3.1.

Integral Control Law

Operating at 1 kHz with 21 actuator across the aperture in H-band allows for the PyWFS to achieve good performance with the standard AO control law: a leaky integrator. In Eq. (1), the leaky integrator calculates the DM commands, $V^{\mathrm{int}}(t)$, at time step $t$:

3.2.

Predictive Control

Using the separation principle, our predictive controller is implemented in two steps: (1) predict the input disturbance (i.e., wavefront error due to atmospheric turbulence above the telescope) and (2) control the system plant (i.e., DM). Within this flexible framework, we can implement different predictive algorithms. We are also able to pick various control laws including open-loop (OL) control or the already existing leaky integrator implemented for the PyWFS.

In this work, we predict the DM commands one lag time into the future by finding a filter that uses a subset of previous measurements subject to some cost function. We focus on data-driven approaches where we use recent measurements to determine the future state of the input disturbance. With that, we rely on spatial and temporal relationships of the data that not only allow for us to predict but also help reduce other error terms in the system that is present in the data. Since we aim to predict the full input disturbance (e.g., the wavefront due to turbulence), we must first reconstruct the pseudo-open loop (POL) wavefront as the PyWFS is downstream from the DM measuring the residual wavefront error.

3.3.

Residual Wavefront Error

To determine the performance of the AO system, we look at the RMS calculated from the residual (close-loop; CL) reconstructed wavefronts. Note we use the RMS as an intermediate performance metric and then use the measured contrast as our final metric. We make use of two different CL wavefront reconstructed values: filtered and full CL wavefronts. For both cases, the CL values (i.e., residuals) are calculated in DM units of voltages and then a conversion factor of $0.6\frac{\mu \mathrm{m}}{\mathrm{V}}$ is applied to convert between DM voltages and OPD/wavefront.

The filtered CL DM commands are those reconstructed in real-time by the RTC from the residual/CL PyWFS slopes. Modal filtering is embedded in the CM and so the resulting residual is filtered depending on the applied modal gains and mode cut-off for the leaky integrator. In postprocessing, the saved AO telemetry is used to reconstruct the full CL DM commands using the full command matrix without any modal filtering. The filtered and full wavefront residuals are then computed from their respective CL DM commands using the conversion factor and finally the filtered and full RMS wavefront errors are computed represented by $C{L}_{\text{filt}}=CM\times [S(t)-S_{\mathrm{ref}}]\times 0.6$ and $C{L}_{\text{full}}=C{M}_{\text{full}}\times [S(t)-S_{\mathrm{ref}}]\times 0.6$, respectively.

5.1.

Night of June 20, 2021

We were able to close the AO loop on J21095901+2859392 - a bright star with an H-band magnitude of 4.24. Issues with the secondary mirror of the telescope and the suboptimal performance of the AO system resulted in low SR and the decision to take noncornographic images. These issues mean the results from June 20, 2021, cannot be used conclusively to demonstrate the performance of predictive control compared to the integrator. However, the night allowed us to verify the functionality of our implementation and observe the overall behavior of the predictive controller. We note that during our actual tests no seeing data was available from the Canada-France-Hawaii Telescope’s (CFHT) MASS and DIMM instruments.³⁴

In Fig. 4, we show the filtered RMS wavefront error (i.e., $C{L}_{\text{filt}}$) for the entire night. We also show the SR as measured from NIRC2 images. Studying the time series of the RMS, we can see the improvements provided by prediction as the RMS becomes smaller when predictive control is turned on (shaded orange regions; it is helpful to track the minimum RMS values to see the performance difference and not the maximum RMS values that are contaminated by telescope offloading, DM saturation, and other effects). The time series also reveals the effects of all the changes made to the AO controller while observing. The continued improvement of prediction in a shaded region (i.e., between times 14:40 and 14:45) is a result of the mixing factor and integrator gain being adjusted to find the optimal configuration. During this observation period, we also tested the effects of changing the assumed frame lag in the POL step. Figure 5 shows the results from a lag of 2 frames. With a lag of 1, we are able to improve the performance, but the histogram is skewed with a tail tending toward larger RMS values. This tail is removed when a lag of 2 is used by the predictor. We also confirm the lag of 1.7 frames as determined by Cetre et al..³⁰ We will investigate implementing a noninteger lag in the near future for our predictive control filter for further improvement. Furthermore, the lag of 2 histogram shows a greater improvement in RMS wavefront error. Comparing the NIRC2 SR for the integrator and predictor with lag 2 in Fig. 6, we see a clear shift toward larger SR values when predictive control is turned on. While we do find that a lag of 2 is better for the predictor and the predictor performs better than the integrator, it is difficult to draw concrete conclusions on the provided improvement due to the lack of seeing data and the secondary mirror issues.

