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VST Project Technology Working Group Osservatorio Astronomico di Capodimonte Napoli Date: 14.11.2000 Page: 1 VST VLT SURVEY TELESCOPE CONTROL SYSTEM PERFORMANCE Doc. No.: VST-TRE-OAC-20000-1009 Issue: 1.5 Pages: 43 Date: 14.11.2000 File Name: VST-TRE-OAC-20000-1009-1.5.doc Activity Names System analysis, functionalities & Document design D. Mancini, P. Schipani Contents by D. Mancini, P. Schipani Document Preparation P. Schipani Document Supervision & Check D. Mancini Task Management D. Mancini Documentation and Quality Control Manager V. Fiume Garelli Task Responsibility D. Mancini Signature mancini@na.astro.it G. Sedmak sedmak@ts.astro.it Questions and Remarks on contents P. Schipani - schipani@na.astro.it Phone: ++39 081 5575111 Fax: ++39 081 456710 VST Project Technology Working Group Osservatorio Astronomico di Capodimonte Napoli Date: 14.11.2000 Page: 2 CHANGE RECORD Issue Date Section/Paragraph Affected Reason/Initiation/Documents/Remarks 1.0 30/10/99 First release 1.1 11/02/00 Modified after FDR rixs 1.2 26/05/00 Par. 10.3 and 10.7 Many changes, most of them in paragraphs 10.3 and 10.7 1.3 31/07/00 Par. 10.3 and 10.7 Some integration to paragraph 10.3-10.7 after ESO comments 1.4 11/09/00 Figures 10.1, 10.3, 10.5, 10.7, Modified most of the figures and tables according 10.9,10.10, 10.15, 10.16, 10.17, to the different inertia values expressed by the 10.23 and tables 10.1, 10.2, mechanical design and to the consequent new 10.3, 10.4, 10.5, 10.6, 10.7, 10.8 simulation results. Only few modifications in the text (results reported in the tables). 1.5 14/11/00 Modified most of the figures and tables according to the final inertia values expressed by the mechanical design and to the consequent new simulation results. Only few modifications in the text (results reported in the tables). VST Project Technology Working Group Osservatorio Astronomico di Capodimonte Napoli Date: 14.11.2000 Page: 3 TABLE OF CONTENTS 10 TELESCOPE CONTROL PERFORMANCE ...............................................................................................5 10.1 INTRODUCTION ...........................................................................................................................................5 10.1.1 Purpose .............................................................................................................................................5 10.1.2 Reference Documents ......................................................................................................................5 10.1.3 Abbreviations and Acronyms ............................................................................................................6 10.2 OVERVIEW .................................................................................................................................................7 10.3 AXES MODELING ........................................................................................................................................7 10.3.1 Azimuth axis modeling ......................................................................................................................7 10.3.2 Elevation axis modeling ................................................................................................................. 10 10.4 CONTROLLER DESIGN .............................................................................................................................. 12 10.4.1 Required Control Performance ...................................................................................................... 13 10.5 AZIMUTH AXIS CONTROLLER .................................................................................................................... 14 10.5.1 Speed Loop .................................................................................................................................... 14 10.5.2 Position Loop ................................................................................................................................. 15 10.6 ELEVATION AXIS CONTROLLER ................................................................................................................. 17 10.6.1 Speed Loop .................................................................................................................................... 17 10.6.2 Position Loop ................................................................................................................................. 18 10.7 DISTURBANCE REJECTION ........................................................................................................................ 19 10.7.1 Altitude wind disturbance rejection ................................................................................................ 19 10.7.2 Von Karman Spectrum ................................................................................................................... 21 10.7.3 Altitude time domain analysis ........................................................................................................ 29 10.7.4 Azimuth wind disturbance rejection ............................................................................................... 33 10.7.5 Non-linearities ................................................................................................................................ 37 10.7.6 Preload Control .............................................................................................................................. 41 10.7.7 Encoder error ................................................................................................................................. 42 10.8 SPECIFICATIONS AND GOALS .................................................................................................................... 43 10.9 CONCLUSIONS ......................................................................................................................................... 