Mitigation of vibrations in adaptive optics by minimization of closed-loop residuals

Andres Guesalaga; Benoit Neichel; Jared O’Neal; Dani Guzman

doi:10.1364/OE.21.010676

1. Introduction

Adaptive Optics (AO) is a mature technology that has been successfully implemented to reduce the degrading effects of the Earth’s atmosphere on optical astronomical observations. Almost all modern large telescopes are equipped with AO and it will also be an integral component in the next generation of Extremely Large Telescopes. As AO systems effectively correct the atmospheric turbulence, other factors such as vibrations induced by the instruments or from the telescope become increasingly important [1,2].

Vibrations can originate from effects such as wind-shaking of the telescope structure, moving elements in instruments (e.g. fans, cryo-cooler and motors), or even telescope tracking errors. Identifying vibration sources can be difficult and usually requires extensive measurements and equipment such as accelerometers or dedicated wave-front sensors (WFS). Moreover, implementation of mechanical damping systems can be challenging or impossible. Recent studies suggest the use of control techniques in the AO loop as an alternative method to bypass these difficulties and suppress these perturbations [3]. In other words, the same AO system that corrects for atmospheric turbulence can also compensate for vibrations. These techniques have been successfully implemented and tested at the laboratory level [4] and have recently been introduced in operational systems [5,6]. For the future Extremely Large Telescopes, advanced controllers are considered as the baseline for vibration rejection [5–8].

While most of the recent research effort to cope with vibrations has concentrated on LQG control laws [9,10], an alternative approach using H_∞ synthesis methods have been proposed [11], where a comparison of the classical integrator, LQG and H_∞ controllers is carried out and the following general conclusions reached:

i) Advanced controllers (LQG & H_∞) perform similarly. This is not surprising since both techniques use models for vibrations and turbulence and the design of the controllers is based on the minimization of the closed-loop residuals
ii) Although frequency based approaches (H_∞ / H₂) and LQG solutions are equivalent provided that dynamics of the mirror and resonances can be introduced in the problem formulation, the main difference between these two approaches lies in the way these dynamical modes are introduced in the modeling. In [12] a method of including mirror dynamics has been proposed, but its implementation in an unsupervised procedure like the one presented in this article is not clear.
iii) Compared to the standard integrator, substantial reductions (20 to 30% for the cited work) in the variance seems plausible using advanced controllers such as LQG or H_∞, but only if they are correctly matched to the actual disturbances
iv) When large model errors are present in these techniques, the gains in performance provided by the advanced controllers are lost, showing no clear difference with the classical integrator. Hence, a regular updating of the controller is mandatory

This emphasizes the need for on-line identification or adjustment procedures that would ensure the best possible performance. We think that together with standard identification tools, the variance of the slope residuals can be used as a minimization index to tune the controllers based on two facts:

(i) In spite of identifying the disturbance models correctly, very often the power spectral density (PSD) of the AO loop residuals differs from the expected flat response of LQG, H₂ or H_∞ methods. This is due not only to the varying characteristics of the disturbance, but also to unmodeled dynamics or non-linearities in the AO components.
(ii) According to Bode’s theorem and H₂/H_∞ theory, imbalances in the closed-loop residual spectrum will drive the performance away from the optimum. For instance, over-rejected frequencies worsen the performance in other parts of the spectrum.

This article proposes a solution to this problem using controllers synthesized with an H₂ method that minimizes the variance of slope residuals instead of trying the classical two-stage approach of first identifying the structure and parameters of the disturbance model and then, the design of the associated controller using the Kalman filter [9,10].

Our approach is aimed at tackling disturbances with frequencies below half of the sampling rate. For higher frequency vibrations the reader is referred to [13].

We study the application of our control technique to a multi-conjugate AO system (GeMS) installed at the Gemini South telescope and also to a single-conjugate AO system (NACO) in operation at the Paranal observatory. We use closed-loop data obtained from on-sky runs to test control algorithms off-line and under different disturbance scenarios. Two control laws have been implemented: the classic integrator [14] and H₂ [15] synthesis methods. The choice of H₂ instead of H_∞ norm minimization will be explained later in the paper.

In [16] we have presented preliminary results of this novel approach for off-line and bench implementation at GeMS. However some important issues remained unanswered: i) validation of the use of pseudo open loop values of tip and tilt for the controller synthesis; ii) transportation of the technique to a different instrument and iii) check the on-sky implementation of the method. The first two tasks have been successfully addressed by doing a detailed off-line analysis of NACO data. Despite having a limited access to on-sky operation of GeMS, the correct implementation of the technique has been verified and important lessons regarding this final phase of development have been learned.

The structure of the paper is as follows: Section 2 deals with the theory behind the controllers implemented here. Section 3 describes the facility instruments used for testing. In section 4 we explain the procedure to synthesize the controllers using recorded data from on-sky observations. Section 5 presents the results from actual on-line implementations for artificially generated disturbances and for on-sky runs. In section 6, important implementation issues are discussed. Finally in section 7, conclusions are presented.

2. Controller theory

In this section, the controllers to be used for later implementations in GeMS and NACO are described. These are: the standard integrator and H₂ controllers.

2.1. Integrator

The current default controller for the tip and tilt loops at GeMS and NACO instruments is the classical integrator given by:

C (z) = \frac{K_{i}}{1 - a z^{- 1}},

where z is the Z-transform operator and a is generally set to unity unless a controller free from winding-up is desired (i.e. a “leaky” integrator). Parameter K_i represents the gain of the loop and is adjusted according to noise and performance requirements.

The integral controller can take two forms here: one using a fixed-gain and the other with a computed gain that minimizes the variance of residuals, as will be explained below.

2.2. H₂ control

Motivated by the desire to find new solutions to the challenging control problem of reducing external disturbances, we propose the use of frequency-based design techniques. These synthesis techniques are based on the minimization of H₂ or H_∞ norms [15] and seem particularly suitable to tackle vibration rejection problems as they can readily account for mirror dynamics and performance requirements during design stages.

