Actuator. Position command. Wind forces Segment (492)

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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 476 actuators and 2772 sensors. In addition to many more actuators 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 the TMT design, but the concepts are applicable to any segmented-mirror design. The key insight is to analyze the structural interaction in a Zernike basis; rapid convergence 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 Sensors (2) Actuators (3) Fig.. Conceptual image of the Thirty Meter Telescope design (left), and detail of one primary mirror segment (right). I. INTRODUCTION Optical telescopes with primary mirror (M) 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 approach pioneered by the Keck telescopes [], [2]. While the Keck telescopes each have 36 segments, the design for the Thirty Meter Telescope (Fig. and 2) has 492 [3], while the 39 m European Extremely Large Telescope (E-ELT) design has 798 [4]. The primary mirror control system (MCS) for these designs builds on the approach used at Keck, with feedback from edge sensors used to control position Manuscript submitted to IEEE TCST. D. MacMartin (formerly MacMynowski) is with Control & Dynamical Systems, California Institute of Technology, Pasadena, CA 925 USA, macmardg@cds.caltech.edu. P. Thompson is with Systems Technology Inc., Hawthorne CA. M. Colavita is with the Jet Propulsion Laboratory, Pasadena CA. M. Sirota is with the TMT Observatory Corporation, Pasadena, CA. Fig. 2. The 492-segment primary mirror of TMT (left), and segment actuator and sensor locations (right). Each segment has three position actuators ( + ) and two sensors on each inter-segment edge ( ) that measure relative displacement, for a total of 476 actuators and 2772 sensors. actuators on each segment (see Fig. 2), with an overall surface precision of order nm rms (though low spatial frequency motion can be larger). However, for future telescopes, the problem is more challenging because of the greater number of segments, sensors and actuators, higher desired control bandwidth, and stringent performance goals. Aubrun et al. [], [5] conducted the dynamic control-structure-interaction (CSI) analysis of the Keck observatory primary mirror control system,

2 and furthermore suggested that for a given structure, the destabilizing effects scale linearly with the number of control loops [6]; a potential concern given the large number of segments in planned optical telescopes. The purpose of this paper is to describe the dynamic analysis of segmented-mirror control for large arrays of segments, and for TMT in particular, 25 years after the corresponding analysis for Keck was published []. In addition to the quasi-static gravity and thermal deformations controlled at Keck, MCS at both TMT and E-ELT will provide some reduction of the response to unsteady wind turbulence forces on the primary mirror. The increased bandwidth required to do so also requires more careful attention to CSI than was required for 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 relatively high-bandwidth servo loop using collocated encoder feedback within the actuator; CSI must also be analyzed for these control loops. Finally, analysis would be incomplete without addressing one further complication that results from the large number of segments. The edge-sensor based feedback relies on knowledge of the interaction matrix that relates sensor response to segment motion, in order to estimate the latter from the former [7]. The condition number of this matrix increases with the number of segments, and thus small errors can result in large uncertainty in the control system gain [8], [9]. Additional analysis is required to ensure simultaneous stability in the presence of both this effect and CSI. Scaling effects for both dynamics and control of large arrays of segments have been addressed in [], [], and multivariable CSI robustness of the global control loop in [2], using a more conservative test than the one applied here (noted later). Progress in CSI analysis for TMT has been described in a sequence of papers [9], [3] [7], and similar analyses for the European ELT in [8] [2]. The key observation that allows for both rapid analysis and design intuition is that the segment dynamics can be analyzed in any basis. For a realistic control bandwidth, the coupling with the telescope structure is primarily an issue at low spatial frequencies. As a result, using a Zernike basis (or something similar) yields rapid convergence of stability and robustness predictions and does not require analysis with all 492 segments of the primary mirror. A higher control bandwidth may require more basis vectors to predict robustness. Several additional aspects to the segmented-mirror control problem are worth noting. For the desired closedloop bandwidths, the computational burden of the real- Position command K global Actuator force K act Wind forces Segment (492) Telescope structure Segment motion Actuator encoder A Edge sensor Fig. 3. Block diagram showing control loops, both local actuator servo loops (K act) and