上海交通大学：《探索物质微观世界》课程教学资源（补充材料）引力波_Damping and local control of mirror suspensions for laser interferometric gravitational wave detectors

AIP Review of Scientific Instruments Damping and local control of mirror suspensions for laser interferometric gravitational wave detectors K.A.Strain and B.N.Shapiro Citation:Review of Scientific Instruments 83,044501(2012);doi:10.1063/1.4704459 View online:http://dx.doi.org/10.1063/1.4704459 View Table of Contents:http://scitation.aip.org/content/aip/journal/rsi/83/4?ver=pdfcov Published by the AlP Publishing Articles you may be interested in Invited Article:CO2 laser production of fused silica fibers for use in interferometric gravitational wave detector mirror suspensions Rev.Sci.Instrum.82,011301(2011);10.1063/1.3532770 An investigation of eddy-current damping of multi-stage pendulum suspensions for use in interferometric gravitational wave detectors Rev.Sci.Instrum.75,4516(2004);10.1063/1.1795192 Monolithic fused silica suspension for the Virgo gravitational waves detector Rev.Sci.Instrum.73,3318(2002;10.1063/1.1499540 Inertial control of the mirror suspensions of the VIRGO interferometer for gravitational wave detection Rev.Sci.Instrum.72,3653(2001;10.1063/1.1394189 An interferometric device to measure the mechanical transfer function of the VIRGO mirrors suspensions Rev.Sci.Instrum.69,1882(1998);10.1063/1.1148858 Recognize Those Utilizing Science to Innovate American Business Call for Nominate Proven Leaders for the 2016 A/P General Prize for Industrial Applications of Physics Motors Nominations More Information /www.aip.org/industry/prize Deadline∥July1,2016 AIP Questions /assoc@alp.org Reuse of AlP Publishing content is subject to the terms at:https://publishing.aip.org/authors/rights-and-permissions.Download to IP:183.195.251.6 On:Fri.22 Apr 2016 00:5549

Damping and local control of mirror suspensions for laser interferometric gravitational wave detectors K. A. Strain and B. N. Shapiro Citation: Review of Scientific Instruments 83, 044501 (2012); doi: 10.1063/1.4704459 View online: http://dx.doi.org/10.1063/1.4704459 View Table of Contents: http://scitation.aip.org/content/aip/journal/rsi/83/4?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Invited Article: CO2 laser production of fused silica fibers for use in interferometric gravitational wave detector mirror suspensions Rev. Sci. Instrum. 82, 011301 (2011); 10.1063/1.3532770 An investigation of eddy-current damping of multi-stage pendulum suspensions for use in interferometric gravitational wave detectors Rev. Sci. Instrum. 75, 4516 (2004); 10.1063/1.1795192 Monolithic fused silica suspension for the Virgo gravitational waves detector Rev. Sci. Instrum. 73, 3318 (2002); 10.1063/1.1499540 Inertial control of the mirror suspensions of the VIRGO interferometer for gravitational wave detection Rev. Sci. Instrum. 72, 3653 (2001); 10.1063/1.1394189 An interferometric device to measure the mechanical transfer function of the VIRGO mirrors suspensions Rev. Sci. Instrum. 69, 1882 (1998); 10.1063/1.1148858 Reuse of AIP Publishing content is subject to the terms at: https://publishing.aip.org/authors/rights-and-permissions. Download to IP: 183.195.251.6 On: Fri, 22 Apr 2016 00:55:49

REVIEW OF SCIENTIFIC INSTRUMENTS 83.044501(2012) Damping and local control of mirror suspensions for laser interferometric gravitational wave detectors K.A.Strain1.a)and B.N.Shapiro2.b) SUPA School of Physics Astronomy,University of Glasgow,Glasgow G12 8QQ.Scotland, United Kingdom 2LIGO-Massachusetts Institute of Technology,Cambridge,Massachusetts 02139,USA (Received 24 February 2012;accepted 2 April 2012;published online 18 April 2012) The mirrors of laser interferometric gravitational wave detectors hang from multi-stage suspensions. These support the optics against gravity while isolating them from external vibration.Thermal noise must be kept small so mechanical loss must be minimized and the resulting structure has high-Q resonances rigid-body modes,typically in the frequency range between about 0.3 Hz and 20 Hz.Op- eration of the interferometer requires these resonances to be damped.Active damping provides the design flexibility required to achieve rapid settling with low noise.In practice there is a compromise between sensor performance,and hence cost and complexity,and sophistication of the control algo- rithm.We introduce a novel approach which combines the new technique of modal damping with methods developed from those applied in GEO 600.This approach is predicted to meet the goals for damping and for noise performance set by the Advanced LIGO project.2012 American Institute of Physics.[http://dx.doi.org/10.1063/1.4704459] I.INTRODUCTION-SUSPENSIONS the optical wavelength:1.064 um.The mirrors must be posi- FOR INTERFEROMETRIC GRAVITATIONAL tioned to <1 pm in distance along the beam direction (modulo WAVE DETECTORS half the wavelength),and of order nanoradians in angle.Sta- Following initial searches for gravitational radiation car- ble,quiet suspensions are needed even to achieve the required ried out in recent years by a network of km-scale laser inter- operating point. ferometric gravitational wave detectors,-3 the detectors are The test-mass suspensions consist of 4 cascaded pendu- currently being upgraded.The sensitivity of the LIGO detec- lum stages,as sketched in Figure 1.The mirror is suspended tors is to be improved by an order of magnitude in the fre- on fused silica fibers for low thermal noise.6 The second stage quency range around 100 Hz,with the lower frequency limit up consists of a fused silica mass supported on loops of high for observing reduced from 40 Hz to 10 Hz.The project is carbon steel wire,while the (2 x 2)upper stages are made called Advanced LIGO(aLIGO),4 