43 VST Project Technology Working Group Osservatorio Astronomico di Capodimonte Napoli Date: 14.11.2000 Page: 4 TABLE & FIGURES INDEX Tab. 10.1 - Inertia of azimuth far parts related to the elevation angle ....................................................................8 Tab. 10.2 - AZ axis: structural parameters .............................................................................................................9 Tab. 10.3 - EL axis: structural parameters .......................................................................................................... 11 Tab. 10.4 - AZ axis - Stability Margins and -3db Bandwidth ............................................................................... 17 Tab. 10.5 - EL axis - Stability Margins and -3db Bandwidth................................................................................ 18 Tab. 10.6 - Axis rotation due to the wind for different control loop bandwidth .................................................... 23 Tab. 10.7 - RMS tracking errors due to wind disturbance ................................................................................... 32 Tab. 10.8 - RMS azimuth tracking errors due to wind disturbance ..................................................................... 35 Tab. 10.9 - Preload Control Parameters .............................................................................................................. 41 Fig. 10.1 - Far inertia FEA values ...........................................................................................................................8 Fig. 10.2 - Azimuth axis modeling ..........................................................................................................................9 Fig. 10.3 - AZ axis - Open Loop Transfer Function Bode Diagram ..................................................................... 10 Fig. 10.4 - Elevation axis modeling ...................................................................................................................... 11 Fig. 10.5 - EL axis - Open loop transfer function of the mechanical model ........................................................ 12 Fig. 10.6 - Simplified control scheme .................................................................................................................. 13 Fig. 10.7 - AZ axis - Closed Speed Loop Transfer Function Bode Diagram ....................................................... 14 Fig. 10.8 - AZ axis - Closed Position Loop Transfer Function Bode Diagram ..................................................... 16 Fig. 10.9 - EL axis - Closed Speed Loop Transfer Function Bode Diagram ....................................................... 17 Fig. 10.10 - EL axis - Closed Position Loop Transfer Function Bode Diagram ................................................... 18 Fig. 10.11 - EL axis – Position/Disturbance Transfer Function Bode Diagram – Open Loop ............................. 19 Fig. 10.12 - EL axis – Position/Disturbance with and without velocity feedback ................................................. 20 Fig. 10.13 - EL axis – Position/Disturbance with position feedback .................................................................... 21 Fig. 10.14 - PSD of the wind speed Sv(f) - R: 1=1, G: 2=0.98, B: 3=0.63 ...................................................... 24 Fig. 10.15 - PSD of the wind speed: comparison - R: Sv(f), G: Sw(f) (3=0.63) .................................................. 25 Fig. 10.16 - PSD of the wind torque S(f) - R: 1=1, G: 2=0.98, B: 3=0.63 ...................................................... 26 Fig. 10.17 - PSD of axis rotation due to the wind disturbance torque S(f) - R: 1=1, G: 2=0.98, B: 3=0.63 ... 27 Fig. 10.18 - Tracking error vs bandwidth of the control loop - R: 1=1, G: 2=0.98, B: 3=0.63 ......................... 28 Fig. 10.19 - Altitude tracking error (windspeed=6m/s, 2=0.63) .......................................................................... 30 Fig. 10.20 - Altitude tracking error (windspeed=12m/s, 2=0.63) ........................................................................ 31 Fig. 10.21 - Wind Speed Reduction Factors - R: 1=0.98, G: 2=0.63 ............................................................... 32 Fig. 10.22 - Time % vs Wind Speed in Paranal .................................................................................................. 33 Fig. 10.23 - Azimuth tracking error (ALT=45°, windspeed=12m/s, 2=0.63)....................................................... 36 Fig. 10.24 - Axis Control Scheme ........................................................................................................................ 37 Fig. 10.25 - Axis Model ........................................................................................................................................ 38 Fig. 10.26 - Motor representation ........................................................................................................................ 39 Fig. 10.27 - Tracking error due to quantization noise .......................................................................................... 40 Fig. 10.28 - Preload Control Scheme .................................................................................................................. 42 VST Project Technology Working Group Osservatorio Astronomico di Capodimonte Napoli Date: 14.11.2000 Page: 5 10 TELESCOPE CONTROL PERFORMANCE 10.1 INTRODUCTION 10.1.1 Purpose This section describes the controller design and the electromechanical dynamical model for the VST telescope. The correct controller design is based on the reliability of the model. The model itself is based on the main physical characteristics of the telescope structure, and takes into account the results of the structural analysis. 