Dynamic Analysis of the Actively-Controlled Segmented Mirror of the Thirty Meter Telescope Douglas G

1 Dynamic analysis of the actively-controlled segmented mirror of the Thirty Meter Telescope Douglas G. MacMartin, Peter M. Thompson, M. Mark Colavita and Mark J. Sirota

Abstract—Current and planned large optical telescopes use a segmented primary mirror, with the out-of-plane degrees of freedom of each segment actively controlled. The primary mirror of the Thirty Meter Telescope (TMT) considered here is composed of 492 segments, with 1476 actuators and 2772 sensors. In addition to many more actuators Sensors (12) and sensors than at existing telescopes, higher bandwidths are desired to partially compensate for wind-turbulence loads on the segments. Control-structure-interaction (CSI) limits the achievable bandwidth of the control system. Robustness can be further limited by uncertainty in the interaction matrix that relates sensor response to segment motion. The control system robustness is analyzed here for Actuators (3) the TMT design, but the concepts are applicable to any segmented-mirror design. The key insight is to analyze the Fig. 1. Conceptual image of the Thirty Meter Telescope design (left), structural interaction in a Zernike basis; rapid convergence and detail of one primary mirror segment (right). with additional basis functions is obtained because the dynamic coupling is much stronger at low spatial-frequency than at high. This analysis approach is both computational efﬁcient, and provides guidance for structural optimization to minimize CSI. Index Terms—Telescopes, Control-structure-interaction

I.INTRODUCTION Optical telescopes with primary mirror (M1) diameters larger than about 8.5 m use a segmented primary mirror, relying on active control of the out-of-plane degrees of freedom to maintain a smooth optical surface; an Fig. 2. The 492-segment primary mirror of TMT (left), and segment approach pioneered by the Keck telescopes [1], [2]. actuator and sensor locations (right). Each segment has three position While the Keck telescopes each have 36 segments, the actuators (‘+’) and two sensors on each inter-segment edge (‘•’) that measure relative displacement, for a total of 1476 actuators and 2772 design for the Thirty Meter Telescope (Fig. 1 and 2) sensors. has 492 [3], while the 39m European Extremely Large Telescope (E-ELT) design has 798 [4]. The primary mirror control system (M1CS) for these actuators on each segment (see Fig. 2), with an overall designs builds on the approach used at Keck, with surface precision of order 10nm rms (though low spatial feedback from edge sensors used to control position frequency motion can be larger). However, for future telescopes, the problem is more challenging because Manuscript submitted to IEEE TCST. D. MacMartin (formerly MacMynowski) is with Control & Dynam- of the greater number of segments, sensors and actu- ical Systems, California Institute of Technology, Pasadena, CA 91125 ators, higher desired control bandwidth, and stringent USA, [email protected]. performance goals. Aubrun et al. [1], [5] conducted the P. Thompson is with Systems Technology Inc., Hawthorne CA. M. Colavita is with the Jet Propulsion Laboratory, Pasadena CA. dynamic control-structure-interaction (CSI) analysis of M. Sirota is with the TMT Observatory Corporation, Pasadena, CA. the Keck observatory primary mirror control system, and furthermore suggested that for a given structure, Position Actuator Wind command force forces Segment Edge the destabilizing effects scale linearly with the number motion sensor Segment A K K of control loops [6]; a potential concern given the global act (492)

large number of segments in planned optical telescopes. The purpose of this paper is to describe the dynamic A# analysis of segmented-mirror control for large arrays of Telescope encoder segments, and for TMT in particular, 25 years after the structure Actuator corresponding analysis for Keck was published [1]. In addition to the quasi-static gravity and thermal de- formations controlled at Keck, M1CS at both TMT and Fig. 3. Block diagram showing control loops, both “local” actuator E-ELT will provide some reduction of the response to servo loops (Kact) and “global” edge-sensor based feedback (Kglobal, # unsteady wind turbulence forces on the primary mirror. A ); the input and output of both Kact and Kglobal have dimension 1476. The dynamics of the segments and control loops will be coupled The increased bandwidth required to do so also requires to the telescope structure (coupling points marked by solid circles) in more careful attention to CSI than was required for a different basis as described in Sec. III and IV. Keck. Furthermore, in addition to the “global” feedback from edge-sensors, TMT will use voice-coil actuators to control each segment; these are stiffened with a time controller is not an issue; if it were, then approaches relatively high-bandwidth servo loop using collocated developed for adaptive optics can easily be extended encoder feedback within the actuator; CSI must also be to this problem, e.g. [22], [23]. The analysis herein analyzed for these control loops. focuses only on the out-of-plane degrees of freedom of each segment; in-plane motion does couple with the Finally, analysis would be incomplete without address- out-of-plane control [24], but the effects are essentially ing one further complication that results from the large quasi-static and can be separately analyzed. Sensor noise number of segments. The edge-sensor based feedback propagation can also be separately understood [7], [25], relies on knowledge of the interaction matrix that relates although this may also limit the desired bandwidth of sensor response to segment motion, in order to estimate poorly observed modes. the latter from the former [7]. The condition number The next section introduces the control problem in of this matrix increases with the number of segments, more detail, followed by analysis in Sec. III of a simpli- and thus small errors can result in large uncertainty in ﬁed problem that contains the most important features the control system gain [8], [9]. Additional analysis is of the full problem. The insights obtained are then used required to ensure simultaneous stability in the presence in Sec. IV to compute CSI robustness for TMT. Finally, of both this effect and CSI. Sec. V introduces interaction-matrix uncertainty and the Scaling effects for both dynamics and control of large analysis required to prove simultaneous stability to this arrays of segments have been addressed in [10], [11], and and CSI. multivariable CSI robustness of the global control loop in [12], using a more conservative test than the one applied here (noted later). Progress in CSI analysis for TMT has II.CONTROLPROBLEM been described in a sequence of papers [9], [13]–[17], A block diagram for the control problem is shown and similar analyses for the European ELT in [18]–[21]. in Fig. 3. Each segment of the mirror is controlled by The key observation that allows for both rapid analysis three position actuators (see Fig. 2), leading to a total of and design intuition is that the segment dynamics can be 1476 actuators for TMT. Several different actuator tech- analyzed in any basis. For a realistic control bandwidth, nologies have been considered, and voice-coils selected the coupling with the telescope structure is primarily based in part on low transmission of higher-frequency an issue at low spatial frequencies. As a result, using vibrations to the mirror surface. Stiffness is obtained a Zernike basis (or something similar) yields rapid using feedback from a local encoder with a bandwidth convergence of stability and robustness predictions and of 8–10 Hz; each actuator uses the same controller. The does not require analysis with all 492 segments of the interaction of these 1476 control loops with the structural primary mirror. A higher control bandwidth may require dynamics is the most challenging CSI concern for TMT. more basis vectors to predict robustness. For an individual segment mounted on a rigid base Several additional aspects to the segmented-mirror (rather than on the telescope structure), the uncontrolled control problem are worth noting. For the desired closed- segment behaves roughly as a mass (mirror segment) loop bandwidths, the computational burden of the real- on a spring (actuator open-loop spring stiffness), with

