Download VLT DSM, the control system of the largest deformable

VLT DSM, the control system of the largest deformable secondary
mirror ever manufactured
Mauro Manettib , Marco Morandinia , Paolo Mantegazzaa ,
Roberto Biasib , Mario Andrighettonib , Daniele Gallienic
a Politecnico
di Milano, Dip. Ing. Aerospaziale, via La Masa 34 , 20156 Milano, Italy;
b Microgate srl, Via Stradivari 4, 39100 Bolzano, Italy;
c ADS International srl, Via Roma 87, 23868 Valmadrera (LC), Italy.
ABSTRACT
A proven technology for the shape control of large secondary deformable mirrors employs a magnetically levitated contactless solution and relies on voice-coil actuators co-located to capacitive position sensors. The present work focuses on the
description of the latest upgrade of this technology, as applied to the Very Large Telescope Deformable Secondary Mirror,
the largest continuous facesheet adaptive mirror ever manufactured. The controller is based on a completely decentralized
high frequency feedback coupled to a lower frequency improved feedforward. The system enhancements and performances
are verified through electromechanical tests.
1. INTRODUCTION
The first operative AO system with voice-coil actuators co-located to capacitive position sensors was mounted on the Multiple Mirror Telescope (MMT)1 at the end of the 90s. Such a system is based on the first generation of the technology
described here and features a 0.64 m shell diameter controlled at 336 points. A second generation realization, fully working since the beginning of 2012, is represented by the twin secondary adaptive mirrors mounted on the Large Binocular
Telescope (LBT),2, 3 which has 672 actuated points and a 0.91 m shell diameter. The LBT can be considered the state
of the art for working adaptive optics applications, with performances never achieved before.4 A further second generation deformable mirror has been mounted on the Magellan Telescope (MAG).5 In the spring 2012 the final design and
manufacturing of the European Southern Observatory (ESO) Very Large Telescope (VLT) secondary adaptive mirror was
completed.6 The third generation VLT AO system features a secondary mirror with 1170 actuators and a 1.12 m diameter
and is the largest adaptive mirror ever manufactured (see Fig. 1). New incoming realizations are envisaged for the Giant
Magellan Telescope (GMT)7 and the European Extremely Large Telescope (E-ELT).8, 9
Magnetically levitated large adaptive mirrors are now a reality, fully proving their potentialities. However they represented and still embody an engineering challenge because of: their deformable structure with a dimension in the order
of a meter, up to a thousand of actuated points, command rates up to 1 kHz, positioning accuracy within a few tens of
nanometers. The contact-less solution provides several advantages: the reduction of assembly distortions, the possibility
of mounting actuators with less stringent tolerances, the capability to operate the system with failed control points without any major performance degradation. Moreover, it allows to command relatively large average strokes, in the order of
100 μm, so that a single unit can carry out both chopping and high order corrections. Its main drawback is represented by
‘null-stiffness’ actuators, i.e. pure force actuators, so that all of the control points are coupled through the mirror stiffness.
Moreover the shape control of large shells is affected by high modal densities, with many thousands resonances very close
to each other, circa six per Hz, which are placed within both the controller and spillover bandwidth. A further critical
point is the difficulty to accurately predict and describe the mirror dynamics because of the intrinsic multiphysics nature of
the whole system. For example, the system performances are dependent on the controller gains, which in turn rely on the
system passive damping to avoid instabilities associated to the control action. The damping, in turn, significantly depends
on the thin air film entrapped between the mirror and its reference plane, whose properties are mainly determined by the
air gap height, that changes because of the imposed shape control. An adequate modeling is required for both the fluid
Further author information: (Send correspondence to Marco Morandini)
Marco Morandini: E-mail: [email protected], Telephone: +39 02 2399 8362
Control
electronics
Cold plate
Actuators
Reference
body
Thin mirror
with
magnets
Figure 1. Very Large Telescope Deformable Secondary Mirror.
dynamics and the structural damping but this is not an easy task since the modal damping is frequency dependent and control spillover affects resonances over a wide and modally dense frequency range. Another possible issue, which can affect
system performances and effectiveness, is represented by the electromagnetic sensors-actuators cross-talk: the voice-coil
electromagnetic fields can disturb sensor measurements, so deteriorating the positioning accuracy and possibly limiting
system stability trough the introduction of a feedback dependence between control forces and sensors noise/disturbance.