10.1.2 Reference Documents [1] VST FDR - Finite Element Analysis – Perrotta F. [2] Ravensbergen, M.: 1994, Main axes servo systems of the VLT, SPIE proc. 2199, 997 [3] Fax ESO 11.2.99 from Marco Quattri, Data to calculate the wind speed used in analyzing the tracking stability of VST. [4] VST FDR – Wind Effect on the Telescope – Mancini, D., Mavar, P. [5] An Introduction to control systems - Warwick - World Scientific Editor [6] Modern Control System Theory and Design - Shinners - Whiley & Sons Editor [7] Encoder Constructors. Quick reference for RMS error calculation. HEIDENHEIN, web site: http://www.heidenhain.com/phaise2/posma.html [8] McGonegal, R.: 1994, Control philosophy of the Gemini 8-m telescopes, SPIE proc. 2199, 783 [9] Burns, M.: 1994, Tracking performance simulation for the Gemini 8-m telescopes, SPIE proc. 2199, 805 [10] Dierickx, P.: 1994, Error budget and expected performance of the VLT unit telescopes, SPIE proc. 2199, 950 [11] Cullum, M., J., Enard, D., Ravensbergen, M.: 1994, Control of image position errors with the VLT, SPIE proc. 2199, 950 [12] Quattri, M, Dimichino, F., Marchiori, G., Piccinini, E.: 1994, VLT 8m unit telescope main structure: design solutions and performance calculation, SPIE proc. 2199, 986 [13] Gilli, B.: 1995, VLT tracking and guiding software, SPIE proc. 2479, 314 [14] Chiozzi, G., Wirenstrand, K., Ravensbergen, M., Gilli, B.: 1995, Integration tests of the VLT telescope control system, SPIE proc. 3112, 141 [15] Wallander, A., Spyromilio, J.: 1997, NTT project: a field test of the VLT software and hardware, SPIE proc. 3112, 9 [16] Ellington, S.: 1995, Disturbance rejection of the WIYN telescope position control servosystem, SPIE proc. [17] Tomelleri, R., Zamperini, E.: 1990, Studio definizione e prestazioni del sistema di movimentazione del TNG [18] Zago, L.: http://web1.hq.eso.org/gen-fac/pubs/astclim/papers/lz-thesis [19] Sarazin, M.: http://www.eso.org/gen-fac/pubs/astclim/paranal/wind/ "Wind at Paranal: Nighttime Velocity Histogram (1985-1999)" [20] Mancini, D.: Main Axes Drive Preload System, 15.04.2000 [21] Andersen, T.: The Servo System of the EISCAT Svalbard Antenna, SPIE proc. 2479, 301 VST Project Technology Working Group Osservatorio Astronomico di Capodimonte Napoli Date: 14.11.2000 Page: 6 10.1.3 Abbreviations and Acronyms A/D Analog-to-Digital conversion AZ Azimuth Axis D/A Digital-to-Analog conversion EL Elevation Axis ESO European Southern Observatory FEA Finite Element Analysis HBS Hydrostatic Bearing System OAC Osservatorio Astronomico di Capodimonte PI Proportional-Integrative Controller RMS Root Mean Square TWG Technology Working Group VLT Very Large Telescope VST VLT Survey Telescope VST Project Technology Working Group Osservatorio Astronomico di Capodimonte Napoli Date: 14.11.2000 Page: 7 10.2 OVERVIEW This document is focused on the study of the electro-mechanical system, the synthesis of the speed and position loop controllers, and the rejection of the controller to the external disturbances. The most influencing factor on the performance of a telescope is the wind. Therefore this document deals very much with the study of the wind torque disturbance effect, both in the time domain and in the frequency domain. Some other error sources originated from non linear effect, such as tacho and torque quantization, have been estimated by a SImulink dynamical model of the telescope. The wind real effect is highly dependent on the direction of the wind with respect to the telescope, the setting of the windscreens, the bandwidth of the control loop. Therefore the error contribution is expected to have a wide range depending on the operating conditions. 10.3 AXES MODELING The telescope coupling dynamic is so slow that it is possible to study independently the two axis behaviors. The coupling of the two axes is usually negligible in a telescope, especially in the most important operating condition, i.e. the tracking phase, in which the axes trajectory is regular and not interested by strong accelerations (the only exception being the zenith approaching). Of course if azimuth axis control at high speeds (which azimuth axis can assume in an Alt-Az mount) would be very poor there should be a negative effect on altitude too, but after a proper tuning of the speed and position controllers this kind of coupling is going to disappear. Both axes structures are modeled by a number of inertias joined together by means of stiffnesses and structural dampings. The axis gear is represented by the motor and teeth contact stiffness and by the damping. The inertia of the motor itself is taken into account, properly scaled by the transmission ratio. The structural data are derived from a FEA of the mechanical structure of the telescope [1]. 10.3.1 Azimuth axis modeling The model structure of azimuth axis changes with the elevation axis angle. A set of F.E.A. data, azimuth inertia vs the elevation angle, has been used and an interpolation law has been derived (Fig. 10.1). J ( ) J 90 sin 2 ( ) J 0 cos 2 ( ) where J90, J0 are respectively the inertia with altitude at 90 and 0 degree, and is the elevation. VST Project Technology Working Group Osservatorio Astronomico di Capodimonte Napoli Date: 14.11.2000 Page: 8 Fig. 10.1 - Far inertia FEA values The far inertia parameters shown in the Fig. 10.1 and schematized by circles are quoted in Tab. 10.1. Parameter Value ALT angle 2 Near inertia (kg·m ) 55286 all 19343 90° 2 Far inertia (kg·m ) 21131 45° 22920 0° Tab. 10.1 - Inertia of azimuth far parts related to the elevation angle The azimuth axis simulation reported in this document has been done in the operating conditions with the EL axis at 45°, i.e. in an intermediate condition. VST Project Technology Working Group Osservatorio Astronomico di Capodimonte Napoli Date: 14.11.2000 Page: 9 far inertia q2 near inertia reduction: 17 q1 gear q1' m otors inertia K12 K23 motors Near Far inertia i nertia inert i a q1 q2 q3 F12 F23 F1 F2 Fig. 10.2 - Azimuth axis modeling Parameter Value Symbol 2 Near inertia [kg·m ] 55286 J2 2 Far inertia (formula) [kg·m ] 21131 J3 2 Motors inertia [kg·m ] 4 * 1.525 J1 HBS viscous friction [Nm/(rad/s)] 763.3 F2 Motors viscous friction [Nm/(rad/s)] 4*22.4689 F1 Transmission stiffness [Nm/rad] 3.2e8 K12 Structural stiffness [Nm/rad] 1.378e9 K23 Transmission damping factor [Nm/(rad/s)] 488680 F12 Structural damping factor [Nm/(rad/s)] 516280 F23 Transmission ratio 16.81 R Tab. 10.2 - AZ axis: structural parameters The dynamic of the mechanical system can be described by second order differential equations in matrix form as: .. . J F K T where J, F, K, T are the inertia, viscous damping, stiffness and torque matrix respectively, and is the angular position vector (degrees of freedom). It is convenient to translate this representation into the space state: . x Ax Bu y Cx Du VST Project Technology Working Group Osservatorio Astronomico di Capodimonte Napoli Date: 14.11.2000 Page: 10 st The space state model is represented by 1 order differential equations. The open loop response of the AZ system, representing the relationship between the axis speed and the applied driver voltage signal is shown in Fig. 10.3. Fig. 10.3 - AZ axis - Open Loop Transfer Function Bode Diagram The notch in the bode gain plot represents the Locked Rotor eigenfrequency (10 Hz), while the peak is the Free -1 Rotor eigenfrequency (61 Hz). They can be obtained from the eigenvalues of the matrix J K, in the two “locked” and “free” rotor cases. 