2.1.

METIS-SCAO System

The METIS-SCAO system depicted in Fig. 2(a) is a cascaded control loop, featuring local control systems for the active mirrors M4 and M5 and rejecting arbitrary wavefront disturbances $d$ to fully exploit the ELT’s spatial resolution capability. Using the infrared wavefront sensor of METIS, the wavefront $y_{\mathrm{wf}}$ is measured, which is the superposition of the disturbance $d$ (caused by, e.g., atmosphere and wind-induced vibrations) and the corrections ${\tilde{y}}_{\mathrm{M}4/\mathrm{M}5}$ of the active mirrors M4 and M5. Subsequently, the SCAO controller computes the modal control inputs ${\mathfrak{u}}_{\mathrm{M}4/\mathrm{M}5}$ for M4 and M5 at a loop rate up to 1 kHz to reduce the control error ${\mathfrak{e}}_{\mathrm{wf}}={\mathfrak{r}}_{\mathrm{wf}}-{\mathfrak{y}}_{\mathrm{wf}}$ between the reference and measured wavefront. The reference wavefront ${\mathfrak{r}}_{\mathrm{wf}}$ is always piece-wise constant and typically zero. Moreover, the corrections ${\tilde{y}}_{\mathrm{M}4/\mathrm{M}5}$ of the active mirrors result from projecting their doubled mirror shapes $2y_{\mathrm{M}4/\mathrm{M}5}$ according to the ELT’s optical layout [see Fig. 1(b) and Ref. 46].

METIS’s wavefront sensor is an infrared pyramid sensor measuring the incoming wavefront at a rate between 100 Hz and 1 kHz. The tip–tilt mirror M5 is a flat monolithic mirror sized $\approx 2.2\mathrm{m}\times 2.7\mathrm{m}$ providing exclusively slow and large tip–tilt corrections. M5 is actuated by three piezo-electric drives, its tip–tilt position is controlled via a local control system also damping M5’s eigenmodes, and its mirror tile is made of silicon carbide.²

2.2.

Deformable Mirror M4

The deformable mirror M4 pictured in Fig. 3 is a flat annular mirror composed of six identical independent plate segments. Its reflective annulus has an inner diameter of $\approx 0.5\mathrm{m}$, an outer diameter of $\approx 2.5\mathrm{m}$, and is coated with a thin aluminum layer.²

Fig. 3

Rendering of M4 including its powered mount. One of the six identical and independent mirror segments is highlighted (adapted from the European Southern Observatory).

Each M4 segment shown in Fig. 4 is flat, uniformly 1.95-mm thick, and made of a glass ceramic. All segments are elastically supported at their outer curved edges via flat springs directly attached to them (see Figs. 3 and 4). These flat springs themselves are clamped to the support structure (dark gray with green lines in Fig. 3), which in turn is attached to M4’s powered mount. Consequently, the springs impede in-plane translations and rotations of the M4 segments while enabling out-of-plane movements on the other hand. Since the M4 segments hover $\approx 100\mu \mathrm{m}$ above the support structure during operation, all other edges of the M4 segments are free and hence M4 consists of six independent annular sectors.²

Fig. 4

Abstracted sketch of an M4 segment elastically supported by flat springs at its outer curved edge. Several actuator-sensor pairs (×) deforming the segment are marked.

The segments can adopt arbitrary mechanical out-of-plane deformations with amplitudes up to $\pm 40\mu \mathrm{m}$ and high spatial frequencies utilizing 892 non-contacting voice coil actuators and collocated capacitive sensors per segment. The actuator-sensor pairs are arranged in a triangular grid across each segment as sketched in Fig. 4 and are spaced $\approx \mathrm{31,5}\mathrm{mm}$ apart.²

We conducted a modal analysis of M4 and presented the obtained eigenmodes ${\mathfrak{m}}_k(r,\theta )$ of an M4 segment in a previous paper (see also Fig. 5).⁴⁷ Due to the finite number of M4 actuators, the modal METIS-SCAO controller can use up to 540 linear independent modes (calculated via a finite element analysis) per segment to correct wavefront disturbances. Among other, the Petersen–Middleton theorem for multidimensional sampling considering M4’s actuator pattern was used to determine the reduced set of applicable M4 modes. A detailed description of all methods and characteristics for determining the reduced modal set will be presented in future publications. Accordingly, the modal basis utilized in the METIS-SCAO controller is the union of the segments’ modal bases totalling 3240 modes.

Fig. 5

First four eigenmodes ${\mathfrak{m}}_k$ of an M4 segment sorted according to their respective eigenfrequencies (cf., Ref. 47).

M4’s shape is controlled via a local control system using the discrete actuator-sensor pairs. Therefore, the spatio-temporal dynamics of the locally controlled M4 segments, modally transformed with their eigenmodes ${\mathfrak{m}}_k$, are identical and decoupled MIMO system. The diagonal elements of these MIMO systems are

To prevent any damage to M4 during operation and windup phenomena due to internal checks of M4, the following (mechatronic) constraints are imposed to M4 at all times and also on the full mirror:

As the M4 segments are independent, the displacement and change constraints apply to each segment as well as M4 as a whole. Furthermore, the IAS is the stroke or shape difference between neighboring actuators of an M4 segment. Moreover, any violation of these constraints, defined in Cartesian coordinates, depends on all eigenmodes of the respective M4 segment, used in the modal METIS-SCAO controller. Acting as an additional protective mechanism besides the STEG, the aforementioned constraints are rechecked in M4’s local control system.