and the work reported here of metal and are also suspended on wires.Each of the wire- is designed to meet the project goals. hung stages includes high-stress,maraging-steel,cantilever- An interferometric gravitational wave detector is based mounted,triangular blade springs to provide a softer system on a set of sufficiently quiet,well-separated test masses whose and hence better isolation.'The upper end of each wire is local horizontal motion follows the oscillating gravitational attached to the tip of such a spring.The thick end of each field.Sensitive interferometric readout allows the evolution spring is attached to the previous suspension stage (or to the of the relative positions of the test masses to be recorded.The mounting structure in the case of the springs supporting the gravitational field acts on the bulk,or equivalently center of top mass).We refer to each set of wires or fibers and the mass mass,of the test masses. they support as a stage,numbered from I at the top to 4 at the To enable sensing,mirror coatings are applied to the sur- bottom. faces of the test masses.It is necessary to minimize thermal The masses are considered to be rigid bodies.We choose noise in the substrate and coatings.3 Low displacement noise Euler-basis local coordinates:x for the direction sensed by the can then be achieved by hanging each test mass on a suspen- interferometer,3 for local vertical,and y orthogonal to these. sion to provide isolation from the environment.Mechanical The angles about these axes are roll,yaw,and pitch.Of these dissipation in the materials of the suspension would also lead x,pitch and yaw are sensed by the interferometer,and the oth- to thermal noise and so must be limited.Low loss materials ers cross-couple weakly into interferometer signals.The max- are employed,such as fused silica for the suspension fibers imum acceptable displacement and angular noise at the test which support the mirrors. mass are given in Table I. Operation of the interferometer requires precise align- We describe the development of control methods and al- ment of its mirrors.The suspensions must allow control of gorithms required to meet the stated performance goals for test mass separation-4 km in aLIGO,to very much less than aLIGO.We show that a combination of two control methods provides best performance.One of these "modal damping" (Sec.V)was newly developed using modern control theory, a)Electronic mail:kenneth.strain@glasgow.ac.uk. the other(Sec.VI)is an extension and refinement of methods b)Electronic mail:bshapiro@MIT.EDU. applied in GEO 600,which has operated for almost a decade. 0034-6748/2012/83(4)/044501/9/$30.00 83,044501-1 2012 American Institute of Physics Reuse of AlP Publishing content is subject to the terms at:https://publishing.aip.org/authors/nghts-and-permis Downlo8dolP:183.195251.60:Fi.22Apr2016 00:5549

REVIEW OF SCIENTIFIC INSTRUMENTS 83, 044501 (2012) Damping and local control of mirror suspensions for laser interferometric gravitational wave detectors K. A. Strain1,a) and B. N. Shapiro2,b) 1SUPA School of Physics & Astronomy, University of Glasgow, Glasgow G12 8QQ, Scotland, United Kingdom 2LIGO – Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA (Received 24 February 2012; accepted 2 April 2012; published online 18 April 2012) The mirrors of laser interferometric gravitational wave detectors hang from multi-stage suspensions. These support the optics against gravity while isolating them from external vibration. Thermal noise must be kept small so mechanical loss must be minimized and the resulting structure has high-Q resonances rigid-body modes, typically in the frequency range between about 0.3 Hz and 20 Hz. Op￾eration of the interferometer requires these resonances to be damped. Active damping provides the design flexibility required to achieve rapid settling with low noise. In practice there is a compromise between sensor performance, and hence cost and complexity, and sophistication of the control algo￾rithm. We introduce a novel approach which combines the new technique of modal damping with methods developed from those applied in GEO 600. This approach is predicted to meet the goals for damping and for noise performance set by the Advanced LIGO project. © 2012 American Institute of Physics. [http://dx.doi.org/10.1063/1.4704459] I. INTRODUCTION—SUSPENSIONS FOR INTERFEROMETRIC GRAVITATIONAL WAVE DETECTORS Following initial searches for gravitational radiation car￾ried out in recent years by a network of km-scale laser inter￾ferometric gravitational wave detectors,1–3 the detectors are currently being upgraded. The sensitivity of the LIGO detec￾tors is to be improved by an order of magnitude in the fre￾quency range around 100 Hz, with the lower frequency limit for observing reduced from 40 Hz to 10 Hz. The project is called Advanced LIGO (aLIGO),4 and the work reported here is designed to meet the project goals. An interferometric gravitational wave detector is based on a set of sufficiently quiet, well-separated test masses whose local horizontal motion follows the oscillating gravitational field. Sensitive interferometric readout allows the evolution of the relative positions of the test masses to be recorded. The gravitational field acts on the bulk, or equivalently center of mass, of the test masses. To enable sensing, mirror coatings are applied to the sur￾faces of the test masses. It is necessary to minimize thermal noise in the substrate and coatings.5 Low displacement noise can then be achieved by hanging each test mass on a suspen￾sion to provide isolation from the environment. Mechanical dissipation in the materials of the suspension would also lead to thermal noise and so must be limited. Low loss materials are employed, such