10.3.2 Elevation axis modeling As for the azimuth modeling, the model's parameters have been obtained from a Finite Element Analysis of the mechanical structure. In order to have a more detailed model a higher number of degrees of freedom has been used with respect to the PDR analysis. A six inertias model has been considered, see Tab. 10.3. VST Project Technology Working Group Osservatorio Astronomico di Capodimonte Napoli Date: 14.11.2000 Page: 11 K23 K12 K2ab F23 K3ab K3bc J1 J2a J2b J3a J3b J3c F12 F2ab F3ab F3bc F1 F2 Fig. 10.4 - Elevation axis modeling Parameter Value Symbol 2 Center Piece inertia [kg·m ] 24000 J2a 2 M1 inertia [kg·m ] 2931 J2b 2 Top Ring Inertia [kg·m ] 10452 J3a 2 M2 box Inertia [kg·m ] 8737 J3b 2 M2 Inertia [kg·m ] 14350 J3c 2 Motors inertia [kg·m ] 4 * 1.525 J1 Motors viscous friction [Nm/(rad/s)] 4*22.4689 F1 Viscous friction [Nm/(rad/s)] 1145.9 F2 Transmission damping [Nm/(rad/s)] 361520 F12 M1 pad damping [Nm/(rad/s)] 1.3846e6 F2ab Structural damping [Nm/(rad/s)] 1.0582e7 F23 Top Ring - M2 box damping [Nm/(rad/s)] 2.5177e5 F3ab M2 box - Mirror damping [Nm/(rad/s)] 1.8588e5 F3bc Transmission stiffness [Nm/rad] 3.2e8 K12 M1 pad stiffness [Nm/rad] 1e11 K2ab Structural stiffness [Nm/rad] 4.886e11 K23 Top Ring - M2 box stiffness [Nm/rad] 4e8 K3ab M2 box - Mirror stiffness [Nm/rad] 3.5e8 K3bc Transmission ratio 16.81 R Tab. 10.3 - EL axis: structural parameters VST Project Technology Working Group Osservatorio Astronomico di Capodimonte Napoli Date: 14.11.2000 Page: 12 The description of the EL mechanical structure model equations is similar to the AZ case. Fig. 10.5 shows the open loop response of the EL axis electromechanical model. The first notch in the bode gain plot represents the Locked Rotor eigenfrequency (~10 Hz). The effect of the higher number of degrees of freedom is visible in the new notch and peak frequencies, at about 20 Hz. As expected the locked rotor frequency is still the lowest and most interesting for control. Fig. 10.5 - EL axis - Open loop transfer function of the mechanical model 10.4 CONTROLLER DESIGN The design of the controller is based on the specifications stated in subparagraph 10.4.1. Furthermore, the controller, joined with electromechanical model, allows the study of linear and non-linear effects (only for stick slip phenomenon). The control scheme (Fig. 10.6) is obtained with two standard PI controllers, respectively for the speed loop and the position loop. VST Project Technology Working Group Osservatorio Astronomico di Capodimonte Napoli Date: 14.11.2000 Page: 13 Ac c eleration ref. Feedforward T orque dis turbanc es Speed ref. ac tions (wind, non linear fric tion,...) T orque Pos ition Pos ition ref. PI position PI s peed Driver and T eles c ope c ontroller c ontroller motor Axis Model Speed - - Speed limits T orque limits T ac hometers s ys tem T ac hometers error Enc oder s ys tem Enc oders error Fig. 10.6 - Simplified control scheme The controller parameters have to be tuned to guarantee the two most important requirements: an extremely low tracking error a good disturbance rejection Of course the stability of the system must be ensured too; also, the tracking quality depends on the speed and (above all) position transducers accuracy. The main external disturbance to consider in this analysis is the wind shake. The wind mainly affects the altitude performance, because the altitude axis is subjected to a greater wind torque. The azimuth rotation is more protected by the dome. The scheme reported in Fig. 10.6 includes some saturation blocks that avoid any violation of the speed and torque limits. 10.4.1 Required Control Performance The servo system required performance have been fixed taking into account the scientific targets, in terms of tracking accuracy, service efficiency, and of pointing capability. The main target to be considered is the required absolute RMS tracking error, which mainly guarantees the image quality during observation. The pointing operation efficiency suggests to fix the maximum axes speed at 1.5°/s and the maximum axes 2 acceleration at 0.5°/s . Higher acceleration can be used in tracking phase to allow for fast corrections. These values can guarantee a short pointing time (probably limited on dome rotation performance) and allow to reach good tracking performance. Furthermore, the stability robustness of the closed loop must be guaranteed with proper gain and phase margins, to reject structural parameter changes of the controlled system and/or environment modifications. Particular attention will be paid in the synthesis of the controllers, in order to improve the guide and to assure a proper rejection against the typical disturbances (wind effect, torque quantization, position, speed sensors noise, stick slip). VST Project Technology Working Group Osservatorio Astronomico di Capodimonte Napoli Date: 14.11.2000 Page: 14 10.5 AZIMUTH AXIS CONTROLLER 10.5.1 Speed Loop In this simulation analysis the speed loop compensator is based on a simple PI controller. The complexity of the nd real telescope controller could be increased by adding a series of Notch 2 order filters in order to attenuate eventual resonant spectral components. The speed loop -3db bandwidth obtained with this simulation choice is 8 Hz. The bode diagram is reported in fig. Fig. 10.7. Fig. 10.7 - AZ axis - Closed Speed Loop Transfer Function Bode Diagram VST Project Technology Working Group Osservatorio Astronomico di Capodimonte Napoli Date: 14.11.2000 Page: 15 10.5.2 Position Loop The position controller chosen for this simulation is a standard PI controller. It ideally guarantees zero error to a ramp input after the transient phase. The real controller parameters will not be constant; pointing and tracking phases need different values for the proportional and integral constants. The problem can be circumvented or with