Abstract

The adaptive optics minimum variance control problem is formulated as a linear-quadratic-Gaussian optimization. The formulation incorporates the wavefront sensor frame integration in discrete-time models of the deformable mirror and incident wavefront. It shows that, under nearly ideal conditions, the resulting minimum variance controller approaches the integral controller commonly used in adaptive optics systems. The inputs to the controller dynamics are obtained from a reconstructor with the maximum a posteriori structure that uses the estimation error covariance of the wavefront error. The ideal conditions assumed to obtain the integral controller are as follows; isotropic first-order (but nonstationary) temporal atmospheric aberrations, no computational loop delay, and no deformable mirror dynamics. The effects of variations in these conditions are examined.

© 2006 Optical Society of America

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References

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  1. J. W. Hardy, Adaptive Optics for Astronomical Telescopes (Oxford, 1998).
  2. M. C. Roggemann and B. M. Welsh, Imaging Through Turbulence (CRC Press, 1996).
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    [CrossRef]
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  5. M. Kasper, 'Optimization of an adaptive optics system and its application to high-resolution imaging spectroscopy of T Tauri,' Ph.D. thesis (University of Heidelberg, Germany, 2000).
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    [CrossRef]
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    [CrossRef]
  8. J.-P. Veran, F. Rigaut, H. Maitre, and D. Rouan, 'Estimation of the adaptive optics long-exposure point-spread function using control loop data,' J. Opt. Soc. Am. A 14, 3057-3069 (1997).
    [CrossRef]
  9. C. Dessenne, P.-Y. Madec, and G. Rousset, 'Modal prediction for closed-loop adaptive optics,' Opt. Lett. 22, 1535-1537 (1997).
    [CrossRef]
  10. 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 (2003).
    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
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  17. D. M. Wiberg, C. E. Max, and D. T. Gavel, 'A spatial non-dynamic LQG controller: Part 2, Theory,' in Proceedings of the 2004 IEEE Conference on Decision and Control (IEEE, 2004), pp. 3333-3338.
  18. R. Ragazzoni, 'Pupil plane wavefront sensing with an oscillating prism,' J. Mod. Opt. 43, 289-293 (1996).
    [CrossRef]
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    [CrossRef]
  20. H. Kwakernaak and R. Sivan, Linear Optimal Control Systems (Wiley-Interscience, 1972).
  21. B. D. O. Anderson and J. B. Moore, Optimal Control: Linear Quadratic Methods (Prentice-Hall, 1990).
  22. MATLAB: Control System Toolbox (Mathworks, 1996).
  23. A. Oppenheim, A. S. Willsky, and I. T. Young, Signals and Systems (Prentice-Hall, 1983).
  24. A. Wirth, J. Navetta, D. P. Looze, S. Hippler, A. Glindemann, and D. Hamilton, 'Real-time modal control implementation for adaptive optics,' Appl. Opt. 37, 4586-4597 (1998).
    [CrossRef]
  25. S. Hippler, D. P. Looze, M. Feldt, E. Masciadri, R. Stuik, and W. Brandner, 'CHEOPS adaptive optics white book,' MPIA Doc. CHEOPS-TRE-MPI-00026 (Max-Planck: Institut für Astronomic, 2004).
  26. R. G. Lane, A. Glindemann, and J. C. Dainty, 'Simulation of a Kolmogorov phase screen,' Waves Random Media 2, 209-224 (1992).
    [CrossRef]

2005 (1)

D. P. Looze, 'Realization of systems with CCD-based measurements,' Automatica 41, 2005-2009 (2005).
[CrossRef]

2004 (1)

2003 (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 (2003).
[CrossRef]

2002 (1)

G. Duchene, A. M. Ghez, and C. McCabe, 'Resolved near-infrared spectroscopy of the mysterious pre-main-sequence binary system T Tauri S,' Astrophys. J. 568, 771-778 (2002).
[CrossRef]

2000 (2)

W. Hackenberg, A. Eckart, R. J. Davies, S. Rabien, T. Ott, M. Kasper, S. Hippler, and A. Quirrenbach, 'Near-infrared adaptive optics observations of galaxy clusters: Abell 262 at z=0.0157, J1836.3CR at z=0.414, and PKS 0743-006 at z=0.994,' Astron. Astrophys. 363, 41-61 (2000).