as fused silica for the suspension fibers which support the mirrors. Operation of the interferometer requires precise align￾ment of its mirrors. The suspensions must allow control of test mass separation—4 km in aLIGO, to very much less than a)Electronic mail: kenneth.strain@glasgow.ac.uk. b)Electronic mail: bshapiro@MIT.EDU. the optical wavelength: 1.064μm. The mirrors must be posi￾tioned to <1 pm in distance along the beam direction (modulo half the wavelength), and of order nanoradians in angle. Sta￾ble, quiet suspensions are needed even to achieve the required operating point. The test-mass suspensions consist of 4 cascaded pendu￾lum stages, as sketched in Figure 1. The mirror is suspended on fused silica fibers for low thermal noise.6 The second stage up consists of a fused silica mass supported on loops of high carbon steel wire, while the (2 × 2) upper stages are made of metal and are also suspended on wires. Each of the wire￾hung stages includes high-stress, maraging-steel, cantilever￾mounted, triangular blade springs to provide a softer system and hence better isolation.7 The upper end of each wire is attached to the tip of such a spring. The thick end of each spring is attached to the previous suspension stage (or to the mounting structure in the case of the springs supporting the top mass). We refer to each set of wires or fibers and the mass they support as a stage, numbered from 1 at the top to 4 at the bottom. The masses are considered to be rigid bodies. We choose Euler-basis local coordinates: x for the direction sensed by the interferometer, z for local vertical, and y orthogonal to these. The angles about these axes are roll, yaw, and pitch. Of these x, pitch and yaw are sensed by the interferometer, and the oth￾ers cross-couple weakly into interferometer signals. The max￾imum acceptable displacement and angular noise at the test mass are given in Table I. We describe the development of control methods and al￾gorithms required to meet the stated performance goals for aLIGO. We show that a combination of two control methods provides best performance. One of these “modal damping” (Sec. V) was newly developed using modern control theory, the other (Sec. VI) is an extension and refinement of methods applied in GEO 600, which has operated for almost a decade. 0034-6748/2012/83(4)/044501/9/$30.00 © 2012 American Institute of Physics 83, 044501-1 Reuse of AIP Publishing content is subject to the terms at: https://publishing.aip.org/authors/rights-and-permissions. Download to IP: 183.195.251.6 On: Fri, 22 Apr 2016 00:55:49

044501-2 K.A.Strain and B.N.Shapiro Rev.Sci.Instrum.83,044501(2012) (15 Hz mode)of the lowest stage,and as they are well isolated from external vibration and nearly orthogonal to x,they do not require to be damped. Active or "cold"damping has been chosen instead of steel eddy-current dampers.The noise associated with active damping can be improved with a better sensor,or increased wires filtering,whereas with passive damping there is a fixed rela- -stage 1 tionship between damping and noise.The active approach can more easily provide variable damping to suit environmental -stage 2 conditions. By choice of masses,moments of inertia,and length and attachment points of wires and fibers,the 22 rigid-body +-stage 3 modes that require to be damped were arranged to lie below 5 Hz,to leave a gap for the application of filtering to take effect below the lower end of the observing band at 10 Hz. steel Each suspension stage provides low pass filtering above silica wires the rigid-body resonances of the coupled system.This deter- fibers mines our approach to the application of control forces.As +-stage 4 in previous applications,such as in GE600,0 we apply damping feedback at the top mass,to maximize the low-pass filtering of associated noise.Therefore,6 one-axis displace- ment sensors are fitted around each top mass.An actuator, consisting of a coil which acts on a magnet fixed to the top FIG.1.Modified rendering of a CAD model of the main elements of an stage,is co-located with every sensor.The resulting sensor- aLIGO quadruple suspension.There are two chains of 4 stages,numbered as shown.One supports the mirror (lowest mass in the front chain),the other actuator unit is called an optical sensor electro-magnetic ac- provides a quiet platform at each level for actuation.The top 3 stages are tuator (OSEM). supported on springs to improve vertical isolation.Stages 1 and 2(and stage To allow damping of all stages from the top mass 3 of the reaction chain)incorporate adjustable/moveable mass to trim and balance the suspension.Stages 3 and 4 of the main chain are formed from (stage 1)a"marionette"-like arrangement is employed,with fused silica and weigh 40 kg each.The test mass is polished and coated to the masses coupled so that movement of the top-most stage form a mirror which hangs on 4 fused silica fibers of 0.2 mm radius and leads to motion,primarily in the same direction,of all the 600 mm length,to provide low thermal noise masses,and vice versa.This is achieved by linking the masses with 4 wires (or fibers).with suitably chosen points of attach- ment at each mass.In the language of control theory,all the II.LOCAL CONTROL OF THE ALIGO modes to be damped are observable and controllable at the QUADRUPLE SUSPENSIONS top mass.A fuller description of the mechanical design of the The test mass suspension has rigid-body resonances from suspension may also be found in Ref.11. the lowest x mode at 0.41 Hz to the highest roll mode near The desire for a simple,reliable sensor led to the se- 13 Hz.In the region of 0.41 Hz the residual motion of the lection of a"shadow"sensor.The displacement is measured supporting isolation table is ~2nm/VHz.8 The