## Frequency based design of modal controllers for adaptive optics systems |

Optics Express, Vol. 20, Issue 24, pp. 27108-27122 (2012)

http://dx.doi.org/10.1364/OE.20.027108

Acrobat PDF (957 KB)

### Abstract

This paper addresses the problem of reducing the effects of wavefront distortions in ground-based telescopes within a “Modal-Control” framework. The proposed approach allows the designer to optimize the Youla parameter of a given modal controller with respect to a relevant adaptive optics performance criterion defined on a “sampled” frequency domain. This feature makes it possible to use turbulence/vibration profiles of arbitrary complexity (even empirical power spectral densities from data), while keeping the controller order at a moderate value. Effectiveness of the proposed solution is also illustrated through an adaptive optics numerical simulator.

© 2012 OSA

## 1. Introduction

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

2. C. Kulcsár, H.-F. Raynaud, C. Petit, J.-M. Conan, and P. Viaris de Lesegno, “Optimal control, observers and integrators in adaptive optics,” Opt. Express **14**(17), 7464–7476 (2006). [CrossRef] [PubMed]

6. K. Hinnen, M. Verhaegen, and N. Doelman, “A data-driven *H*_{2}-optimal control approach for adaptive optics,” IEEE Trans. Control Syst. Technol , **16**(3), 381–395 (2008). [CrossRef]

8. G. Agapito, F. Quiros-Pacheco, P. Tesi, A. Riccardi, and S. Esposito, “Observer-based control techniques for the LBT adaptive optics under telescope vibrations,” Eur. J. Control **17**(3), 316–326 (2011). [CrossRef]

10. G. Agapito, F. Quiros-Pacheco, P. Tesi, S. Esposito, and M. Xompero, “Optimal control techniques for the adaptive optics system of the LBT,” Proc. SPIE **7015**, 70153G (2008). [CrossRef]

11. L. Close, V. Gasho, D. Kopon, J. Males, K.B. Follette, K. Brutlag, A. Uomoto, and T. Hare, “The Magellan telescope adaptive secondary AO system: a visible and mid-IR AO facility,” Proc. SPIE **7736**, 773605 (2010). [CrossRef]

13. S. Esposito, E. Pinna, F. Quirós-Pacheco, A. Puglisi, L. Carbonaro, M. Bonaglia, V. Biliotti, R. Briguglio, G. Agapito, C. Arcidiacono, L. Busoni, M. Xompero, A. Riccardi, L. Fini, and A. Bouchez, “Wavefront sensor design for the GMT natural guide star AO system,” Proc. SPIE **8447**, 84471L (2012).

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

15. C. Dessenne, P.-Y. Madec, and G. Rousset, “Optimization of a predictive controller for closed-loop adaptive optics,” Appl. Optim. , **37**(21), 4623–4633 (1998). [CrossRef]

*H*

_{2},

*H*

_{∞}and LQG control. [2

2. C. Kulcsár, H.-F. Raynaud, C. Petit, J.-M. Conan, and P. Viaris de Lesegno, “Optimal control, observers and integrators in adaptive optics,” Opt. Express **14**(17), 7464–7476 (2006). [CrossRef] [PubMed]

3. C. Petit, J.-M. Conan, C. Kulcsár, H.-F. Raynaud, and T. Fusco, “First laboratory validation of vibration filtering with LQG control law for adaptive optics,” Opt. Express **16**(1), 87–97 (2008). [CrossRef] [PubMed]

8. G. Agapito, F. Quiros-Pacheco, P. Tesi, A. Riccardi, and S. Esposito, “Observer-based control techniques for the LBT adaptive optics under telescope vibrations,” Eur. J. Control **17**(3), 316–326 (2011). [CrossRef]

17. C. Petit, F. Quiros-Pacheco, J.-M. Conan, C. Kulcsár, H.-F. Raynaud, T. Fusco, and G. Rousset, “Kalman filter based control for adaptive optics,” Proc. SPIE **5490**, 1414–1425 (2004). [CrossRef]

18. C. Correia, H.F. Raynaud, C. Kulcsar, and J.M. Conan, “Minimum-variance control for astronomical adaptive optics with resonant deformable mirrors,” Eur. J. Control **17**(3), 222–236 (2011). [CrossRef]

*optimality*in the sense of performance (Strehl ratio) ideally achievable for the identified process model. Achieving an exact modelling of turbulence and vibrations is however a practically impossible task,

*i.e.*models will necessarily be approximate. It is therefore important that the models used to synthesize the controller are sufficiently accurate at the frequencies of concern. Herein lies the difficulty: model-based approaches provide controllers whose order is essentially determined by the order of the underlying models. High-order turbulence/vibrations models, in addition to requiring non-negligible identification effort, will therefore lead to high-order controllers, which may be infeasible (or simply not desired) from an implementation viewpoint. On the other hand, models with reduced complexity may fail to capture well the behavior of turbulence and/or vibrations at the frequencies of concern.