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Fig. 4. Open-loop actuator frequency response (force to collocated encoder position) for a segment mounted on a rigid base, with (dashed) and without passive damping. The high frequency resonance results from internal dynamics within the segment assembly. The largest compliance that determines the lower resonant frequency comes from an offload spring within the voice-coil actuator. The piston response Fig. 5. Schematic (a) of n identical oscillators coupled through a (three actuators on a segment driven together) is shown; the tip and supporting structure, with disturbance forces fi and control inputs ui; tilt responses are similar. this simplified system captures important features of the full telescope problem. With simplifying assumptions, a change of basis leads to n decoupled systems of the form (b), where M and Ki are associated with the support structure. a resonance near 8Hz (the segment piston and tip/tilt resonances are not quite at the same frequency), with the frequency response shown in Fig. 4. The addition of Any sensor set that measures relative segment motion eddy-current based passive damping within the actuator results in the global rigid-body motion of the full mirror makes control design much more straightforward, as will (piston, tip and tilt) being unobservable (A is rank be seen when the dynamics of the telescope structure are deficient). The edge sensors at TMT are also sensitive accounted for. The segment first resonance damping ratio to the dihedral angle θ between segments (rotation about in Fig. 4 is ζ =0.75. the shared edge): the sensor output is a linear combina- With the local actuator control loops closed, they tion of the relative height between segments, and Leff θ, behave as position actuators for the global control loop. where the effective moment arm Leff has units of length. The global control uses feedback from edge-sensors Without dihedral sensitivity, “focus-mode” would also between neighbouring segments to maintain the optical be unobservable; this pattern corresponds to a uniform continuity of the mirror, with a bandwidth of order 1Hz. dihedral change for all segments, resulting in a change in Differential capacitive sensors [26] measure the relative the overall M1 radius of curvature. With practical values edge height discontinuity, similar to the approach used at of Leff and with many segments, focus-mode estimation Keck; with two segments per edge there are 2772 sensors in particular is sensitive to uncertainty in the matrix for TMT. A, and thus control of this mode in particular will be The relationship between the segment motion at the constrained to a lower bandwidth. actuator locations, x, and the edge-sensor response y, can be expressed through geometry [7] as III. PRELIMINARY ANALYSIS y = Ax + η (1) Before considering the control system dynamics with with sensor noise η. The global control loop involves the full telescope structural model, it is useful to first first estimating x from y using the pseudo-inverse of use some simplifying approximations to explore some A, and then computing control commands. At Keck the general characteristics of the problem. The schematic in control is calculated for each actuator (“zonal control”); Fig 5(a) illustrates important features of the dynamics: for future telescopes, control will be calculated in a there are many identical subsystems (mirror segments) modal basis such as that obtained from the singular value coupled to each other through the telescope structure. decomposition of A (e.g. [12], [14]). The key observation that simplifies analysis is that a