mechanical through the motion of a"flag,"attached to the mass,by its quality factor of the mode is perhaps ~106,so the resulting modulating effect on a beam passing from an IR-emitter to rms motion,atO times the input spectral density,is of order a silicon photodetector.The sensor has a range sufficient to 2 um.This motion is too large and must be damped.as must cope with largest drifts and construction tolerances,with typ- most of the other 23 modes. ical noise performance of 0.07 nm/Hz at 10 Hz The two highest-frequency modes are exceptions.These Damping a simple pendulum,for instance,requires the are dominated by the vertical bounce (9 Hz mode)and roll application of a feedback force proportional to the velocity of the bob.Displacement signals may be differentiated to pro- duce velocity,but this boosts sensor noise at high frequencies. TABLE I.Noise amplitude spectral density limits for the aLIGO test We require the more sophisticated approach described below. masses.Upper limits are set a factor of 10 below the intended instrumen- In summary,a damping system is required to reduce the tal noise floor.allowing for cross-coupling to the sensitive direction.Each limit falls as 1/f from 10 Hz to 30 Hz.The interferometer is insensitive to O of the rigid-body resonances from perhaps 10 due to nat- roll,though roll noise can couple into,e.g..x in the mechanical system. ural damping down to of order 10.We restate this goal:the 1/e damping time is to be no more than about 10 s for fast Coordinate Noise limit at 10 Hz Units recovery from disturbance. 10-20 m/W亚 y 1017 m/√Hz IIl.SENSOR AND ACTUATOR PLACEMENT,AND 10-17 m/VHz IMPLICATIONS FOR CONTROLLER TOPOLOGY Yaw 1017 rad/VHz Pitch 10~17 rad/W伍 Cross-coupling is unavoidable in a marionette-like sus- pension with chained stages:x and pitch are mutually Reuse of AlP Publishing content is subject to the terms at:https://publishing.aip.org/authors/nights-and-pemm ssions.Download to IP:183.195.251.6 On:Fri.22 Apr 2016 00:5549

044501-2 K. A. Strain and B. N. Shapiro Rev. Sci. Instrum. 83, 044501 (2012) FIG. 1. Modified rendering of a CAD model of the main elements of an aLIGO quadruple suspension. There are two chains of 4 stages, numbered as shown. One supports the mirror (lowest mass in the front chain), the other provides a quiet platform at each level for actuation. The top 3 stages are supported on springs to improve vertical isolation. Stages 1 and 2 (and stage 3 of the reaction chain) incorporate adjustable/moveable mass to trim and balance the suspension. Stages 3 and 4 of the main chain are formed from fused silica and weigh 40 kg each. The test mass is polished and coated to form a mirror which hangs on 4 fused silica fibers of 0.2 mm radius and 600 mm length, to provide low thermal noise. II. LOCAL CONTROL OF THE ALIGO QUADRUPLE SUSPENSIONS The test mass suspension has rigid-body resonances from the lowest x mode at 0.41 Hz to the highest roll mode near 13 Hz. In the region of 0.41 Hz the residual motion of the supporting isolation table is ∼2 nm/ √Hz.8 The mechanical quality factor of the mode is perhaps Q ∼ 106, so the resulting rms motion, at √Q times the input spectral density, is of order 2μm. This motion is too large and must be damped, as must most of the other 23 modes. The two highest-frequency modes are exceptions. These are dominated by the vertical bounce (9 Hz mode) and roll TABLE I. Noise amplitude spectral density limits for the aLIGO test masses. Upper limits are set a factor of 10 below the intended instrumen￾tal noise floor, allowing for cross-coupling to the sensitive direction. Each limit falls as 1/f 2 from 10 Hz to 30 Hz. The interferometer is insensitive to roll, though roll noise can couple into, e.g., x in the mechanical system. Coordinate Noise limit at 10 Hz Units x 10−20 m/ √Hz y 10−17 m/ √Hz z 10−17 m/ √Hz Yaw 10−17 rad/ √Hz Pitch 10−17 rad/ √Hz (15 Hz mode) of the lowest stage, and as they are well isolated from external vibration and nearly orthogonal to x, they do not require to be damped. Active or “cold” damping has been chosen instead of eddy-current dampers.9 The noise associated with active damping can be improved with a better sensor, or increased filtering, whereas with passive damping there is a fixed rela￾tionship between damping and noise. The active approach can more easily provide variable damping to suit environmental conditions. By choice of masses, moments of inertia, and length and attachment points of wires and fibers, the 22 rigid-body modes that require to be damped were arranged to lie below ≈5 Hz, to leave a gap for the application of filtering to take effect below the lower end of the observing band at 10 Hz. Each suspension stage provides low pass filtering above the rigid-body resonances of the coupled system. This deter￾mines our approach to the application of control forces. As in previous applications, such as in GEO 600,10 we apply damping feedback at the top mass, to maximize the low-pass filtering of associated noise. Therefore, 6 one-axis displace￾ment sensors are fitted around each top mass. An actuator, consisting of a coil which acts on a magnet fixed to the top stage, is co-located with every sensor.11 The resulting sensor￾actuator unit is called an optical sensor electro-magnetic ac￾tuator (OSEM). To allow damping of all stages from the top mass (stage 1) a “marionette”-like arrangement is employed, with the masses coupled so that movement of the top-most stage leads to motion, primarily in the same direction, of all the masses, and vice versa. This is achieved by linking the masses with 4 wires (or fibers), with suitably chosen points of attach￾ment at each mass. In the language of control theory, all the modes to be damped are observable and