Fig. 4

(a) A time series RMS wavefront error ($C{L}_{\text{filt}}$) for the entire testing period taken starting the night of June 20, 2021. Throughout the night different parameters are changed including the lag and the gain, which is changed continuously during prediction to maximize (although it sometimes degrades) the performance of the predictor. Looking at the minimum RMS values will help the reader to see the performance differences as the spikes occur from telescope offloading and DM saturation. (b) The SR as measured from the NIRC2 images in Brackett Gamma. The shaded orange region indicates when prediction was turned on.

Fig. 5

Analysis of night time tests on June 20, 2021. (a) Data for 1 frame lag and (b) for 2 frame lag predictor compared to the integrator. The integrator data were taken before and after prediction was turned on and off. The probability is the counts per bin normalized to the total counts in each dataset such that the sum of all the bins in each histogram is 1.

Fig. 6

The histogram shows the SR in Brackett Gamma for predictive control with a lag of 2 and for the leaky integrator for data taken on June 20, 2021. The solid lines indicate the kernel densities for the data.

5.2.

July 27, 2021

On the night of July 27, 2021, we received a half night of engineering time for predictive wavefront control tests during which the performance of the AO system was typical. In this section, we present the results for one of our targets, HIP 117578. With an H-band magnitude of 6.1, HIP 117578 is bright enough to take advantage of the 1 kHz frame rate of the PyWFS camera. We took NIRC2 images with the L-band coronagraph in place and made use of QACITS to keep the point-spread-function (PSF) centered on the vortex coronagraph.³⁵

Figure 7 provides an overview of the observations with the RMS wavefront error being plotted as a function of time for both the full and filtered RMS values (i.e., $C{L}_{\text{full}}$ and $C{L}_{\text{filt}}$). We also plot the seeing as measured by the DIMM from CFHT. The figure indicates four data sets that were taken for this target. For each data set, we configured the AO system as desired (see Table 1) and then started the QACITS sequence, which first runs an optimization sequence to center the vortex and then takes a predetermined number of science images. Half way through this set of science images, we switch from the integrator controller to the predictive controller. These full QACITS sequences, which include NIRC2 frames taken using the integrator followed by predictive control, constitute a “dataset” (Table 1). The NIRC2 images were all taken with an exposure time of 0.3 s and 90 coadds at the full frame of 1024-by-1024 pixels. When we swap controllers, one to two NIRC2 frames are contaminated and hence dropped from this analysis. Therefore, we compare the predictor to integrator data that was taken just before the predictor data set. In Fig. 8, we plot the histogram of the RMS wavefront error for the four different data sets once again for the filtered and full RMS values. These histograms only include data that were taken during the NIRC2 exposures that are used for our contrast analysis. The ratio of the medians and standard deviations (std) of the histogram for each dataset are presented in Table 1 for the filtered and full RMS values. From these values, we see that the predictor provides an improvement in median wavefront error for all cases when looking at the filtered wavefront error, which makes sense as the LMMSE aims to minimize the residual wavefront error and we trained on the filtered wavefront. For the full wavefront, the predictor’s median is worse than that of the integrator for dataset 3. The standard deviation for dataset 2 using both the filtered and full residuals wavefronts is improved by the predictor, which indicates that the predictor provided a more stable AO correction than the integrator during this time.

Fig. 7

(a) $C{L}_{\text{full}}$ and (b) $C{L}_{\text{filt}}$ time series for observations for data taken on July 27, 2021, of HIP 117578 with the shaded regions indicating when prediction was turned on. The seeing in both panels is the DIMM as measured by CFHT.³⁴ We indicate four different datasets that were taken with various configurations as outlined in Table 1.

