## Optimal control, observers and integrators in adaptive optics

Optics Express, Vol. 14, Issue 17, pp. 7464-7476 (2006)

http://dx.doi.org/10.1364/OE.14.007464

Acrobat PDF (152 KB)

### 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

## 1. Introduction

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

2. T. Fusco, G. Rousset, J.-L. Beuzit, D. Mouillet, K. Dohlen, R. Conan, C. Petit, and 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]

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

## 2. AO closed-loop: continuous or discrete time?

^{tur}, ϕ

^{res}and ϕ

^{cor}represent respectively the turbulent, residual and correction phases,

*w*the measurement noise,

*y*the measurements and

*u*the control voltages. In this setup, one has to define an optimality criterion, to be minimized by the controller. In classical AO, one usually aims at minimizing the residual phase variance, that is, a quadratic criterion on the phase. For the sake of simplicity the formalism presented in this paper is restricted to this case. Nevertheless, it can be easily generalized to MCAO, where the relevant criterion is usually the minimization of the residual phase variance in a given field of view of interest [7

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

^{res}= ϕ

^{tur}-ϕ

^{cor}, averaged over a sufficiently large exposure time, has thus to be minimized by the controller, which is realized by minimizing criterion

*J*

^{c}(

*u*) with respect to the control

*u*,

^{res}(

*t*) actually depends on

*u*(omitted for the sake of notation simplicity), and ∥ ∙ ∥

^{2}is the Euclidean norm, assuming that all phases are expanded on a suitable basis. This minimization is done by adjusting the mirror voltages

*u*according to noisy measurements

*y*provided by the wave-front sensor (WFS) from integrated and delayed ϕ

^{res}. Integration is assumed to be performed during a time interval of length Δ

*T*. Moreover, because AO devices are computer-controlled, the control

*u*remains constant over time intervals of length Δ

*T*′ ≤ Δ

*T*. Time intervals Δ

*T*and Δ

*T*′ are usually chosen equal, and we shall not depart from the rule. (Note however that the case Δ

*T*≠ Δ

*T*′ could be considered as well, provided that there exists some integers ℓ

_{1},ℓ

_{2}> 0 and a time period Δ

*T*″ such that Δ

*T*= ℓ

_{1}Δ

*T*″ and Δ

*T*′ = ℓ

_{2}Δ

*T*″, which means that Δ

*T*and Δ

*T*′ are commensurable.) The fact that

*u*is piecewise constant induces a loss of optimality, as the turbulent phase evolves over Δ

*T*. What we prove in this section is that there is no additional loss of optimality by considering a complete AO system in discrete time, with discrete variables corresponding to their temporal average over a time interval of length Δ

*T*.

^{res}during a time interval Δ

*T*, and that WFS measurements are deduced linearly from this with some additional delay, leading to a total discrete measurement delay

*d*

_{m}≥ 1 (in time unit Δ

*T*). This corresponds for example to a Shack-Hartmann WFS in linear regime. The overall operation produces noisy measurements, where the measurement noise

*w*is supposed to be additive. Furthermore, using the following notation for any averaged value of a continuous variable, e.g. for the residual phase

*D*is the WFS matrix and

*w*

_{k}a discrete zero-mean white noise.

^{cor}is assumed to be a linear function of the control input

*u*with a delay

*d*

_{c}≥ 1,

*i.e*

*N*stands for the influence matrix (interaction matrix is thus

*DN*), and

*u*

_{k}is the control computed from

*y*

_{k-dc +1}and applied on time interval [

*k*Δ

*T*, (

*k*+1)Δ

*T*] (a chronogram for

*d*

_{m}= 1 and

*d*

_{c}= 1 is given in Fig. 2). We consider here that the mirror’s response time is negligible (which is justified as soon as its response time is small compared to Δ

*T*), so there is no mirror’s dynamics.

*J*

^{c}, Eq. (1). It can be equivalently written by replacing

*τ*with

*n*Δ

*T*:

^{res}(

*t*) = ϕ

^{tur}(

*t*)-

*t*∊ [(

*k*-1)Δ

*T*,

*k*Δ

*T*], and using Eq. (4) and the fact that ϕ

^{tur}(

*t*)-

*u*and the other that only depends on discrete variables, minimizing

*J*

^{c}(

*u*) leads exactly to the same control value as minimizing

*T*. In the sequel, we shall thus describe in discrete-time all the constitutive elements and processes that make up the AO closed-loop.

