Abstract

The woofer–tweeter concept in adaptive optics consists in correcting for the turbulent wavefront disturbance with a combination of two deformable mirrors (DMs). The woofer corrects for temporally slow-evolving, spatially low-frequency, large-amplitude disturbances, whereas the tweeter is generally its complement, i.e., corrects for faster higher-order modes with lower amplitude. A special feature is that in general both are able to engender a common correction space. In this contribution a minimum-variance solution for the double stage woofer–tweeter concept in adaptive optics systems is addressed using a linear-quadratic-Gaussian approach. An analytical model is built upon previous developments on a single DM with temporal dynamics that accommodates a double-stage woofer–tweeter DM. Monte Carlo simulations are run for a system featuring an 8×8 actuator DM (considered infinitely fast), mounted on a steering tip/tilt platform (considered slow). Results show that it is essential to take into account temporal dynamics on the estimation step. Besides, unlike the other control strategies considered, the optimal solution is always stable.

© 2010 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. F. Roddier, Adaptive Optics in Astronomy (Cambridge University Press, 1999).
    [Crossref]
  3. J. Herrmann, “Phase variance and Strehl ratio in adaptive optics,” J. Opt. Soc. Am. A 9, 2257–2258 (1992).
    [Crossref]
  4. T. J. Brennan and T. A. Rhoadarmer, “Performance of a woofer-tweeter deformable mirror control architecture for high-bandwidth high-spatial resolution adaptive optics,” Proc. SPIE 6306, 63060B (2006).
    [Crossref]
  5. P. J. Hampton, R. Conan, C. Bradley, and P. Agathoklis, “Control of a woofer tweeter system of deformable mirrors,” Proc. SPIE 6274, 62741Z (2006).
    [Crossref]
  6. R. Conan, C. Bradley, P. Hampton, O. Keskin, A. Hilton, and C. Blain, “Distributed modal command for a two-deformable-mirror adaptive optics system,” Appl. Opt. 46, 4329–4340 (2007).
    [Crossref] [PubMed]
  7. J.-F. Lavigne and J.-P. Véran, “Woofer-tweeter control in an adaptive optics system using a Fourier reconstructor,” J. Opt. Soc. Am. A 25, 2271–2279 (2008).
    [Crossref]
  8. J.-P. Véran and G. Herriot, “Type-II woofer-tweeter control for NFIRAOS on TMT,” in Adaptive Optics: Methods, Analysis and Applications, OSA Technical Digest,” CD (Optical Society of America, 2009), paper JTuC2.
  9. J. J. Perez, G. J. Toussaint, and J. D. Schmidt, “Adaptive control of woofer-tweeter adaptive optics,” Proc. SPIE 7466, 74660B (2009).
    [Crossref]
  10. T. Söderström, Discrete-time Stochastic Systems, Advanced Textbooks in Control and Signal Processing (Springer-Verlag, 2002).
    [Crossref]
  11. C. Correia, C. Kulcsár, J.-M. Conan, and H.-F. Raynaud, “On the optimal reconstruction and control of adaptive optical systems with mirroir dynamics,” J. Opt. Soc. Am. A 27, 333–349 (2010).
    [Crossref]
  12. Y. Bar-Shalom and E. Tse, “Dual effect, certainty equivalence, and separation in stochastic control,” IEEE Trans. Autom. Control 19, 494–500 (1974).
    [Crossref]
  13. P. Joseph and J. Tou, “On linear control theory,” IEEE Trans. Appl. Ind. 80, 193–196 (1961).
  14. C. Kulcsár, H.-F. Raynaud, C. Petit, J.-M. Conan, and P. V. de Lesegno, “Optimal control, observers and integrators in adaptive optics,” Opt. Express 14, 7464–7476 (2006).
    [Crossref] [PubMed]
  15. B. D. O. Anderson and J. B. Moore, Optimal Control, Linear Quadratic Methods (Dover, 1995).
  16. B. D. O. Anderson and J. B. Moore, Optimal Filtering (Dover, 1995).
  17. C. Kulcsár, H.-F. Raynaud, C. Petit, and J.-M. Conan, “Minimum variance control in presence of actuator saturation in adaptive optics,” Proc. SPIE 7015, 70151G (2008).
    [Crossref]
  18. H.-F. Raynaud, C. Correia, C. Kulcsár, and J.-M. Conan, “Minimum-variance control of astronomical adaptive optics systems with actuator dynamics under synchronous and asynchronous sampling,” Int. J. Robust Nonlinear Control (2010); http://dx.doi.org/10.1002/rnc./625.
  19. C. Correia, H.-F. Raynaud, C. Kulcsár, and J.-M. Conan, “Minimum variance control for the woofer-tweeter concept,” in Adaptive Optics: Methods, Analysis and Applications, OSA Technical Digest (CD) (Optical Society of America, 2009), paper AOWB4.
  20. R. J. Noll, “Zernike polynomials and atmospheric turbulence,” J. Opt. Soc. Am. A 66, 207–211 (1976).
    [Crossref]
  21. J.-M. Conan, G. Rousset, and P.-Y. Madec, “Wavefront temporal spectra in high-resolution imaging through turbulence,” J. Opt. Soc. Am. A 12, 1559–1570 (1995).
    [Crossref]
  22. I. I. Gikhman and A. V. Skorokhod, Introduction to the Theory of Random Processes (Dover, 2006).