Doyle et al. [15] demonstrate that the computation for H₂ (reminiscent of the classical LQG problem) and the H_∞ solutions (minimization of the supreme value of a variable over the frequencies of interest) follow the same path that essentially consists of solving two Riccati equations in their static form (optimal estimation and optimal control problems).

In [11], we have analyzed the use of H_∞ and LQG controllers in GeMS’ AO bench and no significant differences have been found, save for some advantages regarding practical implementation and an easier inclusion of mirror dynamics in the problem formulation. This similarity in performance also applies to the H₂ synthesis. In this current paper we have opted for the H₂ approach based on the following:

- H₂ is intuitive since it minimizes a quadratic norm containing the variance of residuals
- H_∞ seeks to flatten a mixed norm of residuals and noise when this may be difficult to achieve due to the mirror’s limited bandwidth.
- H₂ is more efficient in terms of computation since it does not require a search for the optimum as required in the H_∞ case [15]. Our first approximation to this method was carried out using the H_∞ synthesis, but it proved inadequate due to the excessive processing time required. The H_∞ approach involves a binary search with unknown convergence speed, which translates into an unpredictable (and usually high) number of Riccati solution calculations.

The term H₂ comes from the name of the space over which the optimization is pursued, i.e. the space of matrix-valued functions that are analytic and bounded in the open right-half of the complex plane defined by Re(s) > 0. Here, the control problem is presented as a mathematical optimization problem that the H₂ synthesis technique solves.

In the problem formulation, for a continuous-time representation, the following standard configuration is used in Fig. 1:

Fig. 1 Standard configuration for H_∞/H₂ synthesis [15].

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The block P(s) has two inputs, the exogenous input w that includes reference signal and disturbances, and the manipulated variables u. The outputs are the signals contained in vector z, which we want to minimize, and the feedback error e, which are used to control the system. The input e is used by G(s) to calculate the manipulated variable u, i.e. $u = G (s) e$ . Notice that G(s) is a scalar function.

In matrix form, but still in the Laplace domain, matrix P(s) can be represented by

[\begin{matrix} z \\ e \end{matrix}] = P (s) [\begin{matrix} w \\ u \end{matrix}] = [\begin{matrix} P_{11} (s) & P_{12} (s) \\ P_{21} (s) & P_{22} (s) \end{matrix}] [\begin{matrix} w \\ u \end{matrix}],

By defining w = r – n, our tilt servo-control problem described in Fig. 2-left can be re-arranged to form what is called the augmented representation [15] shown in Fig. 2-right. The mirror transfer function M(s) contains the dynamics of the actuators and the two-frame delay of a standard Shack-Hartmann loop. In the augmented configuration, two weighting functions are added to the outputs. Function W_e(s) penalizes control errors and W_y(s) weights the mirror output. The latter can be used to attenuate the effect of noise amplification in the loop.

Fig. 2 Left: servo-control problem. Right: the rearrangement of the servo loop into the augmented configuration used to synthesize the H₂ controller. Functions W_e(s) and W_y(s) have been introduced to penalize control errors and weigh the noise, respectively.

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In the starting configuration (Fig. 2-left), the setpoint r is formed by the incident tilt and disturbances (e.g. vibrations). The residual e enters the controller G(s) to generate the manipulated variable u to move the tip-tilt mirror (TTM) whose output y is contaminated with noise n and later subtracted from the incoming r. In the literature [3] the noise is generally introduced right before the controller (where the wavefront sensor is). In our case, this signal is added to the mirror output. It is easy to verify that both configurations are equivalent as the noise insertion can be shifted to the controller entrance after passing through the subtraction circle, i.e. changing its sign. Since we assume that the white noise has a zero mean value, the minus sign has no effect in the modeling.

From the general structure in Fig. 2-right and Eq. (2),

P (s) = [\begin{matrix} P_{11} (s) & P_{12} (s) \\ P_{21} (s) & P_{22} (s) \end{matrix}] = [\begin{matrix} W_{e} (s) & - W_{e} (s) M (s) \\ 0 & W_{y} (s) M (s) \\ 1 & - M (s) \end{matrix}],

or in its compact state-space form:

P (s) : [\begin{matrix} A & B_{1} & B_{2} \\ C_{1} & D_{11} & D_{12} \\ C_{2} & D_{21} & D_{22} \end{matrix}],

where

P_{11} (s) = [\begin{matrix} W_{e} (s) \\ 0 \end{matrix}] = C_{1} {(s I - A)}^{- 1} B_{1} + D_{11},

P_{12} (s) = [\begin{matrix} - W_{e} (s) M (s) \\ W_{y} (s) M (s) \end{matrix}] = C_{1} {(s I - A)}^{- 1} B_{2} + D_{12},

P_{21} (s) = 1 = D_{21},

P_{22} (s) = - M (s) = C_{2} {(s I - A)}^{- 1} B_{2} + D_{22} .

The augmented representation in compact state-space form described by Eq. (4) contains all the spectral characteristics of noise, vibrations and atmospheric turbulence.

Dropping the Laplace operator s for simplicity and a simple algebraic manipulation of the closed-loop system in Fig. 1, the output z can be expressed in terms of the input w by:

z = F_{l} (P, G) w,

where F_l is:

F_{l} (P, G) = P_{11} + P_{12} G {(I - P_{22} G)}^{- 1} P_{21} .

The H₂ synthesis finds a controller G such that the so-called mixed-sensitivity H₂ norm of F_l (P,G) is minimized:

\min_{G} {‖ F_{l} (P, G) ‖}_{2} = \min_{G} {‖ \begin{matrix} W_{e} \cdot S F \\ W_{y} \cdot N S F \end{matrix} ‖}_{2},

where

{‖ F_{l} (P, G) ‖}_{2} = {(\frac{1}{2 π} \int_{- \infty}^{\infty} trace[F_{l} (P, G) {(j ω)}^{*} \times F_{l} (P, G) (j ω)] d ω)}^{1 / 2},

and SF and NTF are the sensitivity function (also known as error transfer function) and noise transfer function given by:

S F (s) = \frac{e (s)}{w (s)} = \frac{1}{1 + M (s) G (s)},

N T F (s) = \frac{y (s)}{w (s)} = \frac{M (s) G (s)}{1 + M (s) G (s)} .

