Abstract

The fundamental issue of residual phase variance minimization in adaptive optics (AO) loops is addressed here from a control engineering perspective. This problem, when suitably modeled using a state-space approach, can be broken down into an optimal deterministic control problem and an optimal estimation problem, the solution of which are a linear quadratic (LQ) control and a Kalman filter. This approach provides a convenient framework for analyzing existing AO controllers, which are shown to contain an implicit phase turbulent model. In particular, standard integrator-based AO controllers assume a constant turbulent phase, which renders them prone to the notorious wind-up effect.

© 2006 Optical Society of America

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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]

2005 (3)

T. Fusco, G. Rousset, J.-L. Beuzit, D. Mouillet, K. Dohlen, R. Conan, C. Petit, G. Montagnier, “Conceptual design of an extreme AO dedicated to extra-solar planet detection by the VLT-planet finder instrument,” Proc. SPIE, 5903, (2005).
[CrossRef]

Comptes Rendus de l’Académie des Sciences, Adaptive Optics in Astronomy, (Elsevier, France, Comptes Rendus Physique, 2005) Vol. 6, Num. 10.

C. Petit, J.-M. Conan, C. Kulcsár, H. F. Raynaud, T. Fusco, J. Montri, D. Rabaud. “Optimal control for Multi-Conjugate Adaptive Optics,” Elsevier, Comptes Rendus Physique 6, 1059–1069 (2005).
[CrossRef]

2004 (3)

C. Petit, F. Quiros-Pacheco, J.-M. Conan, C. Kulcsár, H.-F. Raynaud, T. Fusco, G. Rousset, “Kalman Filter based control loop for Adaptive Optics,” Proc. SPIE, 5490, (2004).
[CrossRef]

D.M. Wiberg, C.E. Max, D.T. Gavel, “A special non-dynamic LQG controller: part I, application to adaptive optics,” Proceedings of the 43th IEEE Conference on Decision and Control 3, 3333–3338 (2004).

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

2003 (1)

B. Le Roux, J.-M. Conan, C. Kulcsár, H. F. Raynaud, L. M. Mugnier, T. Fusco, “Optimal control law for multiconjugate adaptive optics,” Proc. SPIE, 4839, (2003).
[CrossRef]

2002 (2)

M.W. Oppenheimer, M. Pachter, “Adaptive optics for airbone platforms—part 2: controller design,” Opt. Laser Technol. 34, 159–176 (2002).
[CrossRef]

G. Rousset, F. Lacombe, et al. “NAOS, the first AO system of the VLT: on sky performance,” Proc. SPIE 4839, 140–149 (2002).
[CrossRef]

2000 (1)

1998 (2)

1995 (1)

1994 (2)

D.C. Johnson, B.M. Welsh, “Analysis of multiconjugate adaptive optics,” J. Opt. Soc. Am. A 11, 394–408 (1994).
[CrossRef]

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

1993 (1)

1990 (1)

C. Mohtadi, “Bode’s integral theorem for discrete-time systems,” Proc. IEE 137, 57–66 (1990).

1975 (1)

R.H. Dicke, “Phase-contrast detection of telescope seeing and their correction,” Astron. J. 198, 605–615 (1975).

1974 (1)

Y. Bar-Shalom, E. Tse, “Dual effect, certainty equivalence and separation in stochastic control,” IEEE Trans. Automat. Contr. 19, 494–500 (1974).
[CrossRef]

1971 (1)

R. N. Patchell, Jacobs, “Separability, neutrality and certainty equivalence,” Int. J. Control 13, (1971).
[CrossRef]

1970 (1)

H. W. Sorenson, “Least-square estimation: from Gauss to Kalman,” IEEE Spectrum 7, 63–68 (1970).
[CrossRef]

1961 (1)

P.D. Joseph, J.T. Tou, “On linear control theory,” AIEE Trans. Applications in Industry, pgs. 193–196 (1961).

Anderson, B.D.O

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

Anderson, D.J.

Bar-Shalom, Y.

Y. Bar-Shalom, E. Tse, “Dual effect, certainty equivalence and separation in stochastic control,” IEEE Trans. Automat. Contr. 19, 494–500 (1974).
[CrossRef]

Beuzit, J.-L.

