Disturbance Modelling for Minimum Variance Control in Adaptive Optics Systems Using Wavefront Sensor Sampled-Data

Abstract

Modern large telescopes are built based on the effectiveness of adaptive optics systems in mitigating the detrimental effects of wavefront distortions on astronomical images. In astronomical adaptive optics systems, the main sources of wavefront distortions are atmospheric turbulence and mechanical vibrations that are induced by the wind or the instrumentation systems, such as fans and cooling pumps. The mitigation of wavefront distortions is typically attained via a control law that is based on an adequate and accurate model. In this paper, we develop a modelling technique based on continuous-time damped-oscillators and on the Whittle’s likelihood method to estimate the parameters of disturbance models from wavefront sensor time-domain sampled-data. On the other hand, when the model is not accurate, the performance of the minimum variance controller is affected. We show that our modelling and identification techniques not only allow for more accurate estimates, but also for better minimum variance control performance. We illustrate the benefits of our proposal via numerical simulations.

An Optimal Integral Controller for Adaptive Optics Systems

“…In AO systems, disturbances such as turbulence and vibrations are typically described using discrete-time second-order autoregressive (AR) models, as shown in [17,29,37]. However, in this study we consider an alternative approach utilizing a sum of continuous-time oscillators, as proposed in [22]. Consequently, the continuous-time model for these disturbances is described as follows:…”

“…Figure 2 represents an equivalent block diagram model for the AO system presented in Figure 1. As presented in [22], the AO disturbances in discrete time can be obtained by solving a Riccati equation [26,33,38] to compute the spectral factorization H(z) [33] via numerical approximation. Thus, the discrete-time model of the disturbances is as follows (for more details, see Appendix B and [22]):…”

“…In Figure 2, it can be observed that the plant is G(z) = M(z)D(z). Here, it is assumed that the plant is known and that the disturbances are identified using the identification method presented in [22].…”

“…The Whittle likelihood function depends on the discrete-time Power Spectral Density (PSD) of the disturbances Φ(e iω j ∆ ) and the periodogram of the sampled-data series of the measured ϕ Tot , I(e iω j ∆ ). This likelihood function for sampled data and finite data length was presented in [22], and is provided by…”

“…This work proposes a novel tuning procedure for integral controllers in astronomical AO systems. The proposed tuning procedure utilizes wavefront sensor measurements recorded over an interval of 3 s to estimate the disturbance model using the modeling method in [22], which represents the operational conditions of the adaptive optics system. This approach aims to minimize the output variance, which directly impacts the quality of astronomical images.…”