It is worth noting that this paper is a conceptual study and the presented values of the constraints ${\mathbb{L}}_d$, ${\mathbb{L}}_c$, and ${\mathbb{L}}_{\mathrm{ias}}$ are tentative data. Additionally, the constraints to be fulfilled or their combination will likely change in the further course of the ELT project, for instance, the IAS will be replaced by the closely related applied forces of M4’s actuators in the near future. The STEG and its corresponding algorithms can be easily adapted to accommodate force instead of IAS constraints.

2.3.

METIS-SCAO Control Concept

Our overarching goal for the METIS-SCAO system is the development of a SCAO controller providing the best wavefront correction to fully exploit the ELT’s spatial resolution capability. Since, e.g., METIS-SCAO system features non-negligible spatio-temporal dynamics of M4 and M5 as well as a high number of degrees of freedom (cf., Secs. 2.2 and 4.2), we use the METIS-SCAO controller shown in Fig. 2(b). This SCAO controller was extensively explained in Ref. 43 and it is the outer controller for the cascaded SCAO control loop. The numeric evaluation of the SCAO controller is partitioned among the real-time computers of the ELT (e.g., ${\mathcal{T}}_{\mathrm{tt}}$ and ${\mathcal{K}}_{\mathrm{tt},\mathrm{s}}$) and METIS (e.g., ${\mathcal{E}}_{\mathrm{tt}}$, ${\mathcal{K}}_{\mathrm{hm}}$, and $\mathcal{G}$).

First, in the SCAO controller, the wavefront error ${\mathfrak{e}}_{\mathrm{wf}}$ is split into its tip–tilt ${\mathfrak{e}}_{\mathrm{tt}}$ and higher modal (tip–tilt free) ${\mathfrak{e}}_{\mathrm{hm}}$ components via linear transformations (red blocks). Based on M4’s current tip–tilt correction ${\tilde{\mathfrak{y}}}_{\mathrm{M}4,\mathrm{tt}}$ and the error ${\mathfrak{e}}_{\mathrm{tt}}$, a modal main-secondary control, featuring decoupled I and PI controllers, computes the tip–tilt control inputs ${\mathfrak{u}}_{\mathrm{M}5}$ and ${\mathfrak{u}}_{\mathrm{M}4,\mathrm{tt}}$, represented in M4’s eigenmodes via ${\mathcal{T}}_{\mathrm{tt}}$, for the active mirrors (blue blocks, the main-secondary control has been usually referred to as master-slave control in the past.) In parallel, the modal control input ${\mathfrak{u}}_{\mathrm{hm}}$ for M4 correcting the higher modal errors ${\mathfrak{e}}_{\mathrm{hm}}$ is calculated via the STEG $\mathcal{G}$ and decoupled modal PI controllers (green blocks).

The STEG is located in the controller branch responsible for the higher modal error ${\mathfrak{e}}_{\mathrm{hm}}$ and modifies just this error. Moreover, the STEG takes into account M4’s tip–tilt command ${\mathfrak{u}}_{\mathrm{M}4,\mathrm{tt}}$ in addition to its shape ${\mathfrak{y}}_{\mathrm{M}4}$ and sends a freeze command to M4’s tip–tilt controller ${\mathcal{K}}_{\mathrm{tt},m}$ if the STEG is active. Hence, the STEG manipulates the tip–tilt main-secondary control as little as possible while enforcing the constraints of M4 and complies with all technical requirements for the METIS-SCAO control. Additionally, the tip–tilt controllers ${\mathcal{K}}_{\mathrm{tt},m}$ and ${\mathcal{K}}_{\mathrm{tt},s}$ feature integrator and output clipping to prevent excessive control inputs.

Since the M4 segments are independent, the STEG’s calculations are performed per segment. Therefore, the higher modal control input ${\mathfrak{u}}_{\mathrm{hm}}$ may contain a synthetic and quasi-static tip–tilt component if the STEG was active, which is gradually shifted to the main-secondary control via the bleed-off ${\mathfrak{e}}_{\mathrm{tt},\mathrm{bo}}$. Furthermore, the STEG is a central component of the METIS-SCAO system and all components of the SCAO controller are numerically evaluated at a loop rate $\le 1\mathrm{kHz}$.

2.4.

Reasoning and Objectives

The goal for this conceptual study is the consideration of M4’s constraints in the modal METIS-SCAO controller to prevent any damage to M4 during operation and windup phenomena due to internal checks of M4. Moreover, the corresponding strategy shall be a modular add-on to the previously developed SCAO controller (cf., Sec. 2.3) and shall not require a permanent trade-off between control performance and constraint compliance. In the following, we will summarize the reasons to use the STEG (cf., Sec. 1).

We dismissed an anti-windup for the METIS-SCAO controller, because these strategies focus on input constraints, whereas M4’s constraints are output constraints, and strongly dependent on the used controller (e.g., modification of controller states). Furthermore, we opted against LQ controllers since they incorporate soft constraints resulting in a permanently reduced METIS-SCAO performance. Moreover, we decided to not use MPC, as the design and real-time implementation of any variant of MPC is challenging for the METIS-SCAO system, being a high-dimension MIMO system running at high loop rate. Additionally, MPC is a fully integrated and non-modular control architecture. Since the METIS-SCAO system rejects arbitrary disturbances and its reference is always piece-wise constant, a reference governor is inapplicable.

An error governor is a promising concept to consider M4’s constraint, because it is an disentangled add-on to the pre-existing controller and requires no trade-off between performance and constraint fulfillment, for example. Nevertheless, the well-known Kapasouris–Tan type error governor features several drawbacks. First and similar to MPC, the design and real-time implementation of the required online optimization and evaluation of high-dimensional sets and control loop models is challenging. Moreover, a scalar scaling of the control error is detrimental in general, as a potential local constraint violations results in a global scaling of the relevant M4 segment’s control error and hence a serve global modification of the segment’s shape. Extending this type of error governor to a element-wise modification of the control error would increase the associated numeric workload tremendously. Furthermore, the Kapasouris–Tan type governor requires a precise model of the control loop and its state information. To circumvent the aforementioned disadvantages and exploit the general advantages of the error governor, we developed the STEG being presented subsequently.