The controller G(s) is synthesized in its state-space form ([A_G,B_G,C_G,D_G]), by the following formulae [15]:

A_{G} = A - B_{2} B_{2}^{'} X_{2} - Y_{2} C_{2}^{'} C_{2}, B_{G} = B_{2}^{'} X_{2}, C_{G} = Y_{2} C_{2}^{'}, D_{G} = 0,

where

X_{2}

and

Y_{2}

are the solutions of two Riccati equation given by:

X_{2} = Ric [\begin{matrix} A & - B_{2} B_{2}^{'} \\ B_{1} B_{1}^{'} & - A \end{matrix}] and Y_{2} = Ric [\begin{matrix} A' & - C_{2} C_{2}^{'} \\ - B_{1} B_{1}^{'} & - A \end{matrix}] .

One of the unique features of the H₂ synthesis is that the controller obtained from Eqs. (15) and (16) contains the optimal state observer and the optimal controller (this is why it requires two Riccati solutions). Transforming to the Laplace domain, the controller becomes:

G (s) \equiv C_{G} {(s I - A_{G})}^{- 1} B_{G} + D_{G} .

The H₂ mixed-sensitivity optimization penalizes the mirror output by function W_y(s). In our case the penalty is (but not necessarily need be) a constant, which results in an equal cost for each frequency in y. Due to the lack of access to control frequencies above the loop bandwidth and the importance of lower frequencies or those of vibrations, some kind of weighting must be applied to those frequencies above this bandwidth. The way this is accomplished is via the other component in the z vector, which weights the error (residuals) according to the turbulence and vibration intensity (function W_e(s)). The transfer function of the mirror, M(s), also affects the mirror output by attenuating those frequencies above the mirror cut-off frequency. This will cause the synthesized controller to devote more energy to correct those frequencies related to turbulence or vibrations compared to those with a base cost given by W_y(s). These weighting functions are complementary, so that the competing requirements of good accuracy and noise rejection can be met by the resulting controller.

For implementation in the real time controller (RTC), G(s) is digitized using zero-order-hold transformations. . While the input-output is exactly the same as in the classical integrator, the approaches are slightly programmatically different in that the full state-space vector must be saved for the next cycle.

The noise W_y(s) is modeled with a flat spectrum and the turbulence and vibration amplitude information is contained in W_e(s) so each frequency is weighted according to its intensity during the controller synthesis. This will become apparent in section 4.

3. Astronomical adaptive optics systems used

Atmospheric turbulence telemetry data has been gathered during closed-loop operation of the GeMS system and of NACO, which is an instrument installed on Paranal’s unit telescope UT4. These 8-meter class telescopes are located, respectively, on the summit of Cerro Pachón and Cerro Paranal in northern Chile.

3.1. GeMS

GeMS is the Gemini multi-conjugate AO system. It uses 5 artificial laser guide stars (LGS) with their associated LGS wavefront sensors (LGSWFS) and 3 deformable mirrors to compensate the turbulence over a field of view of 2 arcmin. Besides this, 3 natural guide stars (NGS) are required for the control of the tip-tilt and plate scale modes. The NGS consists of 3 probes, each containing a reflective pyramid that acts like a quad-cell feeding a set of 4 fibers and associated avalanche photodiodes.

Three NGS wavefront sensors (NGSWFS) provide six X-Y slopes necessary to generate global tip and tilt residuals that feed a TTM controller residing in the real time controller (RTC). Plate scale modes can also be estimated from this set of slopes [17] but are not considered in this paper. The laser loop and the NGS loop can be driven at a rate of up to 800Hz. GeMS delivers a corrected wavefront to the near-infrared imager (GSAOI) that generate diffraction-limited images. More details about GeMS can be found elsewhere [18].

3.2. NACO

NACO is a versatile instrument that has been in operation at ESO’s Paranal observatory since 2002 and is the combination of the two instruments NAOS and CONICA. CONICA is a high-resolution near-IR camera that is capable of executing a wide variety of different scientific observations such as imaging, polarimetry, coronography, sparse aperture masking and spectroscopy. NAOS is a single-conjugate AO system that provides CONICA with a turbulence-compensated beam using either an NGS or an LGS and NGS tip-tilt star combination. In the latter mode, the artificial LGS is positioned on-sky on-axis with the science target and is used for high-order AO corrections while the NGS is used for correcting atmospheric tip-tilt motions, which cannot be sensed with the LGS. NAOS has two separate paths for wavefront sensing in either the visible (0.46-0.95 μm) or infrared (0.8-2.5 μm) bands. Each wavefront sensor has three, remote-exchangeable micro-lens arrays. Wavefront correction is accomplished with a fast tip-tip mirror and a deformable mirror with 185 piezo stack actuators. A more detailed description of the instrument can be found at [19].

The NAOS on-sky tip-tilt residual slopes data used for this study were computed by the NAOS RTC from the set of X-Y slopes measured directly by the 14x14 visible wavefront sensor. This NGS setup is based on a low noise fast readout 128x128 pixel CCD camera located in the focal plane of the micro-lens array and can be run at loop rates from 30 to 444 Hz based on different subaperture windowing and binning configurations. For optimal sampling of vibrations with the wavefront sensor, data was acquired at a rate of 444Hz with 8x8 pixels per subaperture and no binning.

4. Procedure to tune the tip-tilt loop

In this section we describe a new method that consists of a sequence of steps that search for a loop shaping based on the H₂ method described above. To illustrate the technique, we use on-sky data from the GeMS system for two distinctive cases of turbulence characteristics.