a rough switch between different controller structure for small, medium and large errors, or better with a variable structure controller in which the parameters Kp(e), Ki(e) depend on the instantaneous position error value. A detailed analysis of the dependence of the position controller on the tracking error is beyond the scope of the present simulation, which is limited to the "small" error case, i.e. to the tracking phase control. The azimuth axis in an Alt-Az telescope can assume very high speeds when altitude axis is close to zenith, especially when crossing the meridian. On the contrary when altitude angle is low azimuth moves very slow. Therefore the position control is much more difficult for this axis, since the dynamic speed range is wide. The same thing does not apply to altitude axis, whose speed range is upper bounded to a low value. Usually when the speed is higher the axis error is higher too, but fortunately the influence of azimuth error on the sky is lessened by the cos(alt) factor which becomes dominant when altitude angle approaches 90°. After the tuning of the position control parameters the closed position loop transfer function has a -3db bandwidth of 2.3 Hz. VST Project Technology Working Group Osservatorio Astronomico di Capodimonte Napoli Date: 14.11.2000 Page: 16 Fig. 10.8 - AZ axis - Closed Position Loop Transfer Function Bode Diagram VST Project Technology Working Group Osservatorio Astronomico di Capodimonte Napoli Date: 14.11.2000 Page: 17 Parameter Values Phase margin (deg) 166 Gain margin (dB) 3 Bandwidth -3dB (Hz) 2.3 Tab. 10.4 - AZ axis - Stability Margins and -3db Bandwidth 10.6 ELEVATION AXIS CONTROLLER 10.6.1 Speed Loop As for the azimuth axis, in this simulation analysis the speed loop compensator is based on a simple PI controller. The speed loop -3db bandwidth obtained with the simulation choice is about 7 Hz. The bode diagram is reported in fig. Fig. 10.9. Fig. 10.9 - EL axis - Closed Speed Loop Transfer Function Bode Diagram VST Project Technology Working Group Osservatorio Astronomico di Capodimonte Napoli Date: 14.11.2000 Page: 18 10.6.2 Position Loop As for azimuth, the position controller chosen for this simulation is a standard PI controller. The same considerations on a different choice for pointing and tracking controllers of 10.5.2 apply also to the altitude axis case. The altitude speed range is upper bounded to a value depending on the latitude of the site. Therefore there are not the same problems of azimuth due to a wide operating speed range, and it is easier to obtain good performance in undisturbed conditions (absence of wind). Nevertheless the altitude axis is much more sensitive to the wind shake external disturbance than azimuth. After the tuning of the position control parameters the closed position loop transfer function has a -3db bandwidth of about 3.5 Hz. The bode diagram of the transfer function is reported in fig. Fig. 10.10. Parameter Values Phase margin (deg) 160 Gain margin (dB) 8 Bandwidth –3dB (Hz) 3.5 Tab. 10.5 - EL axis - Stability Margins and -3db Bandwidth Fig. 10.10 - EL axis - Closed Position Loop Transfer Function Bode Diagram VST Project Technology Working Group Osservatorio Astronomico di Capodimonte Napoli Date: 14.11.2000 Page: 19 10.7 DISTURBANCE REJECTION The main external disturbance for a telescope is certainly the wind. Therefore the performance of a telescope position control servo system depends on its ability to minimize changes in position due to the wind. In the following an analysis of the effect of the wind disturbance on the axes behavior is detailed. The altitude axis is certainly the most influenced by the wind disturbance and so most of the analysis will be focused on it. 10.7.1 Altitude wind disturbance rejection The wind shake disturbance is modeled as a torque input in the space state model of the electro-mechanical system. Therefore it is possible to extract from the model the transfer function: ( s) D ( s) ( s) Fig. 10.11 - EL axis – Position/Disturbance Transfer Function Bode Diagram – Open Loop VST Project Technology Working Group Osservatorio Astronomico di Capodimonte Napoli Date: 14.11.2000 Page: 20 i.e. the ratio of axis position angle to applied disturbance torque. To evaluate the disturbance rejection goodness, it is necessary to examinate D(s) magnitude, which must be small. The open loop D(s) transfer function (without speed and position feedbacks) presents in our model an antiresonance at 66 Hz and a resonance at 70 Hz. They are not so relevant since the wind disturbance spectral components are very low at so high frequencies. Closing the speed loop there is a clear advantage at low frequencies in disturbance rejection. The new situation compared to the open loop one is in fig. Fig. 10.12. At last, closing also the position loop, the disturbance rejection at low frequencies is more and more increased (fig. Fig. 10.13). From f=6Hz on the disturbance rejection becomes comparable or even slightly lower, in a limited spectral range. This could be avoided by complicating a little the design of the controller, but it is not necessary for the purposes of the present analysis: in fact the wind disturbance is mostly present at frequencies up to 1 Hz and below this value the disturbance is clearly attenuated by the speed and position loops used for the VST simulation. Fig. 10.12 - EL axis – Position/Disturbance with and without velocity feedback VST Project Technology Working Group Osservatorio Astronomico di Capodimonte Napoli Date: 14.11.2000 Page: 21 10.7.2 Von Karman Spectrum Another approach to evaluate the effect of the wind on the telescope performance ([2], [3]) is based on the description of the power spectral density of the wind by the Von Karman spectrum: S v f 4Iv 2 L 1 5 v 2 6 1 70 .78 f L v where v is the mean wind speed, I is the turbulence intensity, f the frequency, L the outer scale of turbulence. The Von Karman spectrum must be multiplied by the square of the aerodynamic correction factor (f): f 1 4 A 3 1 2 f v Fig. 10.13 - EL axis – Position/Disturbance with position feedback VST Project Technology Working Group Osservatorio Astronomico di Capodimonte Napoli Date: 14.11.2000 Page: 22 S w f S v f 2 f The effect of the fluid dynamic attenuation factor is to reduce the disturbance (fig. 10.20). The PSD of the torque is given by: 2 S f 4 S w f v The PSD of the altitude axis rotation due to the disturbance torque can be obtained by multiplying this spectrum with the square of the disturbance transfer function D(f): ( f ) D( f ) ( f ) S f D f S f 2 The following choice of parameters has been used for the simulation: 3 p = 1 kg/m (air density) 2 A=2.3 m Cd = 1.2 (drag coefficient) v=18 m/s Three simulations have been done with the following wind speed reduction factor inside the enclosure: 1=1 (no reduction - open air) 2=0.98 3=0.63 The turbulence intensity used in the three cases are: I1=0.15 I2=0.15 I3=0.12 The length scale of turbulence is much higher in open air, i.e. in the first case: L1=79 m L2=3.2 m L3=3.2m These data have been suggested by ESO on the base of previous experiences with the VLT. VST Project Technology Working Group Osservatorio Astronomico di Capodimonte Napoli Date: 14.11.2000 Page: 23 The results are shown in figures from Fig. 10.14 to Fig. 10.18. The amplitude of the disturbance as calculated with this approach depends very much on the applicability of Von Karman spectrum and on data mutuated from VLT wind tunnel tests and related experience, since for VST no wind tunnel data are available neither for the telescope nor for the dome. The RMS displacement error due to the torque disturbance induced by the wind is represented as a function of the bandwidth of the control loop in Fig. 10.18. In Tab. 10.6 the altitude rotation due to the wind disturbance is tabulated for different bandwidth of the control loop, in the two practical cases (the open air condition being only for comparison). Higher is the bandwidth, better is the disturbance rejection. With a bandwidth of 3 Hz the error roughly estimated by these frequency domain considerations is 0.14 arcsec RMS. A better error estimation is obtained from the time domain analysis reported in the following. Bandwidth [Hz] RMS error [arcsec] RMS Error [arcsec] Worst case (2=0.98) Worst Most Favorable Case (3=0.63) 1 0.23 0.06 2 0.18 0.04 3 0.14 0.03 4 0.12 0.02 5 0.11 0.02 6 0.10 0.02 7 0.09 0.02 8 0.08 0.02 9 0.08 0.02 Tab. 10.6 - Axis rotation due to the wind for different control loop bandwidth VST Project Technology Working Group Osservatorio Astronomico di Capodimonte Napoli Date: 14.11.2000 Page: 24 Fig. 10.14 - PSD of the wind speed Sv(f) - R: 1=1, G: 2=0.98, B: 3=0.63 VST Project Technology Working Group Osservatorio Astronomico di Capodimonte Napoli Date: 14.11.2000 Page: 25 Fig. 10.15 - PSD of the wind speed: comparison - R: Sv(f), G: Sw(f) (3=0.63) VST Project Technology Working Group Osservatorio Astronomico di Capodimonte Napoli Date: 14.11.2000 Page: 26 Fig. 10.16 - PSD of the wind torque S(f) - R: 1=1, G: 2=0.98, B: 3=0.63 VST Project Technology Working Group Osservatorio Astronomico di Capodimonte Napoli Date: 14.11.2000 Page: 27 Fig. 10.17 - PSD of axis rotation due to the wind disturbance torque S(f) - R: 1=1, G: 2=0.98, B: 3=0.63 VST Project Technology Working Group Osservatorio Astronomico di Capodimonte Napoli Date: 14.11.2000 Page: 28 Fig. 10.18 - Tracking error vs bandwidth of the control loop - R: 1=1, G: 2=0.98, B: 3=0.63 VST Project Technology Working Group Osservatorio Astronomico di Capodimonte Napoli Date: 14.11.2000 Page: 29 10.7.3 Altitude time domain analysis The disturbance effect has been studied simulating a slow tracking at a speed of 10 arcsec/sec. The wind conditions chosen for the simulation have been suggested by ESO, which did for VLT similar studies based on wind tunnel tests. For VST no wind tunnel test has been done or is foreseen. Therefore the reliability of the analysis is based on the extension of some results obtained experimentally for VLT to the VST case. The maximum operational wind speed is v=18 m/s, which represents a very stressing condition for the servo loop. The turbulence intensity percentage value used in the simulation is in the range 12-15%. Inside the enclosure there is a reduction of the wind speed by a factor in the range 0.63-0.98. The top ring with the spiders represents the largest aerodynamic drag contribution to the torque. SImulations have been done in the time domain simulating a tracking at a speed of 10 arcsec/sec (almost the maximum velocity), generating torque disturbance time series as a sum of sine waves with amplitudes determined from the power spectral density and with random phases: N t 2 S f k f cos2f k t k 1 where f = 0.01 Hz is the frequency resolution, N = 10000 the number of frequency samples, k random phase angles. The parameters for this case are: 3 p = 1 kg/m (air density) 2 A = 2.3 m (exposed area, sum of areas n.1-2-3-4 – see [4]) Cd = 1.2 (drag coefficient) v =0 - 18 m/s Two simulations have been done with the following wind speed reduction factor inside the enclosure: 1=0.98 2=0.63 The turbulence intensities used in the two cases are: I1=0.15 I2=0.12 VST Project Technology Working Group Osservatorio Astronomico di Capodimonte Napoli Date: 14.11.2000 Page: 30 Fig. 10.19 - Altitude tracking error (windspeed=6m/s, 2=0.63) Simulations have been done at increasing wind speeds. It should be considered that since observing in the direction of the wind is not a preferred scenario the 2=0.63 case can be considered a more realistic observation condition, taking care to an accurate setting of the enclosure. A better estimation of the wind reduction factor would require wind tunnel analysis and it is impossible at this stage. Some of the results are shown in Fig. 10.19 and Fig. 10.20. The interpolation of the data results is reported in Fig. 10.21. Tab. 10.7 reports also the percentage of time in which the wind in Paranal is above the selected wind speed (approximated data taken from [19], see also Fig. 10.22). VST Project Technology Working Group Osservatorio Astronomico di Capodimonte Napoli Date: 14.11.2000 Page: 31 Fig. 10.20 - Altitude tracking error (windspeed=12m/s, 2=0.63) VST Project Technology Working Group Osservatorio Astronomico di Capodimonte Napoli Date: 14.11.2000 Page: 32 Fig. 10.21 - Wind Speed Reduction Factors - R: 1=0.98, G: 2=0.63 V [m/s] Time % with RMS Tracking RMS Tracking Wind Speed > V Error Error (1=0.98) [arcsec] (2=0.63) [arcsec] 3 77 0.02 0.005 6 50 0.10 0.03 9 24 0.30 0.09 12 11 0.54 0.16 15 5 0.80 0.26 18 