4. E. Fedrigo, R. Muradore, and D. Zillo, “High performance adaptive optics system with fine tip/tilt control,” Control Eng. Pract. **17**(1), 122–135 (2009). [CrossRef]

8. G. Agapito, F. Quiros-Pacheco, P. Tesi, A. Riccardi, and S. Esposito, “Observer-based control techniques for the LBT adaptive optics under telescope vibrations,” Eur. J. Control **17**(3), 316–326 (2011). [CrossRef]

18. C. Correia, H.F. Raynaud, C. Kulcsar, and J.M. Conan, “Minimum-variance control for astronomical adaptive optics with resonant deformable mirrors,” Eur. J. Control **17**(3), 222–236 (2011). [CrossRef]

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

*H*

_{2},

*H*

_{∞}and LQG control design). This makes it possible to use turbulence/vibration profiles of arbitrary complexity, even empirical power spectral densities (PSDs) from data, while keeping the controller order at a desired (manageable) value. As elaborated next in detail, the proposed approach relies on the idea of optimizing the Youla parameter starting from a given stabilizing controller which can be either: i) a non-model-based controller, avoiding any identification effort; or ii) a model-based (

*H*

_{2},

*H*

_{∞}or alike) controller synthesized in accordance with turbulence and vibration models of limited complexity, avoiding to identify mathematical models of turbulence and vibrations of increased complexity. Because of this, the proposed approach is best viewed as cooperative with (rather than as alternative to) other control design approaches. Also, the problem can be cast as a quadratic programming problem (unconstrained, or with linear constraints if the controller is required to be stable), which can be efficiently solved by means of standard optimization routines.

## 2. AO control system architecture

*external*AO control system (working at the AO sampling time

*T*= 1 ms), whose task is to determine the commands to the ASM, and an

_{s}*internal*ASM control system (with sampling time

*T*≈ 10

_{ASM}^{−2}

*T*). The principal devices characterizing the control loop are the AO controller, the Pyramid WFS, and the ASM. The AO controller receives at time

_{s}*kT*the measurement vector

_{s}*y*(

*k*) ∈ ℝ

*, with*

^{q}*q*the number of measurements, from the Pyramid WFS and computes the command vector

*u*(

*k*) ∈ ℝ

*, with*

^{p}*p*the number of actuators, so as to pilot the actuators acting on the ASM shell. The command vector provides the information about the shape to be reproduced by the shell in order to compensate for the wavefront distortions. Due to the non-negligible dynamics of the shell, a dedicate control loop is needed to get the desired shape within

*T*. Such a control loop is realized through the use of

_{s}*p*capacitive sensors co-located with the actuators and placed at the back of the shell. Specifically, the control law piloting each actuator has a decentralized component obtained by a Proportional-Derivative feedback action which depends on the measurement of the co-located sensor, and a centralized component implemented through a feed-forward action which equals the force needed to statically deform the shell as indicated by the command vector

*u*(

*k*): this latter action is obtained by multiplying

*u*(

*k*) with the estimate of a shell stiffness matrix of dimensions

*p*×

*p*, which is a-priori calibrated. For more details about the ASM internal position control, the interested reader is referred to [9

9. G. Agapito, S. Baldi, G. Battistelli, D. Mari, E. Mosca, and A. Riccardi, “Automatic tuning of the internal position control of an adaptive secondary mirror,” Eur. J. Control **17**(3), 273–289 (2011). [CrossRef]

22. A. Riccardi, G. Brusa, P. Salinari, D. Gallieni, R. Biasi, M. Andrighettoni, and H. M. Martin, “Adaptive secondary mirrors for the Large Binocular Telescope,” Proc. SPIE **4839**, 721–732 (2003). [CrossRef]

*ϕ*(

^{tot}*k*). Such a distortion has to be corrected by the shell deformations which provide the so called correction phase

*ϕ*(

^{cor}*k*). The difference

*ϕ*(

^{res}*k*) =

*ϕ*(

^{tot}*k*)−

*ϕ*(

^{cor}*k*) is the residual phase after the ASM correction. The objective of the external control loop therefore consists in regulating the residual phase

*ϕ*(

^{res}*k*) about 0.

*Zernike*and

*Karhunen-Loève*basis [1

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

*y*(

*k*) from the Pyramid WFS as described by the following where

*w*(

*k*) ∈ ℝ

*is the measurement noise vector, which can be assumed to be a zero-mean white noise. The matrix*

^{q}*𝒟*characterizes the WFS and describes the geometric relationship between the modal space and the measurement space. Notice that the modal space depends on the selected modal basis and all the phase variables represent coefficient vectors belonging to ℝ

*, where*

^{p}*p*denotes the dimension of the modal space.

