Optimal control law for classical and multiconjugate adaptive optics

Fractional loop delays in adaptive optics modeling and control

This paper revisits the problem of optimal (minimum variance) control for adaptive optics (AO) systems when measurement and command applications are asynchronous, resulting in a non-integer servo loop delay. When not properly accounted for, such fractional delays may severely degrade the AO performance, especially in the presence of high-frequency vibrations. We present evidence of this performance degradation thanks to in-lab experimental measurements on the Gran Telescopio Canarias Adaptive Optics (GTCAO) system controlled with standard suboptimal linear quadratic Gaussian (LQG) controllers. A constructive, easy to implement LQG control design is then proposed and validated in a simulation for vibrations affecting the tip-tilt modes. Our methodology is very interesting because it allows a performance assessment for any linear controller in terms of variance, rejection transfer functions, power spectral densities, and stability margins. We also show how the continuous-time disturbance model can be derived from standard discrete-time disturbance data-based modeling.

Laboratory demonstration of the prediction of wind-blown turbulence by adaptive optics at 8 kHz with use of LQG control

The low-latency adaptive optical mirror system (LLAMAS) is designed to push the limits on achievable latencies and frame rates. It has 21 subapertures across its pupil. A reformulated version of the linear quadratic Gaussian (LQG) method predictive Fourier control is implemented in LLAMAS; for all modes, it takes just 30 µs to compute. In the testbed, a turbulator mixes hot and ambient air to produce wind-blown turbulence. Wind prediction clearly improves correction when compared to an integral controller. Closed-loop telemetry shows that wind-predictive LQG removes the characteristic "butterfly" and reduces temporal error power by up to a factor of three for mid-spatial frequency modes. Strehl changes seen in focal plane images are consistent with telemetry and the system error budget.

Adaptive optics control with multi-agent model-free reinforcement learning

We present a novel formulation of closed-loop adaptive optics (AO) control as a multi-agent reinforcement learning (MARL) problem in which the controller is able to learn a non-linear policy and does not need a priori information on the dynamics of the atmosphere. We identify the different challenges of applying a reinforcement learning (RL) method to AO and, to solve them, propose the combination of model-free MARL for control with an autoencoder neural network to mitigate the effect of noise. Moreover, we extend current existing methods of error budget analysis to include a RL controller. The experimental results for an 8m telescope equipped with a 40x40 Shack-Hartmann system show a significant increase in performance over the integrator baseline and comparable performance to a model-based predictive approach, a linear quadratic Gaussian controller with perfect knowledge of atmospheric conditions. Finally, the error budget analysis provides evidence that the RL controller is partially compensating for bandwidth error and is helping to mitigate the propagation of aliasing.

Sparse data-driven wavefront prediction for large-scale adaptive optics

This paper presents a computationally efficient wavefront aberration prediction framework for data-driven control in large-scale adaptive optics systems. Our novel prediction algorithm splits prediction into two stages: a high-resolution and a low-resolution stage. For the former, we exploit sparsity structures in the system matrices in a data-driven Kalman filtering algorithm and constrain the identified gain to be likewise sparse; for the latter, we identify a dense Kalman gain and perform corrections to the suboptimal predictions of the former on a smaller grid. This novel prediction framework is able to retain the robustness to measurement noise of the standard Kalman filter in a much more computationally efficient manner, in both its offline and online aspects, while minimally sacrificing performance; its data-driven nature further compensates for modeling errors. As an intermediate result, we present a sparsity-exploiting data-driven Kalman filtering algorithm able to quickly estimate an approximate Kalman gain without solving the Riccati equation.

