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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 20, Iss. 24 — Nov. 19, 2012
  • pp: 27108–27122
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Frequency based design of modal controllers for adaptive optics systems

Guido Agapito, Giorgio Battistelli, Daniele Mari, Daniela Selvi, Alberto Tesi, and Pietro Tesi  »View Author Affiliations


Optics Express, Vol. 20, Issue 24, pp. 27108-27122 (2012)
http://dx.doi.org/10.1364/OE.20.027108


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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

A central problem when dealing with ground-based telescopes arises from deformations on the light wavefront caused by the atmosphere. It is well known that the maximum resolution of any optical device is due to diffraction; in order to recover diffraction limited resolution, modern ground-based telescopes are equipped with adaptive optics (AO) devices, which aim at reducing, thanks to the use of deformable mirrors, the aforementioned effects of wavefront distortion. In addition to the atmospheric turbulence [1

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

], telescopes also suffer from structural vibrations due to situations such as telescope orientation, telescope tracking errors, and wind shaking. Due to their potential impact, considerable attention has been devoted over the last decades to the analysis and design of AO systems. Together with developments of physics, mathematics and technology, contributions to the subject have been recently proposed also from a control engineering perspective, as witnessed by [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]

6

6. 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]

] and the special issue on AO for ground-based telescopes organized by the European Journal of Control [7

7. “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).

].

In this work, we focus on the control architecture installed on the Large Binocular Telescope (LBT, located on Mt. Graham, Arizona, USA) [8

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

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]

], and one of the two Magellan Telescopes (Las Campanas Observatory, Chile) [11

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]

]. A similar architecture will be also adopted for the upgrading of one of the four Very Large Telescopes (Cerro Paranal, Chile) [12

12. E. Marchetti, M. Le Louarn, C. Soenke, E. Fedrigo, P.-Y. Madec, and N. Hubin, “ERIS adaptive optics system design,” Proc. SPIE 8447, 84473M (2012).

] and the Giant Magellan Telescope (Las Campanas, Chile), which is expected to be operative by the end of 2024 [13

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).

]. The considered AO unit comprises a wavefront sensor (WFS) - in this work a pyramid one has been taken into account - an adaptive secondary mirror (ASM), and a real-time computer (RTC). Roughly speaking, the pyramid WFS delivers a signal that is proportional, by a first-order approximation, to the first derivative of the incoming wavefront. The RTC, which implements the AO controller, computes the command vector driving the actuators of the ASM. Through the use of hundreds of voice-coil (electro-magnetic force) actuators, distributed on the mirror shell, the ASM corrects the wavefront distortion according to the RTC command vector. The correction is realized in such a way that the shape of the shell of the ASM becomes, as closely as possible, opposite to that of the wavefront distortion [1

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

, 14

14. M. Born and E. Wolf, Principles of Optics (Cambridge University Press, Cambridge, U.K, 1980).

].

Classic approaches to AO control can be subdivided into two main groups: those which do not employ any form of identification of turbulence and vibrations models [15

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]

, 16

16. E. Gendron and P. Lena, “Modal control optimization,” Astron. Astrophys. , 291, 337–347 (1994).

]; and those which employ models of turbulence and/or vibrations in combination with model-based control design techniques, such as H2, 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

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

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

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

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]

]. Model-based approaches have the intuitive advantage of including a notion of 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.

The remainder of the paper is as follows. Section 2 describes the problem from a control perspective, along with the control system architecture. Sections 3 and 4 are devoted to the explanation of the proposed control synthesis strategy, with particular emphasis on the choice of the class of controllers and the description of the optimization problem to be solved. Simulation results are presented in Section 5 using an End-to-End Simulator for the performance analysis of the First Light Adaptive Optics (FLAO) system of the LBT. Finally, concluding remarks are provided in Section 6.

2. AO control system architecture

The architecture of the AO control system we consider is depicted in Fig. 1, and is composed of an external AO control system (working at the AO sampling time Ts = 1 ms), whose task is to determine the commands to the ASM, and an internal ASM control system (with sampling time TASM ≈ 10−2 Ts). The principal devices characterizing the control loop are the AO controller, the Pyramid WFS, and the ASM. The AO controller receives at time kTs the measurement vector y(k) ∈ ℝq, with q the number of measurements, from the Pyramid WFS and computes the command vector u(k) ∈ ℝp, with 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 Ts. Such a control loop is realized through the use of 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

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]

].