global edge-sensor based feedback (K global, A # ); the input and output of both K act and K global have dimension 476. The dynamics of the segments and control loops will be coupled to the telescope structure (coupling points marked by solid circles) in a different basis as described in Sec. III and IV. time controller is not an issue; if it were, then approaches developed for adaptive optics can easily be extended to this problem, e.g. [22], [23]. The analysis herein focuses only on the out-of-plane degrees of freedom of each segment; in-plane motion does couple with the out-of-plane control [24], but the effects are essentially quasi-static and can be separately analyzed. Sensor noise propagation can also be separately understood [7], [25], although this may also limit the desired bandwidth of poorly observed modes. The next section introduces the control problem in more detail, followed by analysis in Sec. III of a simplified problem that contains the most important features of the full problem. The insights obtained are then used in Sec. IV to compute CSI robustness for TMT. Finally, Sec. V introduces interaction-matrix uncertainty and the analysis required to prove simultaneous stability to this and CSI. II. CONTROL PROBLEM A block diagram for the control problem is shown in Fig. 3. Each segment of the mirror is controlled by three position actuators (see Fig. 2), leading to a total of 476 actuators for TMT. Several different actuator technologies have been considered, and voice-coils selected based in part on low transmission of higher-frequency vibrations to the mirror surface. Stiffness is obtained using feedback from a local encoder with a bandwidth of 8 Hz; each actuator uses the same controller. The interaction of these 476 control loops with the structural dynamics is the most challenging CSI concern for TMT. For an individual segment mounted on a rigid base (rather than on the telescope structure), the uncontrolled segment behaves roughly as a mass (mirror segment) on a spring (actuator open-loop spring stiffness), with A # 2

3 Magnitude (m/n) (a) Phase (deg) (b) m k p i q i 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 (three actuators on a segment driven together) is shown; the tip and tilt responses are similar. a resonance near 8 Hz (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 eddy-current based passive damping within the actuator makes control design much more straightforward, as will be seen when the dynamics of the telescope structure are accounted for. The segment first resonance damping ratio in Fig. 4 is ζ =.75. With the local actuator control loops closed, they behave as position actuators for the global control loop. The global control uses feedback from edge-sensors between neighbouring segments to maintain the optical continuity of the mirror, with a bandwidth of order Hz. Differential capacitive sensors [26] measure the relative edge height discontinuity, similar to the approach used at Keck; with two segments per edge there are 2772 sensors for TMT. The relationship between the segment motion at the actuator locations, x, and the edge-sensor response y, can be expressed through geometry [7] as y = Ax + η () with sensor noise η. The global control loop involves first estimating x from y using the pseudo-inverse of A, and then computing control commands. At Keck the control is calculated for each actuator ( zonal control ); for future telescopes, control will be calculated in a modal basis such as that obtained from the singular value decomposition of A (e.g. [2], [4]). Fig. 5. Schematic (a) of n identical oscillators coupled through a supporting structure, with disturbance forces f i and control inputs u i ; 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 K i are associated with the support structure. Any sensor set that measures relative segment motion results in the global rigid-body motion of the full mirror (piston, tip and tilt) being unobservable (A is rank deficient). The edge sensors at TMT are also sensitive to the dihedral angle θ between segments (rotation about the shared edge): the sensor output is a linear combination of the relative height between segments, and L eff θ, where the effective moment arm L eff has units of length. Without dihedral sensitivity, focus-mode would also be unobservable; this pattern corresponds to a uniform dihedral change for all segments, resulting in a change in the overall M radius of curvature. With practical values of L eff and with many segments, focus-mode estimation in particular is sensitive to uncertainty in the matrix A, and thus control of this mode in particular will be constrained to a lower bandwidth. III. PRELIMINARY ANALYSIS Before considering the control system dynamics with the full telescope structural model, it is useful to first use some simplifying approximations to explore some general characteristics of the problem. The schematic in Fig 5(a) illustrates important features of the dynamics: there are many identical subsystems (mirror segments) coupled to each other through the telescope structure. The key observation that simplifies analysis is that a 3