A. Tokovinin, M. Le Louarn, and M. Sarazin, 'Isoplanatism in a multiconjugate adaptive optics system,' J. Opt. Soc. Am. A 17, 1819-1827 (2000).
[CrossRef]

1998 (1)

1997 (2)

1996 (2)

R. Ragazzoni, 'Pupil plane wavefront sensing with an oscillating prism,' J. Mod. Opt. 43, 289-293 (1996).
[CrossRef]

N. F. Law and R. G. Lane, 'Wavefront estimation at low light levels,' Opt. Commun. 126, 19-24 (1996).
[CrossRef]

1995 (1)

E. Gendron and P. Léna, 'Astronomical adaptive optics. II. Experimental results of an optimized modal control,' Astron. Astrophys., Suppl. Ser. 111, 153-167 (1995).

1994 (1)

E. Gendron and P. Léna, 'Astronomical adaptive optics. I: Modal control optimization,' Astron. Astrophys. 291, 337-347 (1994).

1993 (1)

1992 (1)

R. G. Lane, A. Glindemann, and J. C. Dainty, 'Simulation of a Kolmogorov phase screen,' Waves Random Media 2, 209-224 (1992).
[CrossRef]

1983 (1)

Anderson, B. D.

B. D. O. Anderson and J. B. Moore, Optimal Control: Linear Quadratic Methods (Prentice-Hall, 1990).

Anderson, D. J.

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 (2003).
[CrossRef]

Brandner, W.

S. Hippler, D. P. Looze, M. Feldt, E. Masciadri, R. Stuik, and W. Brandner, 'CHEOPS adaptive optics white book,' MPIA Doc. CHEOPS-TRE-MPI-00026 (Max-Planck: Institut für Astronomic, 2004).

Conan, J.-M.

Dainty, J. C.

R. G. Lane, A. Glindemann, and J. C. Dainty, 'Simulation of a Kolmogorov phase screen,' Waves Random Media 2, 209-224 (1992).
[CrossRef]

Davies, R. J.

W. Hackenberg, A. Eckart, R. J. Davies, S. Rabien, T. Ott, M. Kasper, S. Hippler, and A. Quirrenbach, 'Near-infrared adaptive optics observations of galaxy clusters: Abell 262 at z=0.0157, J1836.3CR at z=0.414, and PKS 0743-006 at z=0.994,' Astron. Astrophys. 363, 41-61 (2000).

Dessenne, C.

Duchene, G.

G. Duchene, A. M. Ghez, and C. McCabe, 'Resolved near-infrared spectroscopy of the mysterious pre-main-sequence binary system T Tauri S,' Astrophys. J. 568, 771-778 (2002).
[CrossRef]

Eckart, A.

W. Hackenberg, A. Eckart, R. J. Davies, S. Rabien, T. Ott, M. Kasper, S. Hippler, and A. Quirrenbach, 'Near-infrared adaptive optics observations of galaxy clusters: Abell 262 at z=0.0157, J1836.3CR at z=0.414, and PKS 0743-006 at z=0.994,' Astron. Astrophys. 363, 41-61 (2000).

Feldt, M.

S. Hippler, D. P. Looze, M. Feldt, E. Masciadri, R. Stuik, and W. Brandner, 'CHEOPS adaptive optics white book,' MPIA Doc. CHEOPS-TRE-MPI-00026 (Max-Planck: Institut für Astronomic, 2004).

Fusco, T.

Gavel, D. T.

D. M. Wiberg, C. E. Max, and D. T. Gavel, 'A spatial non-dynamic LQG controller: Part 1, Application to adaptive optics,' in Proceedings of the 2004 IEEE Conference on Decision and Control (IEEE, 2004), pp. 3326-3332.

D. M. Wiberg, C. E. Max, and D. T. Gavel, 'A spatial non-dynamic LQG controller: Part 2, Theory,' in Proceedings of the 2004 IEEE Conference on Decision and Control (IEEE, 2004), pp. 3333-3338.

Gendron, E.