2010 (2)

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

H.-F. Raynaud, C. Correia, C. Kulcsár, and J.-M. Conan, “Minimum-variance control of astronomical adaptive optics systems with actuator dynamics under synchronous and asynchronous sampling,” Int. J. Robust Nonlinear Control (2010); http://dx.doi.org/10.1002/rnc./625.

2009 (3)

C. Correia, H.-F. Raynaud, C. Kulcsár, and J.-M. Conan, “Minimum variance control for the woofer-tweeter concept,” in Adaptive Optics: Methods, Analysis and Applications, OSA Technical Digest (CD) (Optical Society of America, 2009), paper AOWB4.

J.-P. Véran and G. Herriot, “Type-II woofer-tweeter control for NFIRAOS on TMT,” in Adaptive Optics: Methods, Analysis and Applications, OSA Technical Digest,” CD (Optical Society of America, 2009), paper JTuC2.

J. J. Perez, G. J. Toussaint, and J. D. Schmidt, “Adaptive control of woofer-tweeter adaptive optics,” Proc. SPIE 7466, 74660B (2009).
[Crossref]

2008 (2)

J.-F. Lavigne and J.-P. Véran, “Woofer-tweeter control in an adaptive optics system using a Fourier reconstructor,” J. Opt. Soc. Am. A 25, 2271–2279 (2008).
[Crossref]

C. Kulcsár, H.-F. Raynaud, C. Petit, and J.-M. Conan, “Minimum variance control in presence of actuator saturation in adaptive optics,” Proc. SPIE 7015, 70151G (2008).
[Crossref]

2007 (1)

2006 (4)

T. J. Brennan and T. A. Rhoadarmer, “Performance of a woofer-tweeter deformable mirror control architecture for high-bandwidth high-spatial resolution adaptive optics,” Proc. SPIE 6306, 63060B (2006).
[Crossref]

P. J. Hampton, R. Conan, C. Bradley, and P. Agathoklis, “Control of a woofer tweeter system of deformable mirrors,” Proc. SPIE 6274, 62741Z (2006).
[Crossref]

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

I. I. Gikhman and A. V. Skorokhod, Introduction to the Theory of Random Processes (Dover, 2006).

2002 (1)

T. Söderström, Discrete-time Stochastic Systems, Advanced Textbooks in Control and Signal Processing (Springer-Verlag, 2002).
[Crossref]

1999 (1)

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

1998 (1)

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

1995 (3)

B. D. O. Anderson and J. B. Moore, Optimal Control, Linear Quadratic Methods (Dover, 1995).

B. D. O. Anderson and J. B. Moore, Optimal Filtering (Dover, 1995).

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

1992 (1)

1976 (1)

R. J. Noll, “Zernike polynomials and atmospheric turbulence,” J. Opt. Soc. Am. A 66, 207–211 (1976).
[Crossref]

1974 (1)

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

1961 (1)

P. Joseph and J. Tou, “On linear control theory,” IEEE Trans. Appl. Ind. 80, 193–196 (1961).

Agathoklis, P.

P. J. Hampton, R. Conan, C. Bradley, and P. Agathoklis, “Control of a woofer tweeter system of deformable mirrors,” Proc. SPIE 6274, 62741Z (2006).
[Crossref]

Anderson, B. D. O.

