Dual-Update Data-Driven Control of Deformable Mirrors Using Walsh Basis Functions

Abstract

In this paper, we develop a novel data-driven method for Deformable Mirror (DM) control. The developed method updates both the DM model and DM control actions that produce desired mirror surface shapes. The novel method explicitly takes into account actuator constraints and couples a feedback control algorithm with an algorithm for recursive estimation of DM influence function models.In addition to this, we explore the possibility of using Walsh basis functions for DM control. By expressing the desired and observed mirror surface shapes as sums of Walsh pattern matrices, we formulate the control problem in the 2D Walsh basis domain. We thoroughly experimentally verify the developed approach on a 140-actuator MEMS DM, developed by Boston Micromachines. Our resultsshow that the novel method produces the root-mean-square surface error in the 14 − 40 nanometer range. These results can additionally be improved by tuning the control and estimation parameters. The developed approach is also applicable to other DM types, such as for example, segmented DMs. I. INTRODUCTION Deformable Mirrors (DMs) are one of the main components of Adaptive Optics (AO) systems [1], [2]. A typical DM consist of a reflective optical surface that is deformed by a set of actuators. By precisely shaping the surface of a DM, we can compensate for wave-front aberrations in AO systems [3]-[16].In this paper, we consider the problem of developing control algorithms for DMs. There are a large number of approaches for DM control. A complete survey of all the methods goes well beyond the length limits of this manuscript. Consequently, we briefly mention only the most

A novel method for adaptive control of deformable mirrors

“…The mirror has a pitch of 400 [µm] and a stroke of about 2 [µm]. For more information about the used DM, see 18,43 and references therein. The mirror surface shape is observed by using the Partitioned Aperture Wavefront (PAW) sensor.…”

“…That is, we need to estimate the initial values of the influence matrix parameters. We use a multivariable least-squares method explained in 18,30,43 to estimate these parameters. For the time being, we assume that over a discretetime horizon k = 0, 1, 2, .…”

“…Furthermore, the adaptive control method proposed in 41 works completely in closed-loop, and consequently, special care needs to be dedicated to proper parameter tuning and the problems of persistency of excitation and estimation consistency. 42 Recently, in 18,43 we have proposed adaptive control algorithms for DM control. The approach presented in 18 is combining an open-loop control approach with a batch-data multivariable least-square problem for dynamically updating the DM influence matrix.…”

“…Consequently, since the control process is performed in open loop and since only the parameters of the recursive least-squares method need to be tuned, the tuning of this method is simpler compared to the tuning of the adaptive feedback control approach proposed in. 43 On the other hand, the advantage of the developed method over the method proposed in, 18 is that it requires fewer samples to converge. This is because the adaptive estimation of the influence matrix is performed by using the recursive least-squares method, instead of using the batch-data multivariable least-squares method.…”