INITIAL CONTROL RESULTS FOR THE THIRTY METER TELESCOPE

Transcription

1 AIAA INITIAL CONTROL RESULTS FOR THE THIRTY METER TELESCOPE Douglas G. MacMynowski Control and Dynamical Systems California Institute of Technology Pasadena, CA Carl Blaurock Nightsky Systems Raleigh, NC George Z. Angeli Systems Engineering Thirty Meter Telescope Project Pasadena, CA Abstract The next generation of large ground-based optical telescopes are likely to involve a highly segmented primary mirror that must be controlled in the presence of wind and other disturbances, resulting in a new set of challenges for control. The current design concept for the Thirty Meter Telescope (TMT) has 738 segments in the primary mirror, with the out-of-plane degrees of freedom actively controlled. The secondary mirror also requires at least 5 degree of freedom control. We discuss control issues for extremely large segmented-mirror telescopes and present preliminary simulation results for the current TMT design. The most significant departure from existing telescope control systems is that wind buffeting due to turbulence inside the telescope enclosure drives the desired control bandwidth higher, and hence limitations resulting from control-structure-interaction must be understood. The bandwidth of the main telescope elevation drive is limited by interaction with structural modes. In order to achieve a significant reduction in image motion, a fast tip/tilt control of the secondary requires momentum compensation to mitigate interaction with structural dynamics. Control of the primary mirror segments is limited only by interaction with the segment support resonant frequencies, and not with the global modes of the telescope structure. 1 Introduction and Overview The current generation of ground-based optical telescopes have primary mirrors with an effective aperture of 8-10 m. The largest monolithic primary mirrors are 8.2 m in diameter; to exceed this size, the twin 10 m Keck telescopes use segmented primary mirrors whose alignment is actively controlled. 1 3 Various designs are underway for the next generation of optical telescopes with effective apertures of 30 m or more. All of these designs involve a highly macmardg@cds.caltech.edu Copyright c 2005 by the authors. Published by the American Institute of Aeronautics and Astronautics with permission. Figure 1: The arrangement of 738 segments of circumscribed radius a =0.6 m forming a 30 m primary mirror. The central segments are obscured by the secondary mirror and are removed segmented primary mirror, resulting in a new set of control challenges. The number of hexagonal segments being considered for the primary mirror of the Thirty Meter Telescope (TMT) ranges between the 738 used in the current point design (shown in Figure 1) to as high as 1080; the latter was proposed for the 30-meter diameter California Extremely Large Telescope 4 (CELT). The out-of-plane degrees of freedom are actively controlled by 2214 actuators using feedback from 4212 edge sensors; the geometry is similar to that of the Keck telescopes, but with more than 20 times as many actuators. The point design for the 50-meter Euro50 5 uses 618 segments. Even larger telescopes are being considered such as OWL, 6 and future designs with many more segments have also been considered. 7 The ability to control thousands of mirror degrees of freedom with extremely tight performance specifications is a fundamental enabler of the next generation of ground- 1

2 Figure 2: Thirty Meter Telescope (TMT) f/1 reference design. The secondary mirror is Gregorian, and a tertiary mirror at the elevation axis (in front of M1) feeds instruments on Nasmyth platforms. based optical telescopes. Current generation telescopes use active control to maintain the figure of the primary mirror (M1), whether it is monolithic or segmented. However, the bandwidth of these control systems is intended to compensate only for gravity- and thermal-induced deformations. Dynamic analysis of the Keck telescopes during their design predicted a maximum achievable control bandwidth of 0.5 Hz, limited by structural interactions, 3 but the actual implemented bandwidth is roughly 1/10 of this. Wind-induced deformations may be larger for the larger telescopes, 8 11 and thus the bandwidth of the control system will need to increase to compensate. Furthermore, the larger structures will have lower resonant frequencies. Thus, in addition to having many more actuators and sensors, the control bandwidth may be sufficient to interact with flexible structural modes. All of the telescope designs mentioned above also differ from current generation optical telescope designs in the use of a tripod or quadrupod feedleg support structure for the secondary mirror (M2) in place of a spider supported secondary mounted on a telescope tube. For example, the current reference design for the Thirty Meter Telescope (TMT) is shown in Fig. 2. The secondary position will need to be actively controlled to compensate for both gravity and wind loads. Furthermore, the loads on the secondary structure can couple into the primary mirror causing further optical distortions. 