3 diagonal system of identical subsystems remains diag- That is, the dynamics decouple into n independent onal under any change of basis. Thus if the dynamics of coupled-oscillator systems, as shown in Fig. 5(b). an individual segment are written as g(s), then for any Define ω = k/m as the oscillator natural frequency unitary matrix φ: if mounted onp a rigid support, and the mass and frequency ratios g(s) 0 ··· 1/2 . nm (Ki/M) φT G(s)φ = G(s) where G(s) =  0 ..  µ = and Ω = (6) M ω  .   . g(s)    Then for each basis function i (dropping the subscript for clarity) we have: where we assume that the dynamics of each segment are identical (this is a very good approximation). p¨ 2 1 −1 p As an example, consider the case where the dynamics + ω 2 q¨ −µ µ +Ω q of the support structure can be described solely by the 1 1 1 1 n displacements z at the segment locations (so that = f˜ + u˜ (7) i 0 −µ it has exactly n degrees of freedom and n structural m m modes), and has uniform mass distribution. While this Scaling frequency by ω, the transfer function from a is not a realistic assumption for design, it is sufficient displacement input (˜u/k) to output p is: to illustrate some key scaling laws. For this case, the s2 +Ω2 modes of the structure evaluated at the segment mounting 4 2 2 2 (8) locations provide an orthogonal basis for transforming s +(1+Ω + µ) s +Ω the segment dynamics. The transformation results in and the two modes are at frequencies n decoupled systems that each describe the coupling 1 2 2 / between one structural mode and the corresponding 1+ µ +Ω ± µ2 + 2µ + 2µΩ2 + 1 − 2Ω2 +Ω4 2 pattern of segment motion. That is, in this case, there p ! exists a basis that simultaneously diagonalizes both the (9) supporting structure and the segment dynamics. corresponding to in-phase and out-of-phase oscillation We start by ignoring damping for simplicity, although between the structural mode and the corresponding pat- it will of course be critical to the control design problem, tern of oscillator motion. If the support structure stiffness and we represent each segment by a single degree of is small compared to the oscillator stiffness (Ki nk), freedom rather than three. Define x, z ∈ Rn as the then to first order the lower resonant frequency (normal- vectors of segment and structure displacement, and u, ized by ω) is Ω f ∈ Rn the control inputs and disturbance forces. The (10) dynamics of the ith segment are described by (1 + µ) which is just mass-loadingp of the telescope structure mx¨ + kx = f + u + kz (2) i i i i i resonance. For small mass ratio µ (support structure The coupling structure dynamics are described by massive compared to the total mass of the oscillators), then the systems decouple. With damping b added in Mz¨ + Kz = −u + k(x − z) (3) parallel with the actuator, as in the TMT actuator design, then the zeros of the transfer function are unaffected where K is the stiffness matrix, and the mass matrix (these correspond to zero motion across the actuator). An M = (M/n)In×n because of the assumed uniform mass approximate formula for the damping of the two modes distribution, with M the total support structure mass, n can be derived by neglecting the shift in the imaginary the number of segments, and In×n the identity matrix. part of the eigenvalues relative to their undamped values: For any orthonormal basis φ ∈ Rn×n, with p = φT x, q = φT z, f˜ = φT f and u˜ = φT u, then b 1 − µ − Ω2 2ζ ' 1 ± (11) 2 µ2 + 2µ + 2µΩ2 + 1 − 2Ω2 +Ω4 ! mpï + kpi = f˜i +˜ui + kqi (4) For small µ, thep mode involving mostly segment mo- Furthermore, if φ are the modes shapes of the support tion is significantly damped, while the mode involving T structure, so that φ diagonalizes K, then φi Kφi = Ki primarily mirror cell motion is only slightly damped. and Fig. 6 compares the frequency response from eq. (8) Mqï + Kiqi + nkqi = −nu˜i + nkpi (5) with the frequency response for Zernike focus for TMT,

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Phase (deg) −150 −60 −200 0 1 2 10 10 10 −80 25 Frequency (Hz) −2 −1 0 1 −100 −80 −60 −40 −20 0 Fig. 6. Actuator open-loop frequency response for TMT focus Real part of pole mode (without added passive damping, solid), compared with the approximate response from eq. (8) (dashed); the amplitude of the latter 100 is scaled to match the static gain. 80

60 using the models described in the next section. Actuator 40 damping is not included for ease of comparing the 20 resonances. There is a single resonance of the telescope 0 structure (shown in Fig. 9) that predominantly projects 35 −20 onto Zernike focus. The mass and stiffness values Ω = −40 0.81 and µ =0.26 provide a good fit to the behavior for Imaginary part of pole 30 the projection onto this Zernike. −60 A representative root locus for these values of Ω −80 25 −2 −1 0 1 and µ is sketched in Fig. 7, using a PID controller. −100 Control design is straightforward for the uncoupled −80 −60 −40 −20 0 system, however this controller destabilizes the coupled Real part of pole structural mode when the segment is mounted on the Fig. 7. Root locus for actuator servo loop, using mass and stiffness flexible telescope structure. The extent of destabilization from TMT focus mode, and a PID controller (which yields the damped depends on the frequency separation of the pole and zero, zeros). Without any passive damping added, the closed-loop system with these parameters would be unstable (see inset). The addition which again depends on the mass ratio (Eq. 10). Adding of passive damping in parallel with the actuator makes the control passive damping to the actuator damps both modes and problem easier (bottom panel). increases the maximum stable gain of simple controllers, but the gain will always be limited by the destabilizing interaction with the coupled structural dynamics. challenge than lower frequency resonances. The case in Fig. 7 corresponds to Ω = 0.82, for The main observations from this simple analysis are as a structural resonance relatively close to the segment follows. First, that much can be gained by analysis in an resonance. Fig. 8 illustrates the behaviour for higher appropriate basis set (as opposed to considering individ- frequency structural modes (using Ω=1.6). With no ual segment motion). Second, recall that the analysis in damping, the root locus topology is similar to before, [6] suggested that destabilization due to CSI was approx- although now it is the lower frequency pole that involves imately linear in the number of control loops. While this more segment motion, and thus the order of the pole is true for a given structure, it is the mass ratio (nm)/M and zero introduced by the coupling to the structure is that is the relevant parameter. Increasing the number of flipped relative to before. With passive damping added, segments while keeping the areal density constant does both modes now have more damping, following from not affect stability. Third, the lowest frequency support Eq. (11) and the higher value of Ω. The added damping structure resonances will decrease in frequency relative and the shift in pole-zero order lead to resonances above to their uncoupled values by an amount that again the segment support resonance being less of a robustness depends on the mass ratio, leading to a pole-zero pair that

5 the main telescope model. This approach ensures that 100 the desired segment dynamics are retained regardless 80 of any model reduction performed on the main tele- 60 scope structural model, and allows ﬂexibility in choosing

