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

044501-5 K.A.Strain and B.N.Shapiro Rev.Sci.Instrum.83,044501(2012) The estimator equation in (8)receives the control input i that minimizes this cost function in the chosen interval is opti- and the stage 1 sensor signals y in the Euler frame and esti- mal.For x,R is a scalar because it is a decoupled DOF by the mates the corresponding modal signals g using the modeled symmetry of the pendulum.Coupled DOFs such as y,pitch, state space matrices Am,Bm,Cm.The subscript m indicates and roll must be considered simultaneously resulting in a ma- the matrices refer to the modal frame.The goal is now to de- trix. sign the estimator feedback gain L. The two competing performance criteria of damping time The observer separation principle allows us to keep the and technical noise are stated above.The sensor noise is a already established control scheme,provided that the estima- known measured quantity approximately 7x 10-11 m/Hz tor has an accurate model of the pendulum,which is the as- beyond 10Hz.Since the contribution to mirror motion drops sumption here.Note that the estimator replaces the direct sig- off quickly in the frequency domain,we will only consider nal transformation in (5). the contribution at 10Hz,the start of the gravitational wave The estimator is designed using the Linear Quadratic detection band. Regulator(LQR)technique which solves the cost function in These two performance criteria are represented in the op- Egs.(9)and(10): timization routine with the cost function: =( Jg(R)=max(T2)+max(N2). (12) (9) DOF DOF with R=argmin(J&). (13) with Lm=argmin(). (10) R Here,T,is the normalized stage 4 settling time;chosen The Q matrix weights the accuracy of the modal state es- as the maximum settling time of all the DOFs normalized by timation while the R matrix weights the cost of using a noisy the goal,10s.Similarly,Nis the maximum sensor noise con- measurement.Here 3 is defined as tribution to the mirror relative to the respective requirement at 10 Hz for each DOF considered. (11) As an example,the results of Eg.(12)are plotted in Figure 4 for the x DOF where R is scalar valued.The opti- where Lm is the estimator feedback matrix determined mal value occurs at R=0.06.Since the total cost is less than by(10). 1,both design requirements are met at this optimal point.The An estimator could be generated in the form of a Kalman spectral density of the mirror displacement under this opti- filter for this application.However,a Kalman filter is con- mized modal damping control is plotted in Figure 5. cerned with optimal sensor data recovery whereas our goal is The need to include a model of the suspension within the to optimize the noise reaching the suspended mirror(stage 4). control algorithm suggests a limitation of the modal damping Sacrificing sensor signal accuracy and damping is acceptable approach.If the model is not accurate the inferred motion of to a certain degree if the noise performance is improved.Con- the lowest three stages may differ from reality.The dynamics sequently,a more direct approach is taken to obtain the values of the suspension depends on the positions of,for example, of Q and R. the attachment points relative to the masses and/or springs First,Q is set by placing the square of the inverse of the on each stage.In the aLIGO quadruple suspension,an error resonance frequencies on the diagonal.In this way lower res- of order 100 um in the vertical coordinate of an attachment onance frequencies,which have more mechanical energy.will point can affect the pitch modes significantly.This requires be damped more efficiently.Only the value of R remains to be the controller to be tuned to the suspension,at least in some optimized. If we restrict R to be diagonal,a common assumption, 10 we are left with six parameters to optimize simultaneously, relating to six Euler stage 1 measurement signals.To sim- 10 plify this problem one can invoke the symmetry of the pendu- 10 lum.For example,all the modes that represent vertical motion 10 (z)are decoupled from the others.Thus,these modes can be thought of as representing a separate system which has only 10 four DOFs and a single sensor signal.The same decompo- 10 sition can be made for yaw and x motion.The remaining 3 10 directions are inseparably coupled. settling cost,max(T) To choose the best value of R the control design is first 10 sensor noise cost,max(N total cost,Jg set so that the damping time requirement is met assuming full state information.An optimization routine in MATLAB then 10 10 10 R simulates the performance of the closed loop system for es- timators designed using many values of R over a sufficiently FIG.4.The components of the cost function Eq.(12)for the x DOF as a function of R calculated by the optimization routine.At each value of R the large space.A cost function,Eq.(12),dependent on the per- closed loop system performance is simulated using the estimator design based formance criteria is calculated for each value of R.The value on the LOR solution with that particular R value. 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044501-5 K. A. Strain and B. N. Shapiro Rev. Sci. Instrum. 