Table 1

The ratios of the median (med) and standard deviation (std) of CLfilt and CLfull (indicated by “filter” and “full,” respectively) for each data set along with the ratio of contrast (C, contrast gain) at a separation of 3 λ/D. For each measured performance metric a value greater than 1 indicates the predictor is performing better than the integrator. The controller settings for each of the data sets are shown and the number of modes controlled for all data sets was 300 modes with the LO gain being set to 1. For the NIRC2 images, the exposure was set to 0.3 s and the co-adds to 90 for all images. The temporal order (Order) and CoF are also given for each data set. These values correspond to the histograms shown in Fig. 8 showing data taken the night of July 27, 2021.

Fig. 8

Probability (i.e., counts normalized by total number of data points) for each of the different data sets separated by controller type. The first two rows are for the $C{L}_{\text{full}}$ data and the bottom two rows are for the $C{L}_{\text{filt}}$ data. We compare the predictor with the integrator data taken just before the predictor is turned on. The different parameters for the data sets taken on the night of July 27, 2021, are provided in Table 1.

We present the contrast curves for the four different data sets taken while observing HIP 117578 in Fig. 9. The NIRC2 data were first bad pixel, flat-field, and sky corrected, and registered, using the automated pipeline described in Ref. 26. We used VIP to derotate, median subtract, and median combine the data, modifying the number of images in each dataset such that the integrator and predictive components had the same total parallactic rotation. We then used VIP to create student-t corrected, algorithmic throughput corrected contrast curves, as detailed in Ref. 33. From the contrast curves, we find a contrast gain of 2 for dataset 2 at a separation of $3\lambda /D$ as reported in Table 1. In all of the datasets, however, we see an improvement in contrast located at different separations, which can be seen in Fig. 9. Figure 15 in Appendix A shows the contrast gain (ratio of the integrator and predictor contrast curves) for the four data sets, clearly showing the achieved improvement with predictive control. In data set 2, we see contrast gains close to 3 at separations of $4-6\lambda /D$ while we see more modest gains around 1.5 for data sets 0 and 3. From these contrast curves, we have repeatable improvement in contrast with EOF predictive control take over a period of 1.5 h while the only time we do not have a contrast improvement (dataset 3) we made a significant change to order of the predictive filter, which we had not previously tested. We also plot the postprocessed corongraphic PSFs for dataset 4 in Fig 10, further illustrating the spatial improvement provided by the predictor.

Fig. 9

Angular differential imaging contrast curves for HIP 117578 taken on the night of July 27, 2021, for the four different datasets as outline in Table 1.

Fig. 10

Postprocessed coronagraphic PSFs for the integrator and predictor corresponding to the contrast curves in Fig. 9 for dataset 2 taken the night of July 27, 2021, for HIP 117578.

Studying all the data, it appears that the ratio of the full std of the RMS wavefront error is the best indicator for contrast improvement as shown in Table 1. When the full std ratio is less than 1 the contrast is worse, when it is equal to 1 we have slight improvement and for larger than 1 we have the greatest contrast improvement.

5.3.

October 25, 2021

We did more testing with EOF predictive control during science observations on the night of October 25, 2021, for which we had typical AO performance. Not only were we able to demonstrate an improvement in residual wavefront error again but also an improvement in contrast. Figure 11 shows the ADI median subtracted contrast curve for HIP 25645, which has an H-band magnitude of 6.77. The contrast is improved between 5 and $8\lambda /D$, which is in agreement with the results from July and also close between 2 and $3\lambda /D$. For these observations, we corrected 212 modes with an HO and integrator gain of 0.28 and 0.4, respectively. These parameters were set by Keck personnel following standard AO operations. A mixing factor of 0.5 and gain of 0.23 were used for predictive control.

Fig. 11

Angular differential imaging contrast curves for HIP25645 taken on the night of October 25, 2021.

5.4.