*𝒩*is the so-called

*Commands-to-Modes*matrix, which describes the geometric relationship between the command space and the modal space. An estimate of

*𝒩*can be obtained via finite elements analysis of the mirror shell, once chosen the modal basis to be used.

*U*(

*z*) and

*Y*(

*z*) are the

*Z*-transforms of

*u*(

*k*) and

*y*(

*k*), respectively (hereafter we will use the corresponding capital letter to denote the

*Z*-transform of a lower-case letter signal).

*ℳ*is the

*Modes-to-Commands*(or

*Projection*) matrix, represented by the

*From modes to commands*block in Fig. 1, and

*ℛ*is the

*Reconstruction*matrix, which yields the geometric relationship between the WFS measurements and the deformations of the ASM (

*Modal reconstruction*block in Fig. 1). The transfer matrix

*𝒦*(

_{M}*z*) has dimension

*p × p*and has to be designed in order to satisfy the control requirements. In the proposed setting,

*𝒦*(

_{M}*z*) is chosen as a diagonal matrix,

*𝒦*(

_{M}*z*) = diag{

*C*

_{1}(

*z*), ⋯ ,

*C*(

_{p}*z*)}, as the ideal goal would be to control the

*i*–th mode by means of controller

*C*(

_{i}*z*).

*z*) := (

*I*+

*z*

^{−2}

*𝒩ℳ𝒦*(

_{M}*z*)

*ℛ𝒟*) and Ξ(

*z*) :=

*z*

^{−1}

*𝒩ℳ𝒦*(

_{M}*z*)

*ℛ*. The matrix

*ℛ*is then selected as the Moore-Penrose inverse of

*𝒟*,

*i.e. ℛ*:= (

*𝒟*

^{T}*𝒟*)

^{−1}

*𝒟*. Accordingly, Δ(

^{T}*z*) simplifies to Δ(

*z*) = (

*I*+

*z*

^{−2}

*𝒩ℳ𝒦*(

_{M}*z*)). The matrices

*𝒩*and

*ℳ*are instead calibrated so as to yield

*𝒩ℳ*≈

*I*(more precisely for the LBT,

*𝒩ℳ*is a matrix with elements on the diagonal equal to 1 and extra-diagonal elements with absolute values ≈ 2

^{−15}). Thus, by neglecting the measurement noise, we can therefore select a diagonal controller

*𝒦*(

_{M}*z*) and achieve modal decoupling,

*i.e.*Φ

*(*

^{res}*z*) ≈ (

*I*+

*z*

^{−2}

*𝒦*(

_{M}*z*))

^{−1}Φ

*(*

^{tot}*z*).

*C*(

_{i}*z*) for each mode. However, in practice, it is convenient to consider advanced control design techniques only for those modes having more influence on the value of Φ

*(*

^{res}*k*). Within the addressed framework, a dedicated controller

*C*(

_{D}*z*) is synthesized only for tip and tilt modes [4

4. E. Fedrigo, R. Muradore, and D. Zillo, “High performance adaptive optics system with fine tip/tilt control,” Control Eng. Pract. **17**(1), 122–135 (2009). [CrossRef]

**17**(3), 316–326 (2011). [CrossRef]

10. G. Agapito, F. Quiros-Pacheco, P. Tesi, S. Esposito, and M. Xompero, “Optimal control techniques for the adaptive optics system of the LBT,” Proc. SPIE **7015**, 70153G (2008). [CrossRef]

*C*(

_{I}*z*) =

*gz*/(

*z*− 1), where

*g*is a gain to be set. In fact, it has been observed by means of experimental studies on LBT that turbulence and vibrations affect mainly tip/tilt modes. In turn, the experimental studies have revealed that in the considered operating setting tip/tilt modes yield more than 80% of the overall atmospheric turbulence variance. The choice of considering only two dedicated controllers has been mainly motivated by the memory constraints of the RTC which, as detailed in [8

**17**(3), 316–326 (2011). [CrossRef]

*𝒦*(

_{M}*z*) is 672. It is however important to point out that the design procedure proposed in the following sections is well-suited to being applied to each modal controller

*C*(

_{i}*z*).

## 3. Modal control design: a frequency based approach

*C*(

*z*,

*ρ*) represent the transfer function of an arbitrary controller with fixed structure depending on a parameter vector

*ρ*which has to be tuned so as to achieve the desired performance, where the choice of

*C*(

*z*,

*ρ*) will be discussed more accurately in the sequel of the section. Hereafter, we will denote by

*P*(

*z*) the transfer function of the plant to be controlled through

*C*(

*z*,

*ρ*). By referring to Fig. 2,

*P*(

*z*) consists of the cascade of the blocks

*M*(

*z*) and

*H*(

*z*), representing the dynamics of the ASM and WFS, respectively. It is to be pointed out that, within our setting, both

*M*(

*z*) and

*H*(

*z*) can be assumed to behave as a unit delay, in that their tasks are executed within one sampling time of the external loop. Specifically, the internal position control acting on the mirror shell, see Fig. 1, works with a sampling time much lower than the one of the external AO loop (in LBT, for example, it is about 0.01 ms), and thus it allows the mirror shell to achieve the desired shape within one AO loop sampling time. Accordingly, we can assume

*M*(

*z*) = 1/

*z*and

*H*(

*z*) = 1/

*z*.