This problem has been substantially eliminated by means of proper electromagnetic and electrostatic design of the actuator. Other practical problems, like the entrapping of dust between the shell and reference body and the system survival
against extreme environmental conditions (wind, earthquake) have been solved by means of appropriate devices, applied
and verified to the already deployed units.
The stringent system requirements, along with the above cited critical issues, make the controller design of large adaptive mirrors highly challenging. At the very beginning, the coupled nature of the problem suggested the use of centralized
control systems. Nevertheless, the time needed to acquire, condition and process a large number of signals, together with
the high computational power required to implement a centralized controller, hindered the adoption of such a solution.
Moreover, even with centralized controllers, the system stability constraint prevents the possibility of achieving the commanded positions within the required bandwidth. The key difficulties of this kind of control problem are the high modal
density which characterizes large shells and the impossibility of applying an adequate attenuation to a significant portion
of the modes across and beyond the control bandwidth, because of the well-known relation between signal attenuation and
phase lag and the lower effectiveness of the fluid dynamics damping on the highest modes.
After several attempts, a satisfactory solution has been implemented on the MMT secondary mirror. It is based on a two
degrees of freedom controller based on a static feedforward (FF) combined with a completely decentralized proportionalderivative (PD) feedback (FB).1 Acknowledging the stiffness dominated nature of the control problem, the FF is designed
to provide the static forces needed to deform the shell to a required steady shape. Such a term relies on an in the field
identification of the shell stiffness matrix condensed at the actuation points, whose accuracy is a key element in granting
a precise positioning. The computation of the FF contribution requires a complete centralization, but is carried out at a
relatively low command frequency (250-1000 Hz), so becomes feasible even if it introduces a moderate phase lag in the
optical control loop. On the contrary the FB contribution is digitally realized at a far higher frequency (40-100 kHz),
so to allow the addition of a sizable electronic damping of the modal resonances, within both the command bandwidth
(up to 250-1000 Hz) and the spillover frequency region. The FB is also useful for improving the tracking capabilities
and the rejection of external disturbances, e.g. wind and gravity effects, while the authority of its proportional gain is
limited to the low stiffness/frequency modes. Such a control strategy demonstrated to be reliable and effective, so that
it has been left substantially unchanged up to the VLT secondary deformable mirror. Meanwhile possible improvements
for both the FF and FB paths have been suggested in the literature by several authors, e.g. the possibility of exploiting
a distributed implementation of the static FF10, 11 or the modification of the static FF term computation to obtain at the
same time an efficient centralized integral action,12, 13 along with a FF completion through the introduction of a system
dynamics compensation.12, 14, 15 Within the area of distributed controls some applications proposed for large deformable
mirrors can be found in.16–21 Nevertheless, despite the new technology potentialities available with respect to the first
MMT implementation, a complete centralization of the FB remains quite cumbersome to apply, because of the intervened
significant increase of controlled points.
The quite long experience acquired with thin shells deformable mirrors clearly demonstrated the leading role of the FF
contribution in determining system performances. In practice the system is controlled mainly through FF, while the FB is
required to condition the behavior of lightly damped modal shapes, which can be excited during the FF force application.
