geminisummary of gemini control simulations tn-c-g0015 page 5 this assumes a seeing layer velocity...

TN-C-G0015

Summary of Gemini Control Simulations

Mike Burns Controls Group September 13, 1993

GEMINI 8-M Telescopes Project

Summary of Gemini Control Simulations TN-C-G0015

INTRODUCTION and

SUMMARY

The following reports have been produced or are in progress:

1. Interim Servo System Performance Analysis Report: RPT-C-G0004, Appendix A.

2. Control System Design Study Report: PPT-UBC-G0014, Appendix B.

3. Chopping Secondary Control Study: TN-C-G0006, Appendix C. Can get the required 80% duty cycle and accuracy. Requires 75W power and high closed-loop bandwidth of 60OHz.

4. A Method for Determining Tip-Tilt Secondary Bandwidth and Power Requirements:

TN-C-G0007, Appendix D. Can remove more than 90% of tip-tilt noise power with a bandwidth= 15Hz and sampling rate= 100HZ.

5. Image Smear Error Budget with Required Servo Bandwidth and Sampling Rate: TN-C-

G0008, Appendix E. Can bring windshake RMS error from 1.16micro-rad to less than 0.1 micro-rad with BW = 40Hz and sampling rate = 200Hz.

6. Effect of filtering on open-loop tracking errors: Appendix F. A lookup table provides

some attenuation of the error associated with raw tracking. A sampled data filter provides more improvement.

7. Restriction Imposed on Tip-Tilt for an Off-Axis Guide Star: TN-C-G0011, Appendix

G. The distance between guide star and science object must be much smaller than the angular correlation length in order to permit significant reduction of tip-tilt noise.

8. SNR vs. Sample Rate for Tip-Tilt Using Off-Axis Guide Star: TN-C-G0012, Appendix

H.

9. Windshake vs. Sample Rate and Centroid Error vs. Sample Rate for Tip-Tilt Using an Off-Axis Guide Star: TN-C-G0013, Appendix I.

10. Meeting Atmospheric Tip-Tilt Requirement by Reducing Wind Shake: in progress

11. Baseline Non-linear Simulation: in progress - Can tolerate TBD angular quantization in azimuth during tracking. - Can tolerate TBD angular quantization in azimuth during slewing. - Can tolerate TBD coulomb friction in azimuth. - Can tolerate TBD static friction in azimuth. - Can tolerate TBD viscous friction in azimuth. - Can tolerate TBD rate quantization in ring-laser-gyro (RLG). - Can tolerate TBD other noise in RLG.


- Can tolerate TBD angular quantization in altitude. - Can tolerate TBD friction in altitude. - Can tolerate TBD angular quantization in cassegrain. - Can tolerate TBD friction in cassegrain. - Can tolerate TBD momentum disturbance spectrum from secondary chopping. - Can tolerate TBD momentum disturbance spectrum from cassegrain drive. - Can tolerate TBD wind spectrum. - Can tolerate TBD bearing disturbance spectra.


1. Interim Servo System Performance Analysis Report

A linear state-space model of the telescope is formulated. A locked rotor model is developed and the results compared with the detailed FEA analysis of the structure. This model is then expanded to include a drive system, a servo system is designed around the drive, and the effects of wind loading and step response are investigated. 2. Control System Design Study Report

This report describes the results of a three mass simulation of the structural dynamics and control response of the Gemini 8-M telescope. The pier, mount, and telescope tube are each modeled as rigid bodies with appropriate three dimensional inertia characteristics. The model includes a detailed representation of the pier/soil interface, and the stiffness and damping linking the three components. The primary purpose of the model is to determine the effects of three different pier designs on dynamic response, and on the ability of the attitude control system to compensate for wind-induced vibration of the structure. 3. Chopping Secondary Control Study

The chopping secondary servo system is required to provide 56.4 arcsecond (=270 micro-rad) motion at a rate of 10HZ. The required settling band of 0.485 micro-rad must be reached in 0.01 seconds to provide a usable duty-cycle of 80%.

Figure 1 of Appendix C shows the block diagram used to study some aspects of the chopping secondary control system. From the far left of the figure, the step command of 270 micro-rad is compared to the modeled position of the secondary to get an angular error, theta-err. The controller receives this error signal and generates a voltage conunand to the servo. After some lag, the servo produces a torque which drives the model secondary producing the angle theta an rate theta-dot. The current consumed by the servo is used to compute power used.

The servo model includes a number of parasitic elements which detract from performance: current and voltage limits, back emf, and servo electrical lag. The numerical values were taken from the manufacturers data sheets of a voice coil actuator.