9 Herein we assume that the optical tip/tilt caused by motion of either M1 or M2 is controlled via tilting M2, although the tertiary (M3) or a further downstream mirror could be used for this purpose. In addition to the active optics problem of controlling tip/tilt and figure errors of the primary and secondary mirrors, adaptive optics (AO) will be used to compensate for atmospheric distortion and enable near-diffraction-limited resolution. Adaptive optics for extremely large telescopes brings its own set of control challenges that are documented elsewhere. The initial control strategy treats the 12, 13 active and adaptive optics systems independently; future work could evaluate the benefit of a simultaneous integrated design. The telescope will operate in two modes, seeinglimited (without adaptive optics), and (nearly) diffraction-limited (with adaptive optics). The goal for seeing-limited observations is chosen so that the telescope does not degrade the 90 th percentile atmospheric seeing by less than 10%. 4 This translates approximately into an rms segment rotation of less than 20 milli-arcseconds (mas) for control errors. These include actuator noise, the error in desired sensor readings, sensor noise, and residual vibration above the control bandwidth. Similarly, to not significantly degrade adaptive optics performance, the diffraction-limited error budget is <20 nm rms of uncorrectable wavefront error due to errors from the active control system. Low wavenumber distortions of the primary mirror can be corrected by the adaptive optics system provided that this does not result in saturation of the AO actuators, while segment edge discontinuities cannot be well corrected by a smooth deformable mirror. The control challenges involved in meeting these stringent performance targets for telescope active optics are discussed in the subsequent sections, using the initial point design for TMT. Previous papers describe the preliminary design concepts for the active control hardware for CELT, 15 and preliminary analysis of the control problem for CELT 14, and the Giant Segmented Mirror Tele- 19, 20 The scope (GSMT). control challenges for active optics result primarily from compensating for wind buffeting. Previous work 9 relied on parametric structural models that were validated against finite element models. Herein, the finite element model of a particular 30 m telescope design is used both to improve predictions of the dynamic response, and to provide a more rigourous assessment of the feasibility of active control of the primary mirror to reduce wind-buffeting figure errors. A model for wind loads on telescopes has been de- 2

3 Distortion (turbulence) Gravity Thermal Wind Gravity Wind Science object wavefront Atmosphere Primary Mirror (M1) Other Secondary Mirror (M2) FSM K TT AO Def. Mirror(s) Scientific Instruments Natural, Laser guide stars Segment Actuators K PM Edge Sensors K SM DM Actuators K AO Wavefront Sensors Drive motors El/Az drive Encoders Figure 3: Block diagram for wavefront propagation indicating disturbances (entering from the top of the figure) and control loops (mostly shown below the wavefront propagation path), adapted from Ref. 14. Active optics loops are shown shaded. veloped 11, 21 based on combining information from full-scale measurements at the Gemini South Observatory, 22 scale wind-tunnel experiments, 23 and computational fluid dynamics. 24 Applying worstcase wind loads to the telescope structure results in significant deflections of both M1 and M2. The resulting degradation in image motion can be compensated through control of the main axis drives and M2 tip/tilt, while the degradation in image quality or image blur can be compensated for with control of the individual segments of the M1. Furthermore, maximizing the achievable bandwidth of active control represents an economical risk mitigation strategy should the dynamic response be worse than predicted. The next section describes the control problem in detail; including the geometry, hardware, structure, and disturbance environment. Control strategies are laid out in Section 3 and the performance explored through simulation. 