335 (2004).
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  3. M. Le Louarn, N. Hubin, M. Sarazin, and A. Tokovinin, “New challenges for adaptive optics: Extremely large telescopes,” Mon. Not. R. Astron. Soc. 317, 535–544 (2000).
    [CrossRef]
  4. B. Ellerbroek, “Efficient computation of minimum variance wavefront reconstructors using sparse matrix techniques,” J. Opt. Soc. Am. A 19, 1803–1816 (2002).
    [CrossRef]
  5. L. Gilles, C. Vogel, and B. Ellerbroek, “Multigrid preconditioned conjugate-gradient method for large-scale wave-front reconstruction,” J. Opt. Soc. Am. A 19, 1817–1822 (2002).
    [CrossRef]
  6. L. Poyneer, D. Gavel, and J. Brase, “Fast wave-front reconstruction in large adaptive optics systems with use of the Fourier transform,” J. Opt. Soc. Am. A 19, 2100–2111 (2002).
    [CrossRef]
  7. M. Tallon, E. Thiébaut, and C. Béchet, “A fractal iterative method for fast wavefront reconstruction for extremely large telescopes,” in Proceedings of Adaptive Optics: Analysis and Methods (2007), pp. 1–3.
  8. L. Lessard, M. West, D. MacMynowski, and S. Lall, “Warm-started wavefront reconstruction for adaptive optics,” J. Opt. Soc. Am. A 25, 1147–1155 (2008).
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  11. D. Gavel and D. Wiberg, “Toward Strehl-optimizing adaptive optics controllers,” Proc. SPIE 4839, 890–901 (2003).
    [CrossRef]
  12. B. L. 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, 1261–1276 (2004).
    [CrossRef]
  13. K. Hinnen, M. Verhaegen, and N. Doelman, “Robust spectral factor approximation of discrete-time frequency domain power spectra,” Automatica 41, 1791–1798 (2005).
    [CrossRef]
  14. A. Beghi, A. Cenedese, and A. Madiero, “Atmospheric turbulence prediction: a PCA approach,” in Proceedings of the 46th IEEE Conference on Decision and Control (IEEE, 2007), pp. 566–571.
    [CrossRef]
  15. A. Beghi, A. Cenedese, and A. Masiero, “Stochastic realization approach to the efficient simulation of phase screens,” J. Opt. Soc. Am. A 25, 515–525 (2008).
    [CrossRef]
  16. K. Hinnen, M. Verhaegen, and N. Doelman, “Exploiting the spatiotemporal correlation in adaptive optics using data-driven H2-optimal control,” J. Opt. Soc. Am. A 24, 1714–1725 (2007).
    [CrossRef]
  17. N. Doelman, R. Fraanje, I. Houtzager, and M. Verhaegen, “Real-time optimal control for adaptive optics systems,” Eur. J. Control 15, 480–488 (2009).
    [CrossRef]
  18. L. Poyneer, B. Macintosh, and J.-P. Véran, “Fourier transform wavefront control with adaptive prediction of the atmosphere,” J. Opt. Soc. Am. A 24, 2645–2660 (2007).
    [CrossRef]
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    [CrossRef]
  20. J. Rice and M. Verhaegen, “Distributed control: A sequentially semi-separable approach for spatially heterogeneous linear systems,” IEEE Trans. Autom. Control 54, 1270–1283 (2009).
    [CrossRef]
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  22. J. Rice and M. Verhaegen, “Distributed control of spatially invariant systems in multiple dimensions: A structure preserving computational technique” (submitted for publication, http://www.dcsc.tudelft.nl/jrice/Publications.htm).
  23. T. Laakso, V. Välimäki, M. Karjalainen, and U. Laine, “Splitting the unit delay—tools for fractional delay filter design,” IEEE Signal Process. Mag. 13, 30–60 (1996).
    [CrossRef]
  24. P. Brockwell and R. Davies, Time Series: Theory and Methods (Springer, 1991).
    [CrossRef]
  25. M. Verhaegen and V. Verdult, Filtering and System Identification—A Least Squares Approach (Cambridge University Press, 2007).
    [CrossRef]
  26. P. Van Overschee and B. De Moor, “Subspace algorithms for the stochastic identification problem,” Automatica 29, 649–660 (1993).
    [CrossRef]
  27. B. Anderson and J. Moore, Optimal Filtering (Prentice-Hall, 1979).
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    [CrossRef] [PubMed]