A controller was found which satisfies the specifications while using a total of 75W for the three actuators. The approach is to build a Kalman filter and fast linear sampled data controller with a bandwidth of approximately 600Hz and a sample rate near 3kHz. 4. A Method for Determining Tip-Tilt Secondary Bandwidth and Power Requirements

Atmospheric tip-tilt errors are modeled by way of a Greenwood type power spectral density. Measured in rad2/Hz, the spectrum falls off like the 2/3rds power of frequency up to some cutoff, then falls off like the 11/3rds power of frequency thereafter. If the spectrum is integrated over frequency and the square root taken, the resulting RMS tip-tilt is found to be 0.98 micro-radians.


This assumes a seeing layer velocity of 20m/s, aperture of 1m, mirror diameter of 8m, and wavelength of 2.2 microns.

The specification for tip-tilt is that the control system shall remove 90% of the power, which is equivalent to bringing the RMS level down to a factor of 0.31 of its original value. Thus, the RMS tip-tilt after compensation may be (0.98 micro-rad)*(0.31)=0.30 micro-rad=0.063 arcsec.

To get the tip-tilt image motion after compensation, it is necessary to multiply the uncompensated spectrum by the square of the transfer function of the compensated system. The approach taken by this study was to vary the compensated system until a minimum bandwidth and sampling rate were found which met the specification on RMS compensated tip-tilt error. The system transfer function includes the degradation associated with digital sampling. Prudent design demands use of a sample rate at least five times the closed loop bandwidth and there are some gains in performance for faster sampling.

Specification was met for a closed loop bandwidth of 15Hz sampled at 100Hz. 5. Image Smear Error Budget with Required Servo Bandwidth and Sampling Rate

The approach used to model compensation for windshake error is very similar to that used for tip-tilt error as described in the earlier section. An input spectrum is passed through a filter and the resulting spectrum is integrated and the square root taken to get the compensated RMS image motion due to wind. The motion due to windshake is reduced from 1.16 micro-rad to 0.10 micro-rad by using a 40Hz filter sampled at 200Hz. A redesigned secondary mirror support is expected to reduce the uncompensated windshake and thus provide some relief from this high bandwidth/sampling rate.

The uncompensated windshake spectrum is obtained from a finite element analysis (FEA) model of the telescope. The wind is input to the telescope model according to a Kolmogorov model at 5m/s, which falls off like frequency to the -5/3rds power. The telescope is divided into many small elements each of which is acted upon by a drag force proportional to the square of wind velocity. The interplay of all of these forces with the telescope produces motion in the image plane, and the spectrum of this image plane motion is used as input to the control simulation to compute the net RMS image motion after compensation. 6. Effect of Filtering on Open-loop Tracking Errors

The telescope control system tends to filter out some of the errors associated with raw tracking coffected with a lookup table. Given a model of the spectrum of errors encountered for raw tracking, and given a model for the effect of a lookup table on these errors, one may obtain a spectrum of tracking errors after the lookup table. The lookup table is sampled at 20Hz, and will introduce some noise in the process. The noise plus errors after the lookup table provides a net spectrum which is acted upon by the telescope control system. For the purpose of this simulation, the telescope control system is modeled as a sampled data filter having a 40Hz bandwidth and 200Hz sampling.


The type of model used for the noise has a strong effect upon the tracking error after control-

system filtering, though it has only a modest effect on the RMS error before filtering. This is because the errors in the lookup table have a low spectral content and are easy for the control system to filter out, while the noise has high spectral content and is thus difficult to filter. 7. Restrictions Imposed Upon Tip-Tilt for an Off-Axis Guide Star

If one attempts to do tip-tilt compensation on a science object by tracking a bright nearby object there will be significant error introduced due to lack of correlation. In order to keep the errors small it is necessary for science object and guide star to be close, but it is difficult to get enough sufficiently bright objects to guarantee good sky coverage near the north galactic pole.

The eff or introduced by using an off-axis guide star is given as an empirical function of the off-axis angular distance. The distance implies a certain number of stars per square arcminute N, which in turn specifies a visual magnitude at the North Galactic Pole V.

If it is desired to get a factor of 2 reduction in noise power on the science object, the science object must be within 0.7 correlation lengths, which in turn requires stars to magnitude 21.5 to be used. At one correlation length there is no reduction in tip-tilt power. 8. SNR vs. Sampling Rate for Tip-Tilt Using an Off-Axis Guide Star

This technical note builds upon the preceding one to derive SNR (signal/noise ratio) vs. sampling rate for several cases of using an off-axis guide star to do correction of atmospheric induced tip-tilt errors. For a given diameter D_90, representing the angular difference between guide star and science object, a visual magnitude was derived which guaranteed that 90% sky coverage was achieved. Based on the visual magnitude calculations, this note computes SNR curves for three different diameters: 2.4, 3.5 and 7.2 arcminutes. These diameters are respectively: the diameter at which tip-tilt power is halved, the diameter of the science field, and the equivalent diameter of an annular region surrounding the science field.

