Control-Structure Interaction in Extremely Large Telescopes

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1 Control-Structure Interaction in Extremely Large Telescopes A. Preumont, B. Mokrani & R. Bastaits Université Libre de Bruxelles (ULB)-Active Structures Laboratory, Brussels, Belgium Abstract: The next generations of earth-based optical telescopes will be based on primary mirrors with diameters of 30m and more, giving access to unprecedented optical performances with respect to the current generation of 10- meter class telescopes. Building on the success of the technologies developed for the 11m Keck Telescopes, telescope designers will resort to dense segmentation to build those large primary mirrors. However, the general change of scale in the whole opto-mechanical system composing the telescope will pose new challenges in maintaining the overall shape of the primary mirror and the relative position of its segments within optical tolerances. Because of the increased control bandwidth and the increased flexibility of the telescope, the control-structure interaction has become a critical issue in the control system design. This paper examines the control-structure interaction within the general robustness theory of multivariable feedback systems, where the dynamic response of the structure is considered as uncertainty. The study is illustrated by the results of a robustness test applied to a numerical model of a primary mirror involving 90 segments. Keywords: Active Optics, Control-Structure Interaction, Extremely Large Telescopes, Scale Effects. Fig. 1: Primary mirror (M1) of the largest earth-based and space telescopes (date of first light). Introduction There has been a change of paradigm in the telescope technology, with the advent of segmented mirrors (first used in the Keck telescope), which, seemingly, make the very large telescopes scalable to infinity, especially with the success of Adaptive Optics which removes the blur due to the atmospheric turbulence. It was once believed that the only clearly identified show stopper seems to be the funding (!) [1]. Several projects are due to see first light during this decade and the European project E-ELT has recently been the first to be approved (M1 diameter initially foreseen of 42 m, and recently reduced to 39 m). E-ELT represents a giant step with respect to existing telescopes, Fig.1. Note that the wave front error allowed for good image quality is only related to the wavelength observed (λ/14), making the ratio ε =precision/size significantly smaller than any existing project. The size of these structures makes them increasingly sensitive to external disturbances, such as the change

2 Fig. 2: Spatial and temporal frequency distribution of the various active control layers of extremely large telescopes (adapted from [5]). Adaptive optics has amplitudes of a few microns. The shape control of M1 involves much larger amplitudes. of the gravity vector due to the earth rotation and wind gusts; the high accuracy requires control systems with larger bandwidth, conflicting with decreasing natural frequencies and very light natural damping. A scale effect analysis [2] shows that the behavior of these complex opto-mechatronics systems is threatened by control-structure interaction, which was so far insignificant, or at least manageable [3,4]. Active Optics Adaptive optics is intended to correct the wavefront errors introduced by the atmospheric turbulence, which exist even in a perfect telescope operating on earth. However, the wavefront sensor cannot separate the wavefront error due to atmospheric turbulence from that due to the telescope imperfections and the adaptive optics will attempt to correct them as well. For large telescopes, however the telescope deformations due to manufacturing errors, thermal gradients and gravity loads are several orders of magnitude larger and cannot be fully corrected by the adaptive optics; they are alleviated with a specific control system called active optics. Figure 2 shows the various control layers of an extremely large telescope and their spatial and temporal frequency distribution [5]. Adaptive optics covers a wide band of temporal frequencies as well as spatial frequencies (Zernike modes of higher orders), but with small amplitudes of a few microns. On the contrary, the control of the rigid body motion of the secondary mirror M2 and the active shape control of the primary mirror M1 must counteract disturbances of very low frequency (changes in gravity loads take place at one cycle per day, that is 1.16x10-5 Hz), but with much larger amplitudes, in millimeters (>>100 λ); this requires larger gains than adaptive optics, and a different technology. Fig. 3: Schematic view of a segmented primary mirror, with the supporting truss, the position actuators and the edge sensors, and the active optics control flow. Segmented Primary Mirrors Figure 3 shows a schematic view of a segmented primary mirror, with the supporting truss and the active optics control flow. Every segment can be regarded as a rigid body; it is supported by 3 position actuators via a whiffle tree. A set of six edge sensors monitor the position of every segment with respect to its neighbors (overall, there will be 2952 position actuators and 5604 edge sensors for E-ELT); the edge sensors play the key role of cophasing the various segments (i.e. making them work as a single, monolithic mirror). If the supporting truss is assumed infinitely stiff, the ACTUATOR 2012, Messe Bremen 2/5