83, 044501 (2012) The estimator equation in (8) receives the control input u and the stage 1 sensor signals y in the Euler frame and esti￾mates the corresponding modal signals ˆ q using the modeled state space matrices Am, Bm, Cm. The subscript m indicates the matrices refer to the modal frame. The goal is now to de￾sign the estimator feedback gain Lm. The observer separation principle allows us to keep the already established control scheme, provided that the estima￾tor has an accurate model of the pendulum, which is the as￾sumption here. Note that the estimator replaces the direct sig￾nal transformation in (5). The estimator is designed using the Linear Quadratic Regulator (LQR) technique which solves the cost function in Eqs. (9) and (10): J =  ∞ 0  [ ˜ qT ˙˜ qT ]Q ˜ q ˙˜ q + zT mRzm  dt, (9) with Lm = argmin Lm (J ). (10) The Q matrix weights the accuracy of the modal state es￾timation while the R matrix weights the cost of using a noisy measurement. Here zm is defined as zm = −LT m ˜ q ˙˜ q , (11) where Lm is the estimator feedback matrix determined by (10). An estimator could be generated in the form of a Kalman filter for this application. However, a Kalman filter is con￾cerned with optimal sensor data recovery whereas our goal is to optimize the noise reaching the suspended mirror (stage 4). Sacrificing sensor signal accuracy and damping is acceptable to a certain degree if the noise performance is improved. Con￾sequently, a more direct approach is taken to obtain the values of Q and R. First, Q is set by placing the square of the inverse of the resonance frequencies on the diagonal. In this way lower res￾onance frequencies, which have more mechanical energy, will be damped more efficiently. Only the value of R remains to be optimized. If we restrict R to be diagonal, a common assumption, we are left with six parameters to optimize simultaneously, relating to six Euler stage 1 measurement signals. To sim￾plify this problem one can invoke the symmetry of the pendu￾lum. For example, all the modes that represent vertical motion (z) are decoupled from the others. Thus, these modes can be thought of as representing a separate system which has only four DOFs and a single sensor signal. The same decompo￾sition can be made for yaw and x motion. The remaining 3 directions are inseparably coupled. To choose the best value of R the control design is first set so that the damping time requirement is met assuming full state information. An optimization routine in MATLABR then simulates the performance of the closed loop system for es￾timators designed using many values of R over a sufficiently large space. A cost function, Eq. (12), dependent on the per￾formance criteria is calculated for each value of R. The value that minimizes this cost function in the chosen interval is opti￾mal. For x, R is a scalar because it is a decoupled DOF by the symmetry of the pendulum. Coupled DOFs such as y, pitch, and roll must be considered simultaneously resulting in a ma￾trix. The two competing performance criteria of damping time and technical noise are stated above. The sensor noise is a known measured quantity approximately 7 × 10−11 m/ √Hz beyond 10 Hz. Since the contribution to mirror motion drops off quickly in the frequency domain, we will only consider the contribution at 10 Hz, the start of the gravitational wave detection band. These two performance criteria are represented in the op￾timization routine with the cost function: JR(R) = max DOF (T 2 s ) + max DOF (N2 ), (12) with R = argmin R (JR). (13) Here, Ts is the normalized stage 4 settling time; chosen as the maximum settling time of all the DOFs normalized by the goal, 10 s. Similarly, N is the maximum sensor noise con￾tribution to the mirror relative to the respective requirement at 10 Hz for each DOF considered. As an example, the results of Eq. (12) are plotted in Figure 4 for the x DOF where R is scalar valued. The opti￾mal value occurs at R = 0.06. Since the total cost is less than 1, both design requirements are met at this optimal point. The spectral density of the mirror displacement under this opti￾mized modal damping control is plotted in Figure 5. The need to include a model of the suspension within the control algorithm suggests a limitation of the modal damping approach. If the model is not accurate the inferred motion of the lowest three stages may differ from reality. The dynamics of the suspension depends on the positions of, for example, the attachment points relative to the masses and/or springs on each stage. In the aLIGO quadruple suspension, an error of order 100μm in the vertical coordinate of an attachment point can affect the pitch modes significantly. This requires the controller to be tuned to the suspension, at least in some FIG. 4. The components of the cost function Eq. (12) for the x DOF as a function of R calculated by the optimization routine. At each value of R the closed loop system performance is simulated using the estimator design based on the LQR solution with that particular R value. 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

044501-6 K.A.Strain and B.N.Shapiro Rev.Sci.Instrum.83,044501(2012) 10 Xno error x with error 30 10 0 10 (s/pe)ued 0 0 -10 ground disturbance (power law fit) -OSEM sensor noise -mirror response to ground disturbance -30 .mirror response to sensor noise 1022 total mirror response 40 10 10 100 101 3 -2.5 -2.1.5 -0.5 0 frequency (Hz) real part(rad/s) FIG.5.An amplitude spectrum showing a simulation of the mirror displace- FIG.6.Complex frequency (s)-plane plot showing the poles x of the closed ment along the x DOF under the influence of the optimized modal damp- loop modal damping system.The reference system is represented by the bold ing loop with R=0.06.The black dashed line is the sensor noise and the (black)symbols,while the 100 trials of perturbed systems are represented green line is its contribution to the mirror displacement.The solid black line by the finer (red)symbols.Each trial represents a system modified from the is the ground disturbance and the blue line is its contribution to the mir- ideal using the random parameters described in the text.In this test 16%of ror displacement.The red line is the uncorrelated stochastic sum of both the cases are unstable. contributions trials,16 were unstable,confirming that,unless the modal subset of the parameters.Un-modeled sensor cross-coupling controller is matched to the suspension,there is a significant is also a potential source of instability.Both aspects of robust- risk of instability. ness of the control scheme to variations within the plant were The second robustness test explores whether unintended tested. cross-coupling of signals leads to instability.If,for example, First we consider robustness against assembly tolerances a sensor were to be mounted at an angle to the intended axis that could lead to changes