E. Gendron and P. Léna, 'Astronomical adaptive optics. II. Experimental results of an optimized modal control,' Astron. Astrophys., Suppl. Ser. 111, 153-167 (1995).

E. Gendron and P. Léna, 'Astronomical adaptive optics. I: Modal control optimization,' Astron. Astrophys. 291, 337-347 (1994).

Ghez, A. M.

G. Duchene, A. M. Ghez, and C. McCabe, 'Resolved near-infrared spectroscopy of the mysterious pre-main-sequence binary system T Tauri S,' Astrophys. J. 568, 771-778 (2002).
[CrossRef]

Glindemann, A.

Hackenberg, W.

W. Hackenberg, A. Eckart, R. J. Davies, S. Rabien, T. Ott, M. Kasper, S. Hippler, and A. Quirrenbach, 'Near-infrared adaptive optics observations of galaxy clusters: Abell 262 at z=0.0157, J1836.3CR at z=0.414, and PKS 0743-006 at z=0.994,' Astron. Astrophys. 363, 41-61 (2000).

Hamilton, D.

Hardy, J. W.

J. W. Hardy, Adaptive Optics for Astronomical Telescopes (Oxford, 1998).

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 (2003).
[CrossRef]

W. Hackenberg, A. Eckart, R. J. Davies, S. Rabien, T. Ott, M. Kasper, S. Hippler, and A. Quirrenbach, 'Near-infrared adaptive optics observations of galaxy clusters: Abell 262 at z=0.0157, J1836.3CR at z=0.414, and PKS 0743-006 at z=0.994,' Astron. Astrophys. 363, 41-61 (2000).

A. Wirth, J. Navetta, D. P. Looze, S. Hippler, A. Glindemann, and D. Hamilton, 'Real-time modal control implementation for adaptive optics,' Appl. Opt. 37, 4586-4597 (1998).
[CrossRef]

S. Hippler, D. P. Looze, M. Feldt, E. Masciadri, R. Stuik, and W. Brandner, 'CHEOPS adaptive optics white book,' MPIA Doc. CHEOPS-TRE-MPI-00026 (Max-Planck: Institut für Astronomic, 2004).

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 (2003).
[CrossRef]

W. Hackenberg, A. Eckart, R. J. Davies, S. Rabien, T. Ott, M. Kasper, S. Hippler, and A. Quirrenbach, 'Near-infrared adaptive optics observations of galaxy clusters: Abell 262 at z=0.0157, J1836.3CR at z=0.414, and PKS 0743-006 at z=0.994,' Astron. Astrophys. 363, 41-61 (2000).

M. Kasper, 'Optimization of an adaptive optics system and its application to high-resolution imaging spectroscopy of T Tauri,' Ph.D. thesis (University of Heidelberg, Germany, 2000).

Kulcsar, C.

Kwakernaak, H.

H. Kwakernaak and R. Sivan, Linear Optimal Control Systems (Wiley-Interscience, 1972).

Lane, R. G.

N. F. Law and R. G. Lane, 'Wavefront estimation at low light levels,' Opt. Commun. 126, 19-24 (1996).
[CrossRef]

R. G. Lane, A. Glindemann, and J. C. Dainty, 'Simulation of a Kolmogorov phase screen,' Waves Random Media 2, 209-224 (1992).
[CrossRef]

Law, N. F.

N. F. Law and R. G. Lane, 'Wavefront estimation at low light levels,' Opt. Commun. 126, 19-24 (1996).
[CrossRef]

Le Louarn, M.

Le Roux, B.

Léna, P.

E. Gendron and P. Léna, 'Astronomical adaptive optics. II. Experimental results of an optimized modal control,' Astron. Astrophys., Suppl. Ser. 111, 153-167 (1995).

E. Gendron and P. Léna, 'Astronomical adaptive optics. I: Modal control optimization,' Astron. Astrophys. 291, 337-347 (1994).

Looze, D. P.