*ω*

_{1},

*ω*

_{2},...,

*ω*of the PSDs of

_{N}*w*(

_{t}*k*). These samples can be obtained either from a model, possibly infinite-dimensional and non parametric, or directly from collected data series. Accordingly, we redefine the performance criterion in Eq. (7) as where ϒ̂

*(*

_{ϕ}*ω*) and ϒ̂

_{f}*(*

_{ϕ}*ω*) denote estimates of the PSDs ϒ

_{f}*(*

_{ϕ}*ω*) and ϒ

_{f}*(*

_{ϕ}*ω*), respectively, at the frequency

_{f}*ω*.

_{f}*C*(

*z*,

*ρ*) with optimized performance with respect to the objective function

*f*(

*ρ*), we resort to the well-known

*Youla*parametrization [24

24. R.A. de Callafon and C.E. Kinney, “Robust estimation and adaptive controller tuning for variance minimization in servo systems,” JSME J. Adv. Mech. Des. Syst. Manuf. , **4**(1), 130–142 (2010). [CrossRef]

*𝒮*be the family of all the proper transfer functions with poles inside the open unit disk, and

*𝒞*(

*P*) the set of all stabilizing controllers, i.e., the set of all controllers which are able to guarantee that the closed-loop is internally stable. Since

*P*(

*z*) ∈

*𝒮*, the set of all stabilizing controllers takes the form ([25], Chapter 5) where

*Q*(

*z*), the so-called

*Youla parameter*, is an arbitrary transfer function belonging to

*𝒮*. It is straightforward to prove that the controllers which can be expressed as shown in Eq. (9) are all and the only ones which guarantee the internal stability ([25], Chapter 5).

*Q*(

*z*) has to be regarded as a free parameter which can be tuned so as to achieve the desired control performance. To this end, we give

*Q*(

*z*) a fixed structure depending on a parameter vector

*ρ*, i.e.,

*Q*(

*z*) =

*Q*(

*z*,

*ρ*), thus considering controllers of the form One of the main positive features of the Youla parametrization is that it allows one to express both

*S*(

*z*,

*ρ*) and

*T*(

*z*,

*ρ*) in a form which is affine in the Youla parameter

*Q*(

*z*,

*ρ*) in that Hence, by expressing

*Q*(

*z*) as a linear combination of functions

*ψ*(

_{i}*z*), each one weighted by a parameter

*ρ*,

_{i}*i.e.*where

*ψ*(

*z*) = [

*ψ*

_{1}(

*z*)

*ψ*

_{2}(

*z*) ⋯

*ψ*(

_{n}*z*)]

*, and*

^{T}*ρ*= [

*ρ*

_{1}

*ρ*

_{2}⋯

*ρ*]

_{n}*, it is possible to make both*

^{T}*S*(

*z*,

*ρ*) and

*T*(

*z*,

*ρ*) affine functions of the parameter vector

*ρ*:

*ψ*(

*z*) is important. For instance, one can choose

*ψ*(

*z*) as a collection of basis functions, through which it is possible to approximate any finite-order stable transfer function with arbitrary accuracy by increasing the value of

*n*[26

26. A. Karimi and G. Galdos, “Fixed-order *H*_{∞} controller design for nonparametric models by convex optimization,” Automatica **46**, 1388–1394 (2010). [CrossRef]

*ψ*(

_{i}*z*), in this work we considered where the denominator

*d*(

*z*) is a fixed polynomial of degree

*n*− 1. As it can be seen from Eqs. (14) and (15), the polynomial

*d*(

*z*) determines the closed-loop poles of the considered modal control loop. Accordingly, the choice of

*d*(

*z*) allows to a priori fix an adequate stability margin. As will be discussed in details in Section 4, in practice

*d*(

*z*) can be chosen on the basis of a given reference controller which represents the starting point of the proposed control design procedure.

*ρ*plays a fundamental role, in that it is used as the decision variable of the optimization problem. In fact, the objective function in Eq. (8) can be equivalently written as a quadratic function of

*ρ*: Thanks to the form of the function

*f*(

*ρ*), it would now be immediate to compute the parameter vector, say

*ρ*°, which minimizes

*f*(

*ρ*). The resulting controller

*C*(

*z*,

*ρ*°) would provide optimized performance in the sense of the minimum sampled-valued variance.