Moreover secondary mirrors upgrades through different technology generations have always been undertaken by following
the so called KISS principle,22 i.e. ‘Keep It Simple, Stupid’. This is because, once the design specs are satisfied, system
reliability has always been favored with respect to pure performances, i.e., referencing the KISS’s author once more:“If it
works do not fix it”. On the basis of the two just mentioned considerations it is easy to understand why, once the decision
to upgrade the VLT mirror control strategy was taken, the attention has been focused on FF improvements, choosing their
most easy and reliable implementation. In practice the new VLT controller exploits the same features of its predecessors
with the addition of a dynamic FF term, designed on the basis of the proposal of,12 with a few further improvements. The
result is a specialization to the problem at hand of the PD control with FF system dynamics compensation developed in the
robotic field during the eighties.23, 24
2. VLT DSM SYSTEM DESCRIPTION
The VLT adaptive secondary has been designed to replace the current telescope secondary unit, so that the optical requirements and the mechanical interface remain unchanged. The adaptive mirror features a 1.12 m diameter and it is aspheric
convex with a 4.55 m radius curvature. The shell is supported through a membrane placed at its central hole with a 96 mm
diameter. It constraints the in-plane mirror displacements, while introducing a negligible out-plane stiffness. The 2 mm
thick continuous facesheet Zerodur mirror is controlled through 1170 actuated points. The contact-less technology is based
on voice-coil motors, with the coils placed at the top of 190 mm long actuator fingers applying their forces to the mirror
through an electromagnetic interaction with permanent magnets glued on the rear mirror face. The actuators bottom side is
rigidly connected to the cold plate, see Fig. 1, which acts both as the support and heat sink for the coils. Capacitive position
sensors are exploited to measure the distance between the reference plane and the rear mirror face, so to well approximate
a co-location with the actuators. The system controller is implemented through the use of an on-board unit, featuring 78
dedicated digital signal processors (DSP), which guarantee an overall computational power of 152 GMACs/s. Every DSP
board handles 16 coils by means of high efficiency switching drives. A dedicated hexapod linked to the cold plate allows
a fine support positioning and switching between different focal stations.
The VLT secondary implements a third technology generation, which ensures several improvements. The cold plate
is connected directly to the hexapod, to obtain a higher mechanical stiffness, the Zerodur reference body has been highly
light-weighted to reduce its mass and further improve the mechanical stability. The switching voice-coil drivers can halve
the required power, so limiting energy consumption and heat dissipation needs at the same time. The computational power
is increased and the all electronic hardware made more compact.
3. CONTROL STRATEGY
On the basis of the wavefront sensor measurements the optimal mirror shape generator sends a step command to the
deformable mirror, with a maximum command frequency of 1 kHz. The (na × 1) step command vector, where na is the
r (t),
number of actuation points, is then shaped in time though a polynomial function which guarantees a C1 acceleration, fsh
so that the reference position signal to be tracked at the (k + 1) command step can be written as
r
d r (t) = d r(k) + (dd r(k+1) − d r(k) ) fsh
(t).
(1)
In this way every step command is split into an initial time varying part, followed by a steady reference position.
The (na × 1) control forces vector is the sum of a FF and a FB contribution
f c = f cf f + f cf b .
(2)
The (na × 1) FB force vector is a completely decentralized PD control realized through a digital implementation at a
relatively high sampling frequency, of 61 kHz.
f cf b = [r G d r ] ṗp + [r G pr ](dd r − p ),
(3)
where [r G d r ] and [r G pr ] are (na × na ) diagonal matrices representing respectively the derivative and the proportional
gains, while p and ṗp are (na × 1) vectors of the control points position and velocity. The position is acquired directly from
the capacitive sensors, while the velocity is estimated through pseudo-derivative filters.
The FF term can be further split into two different contributions
f cf f = f cf fs + f cf fd ,
(4)
where f cf fs is the (na × 1) static FF vector and f cf fd the (na × 1) dynamic FF vector. The static FF provides the static forces
to deform the mirror and requires a completely centralized computation, which is performed at the command frequency of
up to 1 kHz. The static FF force is shaped in time within each command step, in the same way as for the reference signal,
usually through a slower function, fshf (t), to help limiting the excitation of high frequency dynamics. The force applied at
the (k + 1) command step can be written as
f cf fs (t) = f cf fs (k) + K ∗ (dd r(k+1) − s r(k) ) fshf (t)
(5)
where K ∗ is the (na × na ) identified static FF matrix,1, 12 i.e. a close relative of the true mirror stiffness matrix condensed at
the actuation points, see25, 26 for a discussion of the differences between the identified and the true mirror stiffness matrix.