Compensation Techniques Aimed at Mitigating Vibrations in Optical Ground-Based Telescopes: A Systematic Review

The objective of this study was to evaluate the performance of the different systems and techniques aimed at suppressing vibrations on optical ground-based telescopes. We identified the studies by searching three electronic databases (Science Direct, IEEE library and Web of Science) from the year 2000 to December 2020. The studies were eligible if they proposed systems focused on mitigating the effects of vibrations in optical telescopes and brought performance data. A total of nine studies met our eligibility criteria. Current evidence confirms the feasibility of adaptative optics (AO) systems based on closed-loop control to mitigate vibrations, although variations and additions should be made depending on their nature and characteristics in order to improve the performance of the proposed techniques. This systematic review was conducted to provide a state-of-the-art of the methods and techniques that have been developed over the past two decades. The review also points out some issues that demand future research.

Zonal-based high-performance control in adaptive optics systems with application to astronomy and satellite tracking

This paper presents a model-based approach to adaptive optics (AO) control based on a zonal (i.e., pixelized) representation of the incoming atmospheric turbulence. Describing the turbulence on a zonal basis enables the encapsulation of the standard frozen-flow assumption into a control-oriented model. A multilayer zonal model is proposed for single-conjugate AO (SCAO) systems. It includes an edge compensation mechanism involving limited support, which results in a sparser model structure. To further reduce the computational complexity, new resultant zonal models localized in the telescope pupil are proposed, with AR1 or AR2 structures, that match the spatial and temporal cross-correlations of the incoming turbulence. The global performance of the resulting linear quadratic Gaussian (LQG) regulator is evaluated using end-to-end simulations and compared to several existing controllers for two different configurations: a very large telescope SCAO and low earth orbit satellite tracking. The results show the high potential of the new approach and highlight possible trade-offs between the performance and complexity.

Identification of the dynamics of time-varying phase aberrations from time histories of the point-spread function

To optimally compensate for time-varying phase aberrations with adaptive optics, a model of the dynamics of the aberrations is required to predict the phase aberration at the next time step. We model the time-varying behavior of a phase aberration, expressed in Zernike modes, by assuming that the temporal dynamics of the Zernike coefficients can be described by a vector-valued autoregressive (VAR) model. We propose an iterative method based on a convex heuristic for a rank-constrained optimization problem, to jointly estimate the parameters of the VAR model and the Zernike coefficients from a time series of measurements of the point-spread function (PSF) of the optical system. By assuming the phase aberration is small, the relation between aberration and PSF measurements can be approximated by a quadratic function. As such, our method is a blind identification method for linear dynamics in a stochastic Wiener system with a quadratic nonlinearity at the output and a phase retrieval method that uses a time-evolution-model constraint and a single image at every time step.

Performance of the Gemini Planet Imager's adaptive optics system

The Gemini Planet Imager's adaptive optics (AO) subsystem was designed specifically to facilitate high-contrast imaging. A definitive description of the system's algorithms and technologies as built is given. 564 AO telemetry measurements from the Gemini Planet Imager Exoplanet Survey campaign are analyzed. The modal gain optimizer tracks changes in atmospheric conditions. Science observations show that image quality can be improved with the use of both the spatially filtered wavefront sensor and linear-quadratic-Gaussian control of vibration. The error budget indicates that for all targets and atmospheric conditions AO bandwidth error is the largest term.

First on-sky SCAO validation of full LQG control with vibration mitigation on the CANARY pathfinder

Adaptive optics provides real time correction of wavefront disturbances on ground based telescopes. Optimizing control and performance is a key issue for ever more demanding instruments on ever larger telescopes affected not only by atmospheric turbulence, but also by vibrations, windshake and tracking errors. Linear Quadratic Gaussian control achieves optimal correction when provided with a temporal model of the disturbance. We present in this paper the first on-sky results of a Kalman filter based LQG control with vibration mitigation on the CANARY instrument at the Nasmyth platform of the 4.2-m William Herschel Telescope. The results demonstrate a clear improvement of performance for full LQG compared with standard integrator control, and assess the additional improvement brought by vibration filtering with a tip-tilt model identified from on-sky data, thus validating the strategy retained on the instrument SPHERE at the VLT.