*u*(

*k*) to be interrupted for a few time steps: actuators may be required to perform an action that is not compatible with its stroke or with the force they can bear (this phenomenon can be frequent, up to some events per second, in seeing conditions ≥ 1.2”); also, slopes may not be delivered for a frame or two, producing the same effects of command interruptions (though the latter is an extremely rare event).

*individually*are stable ([25], Chapter 5).

*f*(

*ρ*) can result in an unstable (albeit stabilizing) controller

*C*(

*z*,

*ρ*°). A possible way of overcoming such a drawback consists in taking into account the controller stability requirement directly while synthesizing the controller. Accordingly, we propose to solve a

*constrained*optimization problem (

*COP*), defined such that

*f*(

*ρ*) is the objective function and the controller stability requirement characterizes the feasibility domain s.t.

## 4. Modal control design: algorithm details

*ρ*̂ is available corresponding to a stable controller

*C*(

*z*,

*ρ*̂) (in practice

*ρ*̂ can be computed from a given stable and stabilizing controller as will be detailed in the following). In fact, a sufficient condition for the stability of a controller

*C*(

*z*,

*ρ*) is that where

*Re*{·} denotes the real part. Indeed, condition in Eq. (20) ensures that the number of counterclockwise encirclements around the origin of the Nyquist plot of 1−

*P*(

*z*)

*Q*(

*z*,

*ρ*) equals that of the Nyquist plot of 1 −

*P*(

*z*)

*Q*(

*z*,

*ρ*̂). In view of the Nyquist stability criterion ([27], Chapter 5), this ensures that

*C*(

*z*,

*ρ*) and

*C*(

*z*,

*ρ*̂) have the same stability property (recall Eq. (10)).

*ω*,

*i.e.*over a continuum, thus of no practical use. One possible solution in order to avoid this drawback is to impose that Eq. (20) is satisfied only in the sampled-frequency domain {

*ω*

_{1},

*ω*

_{2},...

*ω*}. The resulting condition can be written in the form of linear constraints by conveniently defining matrix

_{N}*A*and vector

*b*. Specifically, matrix

*A*takes the form with elements

*a*(

_{fi}*ρ*̂),

*f*= 1,...,

*N*,

*i*= 1,...,

*n*computed as and vector

*b*(

*ρ*̂) is defined as with elements

*b*(

_{f}*ρ*̂),

*f*= 1,...,

*N*computed as

26. A. Karimi and G. Galdos, “Fixed-order *H*_{∞} controller design for nonparametric models by convex optimization,” Automatica **46**, 1388–1394 (2010). [CrossRef]

*Ĉ*(

*z*) which can be synthesized either by means of a simple non-model based technique or of some model-based design procedure. Then we compute the Youla parameter In order to increase the degrees of freedom of our optimization problem we can introduce

*γ*additional poles in

*z*= 0 and define the polynomial

*d*(

*z*) in Eq. (16) as

*d*(

*z*) =

*d*̂(

*z*)

*z*. Then,

^{γ}*n*is set equal to the degree of

*d*(

*z*) increased by one, while the reference parameter vector

*ρ*̂ is set equal to the vector of the coefficients of the polynomial

*n*̂(

*z*)

*z*. At this point, it is possible to minimize the performance criterion

^{γ}*f*(

*ρ*) under the constraints in Eq. (21). In addition, in order to improve the controller performance one can construct an iterative procedure by using, at every step, the solution of the optimization problem as a novel reference parameter vector. This idea gives rise to the following iterative procedure.

### Iterative design procedure

**Step 1**: given an initial controller*Ĉ*(*z*) and a nonnegative integer*γ*, compute*d*(*z*),*n*, and*ρ*̂ as described above;**Step 2**: set*i*:= 0 and*ρ*^{(0)}:=*ρ*̂;**Step 3**: compute the matrix*A*(*ρ*^{(i)}) and the vector*b*(*ρ*^{(i)});**Step 5**: if the termination criterion is met then return*ρ*^{(i+1)}; otherwise set*i*:=*i*+ 1 and go back to step 3.

*f*(

*ρ*

^{(i+1)}) −

*f*(

*ρ*

^{(i)})| <

*ε*, with

*ε*a given threshold. In this respect, notice that by construction the sequence

*f*(

*ρ*

^{(i)}) is monotonically non-increasing with

*i*. In fact, it can be easily verified that

*ρ*

^{(i)}always satisfies the constraint in Eq. (28) so that

*f*(

*ρ*

^{(i+1)}) ≤

*f*(

*ρ*

^{(i)}). Since

*f*(

*ρ*) is bounded from below, this ensures that the sequence

*f*(

*ρ*

^{(i)}) converges to some finite limit and, hence, that the termination criterion is always eventually met.

## 5. Simulation results

*i.e.*from a controller synthesized without identifying mathematical models of turbulence and vibrations; in the second scenario, the proposed procedure is initialized from a

*H*

_{2}-controller synthesized using mathematical models of turbulence and vibrations of reduced complexity, namely a second-order AR model for the turbulence and two second-order ARMA models for the vibrations.

