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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
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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
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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.
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                                         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.
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                                                                        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
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                                          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.
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                              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
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                                              Technology Working Group
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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.
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    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
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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
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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.
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Fig. 10.8 - AZ axis - Closed Position Loop Transfer Function Bode Diagram
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                                           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
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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
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                                               Date: 14.11.2000
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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
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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
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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   4Iv 
                                                    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
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                                               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.
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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
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Fig. 10.14 - PSD of the wind speed Sv(f) - R: 1=1, G: 2=0.98, B: 3=0.63
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Fig. 10.15 - PSD of the wind speed: comparison - R: Sv(f), G: Sw(f) (3=0.63)
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Fig. 10.16 - PSD of the wind torque S(f) - R: 1=1, G: 2=0.98, B: 3=0.63
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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
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Fig. 10.18 - Tracking error vs bandwidth of the control loop - R: 1=1, G: 2=0.98, B: 3=0.63
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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 cos2f 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
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                       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).
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Fig. 10.20 - Altitude tracking error (windspeed=12m/s, 2=0.63)
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                   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.
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                               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:
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S v  f   4Iv 
                   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 )
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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 cos2f 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
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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)
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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
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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
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      Fig. 10.26 - Motor representation
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Fig. 10.27 - Tracking error due to quantization noise
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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.
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                                     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.
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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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