The above presented terms are already implemented on all the working adaptive secondary mirrors based on the here
considered technology. The last control upgrade on the VLT system introduces the (na × 1) dynamic FF vector
r
r
C ∗ ḋd (t)
f cf fd = M ∗ d̈d (t) +C
=M
∗
(6)
C ∗ (dd r(k+1) − d r(k) ) f˙shf (t),
(dd r(k+1) − d r(k) ) f¨shf (t) +C
whose contribution is based on the (na × na ) matrices M ∗ and C ∗ , which ideally represent the mirror mass and damping
matrices condensed at the actuation points. These matrices could be identified as for K ∗ ,14 but the work12 suggests the
possibility of achieving satisfactory performances by approximating both of them as scalar diagonal matrices
r
r
f cf fd = [r m ∗ r ]d̈d + [r c ∗ r ]ḋd ,
(7)
where the scalar values m∗ and c∗ can be easily tuned through an in the field optimization of the system respons as, for
example, proposed in27 . A possible starting value for m∗ is represented by the mass of the mirror divided by the number
of controlled points, i.e. a lumped approximation. With respect to12 the dynamic FF scheme implemented for the VLT
has been improved by introducing a scheduling of the scalar value c∗ as a function of the air gap between the mirror
controlled points and the reference plane. This improvement is required because the fluid dynamic damping is highly
affected by the air gap height and an improved dynamics compensation requires a scheduled damping FF. So there is an
optimal scalar value c∗ , which is a function of the air gap, retrieved directly through the commanded position, c∗ = c∗ (d r ).
In this way the matrix [r c ∗ (dd r )r ] remains diagonal but no more scalar, because each of its diagonal elements is a function
of the corresponding commanded position. The parameter scheduling as function of d r has been tuned directly on the
system by modifying experimentally the value at different gaps in order to maintain the same dynamic behavior. Then,
the obtained values have been fitted as polynomial function of the gap, thus allowing an effective implementation of the
adaptive scheduler in the real time controller.
Figure 2. Commanded modal shape (mode 100).
4. EXPERIMENTAL SETUP AND ELECTROMECHANICAL TESTS
VLT experimental tests have been wholly carried out by exploiting the dedicated electronics developed for the control
of the secondary adaptive mirror. The decentralized FB and the FF contributions are implemented through a hard-real
time code, partly on the DSPs (see Sec. 2) and partly through the use of field-programmable gate array (FPGA). The D/A
converters and current drivers for the voice-coil motors are mounted on the very same DSPs control boards.
The conditioning of the capacitive position sensors is handled by dedicated boards placed on the actuators fingers as
well. In this way the distance from their pickup points is minimized, so reducing their signals noise while improving its
integrity. The sensors bandwidth is about 26 kHz, with a noise of 3-4 nm RMS. The initial sensors calibration is based
on piston motions of the shell and mechanical references, with the assumption of a uniform force distribution among all
the actuators, so that any significant elastic deformation of the mirror can be avoided.6 The related calibration accuracy
is in the order of 10 nm over the full motion range of 120 μm. This calibration process will be eventually improved by a
similar procedure, but using optical metrology. A key point of the experimental setup is represented by the environmental
stability of the controlled system, which is highly correlated to the stability of the capacitive position sensors. Typical
sensor thermal drifts are in the range of 3 nm/◦ C, while the humidity drift achieves a value of about 15 nm/(g m−3 ).
The actuators have a bandwidth of about 25 kHz. The actual force of each actuator is not acquired directly, but can be
estimated accurately by knowing the driven coil current and the motor current/force constant.
The electronics itself represents a powerful tool to perform static and dynamic experimental tests. The synchronous
excitation of all the actuators and acquisition from all co-located sensors is allowed by dedicated hardware and software
features, so that it is possible to test the system capability to track predefined time histories, with a maximum command rate
equal to the digital control frequency (61 kHz). In this way measurements and commands can be compared directly and
typical performance estimators of the control system, like single actuator step responses, modal step responses or modal
transfer functions can be obtained without the need of any external data acquisition system.