T. Fusco, G. Rousset, J.-L. Beuzit, D. Mouillet, K. Dohlen, R. Conan, C. Petit, G. Montagnier, “Conceptual design of an extreme AO dedicated to extra-solar planet detection by the VLT-planet finder instrument,” Proc. SPIE, 5903, (2005).
[CrossRef]

Conan, J.-M.

C. Petit, J.-M. Conan, C. Kulcsár, H. F. Raynaud, T. Fusco, J. Montri, D. Rabaud. “Optimal control for Multi-Conjugate Adaptive Optics,” Elsevier, Comptes Rendus Physique 6, 1059–1069 (2005).
[CrossRef]

C. Petit, F. Quiros-Pacheco, J.-M. Conan, C. Kulcsár, H.-F. Raynaud, T. Fusco, G. Rousset, “Kalman Filter based control loop for Adaptive Optics,” Proc. SPIE, 5490, (2004).
[CrossRef]

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

B. Le Roux, J.-M. Conan, C. Kulcsár, H. F. Raynaud, L. M. Mugnier, T. Fusco, “Optimal control law for multiconjugate adaptive optics,” Proc. SPIE, 4839, (2003).
[CrossRef]

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

Conan, R.

T. Fusco, G. Rousset, J.-L. Beuzit, D. Mouillet, K. Dohlen, R. Conan, C. Petit, G. Montagnier, “Conceptual design of an extreme AO dedicated to extra-solar planet detection by the VLT-planet finder instrument,” Proc. SPIE, 5903, (2005).
[CrossRef]

Dessenne, C.

Dicke, R.H.

R.H. Dicke, “Phase-contrast detection of telescope seeing and their correction,” Astron. J. 198, 605–615 (1975).

Dohlen, K.

T. Fusco, G. Rousset, J.-L. Beuzit, D. Mouillet, K. Dohlen, R. Conan, C. Petit, G. Montagnier, “Conceptual design of an extreme AO dedicated to extra-solar planet detection by the VLT-planet finder instrument,” Proc. SPIE, 5903, (2005).
[CrossRef]

Ellerbroek, B.L.

Fusco, T.

T. Fusco, G. Rousset, J.-L. Beuzit, D. Mouillet, K. Dohlen, R. Conan, C. Petit, G. Montagnier, “Conceptual design of an extreme AO dedicated to extra-solar planet detection by the VLT-planet finder instrument,” Proc. SPIE, 5903, (2005).
[CrossRef]

C. Petit, J.-M. Conan, C. Kulcsár, H. F. Raynaud, T. Fusco, J. Montri, D. Rabaud. “Optimal control for Multi-Conjugate Adaptive Optics,” Elsevier, Comptes Rendus Physique 6, 1059–1069 (2005).
[CrossRef]

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

C. Petit, F. Quiros-Pacheco, J.-M. Conan, C. Kulcsár, H.-F. Raynaud, T. Fusco, G. Rousset, “Kalman Filter based control loop for Adaptive Optics,” Proc. SPIE, 5490, (2004).
[CrossRef]

B. Le Roux, J.-M. Conan, C. Kulcsár, H. F. Raynaud, L. M. Mugnier, T. Fusco, “Optimal control law for multiconjugate adaptive optics,” Proc. SPIE, 4839, (2003).
[CrossRef]

Gavel, D.T.

D.M. Wiberg, C.E. Max, D.T. Gavel, “A special non-dynamic LQG controller: part I, application to adaptive optics,” Proceedings of the 43th IEEE Conference on Decision and Control 3, 3333–3338 (2004).

Gendron, E.

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

Gibson, J.S.

Glindemann, A.

Guikhman, L.

L. Guikhman, A. Skorokhod, The theory of stochastic processes, (Springer Verlag Ed., Berlin, 1979).

Hamilton, D.

Hippler, S.

Jacobs,

R. N. Patchell, Jacobs, “Separability, neutrality and certainty equivalence,” Int. J. Control 13, (1971).
[CrossRef]

Jazwinski, A.H.

A.H. Jazwinski, Stochastic Processes and Filtering Theory, (Academic Press, 1970).