B. D. O. Anderson and J. B. Moore, Optimal Control, Linear Quadratic Methods (Dover, 1995).

B. D. O. Anderson and J. B. Moore, Optimal Filtering (Dover, 1995).

Bar-Shalom, Y.

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

Blain, C.

Bradley, C.

R. Conan, C. Bradley, P. Hampton, O. Keskin, A. Hilton, and C. Blain, “Distributed modal command for a two-deformable-mirror adaptive optics system,” Appl. Opt. 46, 4329–4340 (2007).
[Crossref] [PubMed]

P. J. Hampton, R. Conan, C. Bradley, and P. Agathoklis, “Control of a woofer tweeter system of deformable mirrors,” Proc. SPIE 6274, 62741Z (2006).
[Crossref]

Brennan, T. J.

T. J. Brennan and T. A. Rhoadarmer, “Performance of a woofer-tweeter deformable mirror control architecture for high-bandwidth high-spatial resolution adaptive optics,” Proc. SPIE 6306, 63060B (2006).
[Crossref]

Conan, J.-M.

H.-F. Raynaud, C. Correia, C. Kulcsár, and J.-M. Conan, “Minimum-variance control of astronomical adaptive optics systems with actuator dynamics under synchronous and asynchronous sampling,” Int. J. Robust Nonlinear Control (2010); http://dx.doi.org/10.1002/rnc./625.

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

C. Correia, H.-F. Raynaud, C. Kulcsár, and J.-M. Conan, “Minimum variance control for the woofer-tweeter concept,” in Adaptive Optics: Methods, Analysis and Applications, OSA Technical Digest (CD) (Optical Society of America, 2009), paper AOWB4.

C. Kulcsár, H.-F. Raynaud, C. Petit, and J.-M. Conan, “Minimum variance control in presence of actuator saturation in adaptive optics,” Proc. SPIE 7015, 70151G (2008).
[Crossref]

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

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

Conan, R.

R. Conan, C. Bradley, P. Hampton, O. Keskin, A. Hilton, and C. Blain, “Distributed modal command for a two-deformable-mirror adaptive optics system,” Appl. Opt. 46, 4329–4340 (2007).
[Crossref] [PubMed]

P. J. Hampton, R. Conan, C. Bradley, and P. Agathoklis, “Control of a woofer tweeter system of deformable mirrors,” Proc. SPIE 6274, 62741Z (2006).
[Crossref]

Correia, C.

H.-F. Raynaud, C. Correia, C. Kulcsár, and J.-M. Conan, “Minimum-variance control of astronomical adaptive optics systems with actuator dynamics under synchronous and asynchronous sampling,” Int. J. Robust Nonlinear Control (2010); http://dx.doi.org/10.1002/rnc./625.

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

C. Correia, H.-F. Raynaud, C. Kulcsár, and J.-M. Conan, “Minimum variance control for the woofer-tweeter concept,” in Adaptive Optics: Methods, Analysis and Applications, OSA Technical Digest (CD) (Optical Society of America, 2009), paper AOWB4.

de Lesegno, P. V.

Gikhman, I. I.

I. I. Gikhman and A. V. Skorokhod, Introduction to the Theory of Random Processes (Dover, 2006).

Hampton, P.

Hampton, P. J.

P. J. Hampton, R. Conan, C. Bradley, and P. Agathoklis, “Control of a woofer tweeter system of deformable mirrors,” Proc. SPIE 6274, 62741Z (2006).
[Crossref]

Hardy, J. W.

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

Herriot, G.

J.-P. Véran and G. Herriot, “Type-II woofer-tweeter control for NFIRAOS on TMT,” in Adaptive Optics: Methods, Analysis and Applications, OSA Technical Digest,” CD (Optical Society of America, 2009), paper JTuC2.

Herrmann, J.

Hilton, A.

Joseph, P.

P. Joseph and J. Tou, “On linear control theory,” IEEE Trans. Appl. Ind. 80, 193–196 (1961).

Keskin, O.

Kulcsár, C.

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

H.-F. Raynaud, C. Correia, C. Kulcsár, and J.-M. Conan, “Minimum-variance control of astronomical adaptive optics systems with actuator dynamics under synchronous and asynchronous sampling,” Int. J. Robust Nonlinear Control (2010); http://dx.doi.org/10.1002/rnc./625.

C. Correia, H.-F. Raynaud, C. Kulcsár, and J.-M. Conan, “Minimum variance control for the woofer-tweeter concept,” in Adaptive Optics: Methods, Analysis and Applications, OSA Technical Digest (CD) (Optical Society of America, 2009), paper AOWB4.