4.1 The cost function: variance of residuals

The method searches for a controller that minimizes the variance of the measured residuals using closed-loop data from on-sky observations. Since actuator values have been recorded together with the residual slopes, we can estimate the turbulence and vibrations from the so-called pseudo-open-loop (POL) slopes, i.e.:

S_{k}^{t u r} = S_{k}^{r e s} + i M a t \cdot u_{k - 1}^{} .

Here,

S_{k}^{t u r}

are the reconstructed slopes for the total turbulence and vibrations at time k,

S_{k}^{r e s}

are the slopes of residuals and iMat is the interaction matrix that models the projection of actuator commands (u_k-1) onto slopes of the mirror surface. Notice that the commands are delayed by one frame due to the computer processing interval and the readout of the detectors.

Once the turbulence is calculated using the POL reconstruction, we can represent the turbulence and residual variables in a form compatible with the input w of Fig. 2 using the pseudo inverse of matrix iMat:

w_{k}^{} = i M a t^{- 1} \cdot S_{k}^{t u r} .

The Fourier transform is applied to vector w to shift the calculation to the frequency domain. Our objective is to design a controller that minimizes the variance of the residuals, i.e.

\underset{G (s)}{M i n} σ_{e}^{2} = \frac{1}{N - 1} \sum_{k = 0}^{N - 1} {| e_{k} |}^{2},

where N is the length of vector e. It is assumed that

\frac{1}{N - 1} \sum_{k = 0}^{N - 1} e_{k} = 0 .

Using the ensemble average notation_{$〈 \cdot 〉$}, an estimate for the variance of the residual is given by:

〈 e^{2} 〉 = {\int_{ω} ‖ \frac{1}{1 + M (j ω) G (j ω)} ‖}^{2} {‖ w (j ω) ‖}^{2} d ω .

This estimate of the variance will be used in our method to assess the performance of the controller G(s) during the search for the optimum parameters.

Since the output weighting function W_y is independent of the controller and can be estimated easily from open-loop or POL slopes (assuming that the slopes components at high frequencies are purely noise), the only remaining step in synthesizing the controller is the estimation of the weighting function W_e.

At this stage of the design process, traditional approaches [8,20] employ model-identification tools to represent the turbulence and vibrations and the controller design is based on the identified models. These approaches generally disregard nonlinearities or require a fixed model structure. In [8] an original alternative is suggested where vibrations and turbulence are treated separately. Although satisfactory results are reported, the technique requires some arbitrary definitions that limit the generalization of the method. In other cases [11], simplification of dynamics or nonlinearities in the modeling can yield closed-loop behavior that tends to either over-reject certain frequencies or does not fully eliminate them.

Our approach skips this step and constructs the controller by evolving a basic configuration (classical integrator). Each iteration involves increasing the controller’s complexity by adding filters to W_e with pre-defined structures and searching for the parameter values that minimize the variance. This approach sequentially constructs a controller with an increasing number of states up to a point where no significant improvements in the closed-loop performance are obtained.

4.2 Loop delay and mirror dynamics

The mirror transfer function, M(s), contains the bandwidth of the actuators and the two-frame delay of the standard Shack-Hartmann loop. While the manufacturer-specified bandwidth of the GeMS tip-tilt mirror is 380 Hz, the experimentally determined value of 400 Hz is used to approximate the dynamics of the mirror as:

M (s) = \frac{e^{- 2 Δ s}}{0.0025 s + 1},

where Δ is the sampling interval of the sensors. In the remainder of the paper, pure delays are approximated by bilinear transformations. Dynamic modeling of the mirror is also necessary when mechanical resonances are important [12].

4.3 The tuning sequence

The sequences of steps that form this method are described in this section. The process begins with optimizing the standard integrator and continues with the synthesis of increasingly complex H₂ controllers.

Two representative reconstructed open-loop tilt PSDs are plotted in Fig. 3. A 55Hz peak is clearly present in both sets of on-sky data. In some examples (not shown here) a broad peak around 14Hz also appears, which is believed to be induced at the top-end of the telescope and at the secondary mirror structure. This peak is not always present and may be excited by wind-shake. These two plots also illustrate the range of variation that can be expected from atmospheric turbulence. Indeed, depending on the turbulence strength and wind speed conditions, the cut-off frequency of the turbulence varies significantly [21]. These two cases have been chosen to highlight the ability of the method to adapt to dynamically distinctive disturbances. Notice that the turbulence energy is concentrated at lower frequencies for the slow wind case (left panel), whereas the right panel shows a displacement of the spectrum toward higher frequencies. In addition, the noise level also changes depending on the guiding star magnitudes. In the following description, only tilt values are analyzed as this is the direction of greatest disturbance.

Fig. 3 On-sky PSDs for the measured tilt from NGSWFS POL slopes. Left: data from March 2011, Right: data acquired in April 2011. The noise level is estimated from the rms values of the residuals above 200 Hz.

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4.3.1 Step 1: the fixed gain integrator

The method’s sequence of steps starts by assessing the performance of an integrator controller with an arbitrarily fixed gain that establishes the worst case performance. This is the default controller of GeMS where the gain K_i normally ranges between 0.2 and 0.4, but it is normally fixed at K_i = 0.2. Parameter a in Eq. (1) is usually set to 1 (pure integrator).

4.3.2 Step 2: the optimal integrator

Fine tuning starts by searching for the value of K_i that yields the minimum variance of the residuals in simulated closed-loop, which is reminiscent of the OMGI controller developed by Gendron and Lena [22] but limited to tip and tilt only in this case. Figure 4 plots the variance for the two types of disturbances shown in Fig. 3. A significant difference exists between the minimum attainable variance for the slow and fast turbulences. This optimal tuning of the integrator controller is also useful for establishing a fair comparison between this classical approach and the more complex controllers we are proposing.

Fig. 4 Step 2: Minimum variance for optimal integrator gain for the slow and fast turbulences.

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4.3.3 Step 3: the first order model or leaky integrator

The following steps in the method consider the use of controllers obtained from the H₂ technique. By defining adequate weighting functions W_e and W_y, and the plant M(s) defined in Eq. (22), the closed-loop is excited with the two types turbulences. From the sensitivity function, the residual can be estimated and using Eq. (21) a search for the controller that minimizes the variance of residual can be implemented.