Johnson, D.C.

Joseph, P.D.

P.D. Joseph, J.T. Tou, “On linear control theory,” AIEE Trans. Applications in Industry, pgs. 193–196 (1961).

Kulcsár, C.

C. Petit, J.-M. Conan, C. Kulcsár, H. F. Raynaud, T. Fusco, J. Montri, D. Rabaud. “Optimal control for Multi-Conjugate Adaptive Optics,” Elsevier, Comptes Rendus Physique 6, 1059–1069 (2005).
[CrossRef]

C. Petit, F. Quiros-Pacheco, J.-M. Conan, C. Kulcsár, H.-F. Raynaud, T. Fusco, G. Rousset, “Kalman Filter based control loop for Adaptive Optics,” Proc. SPIE, 5490, (2004).
[CrossRef]

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

B. Le Roux, J.-M. Conan, C. Kulcsár, H. F. Raynaud, L. M. Mugnier, T. Fusco, “Optimal control law for multiconjugate adaptive optics,” Proc. SPIE, 4839, (2003).
[CrossRef]

Lacombe, F.

G. Rousset, F. Lacombe, et al. “NAOS, the first AO system of the VLT: on sky performance,” Proc. SPIE 4839, 140–149 (2002).
[CrossRef]

Lena, P.

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

Looze, D.

Madec, P.-Y.

Manolakis, D.G.

J.G. Proakis, D.G. Manolakis, Digital Signal Processing - Principles, algorithms and applications, (Prentice Hall, Upper Saddle River, New Jersey, 3rd ed., 1996).

Max, C.E.

D.M. Wiberg, C.E. Max, D.T. Gavel, “A special non-dynamic LQG controller: part I, application to adaptive optics,” Proceedings of the 43th IEEE Conference on Decision and Control 3, 3333–3338 (2004).

Mohtadi, C.

C. Mohtadi, “Bode’s integral theorem for discrete-time systems,” Proc. IEE 137, 57–66 (1990).

Montagnier, G.

T. Fusco, G. Rousset, J.-L. Beuzit, D. Mouillet, K. Dohlen, R. Conan, C. Petit, G. Montagnier, “Conceptual design of an extreme AO dedicated to extra-solar planet detection by the VLT-planet finder instrument,” Proc. SPIE, 5903, (2005).
[CrossRef]

Montri, J.

C. Petit, J.-M. Conan, C. Kulcsár, H. F. Raynaud, T. Fusco, J. Montri, D. Rabaud. “Optimal control for Multi-Conjugate Adaptive Optics,” Elsevier, Comptes Rendus Physique 6, 1059–1069 (2005).
[CrossRef]

Moore, J.B.

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

Mouillet, D.

T. Fusco, G. Rousset, J.-L. Beuzit, D. Mouillet, K. Dohlen, R. Conan, C. Petit, G. Montagnier, “Conceptual design of an extreme AO dedicated to extra-solar planet detection by the VLT-planet finder instrument,” Proc. SPIE, 5903, (2005).
[CrossRef]

Mugnier, L. M.

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

B. Le Roux, J.-M. Conan, C. Kulcsár, H. F. Raynaud, L. M. Mugnier, T. Fusco, “Optimal control law for multiconjugate adaptive optics,” Proc. SPIE, 4839, (2003).
[CrossRef]

Navetta, J.

Oppenheimer, M.W.

M.W. Oppenheimer, M. Pachter, “Adaptive optics for airbone platforms—part 2: controller design,” Opt. Laser Technol. 34, 159–176 (2002).
[CrossRef]

Pachter, M.

M.W. Oppenheimer, M. Pachter, “Adaptive optics for airbone platforms—part 2: controller design,” Opt. Laser Technol. 34, 159–176 (2002).
[CrossRef]

Paschall, R.N.

Patchell, R. N.

R. N. Patchell, Jacobs, “Separability, neutrality and certainty equivalence,” Int. J. Control 13, (1971).
[CrossRef]

Petit, C.