C. Kulcsár, H.-F. Raynaud, C. Petit, and J.-M. Conan, “Minimum variance control in presence of actuator saturation in adaptive optics,” Proc. SPIE 7015, 70151G (2008).
[Crossref]

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

Lavigne, J.-F.

Madec, P.-Y.

Moore, J. B.

B. D. O. Anderson and J. B. Moore, Optimal Filtering (Dover, 1995).

B. D. O. Anderson and J. B. Moore, Optimal Control, Linear Quadratic Methods (Dover, 1995).

Noll, R. J.

R. J. Noll, “Zernike polynomials and atmospheric turbulence,” J. Opt. Soc. Am. A 66, 207–211 (1976).
[Crossref]

Perez, J. J.

J. J. Perez, G. J. Toussaint, and J. D. Schmidt, “Adaptive control of woofer-tweeter adaptive optics,” Proc. SPIE 7466, 74660B (2009).
[Crossref]

Petit, C.

C. Kulcsár, H.-F. Raynaud, C. Petit, and J.-M. Conan, “Minimum variance control in presence of actuator saturation in adaptive optics,” Proc. SPIE 7015, 70151G (2008).
[Crossref]

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

Raynaud, H.-F.

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

H.-F. Raynaud, C. Correia, C. Kulcsár, and J.-M. Conan, “Minimum-variance control of astronomical adaptive optics systems with actuator dynamics under synchronous and asynchronous sampling,” Int. J. Robust Nonlinear Control (2010); http://dx.doi.org/10.1002/rnc./625.

C. Correia, H.-F. Raynaud, C. Kulcsár, and J.-M. Conan, “Minimum variance control for the woofer-tweeter concept,” in Adaptive Optics: Methods, Analysis and Applications, OSA Technical Digest (CD) (Optical Society of America, 2009), paper AOWB4.

C. Kulcsár, H.-F. Raynaud, C. Petit, and J.-M. Conan, “Minimum variance control in presence of actuator saturation in adaptive optics,” Proc. SPIE 7015, 70151G (2008).
[Crossref]

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

Rhoadarmer, T. A.

T. J. Brennan and T. A. Rhoadarmer, “Performance of a woofer-tweeter deformable mirror control architecture for high-bandwidth high-spatial resolution adaptive optics,” Proc. SPIE 6306, 63060B (2006).
[Crossref]

Roddier, F.

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

Rousset, G.

Schmidt, J. D.

J. J. Perez, G. J. Toussaint, and J. D. Schmidt, “Adaptive control of woofer-tweeter adaptive optics,” Proc. SPIE 7466, 74660B (2009).
[Crossref]

Skorokhod, A. V.

I. I. Gikhman and A. V. Skorokhod, Introduction to the Theory of Random Processes (Dover, 2006).

Söderström, T.

T. Söderström, Discrete-time Stochastic Systems, Advanced Textbooks in Control and Signal Processing (Springer-Verlag, 2002).
[Crossref]

Tou, J.

P. Joseph and J. Tou, “On linear control theory,” IEEE Trans. Appl. Ind. 80, 193–196 (1961).

Toussaint, G. J.

J. J. Perez, G. J. Toussaint, and J. D. Schmidt, “Adaptive control of woofer-tweeter adaptive optics,” Proc. SPIE 7466, 74660B (2009).
[Crossref]

Tse, E.

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

Véran, J.-P.

J.-P. Véran and G. Herriot, “Type-II woofer-tweeter control for NFIRAOS on TMT,” in Adaptive Optics: Methods, Analysis and Applications, OSA Technical Digest,” CD (Optical Society of America, 2009), paper JTuC2.