2 Control Problem 2.1 Problem Description The wavefront propagation, disturbances, and (decentralized) control loops are shown in Figure 3. Disturbances include atmospheric turbulence distorting the wavefront; gravity, thermal, seismic and wind influences on the primary mirror; and similar influences on the secondary mirror. Wind loads on the primary and secondary mirrors affect their positions directly, and loads on the secondary affect the primary mirror figure due to torque coupling through the structure. In addition to the main elevation and azimuth axes for pointing of the telescope, there are 3 groups of control actuators: primary mirror segment actuation, secondary mirror rigid body actuation, and deformable mirrors (possibly including the secondary). Low bandwidth actuation of the tertiary mirror will also be required, but has not yet been evaluated. The three out-of-plane degrees of freedom of each mirror segment are actively controlled, resulting in 2214 actuators for the current TMT design, compared with 108 for the 36-segment Keck telescopes. Figure (shape) control uses feedback from edge sensors mounted across each inter-segment gap that measure the relative out-of-plane displacement between neighbouring segments (see Figure 4 for the geometry). Supplemental wavefront information has also been suggested for control of the primary mirror for large telescopes, but as discussed in Sec. 3.4, is unnecessary. The secondary mirror (M2) position must be actively controlled in five degrees of freedom. In addition, the figure of the secondary is likely to be actively controlled at low bandwidth, and may be adaptive (a thin face-sheet mirror controlled by distributed actuation) and used as an element of the adaptive optics system. Although there are many actuators and sensors in the primary mirror control loop, the computational burden is unlikely to be a significant issue due to the relatively low control bandwidth required. However, techniques exist to reduce the computational burden if necessary, 25 thereby allowing more complex controllers and higher bandwidth to be used if desired. 2.2 Control Hardware Actuators for M2 are likely to follow from the voicecoil technology used in recent telescope 26, 27 designs. 3

4 Actuator locations (three per segment) Sensor locations (two per intersegment edge) t Figure 4: Schematic of sensor and actuator arrangement on mirror segments. The upper right segment shows the distribution of forces over the mirror through the whiffle tree. (The inter-segment gap is greatly magnified for clarity.) From Ref. 14. The primary mirror actuators used at Keck 1, 2 use a roller-screw with a 24:1 hydraulic reduction lever to achieve sufficient precision, about 4 nm per step. To improve cost and reliability, two stage actuators are being considered, as in [28]. The displacement of the actuator is measured with a local sensor, and a high gain servo loop provides actuator stiffness. The relative displacement of neighbouring segments can be measured using capacitive edge sensors. This approach has proven successful at Keck, where the sensor noise at low frequencies 17 is less than 1 nm 2 /Hz. A redesigned sensor 15 should be less expensive while still measuring all of the internal modes of the segment array. 2.3 Wind disturbances The largest amplitude disturbance acting on the telescope is due to the changing direction of gravity with respect to the mirror as the telescope tracks. While these deflections are large (a few mm) and set actuator stroke requirements, they are also slow, predictable, and thus easily compensated. Thermal deformations are similarly slow and easily corrected. However, although wind-buffeting is typically not a significant design driver for current generation telescopes, it could be significant for larger telescopes due to the larger cross-sectional area, lower stiffness and lower structural resonant frequencies. Although the telescope enclosure significantly reduces the wind speeds inside the dome relative to those a outside the dome, the residual wind may still lead to telescope vibration and resulting unacceptable image blur if not compensated. The static wind loads can be readily compensated by a low bandwidth active control system, but the dynamic wind-induced vibration due to turbulence drives the control system bandwidth requirements, and/or yields a contribution to image blur due to uncorrected contributions. The wind influences the telescope structure through three distinct paths: (i) loads on the secondary and secondary support structure causing motion of the secondary and primary mirror, (ii) loads on the primary mirror deforming the primary mirror, and (iii) loads on the secondary and secondary support structure that lead to deformations of the primary mirror through structural coupling. Other sources, such as wind loads on instrument platforms or wind on the dome coupling through the pier are currently believed to be small. The primary mechanism of unsteady wind loads 21 is broadband turbulence