Local ensemble transform Kalman filter, a fast non-stationary control law for adaptive optics on ELTs: theoretical aspects and first simulation results

We propose a new algorithm for an adaptive optics system control law, based on the Linear Quadratic Gaussian approach and a Kalman Filter adaptation with localizations. It allows to handle non-stationary behaviors, to obtain performance close to the optimality defined with the residual phase variance minimization criterion, and to reduce the computational burden with an intrinsically parallel implementation on the Extremely Large Telescopes (ELTs).

Experimental study on modified linear quadratic Gaussian control for adaptive optics

To achieve high-resolution imaging the standard control algorithm used for classical adaptive optics (AO) is the simple but efficient proportional-integral (PI) controller. The goal is to minimize the rms error of the residual wave front. However, using the PI controller, it is not possible to do this. One possible way to minimize the rms error is to use linear quadratic Gaussian (LQG) control. In practice, however, this control algorithm still encounters an unexpected problem that leads to the divergence of control in AO. This paper proposes a modified LQG (MLQG) to solve this issue. The controller is analyzed explicitly. Laboratory tests shows strong stability and high precision compared to the classical control.

Static and predictive tomographic reconstruction for wide-field multi-object adaptive optics systems

Multi-object adaptive optics (MOAO) systems are still in their infancy: their complex optical designs for tomographic, wide-field wavefront sensing, coupled with open-loop (OL) correction, make their calibration a challenge. The correction of a discrete number of specific directions in the field allows for streamlined application of a general class of spatio-angular algorithms, initially proposed in Whiteley et al. [J. Opt. Soc. Am. A15, 2097 (1998)], which is compatible with partial on-line calibration. The recent Learn & Apply algorithm from Vidal et al. [J. Opt. Soc. Am. A27, A253 (2010)] can then be reinterpreted in a broader framework of tomographic algorithms and is shown to be a special case that exploits the particulars of OL and aperture-plane phase conjugation. An extension to embed a temporal prediction step to tackle sky-coverage limitations is discussed. The trade-off between lengthening the camera integration period, therefore increasing system lag error, and the resulting improvement in SNR can be shifted to higher guide-star magnitudes by introducing temporal prediction. The derivation of the optimal predictor and a comparison to suboptimal autoregressive models is provided using temporal structure functions. It is shown using end-to-end simulations of Raven, the MOAO science, and technology demonstrator for the 8 m Subaru telescope that prediction allows by itself the use of 1-magnitude-fainter guide stars.

Multiscale phase screen synthesis based on local principal component analysis

Motivated by the increasing importance of adaptive optics (AO) systems for improving the real resolution of large ground telescopes, and by the need of testing the AO system performance in realistic working conditions, in this paper we address the problem of simulating the turbulence effect on ground telescope observations at high resolution. The procedure presented here generalizes the multiscale stochastic approach introduced in our earlier paper [Appl. Opt. 50, 4124 (2011)], with respect to the previous solution, a relevant computational time reduction is obtained by exploiting a local spatial principal component analysis (PCA) representation of the turbulence. Furthermore, the turbulence at low resolution is modeled as a moving average (MA) process, while previously [Appl. Opt. 50, 4124 (2011)] the wind velocity was restricted to be directed along one of the two spatial axes, the use of such MA model allows the turbulence to evolve indifferently in all the directions. In our simulations, the proposed procedure reproduces the theoretical statistical characteristics of the turbulent phase with good accuracy.