4 diagonal system of identical subsystems remains diagonal under any change of basis. Thus if the dynamics of an individual segment are written as g(s), then for any unitary matrix φ: φ T G(s)φ = G(s) where G(s) = g(s).... g(s) where we assume that the dynamics of each segment are identical (this is a very good approximation). As an example, consider the case where the dynamics of the support structure can be described solely by the n displacements z i at the segment locations (so that it has exactly n degrees of freedom and n structural modes), and has uniform mass distribution. While this is not a realistic assumption for design, it is sufficient to illustrate some key scaling laws. For this case, the modes of the structure evaluated at the segment mounting locations provide an orthogonal basis for transforming the segment dynamics. The transformation results in n decoupled systems that each describe the coupling between one structural mode and the corresponding pattern of segment motion. That is, in this case, there exists a basis that simultaneously diagonalizes both the supporting structure and the segment dynamics. We start by ignoring damping for simplicity, although it will of course be critical to the control design problem, and we represent each segment by a single degree of freedom rather than three. Define x, z R n as the vectors of segment and structure displacement, and u, f R n the control inputs and disturbance forces. The dynamics of the i th segment are described by mẍ i + kx i = f i + u i + kz i (2) The coupling structure dynamics are described by M z + Kz = u + k(x z) (3) where K is the stiffness matrix, and the mass matrix M = (M/n)I n n because of the assumed uniform mass distribution, with M the total support structure mass, n the number of segments, and I n n the identity matrix. For any orthonormal basis φ R n n, with p = φ T x, q = φ T z, f = φ T f and ũ = φ T u, then m p i + kp i = f i + ũ i + kq i (4) Furthermore, if φ are the modes shapes of the support structure, so that φ diagonalizes K, then φ T i Kφ i = K i and M q i + K i q i + nkq i = nũ i + nkp i (5) That is, the dynamics decouple into n independent coupled-oscillator systems, as shown in Fig. 5(b). Define ω = k/m as the oscillator natural frequency if mounted on a rigid support, and the mass and frequency ratios µ = nm and Ω = (K i/m) /2 (6) M ω Then for each basis function i (dropping the subscript for clarity) we have: [ ] [ ][ ] + ω p q 2 p µ µ + Ω 2 q = [ ] f + [ ] ũ (7) m m µ Scaling frequency by ω, the transfer function from a displacement input (ũ/k) to output p is: s 2 + Ω 2 s 4 + ( + Ω 2 + µ)s 2 + Ω 2 (8) and the two modes are at frequencies! /2 + µ + Ω 2 ± p µ 2 + 2µ + 2µΩ Ω 2 + Ω 4 2 (9) corresponding to in-phase and out-of-phase oscillation between the structural mode and the corresponding pattern of oscillator motion. If the support structure stiffness is small compared to the oscillator stiffness (K i nk), then to first order the lower resonant frequency (normalized by ω) is Ω () ( + µ) which is just mass-loading of the telescope structure resonance. For small mass ratio µ (support structure massive compared to the total mass of the oscillators), then the systems decouple. With damping b added in parallel with the actuator, as in the TMT actuator design, then the zeros of the transfer function are unaffected (these correspond to zero motion across the actuator). An approximate formula for the damping of the two modes can be derived by neglecting the shift in the imaginary part of the eigenvalues relative to their undamped values:! 2ζ b µ Ω 2 ± p () 2 µ2 + 2µ + 2µΩ Ω 2 + Ω 4 For small µ, the mode involving mostly segment motion is significantly damped, while the mode involving primarily mirror cell motion is only slightly damped. Fig. 6 compares the frequency response from eq. (8) with the frequency response for Zernike focus for TMT, 4

5 Magnitude (m/n) Phase (deg) Fig. 6. Actuator open-loop frequency response for TMT focus mode (without added passive damping, solid), compared with the approximate response from eq. (8) (dashed); the amplitude of the latter is scaled to match the static gain. Imaginary part of pole Real part of pole using the models described in the next section. Actuator damping is not included for ease of comparing the resonances. There is a single resonance of the telescope structure (shown in Fig. 9) that predominantly projects onto Zernike focus. The mass and stiffness values Ω =.8 and µ =.26 provide a good fit to the behavior for the projection onto this Zernike. A representative root locus for these values of Ω and µ is sketched in Fig. 7, using a PID controller. Control design is straightforward for the uncoupled system, however this controller destabilizes the coupled structural mode when the segment is mounted on the flexible telescope structure. The extent of destabilization depends on the frequency separation of the pole and zero, which again depends on the mass ratio (Eq. ). Adding passive damping to the actuator damps both modes and increases the maximum stable gain of simple controllers, but the gain will always be limited by the destabilizing