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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) con- sidered here is composed of 492 segments, with 1476 actua- tors 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. CONTROL PROBLEM 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 2 © −4 (a) 10 ¥ ¦ £ £ −5 £ ¢ 10 £ −6 10 ¤ ¤ ¤ −7 ¡ ¤ 10 § ¨ Magnitude (m/N) −8 10 ¢ 0 1 2 ¢ 10 10 10 0 (b) −50 pi −100 m Phase (deg) −150 k −200 qi 0 1 2 … … 10 10 10 Frequency (Hz) 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.
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