Chapter 4

Chapter 4, The Point Design

Section 4.8: Control Systems

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4.8.1 INTRODUCTION

To meet the performance requirements outlined in the science requirements, a successful GSMT design requires (1) active control of the telescope structure and optical surfaces, and (2) adaptive correction to account both for wavefront distortions wrought by the atmosphere and for the effects of wind-buffeting. Design of a control system represents one of the more significant challenges for the GSMT system, owing to the wide bandwidth requiring correction. We summarize in this chapter and its appendices the challenges and their magnitude, a framework for control system architecture, and a preliminary assessment of how the control system might compensate for known effects.

Active correction of the telescope structure and the optical surfaces is necessary for a 30-m class telescope to meet its specifications. Even using current state-of-the-art materials and structures, significant deformation is expected due to the weight and thermal expansion of the telescope. Furthermore, the drag and lift forces of the wind, as well as turbulent air separation effects, can cause considerable deformations and resonant oscillations in the structure. A telescope control system should (1) acquire a target on the sky, (2) track it during the observation, and (3) correct the wavefront aberrations due to atmospheric effects and telescope deformations. Controlling a large segmented mirror telescope represents several substantial challenges:

To correct the aberrations, the current status of the telescope should be sensed, and then the system should be controlled toward the desired status by means of actuators. Without going into the details, we list the information and stimuli necessary to control the GSMT:

Actuator groups

Sensor groups

Figure 1 Physical structure of GSMT control.

The capabilities of different actuator groups to influence the image quality are well-bounded as shown in Figure 2. The redundancy in image correction is apparent from the overlapping areas of various actuator groups.

In order to present a control architecture capable of keeping the image quality within the specification range, first we should outline a control-oriented model for the telescope. It will allow us to address the issues as well as the solutions in a more specific and detailed manner.

Figure 2 Frequency bands of different actuator groups.

4.8.2 TELESCOPE MODEL

Although the telescope, just like all real-world systems, is inherently nonlinear, the nonlinear effects can well be localized and eliminated. One significant source of nonlinearity is the long-range motion of the telescope, such as slewing and acquisition. However, these operations do not require high-precision shape control. Consequently, they are considered separate operational domains with different specification sets. The control model described here is valid only for natural guide star sidereal tracking during observations.

The imperfections in sensors and actuators constitute another common source of nonlinearity. The actuators can usually be linearized with local high-speed feedback, such as encoder-based servo systems. Time-independent sensor nonlinearities are good candidates for look-up tables (LUT). The saturation (range) issues can be corrected with cascading actuators and sensors of different sensitivity and range. Because these corrections are very local, they are beyond the scope of our current discussion.

4.8.3 MECHANICAL STRUCTURE

A major requirement for the structural design is to avoid hysteresis in the mechanical system. It is a significant consideration throughout the whole design process, from the choice of materials to the types of joints used. We can therefore assume, with good confidence, that the deformations of the structure are repeatable and are linear functions of the forces acting on it.

The customary representation of linear structures is based on structural modes, or eigenvectors.2 The actual displacements (q) are the linear combinations of these modes (qm).

Equation 1

Applying the linear transformation of Phi - which is the system eigenvector matrix - on the Newtonian equations of motion of the structure causes the differential equations to become uncoupled.

Equation 2

The external forces applied on the structure are represented by u. The independence of the modal equations (i.e., the diagonality of the coefficient matrices M, Z and Omega) is the major advantage of modal representation.

A further step toward a higher level of abstraction reduces the differential equations to first order.

Equation 3

This so-called state-space representation of linear time-invariant systems is very visual, and also consistent with all the advanced tools of modern control theory.3 Although these modern control methods are highly model dependent, there are well-developed techniques for addressing parameter uncertainties such as ambiguity in modal damping and frequency.

For simulation and design purposes, the finite element analysis (FEA) of the structure provides the coefficients Omega, Z and M, as well as the eigenvector matrix Phi.