Mitigation of vibrations in adaptive optics by minimization of closed-loop residuals

We describe a new technique to reduce tip and tilt vibrations via the design of adaptive optics controllers in a frequency framework. The method synthesizes controllers by minimizing an H2 norm of the tip and tilt residuals. In this approach, open loop slopes (pseudo-open-loop) are reconstructed from on-sky data and input into off-line simulations of the adaptive optics system. The proposed procedure executes a sequence of off-line closed-loop runs with increasing controller complexity and searches for the controller that minimizes the variance of residuals. Although the method avoids any identification of the vibration and turbulence models during the controller synthesis, the actual models are indirectly constructed as a by-product of the H2 norm minimization. The technique has been implemented on and tested with two operational instruments, namely Paranal's NACO and Gemini-South's GeMS, showing an effective rejection of the main vibrations in the loop and also improving the overall performance of the system over varying turbulence conditions. It is shown that a superior performance is obtained when compared to the standard integrator controller.

Distributed Kalman filtering compared to Fourier domain preconditioned conjugate gradient for laser guide star tomography on extremely large telescopes

This paper discusses the performance and cost of two computationally efficient Fourier-based tomographic wavefront reconstruction algorithms for wide-field laser guide star (LGS) adaptive optics (AO). The first algorithm is the iterative Fourier domain preconditioned conjugate gradient (FDPCG) algorithm developed by Yang et al. [Appl. Opt.45, 5281 (2006)], combined with pseudo-open-loop control (POLC). FDPCG's computational cost is proportional to N log(N), where N denotes the dimensionality of the tomography problem. The second algorithm is the distributed Kalman filter (DKF) developed by Massioni et al. [J. Opt. Soc. Am. A28, 2298 (2011)], which is a noniterative spatially invariant controller. When implemented in the Fourier domain, DKF's cost is also proportional to N log(N). Both algorithms are capable of estimating spatial frequency components of the residual phase beyond the wavefront sensor (WFS) cutoff frequency thanks to regularization, thereby reducing WFS spatial aliasing at the expense of more computations. We present performance and cost analyses for the LGS multiconjugate AO system under design for the Thirty Meter Telescope, as well as DKF's sensitivity to uncertainties in wind profile prior information. We found that, provided the wind profile is known to better than 10% wind speed accuracy and 20 deg wind direction accuracy, DKF, despite its spatial invariance assumptions, delivers a significantly reduced wavefront error compared to the static FDPCG minimum variance estimator combined with POLC. Due to its nonsequential nature and high degree of parallelism, DKF is particularly well suited for real-time implementation on inexpensive off-the-shelf graphics processing units.

Comparison of vibration mitigation controllers for adaptive optics systems

Vibrations are detrimental to the performance of modern adaptive optics (AO) systems. In this paper, we describe new methods tested to mitigate the vibrations encountered in some of the instruments of the Gemini South telescope. By implementing a spectral analysis of the slope measurements from several wavefront sensors and an imager, we can determine the frequencies and magnitude of these vibrations. We found a persistent vibration at 55 Hz with others occurring occasionally at 14 and 100 Hz. Two types of AO controllers were designed and implemented, Kalman and H∞, in the multiconjugate AO tip-tilt loop. The first results show a similar performance for these advanced controllers and a clear improvement in vibration rejection and overall performance over the classical integrator scheme. It is shown that the reduction in the standard deviation of the residual slopes (as measured by wavefront sensors) is highly dependent on turbulence, wind speed, and vibration conditions, ranging--in terms of slopes RMS value--from an almost negligible reduction for high speed wind to a factor of 5 for a combination of low wind and strong vibrations.

Fast computation of an optimal controller for large-scale adaptive optics

The linear quadratic Gaussian regulator provides the minimum-variance control solution for a linear time-invariant system. For adaptive optics (AO) applications, under the hypothesis of a deformable mirror with instantaneous response, such a controller boils down to a minimum-variance phase estimator (a Kalman filter) and a projection onto the mirror space. The Kalman filter gain can be computed by solving an algebraic Riccati matrix equation, whose computational complexity grows very quickly with the size of the telescope aperture. This "curse of dimensionality" makes the standard solvers for Riccati equations very slow in the case of extremely large telescopes. In this article, we propose a way of computing the Kalman gain for AO systems by means of an approximation that considers the turbulence phase screen as the cropped version of an infinite-size screen. We demonstrate the advantages of the methods for both off- and on-line computational time, and we evaluate its performance for classical AO as well as for wide-field tomographic AO with multiple natural guide stars. Simulation results are reported.