2009 (2)

J. Rice and M. Verhaegen, “Distributed control: A sequentially semi-separable approach for spatially heterogeneous linear systems,” IEEE Trans. Autom. Control 54, 1270–1283 (2009).
[CrossRef]

N. Doelman, R. Fraanje, I. Houtzager, and M. Verhaegen, “Real-time optimal control for adaptive optics systems,” Eur. J. Control 15, 480–488 (2009).
[CrossRef]

2008 (2)

2007 (5)

K. Hinnen, M. Verhaegen, and N. Doelman, “Exploiting the spatiotemporal correlation in adaptive optics using data-driven H2-optimal control,” J. Opt. Soc. Am. A 24, 1714–1725 (2007).
[CrossRef]

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

A. Beghi, A. Cenedese, and A. Madiero, “Atmospheric turbulence prediction: a PCA approach,” in Proceedings of the 46th IEEE Conference on Decision and Control (IEEE, 2007), pp. 566–571.
[CrossRef]

M. Verhaegen and V. Verdult, Filtering and System Identification—A Least Squares Approach (Cambridge University Press, 2007).
[CrossRef]

M. Tallon, E. Thiébaut, and C. Béchet, “A fractal iterative method for fast wavefront reconstruction for extremely large telescopes,” in Proceedings of Adaptive Optics: Analysis and Methods (2007), pp. 1–3.

2006 (1)

2005 (1)

K. Hinnen, M. Verhaegen, and N. Doelman, “Robust spectral factor approximation of discrete-time frequency domain power spectra,” Automatica 41, 1791–1798 (2005).
[CrossRef]

2004 (2)

2003 (1)

D. Gavel and D. Wiberg, “Toward Strehl-optimizing adaptive optics controllers,” Proc. SPIE 4839, 890–901 (2003).
[CrossRef]

2002 (4)

2000 (2)

M. Le Louarn, N. Hubin, M. Sarazin, and A. Tokovinin, “New challenges for adaptive optics: Extremely large telescopes,” Mon. Not. R. Astron. Soc. 317, 535–544 (2000).
[CrossRef]

R. Conan, “Modélisation des effects de l’échelle externe de cohérence spatiale du front d’onde pour l’observation à haure résolution angulaire en astronomie,” Ph.D. dissertation (Université de Nice-Sophia Antipolis, 2000).

1996 (1)

T. Laakso, V. Välimäki, M. Karjalainen, and U. Laine, “Splitting the unit delay—tools for fractional delay filter design,” IEEE Signal Process. Mag. 13, 30–60 (1996).
[CrossRef]

1994 (1)

1993 (1)

P. Van Overschee and B. De Moor, “Subspace algorithms for the stochastic identification problem,” Automatica 29, 649–660 (1993).
[CrossRef]

1992 (1)

1991 (1)

P. Brockwell and R. Davies, Time Series: Theory and Methods (Springer, 1991).
[CrossRef]

1989 (1)

M. Morari and E. Zafiriou, Robust Process Control (Prentice Hall, 1989).

1979 (1)

B. Anderson and J. Moore, Optimal Filtering (Prentice-Hall, 1979).

Aitken, G.

Anderson, B.

B. Anderson and J. Moore, Optimal Filtering (Prentice-Hall, 1979).

Baum, G.

Béchet, C.

M. Tallon, E. Thiébaut, and C. Béchet, “A fractal iterative method for fast wavefront reconstruction for extremely large telescopes,” in Proceedings of Adaptive Optics: Analysis and Methods (2007), pp. 1–3.

Beghi, A.

A. Beghi, A. Cenedese, and A. Masiero, “Stochastic realization approach to the efficient simulation of phase screens,” J. Opt. Soc. Am. A 25, 515–525 (2008).
[CrossRef]

A. Beghi, A. Cenedese, and A. Madiero, “Atmospheric turbulence prediction: a PCA approach,” in Proceedings of the 46th IEEE Conference on Decision and Control (IEEE, 2007), pp. 566–571.
[CrossRef]

Brase, J.