## 3. Limitations and mandatory prior information

*z*-transforms can be used to compute the transfer functions (TF) that appear when closing the loop. In the sequel, we shall denote by

*x̃*the

*z*-transform of a discrete temporal process

*x*. Therefore,

^{res}(

*z*) can be written in closed-loop as a function of

^{tur}(

*z*),

*w̃*(

*z*), and as a function of the controller’s

*z*-transform

*C*(

*z*):

*I*

_{d}denotes the identity matrix and

*d*≜

*d*

_{m}+

*d*

_{c}is the total delay in the loop. Leaving aside the influence of measurement noise

*w*, the closed loop performance of this discrete-time system can be analyzed through the closed loop matrix transfer function from ϕ

^{tur}to ϕ

^{res},

*i.e*. the rejection TF

*H*(

*z*) ≜ (

*I*

_{d}+

*L*(

*z*))

^{-1}, where

*L*(

*z*) ≜

*NC*(

*z*)

*Dz*

^{-d}is the open loop TF between ϕ

^{res}and ϕ

^{cor}, including delays.

*L*, and hence

*H*, are diagonal, where the diagonal of

*H*is formed with scalar TFs (as in modal approaches) denoted by

*H*

_{ℓ}. In order to minimize the residual phase variance, the controller’s TF should be selected so as to render each scalar TF

*H*

_{ℓ}as small as possible at all frequencies, while stabilizing the feedback loop. At this critical juncture, Bode’s integral theorem [8] enters the stage; it states that for

*any*choice of stabilizing controller, for

*d*=

*d*

_{m}+

*d*

_{c}> 1, the integral of the logarithm of the modulus of every

*H*

_{ℓ}(

*e*

^{jω}) over the normalized frequency range

*ω*∊ [0, π] is zero:

*z*=

*e*

^{jω}for normalized frequency

*ω*∊ [0, π], see e.g. [9] for more details.)

^{res}smaller than ϕ

^{tur}at all frequencies, and that better attenuation at some frequency will have to be repaid in kind with disturbance amplification in another part of the spectrum. This so-called “water bed effect” is inherent to the feedback loop, whatever the stabilizing controller. In the sequel, we shall not select a particular controller structure, as the goal is to find among all stabilizing controllers the optimal one with respect to performance criterion (7) and hence to (1). This optimal controller will not escape from water bed effect, but will provide an optimal compromise between disturbance rejection and amplification.

*J*in Eq. (7). Parseval’s theorem states that energy is conserved between time and frequency domains, so that

*J*can be equivalently written in the frequency domain as

*x*,

*S*

_{x}stands for the power spectral density (PSD) of

*x*(note that in the case of a vector process of finite dimension,

*S*

_{x}is a matrix-valued function).

^{tur}and

*w*. Assume that ϕ

^{tur}and

*w*are mutually independent zero-mean stationary ergodic processes of finite energy, with PSDs

*S*

_{ϕtur}and

*S*

_{w}. A standard result from stochastic filtering theory (Birkhoff’s theorem, see for example [10]) is that ϕ

^{res}is also stationary with variance almost surely equal to

*J*(

*u*), i.e.

^{*}denotes the conjugate transpose and

*H*

_{w}(

*z*) ≜

*H*(

*z*)

*NC*(

*z*)

*z*

^{-1}is the closed-loop TF from

*w*to ϕ

^{res}(which is obtained from Eq. (8)). The frequency-domain identity (10), when replacing

*S*

_{ϕres}by its expression in (12), leads directly to

*J*(

*u*) requires in one way or another the knowledge of both

*S*

_{ϕtur}and

*S*

_{w}.

^{res})) over the set of all controllers which stabilize the AO loop.