From the previous note, the three different visual magnitudes are:

V(D_90 = 2.4) = 21.5 V(D_90 = 3.5) = 19.4 V(D_90 = 7.2) = 15.4.

The flux is assumed to be related to V by way of

flux = 1.1e12/(l0^(V/2.5)) flux(2.4) = 2.8e3 flux(3.5) = 1.9e4 flux(7.2) = 7.6e5


and SNR is related to flux and sampling rate (fs) by

SNR = sqrt(flux/fs) SNR(D_90=2.4, fs=200Hz) = 3.7 SNR(D_90=3.5, fs=200Hz) = 9.7 SNR(D_90=7.2, fs=200Hz) = 61.6

where the 200Hz sampling rate was chosen rather arbitrarily as that frequency which was

found to satisfy the windshake requirement of section 5 above. For the three diameters, the lower right plot of Figure 2 in Appendix I shows the curve of SNR vs. sampling rate. 9. Windshake vs. Sample Rate and Centroid Error vs. Sampling Rate for Tip-Tilt Using

an Off-Axis Guide Star

It is known that the centroid RMS may be computed from SNR by way of sig-centroid = [4*ln(2)]^( -0.5) * fwhm / SNR where fwhm is the full width half-max which is a constant related to telescope aperture and

wavelength of interest. The resulting centroid RMS is plotted in the upper right of Figure 2 of Appendix 1. So as sampling rate increases, centroid error gets progressively worse, because there is less flux. Compare this to the curve for residual windshake vs. sampling rate (from the methods of section 5) shown in the upper left of Figure 2. The windshake improves with increasing frequency because bandwidth has been assumed to be one fifth of the sample rate. If the centroid eff or and windshake are RSS'ed then the resulting total error, sig-total shown in the lower left of figure 2, has a minimum at some frequency. 10. Meeting Atmospheric Tip-Tilt Requirement by Reducing Windshake

This paper will calculate the reduction in raw windshake required to meet both the atmospheric tip-tilt requirement and the windshake error budget while guaranteeing a 90% change of finding a suitable guide star at the North Galactic Pole.

The paper will first calculate the field diameter over which the anisoplanatism error reduces the effectiveness of atmospheric tip-tilt removal from 90% (at the center of the field) to 50% with the guide star at the edge of the field. Using this field diameter the paper will calculate the V magnitude limit for a 90% probability of having at least one guide star in the field. Using a model for the stellar flux, atmosphere, telescope and bandpass one can calculate the centrold error for different tip-tilt correction bandwidths. For any correction bandwidth, one can calculate the residual windshake, given an input spectrum and a telescope transfer function. The combination of residual windshake and centroid error must RSS to less than 0.1 microradians to meet the error budget. The paper will adjust the wind velocity and the telescope transfer function to bring the overall error within specification.


11. Full Baseline Nonlinear Simulation

The entire telescope and its control systems have been modeled as shown in Figure 1 using the software package Matlab 4.0. The telescope structure is represented by the blocks labeled Pier, Mount, Tube, Casseg, and Second. Each of these may be thought of as a generalized mass, with forces coming in from the left, being integrated twice, and giving positions as outputs. More specifically, each line represents a 6-vector of 3 positions and 3 angles. The telescope structural blocks are obtained from the same FEA program that was used for computing the uncompensated windshake of Section 5. The dynamical models range in size from 12 states for the Pier, Casseg, and Second blocks to 30 states for the Mount and 100 states for the Tube.

Between the structural blocks are drive blocks which represent the drive motors and hearings of the telescope. Figure 2 shows the azimuth drive as an example. The paths at the top of the diagram represent the restoring forces due to the drive motors, and the lower paths represent the effects of the bearing. Most of the axes are not free to rotate very much and are represented by stiff springs and dampers within the state-space blocks. The stiffness and damping coefficients are chosen to be consistent with the FEA telescope natural modes, for example 100 rad/sec and 0.03 damping. For the azimuth drive of the example, the rotation about the z-azis is relatively free to move, so the motor is represented as a PID controller with some nonlinear friction added. The coefficients within the PID controller are chosen to give some reasonable closed-loop servo bandwidth and damping, for example 1Hz and 0.7 damping.