3.1.

Notation

We refer to the area of an M4 segment as ${\mathbb{S}}_{\mathrm{M}4}\subset {\mathbb{R}}^2$, which is most easily expressed in polar coordinates. The generalized coordinates of an M4 segment are named $x=\{x_1,x_2\}\in {\mathbb{S}}_{\mathrm{M}4}$. Moreover, $\mathcal{M}$ is the linear transformation of M4’s shape from its Cartesian to modal (using M4’s eigenmodes ${\mathfrak{m}}_k$) representation and ${\mathcal{M}}^{-1}$ is its inverse. Furthermore, ${\mathcal{A}}^{-1}$ is the linear transformation from the modal coordinates of M4’s shape to the corresponding IAS. We specify the saturation of $f$, an arbitrary matrix or function, with respect to a convex set $\mathbb{L}=[l_{\mathrm{l}},l_u]\subseteq \mathbb{R}$ of constraints as

To reduce the computational costs of the STEG, we establish the operators ${\overline{\mathcal{S}}}_{\mathrm{M}4}$, ${\bar{\mathcal{S}}}_{\mathrm{OL}}$, and ${\overline{\mathcal{S}}}_{\mathrm{OL},c}$ approximating the dynamical behavior of M4 or the interconnection of M4 and ${\mathcal{K}}_{\mathrm{hm}}$. ${\overline{\mathcal{S}}}_{\mathrm{M}4}$ calculates an upper approximation of the M4 segment’s most extreme shape following the current time step $t$ based on the current and past shapes $y_{\mathrm{M}4}(t),y_{\mathrm{M}4}(t-1),\dots$ and tip–tilt control inputs $u_{\mathrm{M}4,\mathrm{tt}}(t),u_{\mathrm{M}4,\mathrm{tt}}(t-1),\dots$, i.e., it is an estimate of M4’s dynamic behavior [see, e.g., Eq. (11a)].

Using ${\overline{\mathcal{S}}}_{\mathrm{M}4}$, we can define the auxiliary variables $c$ and $o$

The operator ${\overline{\mathcal{S}}}_{\mathrm{OL}}$ calculates an upper approximation of the M4 segment’s most extreme shape in the future based on the current and last changes $c(t),c(t-1),\dots$ and offsets $o(t),o(t-1),\dots$, i.e., its an estimate of the open loop interconnection of M4 and ${\mathcal{K}}_{\mathrm{hm}}$ [see, e.g., Eq. (11b)]. Highly related to ${\bar{\mathcal{S}}}_{\mathrm{OL}}$, ${\bar{\mathcal{S}}}_{\mathrm{OL},c}$ computes an upper approximation of the most extreme time-discrete shape change in the future based on the current and past changes $c(t),c(t-1),\dots$ [see, e.g., Eq. (11c)]. The approximating operators ${\bar{\mathcal{S}}}_{\mathrm{M}4}$, ${\bar{\mathcal{S}}}_{\mathrm{OL}}$, and ${\overline{\mathcal{S}}}_{\mathrm{OL},c}$ may be obtained by analytic investigations or heuristic estimations of M4 and ${\mathcal{K}}_{\mathrm{hm}}$ (see e.g., Sec. 4).

We assume that all approximating operators perform a linear combination of their respective inputs, i.e., they are some modified linear difference equations. Additionally ${\overline{\mathcal{S}}}_{\mathrm{M}4}$ includes some linear transformations to merge the different modal spaces used, i.e., M4’s eigenmodes and tip–tilt. Hence, for the operators

3.2.

Working Principle

The STEG consists of two consecutive steps. At first, the STEG checks for violations of M4’s constraints using the approximating operators, the higher modal control error ${\mathfrak{e}}_{\mathrm{hm}}$, M4’s shape ${\mathfrak{y}}_{\mathrm{M}4}$, and M4’s tip–tilt control input ${\mathfrak{u}}_{\mathrm{M}4,\mathrm{tt}}$. If no (upcoming) violation is detected, the STEG just feeds the control error ${\mathfrak{e}}_{\mathrm{hm}}$ through [i.e., ${\mathfrak{e}}_{\mathrm{hm}}(t)={\stackrel{\sim }{\mathfrak{e}}}_{\mathrm{hm}}(t)$] and the SCAO performance is not impaired. Otherwise, the STEG modifies the control error such that all constraints of M4 will be fulfilled using again the approximating operators and the STEG’s input signals. The secondary objective for the modification of the control error is that the measured $e_{\mathrm{hm}}$ and the modified error ${\tilde{e}}_{\mathrm{hm}}$ match as closely as possible. Hereafter, the STEG’s first step is named constraint check, whereas the second is called error modification.

In the following three sections, we will present different algorithms realizing the two-step STEG procedure. These algorithms feature varying computational costs, implementation efforts, and differences between measured and modified control error (resulting in different performances of the METIS-SCAO system). Furthermore, the two step procedure itself reduces the computational costs of the STEG in general since violations of M4’s constraints are rare events.

3.3.

Optimization-Based Algorithm

To establish a reference case for all algorithms implementing the STEG, we present the so-called optimization-based algorithm in this section. The STEG’s constraint check is done by testing the extreme mirror shape and shape change, estimated via the approximating operators, on the M4 segment ${\mathbb{S}}_{\mathrm{M}4}$ (see Sec. 3.3.1). Subsequently, an optimization problem is solved to obtain the modified control error ${\tilde{\mathfrak{e}}}_{\mathrm{hm}}$ ensuring the fulfillment of all M4’s constraints (see Sec. 3.3.2).