4.8.4 OPTICAL SYSTEM

The mechanical deformation of the structure changes the optical path length through the telescope. These changes can be mapped back to the entrance pupil, as if the telescope were perfect but the input wavefront distorted. The optical path difference (OPD) p as a function of position throughout the aperture is a good measure of the image quality of the telescope and, for reasonably small deformations, a linear function of the mechanical displacement q.

p=Pq
An OPD function defined on a circular aperture can be expanded into a discrete Zernike spectrum.4

Equation 4

Zj and aj are the jth Zernike term evaluated at position r or node n and the corresponding coefficient, respectively. Furthermore, a finite subset of the infinite Zernike spectrum can be expressed as a linear transformation on the displacement vector, or on the structural mode vector.

Equation 5
Actually, WPPhi defines a new linearly independent basis for the OPD as a linear combination of Zernike terms. Like the structural eigenvectors for Phi, the basis vectors are the columns of WPPhi. The analytical determination of the P matrix is quite tedious and error prone, especially because it usually contains interpolations (up-sampling) to increase the precision of the calculations. However, a practical way to determine the telescope-specific basis vectors can be through ray tracing the individual unit strength mode shapes of the structure. In this case, the interpolations are carried out on the optical surfaces, and most of the calculations are done by an optical design software.

So far, we have assumed that the optical surfaces are continuous and smooth (continuously differentiable). However, a segmented mirror, even if it is kept continuous by active control, is not necessarily smooth. It can be considered a spatial sample-and-hold on the discrete-space signals of nodal and actuator displacements. In the case of flat segments, it would be similar to a first order sample-and-hold, but for parabolic segments, the optical effect is significantly more complex and needs further investigation.

4.8.5 INTEGRATED MODEL

The control configuration shown in Figure 3 integrates the mechanical, optical, and control subsystems of the telescope, as well as the external disturbances. It is worth mentioning that while the wind and the gravitational and thermal effects affect the image quality through structural deformations, the virtual sky motion and the atmospheric turbulence show up in the light itself.5

The telescope control features two distinct feedback loops based on the two groups of sensors. As mentioned earlier, for now we have neglected the actuator and sensor nonlinearities and limited ranges. The presented model is valid for the most important operation domain: guide star tracking during observation. Figure 3 Control configuration with a single global controller.

The command signal r1(s) for the optical feedback loop is the constant aberration of the undeformed telescope, mainly due to the off-axis position of the guide star. The command signal r2(s) for the edge detector feedback loop corresponds to the perfect positions of the primary mirror segments determined by phasing the mirror.

Other operation domains, like high speed slewing and acquisition, require somewhat different control models to be developed during the telescope design process. However, the specification requirements for these operation domains are much less stringent and the processes themselves are much less complex.

The model also does not contain pertinent information for "boot strapping" the control system; that is, locking it on the guide star. The locking procedure involves nonlinear as well as time- dependent transient processes, and it needs to be investigated further.

4.8.6 CONTROL ARCHITECTURE-DISTURBANCE EVALUATION

In order to define a successful control architecture, first we should investigate the amplitude and frequency range of different disturbances acting on the telescope. There are four major sources of disturbances: (1) the gravitational deformation of the telescope, (2) the thermal expansion of the telescope, (3) the wind induced deformation of the telescope, and (4) the wavefront deformation due to atmospheric turbulence. Fully understanding the effects of these sources on the image quality of the telescope needs significant investigation and simulations. However, at this point, our major concern is the general architecture of the control system and not the detailed design of the individual feedback loops.

The control architecture about to be described here is based on a pivotal assumption; namely, that there is no high frequency disturbance with reasonable power that tends to break the continuity of the segments on the primary mirror. In other words, the segment actuators can be operated slowly enough-below the resonant frequencies of the structure-to avoid interactions with the structural dynamics. Let us see now whether the characteristics of various disturbances support this assumption.

The thermal and gravitational effects are potentially large but tend to be rather slow. The temperature changes on a mountaintop during the night are usually very slow. Considering a 1C/hour maximum temperature gradient and 5C swing, the bandwidth of this disturbance is less than 2*10-5 Hz.

The major gravitational effect on the primary mirror is axial deformation, because the lateral motion of the segments is not going to destroy the continuity of the mirror (as long as the segments are not crushing into each other, of course). These gravity effects due to sidereal tracking repeat at diurnal frequency, so the bandwidth can be estimated as about 10-5 Hz.6

Assuming smooth enough bearings, actuators, and sensors, the correction of thermal and gravitational deformations should not interact with the structural dynamics of the telescope during sidereal tracking. Later in the design process, however, different service bandwidths should be defined that characterize and limit the speed of other telescope operations, such as settling after slewing, pointing, offsetting, and scanning.