in mode frequencies and mode- of sensitivity unwanted signals may pass through a particular coupling.This was informed by measurements that were channel. performed on several examples of the suspension design A 6 x 6 matrix of uniformly distributed random num- (including re-builds of the suspension where the same me- bers was generated,and summed with the identity matrix, chanical parts were partly dismantled and re-assembled to such that there were random elements about unity on the di- test assembly tolerances).A model was built in which the agonal,and random elements about zero elsewhere.This was most important parameters could be varied randomly over then applied to represent errors in the sensing matrix.Again the expected range.In each trial all of the chosen parame- the closed-loop poles,with all 6 damping loops at nominal ters had a random component added or factored in,as ap- gain,were checked for indications of instability.Modal con- propriate.The random elements were generated by scaling trol proved robust against 10%peak uniform random errors in uniformly distributed pseudo-random numbers generated in this test. MATLAB@ The sensitive parameters in the suspension are the z-coordinates of the wire (or fiber)attachment points,mea- VI.MULTI-MODE DAMPING OF MULTI-STAGE SUSPENSIONS sured from the centers of mass-these were varied by up to +1 mm,by adding a suitable random amount.This was done In the multi-mode damping approach up to 4 modes per for the four such attachment planes most subject to assembly coordinate are damped with a single controller.There are 6 errors:that for the wires hanging from the top mass,for the channels each consisting of a damping law and a low-pass fil- wires hanging from the 2nd mass,and for the top and bottom ter in series.The modes in a given channel (x,y,3,pitch,yaw, of the fibers in stage 4. or roll)are spread over about 3 octaves but settling behav- Another class of error arises where production tolerance ior is dominated by the lowest-frequency mode in each co- leads to variations in the moments of inertia of the masses,in ordinate.If a conventional differentiator-law damper is used particular affecting the off-diagonal terms which are relatively these lowest frequency modes are also damped with the least small.The pitch and roll inertias were randomly modified by gain further exaggerating their dominance,nor is noise per- multiplying with a random factor in the range from 0.95 to formance optimal.This motivated a search for alternative 1.05.Finally the off-diagonal term which couples pitch and control laws. roll was varied from 0.5 to 1.5 times the design value (this In the quadruple suspension the shortest settling time large range is required as the nominal value of the associated does not correspond to maximum damping of the top stage: moment of inertia is relatively small). an over-damped top-stage extracts less energy from the lower One hundred random models were generated and com- masses,leading to weaker damping of one or more modes. bined with the modal controller for the reference case.The This behavior was explored in a 2-DOF,2-stage suspension resulting systems were evaluated by plotting the closed-loop model.The transfer functions are quartic in Laplace variable poles in the complex plane,as shown in Figure 6.Of the 100 s and were solved analytically.The solutions are cumbersome 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:Fr.22 Apr 2016 00:5549

044501-6 K. A. Strain and B. N. Shapiro Rev. Sci. Instrum. 83, 044501 (2012) FIG. 5. An amplitude spectrum showing a simulation of the mirror displace￾ment along the x DOF under the influence of the optimized modal damp￾ing loop with R = 0.06. The black dashed line is the sensor noise and the green line is its contribution to the mirror displacement. The solid black line is the ground disturbance and the blue line is its contribution to the mir￾ror displacement. The red line is the uncorrelated stochastic sum of both contributions. subset of the parameters. Un-modeled sensor cross-coupling is also a potential source of instability. Both aspects of robust￾ness of the control scheme to variations within the plant were tested. First we consider robustness against assembly tolerances that could lead to changes in mode frequencies and mode￾coupling. This was informed by measurements that were performed on several examples of the suspension design (including re-builds of the suspension where the same me￾chanical parts were partly dismantled and re-assembled to test assembly tolerances). A model was built in which the most important parameters could be varied randomly over the expected range. In each trial all of the chosen parame￾ters had a random component added or factored in, as ap￾propriate. The random elements were generated by scaling uniformly distributed pseudo-random numbers generated in MATLABR . The sensitive parameters in the suspension are the z-coordinates of the wire (or fiber) attachment points, mea￾sured from the centers of mass—these were varied by up to ±1 mm, by adding a suitable random amount. This was done for the four such attachment planes most subject to assembly errors: that for the wires hanging from the top mass, for the wires hanging from the 2nd mass, and for the top and bottom of the fibers in stage 4. Another class of error arises where production tolerance leads to variations in the moments of inertia of the masses, in particular affecting the off-diagonal terms which are relatively small. The pitch and roll inertias were randomly modified by multiplying with a random factor in the range from 0.95 to 1.05. Finally the off-diagonal term which couples pitch and roll was varied from 0.5 to 1.5 times the design value (this large range is required as the nominal value of the associated moment of inertia is relatively small). One hundred random models were generated and com￾bined with the modal controller for the reference case. The resulting systems were evaluated by plotting the closed-loop poles in the complex plane, as shown