3.4.

Iterative Heuristic Algorithm

Since a real-time implementation of the optimization-based STEG algorithm is challenging for the METIS-SCAO controller, we present the optimization-free so-called iterative heuristic algorithm. This algorithm conducts the STEG’s constraint check by testing M4’s extreme shape and shape change on subsets of its segment area ${\mathbb{S}}_{\mathrm{M}4}$ (see Sec. 3.4.1). Afterwards, the error modification is implemented using a set-based iterative procedure (see Sec. 3.4.2).

3.4.2.

Error modification

If the previous constraint check for the time step $t$ provides a non-empty set ${\mathbb{V}}_r$, a set-based iterative modification of the measured error ${\mathfrak{e}}_{\mathrm{hm}}(t)$ is performed to obtain the modified error ${\tilde{\mathfrak{e}}}_{\mathrm{hm}}(t)$ meeting all constraints of M4. The iterative error modification is subdivided into three steps, which are described below: initialization, iterative modification, and scaling fallback.

Constraint violations usually occur just at one or a few small areas of an M4 segment. Therefore, it is reasonable to just modify the measured error ${\mathfrak{e}}_{\mathrm{hm}}$ or, more specifically, its corresponding change $\frac{1}{2}{\mathcal{M}}^{-1}{\mathfrak{e}}_{\mathrm{hm}}$ in these small areas such that M4’s constraints will fulfilled. This is the basic idea for the presented set-based iterative modification.

Initialization

At first, an initial guess for the set-based iterative modification is calculated based on the required change $\frac{1}{2}{\mathcal{M}}^{-1}{\mathfrak{e}}_{\mathrm{hm}}(t)$. This is done using the sequence of equations

Eq. (15a)

$$c_0\leftarrowtail {\overline{\mathcal{S}}}_{\mathrm{OL}}^{-1}\left({\mathrm{sat}}_{{\mathbb{L}}_d}\right({\overline{\mathcal{S}}}_{\mathrm{OL}}(o,\frac{1}{2}{\mathcal{M}}^{-1}{\mathfrak{e}}_{\mathrm{hm}}(t),c_p)),o,c_p),$$

Eq. (15b)

$$c_0\leftarrowtail {\overline{\mathcal{S}}}_{\mathrm{OL},c}^{-1}({\mathrm{sat}}_{{\mathbb{L}}_c}({\overline{\mathcal{S}}}_{\mathrm{OL},c}(c_0,c_p)),c_p),$$

Eq. (15c)

$${\mathfrak{c}}_0\leftarrowtail \mathcal{M}{c}_0.$$

This sequence provides an initial spatial change $c_0$ that already meets the displacement and change constraints and its corresponding modal representation ${\mathfrak{c}}_0$.

Note that ${\mathcal{M}}^{-1}{\mathfrak{c}}_0\ne c_0$ in general, because the change $c_0$ may not be continuously partially differentiable due to the ${\mathrm{sat}}_{\mathbb{L}}$ operations applied in Eq. (15) and M4’s eigenmodes ${\mathfrak{m}}_k$ cannot represent all such changes $c_0$ exactly.

Iterative modification

After the initialization, the initial change $c_0$ is repeatedly updated to obtain the modified error ${\tilde{\mathfrak{e}}}_{\mathrm{hm}}(t)$, which will not cause any violations of M4’s constraints. This iteration comprises Eqs. (16) and (17) featuring the index $k=1,\dots ,N_{\max }$ and terminal index $N_{\max }$. Hence, the initial change $c_0$ could also be considered as some special ansatz shape used to determine the error ${\tilde{\mathfrak{e}}}_{\mathrm{hm}}$.

At the beginning of the $k$’th step of the iterative modification, a constraint check is done on the full segment area ${\mathbb{S}}_{\mathrm{M}4}$

Eq. (16a)

$${\mathbb{V}}_{k,d}={\overline{\mathbb{V}}}_{k,d}\cup {\underset{\_}{\mathbb{V}}}_{k,d}=\{x\in {\mathbb{S}}_{\mathrm{M}4}|{\mathcal{M}}^{-1}{\overline{\mathcal{S}}}_{\mathrm{OL}}(\mathfrak{o},{\mathfrak{c}}_{k-1},{\mathfrak{c}}_p)\notin {\mathbb{L}}_d\},$$

Eq. (16b)

$${\mathbb{V}}_{k,c}={\overline{\mathbb{V}}}_{k,c}\cup {\underset{\_}{\mathbb{V}}}_{k,c}=\{x\in {\mathbb{S}}_{\mathrm{M}4}|{\mathcal{M}}^{-1}{\overline{\mathcal{S}}}_{\mathrm{OL},c}({\mathfrak{c}}_{k-1},{\mathfrak{c}}_p)\notin {\mathbb{L}}_c\},$$

Eq. (16c)

$${\mathbb{V}}_{k,\mathrm{ias}}={\bar{\mathbb{V}}}_{k,\mathrm{ias}}\cup {\underset{\_}{\mathbb{V}}}_{k,\mathrm{ias}}=\{x\in {\mathbb{S}}_{\mathrm{ias}}|{\mathcal{A}}^{-1}{\bar{\mathcal{S}}}_{\mathrm{OL}}(\mathfrak{o},{\mathfrak{c}}_{k-1},{\mathfrak{c}}_p)\notin {\mathbb{L}}_{\mathrm{ias}}\},$$

using the latest modified change ${\mathfrak{c}}_{k-1}$. The subsets ${\overline{\mathbb{V}}}_{k,\star }$ and ${\underset{\_}{\mathbb{V}}}_{k,\star }$ contain all points $x\in {\mathbb{S}}_{\mathrm{M}4}$ violating the corresponding upper and lower limits of ${\mathbb{L}}_{\star }$, respectively.