interaction with the coupled structural dynamics. The case in Fig. 7 corresponds to Ω =.82, for a structural resonance relatively close to the segment resonance. Fig. 8 illustrates the behaviour for higher frequency structural modes (using Ω =.6). With no damping, the root locus topology is similar to before, although now it is the lower frequency pole that involves more segment motion, and thus the order of the pole and zero introduced by the coupling to the structure is flipped relative to before. With passive damping added, both modes now have more damping, following from Eq. () and the higher value of Ω. The added damping and the shift in pole-zero order lead to resonances above the segment support resonance being less of a robustness Imaginary part of pole Real part of pole Fig. 7. Root locus for actuator servo loop, using mass and stiffness from TMT focus mode, and a PID controller (which yields the damped zeros). Without any passive damping added, the closed-loop system with these parameters would be unstable (see inset). The addition of passive damping in parallel with the actuator makes the control problem easier (bottom panel). challenge than lower frequency resonances. The main observations from this simple analysis are as follows. First, that much can be gained by analysis in an appropriate basis set (as opposed to considering individual segment motion). Second, recall that the analysis in [6] suggested that destabilization due to CSI was approximately linear in the number of control loops. While this is true for a given structure, it is the mass ratio (nm)/m that is the relevant parameter. Increasing the number of segments while keeping the areal density constant does not affect stability. Third, the lowest frequency support structure resonances will decrease in frequency relative to their uncoupled values by an amount that again depends on the mass ratio, leading to a pole-zero pair that 5

6 Imaginary part of pole Real part of pole Fig. 8. Root locus as in Fig. 7 but with structure stiffness increased by a factor of four (corresponding to a structural mode at higher frequency than the segment resonance). The case with no damping is shown in black and is qualitatively similar to before. However, with passive damping (red), then there is now more damping on both modes. is a challenge for robust control design. The addition of passive damping simplifies the control problem. Finally, higher frequency structural resonances are both better damped by added actuator passive damping, and the order of the zero and pole are flipped in frequency, and thus these present less of a challenge for CSI. A. Structural models IV. CSI ANALYSIS FOR TMT We will rely on the previous analysis to provide guidance in understanding the characteristics of the actual telescope system. We first briefly introduce the structural models we use, describe the shift to a different basis for control, and then analyze CSI for both the actuator servo loops and the global control loop. The full CSI analysis for TMT relies on the finiteelement model (FEM) of the telescope structure. For ease of model reduction while retaining both accuracy and flexibility in modeling the segment dynamics, the segments are not included in the telescope FEM. A modal model is obtained from the FEM; 5 modes (up to nearly Hz) are extracted, although only a few dozen low frequency modes matter for CSI. Typically 5 modes (up to 3 Hz) are retained, with the static correction included for truncated modes; convergence with the number of modes retained has been verified. Because the segment model is replicated up to 492 times, a simple lumped-mass model is fit to the detailed FEM of an individual segment before coupling with the main telescope model. This approach ensures that the desired segment dynamics are retained regardless of any model reduction performed on the main telescope structural model, and allows flexibility in choosing what segment dynamics to include only the dynamics associated with retained basis vectors are needed, as described below. Model validation has been conducted by constructing two fully independent models, one interconnecting the component models in state-space, and the other in the frequency domain; both yield identical results for CSI predictions. The structural damping is assumed to be.5% (e.g. Keck damping is in this range [27]). From Fig. 7 this is a critical assumption, since it determines the damping of the zeros, which are unaffected by any actuator passive damping. B. Zernike basis The structural modes of the telescope do not give an orthonormal basis for describing segment dynamics (that approximation might be reasonable if the mirror cell supporting the segments was the only flexible component of the telescope). However, it is still useful to project the dynamics onto a different basis. Instead of modes, we choose a Zernike basis (the natural basis on a circle; polynomials of degree p in radius, and sines 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, there would be no computational savings relative to the original untransformed system. However, the stability characteristics can be