Brockwell, P.

P. Brockwell and R. Davies, Time Series: Theory and Methods (Springer, 1991).
[CrossRef]

Cenedese, A.

A. Beghi, A. Cenedese, and A. Masiero, “Stochastic realization approach to the efficient simulation of phase screens,” J. Opt. Soc. Am. A 25, 515–525 (2008).
[CrossRef]

A. Beghi, A. Cenedese, and A. Madiero, “Atmospheric turbulence prediction: a PCA approach,” in Proceedings of the 46th IEEE Conference on Decision and Control (IEEE, 2007), pp. 566–571.
[CrossRef]

Chandrasekaran, S.

S. Chandrasekaran, P. Dewilde, M. Gu, T. Pals, and A.-J. van der Veen, “Fast stable solvers for sequentially semi-separable linear systems of equations,” in Lecture Notes in Computer Science (Springer Verlag, 2002), pp. 545–554.
[CrossRef]

Conan, J. -M.

Conan, R.

R. Conan, “Modélisation des effects de l’échelle externe de cohérence spatiale du front d’onde pour l’observation à haure résolution angulaire en astronomie,” Ph.D. dissertation (Université de Nice-Sophia Antipolis, 2000).

Davies, R.

P. Brockwell and R. Davies, Time Series: Theory and Methods (Springer, 1991).
[CrossRef]

De Moor, B.

P. Van Overschee and B. De Moor, “Subspace algorithms for the stochastic identification problem,” Automatica 29, 649–660 (1993).
[CrossRef]

Dewilde, P.

S. Chandrasekaran, P. Dewilde, M. Gu, T. Pals, and A.-J. van der Veen, “Fast stable solvers for sequentially semi-separable linear systems of equations,” in Lecture Notes in Computer Science (Springer Verlag, 2002), pp. 545–554.
[CrossRef]

Doelman, N.

N. Doelman, R. Fraanje, I. Houtzager, and M. Verhaegen, “Real-time optimal control for adaptive optics systems,” Eur. J. Control 15, 480–488 (2009).
[CrossRef]

K. Hinnen, M. Verhaegen, and N. Doelman, “Exploiting the spatiotemporal correlation in adaptive optics using data-driven H2-optimal control,” J. Opt. Soc. Am. A 24, 1714–1725 (2007).
[CrossRef]

K. Hinnen, M. Verhaegen, and N. Doelman, “Robust spectral factor approximation of discrete-time frequency domain power spectra,” Automatica 41, 1791–1798 (2005).
[CrossRef]

Ellerbroek, B.

Fraanje, R.

N. Doelman, R. Fraanje, I. Houtzager, and M. Verhaegen, “Real-time optimal control for adaptive optics systems,” Eur. J. Control 15, 480–488 (2009).
[CrossRef]

Fusco, T.

Gavel, D.

Gilles, L.

Gu, M.

S. Chandrasekaran, P. Dewilde, M. Gu, T. Pals, and A.-J. van der Veen, “Fast stable solvers for sequentially semi-separable linear systems of equations,” in Lecture Notes in Computer Science (Springer Verlag, 2002), pp. 545–554.
[CrossRef]

Hinnen, K.

K. Hinnen, M. Verhaegen, and N. Doelman, “Exploiting the spatiotemporal correlation in adaptive optics using data-driven H2-optimal control,” J. Opt. Soc. Am. A 24, 1714–1725 (2007).
[CrossRef]

K. Hinnen, M. Verhaegen, and N. Doelman, “Robust spectral factor approximation of discrete-time frequency domain power spectra,” Automatica 41, 1791–1798 (2005).
[CrossRef]

Houtzager, I.

N. Doelman, R. Fraanje, I. Houtzager, and M. Verhaegen, “Real-time optimal control for adaptive optics systems,” Eur. J. Control 15, 480–488 (2009).
[CrossRef]

Hubin, N.

M. Le Louarn, N. Hubin, M. Sarazin, and A. Tokovinin, “New challenges for adaptive optics: Extremely large telescopes,” Mon. Not. R. Astron. Soc. 317, 535–544 (2000).
[CrossRef]

Jorgenson, M.

Karjalainen, M.