Fig. 1 AO control scheme.

In Fig. 1 the phase aberration of the light wavefront due to turbulence and vibrations is denoted by ϕtot(k). Such a distortion has to be corrected by the shell deformations which provide the so called correction phase ϕcor(k). The difference ϕres(k) = ϕtot(k)−ϕcor(k) is the residual phase after the ASM correction. The objective of the external control loop therefore consists in regulating the residual phase ϕres(k) about 0.

The command vector provided to the ASM is computed by the AO controller as follows
U(z)=𝒦M(z)Y(z),
(3)
where 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). is the Modes-to-Commands (or Projection) matrix, represented by the From modes to commands block in Fig. 1, and 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 𝒦M(z) has dimension p × p and has to be designed in order to satisfy the control requirements. In the proposed setting, 𝒦M(z) is chosen as a diagonal matrix, 𝒦M(z) = diag{C1(z), ⋯ ,Cp(z)}, as the ideal goal would be to control the i–th mode by means of controller Ci(z).

Fig. 2 Modal control scheme.

3. Modal control design: a frequency based approach

Let 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.

The main objective of the AO system is to regulate the residual phase about zero by rejecting the disturbances acting on the system. To this end, a typical AO performance criterion to be minimized is the sampled-valued variance of the residual phase, defined as (see [18

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]

])
f(ρ)=limh1h+1k=0h|ϕtres(k)|2.
(6)
In fact, minimizing f(ρ) amounts to maximizing the Strehl-ratio (SR), which is the ratio of the maximum of the Point Spread Function (PSF) of the distorted image and the maximum of the theoretical diffraction limited image PSF [23

23. J. Herrmann, “Phase variance and Strehl ratio in adaptive optics,” J. Opt. Soc. Am. A, 9, 2257–2258 (1992). [CrossRef]

]. Let now ϒϕ(ω) and ϒw(ω) denote the PSDs of the signals ϕttot(k) and wt(k), respectively. Supposing ϕttot(k) and wt(k) mutually uncorrelated, by virtue of the Parseval’s relationship, one can rewrite f(ρ) in the frequency domain as follows
f(ρ)=12πππ(|S(ejω,ρ)|2ϒϕ(ω)+|T(ejω,ρ)|2ϒw(ω))dω,
(7)
where S(z,ρ), called the sensitivity function, is the transfer function mapping the phase aberration ϕttot(k) into the residual phase ϕtres(k), whereas T(z, ρ) is the transfer function mapping the measurement noise wt(k) into the residual phase ϕtres(k). With reference to Fig. 2, it is immediate to see that S(z,ρ) = (1 +C(z,ρ)P(z))−1 and T(z,ρ) = −M(z)C(z,ρ)(1 +C(z,ρ)P(z))−1.

In practice, it is convenient to approximate the integral in Eq. (7) with a finite sum by considering the samples at certain frequencies ω1, ω2,...,ωN of the PSDs of ϕttot(k) and wt(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
f(ρ)=1Nf=1N(|S(ejωf,ρ)|2ϒ^ϕ(ωf)+|T(ejωf,ρ)|2ϒ^w(ωf)),
(8)
where ϒ̂ϕ(ωf) and ϒ̂ϕ(ωf) denote estimates of the PSDs ϒϕ(ωf) and ϒϕ(ωf), respectively, at the frequency ωf.

In order to derive a controller 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]

]. In particular, let 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

25. J. Doyle, B. Francis, and A. Tannenbaum, Feedback Control Theory (Macmillan Publishing Company, 1992).

], Chapter 5)
𝒞(P)={C(z)=Q(z)1P(z)Q(z),Q(z)𝒮},
(9)
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

25. J. Doyle, B. Francis, and A. Tannenbaum, Feedback Control Theory (Macmillan Publishing Company, 1992).

], Chapter 5).