If ${\mathbb{V}}_k={\mathbb{V}}_{k,\mathrm{d}}\cup {\mathbb{V}}_{k,\mathrm{c}}\cup {\mathbb{V}}_{k,\mathrm{ias}}=Ø$, the iteration breaks and the iterative heuristic STEG algorithms outputs ${\tilde{\mathfrak{e}}}_{\mathrm{hm}}(t)=2{\mathfrak{c}}_{k-1}$. Otherwise, the $k$’th iteration step is continued and finalized with the sequence of equations

Eq. (17a)

$$c_k\leftarrowtail c_{k-1}-{\mathrm{\Delta }}_{d,c}\forall x\in {\overline{\mathbb{V}}}_{k,d}\cup {\overline{\mathbb{V}}}_{k,c},$$

Eq. (17b)

$$c_k\leftarrowtail c_k+{\mathrm{\Delta }}_{d,c}\forall x\in {\underline{\mathbb{V}}}_{k,d}\cup {\underline{\mathbb{V}}}_{k,c},$$

Eq. (17c)

$$c_k\leftarrowtail c_k-{\mathrm{\Delta }}_{\mathrm{ias}}\forall x\in {\mathcal{P}}_{\mathrm{ias}}({\overline{\mathbb{V}}}_{k,\mathrm{ias}}),$$

Eq. (17d)

$$c_k\leftarrowtail c_k+{\mathrm{\Delta }}_{\mathrm{ias}}\forall x\in {\mathcal{P}}_{\mathrm{ias}}({\underset{\_}{\mathbb{V}}}_{k,\mathrm{ias}}),$$

Eq. (17e)

$$c_k\leftarrowtail {\mathrm{sat}}_{{\mathbb{L}}_{\mathrm{im}}}(c_k),$$

Eq. (17f)

$${\mathfrak{c}}_k=\mathcal{M}{c}_k.$$

Here, ${\mathrm{\Delta }}_{d,c}\in {\mathbb{R}}_{>0}$ is the modification step for displacement and change violations, ${\mathrm{\Delta }}_{\mathrm{ias}}\in {\mathbb{R}}_{>0}$ is the modification step for IAS violations, ${\mathcal{P}}_{\mathrm{ias}}$ is the IAS projection operator assigning a small surrounding area to each actuator location, and ${\mathbb{L}}_{\mathrm{im}}⊊\mathbb{R}$ is the limiting set for $c_k$. The application of the operator ${\mathcal{P}}_{\mathrm{ias}}$ is necessary to guarantee a sufficient influence of each IAS violation to $c_k$ and ${\mathfrak{c}}_k$. Furthermore, the saturation of $c_k$ via the chosen set ${\mathbb{L}}_{\mathrm{im}}$ in Eq. (17e) prevents excessive and unreasonable spikes.

It is worth noting that ${\mathcal{M}}^{-1}{\mathfrak{c}}_k\ne c_k$ in general, because $c_k$ is usually not continuously partially differentiable due to Eq. (17e) (see also Fig. 6). Moreover, this is the reason for applying repetitive modal forward and backward transformations in Eqs. (16) and (17). A modification of Eq. (17), which is beneficial for a significantly reduced number of M4’s eigenmodes, is to skip Eqs. (17c) and (17d) if

Eq. (18)

$$\mathcal{F}({\mathbb{V}}_{k,d}\cup {\mathbb{V}}_{k,c})>k_F\mathcal{F}({\mathbb{V}}_{1,d}\cup {\mathbb{V}}_{1,c}),$$

where $\mathcal{F}(\mathbb{V})$ is the operator calculating the area of the set $\mathbb{V}$ and $k_F\in [\mathrm{0,1}]$ is the relative area threshold.

Fig. 6

Iterative modification of the iterative heuristic STEG algorithm for an example featuring a small area with displacement violation. Each lower subplot already displays the violation area, which will be determined in the next iteration step. - - -: border of the area ${\mathbb{V}}_{k,d}$.

Figure 6 shows an example application and the intermediate results of the presented iterative error modification in Eqs. (15)–(17). In this example, the first five eigenmodes ${\mathfrak{m}}_{1,\dots ,5}$ of M4 are used and a single area with a displacement violation exists. The iterative heuristic STEG algorithms successfully terminates during iteration step $k=3$, because ${\mathbb{V}}_3=Ø$, and outputs ${\tilde{\mathfrak{e}}}_{\mathrm{hm}}(t)=2{\mathfrak{c}}_2$.

Scaling fallback

In the rare case that the iterative modification conventionally ends after the final iteration $N_{\max }$ and consequently provides no reasonable output ${\tilde{\mathfrak{e}}}_{\mathrm{hm}}(t)$ for the STEG, the fallback step is executed as explained below. The scaling fallback, as its name already implies, calculates the modified control error ${\tilde{\mathfrak{e}}}_{\mathrm{hm}}(t)$ by scaling the measured error ${\mathfrak{e}}_{\mathrm{hm}}(t)$. The scalar scaling factor $k_{\mathrm{s}}$ is determined via explicit equations, unlike in the optimization-based Kapasouris–Tan type error governor.