caused by flow over and through the dome opening. In addition, there is broadband turbulence caused by flow through open vents used for thermal equilibration, and tonal Rossiter 29 modes. For the 50 th percentile wind on Mauna Kea 21 of 7 m/s, estimated interior rms wind speeds are v 2 < 3 m/s at M2 and v p 0.5 m/s at M1. The temporal spectrum of the wind pressure on any telescope surface is well approximated by a von Karman spectrum k Φ(f) = (f 2 + f0 2 (1) )7/6 where the corner frequency is the ratio of the local mean wind speed to the diameter of the dome opening, f 0 = U local /D s. While the spectrum of pressure has a roll-off slope of 7/3, the effect of integrating the pressure over an area leads to an additional roll-off, with corner frequency dependent on the spatial scale of integration and the local velocity. (This accounts for decorrelation; higher frequency turbulence corresponds to smaller spatial scale. The number of uncorrelated turbulent structures acting on a given area is thus proportional to frequency, and the mean-square force is inversely proportional to frequency.) Details of the model are available in [21], or in preliminary form as [11]. Also note that there is little wind energy at the spatial scales of an individual segment of the primary mirror. Assume frozen turbulence so that the spatial spectrum is similar to the temporal spectrum in Eq n (1). Integrating yields that the energy in all spatial scales shorter than 1 m is roughly 1.5% of the total wind energy. The rms inter-segment edge discontinuity results from the pressure that is decor- 4

5 Mode number Mode number Figure 5: Modal density of preliminary TMT structure with the elevation drive modeled by a stiff spring. The inset shows the frequency of the first 10 modes. related between neighbouring segments and can thus be calculated from the rms pressure ρvp/2 2 and the segment support stiffness. This yields an rms wavefront error of 30 nm, roughly half of which could be corrected by the adaptive optics system. The residual energy at the scale of individual segments is small even for a worst-case wind assumption. 2.4 Telescope structure A finite element model of a possible TMT design has been constructed. The finite element model does not include the individual segment support resonances, which can be separately added as described later. If the drives are locked, then for a zenith angle of zero the first resonant frequency is at roughly 2.4 Hz, involving primarily tilting of the primary mirror and decentering of M2. As the zenith angle increases towards 60, the first resonance drops to 1.8 Hz. Realistic control design must therefore be robust to significant uncertainty in the structural modes. There are close to 80 modes below 20 Hz, as shown in Figure 5, and 450 below 50 Hz. Including the finite compliance associated with control of the drives (discussed in the next section) and soil/pier stiffness (not currently in the model) reduces the stiffness and resonant frequencies further. The dynamic response of M2 and the M1 segment centers to the wind disturbance is obtained through modal superposition using the first 450 modes. Convergence of the modal superposition is validated by comparison with the quasi-static response computed directly from the finite element model. Higher order modes will not be accurate in detail, but the general characteristics are necessary to assess realistic control bandwidth. Also note that many of the modes do not contribute significantly to primary mirror deflections; a model reduction would be straightforward. 30 The response q(t) R p of the (massnormalized) mode amplitudes due to wind forces or torques applied to the optical surfaces is q + C q + Kq = φf (2) and the displacement and rotation of the optical surfaces is y = φ T q. Structural damping of 1% (Q = 50) is assumed. Each segment is supported at three support points a radius t from the segment center, as illustrated in Fig. 4. Each segment thus has a piston and two (equal-frequency) torsional resonances, with frequencies given by 3ki ω p = (3) m 36 5 ( ) 2 k i t ω t = (4) m a where k i is the stiffness of each support, and m is the segment mass. The torsional stiffness of each segment is k r =(3/2)t 2 k i and the moment of inertia is J = (5/24)a 2 m. For the current design choice of t = 0.24 m and segment circumscribed radius a = 0.6 m, the segment piston resonant frequency is 61% higher than the torsional frequencies. The nominal value for each segment torsional resonance is assumed to be 35 Hz, corresponding to a stiffness at each support point of N/m and an assumed segment mass of 130 kg. The actuation model assumes a local servo loop so that at low frequencies, each actuator effectively commands displacement, shown schematically in Fig. 6. Thus with non-zero damping, the displacement of the i th segment is given by m z i + cż i + k p z i = k p (d i + y i ) (5) and similarly for the torsion degrees of freedom. The segment dynamics are coupled to the telescope structure through the applied force f i = k p ( z i +y i +d i ), with similar equations for the torque. Appending the segment motion mz and Jθ to the structure degrees of freedom q gives the mass-normalized stiffness matrix as [ K + kφφ T (k/ ] m)φ K tot = (k/ m)φ T (6) k/m with appropriate choices of k p and m or k r and J for the displacement and rotational degrees of freedom for each segment. The bandwidth limitations 5