D. P. Looze, 'Realization of systems with CCD-based measurements,' Automatica 41, 2005-2009 (2005).
[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 (2003).
[CrossRef]

A. Wirth, J. Navetta, D. P. Looze, S. Hippler, A. Glindemann, and D. Hamilton, 'Real-time modal control implementation for adaptive optics,' Appl. Opt. 37, 4586-4597 (1998).
[CrossRef]

S. Hippler, D. P. Looze, M. Feldt, E. Masciadri, R. Stuik, and W. Brandner, 'CHEOPS adaptive optics white book,' MPIA Doc. CHEOPS-TRE-MPI-00026 (Max-Planck: Institut für Astronomic, 2004).

Madec, P.-Y.

Maitre, H.

Masciadri, E.

S. Hippler, D. P. Looze, M. Feldt, E. Masciadri, R. Stuik, and W. Brandner, 'CHEOPS adaptive optics white book,' MPIA Doc. CHEOPS-TRE-MPI-00026 (Max-Planck: Institut für Astronomic, 2004).

Max, C. E.

D. M. Wiberg, C. E. Max, and D. T. Gavel, 'A spatial non-dynamic LQG controller: Part 1, Application to adaptive optics,' in Proceedings of the 2004 IEEE Conference on Decision and Control (IEEE, 2004), pp. 3326-3332.

D. M. Wiberg, C. E. Max, and D. T. Gavel, 'A spatial non-dynamic LQG controller: Part 2, Theory,' in Proceedings of the 2004 IEEE Conference on Decision and Control (IEEE, 2004), pp. 3333-3338.

McCabe, C.

G. Duchene, A. M. Ghez, and C. McCabe, 'Resolved near-infrared spectroscopy of the mysterious pre-main-sequence binary system T Tauri S,' Astrophys. J. 568, 771-778 (2002).
[CrossRef]

Moore, J. B.

B. D. O. Anderson and J. B. Moore, Optimal Control: Linear Quadratic Methods (Prentice-Hall, 1990).

Mugnier, L. M.

Navetta, J.

Oppenheim, A.

A. Oppenheim, A. S. Willsky, and I. T. Young, Signals and Systems (Prentice-Hall, 1983).

Ott, T.

W. Hackenberg, A. Eckart, R. J. Davies, S. Rabien, T. Ott, M. Kasper, S. Hippler, and A. Quirrenbach, 'Near-infrared adaptive optics observations of galaxy clusters: Abell 262 at z=0.0157, J1836.3CR at z=0.414, and PKS 0743-006 at z=0.994,' Astron. Astrophys. 363, 41-61 (2000).

Paschall, R. N.

Quirrenbach, A.

W. Hackenberg, A. Eckart, R. J. Davies, S. Rabien, T. Ott, M. Kasper, S. Hippler, and A. Quirrenbach, 'Near-infrared adaptive optics observations of galaxy clusters: Abell 262 at z=0.0157, J1836.3CR at z=0.414, and PKS 0743-006 at z=0.994,' Astron. Astrophys. 363, 41-61 (2000).

Rabien, S.

W. Hackenberg, A. Eckart, R. J. Davies, S. Rabien, T. Ott, M. Kasper, S. Hippler, and A. Quirrenbach, 'Near-infrared adaptive optics observations of galaxy clusters: Abell 262 at z=0.0157, J1836.3CR at z=0.414, and PKS 0743-006 at z=0.994,' Astron. Astrophys. 363, 41-61 (2000).

Ragazzoni, R.

R. Ragazzoni, 'Pupil plane wavefront sensing with an oscillating prism,' J. Mod. Opt. 43, 289-293 (1996).
[CrossRef]

Raynaud, H.-F.

Rigaut, F.

Roggemann, M. C.

M. C. Roggemann and B. M. Welsh, Imaging Through Turbulence (CRC Press, 1996).

Rouan, D.

Rousset, G.

Sarazin, M.

Sivan, R.

H. Kwakernaak and R. Sivan, Linear Optimal Control Systems (Wiley-Interscience, 1972).

Stuik, R.

S. Hippler, D. P. Looze, M. Feldt, E. Masciadri, R. Stuik, and W. Brandner, 'CHEOPS adaptive optics white book,' MPIA Doc. CHEOPS-TRE-MPI-00026 (Max-Planck: Institut für Astronomic, 2004).

Tokovinin, A.

Veran, J.-P.

Wallner, E. 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 (2003).
[CrossRef]

Welsh, B. M.