3 Fig. 4: Block diagram of the Singular Value Decomposition (SVD) controller [6]. quasi-static behavior of the system is governed by a purely kinematic relationship y 1 = J e.a where J e is the Jacobian of the edge sensors. Since the lower optical modes (piston, tilt, defocus) are not observable from the edge sensors, another set of sensors measuring the normal to the segments (e.g. one per segment, or just a few for the entire mirror) may be used, leading to y 2 = J n.a where J n is the Jacobian of the normal sensors. Note, however, that the output signal y 2 is also affected by atmospheric turbulence, which is eliminated by time averaging, just as for monolithic mirrors. In order to include the flexibility of the whiffle tree into the model, the position actuator can be modelled with a force actuator F a acting in parallel on a spring k a and dash-pot c a ; the stiffness k a is selected to account for the local modes of the segments and c a to provide the appropriate damping. the force is related to the unconstrained displacement a by F a =a.k a. The position actuators rest on the supporting truss carrying the whole mirror. The disturbances d acting on the system come from thermal gradients, changing gravity vector with the elevation of the telescope, and wind. Control-structure interaction may arise from the force actuator F a exciting the resonances f i of the supporting truss or the local modes of the segments. Before addressing the control-structure interaction, let us discuss the quasistatic shape control of a deformable mirror. Loop shaping via SVD controller The quasi-static behavior of the primary mirror is described by an equation y = Ja where y is the output of a set of sensors (combining y 1 and y 2 above), a is the actuator input, and J is the Jacobian of the system. The block diagram of the Singular Value Decomposition (SVD) controller [6] is shown in Figure 4. The controller essentially inverts the Jacobian, leading to an open loop transfer matrix G 0 (s)k(s) being diagonal. Control-Structure Interaction The controller transfer matrix is essentially the inverse of the quasi-static response of the mirror. However, because the response of the mirror includes a dynamic contribution at the frequency of the lowest structural modes and above, the system behaves according to Figure 5 and the robustness with respect to control-structure interaction must be examined with care. The structure of the control system is that of Figure 5.a, where the primary response G 0 (s) corresponds to the quasi-static response described earlier and the residual response G R (s) is the deviation resulting from the dynamic amplification of the flexible modes; K(s) is the controller, contained in the feedback loop of Figure 4; the design of the compensator H(s) is omitted here [7]. The control-structure interaction may be addressed with the general robustness theory of multivariable feedback systems [8,9,10]; the residual response being considered as uncertainty. The uncertainty may be considered as multiplicative (Figure 5.b) or additive (Figure 5.c); the multiplicative is considered alone here, because of the lack of space; the additive uncertainty is dealt with in [11,12]. If one assumes a multiplicative uncertainty the standard structure of Figure 5.b applies, and one can check that Figure 5.a and b are equivalent with G(s)=K(s)G 0 (s) and L=G -1 0 G R [in all this section, the inverse of rectangular matrices should be understood in the sense of pseudo-inverse]. A sufficient condition for stability is given by: σ L ( jω) < σ I + G ( jω), ω > [ ] [ ] 0 (σ and σ stand respectively for the maximum and the minimum singular value), which is transformed here into σ G G ( jω) < σ I + ( KG ) ( jω), ω [ ] [ ] 0 0 R 0 > This test is quite meaningful; the left hand side represents an upper bound to the relative magnitude of the uncertainty; it is independent of the controller; it starts from 0 at low frequency where the residual dynamics is negligible and increases gradually when the frequency approaches the flexible modes of the mirror structure, which are not included in the ACTUATOR 2012, Messe Bremen 3/5