The time-domain inputs to the model are the commands to the azimuth, altitude, and Cassegrain. These inputs are typically ramps at either sidereal rates or slewing rates. Steps are also used to demonstrate damping, although it might not be desirable to subject the telescope to a step command. The most important outputs are Tx and Ty, the image plane motions in radians. Transient behavior, steady state, limit cycles and statistical measures of these outputs are all useful tools to describe system performance. Image motion is specified in a tracking error budget and may be quantified for various system non-linearities and errors.

Among the non-linearities of interest are quantization of angle measurement for the various axes, frictional characteristics of bearings (as supplied by Kaman Aerospace), and rate quantization and noise in a proposed rate sensing ring laser gyro (RLG). Some of the non-linearities are likely to effect each other and in such cases parametric runs are useful to show cross-dependencies, for example between bearing stiction and angular quantization. Statistics may be compiled for noise sources such as bearing friction and wind.


Appendix A

Interim Servo System Performance Analysis Report

RPT-C-G0004


Appendix B

Control System Design Study Report

RPT-UBC-G0014


Appendix C

Chopping Secondary Control Study

TN-C-G0006


Appendix D

A Method for Determining Tip-Tilt Secondary Bandwidth

and Power Requirements

TN-C-G0007


Appendix E

Image Smear Error Budget with Required Servo Bandwidth and Sampling Rate

TN-C-G0008


Appendix F

Effect of filtering on open-loop tracking errors


GEMINI 8-METRE TELESCOPES PROJECT CONTROLS GROUP

To: Distribution From: Rick McGonegal Date: June 10, 1993 Subject: Open Loop Tracking Action Requested Please forward comments, corrections, etc. to the attached document to R.McGonegal - thank you. We would especially appreciate input on appropriate models for the filtering action of the look up table (LUT) and the noise introduced by the LUT. Introduction The attached mathcad document proposes empirical models for the raw tracking performance of the Gemini telescope and for the filtering action of the look up table used to correct the tracking based on the current position of the telescope. Definition Raw tracking is defined as the deviations in the line of sight of the telescope due to the predictable and unpredictable mechanical flexures of the telescope as it moves across the sky. It does not include the effects of wind or atmosphere. Open loop tracking is defined as raw tracking corrected with LUT but with no focal plane feedback from a guide object. Summary Over a 10 minute/1 hour period the raw tracking performance is calculated as 0.667/2.324 arcsec rms which is within reason. Open loop tracking of 0.1 arcsec nns over a 10 minute interval requires better models for both the fitting action of the LUT and for the noise introduced by the LUT than have been suggested empirically. Problems The following problems have been noted and require further work: • the raw tracking performance, based on the empirical tracking power spectrum, appears too

good • a model of the effective filtering action of the LUT has difficulty in lowering the 10 n-dnute

tracking performance to 0.1 arcsec (expected for a reasonable telescope) unless either:


• the filter effect at high (0.1-1 hz) is greater than expected and there is very little noise introduced by the LUT, or

• the noise is negligible at frequencies above that corresponding to the pointing grid spacing

Derivation Of Models The models were derived in the following sequence: • an estimate of the periods and amplitudes of pointing corrections was received from Pat

Wallace • power spectrum was generated from this • the rms tracking error over a 10 minute interval was calculated by integrating the power

spectrum from 1/600 hz to 1000 hz and taking the square root - this yielded 0.667 arcsec rms; which is slightly better than expected for raw tracking performance

• different look up tables were modeled: • filter reduction decreased with increasing frequency and there was no noise introduced

by the LUT • LUT removed a constant 98% of the error but introduced noise. • filter reduction decreased with increasing frequency and LUT introduced noise.

Effect On Tracking Error Budget For the model where there is little noise the 0.100 arcsec rms tracking error is reduced to effectively zero by the tip/tilt secondary - there is no effect. For the model where the 0. 100 arcsec rms tracking error is mostly noise introduced by the LUT interpolation process this is only reduced -by to 0.027 arcsec by the tip/tilt secondary - this represents 100% of the tracking error budget ! The third model appears to be the best representation of what is happening and will be adopted as the baseline open loop tracking model Caveats / Questions • the noise model and its cutoff at the tracking loop frequency is arbitrary - but what is a better

model • are raw and open loop 10 minute tracking performances of 1 and 0.1 arcsec realistic of

current telescopes • what is a good model of the filtering action of a LUT on the tracking performance • if there is substantial noise introduced by the LUT, and this noise has a substantial component

at the tracking loop frequency - then it will be very difficult to remove as the tracking loop frequencies are at or above the bandwidths of most tip/tilt systems

Work To Do The empirical basis for the tracking model is not based on Gemini simulation data. The static deflections of the telescope from FEA analysis need to be used to update this.