By assuming that the reconstructed slopes at higher frequencies are purely noise, function W_y is estimated from the average rms value of the residuals above 200 Hz. W_y then takes on a constant value, which is 0.026 and 0.017 arcsecs for the two turbulence cases. We then assume that the turbulence can be described by the simplest first order function:

W_{e} (s) = \frac{C_{0}}{s + C_{1}} .

This function is chosen as other more complex models that could represent the turbulence better will include the notch filters defined in the next section and added in series to this function. This will become apparent in section 4.4, where more complex functions are required to represent real turbulences.

A search for C₀ and C₁ to minimize the variance is carried out using the Nelder-Mead nonlinear minimization technique [23]. This algorithm has also been applied to the following stages in the sequence as it proved very reliable and converges quickly.

For the optimum case, the PSD of the residuals differ significantly from that of the optimized integral controller (Fig. 5, right panel); however, it is similar to the integrator in the case of the slow turbulence (Fig. 5, left panel). This is not surprising since the first order function that minimizes the residuals have very small leaky action as confirmed by Fig. 6, where the left panel shows that the W_e function in this case resembles that of a pure integrator (cutoff frequency lower than the displayed range). On the contrary, in the right panel (fast turbulence), the fitted W_e has a cutoff frequency of around 10 Hz.

Fig. 5 Step 3: residual PSD for optimum first order turbulence model. PSDs of integrator residuals are also shown.

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Fig. 6 Step 3: Weighting functions W_e and W_y that result from the optimal C₀ and C₁ values.

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From Fig. 5 it is clear that there is still room for improving the residual response in order to get a more even PSD. For both types of turbulences, there is a strong peak due to the 55 Hz vibration and also some high residual values at lower frequencies that are dealt with in the following steps.

4.3.4 Step 4: first order plus 1 notch

To tackle vibrations, a second order notch filter is added to W_e.

W_{e} (s) = \frac{C_{0}}{s + C_{1}} \frac{s^{2} + 2 η_{1} ω_{o} s + ω_{o}^{2}}{s^{2} + 2 η_{2} ω_{o} s + ω_{o}^{2}} .

These filters are attached to the first order function calculated in the previous step. It will be shown that this function not only rejects vibrations but it can also shape the dynamics of the closed-loop to reduce the turbulence residuals leftover from the previous step.

Examples of the notch function are presented in Fig. 7 for a vibration at ω_o = 2π∙50 rad/s and for different values of η₁ and η₂. This filter allows modeling a vibration in terms of amplitude and width. The left panel shows the responses for different ratios of η₁/η₂, and in the right panel, the ratio η₁/η₂ is kept constant at 100, but three different values of η₁/η₂ are plotted.

Fig. 7 Frequency response of notch filters for ω_o = 2π∙50 rad/s.

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The critical part in searching for optimal parameters is finding the vibration frequencies. We have found that local minima tend to appear in the searching process when using classical identification schemes and as a result the process components of interest are frequently missed. In [8], an original approach to avoid this omission is used whereby a clipping technique separates vibrations from turbulence or noise. Here, we propose a scanning method that despite an increase in processing demand and lack of optimality is robust and ensures the detection of the vibrations of interest.

The method consists of scanning the spectrum within the range where vibrations are likely to exist, i.e. sweeping the complete spectrum with frequency ω_o in Eq. (24), with several values of η₁ and η₂. This process is shown in Fig. 8 for the slow turbulence, where the notch function is swept from left to right, generating the corresponding variance plot (bottom panel). For each scanning value of ω_o, a controller is synthesized and the subsequent closed-loop variance is computed. The frequency value that generates the minimum variance during this scanning process is selected for the notch rejection frequency and a search for the values of C₀, C₁, η₁ and η₂ that minimize the variance are sought.

Fig. 8 Step 4: scanning of the notch filter in search for vibrations. The minimum of the variance (bottom plot) corresponds to the strong vibration shown in the top panel.

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Notice that the scanning process generates a distinctive minimum in the variance at the vibration peak, which is significantly lower than the minimum variance obtained in step 3 (Fig. 8, horizontal line, bottom panel).

If the ratio η₁/η₂ is kept constant through the scanning, the noise would cover the filter peak at higher frequencies. To avoid this, the ratio must be modified to keep a peak of constant amplitude above the sum of W_e and W_y PSDs so that a “tip of the iceberg” effect is obtained in the vibration modeling (see dashed line in Fig. 9). The ratio η₁/η₂ is then:

\frac{η_{1} (j ω)}{η_{2} (j ω)} = A \sqrt{1 + \frac{{‖ W_{y} (j ω) ‖}^{2}}{{‖ W_{e} (j ω) ‖}^{2}}},

where A is the η₁/η₂ ratio for W_e >> W_y .

Fig. 9 Step 4: PSDs of disturbances and noise for minimum variance.

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4.4. Step 5: first order plus 2 notches

In the following steps additional notches are added to W_e, so for step 5, the error weighting function becomes a fourth order transfer function containing two notch filters. A new scanning process is executed keeping the previous parameters fixed and varying the values of ω_o, η₁ and η₂ only. In this case, a minimum is achieved at lower frequencies corresponding to a turbulence component rather than a vibration. Once the value of ω_o generates the minimum variance, a full search for the eight parameters in the W_e that give the minimum variance is performed.

Figure 10 shows the PSDs of residuals for the upgraded controller. Note that not only has the vibration been fully eliminated, but also a flat response is obtained, which is remarkably different from the integrators for the two types of turbulence. Figure 10 also shows a decline in the PSD of the residuals above a certain frequency, especially in the fast turbulence case (right panel). This is caused by the loop bandwidth, which is determined by the mirror dynamics and the wavefront sensor sampling rate that restrict the frequencies that can be effectively rejected.