6 m z d k y Number of modes Figure 6: Schematic of segment actuation model; commanded displacement results in forces on both the segment and mirror cell. The full model also includes torsional degrees of freedom that are similarly modeled. for the segment control system are evaluated with these full segment dynamics, while the performance is evaluated with a quasi-static approximation (i.e. Eq. (2) with output z = y + d). There are 1476 segment torsional resonances at the same nominal frequency, coupled through the mirror cell structural dynamics, and a further 738 resonances at a higher frequency. Because the structure is stiff and massive compared to the segments, the coupling is relatively weak. The spread in system resonances resulting from this coupled oscillator problem (from the eigenvalues of K tot )isshown in Fig. 7. The frequencies shown are for the 1476 system modes with the largest strain energy in segment torsion (the vertical axis is truncated; there are many additional resonances with frequencies in the central bin of the histogram). There is roughly a ±20% spread in frequency for this structure. The apparent spread may be limited by the finite spatial resolution of the finite element model, which may not resolve the extent of coupling between neighbouring segments. 2.5 Optical Consequences The performance metric for seeing-limited observations is quantified in terms of the optical path difference (OPD), computed from a linear optical model. 31 This is projected onto Zernike basis functions. The performance can further be separated in terms of the image jitter, or tip/tilt components of the OPD (n = 1 radial degree) and image quality, composed of higher order OPD distortions (n >1) Figure 7: Histogram of resonant frequencies involving significant segment rotations. The vertical axis is truncated. 3 Control Architecture and Simulation 3.1 Preliminary observations In preliminary estimates of the impact of wind buffeting on the telescope structure the most significant impact was the excitation of M1 due to the wind loads on the secondary support structure. 9 However, with the current structural design, forces on M2 are transferred directly into the elevation journals and do not significantly deform M1. The dominant effects are therefore (i) the overall pointing or image jitter due to the loads on M2 and its support structure, and (ii) image quality degradation due to the deformation of M1 due to loads directly on M1, and also due to the motion of M2 due to loads on M2. The bandwidth of the disturbances is relatively low; for median winds, then the forces on the secondary start to decrease above roughly f 0 = v 2 /D 3/30 1/10 Hz, while those on the primary decrease above 0.02 Hz. With the spectrum in Eq n (1), then roughly half the energy is above the corner frequency, but this fraction is further decreased by correlation effects described earlier. There is still some wind energy at the frequency of the first structural resonance, but it does not contribute a significant fraction of the open-loop OPD error. Thus, subject to the uncertainties in the modeling of the wind and the structure, it is not required to control the response of even the first structural resonance to maintain sufficient image quality, although some control may be necessary to stabilize the image motion. Nonetheless, achieving significant reductions in wind-induced buffeting requires control bandwidths larger than the wind corner frequency. The first few telescope structural modes are below 5 Hz, are 6