Here the modal tests are shortly summarized to highlight the improvements obtained through the introduction of the
dynamic FF term. The commanded modal shapes are computed as the right singular vectors of the identified static FF
matrix K ∗ , ordered for increasing singular values, i.e. stiffness. The system requirements ask for a settling time to 90% of
modal step commands within 1 ms. The performance has to be checked in presence of a mirror tilt, up to ±12 arcsec, as
required for field stabilization. Fig. 2 shows the 100th mirror modal shape; the corresponding modal time step responses,
superimposed to a mirror tilt of 12 arcsec, are reported in Fig. 3. Figure 3 compares two modal step responses obtained with
different control schemes, the first one based only on the static FF (standard FF), the second one exploiting the dynamic FF
5
Modal amplitude (µ m)
4
3
2
Response standard FF
Response dynamic FF
Shaped command standard FF
Shaped command dynamic FF
± 10% step command
1
0
2
3
4
5
Time (ms)
6
7
Figure 3. Modal step response with standard FF and with dynamic FF (mode 100).
contribution. Both the responses fulfill the system requirements in terms of settling time; however, the response obtained
with the standard control requires a smoother shaped command to limit the overshoot within 10% of the modal step
amplitude, so that the resulting settling time remains acceptable but clearly higher than that achievable with the dynamic
FF contribution. In practice, the dynamic FF allows to limit response overshoots, so that the same overshoot obtained with
standard FF is achieved despite a faster raising command, i.e. the dynamic FF improves the system settling time.
The aforementioned comment applies to all of the modal shapes. A summary of the system response behavior with
respect to all the 1170 modal shapes obtained from the static FF matrix can be appreciated in Figs. 4 and 5. These figures
clearly show that the dynamic FF allows to obtain the same responses overshoots with an improved settling time, within
all the considered frequency range. Figs. 4 and 5 confirm that both the control strategies guarantee the system specification
fulfillment, with an overshoot within 10% of modal amplitude and a settling time below 1 ms. Fig. 4 also suggests that
low order modes are almost critically damped, mainly because of the fluid dynamics contribution, so that their overshoot
is limited. The modes up to about the 100th show an increasingly overshoot because the damping is decreasing and they
belongs to a frequency range which is still excited by the control action. Moving to higher order modes the system response
approaches a static behavior and is dominated by static FF forces. The appreciable overshoot increment of the very high
order modes is an artifact due to the small modal step amplitude, which becomes close to sensors noise.
5. FINAL REMARKS
The experimental tests confirm the importance of the dynamic FF in improving the deformable mirror transient behavior,
which is expected to provide better overall optical performances eventually. The upgraded controller allows to obtain
faster raising times to reference step commands, while limiting overshoots, so that an improved settling time is obtained.
This result has been achieved with marginal modifications with respect to the previous controllers and with a negligible
added computational cost. In this way a step toward improved performances has been accomplished without any risk to
compromise the system reliability gained through a 15 years work.
7
Modal response overshoot (%)
6
Standard FF
Dynamic FF
5
4
3
2
1
0
0
200
400
600
Mode number
800
1000
Figure 4. Modal overshoot.
1
0.95
Settling time (ms)
0.9
0.85
Standard FF
Dynamic FF
0.8
0.75
0.7
0.65
0
200
400
600
Mode number
800
Figure 5. Modal settling time.
1000
REFERENCES
[1] Wildi, F. P., Brusa, G., Lloyd-Hart, M., Close, L. M., and Riccardi, A., “First light of the 6.5-m MMT adaptive optics
system,” in [Astronomical Adaptive Optics Systems and Applications ], Tyson, R. and Lloyd-Hart, M., eds., 5169,
17–25, SPIE, San Diego, CA, USA (Dec. 2003).
[2] Riccardi, A., Xompero, M., Briguglio, R., Quirós-Pacheco, F., Busoni, L., Fini, L., Puglisi, A., Esposito, S., Arcidiacono, C., Pinna, E., Ranfagni, P., Salinari, P., Brusa, G., Demers, R., Biasi, R., and Gallieni, D., “The adaptive
secondary mirror for the large binocular telescope: optical acceptance test and preliminary on-sky commissioning results,” in [Adaptive Optics Systems II ], Ellerbroek, B. L., Hart, M., Hubin, N., and Wizinowich, P. L., eds.,
77362C–77362C, SPIE (July 2010).