Fig. 10 Step 5: residual PSD for the fifth order W_e. PSDs of residuals generated by the integrators are also shown.

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An appealing feature of this method is its ability to generate disturbance and noise models as byproducts out of the minimization of residuals. Figure 11 shows that the models for functions W_y and W_e that result from the minimization of the variance fit very well the measured disturbance spectrum.

Fig. 11 Step 5: PSD of tilt slopes and associated model for minimum variance.

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That the optimization process captures the characteristics of the fast turbulence between 2 and 5 Hz (Fig. 11-right panel) means that the values found for η₁ and η₂ for that particular notch resemble a band-pass filter more than an individual vibration (see Fig. 7, right panel). In other models used for modeling vibrations [4,13] broadening the peak is not possible as the numerator of the vibration model has only a constant term. Having η₁ and η₂ to describe a spectrum peak (Eq. (24) allows for modeling a broader range of spectral features.

Figure 12 shows the substantial differences between the sensitivity functions of the H₂ approach and those of the classical integrators. This is a clear example of how Bode’s theorem and the minimization of the quadratic norm shape the SF to achieve a balanced frequency response. These plots also contradict the common belief that higher loop bandwidths can deliver better performances. Although this is true for the high speed turbulence case (right panel), it is not so for the slow turbulence case (left panel).

Fig. 12 Step 5: Sensitivity functions for integral and final H₂ controllers.

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More filters can be added in series to the W_e function as long as further reductions in variance are achieved. We have found that for the turbulences analyzed here, the fifth order system defined in step 5 is sufficient. Although further improvements can be achieved for higher order functions, the gain does not necessarily compensate the higher controller complexity and the increase in the execution time of the full optimization process. This “diminishing returns” is shown in Fig. 13 and can be detrimental to the loop performance when evaluated using turbulence data different from the one used for the synthesis, as it would occur in a real implementation. This will be explained in the next section.

Fig. 13 Progressive reduction of the variance of residuals for the sequence of design steps.

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From Fig. 13, the necessity of a self-tuning controller is evident, even for the integrator controller. As shown in the left panel, a simple tuning of the integrator gain can lead to substantial reductions in the variance (in this case, a reduction of 14%). This is not the case for the fast turbulence (right panel), where the variance for the fixed-gain integrator is similar to that of the optimal integrator (only 1%).

To summarize the procedure described above we list the sequence of calculations required for the implementation of the technique:

a) Estimate function W_y based on the temporal PSD of the tip or tilt at higher frequencies
b) Define a parameterized function W_e formed by the first order function in Eq. (23) with an initial value for C₀ and C₁ (unity for instance).
c) Construct the state space model of P(s) as in Eq. (4). This model contains the dynamics of W_e and W_y defined above.
d) Compute the optimal H₂ controller for P(s) from Eqs. (15) and (16)
e) Repeat steps b), c) and d) for different values of C₀ and C₁ till the variance estimated by Eq. (21) reaches a minimum. This can be done with any standard minimum search techniques. Register the optimal parameters C₀* and C₁*
f) Define a new W_y function as the product of the first order function found in e) with a notch filter as defined in Eq. (24)
g) Construct the state space model of the new P(s) as in Eq. (4).
h) By scanning the valid range of frequencies with varying ω_o (leaving the other parameters fixed), find the value of ω_o* that minimizes the residual variance calculated by (21). This requires computing an optimal H₂ controller (Eqs. (15) and (16)) for each value of ω_o.
i) Using gradient search techniques, find the optimal parameters η₁ and η₂ with ω_o* fixed
j) Repeat steps f) thru i) with increasing number of notch filters until the reduction in the variance of residual reaches a value below a threshold defined by the user
k) Perform a final minimization, this time for all the parameters and starting from the values found in the previous step. This will speed up convergence.

4.5 Validation of the method with NACO

The attachment of new notches to W_e as described above can go on indefinitely. However, as demonstrated in Fig. 13, the variance eventually approaches a minimum with the addition of more filters. Furthermore, since the controller will be applied to turbulence and vibrations whose characteristics change, the impact of adding a high number of notches could be counterproductive. To show this effect clearly, we now analyze the application of the technique to NACO, where a forested spectrum exists (see Fig. 14-left). Several vibrations can be observed between 9 and 100 Hz. In principle, the order of the controller should be determined by the number and strength of the vibrations.

Fig. 14 Left: open loop PSDs of residuals and associated model (coarse line); Right: PSDs of NAOS residuals using the default integrator and H₂ controllers. Data from the night of November 2nd, 2011.

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Next, an off-line analysis of the impact of using the H₂ controller is carried out. To check its robustness, the method is applied to nine sets of data containing open-loop and closed-loop residuals obtained with NACO. Each controller is designed with the POL data generated from a specific set and later evaluated with open-loop data from a subsequent data measurement, as would occur in an operational scenario. Its performance is compared against the closed-loop residuals registered for the integrator controller in the latter set. Figure 14-right shows the PSD of the residuals for a controller synthesized with on-sky data and tested against on-sky open-loop data. Clearly, the vibrations have remained stable between the two sets, making their rejection very effective.

By using a different set of data for the controller validation, the variance will not necessarily be a monotonically decreasing function of the controller’s order as shown in the previous section, but it will depend on how well the validation set matches the data used for controller synthesis. Figure 15 shows the evolution of the variance for data set number 6 as the order of the controller is increased with the addition of notches. Note that for the seventh column the variance increases, which suggest that the choice of the order of W_e is not obvious as small changes in the turbulence or vibration conditions erase the advantage of the additional notches. In this particular case the vibration associated with that notch might not be sufficiently stable in terms of amplitude or frequency and therefore its rejection is questionable. Hence, increases in the variance for a given notch can be used to skip this vibration in the following controller synthesis.

Fig. 15 Reduction in tip-tilt residual variance for increasing complexity of controller.

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This uncertainty at higher orders indicates that there is no point in using controllers with too many notches. In NACO we found that a seventh order W_e function (a leaky integrator plus 3 notches) is an adequate choice.