2 1.10 0.38 Tab. 10.7 - RMS tracking errors due to wind disturbance According to this simulation data the disturbance effect would increase with wind speed as foreseen; up to 12 m/s the effect could be considered not really performance limiting even in good seeing conditions, while at higher wind speeds in very good seeing conditions a negative effect could be noticed. The problem should be limited to a low percentage of telescope usage time; the wind speed is above 12 m/s in about 11% of the time, above 15 m/s only in the 5% of the time. VST Project Technology Working Group Osservatorio Astronomico di Capodimonte Napoli Date: 14.11.2000 Page: 33 Fig. 10.22 - Time % vs Wind Speed in Paranal 10.7.4 Azimuth wind disturbance rejection The azimuth axis rotation is usually much more protected by the enclosure than the altitude rotation. Most of the wind disturbance effect comes from the open slit in the observation direction, which affects mainly the altitude axis. Usually the influence on azimuth of the wind disturbance is neglected. Nevertheless when a lateral disturbance torque is applied, as in the case of side ventilation doors opened, also azimuth rotation performance can be affected, in a way depending on the amount of torque disturbance applied. For altitude the wind disturbance coming from side ventilation doors is supposed to be negligible in comparison with the one coming from the open slit in the observation direction. Therefore the study on the sideway wind loading effect on the telescope performance is here limited to the azimuth axis. In the case of VST the maximum height of the side ventilation doors is supposed to be 4.5 m. The effect of the wind coming from these doors in terms of torque applied to the telescope axis could be exactly evaluated only after a detailed study of wind tunnel tests. These data are not available for VST; the following analysis is based on conservative assumptions and data mutuated from VLT experience. The approach here used to evaluate the effect of the lateral wind on the azimuth performance is based again on the description of the wind power spectral density by the Von Karman spectrum: VST Project Technology Working Group Osservatorio Astronomico di Capodimonte Napoli Date: 14.11.2000 Page: 34 S v f 4Iv 2 L 1 5 v 2 6 1 70 .78 f L v where v is the mean wind speed, I is the turbulence intensity, f the frequency, L the outer scale of turbulence. The Von Karman spectrum must be multiplied by the square of the aerodynamic correction factor (f): f 1 4 A 3 1 2 f v S w f S v f 2 f The effect of the fluid dynamic attenuation factor is to reduce the disturbance. The static force acting on a surface A is: A C D (v) p 1 2 F 2 The top ring with the spiders represents the largest aerodynamic drag contribution to the torque. The torque is obtained as: FL The following choice of parameters has been used: 3 p = 1 kg/m (air density) 2 A = 2.3 m (surface) Cd = 1.2 (drag coefficient) v =6 -18 m/s (wind speed) L = 3.5 sin(90-ALT) m The PSD of the torque is given by : 2 S f 4 S w f v The PSD of the altitude axis rotation due to the disturbance torque can be obtained by multiplying this spectrum with the square of the disturbance transfer function D(f): ( f ) D( f ) ( f ) VST Project Technology Working Group Osservatorio Astronomico di Capodimonte Napoli Date: 14.11.2000 Page: 35 S f D f S f 2 SImulations have been done in the time domain simulating a tracking at a speed of 40 arcsec/sec, generating torque disturbance time series as a sum of sine waves with amplitudes determined from the power spectral density and with random phases: N t 2 S f k f cos2f k t k 1 where f = 0.01 Hz is the frequency resolution, N = 10000 the number of frequency samples, k random phase angles. The turbulence intensity percentage value used in the simulation is in the range 12-15%. Inside the enclosure there is a supposed reduction of the wind speed by a factor in the range 0.63-0.98. Two sets of simulations have been done with the following wind speed reduction factor inside the enclosure: 1=0.98 2=0.63 (hopefully more realistic) The turbulence intensity used in the two cases are: I1=0.15 I2=0.12 The length scale of turbulence is : L1=3.2 m L2=3.2 m Simulations have been done at three different altitude angles (30, 45, 80 deg), for different wind speeds (6, 12, 18 m/s), and for two different wind speed reduction factors inside the enclosure (0.98 and 0.63). Fig. 10.23 reports one of the simulations time series. The results are reported in Tab. 10.8: V [m/s] Alt [deg] RMS Tracking Error RMS Tracking Error (1=0.98) [arcsec] (2=0.63) [arcsec] 6 30 0.06 0.02 6 45 0.04 0.02 6 80 0.02 0.01 12 30 0.28 0.08 12 45 0.21 0.07 12 80 0.05 0.02 18 30 0.59 0.20 18 45 0.44 0.16 18 80 0.10 0.04 Tab. 10.8 - RMS azimuth tracking errors due to wind disturbance VST Project Technology Working Group Osservatorio Astronomico di Capodimonte Napoli Date: 14.11.2000 Page: 36 According to this simulation the influence of the wind coming from the sideway doors on the performance increases with the wind speed and the zenithal distance. At the highest wind speed (18 m/s) there is a clear degradation, so the side doors should be kept closed. These results confirm that the RMS tracking error due to the wind shake is essentially due to altitude, with a much less effect of azimuth (apart from strong wind conditions with lateral doors open, which is certainly a not preferred operational condition). Fig. 10.23 - Azimuth tracking error (ALT=45°, windspeed=12m/s, 2=0.63) VST Project Technology Working Group Osservatorio Astronomico di Capodimonte Napoli Date: 14.11.2000 Page: 37 10.7.5 Non-linearities The effect of quantization on tacho and encoder feedback and on torque command signals have been studied by a Simulink model of the telescope axes, developed to analyze all the non linear disturbance effects on the telescope tracking. Some quantizer blocks have been added to the scheme (Fig. 10.24) and a position ramp input has been used to evaluate the effect of the quantization at low speeds (10 arcsec/sec in this simulation), where most of the influence is expected. In terms of RMS tracking error the influence of the quantization is evaluated about 0.01-0.02 arcsec (by repeating the same simulation with and without the quantizers, with no other