To calculate the scaling factor $k_{\mathrm{s}}$ for the fallback at the current time step $t$, we first calculate the auxiliary shapes

Eq. (19a)

$$C_d={\mathcal{M}}^{-1}\left({\overline{\mathcal{S}}}_{\mathrm{OL}}\right(\mathfrak{o},\frac{1}{2}{\mathfrak{e}}_{\mathrm{hm}}(t),{\mathfrak{c}}_p)-{\overline{\mathcal{S}}}_{\mathrm{OL}}(\mathfrak{o},0,{\mathfrak{c}}_p)),$$

Eq. (19b)

$$O_d={\mathcal{M}}^{-1}{\overline{\mathcal{S}}}_{\mathrm{OL}}(\mathfrak{o},0,{\mathfrak{c}}_p),$$

Eq. (19c)

$$C_c={\mathcal{M}}^{-1}\left({\overline{\mathcal{S}}}_{\mathrm{OL},c}\right(\frac{1}{2}{\mathfrak{e}}_{\mathrm{hm}}(t),{\mathfrak{c}}_p)-{\overline{\mathcal{S}}}_{\mathrm{OL},c}(0,{\mathfrak{c}}_p)),$$

Eq. (19d)

$$O_{\mathrm{c}}={\mathcal{M}}^{-1}{\overline{\mathcal{S}}}_{\mathrm{OL},c}(0,{\mathfrak{c}}_p),$$

Eq. (19e)

$$C_{\mathrm{ias}}={\mathcal{A}}^{-1}\left({\overline{\mathcal{S}}}_{\mathrm{OL}}\right(\mathfrak{o},\frac{1}{2}{\mathfrak{e}}_{\mathrm{hm}}(t),{\mathfrak{c}}_p)-{\overline{\mathcal{S}}}_{\mathrm{OL}}(\mathfrak{o},0,{\mathfrak{c}}_p)),$$

Eq. (19f)

$$O_{\mathrm{ias}}={\mathcal{A}}^{-1}{\overline{\mathcal{S}}}_{\mathrm{OL}}(\mathfrak{o},0,{\mathfrak{c}}_p),$$

using Eqs. (3) and (4), and the approximating operators. In Eq. (19) $O_{\star }$ corresponds to the approximated behavior of M4 without the influence of the control error ${\mathfrak{e}}_{\mathrm{hm}}$ and $C_{\star }$ to the respective influence of ${\mathfrak{e}}_{\mathrm{hm}}$, both defined in Cartesian coordinates on M4’s segment area ${\mathbb{S}}_{\mathrm{M}4}$.

Subsequently, the intermediate scaling factors ${\overline{k}}_{\star }\in \mathbb{R}$ and ${\underset{\_}{k}}_{\star }\in \mathbb{R}$ for the upper and lower limits of each violation type are computed

Eq. (20a)

$${\overline{k}}_{\star }=\frac{{\max }_{x_1{x}_2}(C_{\star })}{|\max ({\mathbb{L}}_{\star })-{\max }_{x_1{x}_2}(O_{\star })|},$$

Eq. (20b)

$${\underset{\_}{k}}_{\star }=-\frac{{\min }_{x_1{x}_2}(C_{\star })}{|\min ({\mathbb{L}}_{\star })-{\min }_{x_1{x}_2}(O_{\star })|},$$

with ${\max }_{x_1{x}_2}$ and ${\min }_{x_1{x}_2}$ providing the global maximum and minimum of their corresponding input, respectively. Based on the intermediate scaling factors, the overall scaling factor

Eq. (21)

$$k_s=\max (\{{\overline{k}}_d,{\underset{\_}{k}}_d,{\overline{k}}_c,{\underset{\_}{k}}_c,{\overline{k}}_{\mathrm{ias}},{\underset{\_}{k}}_{\mathrm{ias}},1\}),$$

for the fallback at the time step $t$ is calculated. The basic idea for this derivation of the scaling factor $k_s$ is that Eq. (20) provides the intermediate factors ${\bar{k}}_{\star }$ and ${\underset{\_}{k}}_{\star }$ required to meet the corresponding constraints of M4 and thus computing $k_s$ is straightforward. The tuple of intermediate scaling factors in Eq. (21) is augmented with 1 to prevent any amplification of the control error ${\mathfrak{e}}_{\mathrm{hm}}$, if all factors ${\overline{k}}_{\star }<1$ and ${\underset{\_}{k}}_{\star }<1$ [e.g., for a false positive detection of violations in Eq. (13)].

After determining the scaling factor $k_s$, the modified error ${\tilde{\mathfrak{e}}}_{\mathrm{hm}}(t)$ output by the iterative heuristic STEG algorithm is given by

Eq. (22)

$${\tilde{\mathfrak{e}}}_{\mathrm{HM}}(t)=\{\begin{array}{ll}0& \text{under the condition of Eq}.(23)\\ \frac{{\mathfrak{e}}_{\mathrm{HM}}(t)}{k_s}& \text{else}\end{array},$$

and

Eq. (23)

$$\exists x\in {\mathbb{S}}_{\mathrm{M}4}:O_{\star }(x)+V_{\star }(x)\notin [{\min }_{x_1{x}_2}(O_{\star }),{\max }_{x_1{x}_2}(O_{\star })]\wedge O_{\star }(x)\notin {\mathbb{L}}_{\star },$$

finalizing the scaling fallback at the current time step $t$. If the algorithm ends with the upper case of Eq. (22) active, we call its status “failed.”

Remarks

In the iterative heuristic STEG algorithm, neither the difference between the measured $e_{\mathrm{hm}}$ and modified ${\tilde{e}}_{\mathrm{hm}}$ control error nor the change difference $\mathrm{\Delta }c$ are considered explicitly. Nevertheless, the presented algorithm can provide a nearly optimal change difference $\mathrm{\Delta }c$ with respect to Eq. (12a) (see Sec. 4.1), depending on the selected tuning parameters (${\mathrm{\Delta }}_{d,c}$, ${\mathrm{\Delta }}_{\mathrm{ias}}$, ${\mathcal{P}}_{\mathrm{ias}}$, ${\mathbb{L}}_{\mathrm{im}}$, $k_F$).