[3] Esposito, S., Riccardi, A., Fini, L., Puglisi, A. T., Pinna, E., Xompero, M., Briguglio, R., Quirós-Pacheco, F., Stefanini, P., Guerra, J. C., Busoni, L., Tozzi, A., Pieralli, F., Agapito, G., Brusa-Zappellini, G., Demers, R., Brynnel, J.,
Arcidiacono, C., and Salinari, P., “First light AO (FLAO) system for LBT: final integration, acceptance test in europe,
and preliminary on-sky commissioning results,” in [Adaptive Optics Systems II ], Ellerbroek, B. L., Hart, M., Hubin,
N., and Wizinowich, P. L., eds., 773609–773609, SPIE (July 2010).
[4] Green, R., Smith, M., and McCarthy, D., “Large binocular telescope brings the universe into sharper focus,” LBT
Official Press Release (Mar. 2012).
[5] Kopon, D., Close, L. M., Males, J., Gasho, V., and Follette, K., “The magellan adaptive secondary VisAO camera:
Diffraction-limited broadband visible imaging and 20mas fiber array IFU,” in [Adaptive Optics Systems II ], Ellerbroek, B., Hart, M., Hubin, N., and Wizinowich, P., eds., 7736, 77362V–77362V–11, SPIE, San Diego, California,
USA (July 2010).
[6] Biasi, R., Andrighettoni, M., Angerer, G., Mair, C., Pescoller, D., Lazzarini, P., Anaclerio, E., Mantegazza, M.,
Gallieni, D., Vernet, E., Arsenault, R., Madec, P.-Y., Duhoux, P., Riccardi, A., Xompero, M., Briguglio, R., Manetti,
M., and Morandini, M., “VLT deformable secondary mirror: integration and electromechanical tests results,” in
[Adaptive Optics Systems III ], Ellerbroek, B. L., Marchetti, E., and Véran, J.-P., eds., 8447, 84472G–84472G, SPIE
(Sept. 2012).
[7] Bouchez, A. H., Acton, D. S., Agapito, G., Arcidiacono, C., Bennet, F., Biliotti, V., Bonaglia, M., Briguglio, R.,
Brusa-Zappellini, G., Busoni, L., Carbonaro, L., Codona, J. L., Conan, R., Connors, T., Durney, O., Espeland, B.,
Esposito, S., Fini, L., Gardhouse, R., Gauron, T. M., Hart, M., Hinz, P. M., Kanneganti, S., Kibblewhite, E. J., Knox,
R. P., McLeod, B. A., McMahon, T., Montoya, M., Norton, T. J., Ordway, M. P., d’Orgeville, C., Parcell, S., Piatrou,
P. K., Pinna, E., Price, I., Puglisi, A., Quiros-Pacheco, F., Riccardi, A., Roll, J. B., Trancho, G., Uhlendorf, K.,
Vaitheeswaran, V., van Dam, M. A., Weaver, D., and Xompero, M., “The Giant Magellan Telescope adaptive optics
program,” in [Adaptive Optics Systems III ], 8447, 84471I–84471I–12, SPIE, Amsterdam (2012).
[8] Gallieni, D., Tintori, M., Mantegazza, M., Anaclerio, E., Crimella, L., Acerboni, M., Biasi, R., Angerer, G., Andrigettoni, M., Merler, A., Veronese, D., Carel, J., Marque, G., Molinari, E., Tresoldi, D., Toso, G., Spano, P., Riva, M.,
Mazzoleni, R., Riccardi, A., Mantegazza, P., Manetti, M., Morandini, M., Vernet, E., Hubin, N., Jochum, L., Madec,
P., Dimmler, M., and Koch, F., “Voice-coil technology for the E-ELT M4 adaptive unit,” in [International Conference
on Adaptive Optics for Extremely Large Telescopes ], (June 2009).
[9] Biasi, R. and Gallieni, D., “Contactless large deformable mirrors: ELT AO corrector technology available now,” in
[Second International Conference on Adaptive Optics for Extremely Large Telescopes ], (2011).