Problem Description Given what we know about raw pointing and tracking and predictive (based on a lookup table) can we predict what the effective filter and noise funciton of a LUT are as a function of frequency ? Units

uradrad

arc micronmm

db: sec:deg

: := = = =10 3600 1000

16

Assumptions fp f fp hz TOLfp: , . , ( ): := − − = ⋅ = −5 4 9 3 10 10 5L

for logarithmically spaced variables need change of variable for accurate integration

normint[ ]f f F F f dff

fmin max, , : ( )

min

max= ∫

logint[ ] [ ]f f F F hzhz

raddx radx x

fhz

fhzmin max

log

log, , : ln( )

min

max

= ⋅ ⋅ ⋅ ⋅ ⋅

∫ 10 10 102

2

Empiricial Raw Tracking Spectrum (from Pat Wallace)

Per FreqPer

Freq hzemp empemp

emp:

..

deg :sec

...

..

.

.

.

deg=

⋅ =⋅ ⋅ ⋅

⋅ →

=

⋅⋅⋅⋅⋅

⋅⋅

−

−

−

−

−

3601809030101

010 01

124 60 60

1157 102 315 104 63 10

1389 104167 10

0 0040 0420 417

360

5

5

5

4

4

Amp arc PowerFreq

Powerrad

hz

i

emp empemp

empAmpemp:

.

..

sec :

......

.

.

: ..

=

⋅ =

→

=

⋅⋅⋅⋅

⋅⋅⋅

⋅

=

→

−

−

−

−

−

−

−

50201021

0501

0 02

0 0054 062 105 077 106 769 105641 10141 10

5641 102 256 10

0 7

2

4

5

7

8

9

12

14

2


Fit to Empiricial Tracking Spectrum

Mag Powerhz

raddbemp emp: log= ⋅ ⋅

⋅

→

20 2

[ ]Log freq Freqemp:

log sec= →

⋅

Slopeemp:=slope[Logfreq,Magemp] Slopeemp = -47.859 Intemp:=intercept[Logfreq,Magemp] Intemp = -294.121 · db Magfit(f):=Intemp+Slopeemp · log(f · sec)

Power frad

hzfit

Mag ffit

( ):( )

= ⋅10 202

So we can fit the empirical tracking spectrum as a Db/decade rolloff type function.

Predicted Empirical Tracking Spectrum For comparison with expected raw tracking performance we will compare the predicted rms tracking over a 10 minute interval

[ ]

T

fT

f hz f

E f hz Power E arc

hour

fit tracking fit fit tracking

10

1010

10 1

103

600

10 002

13600

10 0 667

min

minmin

min

_ min _

: sec

: . :sec

: logint , , . sec

= ⋅

= = ⋅ =⋅

= ⋅ = ⋅


this is somewhat better than we would expect. The rms tracking performance "missed" because we limited the integral to 10 minutes and shorter timescales is

[ ]E hz f Power E arcmissed fit missed: log int , , . secmin= ⋅ = ⋅−10 23528510

so one reason for this apparently good raw tracking performance is that most of the power is below the cutoff frequency corresponding to 10 minutes.

E E E E arctotal fit tracking missed total: . sec_= + = ⋅2 2 23537


LUT Tracking Error Filter Now that we have an empirical raw tracking power spectrum we want to find an empirical function representing the filter action of the LUT. We expect the 10 minute predictive tracking to have an rms error of 0.1 arcsec. The look up table for tracking will act as a high pass filter and remove the low frequency components of the raw tracking error. It will also, above a certain frequency, inject noise as the interpolation procedure fails to represent the underlying tracking errors. Empirical LUT Filter (from Pat Wallace) The following fractional residual errors are expected at these frequencies.

Freqhz

hzRedfilt filt:

.:

.

.=

⋅ ⋅⋅

=

−5 100 004

0 02050

5

we must remember that the above is a prediction in the reduction in the rms level of the signal. This must be squared to deal in the power reduction Fit to Empirical LUT Filter Slopefilt := slope[Freqfilt , Redfilt] Slopefilt = 121.519·sec Intfilt := intercept[Freqfilt, Redfilt] Intfilt =0.014 Ffilt(f):=Intfilt+Slopefilt ·f

Filterfilt(f):=if[Ffilt (f)>1,1,Ffilt (f)]

We can estimate frequency break as frequency where Filter value is unity.

x := 0.5·hz fLUT := root[Ffilt (X)- 1,x]

fLUT =0.008·hz

Modelling the LUT Filter We will model the LUT filter as a lag/lead filter with gains matching those of the empirical filter. In order to calculate the large time constant we will ignore the smaller (in frequency) time constant. we will assume that the error filter can be represented as:

x := 20·sec T root Error x Freq d xLUT filter filt filt2 11 1

: [ [ , ] Re , ]= −

TLUT2 =39.788·sec

this does not fit at low frequencies so we will add a second break frequency.

