# Closed-Loop Calibration of the Interaction Matrix for New Generation Adaptive Optics Systems

### A Least Cost Identification Approach

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## Abstract

Recently, with the switch to the new generation adaptive optics systems like the VLT and ELT an important identification problem emerged. These telescopes are so-called integrated systems, which means that open-loop calibration of the interaction matrix is not possible anymore. The first and foremost reason for this is that the new generation of integrated adaptive optics (AO) systems are much more sensitive to external factors like gravity causing misregistrations and system parameter changes. This sensitivity is due to the integrated design, because each time that the telescope will change from operation setup, such misregistrations will be created. Unfortunately, this can lead to performance loss or closed-loop instability very fast. Even misregistrations as big as 10% of a subaperture of the Shack-Hartmann wavefront sensor lead to significant performance loss.

For this reason, the identification and thereby correction for parameter changes in the deformable mirror (DM) have to be done during operation. Since the identification needs to be done during operation, it becomes much more important to have an identification method that does not disturb the closed-loop operation of the plant. Therefore, in this thesis, we will focus on a least cost method, preferably even a costless method for identification purposes.

Since the concept of integrated AO systems with a deformable secondary mirror (DSM) is still relatively new, there has been only one serious proposal for the solution of the problem by Béchet and Kolb. They falsely claim to have developed a non-parametric costless method, i.e. a method without excitation, for the calibration of the IM and achieved ”reasonable” results.

In our work, we will first of all prove why their method is fundamentally wrong. Furthermore, we will also prove that their approach will result in the identification of the inverse controller instead of the plant. This is probably also the reason why they falsely suppose to have achieved ”reasonable” results, because the dc gain of the inverse controller resembles the plant itself, which we will show as well.

After explanation and comparison of several methods, we will propose the recently developed least cost identification paradigm as the best solution. This paradigm is a framework for parametric prediction error identification and is meant to minimize the impact of the experiment on the underlying system in terms of experiment duration, distortion of the closed-loop operation, power of the input signal or a combination of these, while at the same time guaranteeing a predefined level of accuracy. Different bottlenecks that need to be overcome to apply this method will be taken into account in order to adapt the least cost identification paradigm for the calibration of the interaction matrix (IM).

Using the expression for the information matrix we will first of all derive and proof that, theoretically, and if the experiment duration time allows, unlimited accuracy is achievable with zero cost in closed-loop using the excitation coming from the disturbance.

Furthermore, we will prove in our study that, when the cost is solely defined as the impact of the excitation signal on the normal closed-loop operation, the information-to-cost ratio per frequency is inversely proportional with the squared magnitude of the disturbance H(ω), i.e. proportional with 1/|H(ω)|² . Therefore, this means that in our specific case, in which we try to find the static gain matrix for an underdamped second order resonance system with a low frequency disturbance originating from atmospheric turbulence, the Nyquist frequency will always be the optimal excitation frequency.