in Figure 6. Of the 100 FIG. 6. Complex frequency (s)-plane plot showing the poles × of the closed loop modal damping system. The reference system is represented by the bold (black) symbols, while the 100 trials of perturbed systems are represented by the finer (red) symbols. Each trial represents a system modified from the ideal using the random parameters described in the text. In this test 16% of the cases are unstable. trials, 16 were unstable, confirming that, unless the modal controller is matched to the suspension, there is a significant risk of instability. The second robustness test explores whether unintended cross-coupling of signals leads to instability. If, for example, a sensor were to be mounted at an angle to the intended axis of sensitivity unwanted signals may pass through a particular channel. A 6 × 6 matrix of uniformly distributed random num￾bers was generated, and summed with the identity matrix, such that there were random elements about unity on the di￾agonal, and random elements about zero elsewhere. This was then applied to represent errors in the sensing matrix. Again the closed-loop poles, with all 6 damping loops at nominal gain, were checked for indications of instability. Modal con￾trol proved robust against 10% peak uniform random errors in this test. VI. MULTI-MODE DAMPING OF MULTI-STAGE SUSPENSIONS In the multi-mode damping approach up to 4 modes per coordinate are damped with a single controller. There are 6 channels each consisting of a damping law and a low-pass fil￾ter in series. The modes in a given channel (x, y, z, pitch, yaw, or roll) are spread over about 3 octaves but settling behav￾ior is dominated by the lowest-frequency mode in each co￾ordinate. If a conventional differentiator-law damper is used these lowest frequency modes are also damped with the least gain further exaggerating their dominance, nor is noise per￾formance optimal. This motivated a search for alternative control laws. In the quadruple suspension the shortest settling time does not correspond to maximum damping of the top stage: an over-damped top-stage extracts less energy from the lower masses, leading to weaker damping of one or more modes. This behavior was explored in a 2-DOF, 2-stage suspension model. The transfer functions are quartic in Laplace variable s and were solved analytically. The solutions are cumbersome 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

044501-7 K.A.Strain and B.N.Shapiro Rev.Sci.Instrum.83,044501(2012) 60 -mode 1 10 -mode 2 50 mode 3 -mode 4 40 10 10 27 80 10 10 10 10 frequency (Hz) damping factor(kg/s) FIG.8.Open loop Bode plot comparing damping laws for yaw.The damp- FIG.7.Settling times for the four x modes of a quadruple suspension with ing law,low-pass filter,and mechanical plant are combined.The solid (blue) pure velocity damping of variable strength at stage 1.The lowest mode (1) curves represent the differentiator law,with a suitable low-pass filter.The dominates yielding a shortest settling time of 18s(to 2%).with damping dashed (green)line shows the truncated differentiator with the pole at 3.5 Hz. strength 55 kg/s. The dotted(red)curve represents the interrupted differentiator.Finally the dashed-dotted(cyan)curve shows an example with resonant zeros and poles. The gains are adjusted to match at 10 Hz.Other filter parameters are given in the text. and,apart from the special case with the masses and stage lengths set equal,numerical methods were more convenient. In the equal-mass,equal-length case the minimum O for the same attenuation at 10 Hz,provides a slightly more favor- lower mode is 2.As the mass or length ratios deviate from able balance of phase margin and damping.The transfer func- unity,keeping the same total mass and length,the minimum tion of the resulting loop,with the pole at 3.5 Hz,is added to O and settling time tend to increase.This trend continues as Figure 8. more stages are added,conserving total length.The quadruple A filter with approximately f amplitude response has suspensions have transfer functions that are octic in s requir- /4 phase lead so at a given gain the velocity component is ing numerical methods. reduced by a factor cos(-/4)=1/v2.However due to the To estimate the shortest settling time,a model of the four lower gradient,the gain is increased at the lowest mode by a x DOFs was built with velocity damping applied at stage 1. factor /L:a little over 2 in this case,so there is net ben- The results are shown in Figure 7.As we are interested in efit.The f law can be approximated by inserting an extra the behavior over many 1/e-fold decay periods the slowest- pole and zero to reduce the slope of the differentiator (hence decaying mode dominates.The shortest 1/e decay constant is this filter is called the interrupted differentiator).Performance 4.5s. can be optimized by adjusting the frequencies of these new In the multi-mode system the most important attributes elements.The result,with zeros at 0.04 and 2 Hz and poles at of the controller are:(i)the gain and phase at the lowest- 0.75 and 4.5 Hz,is added to Figure 8. frequency mode,at frequency fi-these determine the settling The final control law considered has a resonant pair of time,(ii)the phase margin at the highest crossing of unity gain zeros and a resonant pair of poles replacing the real zero and immediately above the highest mode,at frequency fi.and(iii) pole of the simplest filter.The idea was to introduce addi- the noise attenuation at 10 Hz.To avoid excessive ringing and tional parameters,namely the two Os,but no performance ad- risk of instability,we aim for ~30 phase margin. vantage was found in increasing either O beyond 0.5,so this The MATLAB model corresponding to a particular in- filter has just two useful parameters.A version with zeros at stance of a suspension from the production series was em- 0.2 Hz and poles at 1.5 Hz,all with =0.5,is added to the ployed.We present results for yaw in detail. comparison graph. In the simplest filter a zero is placed no higher than The low-pass filters