The refraction index fluctuations in the atmosphere don't influence the shape of the telescope, but their effects appear in the optical measurements meant to determine that shape. Some of the atmospheric effects will be corrected by the telescope control system due to their inseparability from telescope deformations. Because the telescope deformations can be measured with a single (most likely natural) guide star, the refractive wavefront deformations will be corrected only in a limited field of view around that guide star.

In order to investigate the effect of wind forces on the telescope structure, we have simulated the telescope deformations using Matlab. As input we applied real wind velocities and pressures recorded at the Gemini South telescope in Chile.7 The mean wind velocity at the secondary mirror was about 5 m/s. Our analysis indicates (see Appendix 5.5.D) that the data collected by the different pressure sensors on the Gemini dummy primary mirror are practically independent. Consequently, these pressure time series could be patched together (repeated for different nodes) to yield the simulation pressures on the 30-m GSMT primary mirror. The resulting deformation, which was calculated as the temporal RMS (root mean square) of the instantaneous displacements of each node, is shown in Figure 4.

The shape of the primary mirror deformation indicates that the bend of the secondary structure has a major contribution, while pressure variations on the primary mirror are less influential. The other important conclusion is that the deformation is dominated by low-order modes. These are even more obvious from the Zernike expansion of the deformation, as shown in Figure 5.

Figure 4 GSMT primary mirror deformation due to telescope wind load.

Figure 5 Zernike expansion of the primary mirror deformation.

The power spectral density (PSD) of the primary mirror deformation, shown in Figure 6, indicates that the bandwidth of the spatial RMS deformation is reasonably narrow. The majority of wind power is concentrated below 0.5 Hz. After removing the first 16 Zernike components of the deformation, the residual RMS deformation with frequency content above 0.5 Hz is about 70 nm. This value does not contradict our pivotal assumption.

Figure 6 PSD of primary mirror deformation due to wind load (spatial RMS).

In the simulation described above we didn't take into account the dynamics of the rafts and segment support structures. Because the lowest resonance frequency of the raft (5.8 Hz) is well above the lowest resonant frequency of the structure (about 2 Hz), the rafts do not impose additional requirements on validating our assumption.

4.8.7 ARCHITECTURE

Based on the assumption of separability of segment alignment and structural dynamics, a distributed telescope control architecture can be outlined. The continuity of the segmented primary mirror is maintained by a system utilizing edge detector information about the shape of the mirror. Because the lowest order modes of the structure have resonant frequencies around 2 Hz, the bandwidth of the continuity maintenance system should be less than 0.5 Hz in order to avoid interactions with telescope resonances.

A continuous primary mirror does not guarantee an aberration-free telescope, though. The shape of the primary mirror still can be deformed just as other parts of the telescope structure can. Our choices are either further observing the structural deformations and correcting them when and where they occur, or extracting information from the aberrated wavefront and arranging a hierarchical correction scheme mostly independent of the actual location of the deformation. The latter method has the advantage of directly observing and controlling the single most important characteristics of the telescope: its optical quality.

The first line of correction should be the deformable secondary mirror. It has the widest bandwidth in both the spatial and temporal sense. However, its stroke is limited, so the slower, lower order but larger amplitude errors due to piston, tip, tilt and de-center should cascade down to the secondary rigid body motion and finally to the main axes (see Figure 9).

From the point of view of control architecture, the AO system has two major actuator groups: the deformable secondary mirror and the higher-order deformable mirrors of MCAO and adaptive coronagraph. The deformable secondary mirror is inherently part of the telescope control system as well, because any wavefront sensor detecting the telescope deformation is also detecting the wavefront correction imposed by the secondary. On the other hand, the higher-order mirrors can be de-coupled from the telescope control by a sensible arrangement of WFSs, where the first high-order AO sensor is downstream of the last telescope control actuator in the light path. As long as the residual OPD is small enough in amplitude after all the telescope shape corrections, the high-order AO system-with its superior resolution and bandwidth-can certainly handle any misalignment possibly generated by the rest of the control system. Consequently, the high-order AO systems (MCAO and adaptive coronagraph) can be considered independent of the telescope control and are beyond the focus of this discussion.