*First scenario:*Let with

*g*such that the closed-loop is stable. By the procedure described in Section 4,

*Q*̂(

*z*) takes the form As previously indicated, in order to increase the degrees of freedom of our optimization problem we can introduce

*γ*additional poles in

*z*= 0 and define the polynomial

*d*(

*z*) in Eq. (16) as

*d*(

*z*) =

*d*̂(

*z*)

*z*. Accordingly, and the algorithm is then initialized by letting As for the simulation, we set

^{γ}*g*= 0.77. Compared with

*Ĉ*(

*z*), which achieves a residual phase variance (

*VAR*)

*VAR*= 0.2528 (

*SR*= 58.51%), Fig. 4 shows a definite performance improvement even for small values of

*γ*, namely starting from

*γ*= 3. In particular, for

*γ*≥ 7 we obtain

*VAR*< 0.0209 (

*SR*> 86.22%).

*Second scenario:*As initial controller, we now consider a

*H*

_{2}-controller synthesized using mathematical models of turbulence and vibrations of reduced complexity. Specifically, we adopted the design procedure described in [8

**17**(3), 316–326 (2011). [CrossRef]

*ν*and output

*ξ*, and described by the difference equation where

*ν*is a white-noise process. As for the structural vibrations, we considered two second-order ARMA models having input

*μ*and output

_{i}*ζ*,

_{i}*i*= 1, 2. They are described by the difference equations for the model having a vibration peak around 13Hz, and for the model having a vibration peak around 22Hz, where

*μ*

_{1}and

*μ*

_{2}are both white-noise processes.

*H*

_{2}-controller is 9 and we set

*γ*= 0 so as to keep the order of the new controller at a moderate level. In this case, the simulation results show a performance improvement from

*VAR*= 0.0393 (

*SR*= 84.02%) to

*VAR*= 0.0152 (

*SR*= 86.61%). The performance obtained for larger values of

*γ*is very similar to the one obtained for

*γ*= 0.

## 6. Conclusions

*H*

_{2},

*H*

_{∞}or LQG) synthesized in accordance with turbulence and vibration models of limited complexity. Also, the problem can be cast as a quadratic programming problem, which can be efficiently solved by means of standard optimization routines. A number of simulation results have been reported using an End-to-End Simulator of the FLAO system of the LBT. The results show that it is possible to obtain a good trade-off between performance and controller complexity by optimizing model-based controllers which are synthesized in accordance with turbulence and vibration models of reduced order. Such a feature suggests that a good trade-off can be obtained without identifying high-order turbulence and vibration mathematical models. This makes the proposed approach a promising technique to be used in combination with classical approaches to AO control design.

*H*

_{2}/

*H*

_{∞}approach. This could be useful in the presence of system uncertainties or to deal with disturbances of intensity and/or frequency range varying with time.

## References and links

1. | F. Roddier, |

2. | C. Kulcsár, H.-F. Raynaud, C. Petit, J.-M. Conan, and P. Viaris de Lesegno, “Optimal control, observers and integrators in adaptive optics,” Opt. Express |

3. | C. Petit, J.-M. Conan, C. Kulcsár, H.-F. Raynaud, and T. Fusco, “First laboratory validation of vibration filtering with LQG control law for adaptive optics,” Opt. Express |

4. | E. Fedrigo, R. Muradore, and D. Zillo, “High performance adaptive optics system with fine tip/tilt control,” Control Eng. Pract. |

5. | L.A. Poyneer, B.A. Macintosh, and J.-P. Véran, “Fourier transform wavefront control with adaptive prediction of the atmosphere,” J. Opt. Soc. Am. A, |

6. | K. Hinnen, M. Verhaegen, and N. Doelman, “A data-driven |

7. | “Adaptive optics control for ground based telescopes,” in |

8. | G. Agapito, F. Quiros-Pacheco, P. Tesi, A. Riccardi, and S. Esposito, “Observer-based control techniques for the LBT adaptive optics under telescope vibrations,” Eur. J. Control |

9. | G. Agapito, S. Baldi, G. Battistelli, D. Mari, E. Mosca, and A. Riccardi, “Automatic tuning of the internal position control of an adaptive secondary mirror,” Eur. J. Control |

10. | G. Agapito, F. Quiros-Pacheco, P. Tesi, S. Esposito, and M. Xompero, “Optimal control techniques for the adaptive optics system of the LBT,” Proc. SPIE |

11. | L. Close, V. Gasho, D. Kopon, J. Males, K.B. Follette, K. Brutlag, A. Uomoto, and T. Hare, “The Magellan telescope adaptive secondary AO system: a visible and mid-IR AO facility,” Proc. SPIE |