already shown are of one generic 0.04 Hz,and a pole is placed at 40 Hz or higher.Note that type but differ in detail to optimize phase margin and atten- real poles and zeros are indicated unless stated explicitly.A uation in each case.The steeply falling response of the sus- low-pass filter (see below)is added and the result is shown in pension implies that strongest attenuation is needed at 10 Hz Figure 8. and immediately above.The key to achieving better perfor- Damping requires a significant feedback component in mance than any of the standard (e.g.,elliptical)filter topolo- the velocity quadrature.We look for control laws with a shal- gies is found in two aspects of the steeply falling frequency lower response slope,trading-off phase lead at fi for increased response of the mechanical system.Even below 10 Hz there gain,and a larger useful feedback component.Modal damp- is a downward slope in the magnitude of the transfer function ing provides a solution matched to the suspension,but we from force applied to the top stage to its displacement.The search for methods that may be more tolerant of variation in same is true of the transfer function from force applied to the the plant. top stage to mirror motion.The former behavior permits Placing a pole to truncate the rising response of the dif- the insertion of complex poles of moderately high into the ferentiator,and modifying the low-pass filter to provide the loop while the latter allows complex zeros to be added. 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044501-7 K. A. Strain and B. N. Shapiro Rev. Sci. Instrum. 83, 044501 (2012) FIG. 7. Settling times for the four x modes of a quadruple suspension with pure velocity damping of variable strength at stage 1. The lowest mode (1) dominates yielding a shortest settling time of 18 s (to 2%), with damping strength 55 kg/s. and, apart from the special case with the masses and stage lengths set equal, numerical methods were more convenient. In the equal-mass, equal-length case the minimum Q for the lower mode is 2. As the mass or length ratios deviate from unity, keeping the same total mass and length, the minimum Q and settling time tend to increase. This trend continues as more stages are added, conserving total length. The quadruple suspensions have transfer functions that are octic in s requir￾ing numerical methods. To estimate the shortest settling time, a model of the four x DOFs was built with velocity damping applied at stage 1. The results are shown in Figure 7. As we are interested in the behavior over many 1/e-fold decay periods the slowest￾decaying mode dominates. The shortest 1/e decay constant is 4.5 s. In the multi-mode system the most important attributes of the controller are: (i) the gain and phase at the lowest￾frequency mode, at frequency fL—these determine the settling time, (ii) the phase margin at the highest crossing of unity gain immediately above the highest mode, at frequency fH, and (iii) the noise attenuation at 10 Hz. To avoid excessive ringing and risk of instability, we aim for ∼30◦ phase margin. The MATLABR model corresponding to a particular in￾stance of a suspension from the production series was em￾ployed. We present results for yaw in detail. In the simplest filter a zero is placed no higher than 0.04 Hz, and a pole is placed at 40 Hz or higher. Note that real poles and zeros are indicated unless stated explicitly. A low-pass filter (see below) is added and the result is shown in Figure 8. Damping requires a significant feedback component in the velocity quadrature. We look for control laws with a shal￾lower response slope, trading-off phase lead at fL for increased gain, and a larger useful feedback component. Modal damp￾ing provides a solution matched to the suspension, but we search for methods that may be more tolerant of variation in the plant. Placing a pole to truncate the rising response of the dif￾ferentiator, and modifying the low-pass filter to provide the FIG. 8. Open loop Bode plot comparing damping laws for yaw. The damp￾ing law, low-pass filter, and mechanical plant are combined. The solid (blue) curves represent the differentiator law, with a suitable low-pass filter. The dashed (green) line shows the truncated differentiator with the pole at 3.5 Hz. The dotted (red) curve represents the interrupted differentiator. Finally the dashed-dotted (cyan) curve shows an example with resonant zeros and poles. The gains are adjusted to match at 10 Hz. Other filter parameters are given in the text. same attenuation at 10 Hz, provides a slightly more favor￾able balance of phase margin and damping. The transfer func￾tion of the resulting loop, with the pole at 3.5 Hz, is added to Figure 8. A filter with approximately √ f amplitude response has π/4 phase lead so at a given gain the velocity component is reduced by a factor cos(−π/4) = 1/ √2. However due to the lower gradient, the gain is increased at the lowest mode by a factor √ fH/ fL: a little over 2 in this case, so there is net ben￾efit. The √ f law can be approximated by inserting an extra pole and zero to reduce the slope of the differentiator (hence this filter is called the interrupted differentiator). Performance can be optimized by adjusting the frequencies of these new elements. The result, with zeros at 0.04 and 2 Hz and poles at 0.75 and 4.5 Hz, is added to Figure 8. The final control law considered has a resonant pair of zeros and a resonant pair of poles replacing the real zero and pole of the simplest filter. The idea was to introduce addi￾tional parameters, namely the two Qs, but no performance ad￾vantage was found in increasing either Q beyond 0.5, so this filter has just two useful parameters. A version with zeros at 0.2 Hz and poles at 1.5 Hz, all with Q = 0.5, is added to the comparison graph. The low-pass filters already shown are of one generic type but differ in detail to optimize phase margin and atten￾uation in each case. The steeply falling response of the sus￾pension implies that strongest attenuation is needed at 10 Hz and immediately above. The key to achieving better perfor￾mance than any of the standard (e.g., elliptical) filter topolo￾gies is found in two aspects of the steeply falling frequency response of the mechanical system. Even below 10 Hz there is a downward slope in the magnitude of the transfer function from force applied to the top stage to its displacement. The same is true of the transfer function from force applied to the top stage to mirror motion. The former behavior permits the insertion of complex poles of moderately high Q into the loop while the latter allows complex zeros to be added. 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