Generalized aliasing and its implications in modal gain optimization for multi-conjugate adaptive optics

The error of generalized aliasing associated with the limited sampling of the atmospheric turbulence volume due to the finite number of wavefront sensing directions in wide-field-of-view adaptive optics is formally defined. Following a modal approach, we extend the direct problem formulation of star-oriented multi-conjugate adaptive optics (MCAO) to model and quantify this error analytically. We show that the turbulence estimation with the least-squares reconstructor is subject to strong generalized aliasing, in particular affecting the badly seen modes, whereas with the minimum-mean-square-error reconstructor the estimation is little affected. Finally, we show that the application of modal gain optimization techniques in closed-loop MCAO systems is jeopardized by the generalized aliasing error.

Tomography approach for multi-object adaptive optics

Multi-object adaptive optics (MOAO) is a solution developed to perform a correction by adaptive optics (AO) in a science large field of view. As in many wide-field AO schemes, a tomographic reconstruction of the turbulence volume is required in order to compute the MOAO corrections to be applied in the dedicated directions of the observed very faint targets. The specificity of MOAO is the open-loop control of the deformable mirrors by a number of wavefront sensors (WFSs) that are coupled to bright guide stars in different directions. MOAO calls for new procedures both for the cross registration of all the channels and for the computation of the tomographic reconstructor. We propose a new approach, called "Learn and Apply (L&A)", that allows us to retrieve the tomographic reconstructor using the on-sky wavefront measurements from an MOAO instrument. This method is also used to calibrate the registrations between the off-axis wavefront sensors and the deformable mirrors placed in the science optical paths. We propose a procedure linking the WFSs in the different directions and measuring directly on-sky the required covariance matrices needed for the reconstructor. We present the theoretical expressions of the turbulence spatial covariance of wavefront slopes allowing one to derive any turbulent covariance matrix between two wavefront sensors. Finally, we discuss the convergence issue on the measured covariance matrices, we propose the fitting of the data based on the theoretical slope covariance using a reduced number of turbulence parameters, and we present the computation of a fully modeled reconstructor.

Kalman filtering to suppress spurious signals in adaptive optics control

In many scenarios, an adaptive optics (AO) control system operates in the presence of temporally non-white noise. We use a Kalman filter with a state space formulation that allows suppression of this colored noise, hence improving residual error over the case where the noise is assumed to be white. We demonstrate the effectiveness of this new filter in the case of the estimated Gemini Planet Imager tip-tilt environment, where there are both common-path and non-common-path vibrations. We discuss how this same framework can also be used to suppress spatial aliasing during predictive wavefront control assuming frozen flow in a low-order AO system without a spatially filtered wavefront sensor, and present experimental measurements from Altair that clearly reveal these aliased components.

Tip-tilt disturbance model identification for Kalman-based control scheme: application to XAO and ELT systems

Adaptive optics (AO) systems have to correct tip-tilt (TT) disturbances down to a fraction of the diffraction-limited spot. This becomes a key issue for very or extremely large telescopes affected by mechanical vibration peaks or wind shake effects. Linear quadratic Gaussian (LQG) control achieves optimal TT correction when provided with the temporal model of the disturbance. We propose a nonsupervised identification procedure that does not require any auxiliary system or loop opening and validate it on synthetic profile as well as on experimental data.