J.-F. Lavigne and J.-P. Véran, “Woofer-tweeter control in an adaptive optics system using a Fourier reconstructor,” J. Opt. Soc. Am. A 25, 2271–2279 (2008).
[Crossref]

Appl. Opt. (1)

IEEE Trans. Appl. Ind. (1)

P. Joseph and J. Tou, “On linear control theory,” IEEE Trans. Appl. Ind. 80, 193–196 (1961).

IEEE Trans. Autom. Control (1)

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

J. Opt. Soc. Am. A (5)

Opt. Express (1)

Proc. SPIE (4)

C. Kulcsár, H.-F. Raynaud, C. Petit, and J.-M. Conan, “Minimum variance control in presence of actuator saturation in adaptive optics,” Proc. SPIE 7015, 70151G (2008).
[Crossref]

T. J. Brennan and T. A. Rhoadarmer, “Performance of a woofer-tweeter deformable mirror control architecture for high-bandwidth high-spatial resolution adaptive optics,” Proc. SPIE 6306, 63060B (2006).
[Crossref]

P. J. Hampton, R. Conan, C. Bradley, and P. Agathoklis, “Control of a woofer tweeter system of deformable mirrors,” Proc. SPIE 6274, 62741Z (2006).
[Crossref]

J. J. Perez, G. J. Toussaint, and J. D. Schmidt, “Adaptive control of woofer-tweeter adaptive optics,” Proc. SPIE 7466, 74660B (2009).
[Crossref]

Other (9)

T. Söderström, Discrete-time Stochastic Systems, Advanced Textbooks in Control and Signal Processing (Springer-Verlag, 2002).
[Crossref]

J.-P. Véran and G. Herriot, “Type-II woofer-tweeter control for NFIRAOS on TMT,” in Adaptive Optics: Methods, Analysis and Applications, OSA Technical Digest,” CD (Optical Society of America, 2009), paper JTuC2.

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

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

H.-F. Raynaud, C. Correia, C. Kulcsár, and J.-M. Conan, “Minimum-variance control of astronomical adaptive optics systems with actuator dynamics under synchronous and asynchronous sampling,” Int. J. Robust Nonlinear Control (2010); http://dx.doi.org/10.1002/rnc./625.

C. Correia, H.-F. Raynaud, C. Kulcsár, and J.-M. Conan, “Minimum variance control for the woofer-tweeter concept,” in Adaptive Optics: Methods, Analysis and Applications, OSA Technical Digest (CD) (Optical Society of America, 2009), paper AOWB4.

B. D. O. Anderson and J. B. Moore, Optimal Control, Linear Quadratic Methods (Dover, 1995).

B. D. O. Anderson and J. B. Moore, Optimal Filtering (Dover, 1995).

I. I. Gikhman and A. V. Skorokhod, Introduction to the Theory of Random Processes (Dover, 2006).

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Figures (10)

Fig. 1
Fig. 1

Closed-loop control architecture. The correction phase is the joint contribution of the woofer’s and tweeter’s phases. Solid/dashed lines represent continuous/discrete-time variables.

Fig. 2
Fig. 2

Woofer and tweeter spanned spaces. Among the individual spatial correction spaces engendered by the woofer and the tweeter, a common space can be generated by both at the same time.

Fig. 3
Fig. 3

Chronogram of operations.

Fig. 4
Fig. 4

PSD of the disturbance with a total of 21.4 mas rms. Second-order model (normalized amplitude) used based on fitting the auto-correlation of the disturbance at scales in the vicinity of zero. D r 0 = 420 @ 0.5 μ m , L 0 37 m , V i = [ 13.2 ; 8.6 ; 7.1 ] m s , θ i = [ 0 , 45 , 90 ] deg , for a vertical turbulence profile with relative strengths [0.67 0.22 0.11].

Fig. 5
Fig. 5

(Color online) Bode diagram of the DM transfer function (TF). A resonance of a factor 1.525 is observable at the frequency of f r = 13.1 Hz . Model parameters are ξ = 0.35 and ω n = 88 rad s .

Fig. 6
Fig. 6

Actuator location. Circles, actuators; solid line, telescope aperture.

Fig. 7
Fig. 7

Bi-cubic-spline influence function with 20% cross-coupling. Circles are the actuator locations, normalized by the telescope radius R.

Fig. 8
Fig. 8

Performance of the optimal and the sub-optimal controller as a function of the measurement noise. Five frame rates are used. Solid lines are for the optimal, dot-dashed lines are for the intermediate, and dashed lines are for the sub-optimal controller. The ratio of penalties is ϵ t ϵ w = 100 . Vertical dotted lines indicate the stability limits of the sub-optimal solution.

Fig. 9
Fig. 9

Performance as a function of the ratio of control energy assigned to the woofer and to the tweeter. Noise at the telescope’s edge is 200 nm rms, which is in terms of TT 3.9 mas rms. On the abscissa, the penalties are taken as ϵ w = 1 and ϵ t variable. Solid lines are for the optimal, dot-dashed lines are for the intermediate, and dashed lines are for the sub-optimal controller.