Table 1 presents a summary of the 9 runs for the 10 sets of data using the variance and RMS values of residuals as figures of merit. The analysis has been made for controllers synthesized for a W_e function of order 7. Substantial reductions of over 50% in the variance are achieved in all cases.

Table 1. Reduction in residuals for the classical integrator and H₂ controllers

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5. On-line implementation at GeMS

Having tested the method extensively off-line, an on-line implementation was verified with GeMS. This facility instrument is equipped with a calibration source for generating artificial natural and laser guide stars at the AO bench level. When combined with simulated vibrations added to the mirror commands, the instrument can generate realistic disturbance conditions.

In addition to these experiments, the technique was also tested with on-sky runs. Due to the limited availability of the GeMS system, only a few runs were executed. However, some relevant conclusions regarding implementation were drawn.

5.1. Results from calibration source

Figure 16 shows the results from an experiment carried out with the GeMs bench and its calibration source. An example of a typical turbulence plus a single vibration mode is shown in the left panel for the tilt POL residuals, artificially generated by excitation of the deformable mirrors. Applying the method described above for a W_e constructed with two notches, the response of the closed-loop system is shown in Fig. 16 (thick line in left panel) together with its integrator counterpart. For the H₂ controller, the vibration has been completely removed and an effective control of the turbulence has also been achieved, resulting in a balanced spectrum of residuals. Their associated sensitivity functions are presented in Fig. 16, right panel.

Fig. 16 Result from optical bench experiments using artificially generated turbulence and vibrations. Left: PSDs of POL slopes and the associated tilt residuals for the H₂ and integrator controllers. Right: sensitivity functions for integral and H₂ controllers.

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These results allowed us to confidently test the technique in real on-sky observations.

5.2. On-sky implementation

The H₂ controller was tested on-line with on-sky data. Due to the faint intensity of the stars, the closed-loop had to be run at a reduced rate of 300 Hz. The loop was first run using an integrator tuned to give the best possible performance for the observation conditions during the night of May 10th, 2012. Using these data, POL slopes were reconstructed for the off-line synthesis of the controller as described above. After calculation, the H₂ controller was loaded into the GeMS’ RTC system in place of the integrator.

Due to few runs executed, results can only be used as a confirmation that the technique works. Further tests will be necessary in the future for a more comprehensive evaluation of its benefits. Figure 17-left shows the PSDs for the tilt POL slopes and estimated noise that were used for the synthesis of the controller. Two minor vibrations are observed in the uncorrected spectrum between 10 and 14 Hz that are effectively rejected by H₂ controller as shown in Fig. 17-left (thick line). In this case the variance of the residuals was reduced from 9.7e-4 arcsec² for the best integrator controller to 7.5e-4 arcsecs² for the H₂ controller. The accompanying sensitivity function is presented in Fig. 17-right.

Fig. 17 Results from on-line implementation for on-sky run (May 10th, 2012). Left: PSDs of POL tilt and residuals for the H₂ and integrator controllers. The variance for the integrator and H₂ controllers are 9.7e-4 arcsec² and 7.5e-4 arcsec², respectively. Right: sensitivity functions for integral and H₂ controllers.

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In spite of the reduced loop rate and the unusually weak vibrations encountered during the runs, the method performed according to expectations. We think that for higher sampling rates and stronger disturbances, the benefits of using this approach should be even more apparent.

6. Implementation issues

For an unsupervised operation of the method, a mechanism to stop the sequence of steps is needed. This can be absolutely arbitrary and it will very much depend on the available time in both the processor that will calculate the controller and the RTC that will perform the actuators’ values from the loaded controller. We have found that stopping the sequence when the reduction in the variance between one step to the next drops below 1% is an adequate rule.

From the implementation point of view, the time required to execute the complete off-line sequence using MATLAB software tools is about 2 minutes for a fifth order W_e using a commercial PC.

Table 2 presents the required processing time to calculate a controller for different complexities of W_e and the order of the resulting controller to be loaded into the RTC. Here, for a given turbulence modeled by a first order function and 4 notches to reject the same number of vibrations, a current commercial computer seems perfectly capable of doing the calculations in a reasonable time (7.5 minutes). These processing times can be substantially reduced with code optimization.

Table 2. Processing time (per axis) required for different size of disturbance models and resulting controller size

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To date, we have updated the controller using a guided procedure where data is collected from the RTC, transferred to an independent computer where the controller is calculated and the result uploaded into the RTC. For real implementations, we think that it would not be sensible to load the RTC with such calculations. Instead, a stand-alone computer connected via a high speed socket to the RTC is recommended. This configuration would allow dumping sufficient data from the RTC to the external PC without stealing time from the RTC.

How much data? We have used the full length of GeMS’ circular buffer, which is 48,000 frames or 60 seconds at 800 Hz. For NACO, the length of the buffer is 4,096 or 9.2 seconds at 444 Hz frame rate. One of the factors that should determine the length of the vectors is the minimum frequency to be controlled. For these default parameters the lower end of the spectrum would be 0.033 Hz and 0.21 Hz respectively. To avoid distortions due to leakage caused by low frequency components on the estimation of the lower end of the spectrum, the first two terms of the frequency vector were discarded when computing the variance.

Commercial communication boards easily have 500Mbit/sec transfer rates so that the required data could be dumped in 6ms to the computer running the method for GeMS’ slopes and command vectors. Loading the resulting controller from the PC to the RTC would be negligible. This would allow carrying out constant updates of the controller during the night at the rate presented in Table 2, last column.

Based on our experience with NACO and GeMS data, we don't expect the turbulence and vibrations to change significantly over the time required to update the controller. If the controller used does not match with the input perturbation, a loss of performance is to be expected. This has been studied in detail in [11], and the H₂ controller would never perform worse than the integrator. If the conditions are changing too fast, one could explore the possibility of updating only a first order, or first order plus one notch at a high rate (defined in Table 2), while updating the full controller parameters including more notches at a lower rate. This should ensure to have an optimal rejection of the disturbances that contain most of the energy.