Fast reconstruction and prediction of frozen flow turbulence based on structured Kalman filtering

Efficient and optimal prediction of frozen flow turbulence using the complete observation history of the wavefront sensor is an important issue in adaptive optics for large ground-based telescopes. At least for the sake of error budgeting and algorithm performance, the evaluation of an accurate estimate of the optimal performance of a particular adaptive optics configuration is important. However, due to the large number of grid points, high sampling rates, and the non-rationality of the turbulence power spectral density, the computational complexity of the optimal predictor is huge. This paper shows how a structure in the frozen flow propagation can be exploited to obtain a state-space innovation model with a particular sparsity structure. This sparsity structure enables one to efficiently compute a structured Kalman filter. By simulation it is shown that the performance can be improved and the computational complexity can be reduced in comparison with auto-regressive predictors of low order.

Fast minimum variance wavefront reconstruction for extremely large telescopes

We present what we believe to be a new algorithm, FRactal Iterative Method (FRiM), aiming at the reconstruction of the optical wavefront from measurements provided by a wavefront sensor. As our application is adaptive optics on extremely large telescopes, our algorithm was designed with speed and best quality in mind. The latter is achieved thanks to a regularization that enforces prior statistics. To solve the regularized problem, we use the conjugate gradient method, which takes advantage of the sparsity of the wavefront sensor model matrix and avoids the storage and inversion of a huge matrix. The prior covariance matrix is, however, non-sparse, and we derive a fractal approximation to the Karhunen-Loève basis thanks to which the regularization by Kolmogorov statistics can be computed in O(N) operations, with N being the number of phase samples to estimate. Finally, we propose an effective preconditioning that also scales as O(N) and yields the solution in five to ten conjugate gradient iterations for any N. The resulting algorithm is therefore O(N). As an example, for a 128 x 128 Shack-Hartmann wavefront sensor, the FRiM appears to be more than 100 times faster than the classical vector-matrix multiplication method.

Wide field adaptive optics laboratory demonstration with closed-loop tomographic control

HOMER, the new bench developed at ONERA devoted to wide field adaptive optics (WFAO) laboratory research, has allowed the first experimental validations of multi-conjugate adaptive optics (MCAO) and laser tomography adaptive optics (LTAO) concepts with a linear quadratic Gaussian (LQG) control approach. Results obtained in LTAO in closed loop show the significant gain in performance brought by LQG control, which allows tomographic reconstruction. We present a calibration and model identification strategy. Experimental results are shown to be consistent with end-to-end simulations. These results are very encouraging and demonstrate robustness of performance with respect to inevitable experimental uncertainties. They represent a first step for the study of very large telescope (VLT) and extremely large telescopes (ELT) instruments.

On the optimal reconstruction and control of adaptive optical systems with mirror dynamics

In adaptive optics (AO) the deformable mirror (DM) dynamics are usually neglected because, in general, the DM can be considered infinitely fast. Such assumption may no longer apply for the upcoming Extremely Large Telescopes (ELTs) with DM that are several meters in diameter with slow and/or resonant responses. For such systems an important challenge is to design an optimal regulator minimizing the variance of the residual phase. In this contribution, the general optimal minimum-variance (MV) solution to the full dynamical reconstruction and control problem of AO systems (AOSs) is established. It can be looked upon as the parent solution from which simpler (used hitherto) suboptimal solutions can be derived as special cases. These include either partial DM-dynamics-free solutions or solutions derived from the static minimum-variance reconstruction (where both atmospheric disturbance and DM dynamics are neglected altogether). Based on a continuous stochastic model of the disturbance, a state-space approach is developed that yields a fully optimal MV solution in the form of a discrete-time linear-quadratic-Gaussian (LQG) regulator design. From this LQG standpoint, the control-oriented state-space model allows one to (1) derive the optimal state-feedback linear regulator and (2) evaluate the performance of both the optimal and the sub-optimal solutions. Performance results are given for weakly damped second-order oscillatory DMs with large-amplitude resonant responses, in conditions representative of an ELT AO system. The highly energetic optical disturbance caused on the tip/tilt (TT) modes by the wind buffeting is considered. Results show that resonant responses are correctly handled with the MV regulator developed here. The use of sub-optimal regulators results in prohibitive performance losses in terms of residual variance; in addition, the closed-loop system may become unstable for resonant frequencies in the range of interest.