T. Laakso, V. Välimäki, M. Karjalainen, and U. Laine, “Splitting the unit delay—tools for fractional delay filter design,” IEEE Signal Process. Mag. 13, 30–60 (1996).
[CrossRef]

Kulcsár, C.

Laakso, T.

T. Laakso, V. Välimäki, M. Karjalainen, and U. Laine, “Splitting the unit delay—tools for fractional delay filter design,” IEEE Signal Process. Mag. 13, 30–60 (1996).
[CrossRef]

Laine, U.

T. Laakso, V. Välimäki, M. Karjalainen, and U. Laine, “Splitting the unit delay—tools for fractional delay filter design,” IEEE Signal Process. Mag. 13, 30–60 (1996).
[CrossRef]

Lall, S.

Lessard, L.

Louarn, M. Le

M. Le Louarn, N. Hubin, M. Sarazin, and A. Tokovinin, “New challenges for adaptive optics: Extremely large telescopes,” Mon. Not. R. Astron. Soc. 317, 535–544 (2000).
[CrossRef]

Macintosh, B.

MacMynowski, D.

Madiero, A.

A. Beghi, A. Cenedese, and A. Madiero, “Atmospheric turbulence prediction: a PCA approach,” in Proceedings of the 46th IEEE Conference on Decision and Control (IEEE, 2007), pp. 566–571.
[CrossRef]

Masiero, A.

Moore, J.

B. Anderson and J. Moore, Optimal Filtering (Prentice-Hall, 1979).

Morari, M.

M. Morari and E. Zafiriou, Robust Process Control (Prentice Hall, 1989).

Mugnier, L.

Pals, T.

S. Chandrasekaran, P. Dewilde, M. Gu, T. Pals, and A.-J. van der Veen, “Fast stable solvers for sequentially semi-separable linear systems of equations,” in Lecture Notes in Computer Science (Springer Verlag, 2002), pp. 545–554.
[CrossRef]

Poyneer, L.

Raynaud, H. -F.

Ribak, E.

Rice, J.

J. Rice and M. Verhaegen, “Distributed control: A sequentially semi-separable approach for spatially heterogeneous linear systems,” IEEE Trans. Autom. Control 54, 1270–1283 (2009).
[CrossRef]

J. Rice and M. Verhaegen, “Distributed control of spatially invariant systems in multiple dimensions: A structure preserving computational technique” (submitted for publication, http://www.dcsc.tudelft.nl/jrice/Publications.htm).

Roux, B. L.

Sarazin, M.

M. Le Louarn, N. Hubin, M. Sarazin, and A. Tokovinin, “New challenges for adaptive optics: Extremely large telescopes,” Mon. Not. R. Astron. Soc. 317, 535–544 (2000).
[CrossRef]

Schwartz, C.

Tallon, M.

M. Tallon, E. Thiébaut, and C. Béchet, “A fractal iterative method for fast wavefront reconstruction for extremely large telescopes,” in Proceedings of Adaptive Optics: Analysis and Methods (2007), pp. 1–3.

Thiébaut, E.

M. Tallon, E. Thiébaut, and C. Béchet, “A fractal iterative method for fast wavefront reconstruction for extremely large telescopes,” in Proceedings of Adaptive Optics: Analysis and Methods (2007), pp. 1–3.

Tokovinin, A.

M. Le Louarn, N. Hubin, M. Sarazin, and A. Tokovinin, “New challenges for adaptive optics: Extremely large telescopes,” Mon. Not. R. Astron. Soc. 317, 535–544 (2000).
[CrossRef]

Välimäki, V.

T. Laakso, V. Välimäki, M. Karjalainen, and U. Laine, “Splitting the unit delay—tools for fractional delay filter design,” IEEE Signal Process. Mag. 13, 30–60 (1996).
[CrossRef]

van der Veen, A. -J.

S. Chandrasekaran, P. Dewilde, M. Gu, T. Pals, and A.-J. van der Veen, “Fast stable solvers for sequentially semi-separable linear systems of equations,” in Lecture Notes in Computer Science (Springer Verlag, 2002), pp. 545–554.
[CrossRef]

Van Overschee, P.

P. Van Overschee and B. De Moor, “Subspace algorithms for the stochastic identification problem,” Automatica 29, 649–660 (1993).
[CrossRef]

Véran, J. -P.