4.8.7.1 Feedback of Mechanical Sensors

An influence function can be defined for the primary mirror as the assembly of edge detector responses, y, to actuator pokes, u. Our pivotal assumption implies that the influence function G of the primary mirror is independent of frequency (quasi-static).

The singular value decomposition of the influence function yields actuator and sensor modes as basis vectors to any actuator command set and sensor reading group.8 Each actuator mode, consisting of distinct positions for each actuator, has a corresponding sensor mode, (i.e., a unique set of sensor readings). There is a single scalar gain sigmai for each actuator-sensor mode pair that characterizes the strength of the sensor response.

Figure 7 Primary mirror continuity control configuration.

The global controller that maintains the continuity and shape of the primary mirror performs two functions. First, as a reconstructor, it estimates hypothetical actuator forces that would result in the actual sensor readings. Then, as a controller, it feeds back the inverse of these forces to eliminate the deformation. The control configuration in Figure 7 also shows the disturbance forces d(s) and sensor noise n(s). In the absence of strong noise and disturbances, the optimal reconstructor is the pseudo-inverse of the influence function. The command signal r(s) is the optimal segment position determined by mirror phasing. Figure 8 Modal control loop.

Although the controller is global in the sense that it collects all the edge sensor information and commands all the actuators according to a global control law, it can be conceived as a set of independent modal controllers. Because any actuator command set can be expressed as the linear combination of actuator modes, and the resulting sensor output can be assembled as the same linear combination of the corresponding sensor modes, we are concerned only with the mode propagation through the system. The assembly of the individual modal controllers is the control law realized by the global controller. The ith modal loop is illustrated in Figure 8, which shows the modal contributions of noise and disturbance. The output for the ith sensor mode can be derived from the configuration in a straightforward manner.

Equation 6
One of the major concerns is the propagation of sensor noise through the system. It is apparent from the configuration that the larger the estimator gain, the larger the noise appearing on the actuators. Consequently, the insensitive modes where sigmai is low (i.e., the estimator gain is high) are carrying significant noise from the sensors to the actuators.

It might be assumed that the larger the noise on the actuators, the larger the continuity error on the mirror surface. However, the continuity error, appearing as the sensor reading, is always about equal to the sensor noise and independent of the loop gain or its distribution throughout the loop. To see that, one may look at the above expression for sensor modes.

Although the barely observable modes are not causing excessive discontinuities, they are still degrading the shape of the mirror through noise propagation. Therefore, these modes should be rejected in the continuity maintenance system by setting their gain to zero. Fortunately, the insensitive modes are associated with smooth mirror shapes8,9 and result in low-order optical aberrations that can be corrected on the deformable secondary, based on WFS measurements. If the stroke of the deformable secondary mirror appears to be insufficient for correcting large, low-order, and slow gravitational and thermal deformations, the time average of these deformations can be off-loaded to the segmented primary. A bandwidth as low as 0.1 Hz, which is still appropriate for gravitational correction, can ensure that this additional control will not disturb the fundamental continuity maintenance system that has a 0.5 Hz bandwidth.

4.8.7.2 Feedback of Wavefront Sensors

The optical feedback system has four actuator groups: (1) the deformable secondary mirror, (2) the secondary rigid body motion, (3) the primary mirror segments, and (4) the main axes. The somewhat arbitrarily defined separation of these subsystems in spatial and temporal frequencies is shown in Figure 9. The bandwidths for aO and the main axes were established so that their separation from the structural dynamics is ensured as much as possible.

Figure 9 Frequency band separation of wavefront correction.

The deformable secondary mirror constitutes a major technological challenge. The University of Arizona is building a large (0.6 m), high-resolution (> 300 actuators) deformable mirror capable of acting as a secondary for a large astronomical telescope.10 Although the actual architecture and realization path of the GSMT secondary mirror is open at this point, we assume a similar configuration for this discussion.

