Paolo Massioni

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.

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). FDPCGs 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, DKFs 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 DKFs 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.

Vibration mitigation in adaptive optics control

Perturbations affecting image formation on ground-based telescopes are composed of signals that are not only generated by the atmosphere. They often include vibrations induced by wind excitation on the systems structure, or induced by other sources of excitation like cryo-coolers, shutters, etc. Using state-space control design techniques (e.g., LQG control), efficient perturbation compensation can be obtained in adaptive optics systems. This requires in return an accurate dynamical perturbation model with manageable complexity. The purpose of this paper is to investigate how tip/ tilt state-space models can be constructed and identified from wavefront sensor (WFS) measurements and used for tip/ tilt correction. Several off-the-shelf time-domain identification approaches are considered, ranging from techniques such as subspace identification to extended Kalman filter. Results are compared with controllers that do not account for vibrations, like an integrator or an MMSE reconstructor. Performance improvement is illustrated by replay with on-sky data sets from Gemini South (GeMS and Altair).

Lyapunov Stability Analysis of Switching Controllers in Presence of Sliding Modes and Parametric Uncertainties With Application to Pneumatic Systems

This paper concerns the control and the stability analysis of pneumatic actuators, which are nowadays of widespread use in the industry. A problem related to the use of such actuators is the so-called stick-slip, due to the presence of dry friction in the system. In this paper, we provide an empirical switching control law that avoids this phenomenon as well as a general approach to the stability analysis of nonlinear systems that will let us prove the stability of the closed-loop system. The approach is based on casting the closed-loop system into a piecewise-affine form and finding a Lyapunov function for it. Such an approach will be able to cope with the special features of the controlled pneumatic system model, namely, the presence of sliding modes, a whole equilibrium set, and uncertainties on the values of a few parameters. At the end of this paper, we will show how such a method can be successfully applied to our experimental setup under several hypotheses.

Approximating the Riccati Equation solution for optimal estimation in large-scale Adaptive Optics systems

Adaptive Optics (AO) is a technique that allows the compensation of the atmospheric turbulence effects on ground-based telescopes by means of an actively controlled deformable mirror (DMs), fed back based on the measurements obtained with one or more wavefront sensors (WFSs). For extremely large telescope (more than 20 m in diameter) the number of input and output channels can be in the range of the thousands or tens of thousands, making it problematic to apply optimal control solutions due to the heavy computational load. In this paper we show how it is possible to obtain a quick approximation of the solution of the Discrete Algebraic Riccati Equation (DARE) associated to a certain class of AO optimal control problems, and how the performance are affected by the use of such approximations.

Incremental ℒ 2 -gain analysis of piecewise-affine systems using piecewise quadratic storage functions

In this paper we are interested in the incremental stability of piecewise-affine (PWA) systems. We present conditions to compute an upper bound on the incremental ℒ2-gain based on dissipativity analysis. These conditions are expressed as linear matrix inequalities (LMI) allowing the construction of a continuous piecewise quadratic storage function. It is also shown that these conditions imply incremental asymptotic stability of the system. The result is illustrated with numerical examples.

Fast finite-horizon kalman filter in wavefront estimation for adaptive optics

We consider the problem of wavefront estimation for the control of adaptive optics systems. It has been shown that an improvement in the performance can be achieved by assuming a turbulence dynamical model which allows the use of a Kalman filter; nevertheless, the implementation of such filters has proven to be difficult in real-time large-scale systems due to memory, computational costs and latency issues. In this article we propose a possible work around of such problem, which consists in using the non-recursive expression of a finite-horizon Kalman filter, which only requires a matrix-vector multiplication at run-time. This allows the use of complex state-space models for the estimation, without the need of explicitly computing the state vector at each time step, yielding a significant gain in term of computational time.

Fast Off-Line Kalman Filter Gain Computation for Astronomical Adaptive Optics Systems

We introduce a new procedure for quickly approximating the Kalman gain for the optimal control of large astronomical adaptive optics systems. A computational simplification is obtained in Fourier domain by working on infinite-size phase screens.