accurately predicted with relatively few basis vectors because the coupling is dominated by the most compliant and hence lowest frequency modes of the supporting structure. These are also the lowest wavenumber modes, and thus predominantly project onto the lowest order Zernike basis vectors. Fig. 9 shows the mode shape for a representative low-frequency (9.3 Hz) structural mode. Although this particular mode is not exactly Zernike-focus, the mode is extremely well captured by its projection onto the lowest 5 Zernike basis vectors (up to radial degree 4). For high wavenumber motion that involves significant relative motion between neighbouring segments, the support structure is relatively stiff (see Fig. ). Note that, as in Fig. 9, any structural mode will project onto multiple basis vectors, and conversely, any basis vector will include dynamics associated with multiple modes, and thus multivariable analysis is still required. Although we do not rely on this, for TMT the Zernikebasis nearly diagonalizes the structural dynamics, and 6

7 Projection amplitude Projection amplitude Zernike radial degree Zernike radial degree Compliance (m/n) Structure Segment support Zernike basis function radial degree Fig. 9. Mode shape, evaluated on the primary mirror, of two representative structural modes of the telescope, and their projection onto a Zernike basis (rms of each component normalized by the overall rms across M). The first is the predominant mode associated with 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 and lower. The second illustrates that not all modes project entirely onto a single Zernike, nonetheless over 9% of the rms is associated with either astigmatism or coma, and again, 99% of the rms is captured by the projection onto basis vectors of radial degree 4 and lower. indeed SISO analysis for each Zernike is a good predictor of the multivariable analysis. It is not immediately obvious why this should be true. However, the mirror cell that supports the segments is a truss that can be reasonably approximated at low spatial frequencies as a uniform circular plate, with corresponding flexible mode shapes similar to Zernike basis functions. C. Actuator servo loop The transfer function between voice-coil force and the nearly-collocated encoder position for a segment on a rigid base was shown in Fig. 4, with and without additional passive damping. The damping results in a 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 the actuator will be at least slightly damped (and those modes that do not, do not matter). For a single segment mounted on the telescope structure, the transfer function is similar to the rigid-base case, and indeed it might not be obvious that there is any potential stability problem. However, the coupling is clear when the dynamics are transformed into a Zernike basis, as shown in Fig.. The multivariable robustness metric used here is to require the maximum singular value of the sensitivity Fig.. Static compliance of telescope structure on Zernike basis. The horizontal line illustrates the segment static compliance for comparison; at low spatial frequencies the structure is soft compared to the segments, at high the reverse is true and the coupling is small. Magnitude Phase (deg) Fig.. Actuator frequency response on telescope structure, Zernike basis, including the first 2 basis elements (up to radial degree 5). This includes focus-mode, for which the frequency response without actuator damping was shown in Fig. 6. The solid black line corresponds to a segment mounted on a rigid base (the damped case from Fig. 4). to be less than two; this is a reasonable margin in the absence of a specific understanding of the structure and magnitude of the uncertainty (e.g. gain margin of two). Note that [2] considers the dynamics to be an uncertain perturbation on the static response, and uses the dynamic model to estimate the size of the uncertainty bound, while here we include the dynamics as part of the best estimate of the plant, and require robustness to additional uncertainty on the model. Either approach is reasonable for the global control loop (considered in [2]) where the 7

8 bandwidth is much lower than the structural resonances, but the approach of [2] is too conservative to allow any control design for the higher bandwidth servo loop [2]. Because the encoder is nearly collocated with the 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 5 and 3 Hz, and a high gain margin above 3 Hz 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 the details of gain choices that is important, but rather the lessons learned. With any particular choice of controller, nominal stability could be established by taking eigenvalues of the full system with all segments, but this is computationally intensive and does not provide useful design guidance. Sedghi et al. [2] instead use characteristic transfer functions or CTFs [28] to prove stability: taking the eigenvalues q i (jω) of the transfer function matrix at each frequency, then the multivariable system is stable if the closed-loop system is stable for each q i. However, rather than computing eigenvalues of the