12. | E. Marchetti, M. Le Louarn, C. Soenke, E. Fedrigo, P.-Y. Madec, and N. Hubin, “ERIS adaptive optics system design,” Proc. SPIE |

13. | S. Esposito, E. Pinna, F. Quirós-Pacheco, A. Puglisi, L. Carbonaro, M. Bonaglia, V. Biliotti, R. Briguglio, G. Agapito, C. Arcidiacono, L. Busoni, M. Xompero, A. Riccardi, L. Fini, and A. Bouchez, “Wavefront sensor design for the GMT natural guide star AO system,” Proc. SPIE |

14. | M. Born and E. Wolf, |

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

16. | E. Gendron and P. Lena, “Modal control optimization,” Astron. Astrophys. , |

17. | C. Petit, F. Quiros-Pacheco, J.-M. Conan, C. Kulcsár, H.-F. Raynaud, T. Fusco, and G. Rousset, “Kalman filter based control for adaptive optics,” Proc. SPIE |

18. | C. Correia, H.F. Raynaud, C. Kulcsar, and J.M. Conan, “Minimum-variance control for astronomical adaptive optics with resonant deformable mirrors,” Eur. J. Control |

19. | D. P. Looze, M. Kasper, S. Hippler, O. Beker, and R. Weiss, “Optimal compensation and implementation for adaptive optics systems,” Exp. Astron. , |

20. | L. A. Poyneer and J.-P. Véran, “Optimal modal Fourier transform wavefront control,” J. Opt. Soc. Am. A, |

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

22. | A. Riccardi, G. Brusa, P. Salinari, D. Gallieni, R. Biasi, M. Andrighettoni, and H. M. Martin, “Adaptive secondary mirrors for the Large Binocular Telescope,” Proc. SPIE |

23. | J. Herrmann, “Phase variance and Strehl ratio in adaptive optics,” J. Opt. Soc. Am. A, |

24. | R.A. de Callafon and C.E. Kinney, “Robust estimation and adaptive controller tuning for variance minimization in servo systems,” JSME J. Adv. Mech. Des. Syst. Manuf. , |

25. | J. Doyle, B. Francis, and A. Tannenbaum, |

26. | A. Karimi and G. Galdos, “Fixed-order |

27. | G. Goodwin, S. Graebe, and M. Salgado, |

28. | R. Ragazzoni, “Pupil plane wavefront sensing with an oscillating prism,” J. Mod. Optic. |

29. | C. Vérinaud, “On the nature of the measurements provided by a pyramid wave-front sensor,” Opt. Commun. |

30. | B. L. McGlamery, “Computer simulation studies of compensation of turbulence degraded images,” Proc. SPIE , |

**OCIS Codes**

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

(010.1330) Atmospheric and oceanic optics : Atmospheric turbulence

**ToC Category:**

Adaptive Optics

**History**

Original Manuscript: August 2, 2012

Revised Manuscript: October 12, 2012

Manuscript Accepted: October 14, 2012

Published: November 16, 2012

**Citation**

Guido Agapito, Giorgio Battistelli, Daniele Mari, Daniela Selvi, Alberto Tesi, and Pietro Tesi, "Frequency based design of modal controllers for adaptive optics systems," Opt. Express **20**, 27108-27122 (2012)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-24-27108