controllable at the top mass. A fuller description of the mechanical design of the suspension may also be found in Ref. 11. The desire for a simple, reliable sensor led to the se￾lection of a “shadow” sensor. The displacement is measured through the motion of a “flag,” attached to the mass, by its modulating effect on a beam passing from an IR-emitter to a silicon photodetector. The sensor has a range sufficient to cope with largest drifts and construction tolerances, with typ￾ical noise performance of 0.07 nm/ √Hz at 10 Hz. Damping a simple pendulum, for instance, requires the application of a feedback force proportional to the velocity of the bob. Displacement signals may be differentiated to pro￾duce velocity, but this boosts sensor noise at high frequencies. We require the more sophisticated approach described below. In summary, a damping system is required to reduce the Q of the rigid-body resonances from perhaps 106 due to nat￾ural damping down to of order 10. We restate this goal: the 1/e damping time is to be no more than about 10 s for fast recovery from disturbance. III. SENSOR AND ACTUATOR PLACEMENT, AND IMPLICATIONS FOR CONTROLLER TOPOLOGY Cross-coupling is unavoidable in a marionette-like sus￾pension with chained stages: x and pitch are mutually Reuse of AIP Publishing content is subject to the terms at: https://publishing.aip.org/authors/rights-and-permissions. Download to IP: 183.195.251.6 On: Fri, 22 Apr 2016 00:55:49

044501-3 K.A.Strain and B.N.Shapiro Rev.Sci.Instrum.83,044501(2012) coupled as are y and roll.On the other hand,if the suspension latter model includes a facility to export numerical matrices is perfectly balanced about the vertical axis,vertical and yaw compatible with the earlier MATLAB models (but free of motion are separable from the others,and the x-pitch and symmetry constraints).This method provides rapid and con- y -roll systems are separate from each other.In practice,un- venient evaluation of numerical results and was used to gen- predictable degrees of cross-coupling emerge due to construc- erate the plots presented in this paper. tion tolerances or minor asymmetry in the mechanical assem- The Mathematica model and its exported numerical bly.Further,angles are measured as off-axis displacements, solutions were extensively tested against several prototype and the sensitive axis of each sensor is likely to be slightly suspensions.14 This underpins our use of the model to eval- off.So we anticipate a mixture of information,not fully pre- uate local damping methods. dictable,in each of the 6 sensor outputs. In GEO 600 point-to-point feedback was employed with the 6 channels sharing the same control law,and the result fed V.MODAL DAMPING back only to the co-located actuator.Each channel can damp any mode of the suspension,so the result is robust.A disad- A novel approach to low-noise damping for the quadru- vantage of this approach is that the gain cannot be varied sepa- ple and other suspensions was developed by Rueti5 and rately for displacement and angle signals.This is dealt with by expanded on by Shapiro.16 This exploits modern control placing the sensors offset from the corresponding symmetry theory to convert the control problem for the multi-input- axis by a distance approximately equal to the relevant radius multi-output (MIMO)24 degree of freedom (DOF)system of gyration,rg,to provide the desired balance of gain for the into 24 simpler single-input-single-output (SISO)systems. linear and rotational modes. Each of these systems can be damped independently using The choice of rg is justified because each mode arises as a relatively simple filter of uniform transfer function shape a solution of a second order equation of the form with parameters adjusted to suit each mode (as noted above, the two highest-frequency modes need not be damped and are M=-K, (1) henceforth ignored). where M represents either a mass m or a moment of inertia One of the major goals of modal damping is to sim- plify control design by decomposing many coupled DOFs I=r2m,xi represents any of the 6 coordinates,and K is a generalized spring constant.If there is a pair of modes of sim- into an equal number of independent DOFs in a modal co- ilar frequency,one in a linear and one in a rotational coordi- ordinate system.Each of the resulting independent modal el- nate,the ratio of the K values must be the same as the ratio of ements is a 2nd order (mass-spring)resonance,the modal the mass to moment of inertia,which is by definitionr2.Pro- mass and modal stiffness of which determine the resonant vided that the modes are in the appropriate frequency range, frequency. this provides a starting point for placing the sensors,and this Figure 2 provides a block diagram of the modal damping aspect of the approach was retained for aLIGO signal flow for the 4 modes that dominate the x direction dy- namics,which will be referenced throughout the remainder of There remains the question of how to distribute the re- quired set of 6 sensors around the mass.The most obvious this section.These x modes are already sufficiently decoupled arrangement with two sensors facing each of 3 orthogonal from the other 20 by the symmetry of the system,allowingx faces was discouraged by mechanical constraints arising from displacement of the 4 stages to be handled independently. relatively complicated designs for the top mass to incorpo- rate springs,adjustments,and other features.This led,for the quadruple suspension,to placing 3 sensors at the front face to cover x,yaw,and pitch;one for y on one side of Euler Pendulum the mass,and the final two on top,for z and roll.The radii coordinates of gyration were used as guides for the positioning of these sensors. To allow separate optimization