Verdult, V.

M. Verhaegen and V. Verdult, Filtering and System Identification—A Least Squares Approach (Cambridge University Press, 2007).
[CrossRef]

Verhaegen, M.

J. Rice and M. Verhaegen, “Distributed control: A sequentially semi-separable approach for spatially heterogeneous linear systems,” IEEE Trans. Autom. Control 54, 1270–1283 (2009).
[CrossRef]

N. Doelman, R. Fraanje, I. Houtzager, and M. Verhaegen, “Real-time optimal control for adaptive optics systems,” Eur. J. Control 15, 480–488 (2009).
[CrossRef]

M. Verhaegen and V. Verdult, Filtering and System Identification—A Least Squares Approach (Cambridge University Press, 2007).
[CrossRef]

K. Hinnen, M. Verhaegen, and N. Doelman, “Exploiting the spatiotemporal correlation in adaptive optics using data-driven H2-optimal control,” J. Opt. Soc. Am. A 24, 1714–1725 (2007).
[CrossRef]

K. Hinnen, M. Verhaegen, and N. Doelman, “Robust spectral factor approximation of discrete-time frequency domain power spectra,” Automatica 41, 1791–1798 (2005).
[CrossRef]

J. Rice and M. Verhaegen, “Distributed control of spatially invariant systems in multiple dimensions: A structure preserving computational technique” (submitted for publication, http://www.dcsc.tudelft.nl/jrice/Publications.htm).

Vogel, C.

West, M.

Wiberg, D.

D. Gavel and D. Wiberg, “Toward Strehl-optimizing adaptive optics controllers,” Proc. SPIE 4839, 890–901 (2003).
[CrossRef]

Yang, Q.

Zafiriou, E.

M. Morari and E. Zafiriou, Robust Process Control (Prentice Hall, 1989).

Appl. Opt. (1)

Automatica (2)

P. Van Overschee and B. De Moor, “Subspace algorithms for the stochastic identification problem,” Automatica 29, 649–660 (1993).
[CrossRef]

K. Hinnen, M. Verhaegen, and N. Doelman, “Robust spectral factor approximation of discrete-time frequency domain power spectra,” Automatica 41, 1791–1798 (2005).
[CrossRef]

Eur. J. Control (1)

N. Doelman, R. Fraanje, I. Houtzager, and M. Verhaegen, “Real-time optimal control for adaptive optics systems,” Eur. J. Control 15, 480–488 (2009).
[CrossRef]

IEEE Signal Process. Mag. (1)

T. Laakso, V. Välimäki, M. Karjalainen, and U. Laine, “Splitting the unit delay—tools for fractional delay filter design,” IEEE Signal Process. Mag. 13, 30–60 (1996).
[CrossRef]

IEEE Trans. Autom. Control (1)

J. Rice and M. Verhaegen, “Distributed control: A sequentially semi-separable approach for spatially heterogeneous linear systems,” IEEE Trans. Autom. Control 54, 1270–1283 (2009).
[CrossRef]

J. Opt. Soc. Am. A (9)

B. Ellerbroek, “Efficient computation of minimum variance wavefront reconstructors using sparse matrix techniques,” J. Opt. Soc. Am. A 19, 1803–1816 (2002).
[CrossRef]

L. Gilles, C. Vogel, and B. Ellerbroek, “Multigrid preconditioned conjugate-gradient method for large-scale wave-front reconstruction,” J. Opt. Soc. Am. A 19, 1817–1822 (2002).
[CrossRef]

L. Poyneer, D. Gavel, and J. Brase, “Fast wave-front reconstruction in large adaptive optics systems with use of the Fourier transform,” J. Opt. Soc. Am. A 19, 2100–2111 (2002).
[CrossRef]

B. L. 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, 1261–1276 (2004).
[CrossRef]

C. Schwartz, G. Baum, and E. Ribak, “Turbulence-degraded wave fronts as fractal surfaces,” J. Opt. Soc. Am. A 11, 444–451 (1994).
[CrossRef]

K. Hinnen, M. Verhaegen, and N. Doelman, “Exploiting the spatiotemporal correlation in adaptive optics using data-driven H2-optimal control,” J. Opt. Soc. Am. A 24, 1714–1725 (2007).
[CrossRef]