40 what segment dynamics to include – only the dynamics associated with retained basis vectors are needed, as 20 described below. Model validation has been conducted 0 by constructing two fully independent models, one in- −20 terconnecting the component models in state-space, and the other in the frequency domain; both yield identical

Imaginary part of pole −40 results for CSI predictions. −60 The structural damping is assumed to be 0.5% (e.g. −80 Keck damping is in this range [27]). From Fig. 7 this is −100 a critical assumption, since it determines the damping of −80 −60 −40 −20 0 the zeros, which are unaffected by any actuator passive Real part of pole damping. Fig. 8. Root locus as in Fig. 7 but with structure stiffness increased by a factor of four (correspondingto a structural mode at higher frequency than the segment resonance). The case with no damping is shown B. Zernike basis in black and is qualitatively similar to before. However, with passive The structural modes of the telescope do not give an damping (red), then there is now more damping on both modes. orthonormal basis for describing segment dynamics (that approximation might be reasonable if the mirror cell is a challenge for robust control design. The addition of supporting the segments was the only flexible component passive damping simplifies the control problem. Finally, of the telescope). However, it is still useful to project higher frequency structural resonances are both better the dynamics onto a different basis. Instead of modes, damped by added actuator passive damping, and the we choose a Zernike basis (the natural basis on a order of the zero and pole are flipped in frequency, and circle; polynomials of degree p in radius, and sines thus these present less of a challenge for CSI. or cosines azimuthally), which we modify slightly to orthonormalize at the 492 segment locations to give a unitary transformation. If we included 492 basis vectors, IV. CSI ANALYSIS FOR TMT there would be no computational savings relative to the A. Structural models original untransformed system. However, the stability We will rely on the previous analysis to provide guid- characteristics can be accurately predicted with relatively ance in understanding the characteristics of the actual few basis vectors because the coupling is dominated by telescope system. We first briefly introduce the structural the most compliant and hence lowest frequency modes models we use, describe the shift to a different basis for of the supporting structure. These are also the lowest control, and then analyze CSI for both the actuator servo wavenumber modes, and thus predominantly project onto loops and the global control loop. the lowest order Zernike basis vectors. Fig. 9 shows the The full CSI analysis for TMT relies on the finite- mode shape for a representative low-frequency (9.3 Hz) element model (FEM) of the telescope structure. For structural mode. Although this particular mode is not ease of model reduction while retaining both accuracy exactly Zernike-focus, the mode is extremely well cap- and flexibility in modeling the segment dynamics, the tured by its projection onto the lowest 15 Zernike basis segments are not included in the telescope FEM. A vectors (up to radial degree 4). For high wavenumber modal model is obtained from the FEM; 5000 modes motion that involves significant relative motion between (up to nearly 100Hz) are extracted, although only a few neighbouring segments, the support structure is relatively dozen low frequency modes matter for CSI. Typically stiff (see Fig. 10). 500 modes (up to 30Hz) are retained, with the static Note that, as in Fig. 9, any structural mode will project correction included for truncated modes; convergence onto multiple basis vectors, and conversely, any basis with the number of modes retained has been verified. vector will include dynamics associated with multiple Because the segment model is replicated up to 492 modes, and thus multivariable analysis is still required. times, a simple lumped-mass model is fit to the detailed Although we do not rely on this, for TMT the Zernike- FEM of an individual segment before coupling with basis nearly diagonalizes the structural dynamics, and

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Fig. 9. Mode shape, evaluated on the primary mirror, of two Fig. 10. Static compliance of telescope structure on Zernike basis. representative structural modes of the telescope, and their projection The horizontal line illustrates the segment static compliance for com- onto a Zernike basis (rms of each component normalized by the overall parison; at low spatial frequencies the structure is soft compared to the rms across M1). The ﬁrst is the predominant mode associated with segments, at high the reverse is true and the coupling is small. Zernike focus (e.g. in Fig. 6); over 93% of the rms displacement is captured by the projection onto Zernike-focus, and over 99% of the rms captured by the projection onto basis vectors of radial degree 4 −5 and lower. The second illustrates that not all modes project entirely 10 onto a single Zernike, nonetheless over 90% of the rms is associated −6 with either astigmatism or coma, and again, 99% of the rms is captured 10 by the projection onto basis vectors of radial degree 4 and lower. −7 10 Magnitude −8 10 0 1 2 indeed SISO analysis for each Zernike is a good predic- 10 10 10 tor of the multivariable analysis. It is not immediately Frequency (Hz) obvious why this should be true. However, the mirror 0 cell that supports the segments is a truss that can be −50 reasonably approximated at low spatial frequencies as a −100 uniform circular plate, with corresponding ﬂexible mode

Phase (deg) −150 shapes similar to Zernike basis functions. −200 0 1 2 10 10 10 Frequency (Hz) C. Actuator servo loop The transfer function between voice-coil force and Fig. 11. Actuator frequency response on telescope structure, Zernike the nearly-collocated encoder position for a segment on basis, including the first 21 basis elements (up to radial degree 5). This includes focus-mode, for which the frequency response without a rigid base was shown in Fig. 4, with and without actuator damping was shown in Fig. 6. The solid black line corresponds additional passive damping. The damping results in a to a segment mounted on a rigid base (the damped case from Fig. 4). significantly easier problem for control, as suggested by the root locus for the simplified system in Fig. 7; any structural mode that has non-zero motion across to be less than two; this is a reasonable margin in the the actuator will be at least slightly damped (and those absence of a specific understanding of the structure and modes that do not, do not matter). For a single segment magnitude of the uncertainty (e.g. gain margin of two). mounted on the telescope structure, the transfer function Note that [12] considers the dynamics to be an uncertain is similar to the rigid-base case, and indeed it might not perturbation on the static response, and uses the dynamic be obvious that there is any potential stability problem. model to estimate the size of the uncertainty bound, However, the coupling is clear when the dynamics are while here we include the dynamics as part of the best transformed into a Zernike basis, as shown in Fig. 11. estimate of the plant, and require robustness to additional The multivariable robustness metric used here is to uncertainty on the model. Either approach is reasonable require the maximum singular value of the sensitivity for the global control loop (considered in [12]) where the

7 bandwidth is much lower than the structural resonances, but the approach of [12] is too conservative to allow any control design for the higher bandwidth servo loop [20].