7 lightly damped, and are uncertain. Actuation of M1 segment motion does not strongly couple with these modes. Actuation of the elevation-axis drive and of M2, however, do couple. Because of the variation in the mode frequency with orientation, the control algorithm must be low bandwidth, robust, or adaptive to avoid deleterious structural interaction. 3.2 Control of Elevation Axis The dominant flexibility contributing to windinduced image jitter is due to the finite-bandwidth and therefore finite compliance of the telescope main elevation-axis control loop. The azimuth control is assumed to be less important and is not modeled herein. The upper telescope structure, including M1, M2 and M3, is mounted on two large elevation journals that are supported with two bearing pads each on the lower azimuth structure. At each support point, the drive is assumed to produce an equal tangential force so as to create a net moment about the elevation axis (and a corresponding force on the azimuth structure). The control uses feedback of the relative tangential displacement between the upper (elevation) and lower (azimuth) structures. Flexibility within the drive system is not modeled; this is therefore a best-case analysis corresponding to a high-stiffness direct drive system. The resulting transfer function between torque (about the elevation axis) and rotation is shown in Fig. 8 for both a0 and 60 angle with respect to zenith. The first zero, or Locked Rotor Frequency (LRF) occurs at what would be the resonant frequency if the drive was infinitely stiff. For this structural model, this frequency varies between 2.4 and 1.8 Hz over the possible range of zenith angles. The elevation axis controller uses a Proportional- Integral-Derivative (PID) design with a bandwidth of 0.5 Hz. An integrator with a ten second time constant (0.1 Hz bandwidth) is used to suppress lowfrequency errors, a lead network is used to add phase at crossover, and an elliptical filter is used to gainstabilize structural modes. The loop response is plotted in Fig. 9. Note that the controller is conditionally stable. 3.3 Control of M2 Achieving a high control bandwidth on the displacement (decenter and despace) degrees of freedom of M2 will be difficult due to the large mass. Furthermore, no mechanical sensors are envisioned to measure this displacement. Compensation for gravity is assumed to be open-loop, with no closed-loop control on these degrees of freedom. Magnitude (rad / Nm) degree ZA 60 degree ZA 1/Js Figure 8: Elevation axis transfer function magnitude from applied torque about the elevation axis (in front of M1) to rotation, for 0 zenith angle (solid), 60 zenith angle (dashed), and compared with the inertia transfer function 1/(Js 2 ). Elev Rx/Elev Rx Phase [deg] Elevation axis loop transfer function Log Frequency [Hz] Figure 9: Elevation axis loop transfer function magnitude, for 0 zenith angle. In order to achieve a significant reduction in windinduced image motion, the tip/tilt motion of M2 be controllable at a bandwidth higher than the first structural mode. This requires moving a reaction mass appropriately (momentum compensation) so that the net forces into the structure are reduced by the extent to which the reaction compensation is accurate; herein we assume 95% of the momentum is compensated. This is sufficient so that the low frequency structural modes are not destabilized by the control. As with the displacement degrees of freedom, there is no planned mechanical sensor for 7

8 M2 tip/tilt. The feedback signal will instead be an optical wavefront measurement, using star light to determine the entire optical path tip/tilt. As a result, this control loop automatically compensates for the tip/tilt of the primary (which cannot be sensed by the edge sensors), as well as the optical tip/tilt introduced by decenter of M2. Correcting M2 decenter using M2 tip/tilt also introduces higher order wavefront deformations (particularly coma). Using reactuation, a 10 Hz bandwidth on the M2 tip/tilt degrees of freedom is achievable in the model with a simple integral control. 3.4 Control of M1 The primary mirror control system uses intersegment relative motion sensors and three actuators behind each mirror segment. The relationship between the displacement z R n of the mirror segments at the actuator locations and the sensor measurements s R m can be obtained from geometry, 18 so s = Cz + δ + η (7) for sensor noise η and a desired sensor reading δ that must be determined optically. 18 Real-time control consists of two steps: obtaining the estimate of the positions ẑ and then choosing a suitable control to minimize it. The former problem is easily solved, and this paper focuses on the latter. The displacement estimate is ẑ = K(s δ) for some K; the least-squares pseudo-inverse is used at Keck since sensor noise is small. Smooth modes of the mirror generate less edge discontinuity for a given mode amplitude, and thus are poorly observable, however, the resulting contribution to image quality is small. 17 Supplemental real-time wavefront information is therefore not required for figure control. Additional wavefront information is required to control pointing (tip/tilt) of the telescope which cannot be sensed by internal relative displacement sensors. The focus mode of the primary mirror involving changes only in the dihedral angle between every segment is not observable with idealized displacement sensors, but is observable with actual sensor designs. 