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

A. Beghi, A. Cenedese, and A. Masiero, “Stochastic realization approach to the efficient simulation of phase screens,” J. Opt. Soc. Am. A 25, 515–525 (2008).
[CrossRef]

L. Lessard, M. West, D. MacMynowski, and S. Lall, “Warm-started wavefront reconstruction for adaptive optics,” J. Opt. Soc. Am. A 25, 1147–1155 (2008).
[CrossRef]

Mon. Not. R. Astron. Soc. (1)

M. Le Louarn, N. Hubin, M. Sarazin, and A. Tokovinin, “New challenges for adaptive optics: Extremely large telescopes,” Mon. Not. R. Astron. Soc. 317, 535–544 (2000).
[CrossRef]

Opt. Lett. (1)

Proc. SPIE (2)

D. Gavel and D. Wiberg, “Toward Strehl-optimizing adaptive optics controllers,” Proc. SPIE 4839, 890–901 (2003).
[CrossRef]

C. Vogel, “Sparse matrix methods for wavefront reconstruction revisited,” Proc. SPIE 5490, 1327–1335 (2004).
[CrossRef]

Other (9)

M. Morari and E. Zafiriou, Robust Process Control (Prentice Hall, 1989).

M. Tallon, E. Thiébaut, and C. Béchet, “A fractal iterative method for fast wavefront reconstruction for extremely large telescopes,” in Proceedings of Adaptive Optics: Analysis and Methods (2007), pp. 1–3.

S. Chandrasekaran, P. Dewilde, M. Gu, T. Pals, and A.-J. van der Veen, “Fast stable solvers for sequentially semi-separable linear systems of equations,” in Lecture Notes in Computer Science (Springer Verlag, 2002), pp. 545–554.
[CrossRef]

R. Conan, “Modélisation des effects de l’échelle externe de cohérence spatiale du front d’onde pour l’observation à haure résolution angulaire en astronomie,” Ph.D. dissertation (Université de Nice-Sophia Antipolis, 2000).

J. Rice and M. Verhaegen, “Distributed control of spatially invariant systems in multiple dimensions: A structure preserving computational technique” (submitted for publication, http://www.dcsc.tudelft.nl/jrice/Publications.htm).

P. Brockwell and R. Davies, Time Series: Theory and Methods (Springer, 1991).
[CrossRef]

M. Verhaegen and V. Verdult, Filtering and System Identification—A Least Squares Approach (Cambridge University Press, 2007).
[CrossRef]

A. Beghi, A. Cenedese, and A. Madiero, “Atmospheric turbulence prediction: a PCA approach,” in Proceedings of the 46th IEEE Conference on Decision and Control (IEEE, 2007), pp. 566–571.
[CrossRef]

B. Anderson and J. Moore, Optimal Filtering (Prentice-Hall, 1979).

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Figures (7)

Fig. 1
Fig. 1

Frozen flow propagation by subsystem string interconnection.

Fig. 2
Fig. 2

Frozen flow propagation along a square grid in a circular aperture.

Fig. 3
Fig. 3

SSS matrix-vector multiplication as a subsystem string interconnection.

Fig. 4
Fig. 4

Spectral density for Δ = 2 , SNR = 40   dB .

Fig. 5
Fig. 5

Strehl ratio versus SNR for WFS resolution Δ = 2 .

Fig. 6
Fig. 6

Strehl ratio versus SNR for WFS resolution Δ = 4 .

Fig. 7
Fig. 7

Strehl ratio versus SNR for WFS resolution Δ = 8 .