044501-8 K.A.Strain and B.N.Shapiro Rev.Sci.Instrum.83,044501(2012) For a second-order system resonant at o,the phase of TABLE II.Settling time to 2%resulting from a unit impulse applied to the transfer function is the sensor input of the closed loop system.This is equivalent to the effect of an impulsive motion of the top mass,as observed at the test mass.The interrupted differentiator yields%quicker settling than the mean of the arctan (14) other methods. 0(o-2 Controller Settling time(s) where is the angular frequency.By inspection,increasing the O concentrates the phase change near the resonance,at Simple differentiator 43 the cost of a peak in the response. Truncated differentiator 42 Interrupted differentiator 36 In our application,the downward slope of the mechan- Resonant filter 41 ical transfer function in the relevant frequency range allows the use of such a filter without the resulting peak reaching unity gain and risking instability.According to the noise lim- its in each DOF up to 5 such filter stages may be required. Damping performance is summarized in Table II.The The lowest of these in frequency cannot have very high O initial response of a multi-mode system such as these suspen- and dominates the phase lag,but its inclusion steepens the re- sions depends on the relative excitation of the modes but even- sponse.This allows higher-frequency poles to have higher O. tually the slowest-decaying mode dominates.The l/e-fold de- contributing to the attenuation while adding relatively little cay time was estimated from the settling time to 2%.The phase lag at fu.We aim to create a"plateau"in the gain-curve interrupted differentiator achieves the shortest l/e time of above fi,at a level according to the desired gain margin.For about 9 s-a factor of two longer than the result with pure ve- a given damping strength and gain margin this maximizes the locity damping of ideal strength.Broadly similar results were phase lead atfu. obtained when the controls were optimized to damp pitch, The low-pass design is completed by adding complex ze- and y motion of the suspension. ros at or around 10 Hz.In this way the attenuation in the crit- With the multi-mode approach the damping time could ical band is increased,at the expense of poorer isolation at not be reduced below about 20 s,for x,without compromising higher frequencies (where it is not needed,due to the falling the phase margin (or attenuation,but that is unacceptable).On plant response).The zeros,one per pole,are chosen to have the other hand,the example from Sec.V shows that the modal low O,to help reduce the phase lag at f-low O zeros pro- method can meet all performance requirements in x. vide phase lead over a significant frequency interval.This completes the filter. A.Robustness of the multi-mode technique Parameter optimization was carried out giving priority to filtering performance,then balancing the phase margin and The multi-mode damping technique was tested for ro- settling time.The low-pass filter for the interrupted differen- bustness against parameter variation and construction toler- tiator case consists of three poles at 6 Hz,7 Hz,and 7.7 Hz, ances.The same two tests were applied as described for modal with Os of 3,4.5,and 7,and a zero at 9.5 Hz,plus two at damping.Again the test employed 100 random trials,all of 9.8 Hz,all with Os of 3.Due to the strongly sloping response which remained stable in the parameter variation test. of the plant the zeros require to be placed just below 10 Hz As with modal damping,in the test for the effects to provide greatest attenuation at 10 Hz.Confirmation that of cross-coupling,the multi-mode approach proved stable the noise filtering requirement is met by all 4 control laws is against uniform-random changes of 10%peak in all 36 ele- shown in Figure 9. ments of the sensor matrix. 10 VIl.CONCLUSION:APPLICATION TO ALIGO 10 We have presented two approaches to damping the rigid- body modes of the aLIGO quadruple suspensions.These 10 methods may have application elsewhere that low-noise damping of complicated suspension systems is required.The 10 modal method provides quick settling together with low noise.The multi-mode method can be adjusted to yield ac- 10 ceptable damping,while meeting the noise requirements,but upper lmit in the most important x direction where the most aggressive 10 filtering is required the modal approach produces faster set- tling.The other main difference is that robustness tests indi- 10 2 与 10 cate that the multi-mode method is more tolerant to variation frequency (Hz) in suspension parameters. Advanced LIGO includes 24 quadruple suspensions FIG.9.The magnitude of the transmissibility from the sensor input to mo- tion of the suspended mirror,in rad/m.The 4 curves correspond to the same (4 pairs of chains in each of 3 detectors),and the process 4 control laws as in the previous figure.The"upper-limit"line shows the of tuning each controller would be time consuming.For this maximum value allowed for the transmissibility above 10 Hz. reason it has been decided to apply the multi-mode approach 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