18 All of the deformations other than piston, tip and tilt can be sensed by the sensor array. Therefore, it is sufficient to consider the control of each segment assuming knowledge of the segment displacement. Furthermore, it is straightforward to transform between control of each individual actuator, and control of the piston and two rotational degrees of freedom of each segment. For low bandwidth control such as is currently used at the Keck observatories, then both the structural and actuator dynamics can be ignored. Both Magnitude (rot n/cmd) Collocated Neighbouring Figure 10: Typical transfer function magnitude between commanded rotation on one segment and actual rotation of same segment (solid) and neighbouring segment (dashed). the Keck and planned TMT actuators use local sensors and a (relatively) high bandwidth local servo loop so that for the overall primary mirror control they are effectively displacement actuators. A simple PI loop is then sufficient for low bandwidth control (e.g. [3]). A typical transfer function between applied displacement to a single segment torsional actuation degree of freedom and the resulting rotation of the optical surface for one primary mirror segment is shown in Fig. 10. The response is extremely close to the uncoupled single resonance that would be observed if the structure supporting the segments were rigid. Also shown is the torsional response of one of the neighbouring segments; at zero frequency the difference is a factor of more than 10 4 while at the segment resonance frequency there is still less than 1% excitation of the neighbouring segment. It is therefore reasonable to design each (identical) segment controller assuming no coupling, and verify that the resulting design is stable. With pure integral control of bandwidth g Hz, stability clearly requires that g<f t /Q where f t is the segment torsional frequency and Q is the structural amplification; this is consistent with the analysis for Keck. 3 If a gain margin μ > 1 is required, then g < f t /(μq). With a 35 Hz resonance, Q = 50 and μ = 2, then the maximum bandwidth would be 0.5 Hz. (If the segment control system transformed into piston and torsional degrees of freedom rather than individual actuator degrees of freedom then a higher bandwidth could be achieved on the piston 8

9 degree of freedom.) With a single-pole roll-off at f r =2g (to maintain 60 phase margin), the achievable bandwidth becomes 1 g<f t (8) 2μQ or roughly 2.5 Hz for our parameters. Higher order filters allow still higher bandwidth, however, this is sufficient to result in minimal residual performance errors due to our assumed wind model. Robustness to structural coupling can readily be verified at low frequencies using a small gain argument and computing the H -norm of the structure in Eq. (2). 4 Simulation An integrated model of the telescope performance has been created using the DOCS toolbox from Nightsky Systems. Wind force spectra are computed and applied to structural nodes, control of the elevation-axis, re-actuated M2 tip/tilt, and M1 degrees of freedom are included, and the optical consequences computed. Of primary interest is the control of global (long wavelength, low wavenumber) primary mirror deformations that both couple well into the lowest structural resonances and dominate the seeing-limited error budget. Performance analysis quoted herein considers only the 738 degrees of freedom associated with the segment centres, and ignores the torsional motion of each segment, although these are included in estimating achievable control bandwidth. As noted earlier, there is less wind energy on the smaller spatial scales, and this is a reasonable approximation for large-scale motion. The spectrum of the simulated image motion (OPD tip/tilt in nm) due to wind-buffeting is shown in Fig. 11 with the main elevation axis control loop closed, and with and without the fast tip/tilt M2 loop closed. Most of the response is due to the finite (dynamic) stiffness of the elevation-axis control system, and a relatively high bandwidth tip/tilt control is necessary to reduce the image motion to acceptable levels. The image motion is not affected by the M1 segment control, as the primary mirror edge sensors are not sensitive to overall tip/tilt of M1. The image motion in milli-arcseconds with and without the M2 loop closed is 472 mas and 7 mas. The small tone in this and the next figure near 0.1 Hz is due to a small Rossiter mode in the wind model. The spectrum of the simulated image quality (OPD, with tip/tilt removed, in nm rms) is shown in Fig. 12 with the elevation axis and M2 tip/tilt control loops closed, and with and without the