Although reasonable AO correction for a 30-m aperture certainly requires more than 300 actuators, the corresponding subaperture density is high enough to correct up to several tens of Zernike modes, which is our requirement to correct telescope (mainly primary mirror) deformations. A bandwidth of 100 Hz is suitable, because even the fastest disturbance source, the wind, has only negligible power above that.

The thin glass facesheet of the deformable secondary mirror has negligible mass compared to the whole secondary assembly containing the reference body, support structures, and electronics. Thus, even the fast motions of the facesheet are not transferred to the telescope structure. On the other hand, there is an extensive local control system which transforms the deformable secondary to position actuator with response time well below 0.01 seconds.11 Consequently, from the telescope control point of view, the deformable secondary as a unit can be considered as a quasi-static plant, and the same control configuration can be applied as the continuity maintenance system on the primary mirror.

The main axes are restricted to slow corrections by design, so they also constitute a quasi-static plant. The only control system having significant interaction with structural dynamics is the secondary rigid body motion. The objective of this control is to reduce the optical aberrations to a level correctable by the deformable secondary. Assuming the same 5 m/s wind load as in the Disturbance Evaluation section, the secondary mirror should be tilted about 5 arcsecRMS in about 5 Hz bandwidth (see Figure 10) to achieve less then 0.5 mRMS OPD.

Figure 10 Secondary mirror tip-tilt correcting wind-induced image motion.

The actual design of the feedback for secondary control is outside of the scope of this report; it will be a significant challenge later in the design process.

The control configuration for the optical feedback is shown in Figure 11, while Figure 12 illustrates the corresponding physical configuration. In Figure 11, a(s), d(s) and n(s) stand for the incoming wavefront errors (atmospheric turbulence and virtual sky motion), the mechanical disturbances (wind, thermal, and gravitational effects), and the measurement noise, consecutively.

Figure 11 Wavefront control configuration.

Figure 12 Wavefront control physical configuration.

4.8.8 REFERENCES

  1. G. Chanan, M. Troy, and C. Ohara, "Phasing the Primary Mirror Segments of the Keck Telescopes: A Comparison of Different Techniques," Proc. SPIE 4003, 188 (2000).

  2. W. K. Gawronski, Dynamics and Control of Structures (A Modal Approach), (Springer-Verlag, New York, 1998).

  3. S. M. Shinners, Advanced Modern Control System Theory and Design, (John Wiley & Sons, Inc., New York, 1998).

  4. J. C. Wyant, K. Creath "Basic Wavefront Aberration Theory for Optical Metrology" in Applied Optics and Optical Engineering XI, (Academic Press, Boston, 1992).

  5. R. J. Noll, "Zernike polynomials and atmospheric turbulence," JOSA 66, 207 (1976).

  6. J. Meeus, Astronomical algorithm (Willmann-Bell, Richmond, 1998).

  7. M. K. Cho, L. Stepp, and S. Kim, "Wind buffeting effects on the Gemini 8m primary mirrors," Proc. SPIE 4444, 302 (2001).

  8. G. Chanan, J. E. Nelson, C. Ohara, and E. Sirko, "Design Issues for the Active Control System of the California Extremely Large Telescope (CELT)," Proc. SPIE 4004, 363 (2000).

  9. M. Troy, G. Chanan, E. Sirko, and E. Leffer, "Residual Misalignments of the Keck Telescope Primary Mirror Segments: Classification of Modes and Implications for Adaptive Optics," Proc. SPIE 3352, 307 (1998).

  10. M. Lloyd-Hart, F. Wildi, H. Martin, P. McGuire, M. Kenworthy, R. Johnson, B. Fitz-Patrick, G. Z. Angeli, S. Miller, and R. Angel, "Adaptive Optics for the 6.5 m MMT," Proc. SPIE 4007, 167 (2000).

  11. A. Riccardi, G. Brusa, V. Biliotti, C. Del Vecchio, P. Salinari, P. Stefanini, P. Mantegazza, R. Biasi, M. Andrighettoni, C. Franchini, M. Lloyd-Hart, P. McGuire, S. Miller, and H. Martin, "The Adaptive Secondary Mirror for the 6.5m Conversion of the Multiple Mirror Telescope: Latest Laboratory Test Results of the P36 Prototype," Proc. SPIE 4007, 524 (2000).


October 2002