Linear quadratic Gaussian control for adaptive optics and multiconjugate adaptive optics: experimental and numerical analysis

We present a comprehensive analysis of the linear quadratic Gaussian control approach applied to adaptive optics (AO) and multiconjugated AO (MCAO) based on numerical and experimental validations. The structure of the control law is presented and its main properties discussed. We then propose an extended experimental validation of this control law in AO and a simplified MCAO configuration. Performance is compared with end-to-end numerical simulations. Sensitivity of the performance regarding tuning parameters is tested. Finally, extension to full MCAO and laser tomographic AO (LTAO) through numerical simulation is presented and analyzed.

Experimental verification of the frozen flow atmospheric turbulence assumption with use of astronomical adaptive optics telemetry

We use closed-loop deformable mirror telemetry from Altair and Keck adaptive optics (AO) to determine whether atmospheric turbulence follows the frozen flow hypothesis. Using telemetry from AO systems, our algorithms (based on the predictive Fourier control framework) detect frozen flow >94% of the time. Usually one to three layers are detected. Between 20% and 40% of the total controllable phase power is due to frozen flow. Velocity vector RMS variability is less than 0.5 m/s (per axis) on 10-s intervals, indicating that the atmosphere is stable enough for predictive control to measure and adapt to prevailing atmospheric conditions before they change.

Comparison of minimum-norm maximum likelihood and maximum a posteriori wavefront reconstructions for large adaptive optics systems

The performances of various estimators for wavefront sensing applications such as adaptive optics (AO) are compared. Analytical expressions for the bias and variance terms in the mean squared error (MSE) are derived for the minimum-norm maximum likelihood (MNML) and the maximum a posteriori (MAP) reconstructors. The MAP estimator is analytically demonstrated to yield an optimal trade-off that reduces the MSE, hence leading to a better Strehl ratio. The implications for AO applications are quantified thanks to simulations on 8-m- and 42-m-class telescopes. We show that the MAP estimator can achieve twice as low MSE as MNML methods do. Large AO systems can thus benefit from the high quality of MAP reconstruction in O(n) operations, thanks to the fast fractal iterative method (FrIM) algorithm (Thiébaut and Tallon, submitted to J. Opt. Soc. Am. A).

Linear-quadratic-Gaussian control for adaptive optics systems using a hybrid model

This paper presents a linear-quadratic-Gaussian (LQG) design based on the equivalent discrete-time model of an adaptive optics (AO) system. The design model incorporates deformable mirror dynamics, an asynchronous wavefront sensor and zero-order hold operation, and a continuous-time model of the incident wavefront. Using the structure of the discrete-time model, the dimensions of the Riccati equations to be solved are reduced. The LQG controller is shown to improve AO system performance under several conditions.

Predictive wavefront control for adaptive optics with arbitrary control loop delays

We present a modification of the closed-loop state space model for adaptive optics control that allows delays that are a noninteger multiple of the system frame rate. We derive the new forms of the predictive Fourier control Kalman filters for arbitrary delays and show that they are linear combinations of the whole-frame delay terms. This structure of the controller is independent of the delay. System stability margins and residual error variance both transition gracefully between integer-frame delays.

Warm-started wavefront reconstruction for adaptive optics

Future extreme adaptive optics (ExAO) systems have been suggested with up to 10(5) sensors and actuators. We analyze the computational speed of iterative reconstruction algorithms for such large systems. We compare a total of 15 different scalable methods, including multigrid, preconditioned conjugate-gradient, and several new variants of these. Simulations on a 128x128 square sensor/actuator geometry using Taylor frozen-flow dynamics are carried out using both open-loop and closed-loop measurements, and algorithms are compared on a basis of the mean squared error and floating-point multiplications required. We also investigate the use of warm starting, where the most recent estimate is used to initialize the iterative scheme. In open-loop estimation or pseudo-open-loop control, warm starting provides a significant computational speedup; almost every algorithm tested converges in one iteration. In a standard closed-loop implementation, using a single iteration per time step, most algorithms give the minimum error even in cold start, and every algorithm gives the minimum error if warm started. The best algorithm is therefore the one with the smallest computational cost per iteration, not necessarily the one with the best quasi-static performance.