The search for a suitable parameter configuration for the iterative-heuristic STEG algorithm is usually done through parameter simulation studies to benchmark and adjust the overall behavior of the algorithm. Furthermore, the algorithm’s overall behavior and a proper setting of its tuning parameters strongly depends on the number of applicable M4’s eigenmodes ${\mathfrak{m}}_k$. Besides the parallel evaluation of the STEG for all M4 segments, the modification of the control error for one segment can be performed independently with different parameter settings in parallel and the first reasonable result is output by the STEG.

3.5.

Scaling Heuristic Algorithm

To reduce the computational costs of the STEG algorithm as compared to the iterative heuristic algorithm (described in Sec. 3.4), we present the scaling heuristic algorithm. This algorithm is a heavily streamlined version of the iterative heuristic algorithm using its reduced constraint check and the scaling-based error modification.

3.6.

Characteristics of the STEG

The STEG in general provides the following benefits for the METIS-SCAO system:

In particular, the benefits no. 4 to 8 distinguish STEG and the Kapasouris–Tan type error governor.

Moreover, the computational costs and implementation effort of the presented STEG algorithms vary significantly. Regarding both categories, the optimization-based algorithm is very costly (especially for the METIS-SCAO system, cf., Sec. 3.3.2), whereas the iterative heuristic algorithm is moderately complex and the scaling heuristic algorithm is inexpensive. Furthermore, the optimization-based algorithm will usually provide the smallest change differences $\mathrm{\Delta }c$, whereas the scaling heuristic will typically provide the highest differences (cf., Sec. 4). Additionally, the presented STEG algorithms can be easily adapted to accommodate constraints other than the ones considered in this paper, such as limited forces of M4’s actuators.

Since the STEG algorithms operate independently on each M4 segment, their activity might generate some synthetic tip–tilt in the control input ${\mathfrak{u}}_{\mathrm{hm}}$ that is gradually shifted to the main-secondary control of the METIS-SCAO controller via the tip–tilt bleed-off ${\mathfrak{e}}_{\mathrm{tt},\mathrm{bo}}$ (see Fig. 2 and Sec. 2.3). Depending on the actual use case of the STEG, other combinations of the presented constraint checks and error modifications may be beneficial (e.g., check on full segment and scaling-based modification).

4.1.

Standalone Simulations

The purpose of the standalone simulations is to obtain a comprehensive overview of the algorithms’ behavior. In the standalone simulations, the STEG algorithms are supplied with 20,000 random offsets $\mathfrak{o}$ and errors ${\mathfrak{e}}_{\mathrm{hm}}$ for an M4 segment and subsequently the results of the algorithms are compared. The elements of the offsets $\mathfrak{o}$ and errors ${\mathfrak{e}}_{\mathrm{hm}}$ are normally distributed and feature progressively smaller standard deviations for increasing modal indices $k$. Moreover, the offsets $\mathfrak{o}$ are modified to meet the displacement and IAS constraints of M4 before they are fed into the simulation. The synthetically generated offsets $\mathfrak{o}$ and errors ${\mathfrak{e}}_{\mathrm{hm}}$ are not associated with some specific observation conditions at the ELT site, because no representative information on operating conditions resulting in guaranteed violations of M4’s constraints is available. Additionally, these synthetically generated offsets and errors encompass all combinations and highly variable constraint violations corresponding to th aforementioned simulation goal. Additionally, the STEG algorithms use 540 M4’s eigenmodes ${\mathfrak{m}}_k$ for the segment under investigation and are implemented in MATLAB except for the optimization in Eq. (12) solved by Gurobi.^48,49

Table 2 reports the 5%, 50%, and 95% percentiles of the relative objective functions ${\Vert \mathrm{\Delta }c\Vert }_2^2$ [cf., Eq. (12a)], referring to the optimization-based algorithm, obtained in the standalone simulations. The objective function ${\Vert \mathrm{\Delta }c\Vert }_2^2$ is a good indicator for difference between the measured $e_{\mathrm{hm}}$ and modified ${\tilde{e}}_{\mathrm{hm}}$ control error, and hence Table 2 indicate the relative impact of the STEG algorithms on the SCAO performance (see also Sec. 4.2). Furthermore, the standalone simulations revealed that the iterative heuristic algorithm typically completes its computations $\gg 10$ times faster and the scaling heuristic algorithm $\gg 100$ times faster than the optimization-based STEG algorithm on the same computer.

Table 2

The 5%, 50%, and 95% percentiles of the relative objective functions ‖Δc‖22, obtained in the standalone simulations.

The obtained results highlight the importance to consider M4’s spatio-temporal characteristics, because the scaling heuristic algorithm applies a global error modification, whereas the other algorithms modify the control error just locally. Moreover, the standalone simulations indicated that the optimization-based algorithm will introduce the lowest degradation of SCAO performance if active, while the scaling-heuristic algorithms will introduce the highest degradation (see also Sec. 4.2). Furthermore, the simulation results of the iterative heuristic algorithm strongly depend on the tuning parameters (${\mathrm{\Delta }}_{\mathrm{d},\mathrm{c}}$, ${\mathrm{\Delta }}_{\mathrm{ias}}$, ${\mathcal{P}}_{\mathrm{ias}}$, ${\mathbb{L}}_{\mathrm{im}}$, and $k_F$), which we selected to provide a good compromise between the computation time and objective function ${\Vert \mathrm{\Delta }c\Vert }_2^2$.

4.2.