0 Because the encoder is nearly collocated with the 10 actuator, the transfer function will be phase-bounded regardless of the structural coupling. Thus, rather than relying solely on the model-predicted sensitivity, we

rely on collocation and phase stability between 15 and Magnitude 300Hz, and a high gain margin above 300Hz where collocation may not hold. The control design used here is a simple PID with high-frequency roll-off, tuned so that the desired robustness margin is satisfied; it is not −1 the details of gain choices that is important, but rather 10 −220 −200 −180 −160 −140 −120 −100 −80 the lessons learned. Phase (deg) With any particular choice of controller, nominal stability could be established by taking eigenvalues of the Fig. 12. Nichols plot for characteristic transfer functions (CTFs) full system with all segments, but this is computationally of servo loop, illustrating convergence of stability and robustness intensive and does not provide useful design guidance. calculations with Zernike basis. Blue lines show the Nichols plots of the CTF for the full 1476×1476 system, while magenta lines show Sedghi et al. [20] instead use characteristic transfer the CTF Nichols plots calculated only for the first 6 Zernike basis functions or CTFs [28] to prove stability: taking the elements (radial degree p ≤ 2); these are similar for the least-stable elements of the full CTF. The Nichols plot corresponding to a single eigenvalues qi(jω) of the transfer function matrix at each segment mounted on a rigid base is also shown for comparison (black, frequency, then the multivariable system is stable if the thick line). The red oval indicates a peak sensitivity of two. closed-loop system is stable for each qi. However, rather than computing eigenvalues of the full 3nseg × 3nseg system as in [20], in Fig. 12 we show that these CTFs basis, results converge almost immediately, since the converge rapidly if the system is first transformed into a worst-case structural modes project almost entirely onto Zernike basis. This amounts to a two-step procedure for low-order Zernike basis functions (mostly radial degree proving stability: retaining relatively few Zernike basis one, and some onto radial degree two), and there is only elements results in a system with many fewer inputs and a small increase in the peak sensitivity with further basis outputs; a second frequency-dependent diagonalizing functions added. transformation is then used to evaluate nominal stability The multivariable peak sensitivity is shown in Fig. 14 for this smaller subset, since the Zernike-transformed where only Zernike basis vectors up to a given radial system is still not diagonal. The effect of the neglected degree p are included. The peak sensitivity is remarkably higher-order Zernike basis elements on the first few well predicted by SISO analysis with each Zernike CTFs is small (i.e., diagonal dominance is satisfied), separately, shown in Fig. 15. While the system is not and it is these first few CTFs that matter most for sufficiently diagonally dominant to directly infer stability stability. Starting with a Zernike transformation to isolate without relying on the CTFs shown in Fig. 12, it is the structural dynamics that couple most strongly with nonetheless useful to consider SISO analysis of each the segment control system thus results in a substantial Zernike, as the correspondance between each peak in computational savings that is essential during design. the sensitivity and a particular Zernike can be used as The most important result obtained from transforming a guide to optimizing the telescope structural dynamic to the Zernike basis is shown in Fig. 13. If the servo characteristics. loops are closed on a segment by segment basis, taking a If the control bandwidth is increased to 20Hz (requir- subset of segments distributed uniformly over the mirror, ing an increase in the frequency to which collocation then the peak sensitivity increases nearly linearly with is satisfied), then the convergence behavior in Fig. 13 the number of loops closed, as suggested by [6], and remains. The structural modes that result in the peak of control of all 492 segments needs to be simulated in the sensitivity are still the lower spatial frequency and order to accurately predict the peak sensitivity. However, thus also lower temporal frequency modes, which project this simply reflects a gradual increase in the projection of primarily onto the lowest Zernike modes. Not only are the control loops onto the low-spatial-frequency modes higher frequency modes stiffer, and hence couple less that dominate the structural coupling. Using a Zernike with the control, the pole-zero ordering is flipped as

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Fig. 13. The Zernike basis (red squares, and inset) is much more Fig. 15. SISO Nichols plot for servo loop when mounted on a rigid efﬁcient for predicting the maximum over frequency of the largest sin- base (black, thick line) and for each of the ﬁrst 21 Zernike basis vectors gular value of the sensitivity, kSk∞. Results converge with relatively (up to radial degree 5) plotted separately. The peak SISO sensitivity few basis vectors, while simply increasing the number of segments is 1.85, only slightly lower than the peak multivariable sensitivity in considered in the analysis increases the maximum singular value almost Fig. 14. The red oval indicates a peak sensitivity of two. linearly (blue circles).