full 3n seg 3n seg system as in [2], in Fig. 2 we show that these CTFs converge rapidly if the system is first transformed into a Zernike basis. This amounts to a two-step procedure for proving stability: retaining relatively few Zernike basis elements results in a system with many fewer inputs and outputs; a second frequency-dependent diagonalizing transformation is then used to evaluate nominal stability for this smaller subset, since the Zernike-transformed system is still not diagonal. The effect of the neglected higher-order Zernike basis elements on the first few CTFs is small (i.e., diagonal dominance is satisfied), and it is these first few CTFs that matter most for stability. Starting with a Zernike transformation to isolate the structural dynamics that couple most strongly with the segment control system thus results in a substantial computational savings that is essential during design. The most important result obtained from transforming to the Zernike basis is shown in Fig. 3. If the servo loops are closed on a segment by segment basis, taking a subset of segments distributed uniformly over the mirror, then the peak sensitivity increases nearly linearly with the number of loops closed, as suggested by [6], and control of all 492 segments needs to be simulated in order to accurately predict the peak sensitivity. However, this simply reflects a gradual increase in the projection of the control loops onto the low-spatial-frequency modes that dominate the structural coupling. Using a Zernike Magnitude Phase (deg) Fig. 2. Nichols plot for characteristic transfer functions (CTFs) of servo loop, illustrating convergence of stability and robustness calculations with Zernike basis. Blue lines show the Nichols plots of the CTF for the full system, while magenta lines show the CTF Nichols plots calculated only for the first 6 Zernike basis elements (radial degree p 2); these are similar for the least-stable elements of the full CTF. The Nichols plot corresponding to a single segment mounted on a rigid base is also shown for comparison (black, thick line). The red oval indicates a peak sensitivity of two. basis, results converge almost immediately, since the worst-case structural modes project almost entirely onto low-order Zernike basis functions (mostly radial degree one, and some onto radial degree two), and there is only a small increase in the peak sensitivity with further basis functions added. The multivariable peak sensitivity is shown in Fig. 4 where only Zernike basis vectors up to a given radial degree p are included. The peak sensitivity is remarkably well predicted by SISO analysis with each Zernike separately, shown in Fig. 5. While the system is not sufficiently diagonally dominant to directly infer stability without relying on the CTFs shown in Fig. 2, it is nonetheless useful to consider SISO analysis of each Zernike, as the correspondance between each peak in the sensitivity and a particular Zernike can be used as a guide to optimizing the telescope structural dynamic characteristics. If the control bandwidth is increased to 2 Hz (requiring an increase in the frequency to which collocation is satisfied), then the convergence behavior in Fig. 3 remains. The structural modes that result in the peak of the sensitivity are still the lower spatial frequency and thus also lower temporal frequency modes, which project primarily onto the lowest Zernike modes. Not only are higher frequency modes stiffer, and hence couple less with the control, the pole-zero ordering is flipped as 8

9 2.9.8 S Number of segments Fig. 3. The Zernike basis (red squares, and inset) is much more efficient for predicting the maximum over frequency of the largest singular value of the sensitivity, S. Results converge with relatively few basis vectors, while simply increasing the number of segments considered in the analysis increases the maximum singular value almost linearly (blue circles). 2.8 Magnitude Phase (deg) Fig. 5. SISO Nichols plot for servo loop when mounted on a rigid base (black, thick line) and for each of the first 2 Zernike basis vectors (up to radial degree 5) plotted separately. The peak SISO sensitivity is.85, only slightly lower than the peak multivariable sensitivity in Fig. 4. The red oval indicates a peak sensitivity of two. σ max (S) P P P T T F F.6 p= p=,.4 p=,,2.2 p 7 Rigid base 5 5 Fig. 4. Maximum singular value as a function of frequency for servo loop, for increasing number of basis vectors added by Zernike radial degree p; the legend shows the maximum radial degree included, and the dominant peaks are labeled with P if the peak is due to modes that predominantly project onto piston, T if predominantly tip/tilt modes, or F if predominantly focus and astigmatism. seen in Fig. 8 and, and the damping of these modes is higher, leading to the smoother sensitivity at high frequencies in Fig. 4. The low-order basis approach motivated by the simplified analysis in Sec. III is thus useful both for intuition about which structural modes matter, and for fast design iterations enabled by the rapid convergence with the number of bases included. D. Global loop In the design