4 Fig. 5: Block diagram of the control system. (a) G 0 represents the quasi-static behavior (Jacobian) and G R represents the dynamic response. (b) Multiplicative uncertainty. (c) Additive uncertainty. nominal model G 0 ; the amplitude is maximum at the resonance frequencies where it is only limited by the structural damping of the flexible modes. The right hand side starts from unity at low frequency where KG 0 >> 1 (KG 0 controls the performance of the control system) and grows larger than 1 outside the bandwidth of the control system where the system rolls off ( KG 0 <<1). A typical robustness test plot is represented in Fig. 6; the critical point A corresponds to the closest distance between these curves. The vertical distance between A and the upper curve has the meaning of a gain margin GM (if the gain of all the meaning of a gain margin GM (if the gain of all control channels is multiplied by a scalar g, the high frequency part of the upper curve will be lowered by g). When the natural frequency of the structure changes from f 1 to f * 1, point A moves horizontally according to the ratio f * 1 /f 1 (increasing the frequency will move A to the right). Similarly, changing the damping ratio from ξ 1 to ξ * 1 will change the amplitude according to ξ 1 /ξ * 1 (increasing the damping will decrease the amplitude of A). As the telescopes increase in size, so does the gravity sag, requiring higher control gains to maintain the right shape, and increasing the control bandwidth f c ; this means that the curve σ [ I + ( KG0 ) ( jω) ] in Figure 6 is moving to the right. At the same time, the natural frequencies f i of the flexible modes decrease when the size of the structure increases [2], which means that the curve σ [ G0 GR ( jω) ] is moving to the left. The robustness with respect to the control-structure interaction tends to be controlled by the ratio f i /f c between the natural frequency of the critical mode (not necessarily the first: for the Keck telescope, the critical mode turned out to be a local mode of the segments with a frequency of about 25Hz [3,4]) and the control bandwidth. Fig. 6: Block Robustness test assuming multiplicative uncertainty. The upper curve refers to the nominal system used in the controller design. The lower curve is an upper bound to the relative magnitude of the dynamic response (with respect to the quasi-static one). The critical point A corresponds to the closest distance between these curves; the vertical distance between A and the upper curve has the meaning of a gain margin (GM). ACTUATOR 2012, Messe Bremen 4/5

5 Fig. 7: Evolution of the gain margin GM with the frequency ratio f 1 /f c for various values of the damping ratio ξ i. For a given telescope design, using the foregoing robustness tests, it is possible to plot the evolution of the gain margin with the frequency ratio f 1 /f c ; this curve depends strongly on the structural damping ratio, since the amplitude of the various resonance peaks is proportional to ξ -1. Figure 7 shows a typical plot; one sees that if the critical structural mode of the supporting structure has a damping ratio of ξ=0.01, a gain margin GM=10 requires a frequency separation f 1 /f c significantly larger than one decade. This condition may be more and more difficult to fulfill as the size of the telescope grows. The situation can be improved by increasing the structural damping of the supporting truss, possibly actively. Conclusions This paper shows how the general robustness theory of multivariable feedback systems can be used to estimate the stability margin with respect to controlstructure interaction. It is shown that the stability robustness is controlled by (i) the frequency ratio between the control bandwidth and the natural frequency of the critical mode and (ii) the damping ratio of the critical mode. References [1] Gilmozzi, R. Science and technology drivers for future giant telescopes, in Ground-based Telescopes. Proc. SPIE 5489 (2004) [2] Preumont A., Bastaits R., Rodrigues G., Scale Effects in Active Optics of Large segmented Mirrors, Mechatronics, 19, (2009). [3] Aubrun, J.N., Lorell, K.R., Mast, T.S. & Nelson, J.E., Dynamic Analysis of the Actively Controlled Segmented Mirror of the W.M. Keck Ten-Meter Telescope, IEEE Control Systems Magazine, 3-10, December [4] Aubrun, J.N., Lorell, K.R., Havas, T.W., & Henninger, W.C., Performance Analysis of the Segment Alignment Control System for the Ten- Meter Telescope, Automatica, Vol.24, No 4, , [5] Angeli, G.Z., Cho, M.K., Whorton, M.S., Active optics and architecture for a giant segmented mirror telescope, Proc. SPIE 4840, Paper No , pages (2002). [6] Chanan, G. MacMartin, D.G., Nelson, J., & Mast, T., Control and alignment of segmented-mirror telescopes: matrices, modes, and error propagation, Applied Optics, Vol.43, No 6, , Feb [7] Bastaits, R., Rodrigues, G., Mokrani, B. & Preumont, A., Active Optics of Large Segmented Mirrors: Dynamics and Control, AIAA Journal of Guidance, Control and Dynamics, Vol.32, No 6, , Nov.-Dec [8] Doyle, J.C. and Stein, G., Multivariate Feedback Design: Concepts for a Modern/classical Synthesis, IEEE Trans. on Automatic Control, Vol. AC-26, pp.4-17, Feb [9] Maciejowski, J.M. Multivariable Feedback Design, Addison-Wesley, [10] Kosut, R.L., Salzwedel, H., Emami-Naeini, A.,Robust Control of Flexible Spacecraft, AIAA J. of Guidance, 6(2), , March-April [11] Bastaits, R. Extremely Large Segmented Mirrors: Dynamics, Control and Scale Effects. PhD Thesis, Université Libre de Bruxelles, Active Structures Laboratory, June [12] Preumont, A. Vibration Control of Active Structures, An Introduction, 3 rd Edition, Springer ACTUATOR 2012, Messe Bremen 5/5