2 T number of bases included. F 1.8 P T F 1.6 D. Global loop 1.4 In the design of Keck, it was the dynamic stability P 1.2 analysis of the global (edge-sensor feedback) control

(S) 1 P loop that was required [1], [5], and integral control was max

σ assumed. Including additional roll-off above the control 0.8 bandwidth greatly reduces the CSI; here we use 0.6 p=0 k 1 p=0,1 C(s) = i with α ' 4k (12) 0.4 p=0,1,2 s (1 + s/α)2 i ≤ 0.2 p 7 Rigid base If the interaction-matrix (A in Eq. (1)) is known per- 0 0 5 10 15 fectly, then it can be inverted, giving a perfect estimation Frequency (Hz) of segment motion, other than the unobservable piston, tip, and tilt of the overall primary mirror. With this Fig. 14. Maximum singular value as a function of frequencyfor servo assumption, the multivariable peak sensitivity for the loop, for increasing number of basis vectors added by Zernike radial global loop is plotted in Fig. 16, using a Zernike basis degree p; the legend shows the maximum radial degree included, and for the control, with bandwidths indicated in the caption. the dominant peaks are labeled with “P” if the peak is due to modes that predominantly project onto piston, “T” if predominantly tip/tilt (The sensitivity is evaluated at the output; the input modes, or “F” if predominantly focus and astigmatism. sensitivity is indistinguishable.) The peak sensitivity results from the phase lag introduced both by the servo loop command response and by the roll-off in Eq. (12). seen in Fig. 8 and 11, and the damping of these modes The “ripples” near 2Hz result from choosing different is higher, leading to the smoother sensitivity at high bandwidths for different radial degrees. frequencies in Fig. 14. From Fig. 14 for the servo loop, all of the signif- The low-order basis approach motivated by the sim- icant structural modes that cause coupling are above pliﬁed analysis in Sec. III is thus useful both for intuition 5Hz, where the global control loop has small gain, and about which structural modes matter, and for fast design thus there is little interaction with these modes for the iterations enabled by the rapid convergence with the bandwidths considered here. However, robustness cannot

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(L) 0.2 5 6 7 8 max σ 1 (S), max σ

0.5 Fig. 18. Example of the effect of A-matrix uncertainty, for a 0.1% uncorrelated random uncertainty in every sensor gain. The pattern on the left results in the estimated pattern on the right; roughly the correct 0 pattern plus a comparable amplitude of focus-mode. 0 5 10 15 Frequency (Hz)

Fig. 16. Global CSI maximum sensitivity (blue) and principal loop condition number (and thus quantitative results in this gain (red). Interaction-matrix uncertainty results in an uncertain gain, section) also depend on the sensor sensitivity to dihedral indicated here with shaded bands; the inset shows the worst-case principal loop gain after accounting for A-matrix uncertainty. The case angle changes; for TMT, Leﬀ =0.052 m. Uncertainty or plotted here corresponds to 1.5Hz bandwidth on radial degrees 4 and variation in sensor gain we explicitly separate out of A higher, 1.25Hz on radial degree 3, and 0.75Hz on radial degree 2, with a diagonal gain matrix S = I + δ as shown in with the shaded band corresponding to an uncertain gain factor of 1.2 S to account for A matrix uncertainty (see Fig. 19(a)). Fig. 17, for reasons that will be clear. Deﬁne B0 as the

pseudo-inverse of the nominal matrix A0. The product

1 2 3 4 5 6 7 8 9 9 : (a) 0 Q = B0SA is ideally the identity (except for projecting

A = > ? @ < CD B > ? @ ; out global piston/tip/tilt), but will differ for A =6 A or

S =6 I. JKHILKF EFGHI There are several sources of uncertainty. Uncertain

actuator gain has no signiﬁcant effect on robustness.

(b)

However, uncertain sensor gain can have a signiﬁcant

! $ % " ' " # & effect, if the uncertainty is uncorrelated between sensors,

even for gain errors of order 0.1%. With TMT sensors,

+ , / )

( ) * + , -. . the ratio of dihedral and height sensitivity Leff also varies with changes in the gap between segments [26], but the Fig. 17. Interaction matrix uncertainty: The uncertainty in sensor gain sensors also measure gap, and hence this effect is easily S = I+δs and actuator gain X = I+δx are explicitly separated from corrected. Finally, sensor installation tolerances affect A. The plant and control dynamics are G(s) and K(s) respectively, every non-zero element of A independently. with G(0) = I. The unitary matrix Ψ transforms into- and out-of a modal basis with diagonal estimator gain matrix Γ; required stability If Q has an eigenvalue less than zero, then regardless T margins can be reduced by considering the norm of Ψ ΓΨB0SA. of how small the control bandwidth, the closed-loop system will be unstable. Uncertainty in sensor gain alone can never lead to this type of instability (barring a sign yet be concluded without analysis of interaction matrix error); if A = A0 then Q will be positive-semi-definite uncertainty. Very small errors in A can result in large if S is. Similarly, Leff variations cannot cause this type errors in its inverse, and the resulting gain uncertainty of instability as the dihedral and height sensitivity affect needs to be accounted for in evaluating robustness. different singular vectors of A, as shown in [8]. This type of instability can occur for errors that independently V. INTERACTION MATRIX UNCERTAINTY affect every non-zero element of A [8]. This is in Robustness of the global control loop is complicated principle possible for sensor installation errors, as noted by uncertainty in the interaction matrix A in Eq. (1). above, although we have not observed this at realistic The condition number of A scales with the number of tolerances [9]; it is also possible if A is measured rather segments, and thus robustness to small errors has the than calculated. potential to be a larger challenge for large segmented- Of more direct relevance to CSI analysis is that mirror telescopes such as TMT than at Keck. The the maximum singular value σ¯(Q) can be large even