7. Conclusion

We have presented a procedure for tip-tilt control that reduces the vibration in adaptive optics systems as well as improving the overall performance of the closed for varying turbulence characteristics. Although the technique does not need any previous identification of turbulence or vibration models, it actually delivers a set of parameters for the modeled disturbances as a by-product of an iterative minimization of a frequency domain (H₂) control criterion using residual slope data. The criterion is the H₂ norm of the mixed-sensitivity function, which is composed of the sensitivity function weighted by the disturbance spectrum W_e and the noise transfer function weighted by the measurement noise spectrum W_y. A controller is then designed for a given W_e and W_y. The function W_y is easily identified from the higher end of the pseudo-open loop data spectrum. In turn, W_e gradually adopts the disturbance characteristics by defining transfer functions with increasing complexity and estimating the parameters that minimize the residual slope variance. The method is stopped when increasing the complexity of W_e does not lead to significant reduction in the residual variance and a result for the controller and estimation of the perturbation spectrum W_e is achieved.

The procedure has been implemented off-line using on-sky data from the instruments GeMS and NACO and was later implemented on-line in the optical bench of GeMS using the calibration source. Finally, the method has been tested on-sky to check its correct operation.

Reductions in the variance of residuals of up to 50% have been achieved in off-line runs and although inconclusive, the on-line improvements have also shown improvement of around 30%.

The principal benefit of using sophisticated controllers such as H₂ is that their SF is shaped to tackle specific frequencies where disturbances are concentrated. A balanced or nearly flat spectrum of residuals is expected in this optimal case and although the theory behind H₂ controllers ensures this, generally the actual results tend to miss expectations mainly due to inaccurate modeling of disturbances, non-linearities or dynamics present in the loop (e.g. mirror dynamics). This suggests that the approach of relying on accurate identification tools for finding the turbulence and vibration parameters to tune the controllers might not be the right choice. Our approach looks for a controller with the lowest and balanced PSD of the measured residuals. We have shown that there is an alternative to the identification approach which consists in synthesizing controllers that minimize the variance of residuals.

Despite its mathematical complexity, the technique proposed has the potential to become a totally unsupervised procedure. It is robust and simple to use and can automatically adapt the order of the controller according to the existing conditions for turbulence and vibrations, i.e. it is not only a vibration rejection method but also a means to optimize the closed-loop performance by adapting to the changing turbulence conditions. Furthermore, the progressive approach in the complexity of the controller guarantees a minimum performance given by an optimal integrator (worst case).

The computation time for a typical controller structure of a leaky integrator plus two notches is currently in the order of 2 mins for a commercial PC. We think that this is not an issue, as the characteristics of the turbulence and vibrations should be fairly stable over this time scale.

Acknowledgments

This work was supported by the Chilean Research Council (CONICYT) grants Fondecyt 1120626 and Anillo ACT-86.

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Controller	Integrator arcsec²/ mas RMS	H₂ arcsec²/ mas RMS
Average	5.6e-3 / 75	2.1e-3 / 45
Min	4.1e-3 / 64	1.2e-3 / 35
Max	6.2e-3 / 79	3.6e-3 / 60

Type of W_e	Order of W_e(s)	Order of G(s)	Computing time (mins)
First order (Eq. (23)	1	3	0.3
1st order + 1 notch (Eq. (24)	3	5	0.9
1st order + 2 notches	5	7	2.0
1st order + 4 notches	9	11	7.5
1st order + 7 notches	15	17	28.6

Controller	Integrator arcsec²/ mas RMS	H₂ arcsec²/ mas RMS
Average	5.6e-3 / 75	2.1e-3 / 45
Min	4.1e-3 / 64	1.2e-3 / 35
Max	6.2e-3 / 79	3.6e-3 / 60

Type of W_e	Order of W_e(s)	Order of G(s)	Computing time (mins)
First order (Eq. (23)	1	3	0.3
1st order + 1 notch (Eq. (24)	3	5	0.9
1st order + 2 notches	5	7	2.0
1st order + 4 notches	9	11	7.5
1st order + 7 notches	15	17	28.6

Mitigation of vibrations in adaptive optics by minimization of closed-loop residuals

Abstract

1. Introduction

2. Controller theory

2.1. Integrator

2.2. H₂ control

3. Astronomical adaptive optics systems used

3.1. GeMS

3.2. NACO

4. Procedure to tune the tip-tilt loop

4.1 The cost function: variance of residuals

4.2 Loop delay and mirror dynamics

4.3 The tuning sequence

4.3.1 Step 1: the fixed gain integrator

4.3.2 Step 2: the optimal integrator

4.3.3 Step 3: the first order model or leaky integrator

4.3.4 Step 4: first order plus 1 notch

4.4. Step 5: first order plus 2 notches

4.5 Validation of the method with NACO

5. On-line implementation at GeMS

5.1. Results from calibration source

5.2. On-sky implementation

6. Implementation issues

7. Conclusion

Acknowledgments

References

Cited By

Figures (17)

Tables (2)

Equations (25)

Optics Express

Abstract

1. Introduction

2. Controller theory

2.1. Integrator

2.2. H2 control

3. Astronomical adaptive optics systems used

3.1. GeMS

3.2. NACO

4. Procedure to tune the tip-tilt loop

4.1 The cost function: variance of residuals

4.2 Loop delay and mirror dynamics

4.3 The tuning sequence

4.3.1 Step 1: the fixed gain integrator

4.3.2 Step 2: the optimal integrator

4.3.3 Step 3: the first order model or leaky integrator

4.3.4 Step 4: first order plus 1 notch

4.4. Step 5: first order plus 2 notches

4.5 Validation of the method with NACO

5. On-line implementation at GeMS

5.1. Results from calibration source

5.2. On-sky implementation

6. Implementation issues

7. Conclusion

Acknowledgments

References

Cited By

Figures (17)

Tables (2)

Equations (25)

Optics Express

2.2. H₂ control