M. C. Roggemann and B. M. Welsh, Imaging Through Turbulence (CRC Press, 1996).

Wiberg, D. M.

D. M. Wiberg, C. E. Max, and D. T. Gavel, 'A spatial non-dynamic LQG controller: Part 1, Application to adaptive optics,' in Proceedings of the 2004 IEEE Conference on Decision and Control (IEEE, 2004), pp. 3326-3332.

D. M. Wiberg, C. E. Max, and D. T. Gavel, 'A spatial non-dynamic LQG controller: Part 2, Theory,' in Proceedings of the 2004 IEEE Conference on Decision and Control (IEEE, 2004), pp. 3333-3338.

Willsky, A. S.

A. Oppenheim, A. S. Willsky, and I. T. Young, Signals and Systems (Prentice-Hall, 1983).

Wirth, A.

Young, I. T.

A. Oppenheim, A. S. Willsky, and I. T. Young, Signals and Systems (Prentice-Hall, 1983).

Appl. Opt. (2)

Astron. Astrophys. (2)

E. Gendron and P. Léna, 'Astronomical adaptive optics. I: Modal control optimization,' Astron. Astrophys. 291, 337-347 (1994).

W. Hackenberg, A. Eckart, R. J. Davies, S. Rabien, T. Ott, M. Kasper, S. Hippler, and A. Quirrenbach, 'Near-infrared adaptive optics observations of galaxy clusters: Abell 262 at z=0.0157, J1836.3CR at z=0.414, and PKS 0743-006 at z=0.994,' Astron. Astrophys. 363, 41-61 (2000).

Astron. Astrophys., Suppl. Ser. (1)

E. Gendron and P. Léna, 'Astronomical adaptive optics. II. Experimental results of an optimized modal control,' Astron. Astrophys., Suppl. Ser. 111, 153-167 (1995).

Astrophys. J. (1)

G. Duchene, A. M. Ghez, and C. McCabe, 'Resolved near-infrared spectroscopy of the mysterious pre-main-sequence binary system T Tauri S,' Astrophys. J. 568, 771-778 (2002).
[CrossRef]

Automatica (1)

D. P. Looze, 'Realization of systems with CCD-based measurements,' Automatica 41, 2005-2009 (2005).
[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 (2003).
[CrossRef]

J. Mod. Opt. (1)

R. Ragazzoni, 'Pupil plane wavefront sensing with an oscillating prism,' J. Mod. Opt. 43, 289-293 (1996).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (3)

Opt. Commun. (1)

N. F. Law and R. G. Lane, 'Wavefront estimation at low light levels,' Opt. Commun. 126, 19-24 (1996).
[CrossRef]

Opt. Lett. (1)

Waves Random Media (1)

R. G. Lane, A. Glindemann, and J. C. Dainty, 'Simulation of a Kolmogorov phase screen,' Waves Random Media 2, 209-224 (1992).
[CrossRef]

Other (10)

S. Hippler, D. P. Looze, M. Feldt, E. Masciadri, R. Stuik, and W. Brandner, 'CHEOPS adaptive optics white book,' MPIA Doc. CHEOPS-TRE-MPI-00026 (Max-Planck: Institut für Astronomic, 2004).

H. Kwakernaak and R. Sivan, Linear Optimal Control Systems (Wiley-Interscience, 1972).

B. D. O. Anderson and J. B. Moore, Optimal Control: Linear Quadratic Methods (Prentice-Hall, 1990).

MATLAB: Control System Toolbox (Mathworks, 1996).

A. Oppenheim, A. S. Willsky, and I. T. Young, Signals and Systems (Prentice-Hall, 1983).

M. Kasper, 'Optimization of an adaptive optics system and its application to high-resolution imaging spectroscopy of T Tauri,' Ph.D. thesis (University of Heidelberg, Germany, 2000).

J. W. Hardy, Adaptive Optics for Astronomical Telescopes (Oxford, 1998).

M. C. Roggemann and B. M. Welsh, Imaging Through Turbulence (CRC Press, 1996).

D. M. Wiberg, C. E. Max, and D. T. Gavel, 'A spatial non-dynamic LQG controller: Part 1, Application to adaptive optics,' in Proceedings of the 2004 IEEE Conference on Decision and Control (IEEE, 2004), pp. 3326-3332.