of control filters for each Estimator coordinate,the sensor signals are transformed to the Euler- basis by a 6 x 6 matrix,customized to each instance of the suspension.A similar matrix is applied to turn Euler basis feedback signals into the commands for the actuators.This G Modal increases design flexibility as the filtering in each channel can coordinates be matched to the noise limit for that coordinate. IV.THE ALIGO SUSPENSION MODEL FIG.2.A block diagram of a modal damping scheme for the 4 x modes. Suspension modeling for GEO 600 was carried out An estimator converts the incomplete sensor information into modal signals. in MATLAB,following a method developed by Torrie.12 The modal signals are then sent to damping filters,one for each DOF.The The resulting model was restricted to idealized suspensions resulting modal damping forces are brought back into the Euler coordinate symmetrical about z.Subsequently,Barton3 developed an system through the transpose of the inverse of the eigenvector matrix Only stage 1 forces are applied to maximize sensor noise filtering to stage 4.Note approach in Mathematica which allows asymmetry.This that this figure applies to a four DOF system Reuse of AlP Publishing content is subject to the terms at:https://publishing.aip.org/authors/nghts-and-permi sions. D0wmlo8doP:183.195251.60:Fi.22Apr2016 00:5549

044501-3 K. A. Strain and B. N. Shapiro Rev. Sci. Instrum. 83, 044501 (2012) coupled as are y and roll. On the other hand, if the suspension is perfectly balanced about the vertical axis, vertical and yaw motion are separable from the others, and the x−pitch and y −roll systems are separate from each other. In practice, un￾predictable degrees of cross-coupling emerge due to construc￾tion tolerances or minor asymmetry in the mechanical assem￾bly. Further, angles are measured as off-axis displacements, and the sensitive axis of each sensor is likely to be slightly off. So we anticipate a mixture of information, not fully pre￾dictable, in each of the 6 sensor outputs. In GEO 600 point-to-point feedback was employed with the 6 channels sharing the same control law, and the result fed back only to the co-located actuator. Each channel can damp any mode of the suspension, so the result is robust. A disad￾vantage of this approach is that the gain cannot be varied sepa￾rately for displacement and angle signals. This is dealt with by placing the sensors offset from the corresponding symmetry axis by a distance approximately equal to the relevant radius of gyration, rg, to provide the desired balance of gain for the linear and rotational modes. The choice of rg is justified because each mode arises as a solution of a second order equation of the form Mx¨i = −K xi, (1) where M represents either a mass m or a moment of inertia I = r 2 gm, xi represents any of the 6 coordinates, and K is a generalized spring constant. If there is a pair of modes of sim￾ilar frequency, one in a linear and one in a rotational coordi￾nate, the ratio of the K values must be the same as the ratio of the mass to moment of inertia, which is by definition r 2 g . Pro￾vided that the modes are in the appropriate frequency range, this provides a starting point for placing the sensors, and this aspect of the approach was retained for aLIGO. There remains the question of how to distribute the re￾quired set of 6 sensors around the mass. The most obvious arrangement with two sensors facing each of 3 orthogonal faces was discouraged by mechanical constraints arising from relatively complicated designs for the top mass to incorpo￾rate springs, adjustments, and other features. This led, for the quadruple suspension, to placing 3 sensors at the front face to cover x, yaw, and pitch; one for y on one side of the mass, and the final two on top, for z and roll. The radii of gyration were used as guides for the positioning of these sensors. To allow separate optimization of control filters for each coordinate, the sensor signals are transformed to the Euler￾basis by a 6 × 6 matrix, customized to each instance of the suspension. A similar matrix is applied to turn Euler basis feedback signals into the commands for the actuators. This increases design flexibility as the filtering in each channel can be matched to the noise limit for that coordinate. IV. THE ALIGO SUSPENSION MODEL Suspension modeling for GEO 600 was carried out in MATLABR , following a method developed by Torrie.12 The resulting model was restricted to idealized suspensions symmetrical about z. Subsequently, Barton13 developed an approach in MathematicaR which allows asymmetry. This latter model includes a facility to export numerical matrices compatible with the earlier MATLABR models (but free of symmetry constraints). This method provides rapid and con￾venient evaluation of numerical results and was used to gen￾erate the plots presented in this paper. The MathematicaR model and its exported numerical solutions were extensively tested against several prototype suspensions.14 This underpins our use of the model to eval￾uate local damping methods. V. MODAL DAMPING A novel approach to low-noise damping for the quadru￾ple and other suspensions was developed by Ruet15 and expanded on by Shapiro.16 This exploits modern control theory to convert the control problem for the multi-input￾multi-output (MIMO) 24 degree of freedom (DOF) system into 24 simpler single-input-single-output (SISO) systems. Each of these systems can be damped independently using a relatively simple filter of uniform transfer function shape with parameters adjusted to suit each mode (as noted above, the two highest-frequency modes need not be damped and are henceforth ignored). One of the major goals of modal damping is to sim￾plify control design by