disturbance). The quantization due to the encoder does not play a significant role, as expected since the resolution of the Heidenhain encoders chosen for VST is very high (27 bits). On the contrary the most influencing quantization noise is due to the tacho feedback signals. Fig. 10.24 - Axis Control Scheme VST Project Technology Working Group Osservatorio Astronomico di Capodimonte Napoli Date: 14.11.2000 Page: 38 The meaning of the symbols in Fig. 10.24 is: rad2sec = 206265 tau m1 = external disturbance applied to Motor 1 torque tau m2 = external disturbance applied to Motor 2 torque tau m3 = external disturbance applied to Motor 3 torque tau m4 = external disturbance applied to Motor 4 torque tau2 = external torque disturbance applied to mass no.2 tau3 = external torque disturbance applied to mass no.3 teta 2 = angular position of mass no.2 teta 3 = angular position of mass no.3 teta 2' = rotational speed of mass no.2 teta 3' = rotational speed of mass no.3 Fig. 10.25 - Axis Model VST Project Technology Working Group Osservatorio Astronomico di Capodimonte Napoli Date: 14.11.2000 Page: 39 Fig. 10.26 - Motor representation VST Project Technology Working Group Osservatorio Astronomico di Capodimonte Napoli Date: 14.11.2000 Page: 40 Fig. 10.27 - Tracking error due to quantization noise VST Project Technology Working Group Osservatorio Astronomico di Capodimonte Napoli Date: 14.11.2000 Page: 41 10.7.6 Preload Control Both azimuth and altitude axes are driven by four motors coupled to the telescope through the axis gear and preloaded in pairs in order to compensate for backlash. The preload procedure is described in the following. Cps Static Preload Torque Cp Dynamic Preload Torque kp Preload Variation Constant Ct Total torque requested to the 4 motors Cpo Minimum preload torque dt Sampling Time (0.002 sec) Tp Time constant Kt Preload variation constant [Nm/Amp per motor] icmd Current command (speed loop controller output) icp Preload torque needed per motor i1,i2,i3,i4 Currents for the i-th motor Tab. 10.9 - Preload Control Parameters The meaning of the parameters used in the following is reported in Tab. 10.9 (more details about the choice of the parameters can be found in [20]). The static preload torque is dependent on the instantaneous requested torque according to the following linear relationship: Cps = Cp0 + Kp Ct = Cp0 + Kp 4 Kt icmd The dynamic term Cp must be always >= Cps: if( Cp < Cps ) Cp = Cps else Cp = Cp - (Cp - Cps) / (Tp * dt) This way the dynamic preload is adaptively controlled: it instantaneously increases when a higher global torque is needed, and decreases slowly (depending on Tp) when the high torque is not needed anymore. In terms of motor currents: icp = cp / Kt i1 = i3 = icmd - icp i2 = i4 = icmd + icp The preload control block is applied to the speed controller output (Fig. 10.24). A Simulink implementation of the preload controller is detailed in Fig. 10.28. VST Project Technology Working Group Osservatorio Astronomico di Capodimonte Napoli Date: 14.11.2000 Page: 42 Fig. 10.28 - Preload Control Scheme 10.7.7 Encoder error The IK320.1 Heidenhain encoder used for VST are the state of art of optical encoders. They are currently used in some of the newest technology telescopes (NTT, VLT, TNG). They are provided with a multiple reading heads system and an error compensation system. The low error they introduce into the system cannot be reduced by the control loop. It is expected it affects mainly the pointing phase, but the autoguider is expected to compensate for such pointing errors. The high resolution of the encoder + interpolator cards system (27 bits) allows to reduce the quantization noise to negligible values. VST Project Technology Working Group Osservatorio Astronomico di Capodimonte Napoli Date: 14.11.2000 Page: 43 10.8 SPECIFICATIONS AND GOALS The tracking error specification should be set taking into account the maximum operational wind speed of 18 m/s. This is the most critical condition for the telescope main axes control. Nevertheless since the tracking performance of each telescope are heavily dependent on the wind speed, and taking also into account the wind speed statistics in Paranal [19], stating that the wind speed median is about 6 m/s and only in 5% of the time the wind speed is more than 15 m/s, three different specifications are given for different wind conditions: 1. Tracking performance with autoguiding active shall be better than 0.15 arcsec RMS as measured on the guide probe (x-y axes) up to a wind speed of 6 m/s 2. Tracking performance with autoguiding active shall be better than 0.2 arcsec RMS as measured on the guide probe (x-y axes) for a wind speed 6 < v < 12 m/s 3. Tracking performance with autoguiding active shall be better than 0.4 arcsec RMS as measured on the guide probe (x-y axes) for a wind speed 12 < v < 18 m/s All the specifications are intended for observations not in the direction of the wind and with the enclosure set in the best possible way to contrast the wind disturbance. According to the wind speed statistics in Paranal [4], the first specification should be applicable in about 50% of the time, the second in about 39% of the time, the third in about 11% of the time. The goal is to reach in most of the real observation cases (so up to 12 m/s) a tracking error less than 0.1 arcsec RMS. 10.9 CONCLUSIONS The previous experiences on other telescopes have shown that the most critical error source for the telescope tracking is the wind shake. In strong (18 m/s) wind conditions the simulation shows that the wind torque on the altitude axis can introduce a consistent error amount, but with a proper setting of the windscreens in most of the cases it is possible to reduce the wind torque error contribution to more acceptable values. Anyway from the night time wind speed statistics in Paranal it is shown that a wind speed > 15 m/s is present in only 5% of the time. The capability to reject the disturbance is a function of the control loop bandwidth. The -3db bandwidth for the altitude axis is estimated to be about 3 Hz in this simulation. Lower values would introduce higher errors. The real controller will have to be designed to enlarge as much as possible the bandwidth of the control loop.

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