D. M. Wiberg, C. E. Max, and D. T. Gavel, 'A spatial non-dynamic LQG controller: Part 2, Theory,' in Proceedings of the 2004 IEEE Conference on Decision and Control (IEEE, 2004), pp. 3333-3338.

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Figures (5)

Fig. 1
Fig. 1

Adaptive optics system overview. S/H stands for Shack–Hartmann sensor.

Fig. 2
Fig. 2

Discrete-time, discrete-space adaptive optics system model.

Fig. 3
Fig. 3

Plant zero and compensator pole real-axis locations versus loop computation delay τ d normalized by the sample time.

Fig. 4
Fig. 4

Real-axis locations of plant zeros and compensator poles versus computational delay τ d normalized by the sample time.

Fig. 5
Fig. 5

Estimated power spectrum (solid curve) of data obtained from a simulated Kolmogorov phase screen that is representative of the Paranal Observatory site, Chile. Also shown are the theoretical PSD obtained from the Kolmogorov model by using r 0 and V estimated from the data (dashed–dotted curve), and the power spectrum of the first-order model of the paper with γ = 1 and level determined by a best least-squares fit to the data (dashed curve).

Equations (86)

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

s f ( r , k ) = ( k 1 ) T s k T s s ( r , t ) d t ,
w f ( r , k ) = ( k 1 ) T s k T s w ( r , t ) d t ,
e f ( r , k ) = ( k 1 ) T s k T s e ( r , t ) d t ,
s k = [ s 1 , k s n a , k ] T , s i , k = s f ( r i , k ) ,
w k = [ w 1 , k w n a , k ] T , w i , k = w f ( r i , k ) ,
e k = [ e 1 , k e n a , k ] T ,
e i , k = e f ( r i , k ) , e k = w k s k .
x m , k + 1 = A m x m , k + B m u k ,
s k = C m x m , k .
p k = a T x m , k .
x a , k + 1 = A a x a , k + B a d k ,
w k = C a x a , k .
y k = H e k + θ k = H C a x a , k H C m x m , k + θ k .
x k + 1 = Ax k + Bu k + Wd k ,
y k = Cx k + θ k ,
A = [ A m 0 0 A a ] , B = [ B m 0 ] , W = [ 0 B a ] ,
C = [ H C m H C a ] .
σ w 2 = lim T ¯ 1 T ¯ 0 T ¯ ( 4 π D 2 P e 2 ( r , t ) ) d t .
σ w 2 = 4 π D 2 P [ lim T ¯ 1 T ¯ 0 T ¯ e 2 ( r , t ) d t ] d r = 4 π D 2 P [ lim L 1 L l = 0 L ( k 1 ) T s k T s e 2 ( r , t ) d t ] d r = 4 π D 2 P [ lim L 1 L l = 0 L e f 2 ( r , l ) ] d r .
σ w 2 lim L 1 L l = 0 L e l T Q P e l = e k T Q P e k = x k T Q x k ,
Q = [ C m T Q P C m C m T Q P C a C a T Q P C m C a T Q P C a ] .
J = lim k 1 2 x k T Qx k ,
u k = G ( x ̂ k + L f ( y k C x ̂ k ) ) ,
x ̂ k + 1 = A x ̂ k + Bu k + L ( y k C x ̂ k ) ,
K = A T K A + Q A T K B ( B T K B ) 1 B T K A ,
G = ( B T K B ) 1 B T K A ,