decomposing many coupled DOFs into an equal number of independent DOFs in a modal co￾ordinate system. Each of the resulting independent modal el￾ements is a 2nd order (mass-spring) resonance, the modal mass and modal stiffness of which determine the resonant frequency. Figure 2 provides a block diagram of the modal damping signal flow for the 4 modes that dominate the x direction dy￾namics, which will be referenced throughout the remainder of this section. These x modes are already sufficiently decoupled from the other 20 by the symmetry of the system, allowing x displacement of the 4 stages to be handled independently. FIG. 2. A block diagram of a modal damping scheme for the 4 x modes. An estimator converts the incomplete sensor information into modal signals. The modal signals are then sent to damping filters, one for each DOF. The resulting modal damping forces are brought back into the Euler coordinate system through the transpose of the inverse of the eigenvector matrix . Only stage 1 forces are applied to maximize sensor noise filtering to stage 4. Note that this figure applies to a four DOF system. Reuse of AIP Publishing content is subject to the terms at: https://publishing.aip.org/authors/rights-and-permissions. Download to IP: 183.195.251.6 On: Fri, 22 Apr 2016 00:55:49

044501-4 K.A.Strain and B.N.Shapiro Rev.Sci.Instrum.83,044501 (2012) A.Control design open loop system GM=8(2.48Hz),PM=76.6(1.2Hz) Mathematically,the modal decomposition is illustrated 150 100 in Eq.(2)-(4).The Euler equations of motion of the pendu- 50 lum are represented in Eq.(2)where M is the mass matrix,K 0 the stiffness matrix,P the vector of control forces,and the -50 displacement vector.The eigenvector basis of the matrix -100 M-K transforms between and the modal displacement co- -15 0 10 10 10 10 ordinates g.Substituting Eq.(3)into Eq.(2)and multiplying on the left by gives the modal equations,Eq.(4),i.e., M+K=P, (2) 90 -180 元=Φd, (3) -270 -360 Mmg +Kmg Pm (4) 10 10 100 10 10 frequency (Hz) Here the subscript m indicates that the mass matrix,stiffness matrix.and the force vector are now in the modal domain. FIG.3.The loop gain transfer function of an example 1Hz modal oscillator with its damping filter.The plant contributes the large resonant peak and the Note that these modal matrices are diagonal,which is a math- damping filter contributes the remaining poles and zeros.The 10 Hz notch ematical result of the fact that the dynamics of each modal reduces the sensor noise amplification at the start of the gravitational wave DOF are decoupled. detection band,where it is typically the worst.The large phase margin near the resonance permits tuning of the gain k to achieve a significant range of For the application to damping control of a real system closed loop Qs.All the damping loops have the same basic shape but are there are two important transformations required: shifted in frequency and gain (the notch remains at the same frequency). g=Φ-1x, (5) pairs create a notch in the loop gain at 10 Hz to create a more P=(Φ-)TPm. (6) aggressive cutoff in the low pass filtering near the beginning of the GW detection band.This notch is designed to have a Equation (5)transforms the sensor signals into the modal do- main.Control can then be applied in a system where the dy- depth of a factor of 10 and a quality factor of 10.It is real- namics are decoupled.This transformation requires the full ized with the zeros and poles at10Hz,2.87°，and30°off the measured state.As Figure 2 shows,we only measure stage imaginary axis,respectively. 1 of the pendulum,so an estimator is used to perform the Equation(7)is an example of this damping filter transfer change of coordinates.Section V B details this estimator. function for a I Hz mode,where s is the Laplacian variable. k is the gain value that determines the closed loop damping The second transformation,Eg.(6),occurs after the modal damping control filters,labeled as G to G4 in ratio.See Figure 3 for an example Bode plot of a modal loop Figure 2.It converts the modal damping forces Pm generated gain transfer function employing this filter. by these filters into Euler forces that can be applied to the pen- The gains,k,on each filter can be tuned to provide just dulum by the actuators.Note that actuation is only available enough damping so that the minimum amount of sensor noise at stage 1 for this damping control.Although actuators exist is passed through the loop.Since the lower frequency modes at the lower stages,employing them would allow sensor noise inject the least sensor noise and,in general,contribute the to bypass the mechanical filtering of the stages above. most mechanical energy,the gains will tend to be set higher on those. Consequently,the applied damping forces are a projec- tion of the modal damping forces to stage 1.This limita- ks(s2+6.283s+3948) tion means that the damping is not truly modal,resulting in GIHz= (7) (s2+5.455s+246.7)(s2+62.83s+3948) cross coupling between modal feedback loops.Fortunately, this coupling is minimal for closed loop damping ratios less than 0.2.This upper limit is more than enough to meet the B.State estimation damping requirements. The nominal design of the feedback filters is relatively The mathematics of modal damping requires the posi- simple since each modal plant is identical except shifted in tions of all four stages to be measured.However,only stage 1 frequency and magnitude.The filter design has a total of 3 ze- is directly observed,as Figure 2 illustrates.The stages below ros and 4 poles.A zero at 0 Hz ensures the filters meet the AC have effectively no sensors since any measurement would re- coupling requirement.A complex pole pair is placed at 2.5 fer a moving platform with its own dynamics.Consequently times the frequency of the mode for low pass filtering.These an estimator,as in the equation, poles are placed 20 off the imaginary axis to achieve slight enhancements in the filtering and phase margin.The factor of 2.5 was chosen to achieve the most aggressive filtering pos- B.i-I sible while leaving enough phase margin to allow sufficiently high damping ratios.The remaining complex pole and zero must be employed to reconstruct the full dynamics Reuse of AlP Publishing content is subject to the terms at:https://publishing.aip.org/authors/nghts-and-perm Download to IP: 183.195251.60Fi.22Apr2016 00:5549