## 4. Optimal solution is obtained by separating estimation and control

*a priori*bounded, this optimization problem may appear at first sight intractable. Quite understandably, suboptimal approaches have been pursued to select the controller’s TF

*C*(

*z*). A popular one is to use a static decoupling gain multiplied with a diagonal matrix of scalar dynamic compensators

*C*

_{ℓ}with fixed structure, and to separately tune the resulting series of hopefully independent feedback loops (this corresponds to modal approaches evoked in previous section, with diagonal

*H*(

*z*)). Thus, the

*H*

_{ℓ}may be pure integrators, as in [11, 12], or higher order filters as in [13

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

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

15. J.S. Gibson and B.L. Ellerbroek, “Adaptive optics wave-front correction by use of adaptive filtering and control,” Appl. Opt. **39**, 2525–2538 (2000). [CrossRef]

*d*

_{c}= 1, which means that the computation of

*y*

_{k}and

*u*

_{k}can be performed within [(

*k*- 1)Δ

*T*,

*k*Δ

*T*], as illustrated in Fig. 2), so that the DM equation including control delay is

^{cor}(

*z*) =

*Nz*

^{-1}

*ũ*(

*z*). Let us make now the obviously totally unrealistic assumption that future values of ϕ

^{tur}can be predicted with perfect accuracy. Under this fantasy-world “full information” hypothesis, a perfect solution would be to make ϕ

^{cor}equal to ϕ

^{tur}by solving

*Nu*

_{k}=

*N*being generally non-invertible, the optimal control corresponds to the solution of the least-squares minimization of

*Nu*

_{k},

*i.e*.

^{t}stands for transposition. This optimal control

*u*

_{k}corresponds to the orthogonal projection of

*P*defined as

_{k+1|k}

^{tur}, the minimum-variance estimator of

*ℐ*

_{k}, the set of all prior information and measurements available until time

*k*. This optimal estimate is the conditional expectation given

*ℐ*

_{k}[16

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

*ℐ*

_{k}) conditionally to all information available at time

*k*. This result is known in control literature as the stochastic separation theorem [17, 18

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

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

*u*can be constructed by separately solving a deterministic optimal control problem (the full information case) and a stochastic minimum-variance estimation problem (the incomplete information case). Furthermore, the deterministic optimal control subproblem turns out to be quite simple, as seen in Eq. (18).

### 4.1. Equivalent state-space modelization

*S*

_{w}and

*S*

_{ϕtur}have been shown to be necessarily known when dealing with criterion

*J*. This model, called a state-space model, is based on a full description of the system’s state, and should thus include an explicit description of the turbulent phase dynamics that matches the priors.

*k*, denoted by

*x*

_{k}, is generally defined as follows. It represents all the knowledge needed at time

*k*to compute next state

*x*

_{k+1}and output (WFS measurement)

*y*

_{k}, when inputs are known and if noises are neglected. The state vector dynamics correspond thus to an input-output description of the system, that is a set of equations which gives

*x*

_{k+1}and

*y*

_{k}as a function of

*x*

_{k},

*u*

_{k}, and of the noises. Such a state-space model is usually described in the linear time-invariant case in the form

*x*

_{k}? It should fulfill two requirements: firstly, the state must summarize the entire knowledge on the system including turbulence, and secondly, the optimal control law must be a function of the state only. In view of performance criterion (7), the residual phase ϕ

^{res}should then be part of the state vector, or equivalently ϕ

^{tur}and

*u*.

*k*, considering one period latency for read-out and computation time in the WFS (

*d*

_{m}= 1), the measurement

*y*

_{k}is obtained following Eq. (3) by

*u*

_{k-2}enter the state vector, hence

*u*

_{k-1}for memory storing. As said before, this choice of state vector is not the only possibility, but is motivated by the possible and direct extension to MCAO [7

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

*x*

_{k+1}shall be described from

*x*

_{k}and

*u*

_{k}through

*A*and

*B*(the values of which will be given below).

_{ϕ}, defined as

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

20. R.N. Paschall and D.J. Anderson, “Linear Quadratic Gaussian control of a deformable mirror adaptive optics system with time-delayed measurements,” Appl. Opt. **32**, 6347–6358 (1993). [CrossRef] [PubMed]

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

*v*

_{k}is a zero-mean white Gaussian noise with covariance matrix ∑

_{v}, and

*A*is the matrix defining the dynamical characteristics of the turbulent phase (temporal correlations that depend on turbulence speed, see Sect. 5). For any given ∑

_{ϕ}, the model described in Eq. (25) leads to ∑

_{ϕ}=

*A*

^{t}∑

_{ϕ}

*A*+ ∑

_{v}(using definition (24)). So, based on energy conservation principle, the turbulent phase covariance matrix will indeed be equal to ∑

_{ϕ}if we take ∑

_{v}= ∑

_{ϕ}-

*A*

^{t}∑

_{ϕ}

*A*. While simple, this model can thus simultaneously match the spatial correlation structure of the turbulent phase through ∑

_{ϕ}and its short-term temporal correlation through

*A*.