Fig. 10
Fig. 10

Temporal trajectories for the WT for the optimal (top), intermediate (center), and sub-optimal (bottom) solutions, with T s = 1 400 s and ϵ t ϵ w = 100 . From 30 s of simulation, the empirical values were found (in mas rms): Top: ϕ res = 1.742 , u w = 21.46 , u t = 2.42 ; Centre: ϕ res = 2.085 , u w = 21.67 , u t = 2.166 ; Bottom: ϕ res = 2.448 , u w = 21.69 , u t = 2.167 .

Equations (66)

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

J c ( u ) lim τ + 1 τ 0 τ ϕ res ( t ) 2 d t = lim τ + 1 τ 0 τ ϕ tur ( t ) ϕ cor ( t ) 2 d t ,
u k = ( N T N ) 1 N T ϕ ¯ k + 1 tur ,
ϕ ¯ k + 1 tur = 1 T s k T s ( k + 1 ) T s ϕ tur ( t ) d t .
ϕ cor ( t ) = N w u k w + N t u k t = ( N w N t ) ( u k w u k t ) = N u k .
u k w = ( N w T N w ) 1 N w T ϕ ¯ k + 1 tur , u k t = ( N t T N t ) 1 N t T ( ϕ ¯ k + 1 tur N w u k w ) .
J c ( u ) lim τ + 1 τ 0 τ ϕ res ( t ) 2 d t + lim M + 1 M k = 0 M 1 u k T Δ R u k ,
Δ R ( Δ R w 0 0 Δ R t ) .
ϕ w cor = N w p w ( t ) ,
ϕ cor ( t ) = N w p w ( t ) + N t u t ( t ) .
{ x ̇ w ( t ) = A w x w ( t ) + B w u w ( t ) p w ( t ) = C w x w ( t ) } ,
J dyn d ( u ) = lim M + 1 M k = 0 M 1 J dyn d ( u ) k = lim M + 1 M k = 0 M 1 ( ( z k crit ) T Q wt z k crit + u k T R wt u k + 2 ( z k crit ) T S wt u k ) = lim M + 1 M k = 0 M ( z k crit u k ) T ( Q wt S wt S wt T R wt ) ( z k crit u k ) ,
z k crit ( ϕ ¯ k + 1 tur φ ¯ k + 1 tur x k w ) ( 1 T s 0 T s ϕ tur ( k T s + s ) d s 1 T s 0 T s e s A w T C w T N w T ϕ tur ( k T s + s ) d s x w ( k T s ) ) .
1 T s k T s ( k + 1 ) T s ϕ res ( t ) 2 d t = ( 9 ) 1 T s k T s ( k + 1 ) T s ϕ tur ( t ) N w p w ( t ) N t u t ( t ) 2 d t ,
p w ( t ) = C w e ( t k T s ) A w x k w + ( I + C w e ( t k T s ) A w A w 1 B w ) u k w .
Q wt ( λ ( 2 N w N w T + N t N t T ) 0 0 0 λ ( I + 2 B w T A w T A w 1 B w ) I 0 I Q 0 ) 0 ,
S wt ( N w N t A w 1 B w 0 S 0 T wt N w T N t ) ,
R wt ( R 0 + Δ R w ( I + G wt ) N w T N t N t T N w ( I + G wt ) T N t T N t + Δ R t ) > 0 ,
T wt = 1 T s 0 T s ( C w e t A w ) T d t ,
G wt = 1 T s 0 T s ( C w e s A w A w 1 B w ) T d t ,
R 0 1 T s 0 T s ( I + C w e s A w A w 1 B w ) T N w T N w ( I + C w e s A w A w 1 B w ) d s ,
S 0 1 T s 0 T s e s A w T C w T N w T N w ( I + C w e s A w A w 1 B w ) d s ,
Q 0 1 T s 0 T s e s A w T C w T N w T N w C w e s A w d s .
lim θ 0 Q 0 = 0 , lim θ 0 S 0 = 0 , lim θ 0 R 0 = N w T N w .
lim θ 0 R wt = ( N w T N w N w T N t N t T N w N t T N t ) .
Δ R = ( ϵ t σ ϕ 2 N w T N w 0 0 ϵ w σ ϕ 2 N t T N t ) ,