[10] Heimsten, R., MacMynowski, D. G., Andersen, T., and Owner-Petersen, M., “Concept, modeling, and performance
prediction of a low-cost, large deformable mirror,” Applied Optics 51, 515 (Feb. 2012).
[11] Heimsten, R., Owner-Petersen, M., Ruppel, T., MacMynowski, D. G., and Andersen, T., “Suppressing low-order
eigenmodes with local control for deformable mirrors,” Optical Engineering 51(2), 026601–1–026601–9 (2012).
[12] Manetti, M., Morandini, M., and Mantegazza, P., “High precision massive shape control of magnetically levitated
adaptive mirrors,” Control Engineering Practice 18, 1386–1398 (Dec. 2010).
[13] Manetti, M., Morandini, M., and Mantegazza, P., “Servo-fluid-elastic modeling of contactless levitated adaptive
secondary mirrors,” Computational Mechanics 50, 85–98 (July 2012).
[14] Manetti, M., High Precision Shape Control of Masssively Actuated, Magnetically Levitated, Secondary Adaptive
Mirrors for Extremely Large Telescopes, Ph.D. Thesis, Politecnico di Milano, Milano (Feb 2011).
[15] Ruppel, T., Dong, S., Rooms, F., Osten, W., and Sawodny, O., “Feedforward control of deformable membrane mirrors
for adaptive optics,” Control Systems Technology, IEEE Transactions on 21(3), 579–589 (2013).
[16] Stein, G. and Gorinevsky, D., “Design of surface shape control for large two-dimensional arrays,” Control Systems
Technology, IEEE Transactions on 13(3), 422–433 (2005).
[17] Kulkarni, J., D’Andrea, R., Brandl, B., Wizinowich, P., and Bonaccini, D., “Application of distributed control techniques to the adaptive secondary mirror of Cornell’s Large Atacama Telescope,” in [Adaptive Optical System Technologies II ], 4839, 750–756, SPIE, Waikoloa, HI, USA (Feb. 2003).
[18] Massioni, P. and Verhaegen, M., “Distributed control for identical dynamically coupled systems: A decomposition
approach,” Automatic Control, IEEE Transactions on 54(1), 124–135 (2009).
[19] Ellenbroek, R., Verhaegen, M., Doelman, N., Hamelinck, R., Rosielle, N., Steinbuch, M., Ellerbroek, B. L., and
Calia, D. B., “Distributed control in adaptive optics: Deformable mirror and turbulence modeling,” in [Advances in
Adaptive Optics II ], 6272, 62723K–62723K–12, SPIE, Orlando, FL, USA (June 2006).
[20] MacMynowski, D. G., Heimsten, R., and Andersen, T., “Distributed force control of deformable mirrors,” European
Journal of Control 17, 249–260 (June 2011).
[21] Manetti, M., Morandini, M., and Mantegazza, P., “Completely, partially centralized and fully decentralized control
schemes of large adaptive mirrors,” Journal of Vibration and Control in press.
[22] Rich, B. R., “Clarence leonard (kelly) johnson,” National Academy of sciences (1995).
[23] An, C., Atkeson, C., Griffiths, J., and Hollerbach, J., “Experimental evaluation of feedforward and computed torque
control,” IEEE Transactions on Robotics and Automation 5(3), 368–373 (1989).
[24] Santibanez, V. and Kelly, R., “PD control with feedforward compensation for robot manipulators: analysis and
experimentation,” Robotica 19(1), 11–19 (2001).
[25] Manetti, M., Morandini, M., Mantegazza, P., Biasi, R., Gallieni, D., and Riccardi, A., “Experimental validation
of massively actuated deformable adaptive mirror numerical models,” Control Engineering Practice 20, 783–791
(August 2012).
[26] Manetti, M., Morandini, M., Mantegazza, P., Biasi, R., Gallieni, D., and Riccardi, A., “Experimental validation of a
numerical model for non-contact, massively actuated deformable adaptive mirrors,” in [Proc. SPIE], 7736 (2010).
[27] Manetti, M., Morandini, M., and Mantegazza, P., “Self-tuning shape control of massively actuated adaptive mirrors,”
Control Systems Technology, IEEE Transactions on 22, 838–852 (May 2014).