Sort: Year | Journal | Reset

### References

- F. Roddier, Adaptive Optics in Astronomy (Cambridge University Press, 1999). [CrossRef]
- C. Kulcsár, H.-F. Raynaud, C. Petit, J.-M. Conan, and P. Viaris de Lesegno, “Optimal control, observers and integrators in adaptive optics,” Opt. Express14(17), 7464–7476 (2006). [CrossRef] [PubMed]
- C. Petit, J.-M. Conan, C. Kulcsár, H.-F. Raynaud, and T. Fusco, “First laboratory validation of vibration filtering with LQG control law for adaptive optics,” Opt. Express16(1), 87–97 (2008). [CrossRef] [PubMed]
- E. Fedrigo, R. Muradore, and D. Zillo, “High performance adaptive optics system with fine tip/tilt control,” Control Eng. Pract.17(1), 122–135 (2009). [CrossRef]
- L.A. Poyneer, B.A. Macintosh, and J.-P. Véran, “Fourier transform wavefront control with adaptive prediction of the atmosphere,” J. Opt. Soc. Am. A,24(9), 2645–2660 (2007). [CrossRef]
- K. Hinnen, M. Verhaegen, and N. Doelman, “A data-driven H2-optimal control approach for adaptive optics,” IEEE Trans. Control Syst. Technol, 16(3), 381–395 (2008). [CrossRef]
- “Adaptive optics control for ground based telescopes,” in Special Issue of the European Journal of Control, Vol. 17, No. 3, J.-M. Conan, C. Kulcsár, and H.-F. Raynaud, eds. (Lavoisier, 2011).
- G. Agapito, F. Quiros-Pacheco, P. Tesi, A. Riccardi, and S. Esposito, “Observer-based control techniques for the LBT adaptive optics under telescope vibrations,” Eur. J. Control17(3), 316–326 (2011). [CrossRef]
- G. Agapito, S. Baldi, G. Battistelli, D. Mari, E. Mosca, and A. Riccardi, “Automatic tuning of the internal position control of an adaptive secondary mirror,” Eur. J. Control17(3), 273–289 (2011). [CrossRef]
- G. Agapito, F. Quiros-Pacheco, P. Tesi, S. Esposito, and M. Xompero, “Optimal control techniques for the adaptive optics system of the LBT,” Proc. SPIE7015, 70153G (2008). [CrossRef]
- L. Close, V. Gasho, D. Kopon, J. Males, K.B. Follette, K. Brutlag, A. Uomoto, and T. Hare, “The Magellan telescope adaptive secondary AO system: a visible and mid-IR AO facility,” Proc. SPIE7736, 773605 (2010). [CrossRef]
- E. Marchetti, M. Le Louarn, C. Soenke, E. Fedrigo, P.-Y. Madec, and N. Hubin, “ERIS adaptive optics system design,” Proc. SPIE8447, 84473M (2012).
- S. Esposito, E. Pinna, F. Quirós-Pacheco, A. Puglisi, L. Carbonaro, M. Bonaglia, V. Biliotti, R. Briguglio, G. Agapito, C. Arcidiacono, L. Busoni, M. Xompero, A. Riccardi, L. Fini, and A. Bouchez, “Wavefront sensor design for the GMT natural guide star AO system,” Proc. SPIE8447, 84471L (2012).
- M. Born and E. Wolf, Principles of Optics (Cambridge University Press, Cambridge, U.K, 1980).
- C. Dessenne, P.-Y. Madec, and G. Rousset, “Optimization of a predictive controller for closed-loop adaptive optics,” Appl. Optim., 37(21), 4623–4633 (1998). [CrossRef]
- E. Gendron and P. Lena, “Modal control optimization,” Astron. Astrophys., 291, 337–347 (1994).
- C. Petit, F. Quiros-Pacheco, J.-M. Conan, C. Kulcsár, H.-F. Raynaud, T. Fusco, and G. Rousset, “Kalman filter based control for adaptive optics,” Proc. SPIE5490, 1414–1425 (2004). [CrossRef]
- C. Correia, H.F. Raynaud, C. Kulcsar, and J.M. Conan, “Minimum-variance control for astronomical adaptive optics with resonant deformable mirrors,” Eur. J. Control17(3), 222–236 (2011). [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 (2004). [CrossRef]
- L. A. Poyneer and J.-P. Véran, “Optimal modal Fourier transform wavefront control,” J. Opt. Soc. Am. A,22(2), 1515–1526 (2005). [CrossRef]
- B. Le Roux, J.-M. Conan, C. Kulcsár, H.-F. Raynaud, L. Mugnier, and T. Fusco, “Optimal control law for classical and multiconjugate adaptive optics,” J. Opt. Soc. Am. A,21(7), 1261–1276 (2004). [CrossRef]
- A. Riccardi, G. Brusa, P. Salinari, D. Gallieni, R. Biasi, M. Andrighettoni, and H. M. Martin, “Adaptive secondary mirrors for the Large Binocular Telescope,” Proc. SPIE4839, 721–732 (2003). [CrossRef]
- J. Herrmann, “Phase variance and Strehl ratio in adaptive optics,” J. Opt. Soc. Am. A,9, 2257–2258 (1992). [CrossRef]
- R.A. de Callafon and C.E. Kinney, “Robust estimation and adaptive controller tuning for variance minimization in servo systems,” JSME J. Adv. Mech. Des. Syst. Manuf., 4(1), 130–142 (2010). [CrossRef]
- J. Doyle, B. Francis, and A. Tannenbaum, Feedback Control Theory (Macmillan Publishing Company, 1992).
- A. Karimi and G. Galdos, “Fixed-order H∞ controller design for nonparametric models by convex optimization,” Automatica46, 1388–1394 (2010). [CrossRef]
- G. Goodwin, S. Graebe, and M. Salgado, Control System Design (Prentice Hall, 2001).
- R. Ragazzoni, “Pupil plane wavefront sensing with an oscillating prism,” J. Mod. Optic.43, 289–293 (1996). [CrossRef]
- C. Vérinaud, “On the nature of the measurements provided by a pyramid wave-front sensor,” Opt. Commun.233, 27 – 38 (2004). [CrossRef]
- B. L. McGlamery, “Computer simulation studies of compensation of turbulence degraded images,” Proc. SPIE, 74(9), 225–233 (1976). [CrossRef]

## Cited By |
Alert me when this paper is cited |

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

« Previous Article | Next Article »

OSA is a member of CrossRef.