Closed Loop Simulations

In the closed loop simulations, the functionality of the STEG and its algorithms within a METIS-like SCAO system (cf., Fig. 2 and Sec. 2) is verified. These simulations are conducted in a modified version of the AO simulation tool COMPASS,⁵⁰ extended by the METIS-SCAO controller described in Sec. 2.3 and the full STEG for all M4 segments. In particular, the SCAO controller and the STEG are implemented in MATLAB and Gurobi.^48,49 In Ref. 43, we presented similar closed loop simulations without constraints characterizing the general behavior of the METIS-SCAO system.

The closed loop simulations features a round 37 m pupil with a round 11 m central obstruction as well as six equally spaced spiders of 0.5 m width. Furthermore, atmospheric and synthetic wavefront disturbances are present. The atmospheric disturbances are represented by a 35-layer model, corresponding to medium observation condition at the ELT site and resulting in no violations of M4’s constraints. To push M4 toward and beyond its constraints, the synthetic disturbances are applied in the simulations, which correspond to erroneous wavefront reconstructions (e.g., piston) or mechanical shocks of the ELT and METIS (e.g., tip and defocus) defined in Zernike modes (cf., Refs. 44 and 45) over the entire pupil. The disturbed wavefront is sensed and reconstructed at the $\text{wavelength}=2.2\mu \mathrm{m}$ with a normally distributed error featuring $\mathrm{rms}=150\mathrm{nm}$. Moreover, all communication and computation delays of the METIS-SCAO system account for a total time delay $=2\mathrm{ms}$.

We designed the transfer functions of the modal controllers ${\mathcal{K}}_{\mathrm{hm}}$, ${\mathcal{K}}_{\mathrm{tt},m}$, and ${\mathcal{K}}_{\mathrm{tt},s}$ within the METIS-SCAO controller (cf., Fig. 2) for the closed loop simulations as

Figures 7 and 8 show the results of the closed loop simulations. We split the simulations into two batches with IAS constraint disabled (see Fig. 7) and enabled (see Fig. 8), because not all constraints of M4 can be tested in one simulation setup using the low-order disturbances motivated above. Moreover, the depicted results adopt the operators’ coefficient $k_{\mathrm{C}}=0.6$ since the STEG would limit the actual change of M4’s shape very conservatively for $k_{\mathrm{C}}=1$ and $k_{\mathrm{C}}=0.6$ is an adequate choice for all reasonably tuned controllers ${\mathcal{K}}_{\mathrm{hm}}$. The STEG status for the entire mirror M4 is obtained as follows from the algorithm states for the individual M4 segments:

Fig. 7

Results of the closed loop simulations with enabled M4’s constraints except the IAS constraint. Note that no synthetic tip–tilt is calculated if the STEG is on. The most extreme modal amplitudes of the normalized synthetic disturbances are inscribed in the corresponding plot. Blue lines: optimization-based algorithm; red lines: iterative heuristic algorithm; green lines: scaling heuristic algorithm; dash-dotted lines: M4’s constraints. For tip-tilt plots: tip is solid, tilt is dashed, and arcsec stands for arcsecond.

Fig. 8

Results of the closed loop simulations with all M4’s constraints enabled. Note that the displacement and change constraints are always met with large margins and that no synthetic tip–tilt is calculated if the STEG is on. The most extreme modal amplitudes of the normalized synthetic disturbances are inscribed in the corresponding plot. Blue lines: optimization-based algorithm; red lines: iterative heuristic algorithm; green lines: scaling heuristic algorithm; dash-dotted lines: M4’s constraints. For tip-tilt plots: tip is solid, tilt is dashed, and arcsec stands for arcsecond.

The simulation results show that the STEG and its algorithms are working as intended by successfully enforcing M4’s constraints, preventing any windup phenomena and not impairing the SCAO performance if all constraints of M4 are inactive (cf., Sec. 2.4). Furthermore, the wavefront error $e_{\mathrm{wf}}$ and the SCAO performance with active constraints degrades significantly for all STEG algorithms due to the synthetic disturbances. In particular, the scaling heuristic algorithm causes the highest performance degradation, whereas the impact of the iterative heuristic algorithm is marginally worse than that of the optimization-based one (cf., Sec. 4.1).

Some violations of M4’s constraint are perceptible, caused by a high velocity of M4 and the violated assumptions used to derive the STEG’s approximating operators. However, all STEG algorithms recover from the constraint violations caused by the applied synthetic disturbances. Additionally, the used approximating operators can be selected differently to better represent the METIS-SCAO system for such rare circumstances.

Moreover, Fig. 7 presents a much higher and quasi-static extreme displacement of M4 for the optimization-based and iterative heuristic algorithms compared to the scaling heuristic algorithm. This quasi-static displacement appears on the non-illuminated outer region of M4 after the STEG was active and results in high values of the extreme IAS. Since the IAS constraint is disabled in the corresponding batch of simulations and no other constraint is violated, the quasi-static extreme displacement persists on the non-illuminated M4 region in contrast to the simulations with an enabled IAS constraint.

Furthermore, all STEG algorithms properly cooperate with the tip–tilt limits imposed on the tip–tilt controller ${\mathcal{K}}_{\mathrm{tt},m}$. Additionally, only the scaling heuristic algorithm finished with the status “failed” in the simulations, i.e., the upper case of Eq. (22) was activated, and generated significant amounts of synthetic tip–tilt in the (tip–tilt free) control input ${\mathfrak{u}}_{\mathrm{hm}}$. It is also worth mentioning that for the tip–tilt disturbances in Fig. 8, the iterative heuristic algorithm is inactive sooner than the optimization-based algorithm due to the iterative non-optimal error modification.

Combining the depicted results and the previous discussion, the STEG and all three algorithms presented successfully enforce M4’s constraints and are applicable to the METIS-SCAO controller. A recommendation whether and which STEG algorithm should be implemented on the METIS real-time computer depends on trade-off analysis (e.g., computational costs, impact on SCAO performance for active constraints) currently in progress.