We point out that the transfer function 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
C(z,ρ)=Q(z,ρ)1P(z)Q(z,ρ).
(10)
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
S(z,ρ)=1P(z)Q(z,ρ),
(11)
T(z,ρ)=M(z)Q(z,ρ).
(12)
Hence, by expressing Q(z) as a linear combination of functions ψi(z), each one weighted by a parameter ρi, i.e.
Q(z,ρ)=ψT(z)ρ,
(13)
where ψ(z) = [ψ1(z)ψ2(z) ⋯ ψn(z)]T, and ρ = [ρ1 ρ2ρn]T, it is possible to make both S(z,ρ) and T(z,ρ) affine functions of the parameter vector ρ:
S(z,ρ)=1P(z)ψT(z)ρ,
(14)
T(z,ρ)=M(z)ψT(z)ρ.
(15)

The parameter vector 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:
f(ρ)=1Nf=1N{|(1P(ejωf)ψT(ejωf)ρ)|2ϒ^ϕ(ωf)+|M(ejωf)ψT(ejωf)ρ|2ϒ^w(ωf)}.
(17)
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.

Nevertheless, in many situations a controller should fulfill additional requirements beyond stabilization and performance. For instance, while observing an astronomical object, several circumstances may cause the command signal 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).

In this respect, common practice suggests that it is not advisable to make use of unstable controllers, especially if the plant itself is stable, as it is within our setting. In fact, if the feedback loop opens (generalizing, this could happen due to a sensor or actuator failing, or deliberately being turned off during start-up or shutdown), overall stability is maintained if both plant and controller individually are stable ([25

25. J. Doyle, B. Francis, and A. Tannenbaum, Feedback Control Theory (Macmillan Publishing Company, 1992).

], Chapter 5).

Clearly, the Youla parametrization does not ensure by itself that a generic controller of the form shown in Eq. (10) is stable, thus unconstrained minimization of 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
minρf(ρ),
(18)
s.t.
C(z,ρ)𝒮.
(19)

4. Modal control design: algorithm details

Solving an optimization problem is not a trivial numerical task, and requires that the constraints, as well as the objective function, are expressed in the simplest possible form. Unfortunately, the feasibility domain embodied by Eq. (19) is in general non convex and thus it is advisable to consider some kind of convex approximation. This can be done provided that a reference parameter vector ρ̂ is available corresponding to a stable controller C(z,ρ̂) (in practice ρ̂ 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
Re{(1P(ejω)Q(ejω,ρ^))(1P(ejω)Q(ejω,ρ))}>0,ω,
(20)
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,ρ̂). In view of the Nyquist stability criterion ([27

27. G. Goodwin, S. Graebe, and M. Salgado, Control System Design (Prentice Hall, 2001).

], Chapter 5), this ensures that C(z,ρ) and C(z,ρ̂) have the same stability property (recall Eq. (10)).

The condition given above is defined for all 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 {ω1, ω2,...ωN}. The resulting condition can be written in the form of linear constraints
A(ρ^)ρ+b(ρ^)<0
(21)
by conveniently defining matrix A and vector b. Specifically, matrix A takes the form
A(ρ^)=[a11(ρ^)a1n(ρ^)aN1(ρ^)aNn(ρ^)]
(22)
with elements afi(ρ̂), f = 1,...,N, i = 1,...,n computed as
afi(ρ^)=Re{(1P(ejωf)Q(ejωf,ρ^))P(ejωf)ψi(ejωf)},
(23)
and vector b(ρ̂) is defined as
b(ρ^)=[b1(ρ^)bN(ρ^)]T
(24)
with elements bf(ρ̂), f = 1,...,N computed as
bf(ρ^)=Re{1p(ejωf)Q(ejωf,ρ^)}.
(25)

The problem of dealing with an infinite number of frequency constraints (as in Eq. (20)) via a finite number of frequencies has been addressed in [26

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

] (Section 3.3), wherein guidelines on the choice of the sampled-frequency domain can be found.

Summing up, by replacing the original stability constraint in Eq. (19) with the linear constraints in Eq. (21), we obtain a quadratic programming problem which can be solved efficiently by means of several numerical tools. Typical algorithms for solving such a kind of problems are based on Active-Set, Sequential Quadratic Programming (SQM) or Interior-Point methods. In the proposed implementation, the Active-Set method has been adopted.