if the eigenvalues are all stable. While this is not a 1 1.25

stability problem in the absence of dynamics, it can lead 1.3

1.2 to performance issues, and more critically, can couple 0.5 with CSI to result in instability. Large singular values 1.15 relative gain correspond to a large (multi-variable) gain change. To Astigmatism 0 accomodate this uncertainty naively requires a large gain 0 0.2 0.4 0.6 0.8 1 margin and a corresponding limit on control bandwidth Focus relative gain to guarantee stability; see Fig. 17(a). 1 The particular displacement pattern that has the largest

effect depends on the specific errors and is therefore 3.5 0.5 4.5 5.5 4 not predictable; an example is shown in Fig. 18, where 5 relative gain 0.1% uncorrelated uncertainty in the sensor gains gives Astigmatism σ¯(Q)=1.32. However, the error is in the least observ- 0 0 0.2 0.4 0.6 0.8 1 able, most spatially smooth modes, and focus-mode in Focus relative gain particular. The mechanism by which instability is possi- T ble (though unlikely) is if a focus-mode force command Fig. 19. Maximum singular value of Q˜ = Ψ ΓΨB0SA as a function leads to excitation of a structural resonance that also of estimator gain reductions (in Γ) on focus and astigmatism, and for 0.1% (top) and 1% (bottom) uncorrelated uncertainty in sensor gain includes some of this particular high spatial-frequency S. pattern; this in turn would result in a larger erroneous focus-mode estimate and corresponding control system correction, and so forth. in (i) rapid convergence in stability and robustness calcu- To guarantee stability, then rather than constrain the lations with few basis vectors included, reducing com- gain of all patterns of motion by an extra factor of σ¯(Q), putation time and thus time between design iterations, the directionality information can be used, and only the and (ii) provides intuition regarding important aspects to gain of the lowest spatial frequencies reduced. Define the coupling, which can be used for design guidance Q˜ = ΨT ΓΨQ, where Ψ is a unitary transformation into for structural optimization. The telescope structure is a Zernike or similar basis and Γ a diagonal matrix to only soft at low spatial frequencies, and thus CSI is reduce the estimator gain of low spatial frequencies. only significant for low spatial-frequency patterns of Fig. 19 illustrates the dependence of σ¯(Q˜) on focus segment motion. The strength of the coupling depends and astigmatism gain reductions, for 0.1% and 1% on the mass ratio; the total mass of all of the segments uncorrelated uncertainty in sensor gains. The highest compared with the modal mass of flexible modes. For singular value is limited first by the focus gain, next TMT, CSI is primarily a concern for the actuator servo by astigmatism gain, and further reductions below 1.15 loops, since these operate at a higher bandwidth than the or 3.5 in these two cases would require reductions in the global edge-sensor feedback that maintains the optical gain of trefoil and coma. Note that these factors are in performance of the primary mirror segment array. addition to any gain reductions on low order modes that Robustness of the global control loop is also com- are imposed by the dynamics. plicated by uncertainty in the interaction matrix that For TMT we expect that 0.1% uncorrelated sensor relates edge-sensors to segment motion. Because of ill- gain uncertainty is achievable. From Fig. 19, reducing conditioning (low spatial frequency displacement pat- focus-mode gain by a third reduces the maximum sin- terns are less observable), estimation is quite sensitive gular value to σ¯(Q˜)=1.2. This factor can then be used to small errors in this matrix. Once again, analysis in an as an additional uncertain gain in CSI analysis, as in appropriate basis shows that the gain of the estimator Fig. 16. This may still be conservative, but has only a only needs to be reduced for these poorly observed minor impact on the achievable bandwidth of the global low spatial-frequency patterns; this results in only a loop, and hence on the resulting M1CS performance. small increase in the required stability margins for CSI analysis. VI.CONCLUSIONS Planned large optical telescopes are enabled by ac- ACKNOWLEDGMENTS tive control of the segmented mirror, but the control The TMT Project gratefully acknowledges the support bandwidth is limited by control-structure interaction. of the TMT collaborating institutions. They are the Asso- Analyzing the dynamics in an appropriate basis results ciation of Canadian Universities for Research in Astron-