{ x k + 1 = A d x k + B d u k + Γ d v k z k = C d x k + w k z k crit = C d crit x k } ,
{ x ̇ tur ( t ) = A tur x tur ( t ) + v ( t ) ϕ tur ( t ) = C tur x tur ( t ) } .
y k = ( 14 ) D 1 T s ( k 2 ) T s ( k 1 ) T s ( ϕ tur ( t ) N w p w ( t ) N t u k 2 t ) d t + w k ,
= D ϕ ¯ k 1 tur DN t u k 2 t DN w [ C w T s ( e T s A w I ) A w 1 x k 2 w + ( C w T s ( e T s A w I ) A w 2 B w + I ) u k 2 w ] + w k ,
z k = y k + 1 .
x k ( x k + 1 tur ϕ ¯ k + 1 tur φ ¯ k + 1 tur x k w u k 1 t ϕ ¯ k tur x k 1 w u k 1 w ) , A wt ( e T s A tur 0 0 0 0 0 0 0 Ξ 0 0 0 0 0 0 0 Θ 0 0 0 0 0 0 0 0 0 0 e T s A w 0 0 0 0 0 0 0 0 0 0 0 0 0 I 0 0 0 0 0 0 0 0 0 I 0 0 0 0 0 0 0 0 0 0 0 0 ) ,
B wt ( 0 0 0 0 0 0 A w 1 ( e T s A w I ) B w 0 0 I 0 0 0 0 I 0 ) , Γ wt ( I 0 0 0 I 0 0 0 I 0 0 0 0 0 0 0 0 0 0 0 0 ) ,
C wt D ( 0 0 0 0 N t I N w C w T s ( e T s A w I ) A w 1 N w ( C w T s ( e T s A w I ) A w 2 B w + I ) ) ,
C wt crit ( 0 I 0 0 0 0 0 0 0 0 I 0 0 0 0 0 0 0 0 I 0 0 0 0 ) .
u k = K x ̂ k | k 1 ,
K = ( R wt + B d T P B d ) 1 ( B d T P A d + S wt ) ,
P = Q wt + A d T P A d ( A d T P A d + S wt ) ( R wt + B d T P B d ) 1 ( B d T P A d + S wt ) .
( u k w u k t ) = ( N T N + Δ R ) 1 N T ϕ ¯ k + 1 tur = P ϕ ¯ k + 1 tur ,
u k w = ( N w T N w ) 1 N w T ϕ ¯ k + 1 tur = P w ϕ ¯ k + 1 tur
u k t = P t [ ϕ ¯ k + 1 tur ( N w u k w 1 T s k T s ( k + 1 ) T s N w p w ( t ) d t ) ] .
p ̈ w ( t ) + 2 ξ ω n p ̇ w ( t ) + ω n 2 p w ( t ) = ω n 2 u w ( t ) ,
A w = ( 0 1 ω n 2 2 ξ ω n ) , B w = ( 0 ω n 2 )
C w = ( 1 0 ) , D w = 0 ,
IF c ( x ) = { 1 + ( 4 c 2.5 ) | x | 2 + ( 3 c + 1.5 ) | x | 3 if | x | 1 ( 2 c 0.5 ) ( 2 | x | ) 2 + ( c + 0.5 ) ( 2 | x | ) 3 if 1 < | x | < 2 0 otherwise } ,
N t ( : , i ) = 1 S Ω Ω TT ( x , y ) IF i ( x , y ) d x d y ,
{ 1 S Ω Ω IF j ( x , y ) IF i ( x , y ) d x d y = 1 if i = j ; 0 otherwise } .
N t = N t T N t 2 .
1 T s k T s ( k + 1 ) T s ϕ tur ( t ) N w p w ( t ) N t u t ( t ) 2 d t = 1 T s k T s ( k + 1 ) T s ( ϕ tur ( t ) N w p w ( t ) N t u t ( t ) ) T ( ϕ tur ( t ) N w p w ( t ) N t u t ( t ) ) d t = 1 T s k T s ( k + 1 ) T s ϕ tur ( t ) T ϕ tur ( t ) + p w ( t ) T N w T N w p w ( t ) 2 ϕ tur ( t ) N w p ( t ) 2 ϕ tur ( t ) T N t u t + 2 p w ( t ) T N w T N t u t + ( u t ) T N t T N t u t d t .
p w ( t ) = C w e ( t k T s ) A w x k w + ( I + C w e ( t k T s ) A w A w 1 B w ) u k w ,