Σ = A Σ A T + W W T A Σ C T ( C Σ C T + Θ ) 1 C Σ A T ,
L = A Σ C T ( C Σ C T + Θ ) 1 ,
L f = Σ C T ( C Σ C T + Θ ) 1 ,
Σ ( x k x ̂ k k 1 ) ( x k x ̂ k k 1 ) T .
Σ e ( x k x ̂ k k ) ( x k x ̂ k k ) T .
Σ = A Σ e A T + W W T .
x m , k + 1 = M u k .
P m ( z ) = 1 z M , x ̃ m ( z ) = P m ( z ) u ̃ ( z ) ,
x a , k + 1 = γ x a , k + W d k .
γ = e T s τ a .
A = [ 0 0 0 γ I ] , B = [ M 0 ] , W = [ 0 W ] ,
C = [ H H ] , C e = [ I I ] .
Q P = I , Q = C e T C e .
K = [ K 11 K 12 K 21 K 22 ] , Σ = [ Σ 11 Σ 12 Σ 21 Σ 22 ] .
K 11 = K 12 = K 21 T = K 22 = I n a .
G = ( M T M ) 1 [ M T M T ] [ 0 0 0 γ I ] = [ 0 γ M 1 ] .
Σ 11 = 0 , Σ 12 = Σ 21 T = 0 .
0 = ( 1 γ 2 ) Σ ¯ Ξ + γ 2 Σ ¯ H T ( H Σ ¯ H T + Θ ) 1 H Σ ¯ .
L a = Σ ¯ H T ( H Σ ¯ H T + Θ ) 1 .
L = [ 0 γ L a ] = γ L f , L f = [ 0 L a ] .
u k = G ( I L f C ) x ̂ k + G L f y k ,
x ̂ k + 1 = ( A + B G L C B G L f C ) x ̂ k + ( L + B G L f ) y k .
A c = A + B G L C B G L f C = [ γ L a H γ ( I L a H ) γ L a H γ ( I L a H ) ] ,
B c = L + B G L f = [ γ L a γ L a ] ,
C c = G ( I L f C ) = γ M 1 [ L a H ( I L a H ) ] ,
D c = G L f = M 1 L a .
x ̃ k = T e x ̂ k , T e = [ I 0 I I ] , T e 1 = [ I 0 I I ] .
x ̃ k + 1 = A c e x ̃ k + B c e y k ,
u k = C c e x ̃ k + D c e y k ,
A c e = T e A c T e 1 , B c e = T e B c ,
C c e = C T e 1 , D c e = D c .
A c e = [ γ I γ ( I L a H ) 0 0 ] , B c e = [ γ L a 0 ] ,
C c e = γ M 1 [ I I L a H ] , D c e = M 1 L a .
T c ( z ) = γ z z γ M 1 L a ,
R = ( H T Θ 1 H + V 1 ) 1 H Θ 1 ,
R = V H T ( H V H T + Θ ) 1 .
T c ( z ) = z z 1 M 1 L a .
M 1 L a = M 1 Ξ H T ( H Ξ H T ) ,
J Σ ¯ H T + Θ Θ .
0 = Ξ + L a ( H Σ ̃ H T + Θ ) L a T .
L a U W Θ 1 2 ,
P m ( z ) 1 = z M 1 .
γ m = e T s τ m .
x m 1 , k + 1 = γ m x m 1 , k + ( 1 γ m ) M u k ,
x m , k = [ τ m ( 1 γ m ) ] x m 1 , k + [ T s τ m ( 1 γ m ) ] M u k ,
y k = H e k + θ k ,
P m ( z ) = [ T s τ m ( 1 γ m ) ] z + [ τ m ( 1 γ m ) T s γ m ] z ( z γ m ) M = k m z α m z ( z γ m ) M ,
T c ( z ) = z z 1 ( 1 k m z γ m z α m ) M 1 L a .
τ d = l T s + τ f , l 0 , 0 τ f < T s .
x m 1 , k + 1 = x m 2 , k ,
x m 2 , k + 1 = M u k ,
x m , k = [ ( 1 τ d T s ) I ( τ d T s ) I ] [ x m 1 , k x m 2 , k ] ,
y k = H e k + θ k ,
P m ( z ) = ( 1 τ d T s ) z + τ d T s z 2 M .
α d = τ d T s τ d .
T c ( z ) = z z 1 ( 1 k d z z α d ) M 1 L a , k d = 1 1 α d .
P m ( z ) = ( 1 τ f T s ) z + τ f T s z l + 2 M .
P m ( z ) = k m k d ( z α m ) ( z α d ) z 2 ( z γ m ) ,
T c ( z ) = z z 1 ( 1 k d z z α d ) ( 1 k m z γ m z α m ) M 1 L a .
τ a = L 0 2 π V .

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