11 omy (ACURA), the California Institute of Technology, [12] R. Bastaits, G. Rodrigues, B. Mokrani, and A. Preumont, “Active the University of California, the National Astronomical optics of large segmented mirrors: Dynamics and control,” AIAA J. Guid. Control Dyn., vol. 32, no. 6, pp. 1795–1803, 2009. Observatory of Japan, the National Astronomical Obser- [13] D. G. MacMynowski, P. M. Thompson, and M. J. Sirota, “Con- vatories of China and their consortium partners, and the trol of many coupled oscillators and application to segmented- Department of Science and Technology of India and their mirror telescopes,” in AIAA Guidance, Navigation and Control Conference, 2008. supported institutes. This work was supported as well [14] ——, “Analysis of TMT primary mirror control-structure inter- by the Gordon and Betty Moore Foundation, the Canada action,” in Proc. SPIE 7017, 2008. Foundation for Innovation, the Ontario Ministry of Re- [15] P. M. Thompson,D. G. MacMynowski,andM. J. Sirota, “Control analysis of the TMT primary segment assembly,” in Proc. SPIE, search and Innovation, the National Research Council of 2008. Canada, the Natural Sciences and Engineering Research [16] D. G. MacMynowski, P. Thompson, C. Shelton, and L. C. Council of Canada, the British Columbia Knowledge Roberts, Jr., “Robustness of Thirty Meter Telescope primary mirror control,” in Proc. SPIE 7733, 2010. Development Fund, the Association of Universities for [17] P. M. Thompson, D. G. MacMynowski, M. M. Colavita, M. W. Research in Astronomy (AURA) and the U.S. National Regehr, and M. J. Sirota, “Servo design and analysis for the Science Foundation. Thirty Meter Telescope primary mirror actuators,” in Proc. SPIE 7733, 2010. MMC is employed at the Jet Propulsion Laboratory, [18] B. Sedghi, M. Miˇsković, and M. Dimmler, “Perturbation rejection California Institute of Technology, which is operated control strategy for OWL,” in Proc. SPIE 6271, 2006. under contract for NASA. [19] M. Dimmler, T. Erm, B. Bauvir, B. Sedghi, H. Bonnet,M. Müller, and A. Wallander, “E-ELT primary mirror control system,” in Proc. SPIE 7012, 2008. [20] B. Sedghi, M. Müller, M. Dimmler, B. Bauvir, T. Erm, H. Bon- REFERENCES net, and M. Cayrel, “Dynamical aspects in control of E-ELT segmented primary mirror (M1),” in Proc., SPIE 7733, 2010. [1] J.-N. Aubrun, K. R. Lorell, T. S. Mast, and J. E. Nelson, [21] B. Sedghi, M. Müller, and B. Bauvir, “Dynamical simulation of “Dynamic analysis of the actively controlled segmented mirror E-ELT segmented primary mirror,” in Proc. SPIE 8336, 2011. of the W. M. Keck Ten-Meter Telescope,” IEEE Control Systems [22] D. G. MacMynowski, “Hierarchic estimation for control of Magazine, pp. 3–9, Dec. 1987. segmented-mirror telescopes,” AIAA J. Guidance, Control and [2] R. C. Jared, A. A. Arthur, S. Andreae, A. Biocca, R. W. Cohen, Dynamics, vol. 28, no. 5, 2005. J. M. Fuertes, J. Franck, G. Gabor, J. Llacer, T. Mast, J. Meng, [23] L. Lessard, M. West, D. MacMynowski, and S. Lall, “Warm- T. Merrick, R. Minor, J. Nelson, M. Orayani, P. Salz, B. Schaefer, started wavefront reconstruction for adaptive optics,” J. Optical and C. Witebsky, “The W. M. Keck Telescope segmented primary Society of America, A, vol. 25, no. 5, pp. 1147–1155, 2008. mirror active control system,” in Proc. SPIE 1236 Advanced [24] D. G. MacMynowski,L. C. Roberts, Jr., J. C. Shelton, G. Chanan, Technology Optical Telescopes IV, L. D. Barr, Ed., 1990, pp. and H. Bonnet, “In-plane effects on segmented-mirror control,” 996–1008. Applied Optics, vol. 51, no. 12, pp. 1929–1938, 2012. [3] J. Nelson and G. H. Sanders, “The status of the Thirty Meter [25] D. G. MacMartin and G. Chanan, “Measurement accuracy in Telescope project,” in Proc. SPIE 7012, Ground-based and control of segmented-mirror telescopes,” Applied Optics, vol. 43, Airborne Telescopes II, 2008. no. 3, pp. 608–615, 2004. [4] A. McPherson, J. Spyromilio, M. Kissler-Patig, S. Ramsay, [26] C. Shelton, T. Mast, G. Chanan, J. Nelson, L. C. Roberts, Jr., E. Brunetto, P. Dierickx, and M. Cassali, “E-ELT update of M. Troy, M. J. Sirota, B.-J. Seo, and D. R. MacDonald, “Ad- project and effect of change to 39m design,” in Proc. SPIE 8444, vances in edge sensors for the Thirty Meter Telescope primary 2012. mirror,” in Proc. SPIE 7012, 2008. [5] J.-N. Aubrun, K. R. Lorell, T. W. Havas, and W. C. Henninger, [27] M. Sirota, P. M. Thompson, and H. R. Jex, “Azimuth and “Performance analysis of the segment alignment control system elevation servo performance of the W. M. Keck Telescope,” in for the Ten-Meter Telescope,” Automatica, vol. 24, no. 4, pp. Proc. SPIE Vol. 2199, Advanced Technology Optical Telescopes 437–453, 1988. V, L. M. Stepp, Ed., 1994, pp. 126–141. [6] J.-N. Aubrun and K. R. Lorell, “The multi-loop control/structure [28] O. N. Gasparyan, Linear and Nonlinear Multivariable Feedback interaction effect: experimental verification using the ASCIE test Control. John Wiley and Sons Ltd, 2008. bed,” in NASA/DoD CSI Conference, Nov 1990. [7] G. Chanan, D. G. MacMartin, J. Nelson, and T. Mast, “Control and alignment of segmented-mirror telescopes: Matrices, modes, and error propagation,” Applied Optics, vol. 43, no. 6, pp. 1223– 1232, 2004. [8] D. G. MacMynowski, “Interaction matrix uncertainty in active (and adaptive) optics,” Applied Optics, vol. 48, no. 11, pp. 2105– 2114, 2009. [9] D. G. MacMynowski, P. M. Thompson, J. C. Shelton, L. C. Roberts, Jr., M. M. Colavita, and M. J. Sirota, “Control system modeling for the Thirty Meter Telescope primary mirror,” in Proc. SPIE 8336, 2011. [10] A. Preumont, R. Bastaits, and G. Rodrigues, “Scale effects in active optics of large segmented mirrors,” Mechatronics, vol. 19, no. 8, pp. 1286–1293, 2009. [11] R. Bastaits and A. Preumont, “Structural response of extremely large telescopes,” AIAA J. Guid. Control Dyn., vol. 33, no. 5, pp. 1357–1367, 2010.