of Keck, it was the dynamic stability analysis of the global (edge-sensor feedback) control loop that was required [], [5], and integral control was assumed. Including additional roll-off above the control bandwidth greatly reduces the CSI; here we use C(s) = k i s ( + s/α) 2 with α 4k i (2) If the interaction-matrix (A in Eq. ()) is known perfectly, then it can be inverted, giving a perfect estimation of segment motion, other than the unobservable piston, tip, and tilt of the overall primary mirror. With this assumption, the multivariable peak sensitivity for the global loop is plotted in Fig. 6, using a Zernike basis for the control, with bandwidths indicated in the caption. (The sensitivity is evaluated at the output; the input 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. (2). The ripples near 2 Hz result from choosing different bandwidths for different radial degrees. From Fig. 4 for the servo loop, all of the significant structural modes that cause coupling are above 5 Hz, where the global control loop has small gain, and thus there is little interaction with these modes for the bandwidths considered here. However, robustness cannot 9

10 A ' σ max (S), σ max (L) Fig. 6. Global CSI maximum sensitivity (blue) and principal loop gain (red). Interaction-matrix uncertainty results in an uncertain gain, indicated here with shaded bands; the inset shows the worst-case principal loop gain after accounting for A-matrix uncertainty. The case plotted here corresponds to.5 Hz bandwidth on radial degrees 4 and higher,.25 Hz on radial degree 3, and.75 Hz on radial degree 2, with the shaded band corresponding to an uncertain gain factor of.2 to account for A matrix uncertainty (see Fig. 9(a)). (a) E F G H I = : ; < (b)! $ % " " # & ( ) * +, -. +, /. ) C D B J K H I L K F Fig. 7. Interaction matrix uncertainty: The uncertainty in sensor gain S = I+δ s and actuator gain X = I+δ x are explicitly separated from A. The plant and control dynamics are G(s) and K(s) respectively, with G() = I. The unitary matrix Ψ transforms into- and out-of a modal basis with diagonal estimator gain matrix Γ; required stability margins can be reduced by considering the norm of Ψ T ΓΨB SA. yet be concluded without analysis of interaction matrix uncertainty. Very small errors in A can result in large errors in its inverse, and the resulting gain uncertainty needs to be accounted for in evaluating robustness. V. INTERACTION MATRIX UNCERTAINTY Robustness of the global control loop is complicated by uncertainty in the interaction matrix A in Eq. (). The condition number of A scales with the number of segments, and thus robustness to small errors has the potential to be a larger challenge for large segmentedmirror telescopes such as TMT than at Keck. The Fig. 8. Example of the effect of A-matrix uncertainty, for a.% uncorrelated random uncertainty in every sensor gain. The pattern on the left results in the estimated pattern on the right; roughly the correct pattern plus a comparable amplitude of focus-mode. condition number (and thus quantitative results in this section) also depend on the sensor sensitivity to dihedral angle changes; for TMT, L eff =.52 m. Uncertainty or variation in sensor gain we explicitly separate out of A with a diagonal gain matrix S = I + δ S as shown in Fig. 7, for reasons that will be clear. Define B as the pseudo-inverse of the nominal matrix A. The product Q = B SA is ideally the identity (except for projecting out global piston/tip/tilt), but will differ for A A or S I. There are several sources of uncertainty. Uncertain actuator gain has no significant effect on robustness. However, uncertain sensor gain can have a significant effect, if the uncertainty is uncorrelated between sensors, even for gain errors of order.%. With TMT sensors, the ratio of dihedral and height sensitivity L eff also varies with changes in the gap between segments [26], but the sensors also measure gap, and hence this effect is easily corrected. Finally, sensor installation tolerances affect every non-zero element of A independently. If Q has an eigenvalue less than zero, then regardless 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 error); if A = A then Q will be positive-semi-definite if S is. Similarly, L eff variations cannot cause this type of instability as the dihedral and height sensitivity affect different singular vectors of A, as shown in [8]. This type of instability can occur for errors that independently affect every non-zero element of A [8]. This is in principle possible for sensor installation errors, as noted above, although we have not observed this at realistic tolerances [9]; it is also possible if A is measured rather than calculated. Of more direct relevance to CSI analysis is that the maximum singular value σ(q) can be large even