Tables (2)

Tables Icon

Table 1 Computational Complexity of the SSS Kalman 2-Step-Ahead Predictor

Tables Icon

Table 2 Computational Complexity in Floating Point Operations per Sampling Time for the SSS Structured Kalman Predictor, the Various Types of AR-1 Predictors, the Kalman Filter with Unstructured Kalman Gain but Sparse State-Space Matrices, and the Kalman Filter with Unstructured State-Space Matrices

Equations (64)

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

C ϕ ( δ x , δ y ) E [ ϕ ( t , x , y ) ϕ ( t , x δ x , y δ y ) ] = c ( 2 π r / L 0 ) 5 / 6 K 5 / 6 ( 2 π r / L 0 ) ,
c = Γ ( 11 / 6 ) 2 5 / 6 π 8 / 3 ( 24 5 Γ ( 6 / 5 ) ) 5 / 6 ( L 0 r 0 ) 5 / 3 ,
r = ( δ x ) 2 + ( δ y ) 2 ,
ϕ ( t + τ , x , y ) = ϕ ( t , x v x τ , y v y τ ) .
ϕ ( k , i , j ) = ϕ ( k Δ T , i Δ X , j Δ Y ) ,
ϕ ( k + 1 , i , j ) = ϕ ( k , i 1 , j ) .
E [ ϕ ( k , i , j ) ϕ ( k , i m , j n ) ] = C ϕ ( ( m ) Δ X , n Δ Y ) .
φ ( k ) [ ϕ ( k , 1 , 1 ) , , ϕ ( k , 1 , N y ) , , ϕ ( k , N x , N y ) ] T R N x N y .
C φ ( ) E [ φ ( k ) φ ( k ) T ] .
y ( k ) G φ ( k ) + ν ( k ) R N y ,
φ ̂ ( k + 2 k ) = A 1 y ( k ) ,
A 1 = C φ ( 2 ) G T ( G C φ ( 0 ) G T + σ v 2 I N y ) 1 .
φ ̂ ( k + 2 k ) = A k y k ( k ) ,
y k ( k ) = [ y ( k ) T , y ( k 1 ) T , , y ( 1 ) T ] T ,
A 1 = C φ ( 2 ) G T P 1 ,
P 1 = ( G C φ ( 0 ) G T + σ v 2 I N x N y ) 1 ,
S 1 = G C φ ( 1 ) G T ,
A k = [ A k 1 0 ] + E k 1 [ K k 1 T R k 1 1 ] ,
K k 1 = P k 1 S k 1 R k 1 1 ,
R k 1 = G C φ ( 0 ) G T + σ v 2 I N x N y S k 1 T P k 1 S k 1 ,
E k 1 = A k 1 S k 1 C φ ( k + 1 ) T G T ,
P k = [ P k 1 + K k 1 R k 1 K k 1 T K k 1 K k 1 T R k 1 1 ] ,
S k = [ G C ( k ) G T S k 1 ] .
x ( k + 1 ) = A x ( k ) + K e ( k ) ,
φ ( k ) = C x ( k ) + e ( k ) ,
x ̂ ( k + 1 k ) = A x ̂ ( k k 1 ) + K y ( y ( k ) G C x ̂ ( k k 1 ) ) ,
φ ̂ ( k + 1 k ) = C x ̂ ( k + 1 k ) ,
K y = ( A P C T G T + S ) ( G C P C T G T + R ) 1 ,
P = A P A T ( A P C T G T + S ) ( G C P C T G T + R ) 1 ( A P C T G T + S ) T + Q ,
Q = K R e K T ,
R = G R e G T + σ v 2 I N y ,
S = K R e G T .
x ̂ ( k + 2 k ) = A x ̂ ( k + 1 k ) ,
φ ̂ ( k + 2 k ) = C x ̂ ( k + 2 k ) .
φ ( k + 1 , i , j ) = φ ( k , i 1 , j ) .
φ i ( k ) [ φ ( k , i , 1 ) , φ ( k , i , 2 ) , , φ ( k , i , N y ) ] T .
ξ 1 ( k + 1 ) = A ξ 1 ( k ) + K e 1 ( k ) ,
φ 1 ( k ) = C ξ 1 ( k ) + e 1 ( k ) ,