[ ]Error T T fT fT f

TTfilter2

1 21 2

[ , , ]:min maxmin

max

max

min=

+ ⋅ ⋅ ⋅+ ⋅ ⋅ ⋅

⋅ππ

x := 100-sec T root Error x T Freq d xLUT filter LUT filt filt1 22

0 0: [ [ , , ] Re , ]= −

TLUT1 =5.131·10-3·sec


for the LUT tracking filter

FilterLUT(f):=Error2filter[TLUT1, TLUT2, f]

FilterLUT[5·10-5·hz]= 0.02

FilterLUT(0.004·hz)=0.504

verifying the fit to empirical values.

Effect of LUT Filter When the LUT filter is applied to the raw tracking error spectrum (ignoring noise injection) PowerLUT(f):=Powerfit(f)[FilterLUT(f)]2 Mag f Power f

hzradLUT LUT( ): log ( )= ⋅ ⋅

20 2

Powerfilt(f):=Powerfit(f)[Filterfilt(f)]2 Mag f Power fhz

radfilt filt( ): log ( )= ⋅ ⋅

20 2


E hz hz Power E arc

E hz f Power E arcPower Freq

Power Freq

E f hz Power E arc

E hz hz Power E arc

hz f Power arc

f

LUT LUT LUT

LUT LUT LUTLUT filt

fit filt

LUT LUT LUT

filt filt filt

filt

: log int[ , , ] . sec

: log int[ , , ] . sec[ ]

[ ].



log int[ , , ] . sec

log int[ ,

min

min

min

min

= ⋅ ⋅ = ⋅

= ⋅ = ⋅ =

= ⋅ = ⋅

= ⋅ ⋅ = ⋅

⋅ = ⋅

⋅

−

−

−

−

10 10 0572

10 0 473 0504

10 0 323

10 10 0569

10 0 453

10

5 3

15

10 1

2 103

2

5 3

510

103

1

1

hz Power arcfilt, ] . sec= ⋅0 347

This does not reduce the rms tracking error over 10 minutes to the 0.1 arcsec expected.


High Frequency Gain Modification One option is to increase the effectiveness of the LUT filter at high frequencies until the expected 0.1 arcsec rms tracking is reached.

[ ][ ]

[ ][ ]

Freqhz

hzd

x T root Error x Freq d x

T

x T root Error x T Freq d x

T

filt filt

LUT filter filt filt

LUT

LUT filter LUT filt filt

LUT

15 100 004

10 02010

20 1 1 1

4 421

100 2 1 1

237192

5

2 1

2

1 2 0

1

1

0

:.

Re :..

: sec : , Re ,

. sec

: sec : , , Re ,

. sec

=⋅ ⋅

⋅

=

= ⋅ = −

= ⋅

= ⋅ = −

= ⋅

−

for the LUT tracking filter

[ ]

[ ]

Filter f Error T T f

Power f Power f Filter f Mag f Power fhz

rad

LUT filter LUT LUT

LUT fit LUT LUT LUT

1 2

1 1 20 1

1 2

22

( ): , ,

( ): ( ) ( ) ( ): log ( )

=

= ⋅ = ⋅ ⋅

[ ]E hz hz Power E arc

hz f Power arc

f hz Power arc

LUT LUT LUT

LUT

LUT

: log int , , . sec

logint[ , , ] . sec

logint[ , , ] . . sec

min

min

= ⋅ ⋅ = ⋅

⋅ = ⋅

⋅ = ⋅

−

−

10 10 1 0 471

10 1 0 46

10 1 013

5 3

510

103

This implies that the LUT filter must remove 90% rather than 50% of the errors at 0.004 hz.


High Frequency Error Reduction A second option is to assume that above some frequency the tracking errors reduce to near zero. This can be justified in that we are only concerned here with the tracking errors in the absence of wind and in the tracking errors which would be measured as the motion of the image centroid. For instance, if one were to use a TV autoguider at frame rate then motions slower than 1/30 sec would show up as image centroid motion and motions faster than 1/30 would show up as image point spread function broadening. We are only concerned with timescales shorter than 10 minutes so we will see what the smallest timescale must be.

f hz f hzf

f f Power arcLUT

10

10

0 002 0 0031

5556

0178

min maxmax

min max

. : . . min

log int[ , , ] . sec

= ⋅ = ⋅ = ⋅

= ⋅

By examining the plot of the residual power spectrum one can see that a majority of the power is in the region around 0.001 hz. One can calculate the variance contributed by the different frequency bands.