_{ϕ}and ∑

_{v}, the stochastic state-space model is now completely defined, using (22–25), in the form (19–20) with

*u*(see Eq. (18)) has then the general state feedback form

*K*≜ (

*P*, 0, 0, 0) and

*P*is defined in (16).

*A*,

*B*,

*C*depending on time) could be considered as well, leading to the same conclusions.

### 4.2. Kalman’s optimal filter

*x*

_{k+1}using all the measurements until time

*k*is obtained in the Gaussian case as the output of a Kalman filter [22, 23] (if the Gaussian assumption is released, the Kalman filter gives the best linear unbiased estimator). This filter is an observer, that is, has the general structure

*ŷ*

_{k|k-1}is the best estimate of the model output given

*ℐ*

_{k-1}, obtained as

_{k}given by

_{k|k-1}is the covariance matrix of the state vector and is obtained by solving the following Riccati matrix equation:

*L*(by letting ∑

_{k+1|k}in (31) converge to its asymptotical value) with non-significant loss of optimality, as in [24

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

*k*, the coordinates of the state

*x*corresponding to delayed values of

*u*need not be estimated; the optimal gain

*L*has thus corresponding coordinates equal to zero, so that it is computed using a reduced order Riccati equation.

## 5. Illustration of LQG control based on an end-to-end simulator

**21**, 1261–1276 (2004). [CrossRef]

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

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

*D*= 8 m telescope observing in the near-infrared (2.2 μm), equipped with a 8×8 subaperture Shack-Hartmann WFS and a 9×9 actuator Stacked Actuator Mirror. The turbulence

*V*= 9ms

^{-1}. A 250 Hz correction (which corresponds to Δ

*T*= 4.10

^{-3}s) with a total delay

*d*= 2 (

*d*Δ

*T*= 8.10

^{-3}s, with

*d*

_{c}=

*d*

_{m}= 1), is then simulated.

*n*is the radial order of the Zernike number

*i*. This law allows to account for the decrease of the correlation time with radial order [25

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

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

## 6. Integrators: observer form and hidden turbulent phase model

*v*

_{k}is a zero-mean white noise, and consider a state model of the form (19–20)

*G*. This new state model is different than (26) in an important point: measurement delay is not taken into account,

*i.e*.

*d*

_{m}= 0.

*L*that is computed using the new state model defined through (35–36). Simply because

*d*

_{m}= 0, the measurement equation is

*y*

_{k}=

*D*(

*w*

_{k}, leading to

*ŷ*

_{k|k-1}=

*D*(

_{k|k-1}

^{tur}-

*ŷ*

_{k|k-1}does not appear in (37) when comparing with (28).

*P*= (

*N*

^{t}

*N*)

^{-1}

*N*

^{t}(remember that

*u*

_{k}=

*P*

_{k+1|k}

^{tur}) leads exactly to (33) with

*G*satisfying

*G*, there always exists an observer of the form (37), with corresponding turbulent phase stochastic model (34). The observer (and consequently the integral controller) will not be optimal in the minimum variance sense unless

*G*is defined from (38) with optimal

*L*given by (30).

*k*. A nefarious consequence is that an observer built from this model is not stable and may well lead to unbounded trajectories of

^{tur}, and therefore of the control

*u*- the notorious wind-up effect for controllers with integral action.

## 7. Discussion and conclusions

^{cor}could be part of the state vector instead of

*u*, and a state model of smaller dimension could be used. Our choice is motivated by several considerations. Firstly, it limits the influence of matrices that are obtained through calibration process (and thus that contain model errors) mainly to the observation equation (20), so that there is no direct error propagation through the state equation (19). Secondly, it keeps a clear visible physical structure, so that extensions of the model to various situations (presence of vibrations, off-axis AO, MCAO) is easy.

**21**, 1261–1276 (2004). [CrossRef]

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

**21**, 1261–1276 (2004). [CrossRef]

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

*A*,

*B*and

*C*shall be modified according to the description of each element of the system. Also, several types of DM’s nonlinearities, such as saturations or nonlinear influence functions, can be explicitly accounted for in both the full information optimal control computation and the turbulent phase estimation.