C. Petit, J.-M. Conan, C. Kulcsár, H. F. Raynaud, T. Fusco, J. Montri, D. Rabaud. “Optimal control for Multi-Conjugate Adaptive Optics,” Elsevier, Comptes Rendus Physique 6, 1059–1069 (2005).
[CrossRef]

T. Fusco, G. Rousset, J.-L. Beuzit, D. Mouillet, K. Dohlen, R. Conan, C. Petit, G. Montagnier, “Conceptual design of an extreme AO dedicated to extra-solar planet detection by the VLT-planet finder instrument,” Proc. SPIE, 5903, (2005).
[CrossRef]

C. Petit, F. Quiros-Pacheco, J.-M. Conan, C. Kulcsár, H.-F. Raynaud, T. Fusco, G. Rousset, “Kalman Filter based control loop for Adaptive Optics,” Proc. SPIE, 5490, (2004).
[CrossRef]

Proakis, J.G.

J.G. Proakis, D.G. Manolakis, Digital Signal Processing - Principles, algorithms and applications, (Prentice Hall, Upper Saddle River, New Jersey, 3rd ed., 1996).

Quiros-Pacheco, F.

C. Petit, F. Quiros-Pacheco, J.-M. Conan, C. Kulcsár, H.-F. Raynaud, T. Fusco, G. Rousset, “Kalman Filter based control loop for Adaptive Optics,” Proc. SPIE, 5490, (2004).
[CrossRef]

Rabaud, D.

C. Petit, J.-M. Conan, C. Kulcsár, H. F. Raynaud, T. Fusco, J. Montri, D. Rabaud. “Optimal control for Multi-Conjugate Adaptive Optics,” Elsevier, Comptes Rendus Physique 6, 1059–1069 (2005).
[CrossRef]

Raynaud, H. F.

C. Petit, J.-M. Conan, C. Kulcsár, H. F. Raynaud, T. Fusco, J. Montri, D. Rabaud. “Optimal control for Multi-Conjugate Adaptive Optics,” Elsevier, Comptes Rendus Physique 6, 1059–1069 (2005).
[CrossRef]

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

B. Le Roux, J.-M. Conan, C. Kulcsár, H. F. Raynaud, L. M. Mugnier, T. Fusco, “Optimal control law for multiconjugate adaptive optics,” Proc. SPIE, 4839, (2003).
[CrossRef]

Raynaud, H.-F.

C. Petit, F. Quiros-Pacheco, J.-M. Conan, C. Kulcsár, H.-F. Raynaud, T. Fusco, G. Rousset, “Kalman Filter based control loop for Adaptive Optics,” Proc. SPIE, 5490, (2004).
[CrossRef]

Rousset, G.

T. Fusco, G. Rousset, J.-L. Beuzit, D. Mouillet, K. Dohlen, R. Conan, C. Petit, G. Montagnier, “Conceptual design of an extreme AO dedicated to extra-solar planet detection by the VLT-planet finder instrument,” Proc. SPIE, 5903, (2005).
[CrossRef]

C. Petit, F. Quiros-Pacheco, J.-M. Conan, C. Kulcsár, H.-F. Raynaud, T. Fusco, G. Rousset, “Kalman Filter based control loop for Adaptive Optics,” Proc. SPIE, 5490, (2004).
[CrossRef]

G. Rousset, F. Lacombe, et al. “NAOS, the first AO system of the VLT: on sky performance,” Proc. SPIE 4839, 140–149 (2002).
[CrossRef]

C. Dessenne, P.-Y. Madec, G. Rousset, “Optimization of a predictive controller for the closed loop adaptive optics,” Appl. Opt. 37, 4623 (1998).
[CrossRef]

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

Roux, B. Le

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

B. Le Roux, J.-M. Conan, C. Kulcsár, H. F. Raynaud, L. M. Mugnier, T. Fusco, “Optimal control law for multiconjugate adaptive optics,” Proc. SPIE, 4839, (2003).
[CrossRef]

Skorokhod, A.

L. Guikhman, A. Skorokhod, The theory of stochastic processes, (Springer Verlag Ed., Berlin, 1979).

Sorenson, H. W.

H. W. Sorenson, “Least-square estimation: from Gauss to Kalman,” IEEE Spectrum 7, 63–68 (1970).
[CrossRef]

Tou, J.T.