11 3.5.2 if the eigenvalues are all stable. While this is not a stability problem in the absence of dynamics, it can lead to performance issues, and more critically, can couple with CSI to result in instability. Large singular values correspond to a large (multi-variable) gain change. To accomodate this uncertainty naively requires a large gain margin and a corresponding limit on control bandwidth to guarantee stability; see Fig. 7(a). The particular displacement pattern that has the largest effect depends on the specific errors and is therefore not predictable; an example is shown in Fig. 8, where.% uncorrelated uncertainty in the sensor gains gives σ(q) =.32. However, the error is in the least observable, most spatially smooth modes, and focus-mode in particular. The mechanism by which instability is possible (though unlikely) is if a focus-mode force command leads to excitation of a structural resonance that also includes some of this particular high spatial-frequency pattern; this in turn would result in a larger erroneous focus-mode estimate and corresponding control system correction, and so forth. To guarantee stability, then rather than constrain the gain of all patterns of motion by an extra factor of σ(q), the directionality information can be used, and only the gain of the lowest spatial frequencies reduced. Define Q = Ψ T ΓΨQ, where Ψ is a unitary transformation into a Zernike or similar basis and Γ a diagonal matrix to reduce the estimator gain of low spatial frequencies. Fig. 9 illustrates the dependence of σ( Q) on focus and astigmatism gain reductions, for.% and % uncorrelated uncertainty in sensor gains. The highest singular value is limited first by the focus gain, next by astigmatism gain, and further reductions below.5 or 3.5 in these two cases would require reductions in the gain of trefoil and coma. Note that these factors are in addition to any gain reductions on low order modes that are imposed by the dynamics. For TMT we expect that.% uncorrelated sensor gain uncertainty is achievable. From Fig. 9, reducing focus-mode gain by a third reduces the maximum singular value to σ( Q) =.2. This factor can then be used as an additional uncertain gain in CSI analysis, as in Fig. 6. This may still be conservative, but has only a minor impact on the achievable bandwidth of the global loop, and hence on the resulting MCS performance. VI. CONCLUSIONS Planned large optical telescopes are enabled by active control of the segmented mirror, but the control bandwidth is limited by control-structure interaction. Analyzing the dynamics in an appropriate basis results Astigmatism relative gain Astigmatism relative gain Focus relative gain Focus relative gain Fig. 9. Maximum singular value of Q = Ψ T ΓΨB SA as a function of estimator gain reductions (in Γ) on focus and astigmatism, and for.% (top) and % (bottom) uncorrelated uncertainty in sensor gain S. in (i) rapid convergence in stability and robustness calculations with few basis vectors included, reducing computation time and thus time between design iterations, and (ii) provides intuition regarding important aspects to the coupling, which can be used for design guidance for structural optimization. The telescope structure is only soft at low spatial frequencies, and thus CSI is only significant for low spatial-frequency patterns of segment motion. The strength of the coupling depends on the mass ratio; the total mass of all of the segments compared with the modal mass of flexible modes. For TMT, CSI is primarily a concern for the actuator servo loops, since these operate at a higher bandwidth than the global edge-sensor feedback that maintains the optical performance of the primary mirror segment array. Robustness of the global control loop is also complicated by uncertainty in the interaction matrix that relates edge-sensors to segment motion. Because of illconditioning (low spatial frequency displacement patterns are less observable), estimation is quite sensitive to small errors in this matrix. Once again, analysis in an appropriate basis shows that the gain of the estimator only needs to be reduced for these poorly observed low spatial-frequency patterns; this results in only a small increase in the required stability margins for CSI analysis ACKNOWLEDGMENTS The TMT Project gratefully acknowledges the support of the TMT collaborating institutions. They are the Association of Canadian Universities for Research in Astron

12 omy (ACURA), the California Institute of Technology, the University of California, the National Astronomical Observatory of Japan, the National Astronomical Observatories of China and their consortium partners, and the Department of Science and Technology of India and their supported institutes. This work was supported as well by the Gordon and Betty Moore Foundation, the Canada Foundation for Innovation, the Ontario Ministry of Research and Innovation, the National Research Council of Canada, the Natural Sciences and Engineering Research Council of Canada, the British Columbia Knowledge Development Fund, the Association of Universities for Research in Astronomy (AURA) and the U.S. National Science Foundation. MMC is employed at the Jet Propulsion Laboratory, California Institute of Technology, which is operated under contract for NASA. REFERENCES [] J.-N. Aubrun, K. R. Lorell, T. S. Mast, and J. E. Nelson, Dynamic analysis of the actively controlled segmented mirror of the W. M. 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