Exploiting the spatiotemporal correlation in adaptive optics using data-driven H2-optimal control

A recently proposed data-driven H2-optimal control approach is demonstrated on a laboratory setup. Most adaptive optics (AO) systems are based on a control law that neglects the temporal evolution of the wavefront. The proposed control approach is able to exploit the spatiotemporal correlation in the wavefront without assuming any form of decoupling. By analyzing the dynamic behavior of the wavefront sensor (WFS), it is shown that if the wavefront correction device can be considered static, the transfer function from control input to WFS output reduces to a two-tap impulse response and an integer number of samples delay. Considering this model structure, a data-driven identification procedure is developed to estimate the relevant parameters from measurement data. The specific structure allows for an analytical expression of the optimal controller in terms of the system matrices of the minimum-phase spectral factor of the atmospheric disturbance model. The performance of the optimal controller is compared with that of the standard AO control law. An analysis of the dominant error sources shows that optimal control may reduce the temporal error.

Optimal control, observers and integrators in adaptive optics

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.

Experimental results of ground-layer and tomographic wavefront reconstruction from multiple laser guide stars

We describe results from the first multi-laser wavefront sensing system designed to support tomographic modes of adaptive optics (AO). The system, now operating at the 6.5 m MMT telescope in Arizona, creates five beacons by Rayleigh scattering of laser beams at 532 nm integrated over a range from 20 to 29 km by dynamic refocus of the telescope optics. The return light is analyzed by a Shack-Hartmann sensor that places all five beacons on a single detector, with electronic shuttering to implement the beacon range gate. A separate high-order Shack-Hartmann sensor records simultaneous measurements of wavefronts from a natural star. From open-loop measurements, we find the average beacon wavefront gives a good estimate of ground layer aberration. We present results of full tomographic wavefront analysis, enabled by supplementing the laser data with simultaneous fast image motion measurements from three stars in the field. We describe plans for an early demonstration at the MMT of closed-loop ground layer AO, and later tomographic AO.

Minimum variance control structure for adaptive optics systems

The adaptive optics minimum variance control problem is formulated as a linear-quadratic-Gaussian optimization. The formulation incorporates the wavefront sensor frame integration in discrete-time models of the deformable mirror and incident wavefront. It shows that, under nearly ideal conditions, the resulting minimum variance controller approaches the integral controller commonly used in adaptive optics systems. The inputs to the controller dynamics are obtained from a reconstructor with the maximum a posteriori structure that uses the estimation error covariance of the wavefront error. The ideal conditions assumed to obtain the integral controller are as follows; isotropic first-order (but nonstationary) temporal atmospheric aberrations, no computational loop delay, and no deformable mirror dynamics. The effects of variations in these conditions are examined.

Scintillation and phase anisoplanatism in Shack-Hartmann wavefront sensing

Adaptive optics provides a real-time compensation for atmospheric turbulence that severely limits the resolution of ground-based observation systems. The correction quality relies on a key component, that is, the wavefront sensor (WFS). When observing extended sources, WFS precision is limited by anisoplanatism effects. Anisoplanatism induces a variation of the turbulent phase and of the collected flux in the field of view. We study the effect of this phase and scintillation anisoplanatism on wavefront analysis. An analytical expression of the error induced is given in the Rytov regime. The formalism is applied to a solar and an endoatmospheric observation. Scintillation effects are generally disregarded, especially in astronomical conditions. We shall prove that this approximation is not valid with extended objects.