P.D. Joseph, J.T. Tou, “On linear control theory,” AIEE Trans. Applications in Industry, pgs. 193–196 (1961).

Tse, E.

Y. Bar-Shalom, E. Tse, “Dual effect, certainty equivalence and separation in stochastic control,” IEEE Trans. Automat. Contr. 19, 494–500 (1974).
[CrossRef]

Welsh, B.M.

Wiberg, D.M.

D.M. Wiberg, C.E. Max, D.T. Gavel, “A special non-dynamic LQG controller: part I, application to adaptive optics,” Proceedings of the 43th IEEE Conference on Decision and Control 3, 3333–3338 (2004).

Wirth, A.

AIEE Trans. Applications in Industry, (1)

P.D. Joseph, J.T. Tou, “On linear control theory,” AIEE Trans. Applications in Industry, pgs. 193–196 (1961).

Appl. Opt. (4)

Astron. Astrophys. (1)

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

Astron. J. (1)

R.H. Dicke, “Phase-contrast detection of telescope seeing and their correction,” Astron. J. 198, 605–615 (1975).

Elsevier, Comptes Rendus Physique (1)

C. Petit, J.-M. Conan, C. Kulcsár, H. F. Raynaud, T. Fusco, J. Montri, D. Rabaud. “Optimal control for Multi-Conjugate Adaptive Optics,” Elsevier, Comptes Rendus Physique 6, 1059–1069 (2005).
[CrossRef]

IEEE Spectrum (1)

H. W. Sorenson, “Least-square estimation: from Gauss to Kalman,” IEEE Spectrum 7, 63–68 (1970).
[CrossRef]

IEEE Trans. Automat. Contr. (1)

Y. Bar-Shalom, E. Tse, “Dual effect, certainty equivalence and separation in stochastic control,” IEEE Trans. Automat. Contr. 19, 494–500 (1974).
[CrossRef]

Int. J. Control (1)

R. N. Patchell, Jacobs, “Separability, neutrality and certainty equivalence,” Int. J. Control 13, (1971).
[CrossRef]

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

Opt. Laser Technol. (1)

M.W. Oppenheimer, M. Pachter, “Adaptive optics for airbone platforms—part 2: controller design,” Opt. Laser Technol. 34, 159–176 (2002).
[CrossRef]

Proc. IEE (1)

C. Mohtadi, “Bode’s integral theorem for discrete-time systems,” Proc. IEE 137, 57–66 (1990).

Proc. SPIE (1)

G. Rousset, F. Lacombe, et al. “NAOS, the first AO system of the VLT: on sky performance,” Proc. SPIE 4839, 140–149 (2002).
[CrossRef]

Proc. SPIE, (3)

C. Petit, F. Quiros-Pacheco, J.-M. Conan, C. Kulcsár, H.-F. Raynaud, T. Fusco, G. Rousset, “Kalman Filter based control loop for Adaptive Optics,” Proc. SPIE, 5490, (2004).
[CrossRef]

B. Le Roux, J.-M. Conan, C. Kulcsár, H. F. Raynaud, L. M. Mugnier, T. Fusco, “Optimal control law for multiconjugate adaptive optics,” Proc. SPIE, 4839, (2003).
[CrossRef]

T. Fusco, G. Rousset, J.-L. Beuzit, D. Mouillet, K. Dohlen, R. Conan, C. Petit, G. Montagnier, “Conceptual design of an extreme AO dedicated to extra-solar planet detection by the VLT-planet finder instrument,” Proc. SPIE, 5903, (2005).
[CrossRef]

Proceedings of the 43th IEEE Conference on Decision and Control (1)

D.M. Wiberg, C.E. Max, D.T. Gavel, “A special non-dynamic LQG controller: part I, application to adaptive optics,” Proceedings of the 43th IEEE Conference on Decision and Control 3, 3333–3338 (2004).

Other (6)

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

J.G. Proakis, D.G. Manolakis, Digital Signal Processing - Principles, algorithms and applications, (Prentice Hall, Upper Saddle River, New Jersey, 3rd ed., 1996).