044501-4 K. A. Strain and B. N. Shapiro Rev. Sci. Instrum. 83, 044501 (2012) A. Control design Mathematically, the modal decomposition is illustrated in Eq. (2)–(4). The Euler equations of motion of the pendu￾lum are represented in Eq. (2) where M is the mass matrix, K the stiffness matrix, P the vector of control forces, and x the displacement vector. The eigenvector basis of the matrix M−1K transforms between x and the modal displacement co￾ordinates q. Substituting Eq. (3) into Eq. (2) and multiplying on the left by T gives the modal equations, Eq. (4), i.e., M¨ x + Kx = P, (2) x = q, (3) Mm ¨ q + Kmq = P m. (4) Here the subscript m indicates that the mass matrix, stiffness matrix, and the force vector are now in the modal domain. Note that these modal matrices are diagonal, which is a math￾ematical result of the fact that the dynamics of each modal DOF are decoupled. For the application to damping control of a real system there are two important transformations required: q = −1x, (5) P = (−1 ) T P m. (6) Equation (5) transforms the sensor signals into the modal do￾main. Control can then be applied in a system where the dy￾namics are decoupled. This transformation requires the full measured state. As Figure 2 shows, we only measure stage 1 of the pendulum, so an estimator is used to perform the change of coordinates. Section V B details this estimator. The second transformation, Eq. (6), occurs after the modal damping control filters, labeled as G1 to G4 in Figure 2. It converts the modal damping forces Pm generated by these filters into Euler forces that can be applied to the pen￾dulum by the actuators. Note that actuation is only available at stage 1 for this damping control. Although actuators exist at the lower stages, employing them would allow sensor noise to bypass the mechanical filtering of the stages above. Consequently, the applied damping forces are a projec￾tion of the modal damping forces to stage 1. This limita￾tion means that the damping is not truly modal, resulting in cross coupling between modal feedback loops. Fortunately, this coupling is minimal for closed loop damping ratios less than 0.2. This upper limit is more than enough to meet the damping requirements. The nominal design of the feedback filters is relatively simple since each modal plant is identical except shifted in frequency and magnitude. The filter design has a total of 3 ze￾ros and 4 poles. A zero at 0 Hz ensures the filters meet the AC coupling requirement. A complex pole pair is placed at 2.5 times the frequency of the mode for low pass filtering. These poles are placed 20◦ off the imaginary axis to achieve slight enhancements in the filtering and phase margin. The factor of 2.5 was chosen to achieve the most aggressive filtering pos￾sible while leaving enough phase margin to allow sufficiently high damping ratios. The remaining complex pole and zero FIG. 3. The loop gain transfer function of an example 1 Hz modal oscillator with its damping filter. The plant contributes the large resonant peak and the damping filter contributes the remaining poles and zeros. The 10 Hz notch reduces the sensor noise amplification at the start of the gravitational wave detection band, where it is typically the worst. The large phase margin near the resonance permits tuning of the gain k to achieve a significant range of closed loop Qs. All the damping loops have the same basic shape but are shifted in frequency and gain (the notch remains at the same frequency). pairs create a notch in the loop gain at 10 Hz to create a more aggressive cutoff in the low pass filtering near the beginning of the GW detection band. This notch is designed to have a depth of a factor of 10 and a quality factor of 10. It is real￾ized with the zeros and poles at 10 Hz, 2.87◦, and 30◦ off the imaginary axis, respectively. Equation (7) is an example of this damping filter transfer function for a 1 Hz mode, where s is the Laplacian variable. k is the gain value that determines the closed loop damping ratio. See Figure 3 for an example Bode plot of a modal loop gain transfer function employing this filter. The gains, k, on each filter can be tuned to provide just enough damping so that the minimum amount of sensor noise is passed through the loop. Since the lower frequency modes inject the least sensor noise and, in general, contribute the most mechanical energy, the gains will tend to be set higher on those. G1Hz = ks(s2 + 6.283s + 3948) (s2 + 5.455s + 246.7)(s2 + 62.83s + 3948). (7) B. State estimation The mathematics of modal damping requires the posi￾tions of all four stages to be measured. However, only stage 1 is directly observed, as Figure 2 illustrates. The stages below have effectively no sensors since any measurement would re￾fer a moving platform with its own dynamics. Consequently an estimator, as in the equation, ˙ˆ q ¨ˆ q = Am ˆ q ˙ˆ q + Bmu − Lm  Cm ˆ q ˙ˆ q − y  , (8) must be employed to reconstruct the full dynamics. Reuse of AIP Publishing content is subject to the terms at: https://publishing.aip.org/authors/rights-and-permissions. Download to IP: 183.195.251.6 On: Fri, 22 Apr 2016 00:55:49