Herafter, we write the proposed synthesis procedure step by step and provide a schematic summary of the algorithm. As previously mentioned, the procedure starts with a pre-existing stable and stabilizing controller 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
Q^(z)=C^(z)1+P(z)C^(z)=n^(z)d^(z)
(26)
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) = d̂(z)zγ. Then, n is set equal to the degree of d(z) increased by one, while the reference parameter vector ρ̂ is set equal to the vector of the coefficients of the polynomial n̂(z)zγ. At this point, it is possible to minimize the performance criterion 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 ρ̂ as described above;
  • Step 2: set i := 0 and ρ(0) := ρ̂;
  • Step 3: compute the matrix A(ρ(i)) and the vector b(ρ(i));
  • Step 4: compute the parameter vector ρ(i+1) as the solution of the quadratic programming problem
    minρf(ρ)
    (27)
    s.t.
    A(ρ(i))ρ+b(ρ(i))<0;
    (28)
  • Step 5: if the termination criterion is met then return ρ(i+1); otherwise set i := i + 1 and go back to step 3.

As for the termination criterion, we decided to impose an upper bound on the difference between the value functions corresponding to subsequent iterations, thus terminating the algorithm when |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

In this section we provide the results obtained from simulation studies. The simulation results rely on an End-to-End simulator of the FLAO system. Table 1 presents a summary of the simulation parameters.

Table 1. Simulation parameters and conditions.

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Fig. 3 Turbulence and vibration PSD related to tip.

Simulation results reported hereafter consider two scenarios: in the first one, the proposed procedure is initialized from a non-model-based controller, 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 H2-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
C^(z)=gzz0.95,
(30)
with g such that the closed-loop is stable. By the procedure described in Section 4, Q̂(z) takes the form
Q^(z)=gz2z20.95z+g=n^(z)d^(z).
(31)
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) = d̂(z)zγ. Accordingly,
ψi(z)=zi1d(z)=zi1zγ+20.95zγ+1+gzγ,i=1,2,,γ+3
(32)
and the algorithm is then initialized by letting
ρ(0)=ρ^=[00γ+2g].
(33)
As for the simulation, we set 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%).

Fig. 4 Residual phase variance corresponding to controllers synthesized with increasing values of γ.

Second scenario: As initial controller, we now consider a H2-controller synthesized using mathematical models of turbulence and vibrations of reduced complexity. Specifically, we adopted the design procedure described in [8

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]

], with the following choices: the model of the turbulence is taken as a second-order AR model having input ν and output ξ, and described by the difference equation
ξ(k)=1.9774ξ(k1)0.9776ξ(k2)+ν(k).
(34)
where ν is a white-noise process. As for the structural vibrations, we considered two second-order ARMA models having input μi and output ζi, i = 1, 2. They are described by the difference equations
ζ1(k)=1.9917ζ1(k1)0.9984ζ1(k2)+1.8431μ1(k1)0.8493μ1(k2),
(35)
for the model having a vibration peak around 13Hz, and
ζ2(k)=1.9782ζ2(k1)0.9972ζ2(k2)+1.7418μ2(k1)0.7585μ2(k2),
(36)
for the model having a vibration peak around 22Hz, where μ1 and μ2 are both white-noise processes.

The order of the H2-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.

Fig. 5 PSF profiles resulting from the use of the controllers in the control loop.
Fig. 6 PSDs of the residual phase (related to tip/tilt).
Fig. 7 Bode diagram of the sensitivity function related to the controllers.

Table 2. Performance exhibited by the controllers considered in the simulation tests.

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A final point is worth mentioning. One sees from Table 2 that the best peformance (considering also the order of the final controller) is achieved when starting from a model-based controller synthesized in accordance with models of reduced complexity. Such a result, obtained without identifying mathematical turbulence and vibration models of increased complexity, reinforces the idea that the proposed approach is best viewed as cooperative with (rather than as alternative to) classical AO control design approaches.

6. Conclusions

Several extensions are possible: most notably, the problem formulation appears flexible enough so as to take into account additional requirements which can be expressed as linear constraints within the optimization problem. For example, it is possible to impose some bounds on the sensitivity functions in order to improve robustness under perturbed conditions, thus obtaining a mixed H2/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, 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]

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]

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]

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, 24(9), 2645–2660 (2007). [CrossRef]

6.