L. Guikhman, A. Skorokhod, The theory of stochastic processes, (Springer Verlag Ed., Berlin, 1979).

Comptes Rendus de l’Académie des Sciences, Adaptive Optics in Astronomy, (Elsevier, France, Comptes Rendus Physique, 2005) Vol. 6, Num. 10.

A.H. Jazwinski, Stochastic Processes and Filtering Theory, (Academic Press, 1970).

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

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Figures (3)

Fig. 1.
Fig. 1.

Basic bloc-diagram of an AO closed loop

Fig. 2.
Fig. 2.

Chronogram for d m =1 and d c = 1. The available numerical values at each sampling time are indicated by circles.

Fig. 3.
Fig. 3.

Point Spread Functions obtained with an OMGI (left) and a LQG control (right) in logarithmic scales.

Equations (38)

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J c ( u ) lim τ + 1 τ 0 τ ϕ res ( t ) 2 d t ,
ϕ k res 1 Δ T ( k 1 ) Δ T k Δ T ϕ res ( t ) d t ,
y k = D ϕ k d m res + w k
ϕ k cor = N u k d c ,
J c ( u ) = lim n + 1 n ( k = 1 n 1 Δ T ( k 1 ) Δ T k Δ T ϕ res ( t ) 2 d t ) .
1 Δ T ( k 1 ) Δ T k Δ T ϕ res ( t ) 2 d t = 1 Δ T ( k 1 ) Δ T k Δ T ϕ tur ( t ) ϕ k tur 2 d t + ϕ k res 2 .
J ( u ) lim n + 1 n k = 1 n ϕ k res 2 ,
( I d + NC ( z ) D z d ) ϕ ˜ res ( z ) = ϕ ˜ tur ( z ) NC ( z ) z d c w ˜ ( z ) ,
0 π log H ( e j ω ) d ω = 0 .
J ( u ) = 1 2 π 0 2 π trace ( S ϕ res ( e j ω ) ) d ω ,
J ( u ) = trace ( Var ( ϕ res ) ) almost surely .
S ϕ res = H ( z ) S ϕ tur H ( z ) * + H w ( z ) S w H w ( z ) * ,
J ( u ) = 1 2 π 0 2 π trace ( H ( e j ω ) S ϕ tur ( e j ω ) H ( e j ω ) * + H w ( e j ω ) S w ( e j ω ) H w ( e j ω ) * ) d ω .
ϕ k cor = N u k 1 ,
u k = ( N t N ) 1 N t ϕ k + 1 tur
P ( N t N ) 1 N t .
ϕ ̂ k + 1 t tur E [ ϕ k + 1 tur k ] ,
u k = ( N t N ) 1 N t ϕ ̂ k + 1 | k tur
x k + 1 = A x k + B u k + v k
y k = C x k + w k
y k = D ϕ k 1 tur DN u k 2 + w k
y k = D ( I d , N ) ( ϕ k 1 tur u k 2 ) + w k .
x k = ϕ k tur t ϕ k 1 tur t u k 1 t u k 2 t t .
ϕ E [ ϕ k tur ( ϕ k tur ) t ] ,
ϕ k + 1 tur = A ϕ k tur + v k ,
A = ( A 0 0 0 I d 0 0 0 0 0 0 0 0 0 I d 0 ) , B = ( 0 0 I d 0 ) , C = 0 D 0 DN .
u k = K x ̂ k + 1 | k
x ̂ k + 1 | k = A x ̂ k | k 1 + B u k + L k ( y k y ̂ k | k 1 )
y ̂ k | k 1 = C x ̂ k | k 1 .
L k = A k | k 1 C t ( C k | k 1 C t + w ) 1
k + 1 | k = A k | k 1 A t + v A k | k 1 C t ( C k | k 1 C t + w ) 1 C k | k 1 A t .
a i = exp ( 0.3 ( n + 1 ) V Δ T D ) ,
u k = u k 1 + G y k .
ϕ k + 1 tur = ϕ k tur + v k ,
A = ( I d 0 0 0 ) , B = ( 0 I d ) , C = ( D , DN ) ,
x k = ( ϕ k tur u k ) .
ϕ ̂ k + 1 | k tur = ϕ ̂ k | k 1 tur + L y k ,
G = ( N t N ) 1 N t L .

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