for   i = 1 :     Σ i : [ ξ i ( k + 1 ) v i + 1 m ( k ) φ i ( k ) ] = [ A 0 K C 0 I N y C 0 I N y ] [ ξ i ( k ) v i m ( k ) e i ( k ) ] ,
for   i = 2 , , N x :     Σ i : [ ξ i ( k + 1 ) v i + 1 m ( k ) φ i ( k ) ] = [ 0 I N y 0 I N y 0 0 I N y 0 0 ] [ ξ i ( k ) v i m ( k ) e i ( k ) ] .
[ ξ ¯ ( k + 1 ) φ ¯ ( k ) ] = [ A ¯ K ¯ C ¯ D ¯ ] [ ξ ¯ ( k ) e ¯ ( t ) ] ,
A ¯ = [ A C 0 I N y 0 I N y 0 ] ,     K ¯ = [ K I N y 0 0 0 0 0 ] ,
C ¯ = [ C I N y I N y ] ,     D ¯ = [ I N y 0 0 ] .
A ¯ = ( A i , j ) ,     where   A i j R i × m j   satisfies   A i j = { D i , if   i = j U i W i + 1 W j 1 V j , if   j > i P i R i 1 R j + 1 Q j , if   j < i , }
Σ i : [ v i + 1 m v i 1 p y i ] = [ R i 0 Q i 0 W i V i P i U i D i ] [ v i m v i p u i ] ,
y x ( k , i , j ) = 1 Δ n = 1 Δ φ ( k , i Δ , ( j 1 ) Δ + n ) φ ( k , ( i 1 ) Δ + 1 , ( j 1 ) Δ + n ) + ν x ( k , i , j ) ,
y y ( k , i , j ) = 1 Δ n = 1 Δ φ ( k , ( i 1 ) Δ + n , j Δ ) φ ( k , ( i 1 ) Δ + n , ( j 1 ) Δ + 1 ) + ν x ( k , i , j ) ,
φ ¯ i ( k ) = [ φ ( i 1 ) Δ + 1 ( k ) T , , φ i Δ ( k ) T ] T ,
y i x ( k ) = [ y x ( k , i , 1 ) T , , y x ( k , i , N y ) T ] T ,
y i y ( k ) = [ y y ( k , i , 1 ) T , , y y ( k , i , N y ) T ] T ,
y ¯ i ( k ) = [ y i x ( k ) y i y ( k ) ] = [ G x G y ] φ ¯ i ( k ) + ν ¯ i ( k ) ,
Σ i : [ ξ ¯ i ( k + 1 ) v ¯ i + 1 m ( k ) φ ¯ i ( k ) y ¯ i ( k ) ] = [ A i B i m K i e 0 C i m 0 D i m 0 C i φ 0 D i φ 0 C i y 0 D i y I 2 N y / Δ ] [ ξ ¯ i ( k ) v ¯ i m ( k ) e ¯ i ( k ) ν ¯ i ( k ) ] ,
[ ξ ¯ ( k ) φ ¯ ( k ) y ¯ ( k ) ] = [ A ¯ K ¯ 0 C ¯ D ¯ 0 C ¯ y D ¯ y I ] [ ξ ¯ ( k ) e ¯ ( k ) ν ¯ ( k ) ] ,
e ̂ ( k ) = y ¯ ( k ) C ¯ y ξ ̂ ( k k 1 ) ,
ξ ̂ ( k + 1 k ) = A ¯ ξ ̂ ( k k 1 ) + K ¯ y e ̂ ( k ) ,
ξ ̂ ( k + 2 k ) = A ¯ ξ ̂ ( k + 1 k ) ,
φ ̂ ( k + 2 k ) = C ¯ ξ ̂ ( k + 2 k ) ,
K ¯ y = ( A ¯ P ¯ ( C ¯ y ) T + S ¯ ) ( C ¯ y P ¯ ( C ¯ y ) T + R ¯ ) 1 ,
P ¯ = A ¯ P ¯ A ¯ T ( A ¯ P ¯ ( C ¯ y ) T + S ¯ ) ( C ¯ y P ¯ ( C ¯ y ) T + R ¯ ) 1 ( A ¯ P ¯ ( C ¯ y ) T + S ¯ ) T + Q ¯ ,
Q ¯ = K ¯ R ¯ e K ¯ T ,
R ¯ = D ¯ y R ¯ e ( D ¯ y ) T + σ v 2 I ,
S ¯ = K ¯ R ¯ e ( D ¯ y ) T .
[ v i + 1 m ( k ) v i 1 p ( k ) u i ( k ) ] = [ K i m m 0 K i m e 0 K i p p K i p e K i u m K i u p K i u e ] [ v i m ( k ) v i p ( k ) e ̂ i ( k ) ] ,
S r = exp ( σ e 2 ) ,

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