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

*caveat emptor*should be kept in mind at all times in order to achieve a sensible balance between the model’s accuracy and its complexity.

## Acknowledgments

## References and links

1. | F. Roddier (Ed.), |

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

3. | R.H. Dicke, “Phase-contrast detection of telescope seeing and their correction,” Astron. J. |

4. | D.C. Johnson and B.M. Welsh, “Analysis of multiconjugate adaptive optics,” J. Opt. Soc. Am. A |

5. |
Comptes Rendus de l’Académie des Sciences, |

6. | G. Rousset and F. Lacombe, |

7. | B. Le Roux, J.-M. Conan, C. Kulcsár, H. F. Raynaud, L. M. Mugnier, and T. Fusco, “Optimal control law for classical and multiconjugate adaptive optics,” J. Opt. Soc. Am. A |

8. | C. Mohtadi, “Bode’s integral theorem for discrete-time systems,” Proc. IEE |

9. | J.G. Proakis and D.G. Manolakis, |

10. | L. Guikhman and A. Skorokhod, |

11. | E. Gendron and P. Lena, “Astronomical Adaptive optics I. modal control optimization,” Astron. Astrophys. |

12. | D.M. Wiberg, C.E. Max, and D.T. Gavel, “A special non-dynamic LQG controller: part I, application to adaptive optics,” Proceedings of the 43 |

13. | C. Dessenne, P.-Y. Madec, and G. Rousset, “Optimization of a predictive controller for the closed loop adaptive optics,” Appl. Opt. |

14. | A. Wirth, J. Navetta, D. Looze, S. Hippler, A. Glindemann, and D. Hamilton, “Real-time modal control implementation for adaptive optics,” Appl. Opt. |

15. | J.S. Gibson and B.L. Ellerbroek, “Adaptive optics wave-front correction by use of adaptive filtering and control,” Appl. Opt. |

16. | H. W. Sorenson, “Least-square estimation: from Gauss to Kalman,” IEEE Spectrum |

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

18. | R. N. Patchell and Jacobs, “Separability, neutrality and certainty equivalence,” Int. J. Control |

19. | Y. Bar-Shalom and E. Tse, “Dual effect, certainty equivalence and separation in stochastic control,” IEEE Trans. Automat. Contr. |

20. | R.N. Paschall and D.J. Anderson, “Linear Quadratic Gaussian control of a deformable mirror adaptive optics system with time-delayed measurements,” Appl. Opt. |

21. | M.W. Oppenheimer and M. Pachter, “Adaptive optics for airbone platforms—part 2: controller design,” Opt. Laser Technol. |

22. | A.H. Jazwinski, |

23. | B.D.O Anderson and J.B. Moore, |

24. | B. Le Roux, J.-M. Conan, C. Kulcsár, H. F. Raynaud, L. M. Mugnier, and T. Fusco, “Optimal control law for multiconjugate adaptive optics,” Proc. SPIE, |

25. | J.-M. Conan, G. Rousset, and P.-Y. Madec, “Wave-front temporal spectra in high-resolution imaging through turbulence,” J. Opt. Soc. Am. A |

26. | C. Petit, F. Quiros-Pacheco, J.-M. Conan, C. Kulcsár, H.-F. Raynaud, T. Fusco, and G. Rousset, “Kalman Filter based control loop for Adaptive Optics,” Proc. SPIE, |

27. | C. Petit, J.-M. Conan, C. Kulcsár, H. F. Raynaud, T. Fusco, J. Montri, and D. Rabaud. “Optimal control for Multi-Conjugate Adaptive Optics,” Elsevier, Comptes Rendus Physique |

**OCIS Codes**

(010.1080) Atmospheric and oceanic optics : Active or adaptive optics

(010.1330) Atmospheric and oceanic optics : Atmospheric turbulence

**ToC Category:**

Focus Issue: Adaptive Optics

**History**

Original Manuscript: April 3, 2006

Revised Manuscript: June 23, 2006

Manuscript Accepted: August 10, 2006

Published: August 21, 2006

**Citation**

Caroline Kulcsár, Henri-François Raynaud, Cyril Petit, Jean-Marc Conan, and Patrick Viaris de Lesegno, "Optimal control, observers and integrators in adaptive optics," Opt. Express **14**, 7464-7476 (2006)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-14-17-7464

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### References

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