M1 PSD (nm 2 /Hz) Figure 11: Power spectral density of the simulated image motion due to predicted median wind loads on the telescope, with only the elevation-axis drive loop closed (solid), and with a 10 Hz M2 tip/tilt loop closed (dashed). Units are OPD tip/tilt in nm. PSD (nm 2 /Hz) Figure 12: Power spectral density of the simulated image quality due to predicted median wind loads on the telescope structure, with the elevation-axis and M2 tip/tilt loops closed (solid), and with a 2.5 Hz loop closed on the M1 segment actuators (dashed). Units are OPD, tip/tilt removed, in rms nm. segment control loop closed. With the first two loops closed, most of the image quality degradation (46 nm rms) is at low frequencies due to the frequency content of the wind loads on M1. With the M1 loop closed at 2.5 Hz bandwidth, much of the residual degradation of 19 nm rms is due to the motion of M2 and cannot be corrected with further increases in the M1 control bandwidth. 9

10 Increasing the damping of the structural modes does reduce the wind-buffeting, although not strongly since much of the response is not a result of the resonances. However, increased damping will permit higher control bandwidth for the elevation-axis drive control and M2 tip/tilt control. Re-optimizing the control bandwidth would lead to a much more significant improvement in performance with increased damping. It should be emphasized that the specific numbers obtained are based on an initial analysis of a preliminary structural design. The issues and challenges facing the control design will remain true for any design, and the general conclusions regarding the limitations on bandwidth will likely remain true. However, one of the objectives of investigating the response of the initial structural design is to provide guidance for further design iterations that can improve on the performance. Thus, no conclusion should yet be drawn about the exact magnitude of the wind-buffeting problem on the final telescope. 5 Conclusions The next generation of extremely large optical telescopes that are currently being considered are enabled by active control, and will present several control challenges. Unlike current generation optical telescopes, it is expected that wind buffeting of the telescope structure will excite structural resonances and cause sufficient vibration to degrade image quality if left uncompensated. Herein we provide a preliminary exploration of the maximum achievable control bandwidth. The value of this strategy, as opposed to relying only on the control bandwidth sufficient to meet specifications based on current estimates, is as an economic risk reduction against the possibility that either the dynamic loads or dynamic response will differ from current predictions. A model of wind loads developed from a combination of measurements in full-scale, model-scale, and computational models has been applied to the finite element model of a nominal 738-segment structural design, and the optical performance computed under different control designs. The limiting factor on the achievable control bandwidth of the main axis drive system and any fast tip/tilt compensation of the secondary mirror is the interaction with low frequency uncertain structural modes whose mode shapes and frequencies may depend on telescope orientation. The limiting factor on the achievable control bandwidth of the primary mirror segment control system is the interaction with the segment support resonances, and not due to the interaction with global telescope structural modes. These conclusions are likely to hold true as the structural design is refined, although quantitative predictions about the severity of the wind-buffeting will change as the design matures. Acknowledgements Many of the results herein were developed with the entire project team for TMT, led by Gary Sanders, as well as its predecessor programs CELT, GSMT, and VLOT. In particular, Jerry Nelson and Terry Mast at the University of California, Santa Cruz, have contributed to the overall development of the control strategy. Discussions on wind buffeting also involved Stephen Padin, now at the University of Chicago. The Thirty Meter Telescope (TMT) Project is a partnership of the Association of Universities for Research in Astronomy (AURA), the Association of Canadian Universities for Research in Astronomy (ACURA), the California Institute of Technology and the University of California. The partners gratefully acknowledge the support of the Gordon and Betty Moore Foundation, the US National Science Foundation, the National Research Council of Canada, the Natural Sciences and Engineering Research Council of Canada, and the Gemini Partnership. References [1] Jared, R. 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