044501-8 K. A. Strain and B. N. Shapiro Rev. Sci. Instrum. 83, 044501 (2012) For a second-order system resonant at ωo, the phase of the transfer function is − arctan 1 Q ωωo  ω2 o − ω2   , (14) where ω is the angular frequency. By inspection, increasing the Q concentrates the phase change near the resonance, at the cost of a peak in the response. In our application, the downward slope of the mechan￾ical transfer function in the relevant frequency range allows the use of such a filter without the resulting peak reaching unity gain and risking instability. According to the noise lim￾its in each DOF up to 5 such filter stages may be required. The lowest of these in frequency cannot have very high Q and dominates the phase lag, but its inclusion steepens the re￾sponse. This allows higher-frequency poles to have higher Q, contributing to the attenuation while adding relatively little phase lag at fH. We aim to create a “plateau” in the gain-curve above fH, at a level according to the desired gain margin. For a given damping strength and gain margin this maximizes the phase lead at fH. The low-pass design is completed by adding complex ze￾ros at or around 10 Hz. In this way the attenuation in the crit￾ical band is increased, at the expense of poorer isolation at higher frequencies (where it is not needed, due to the falling plant response). The zeros, one per pole, are chosen to have low Q, to help reduce the phase lag at fH—low Q zeros pro￾vide phase lead over a significant frequency interval. This completes the filter. Parameter optimization was carried out giving priority to filtering performance, then balancing the phase margin and settling time. The low-pass filter for the interrupted differen￾tiator case consists of three poles at 6 Hz, 7 Hz, and 7.7 Hz, with Qs of 3, 4.5, and 7, and a zero at 9.5 Hz, plus two at 9.8 Hz, all with Qs of 3. Due to the strongly sloping response of the plant the zeros require to be placed just below 10 Hz to provide greatest attenuation at 10 Hz. Confirmation that the noise filtering requirement is met by all 4 control laws is shown in Figure 9. FIG. 9. The magnitude of the transmissibility from the sensor input to mo￾tion of the suspended mirror, in rad/m. The 4 curves correspond to the same 4 control laws as in the previous figure. The “upper-limit” line shows the maximum value allowed for the transmissibility above 10 Hz. TABLE II. Settling time to 2% resulting from a unit impulse applied to the sensor input of the closed loop system. This is equivalent to the effect of an impulsive motion of the top mass, as observed at the test mass. The interrupted differentiator yields ≈15% quicker settling than the mean of the other methods. Controller Settling time (s) Simple differentiator 43 Truncated differentiator 42 Interrupted differentiator 36 Resonant filter 41 Damping performance is summarized in Table II. The initial response of a multi-mode system such as these suspen￾sions depends on the relative excitation of the modes but even￾tually the slowest-decaying mode dominates. The 1/e-fold de￾cay time was estimated from the settling time to 2%. The interrupted differentiator achieves the shortest 1/e time of about 9 s—a factor of two longer than the result with pure ve￾locity damping of ideal strength. Broadly similar results were obtained when the controls were optimized to damp pitch, z and y motion of the suspension. With the multi-mode approach the damping time could not be reduced below about 20 s, for x, without compromising the phase margin (or attenuation, but that is unacceptable). On the other hand, the example from Sec. V shows that the modal method can meet all performance requirements in x. A. Robustness of the multi-mode technique The multi-mode damping technique was tested for ro￾bustness against parameter variation and construction toler￾ances. The same two tests were applied as described for modal damping. Again the test employed 100 random trials, all of which remained stable in the parameter variation test. As with modal damping, in the test for the effects of cross-coupling, the multi-mode approach proved stable against uniform-random changes of 10% peak in all 36 ele￾ments of the sensor matrix. VII. CONCLUSION: APPLICATION TO ALIGO We have presented two approaches to damping the rigid￾body modes of the aLIGO quadruple suspensions. These methods may have application elsewhere that low-noise damping of complicated suspension systems is required. The modal method provides quick settling together with low noise. The multi-mode method can be adjusted to yield ac￾ceptable damping, while meeting the noise requirements, but in the most important x direction where the most aggressive filtering is required the modal approach produces faster set￾tling. The other main difference is that robustness tests indi￾cate that the multi-mode method is more tolerant to variation in suspension parameters. Advanced LIGO includes 24 quadruple suspensions (4 pairs of chains in each of 3 detectors), and the process of tuning each controller would be time consuming. For this reason it has been decided to apply the multi-mode approach 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

044501-9 K.A.Strain and B.N.Shapiro Rev.Sci.Instrum.83,044501(2012) for all except the x direction,where the model is not over- Losurdo,J.-M.Mackowski,E.Majorana,C.N.Man.M.Mantovani,F. sensitive to the relevant parameters of the suspension and Marchesoni,F.Marion,J.Marque,F.Martelli,A.Masserot,F.Menzinger, the extra performance provided by the modal method is most L.Milano.Y.Minenkov,C.Moins.J.Moreau.N.Morgado,S.Mosca,B. Mours,I.Neri,F.Nocera,G.Pagliaroli,C.Palomba,F.Paoletti,S.Pardi, beneficial. A.Pasqualetti,R.Passaquieti,D.Passuello,F.Piergiovanni,L.Pinard,R. We have presented the rationale and design method for Poggiani,M.Punturo,P.Puppo,P.Rapagnani,T.Regimbau,A.Remil- active damping of the mirror-suspensions of a gravitational lieux,F.Ricci,L.Ricciardi,A.Rocchi,L.Rolland,R.Romano,P.Ruggi wave detector and shown that a combination of approaches G.Russo,S.Solimeno,A.Spallicci,M.Tarallo,R.Terenzi,A.Toncelli,M. Tonelli,E.Tournefier,F.Travasso,C.Tremola,G.Vajente,J.F.J.van den can be expected to provide the required performance.A par- Brand,S.van der Putten,D.Verkindt,F.Vetrano,A.Vicere,J.