[ ][ ][ ]

logint , , . sec

logint , , . sec

logint , , . sec

min

min max

max

10 0 223

0 032

10 0 072

510

2

102

3 2

− ⋅ = ⋅

= ⋅

⋅ = ⋅

hz f Power arc

f f Power arc

f hz Power arc

LUT

LUT

LUT

The problem can be seen as simply reducing the power in the frequency band around 0.001 hz. Only the assumption of 90% filter efficiency at 0.004 hz seems to work. This would predict filter efficiencies of (where the valueis the percentage reduction in the rms error at that frequency) 1 1 0 01 76 8021 1 01 259771 1 1 3 41

− ⋅ = ⋅− ⋅ = ⋅− ⋅ = ⋅

Filter hzFilter hzFilter hz

LUT

LUT

LUT

( . ) . %( . ) . %( ) . %

compared to


1 0 01 28 351 01 38161 1 0 395

− ⋅ = ⋅− ⋅ = ⋅− ⋅ = ⋅

Filter hzFilter hzFilter hz

LUT

LUT

LUT

( . ) . %( . ) . %( ) . %

Conclusion The modified filter gain appears to be a reasonable assumption to proceed with.


An Alternative Approach to the LUT Filter We could assume that the LUT filter removes a constant 98% of the predicted error but that it introduces noise such that the effectiveness of the filter degrades to 50% at 0.004 hz. Also, we run the LUT at a maximum rate depending on the tracking rate loop. So we introduce no noise nor do we reduce the tracking errors at all at frequencies greater than the loop rate.

[ ] [ ]

[ ] [ ]

[ ][ ]

f hz

Power hzurad

hz

Power d Power Freq Powerurad

hz

Power d Power Freq Powerurad

hz

Noise f if f f Power Powerurad

hz

Filter f if f f

Power f Power f Filter f Noise f

Mag f Power

loop

filt

residual filt filt filt residual

actual filt filt filt actual

LUT loop actual residual

LUT loop

LUT fit LUT LUT

LUT LUT

:

( . ) .

: Re .

: Re .

( ): , ,

( ): , . ,

( ): ( ) ( ) ( )

( ): log (

= ⋅

⋅ = ⋅

= ⋅ = ⋅

= ⋅ = ⋅

= ≤ − ⋅

= ≤

= ⋅ +

= ⋅

20

0 004 269143

0108

67 286

2 0

2 0 02 1

2 2 2

2 20 2

2

2 2

2 2

2

2

0 1

1 1

fhz

rad) ⋅

2

[ ][ ][ ][ ]

E hz hz Power

hz f Power arc

f f Power arc

f hz Power arc

LUT LUT

LUT

loop LUT

loop LUT

: log int , ,

log int , , . sec

logint , , . sec

logint , , . sec

min

min

= ⋅ ⋅

⋅ = ⋅

= ⋅

⋅ = ⋅

−

−

10 10 2

10 2 0 476

2 7 56

10 2 1459

5 3

510

10

3

This does not decrease the tracking error to 0.1 arcsec rms over 10 minutes and the predicted tracking accuracy is much larger than experienced on large telescopes.


Reduction of LUT Noise Power What noise power do we need to reduce the tracking error to the expected value of 0.1 arcsec rms ?

[ ][ ]

[ ][ ][ ]

f hz

Noise f if f furad

hzurad

hz

Filter f if f f


E hz hz Power E arc

hz f Power arc

f f Power arc

f hz Power

loop

LUT loop

LUT loop

LUT fit LUT LUT

LUT LUT LUT

LUT

loop LUT

loop

:

( ): , . ,

( ): , . ,

( ): ( ) ( ) ( )

: log int , , . sec

log int , , . sec

log int , , . sec

log int , ,

min

min

= ⋅

= ≤ ⋅ ⋅

= ≤

= ⋅ +

= ⋅ ⋅ = ⋅

⋅ = ⋅

= ⋅

⋅

−

−

20

3 0 01 0

3 0 02 1

3 3 3

3 10 10 3 3 0 483

10 3 0 471

3 0 093

10 3

2 2

2

5 3

510

10

3[ ]LUT arc= ⋅0 018. sec

This represents a substantial reduction in noise power from that predicted by the empirical filter. It one represents the filter efficiency as the reduction in overall power,

[ ] [ ][ ] [ ]

Eff fPower f

Power fEff f

Power fPower f

Eff Freq Eff Freq

Eff Freq Eff Freq

LUTLUT

fitLUT

LUT

fit

LUT filt LUT filt

LUT filt LUT filt

23

1

2 2 1 2

2 2 023 1 50 387

0 0

1 1

( ):( )

( )( ):

( )( )

% %

. % . %

= =

= ⋅ = ⋅

= ⋅ = ⋅

[ ]

[ ]

x hz Freq root Eff x x

Freq hz

Eff Freq

LUT

LUT

: . : ( ) . ,

.

