**4. Results from the adaptive optics setup**

For closing the loop of the AO setup, a stabilizing controller is required. In Ref. [9], an ℋ∞ controller has been synthesized which robustly stabilizes the AO system. In the past, in general, PI(D) controllers have been used which often were tuned by hand. It would go too far to present the method for controller synthesis. So, this is not exposed in this survey. Since the AO setup uses an RCP approach, it is not very time consuming to test other control schemes, as only the Simulink model has to be adjusted accordingly. Of course, the design of a stabilizing controller which is also robust against a set of uncertainties may be rather complicated and time consuming.

**Figure 17.** Application of a 10 Hz step disturbance while controller is switched on at =6s; controller = 1600 Hz and shwfs = 800 Hz [9].

To validate the applicability of the presented approaches, **Figure 17** shows some recorded data. The controller has been switched on at time instance 6000 ms. The disturbance is a 10 Hz rectangular offset that is applied to one actuator of DM1. As the actuator patterns of the DM2 and DM1 are not the same and, additionally, do not have the same number of actuators, the result is that multiple actuators are required for compensating the disturbance. A rectangular disturbance is ideally suited to visualize the power of the controller as the steady‐state error as well as the time required for compensating the disturbance may be analyzed.

The error value is obtained after the multiplication of the control matrix with the centroids (in **Figure 16** the signal after the "Eigen3‐Matrix‐Mult" block). The control matrix itself is the pseudo‐inverse of the actuator influence function [8, 9].

The dimension of the error value is the same as the number of actuators. Nevertheless, the error value itself does not give a direct insight on how the wavefront looks like. Therefore, **Figure 16** visualizes the reconstructed wavefront as a 3D surface. The respective error values are depicted in **Figure 18** separately. To calculate the Strehl values based on the reconstructed wavefront in **Figure 20**, the wavefront at time instance 6320.63 ms has been used as reference; thus, showing a Strehl value of exactly one. Both the three‐dimensional representation and the error value visualize that it took 3–4 frames to reject the disturbance.

uses an RCP approach, it is not very time consuming to test other control schemes, as only the Simulink model has to be adjusted accordingly. Of course, the design of a stabilizing controller which is also robust against a set of uncertainties may be rather complicated and time

**Figure 17.** Application of a 10 Hz step disturbance while controller is switched on at =6s;

To validate the applicability of the presented approaches, **Figure 17** shows some recorded data. The controller has been switched on at time instance 6000 ms. The disturbance is a 10 Hz rectangular offset that is applied to one actuator of DM1. As the actuator patterns of the DM2 and DM1 are not the same and, additionally, do not have the same number of actuators, the result is that multiple actuators are required for compensating the disturbance. A rectangular disturbance is ideally suited to visualize the power of the controller as the steady‐state error

The error value is obtained after the multiplication of the control matrix with the centroids (in **Figure 16** the signal after the "Eigen3‐Matrix‐Mult" block). The control matrix itself is the

The dimension of the error value is the same as the number of actuators. Nevertheless, the error value itself does not give a direct insight on how the wavefront looks like. Therefore, **Figure 16** visualizes the reconstructed wavefront as a 3D surface. The respective error values are depicted in **Figure 18** separately. To calculate the Strehl values based on the reconstructed wavefront in **Figure 20**, the wavefront at time instance 6320.63 ms has been used as reference;

as well as the time required for compensating the disturbance may be analyzed.

controller = 1600 Hz and shwfs = 800 Hz [9].

pseudo‐inverse of the actuator influence function [8, 9].

consuming.

190 Field - Programmable Gate Array

**Figure 18.** Experimental data with zoomed *x*‐axis to highlight the control behavior after a step disturbance, controller = 1600Hz and shwfs = 800Hz [9].

**Figure 19.** Captured image of the SHWFS, having a lenslet array of 14 × 14, during experiments.

Finally, **Figure 19** shows some captured camera image of the SHWFS. The colors have been adjusted for better visualization. The SHWFS has a lenslet array of 14 × 14 while the pixel area is 224 × 224 pixels. The standard approach cannot be used here as the spots are leaving the area on the image sensor. The area on the image sensor for each lens would be 16 × 16 pixels. However, the new approach has no problem and correctly assigns the spots to the lenses and thus correctly measures the wavefront (**Figure 20**).

**Figure 20.** Reconstructed wavefront, rejecting a step disturbance; same data as in **Figure 18** [9].