Summary of Controller Performances

The RMS wavefront error results from our latter two nights (July and October for which we had good AO performance) are summarized in Fig. 12. We plot the seeing against the filtered residual RMS wavefront error. We take the CFHT seeing value and resample the RMS wavefront error to every 10 s to get a mean correction and then take the mean RMS wavefront values within 30 s of the CFHT seeing value. We make use of all the available predictor data that was taken. The upper and side subplots show the kernel density functions. The plots show that distribution of seeing values for all the data is roughly the same for both controllers (with the seeing actually being better in some cases for the October integrator than the predictor) on the two nights, and the distribution in residual wavefront error is different for the two controllers over both nights. This indicates that the difference in the predictor and integrator is not due to a change in seeing. The predictor on both nights shows a consistent reduction in wavefront error compared to the leaky integrator as illustrated by the scatter plots being shifted to the left for the predictor datasets compared to the two integrator datasets. There is also a clear reduction in the spread of wavefront errors for predictive control showing that for different conditions the predictive controller has similar performance resulting in a more reliable and consistent performance. This is in agreement with what was found by van Kooten et al.¹¹

Fig. 12

The seeing and $C{L}_{\text{filt}}$ for the predictor and integrator data for the two nights of July 27, and October 25, 2021. The side panels show the normalized kernel density plots for the seeing (right) and the $C{L}_{\text{filt}}$ (top).

6.1.

Performance Metric

We compare, in this paper, the residual wavefront error and raw contrast of predictive control to the current controller performance at Keck, the leaky integrator. More specifically, we define an improvement compared with the integrator as setup by Keck personnel such as the telescope operator, observing assistant, or staff astronomer who follow guidelines to provide robust, good AO performance depending on the conditions. This protocol was strictly followed for the October 25 night while loosely followed for observations in June and July (e.g., we manually increased the number of modes such that the integrator had the best performance). During June and July, we altered the leaky integrator to perform even better than what was suggested by the operator by mainly correcting for more modes than recommended. In the end, our implementation of predictive control on Keck aims to produce the best performance possible with the current system through a data-drive approach. Therefore, our chosen metric is the current performance of the system.

Under the assumption that the servo-lag error is a large contributor to the wavefront error and dominating the contrast at small angular separations on NIRC2,²⁶ we implemented a data-driven predictor, EOF, to reduce the temporal wavefront error. Since the real goal is to improve the contrast at small angular separations, we, therefore, attribute an improvement in contrast to a reduction in the servo-lag error. Our method, however, relies on both spatial and temporal correlations; therefore, we are also able to correct for other sources of errors in the system (i.e., calibration and registration alignment), making this an attractive method in general for improving the overall performance of an AO system.

6.2.

Discussion of Results

With the our predictive filter in place and demonstration of contrast improvement over a couple of nights and reduction in RMS wavefront error over several nights, we can transition toward making a comprehensive review of optimum parameters (see Sec. 7).

We began such testing in July 2021 where we aggressively varied the HO gain between 0.3 and 0, where 0 value indicates that we are not controlling any HO modes. It is when we are not controlling the HO gains (dataset 2) that we achieved the largest improvement in contrast. Comparing dataset 1 and 2 where the HO gain goes from 0.3 to 0, the contrast curves in Fig. 9 show the integrator is able to perform slightly better at larger separations than the predictor due to HO modes that are being controlled. Comparing then to the data taken in October shown in Fig. 11, where the HO gain value is similar to dataset 1, we are able to perform better with the predictor. We note that the number of total corrected modes are less in October as the number of modes needed to be lowered to provide a stable correction for the leaky integrator. Therefore, at larger separations, it is difficult to know what HO and number modes to use for the best improvement with prediction as it is clearly driven by external factors such as the atmospheric conditions. Close-in to the star, however, the predictor performs better regardless of the HO gains in those datasets. The transition separation between these two regimes appears to be around $4\lambda /D$.