-Y.Vinet,H. ticular advantage of our approach is that relatively simple sen- Vocca,and M.Yvert.Class.Quantum Grav.25,114045(2008). sors are adequate to meet even the exceedingly tight noise tol- 4G.M.Harry and the LIGO Scientific Collaboration,Class.Quantum Grav. 27,084006(2010). erances associated with gravitational wave detection 5S.Rowan,J.Hough,and D.R.M.Crooks,Phys.Lett.A 347,25 (2005). 6M.V.Plissi.K.A.Strain,C.I.Torrie,N.A.Robertson,S.Killbourn,S. ACKNOWLEDGMENTS Rowan,S.M.Twyford,H.Ward,K.D.Skeldon,and J.Hough,Rev.Sci. nstrum.6.3055(1998). The authors would like to thank members of the LSC- 7S.Braccini.C.Casciano,F.Cordero.F.Corvace,M.DeSanctis,R.Franco, Virgo collaboration for their interest in this work.We would F.Frasconi,E.Majorana,G.Paparo,R.Passaquieti,P.Rapagnani,F.Ricci, like to thank the NSF in the USA(Award Nos.PHY-05 02641 D.Righetti,A.Solina,and R.Valentini,Meas.Sci.Technol.11,467 and PHY-07 57896).LIGO was constructed by the California (2000). R.Abbott,R.Adhikari,G.Allen,D.Baglino,C.Campbell,D.Coyne,E. Institute of Technology and Massachusetts Institute of Tech- Daw,D.DeBra,J.Faludi.P.Fritschel,A.Ganguli,J.Giaime,M.Ham- nology with funding from the National Science Foundation mond,C.Hardham,G.Harry,W.Hua,L.Jones,J.Kern,B.Lantz,K. and operates under cooperative agreement PHY-0107417.In Lilienkamp,K.Mailand,K.Mason.R.Mittleman,S.Nayfeh.D.Ott- the UK,we are grateful for the financial support provided by away,J.Phinney,W.Rankin,N.Robertson,R.Scheffler,D.H.Shoe- maker,S.Wen,M.Zucker,and L.Zuo,Class.Quantum Grav.21.915 Science and Technology Facilities Council (STFC)and the (2004) University of Glasgow.This paper has LIGO Document No. 9M.V.Plissi,C.I.Torrie,M.Barton,N.A.Robertson,A.Grant,C.A.Cant- LIGO-P1200009.Public internal LIGO documents are found ley,K.A.Strain,P.A.Willems,J.H.Romie,K.D.Skeldon,M.M.Perreur- at https://dcc.ligo.org/cgi-bin/DocDB/DocumentDatabase/. Lloyd,R.A.Jones,and J.Hough,Rev.Sci.Instrum.75,4516(2004). 10M.V.Plissi,C.I.Torrie,M.E.Husman,N.A.Robertson.K.A.Strain,H. Ward,H.Luick,and J.Hough,Rev.Sci.Instrum.71,2539 (2000). 1B.Willke,Class.Quantum Grav.19,1377 (2002). 1N.A.Robertson,G.Cagnoli,D.R.M.Crooks,E.Elliffe,J.E.Faller,P. 2D.Sigg and the LIGO Scientific Collaboration,Class.Quantum Grav.25 Fritschel,S.GoBler,A.Grant,A.Heptonstall,J.Hough,H.Liick,R.Mit- 114041(2008). tleman,M.Perreur-Lloyd,M.V.Plissi,S.Rowan,D.H.Shoemaker,P.H 3F.Acernese,M.Alshourbagy,P.Amico,F.Antonucci,S.Aoudia,P.As- Sneddon.K.A.Strain,C.I.Torrie.H.Ward,and P.Willems,Class.Quan- tone,S.Avino,L.Baggio,G.Ballardin,F.Barone,L.Barsotti,M.Bar- tum Grav..19,4043(2002). suglia,T.S.Bauer,S.Bigotta,S.Birindelli,M.A.Bizouard,C.Boccara 2C.Torrie,"Development of suspensions for the Geo 600 gravitational wave F.Bondu.L.Bosi,S.Braccini,C.Bradaschia.A.Brillet,V.Brisson,D. detector,"Ph.D.dissertation(University of Glasgow,1999). Buskulic,G.Cagnoli,E.Calloni,E.Campagna,F.Carbognani,F.Cavalier, 13M.Barton,Models of the Advanced LIGO Suspensions in MathematicaTM R.Cavalieri,G.Cella,E.Cesarini,E.Chassande-Mottin,A.-C.Clapson,F. Internal Technical Document T020205-02D (LIGO,2006). Cleva.E.Coccia,C.Corda,A.Corsi,F.Cottone,J.-P.Coulon,E.Cuoco, 4B.Shapiro,Fitting the Quad Noise Prototype Model to Measured Data, S.D'Antonio,A.Dari,V.Dattilo,M.Davier,R.De Rosa,M.DelPrete,L Internal Technical Document T1000458 (LIGO,2010). Di Fiore,A.Di Lieto,M.D.P.Emilio,A.Di Virgilio,M.Evans,V.Fafone 5L.Ruet,"Active control and sensor noise filtering duality application to I.Ferrante,F.Fidecaro,I.Fiori,R.Flaminio,J.-D.Fournier,S.Frasca,F. advanced LIGO suspensions,"Ph.D.dissertation (Institut National des Frasconi,L.Gammaitoni,F.Garufi,E.Genin,A.Gennai,A.Giazotto,L. Sciences Appliques de Lyon (INSA Lyon).2007). Giordano,V.Granata,C.Greverie,D.Grosjean,G.Guidi,S.Hamdani, 16B.Shapiro,"Modal control with state estimation for advanced LIGO S.Hebri,H.Heitmann,P.Hello,D.Huet,S.Kreckelbergh,P.La Penna, quadruple suspensions,"Master's thesis (Massachusetts Institute of M.Laval,N.Leroy,N.Letendre,B.Lopez,M.Lorenzini,V.Loriette,G. Technology,2007). 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

044501-9 K. A. Strain and B. N. Shapiro Rev. Sci. Instrum. 83, 044501 (2012) for all except the x direction, where the model is not over￾sensitive to the relevant parameters of the suspension and the extra performance provided by the modal method is most beneficial. We have presented the rationale and design method for active damping of the mirror-suspensions of a gravitational wave detector and shown that a combination of approaches can be expected to provide the required performance. A par￾ticular advantage of our approach is that relatively simple sen￾sors are adequate to meet even the exceedingly tight noise tol￾erances associated with gravitational wave detection. ACKNOWLEDGMENTS The authors would like to thank members of the LSC￾Virgo collaboration for their interest in this work. We would like to thank the NSF in the USA (Award Nos. PHY-05 02641 and PHY-07 57896). LIGO was constructed by the California Institute of Technology and Massachusetts Institute of Tech￾nology with funding from the National Science Foundation and operates under cooperative agreement PHY-0107417. In the UK, we are grateful for the financial support provided by Science and Technology Facilities Council (STFC) and the University of Glasgow. This paper has LIGO Document No. LIGO-P1200009. Public internal LIGO documents are found at https://dcc.ligo.org/cgi-bin/DocDB/DocumentDatabase/. 1B. Willke, Class. Quantum Grav. 19, 1377 (2002). 2D. Sigg and the LIGO Scientific Collaboration, Class. Quantum Grav. 25, 114041 (2008). 3F. Acernese, M. Alshourbagy, P. Amico, F. Antonucci, S. Aoudia, P. As￾tone, S. Avino, L. Baggio, G. Ballardin, F. Barone, L. Barsotti, M. Bar￾suglia, T. S. Bauer, S. Bigotta, S. Birindelli, M. A. Bizouard, C. Boccara, F. Bondu, L. Bosi, S. Braccini, C. Bradaschia, A. Brillet, V. Brisson, D. Buskulic, G. Cagnoli, E. Calloni, E. Campagna, F. Carbognani, F. 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