.

= ⋅ = −

= ⋅

=

01 2 050

0 284

2 05

50%

50%

50%

The net result of this is that the frequency at which the correction falls to 50% is 0.3 hz rather than 0.004 hz.


A Better Noise Model The assumption of white noise out to the tracking loop frequency can be improved with a more realistic model.

F hz Kurad

hz

Noise f Kf

f

f

f

loop LUT

LUT LUTloop

loop

: : .

( ): sin

= ⋅ = ⋅

= ⋅⋅

⋅

⋅

20 0 03

4

2

2π

π

[ ]

[ ][ ][ ][ ]

Filter f if f f


E hz hz Power E arc

f f Power arc

f f Power arc

f f Noise arc

LUT loop

LUT fit LUT LUT

LUT LUT LUT

loop LUT

hour loop LUT

loop LUT

4 0 02 1

4 4 4

4 10 10 4 4 0 483

4 0108

4 0117

4 0107

2

5 3

10

1

10

( ): [ , . , ]

( ): ( ) ( ) ( )

: log int , , . sec

log int , , . sec

log int , , . sec

log int , , . sec

min

min

= ≤

= ⋅ +

= ⋅ ⋅ = ⋅

= ⋅

= ⋅

= ⋅

−

Almost all the rms error is due to the noise model for the LUT. This is a serious problem and the noise model needs to be re-examined. In this model we have chosen the constant K(LUT) in order to qet 10 minute tracking performance of 0.1 arcsec. It is also a serious problem for the tip/tilt secondary system - there is a lot of power at frequencies where the tip/tilt system has poor rejection.


An Alternative Noise Model It has been empirically established that, if one obtains a fit to a pointing model of 1 arcsec, the on sky pointing performance will be ~ 2 arcsec. This is a better measurement of the fitting noise. If one considers what is going on while tracking one can see the following: There is the actual raw tracking of the telescope which causes the star to appear to move in the focal plane. At the same time the tracking LUT is attempting to make corrections to the position of the telescope. This tracking LUT is based on a grid of stars which have been observed on a grid of order 5 degrees apart. The interpolation function used by the LUT is designed to be smooth and continuous between the grid points.

[ ]

[ ][ ]

f

Kurad

hz

Noise f Kf

f

f

f

Filter f Filter f


E hz hz Power E arc

f hz Power arc

grid

LUT

LUT LUTgrid

grid

LUT LUT

LUT fit LUT LUT

LUT LUT LUT

LUT

:deg

deg sec

: .

( ): sin

( ): ( )

( ): ( ) ( ) ( )

: log int , , . sec

logint , , . sec

log

min

=⋅

⋅⋅

⋅ ⋅ ⋅

= ⋅

= ⋅ ⋅

⋅⋅

=

= ⋅ +

= ⋅ ⋅ = ⋅

⋅ = ⋅

−

3605

124 60 60

0 03

5

5 1

5 5 5

5 10 10 4 5 0 483

10 5 0103

2

2

2

5 3

103

ππ

[ ][ ]

int , , . sec

logint , , . secmin

f f Power arc

f hz Noise arc

hour loop LUT

LUT

1

103 5

5 0128

10 5 9 987 10

= ⋅

⋅ = ⋅ ⋅−

This assumption results in a noise model which contributes negligable tracking noise to the open loop tracking.


This model assumes that: * the raw tracking spectrum can be modelled as Mag(f) = -294.121 -47.859*log(f) * the LUT filter can be modelled as a lag/lead filter

[ ] [ ]Error T T fT fT f

TT

T T TT T T

filter

LUT

LUT

21 21 2

237 1924 421

1

2

min maxmin

max

max

min

min min

max max

, , :

: . sec: . sec

=+ ⋅ ⋅ ⋅+ ⋅ ⋅ ⋅

⋅

= = ⋅= = ⋅

ππ

* the interpolation table injects noise modeled as a sinc^2 function with a break at the grid frequency.


Appendix G

Restriction Imposed on Tip-Tilt for an Off-Axis Guide Star

TN-C-G0011


Appendix H

SNR vs. Sample Rate for Tip-Tilt Using Off-Axis Guide Star

TN-C-G0012


Appendix I Windshake vs. Sample Rate and Centroid Error vs. Sample Rate for Tip-Tilt

Using an Off-Axis Guide Star

TN-C-G0013

geminisummary of gemini control simulations tn-c-g0015 page 5 this assumes a seeing layer velocity...

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