Finally, we point out the behavior of the filtered and full residual wavefront error as shown in Fig. 7. In the bottom panel, we plot $C{L}_{\text{filt}}$, which is used to generate the POL wavefront error that is used to generate the predictive filter by minimizing the mean square error. As we change various parameters, the $C{L}_{\text{filt}}$ varies in clear explainable way where, for example, by increasing the $g$ from dataset 0 and dataset 1 decreases the residual wavefront error. In the top panel of Fig. 7, we plot $C{L}_{\text{full}}$ where we include the HO modes that we are not controlling into the residual wavefront error. Between datasets it varies in the opposite way from $C{L}_{\text{filt}}$. Again taking dataset 0 and dataset 1, we are increasing the gain along with the HO gain such that we are improving the residual wavefront error for the controlled modes. We, however, appear to be increasing the full residual wavefront error. Testing along with end-to-end simulations are needed to better understand this behavior starting with the leaky integrator.

6.3.

Power Spectral Density Analysis

In Fig. 13, we present the estimated power spectral densities (PSDs) for the on-sky July data. From the POL PSDs, we see that temporally the optical turbulence did not vary significantly between datasets and swapping of the controller. In closed-loop, the predictor reduces the wavefront error at small frequencies while also reducing the peak near 60 Hz. While the predictor PSD is more flat compared to the integrator, it also indicates that a greater improvement in performance is still possible by removing the peak around 60 Hz.

Fig. 13

PSDs calculated using the Welch method for the $C{L}_{\text{filt}}$ (indicated by CL) and the corresponding POL integrator and predictor data sets. The four different line styles correspond to the different data sets taken on the night of July 27, 2021, and outlined in Table 1 with the solid line corresponding to data set 2 (which agrees with two other data sets perfectly).

To further improve the system and the performance of the AO system, we must understand where different features of the PSDs originate from. In the CL PSD, the peak at near 60 Hz has the same amplitude as the low frequencies suggesting that this peak is now a main source of the residual wavefront error. The frequency of the peak is similar to one seen in a tip/tilt PSD indicating that both peaks might have the same source but further investigation is needed. Since EOF trains on the POL, the peak is a weak signal compared with the turbulence such that increasing the temporal order does not improve the performance. Therefore, applying EOF to the closed-loop coefficients might help to remove this feature. We outline the steps necessary in Sec. 7.

6.4.

Comparing EOF to Other Controllers

We outlined above our performance metric used in this paper. Understanding the performance of predictive control in comparison to other proposed controllers is outside the scope of this paper. We instead outline work already underway and provide comments on EOF and RLMMSE as well as gain optimization.

Comparing EOF to other controllers is underway in a laboratory setting at the University of California Santa Cruz on the SEAL test bench.³⁶ We will compare predictive control not only to other types of predictive controllers but also to other controllers such as modal gain control. In our own simulations and with telemetry data, the EOF and the RLMMSE algorithms have the same performance when the appropriate regressors and training data are selected. Even with fewer regressors, the RLMMSE can achieve the same performance as shown in Fig. 14 where two different RLMMSE results are shown to have the same performance as EOF. The regressors and amount of training data influence the achievable performance and contribute to over- and underfitting to the data. In the case where we use the most recent measurement only (i.e., temporal order 1) and a spatial order of one (i.e., temporal regressors only), the RLMMSE solution is equivalent to the optimal gain for the integrator for each actuator. If we change our basis set to modes, we will have the optimal gain for each mode. Adding more regressors, depending on the system setup, might not result in better performance and in the case where the lag is one frame, more regressors might result in overfitting. Therefore, care must be taken when setting up the predictive filter. Similarly modal gain optimization relies on POL reconstruction that incorporates the temporal lag. The optimal gains are found by solving a specific cost function and while the optimal gain is not a predictive algorithm, it improves the temporal performance by maximizing the correction bandwidth. The performance of the optimal gain optimization and a data-driven predictor might be indistinguishable especially when the lag is 1 frame and/or for nonstationary input disturbances. The data, regressors, optical setup of the system, and the performance metric will all contribute to how different modal gain and predictive control methods will behave.

Fig. 14

EOF on 30 s of Keck AO telemetry compared to two versions of RLMMSE that use no spatial regressors and have the same temporal order (s1t10) and another that has five temporal orders and spatial order of a $3\times 3$ grid around each actuator so that it correlates to its neighbors (s3t5). With EOF, we train the data in advance on an earlier 30 s of the data while the RLMMSE trains online. The RLMMSE has many fewer regressors allowing it to converge quickly. We also plot the true OL and the Keck integrator (integrator).