p w ( k T s + s ) T N w T N w p w ( k T s + s ) = [ C w e s A w x k w + ( I + C w e s A w A w 1 B w ) u k ] T N w T N w [ C w e s A w x k w + ( I + C w e s A w A w 1 B w ) u k w ] = ( x k w ) T ( e s A w T C w T N w T N w C w e s A w ) x k w + 2 ( x k w ) T [ e s A w T C w T N w T N w ( I + C w e s A w A w 1 B w ) ] u k w + ( u k w ) T [ ( I + C w e s A w A w 1 B w ) T N w T N w ( I + C w e s A w A w 1 B w ) ] u k w .
1 T s k T s ( k + 1 ) T s p w ( t ) T N w T N w p w ( t ) d t = ( x k w ) T Q 0 x k w + 2 ( x k w ) T S 0 u k w + ( u k w ) T R 0 u k w ,
ϕ tur ( k T s + s ) T N w p w ( k T s + s ) = ϕ tur ( k T s + s ) T N w [ C w e s A w x k w + ( I + C w e s A w A w 1 B w ) u k w ] = ϕ tur ( k T s + s ) T N w C w e s A w ( x k w + A w 1 B w u k w u k w ) + ϕ tur ( k T s + s ) T N w u k w .
1 T s k T s ( k + 1 ) T s p w ( t ) T N w T ϕ tur ( t ) d t = ( u k w ) T N w T ϕ ¯ k + 1 tur + ( x k w + A w 1 B w u k w ) T φ ¯ k + 1 tur ,
ϕ ¯ k + 1 tur = 1 T s 0 T s ϕ tur ( k T s + s ) d s
φ ¯ k + 1 tur = 1 T s 0 T s e s A w T C w T N w T ϕ tur ( k T s + s ) d s
1 T s k T s ( k + 1 ) T s ϕ tur ( t ) T N t u k t d t = ( ϕ ¯ k + 1 tur ) T N t u k t ,
1 T s k T s ( k + 1 ) T s [ C w e ( t k T s ) A w x k w + ( I + C w e ( t k T s ) A w A w 1 B w ) u k w ] T N w T N t u k t d t = ( x k w ) T T wt u k t + ( u k w ) T ( N w T N t + G wt ) u k t ,
1 T s k T s ( k + 1 ) T s ( u k t ) T N t T N t u k t d t = ( u k t ) T N t T N t u k t .
J c ( u ) k = 1 T s k T s ( k + 1 ) T s ϕ tur ( t ) 2 d t + ( z k crit u k ) T ( Q wt S wt S wt T R wt ) ( z k crit u k ) d t ,
( Q wt S wt S wt T R wt ) = ( 0 0 0 N w N t 0 0 I A w 1 B w 0 0 I Q 0 S 0 T wt N w T N w N w T B w T A w T S 0 T R 0 + Δ R w ( I + G wt ) N w T N t N t T 0 N w T N w T wt T N t T N w ( I + G wt ) T N t T N t + Δ R t ) .
( Q 0 S 0 T wt N w T N w S 0 T R 0 + Δ R w ( I + G wt ) N w T N t N w T N w T wt T N t T N w ( I + G wt ) T N t T N t + Δ R t ) 1 λ I > 0 .
J dyn d ( u ) k = ( z k crit u k ) T ( Q wt S wt S wt T R wt ) ( z k crit u k ) = ( x k x ̂ k | k 1 ) T ( C d crit T Q wt C d crit C d crit T S wt K K T S wt T C d crit K T R wt K ) ( x k x ̂ k | k 1 ) .
lim τ + 1 τ 0 τ ϕ tur ( t ) 2 d t = a.s. E ( ϕ tur ( t ) 2 ) ,
lim τ + 1 τ 0 τ ϕ tur ( t ) 2 d t = a.s. trace ( C tur C tur T Σ x ) ,
J c ( u ) = a.s. E ( ϕ tur ( t ) 2 ) + E ( ( z k crit ) T Q wt z k crit + u k T R wt u k 2 ( z k crit ) T S wt u k ) .
J c ( u ) = a.s. E ( ( x k x ̂ k | k 1 ) T ( C tur C tur T + C d crit T Q wt C d crit C d crit T S wt K K T S wt T C d crit K T R wt K ) ( x k x ̂ k | k 1 ) ) = a.s. trace ( W Σ ζ ) ,

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