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]

7.

“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).

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]

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]

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]

12.

E. Marchetti, M. Le Louarn, C. Soenke, E. Fedrigo, P.-Y. Madec, and N. Hubin, “ERIS adaptive optics system design,” Proc. SPIE 8447, 84473M (2012).

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).

14.

M. Born and E. Wolf, Principles of Optics (Cambridge University Press, Cambridge, U.K, 1980).

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]

16.

E. Gendron and P. Lena, “Modal control optimization,” Astron. Astrophys. , 291, 337–347 (1994).

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]

19.

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]

20.

L. A. Poyneer and J.-P. Véran, “Optimal modal Fourier transform wavefront control,” J. Opt. Soc. Am. A, 22(2), 1515–1526 (2005). [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]

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]

23.

J. Herrmann, “Phase variance and Strehl ratio in adaptive optics,” J. Opt. Soc. Am. A, 9, 2257–2258 (1992). [CrossRef]

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]

25.

J. Doyle, B. Francis, and A. Tannenbaum, Feedback Control Theory (Macmillan Publishing Company, 1992).

26.

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

27.

G. Goodwin, S. Graebe, and M. Salgado, Control System Design (Prentice Hall, 2001).

28.

R. Ragazzoni, “Pupil plane wavefront sensing with an oscillating prism,” J. Mod. Optic. 43, 289–293 (1996). [CrossRef]

29.

C. Vérinaud, “On the nature of the measurements provided by a pyramid wave-front sensor,” Opt. Commun. 233, 27 – 38 (2004). [CrossRef]

30.

B. L. McGlamery, “Computer simulation studies of compensation of turbulence degraded images,” Proc. SPIE , 74(9), 225–233 (1976). [CrossRef]

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


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References

  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. Express14(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. Express16(1), 87–97 (2008). [CrossRef] [PubMed]
  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]
  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,24(9), 2645–2660 (2007). [CrossRef]
  6. 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]
  7. “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).
  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. Control17(3), 316–326 (2011). [CrossRef]
  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. Control17(3), 273–289 (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. SPIE7015, 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. SPIE7736, 773605 (2010). [CrossRef]
  12. E. Marchetti, M. Le Louarn, C. Soenke, E. Fedrigo, P.-Y. Madec, and N. Hubin, “ERIS adaptive optics system design,” Proc. SPIE8447, 84473M (2012).
  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. SPIE8447, 84471L (2012).
  14. M. Born and E. Wolf, Principles of Optics (Cambridge University Press, Cambridge, U.K, 1980).
  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]
  16. E. Gendron and P. Lena, “Modal control optimization,” Astron. Astrophys., 291, 337–347 (1994).
  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. SPIE5490, 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. Control17(3), 222–236 (2011). [CrossRef]
  19. 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]
  20. L. A. Poyneer and J.-P. Véran, “Optimal modal Fourier transform wavefront control,” J. Opt. Soc. Am. A,22(2), 1515–1526 (2005). [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]
  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. SPIE4839, 721–732 (2003). [CrossRef]
  23. J. Herrmann, “Phase variance and Strehl ratio in adaptive optics,” J. Opt. Soc. Am. A,9, 2257–2258 (1992). [CrossRef]
  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]
  25. J. Doyle, B. Francis, and A. Tannenbaum, Feedback Control Theory (Macmillan Publishing Company, 1992).
  26. A. Karimi and G. Galdos, “Fixed-order H∞ controller design for nonparametric models by convex optimization,” Automatica46, 1388–1394 (2010). [CrossRef]
  27. G. Goodwin, S. Graebe, and M. Salgado, Control System Design (Prentice Hall, 2001).
  28. R. Ragazzoni, “Pupil plane wavefront sensing with an oscillating prism,” J. Mod. Optic.43, 289–293 (1996). [CrossRef]
  29. C. Vérinaud, “On the nature of the measurements provided by a pyramid wave-front sensor,” Opt. Commun.233, 27 – 38 (2004). [CrossRef]
  30. B. L. McGlamery, “Computer simulation studies of compensation of turbulence degraded images,” Proc. SPIE, 74(9), 225–233 (1976). [CrossRef]

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