From 016098b3273dd4ad2163d189c5eb2e94af31c4ab Mon Sep 17 00:00:00 2001 From: Thomas Dehaeze Date: Fri, 20 Sep 2019 11:35:58 +0200 Subject: [PATCH] Add non-repeatability plots --- figs/compare_tracking_error_attocube_meas.png | Bin 0 -> 217406 bytes figs/psd_tracking_error_rad.png | Bin 0 -> 196376 bytes figs/repeat_plot_raw.png | Bin 0 -> 42536 bytes figs/repeat_plot_subtract_mean.png | Bin 0 -> 56306 bytes figs/repeat_time_signals.png | Bin 0 -> 97435 bytes figs/repeat_tracking_errors.png | Bin 0 -> 115000 bytes index.html | 725 +++++++++--------- index.org | 223 +++--- 8 files changed, 465 insertions(+), 483 deletions(-) create mode 100644 figs/compare_tracking_error_attocube_meas.png create mode 100644 figs/psd_tracking_error_rad.png create mode 100644 figs/repeat_plot_raw.png create mode 100644 figs/repeat_plot_subtract_mean.png create mode 100644 figs/repeat_time_signals.png create mode 100644 figs/repeat_tracking_errors.png diff --git 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      Table of Contents

      -
      -

      1 Introduction

      +
      +

      1 Introduction

      -
      -

      1.1 Block Diagram

      +
      +

      1.1 Block Diagram

      -The block diagram of the setup to be controlled is shown in Fig. 1. +The block diagram of the setup to be controlled is shown in Fig. 1.

      -
      +

      cercalo_diagram_simplify.png

      Figure 1: Block Diagram of the Experimental Setup

      @@ -442,10 +441,10 @@ The transfer functions in the system are:

      -The block diagram with each transfer function is shown in Fig. 2. +The block diagram with each transfer function is shown in Fig. 2.

      -
      +

      cercalo_diagram.png

      Figure 2: Block Diagram of the Experimental Setup with detailed dynamics

      @@ -453,14 +452,14 @@ The block diagram with each transfer function is shown in Fig. -

      1.2 Cercalo

      +
      +

      1.2 Cercalo

      -From the Cercalo documentation, we have the parameters shown on table 1. +From the Cercalo documentation, we have the parameters shown on table 1.

      - +
      @@ -518,11 +517,11 @@ The Inductance and DC resistance of the two axis of the Cercalo have been measur

      -Let's first consider the horizontal direction and we try to model the Cercalo by a spring/mass/damper system (Fig. 3). +Let's first consider the horizontal direction and we try to model the Cercalo by a spring/mass/damper system (Fig. 3).

      -
      +

      mech_cercalo.png

      Figure 3: 1 degree-of-freedom model of the Cercalo

      @@ -561,7 +560,7 @@ The current \(I\) is also proportional to the voltage at the output of the buffe

      Let's try to determine the equivalent mass and spring values. -From table 1, for the horizontal direction: +From table 1, for the horizontal direction: \[ \left| \frac{x}{I} \right|(0) = \left| \alpha \frac{x}{F} \right|(0) = 28.4\ \frac{mA}{deg} = 1.63\ \frac{A}{rad} \]

      @@ -622,18 +621,18 @@ This will be done using the Newport.
      -
      -

      1.3 Optical Setup

      +
      +

      1.3 Optical Setup

      -
      -

      1.4 Newport

      +
      +

      1.4 Newport

      -Parameters of the Newport are shown in Fig. 4. +Parameters of the Newport are shown in Fig. 4.

      -It's dynamics for small angle excitation is shown in Fig. 5. +It's dynamics for small angle excitation is shown in Fig. 5.

      @@ -645,14 +644,14 @@ And we have: \end{align*} -

      +

      newport_doc.png

      Figure 4: Documentation of the Newport

      -
      +

      newport_gain.png

      Figure 5: Transfer function of the Newport

      @@ -660,25 +659,25 @@ And we have:
      -
      -

      1.5 4 quadrant Diode

      +
      +

      1.5 4 quadrant Diode

      -The front view of the 4 quadrant photo-diode is shown in Fig. 6. +The front view of the 4 quadrant photo-diode is shown in Fig. 6.

      -
      +

      4qd_naming.png

      Figure 6: Front view of the 4QD

      -Each of the photo-diode is amplified using a 4-channel amplifier as shown in Fig. 7. +Each of the photo-diode is amplified using a 4-channel amplifier as shown in Fig. 7.

      -
      +

      4qd_amplifier.png

      Figure 7: Wiring of the amplifier. The amplifier is located on the bottom right of the board

      @@ -686,8 +685,8 @@ Each of the photo-diode is amplified using a 4-channel amplifier as shown in Fig
      -
      -

      1.6 ADC/DAC

      +
      +

      1.6 ADC/DAC

      Let's compute the theoretical noise of the ADC/DAC. @@ -707,14 +706,14 @@ with \(\Delta V\) the total range of the ADC, \(n\) its number of bits, \(q\) th

      -
      -

      2 Identification of the system dynamics

      +
      +

      2 Identification of the system dynamics

      - +

      -In this section, we seek to identify all the blocks as shown in Fig. 1. +In this section, we seek to identify all the blocks as shown in Fig. 1.

      Table 1: Cercalo Parameters
      @@ -811,8 +810,8 @@ All the files (data and Matlab scripts) are accessible -

      2.1 Calibration of the 4 Quadrant Diode

      +
      +

      2.1 Calibration of the 4 Quadrant Diode

      Prior to any dynamic identification, we would like to be able to determine the meaning of the 4 quadrant diode measurement. @@ -827,8 +826,8 @@ We then should be able to obtain the "gain" of the 4QD in [V/rad].

      -
      -

      2.1.1 Input / Output data

      +
      +

      2.1.1 Input / Output data

      The identification data is loaded @@ -862,7 +861,7 @@ uv.t = uv.t - uv.t +

      calib_4qd_h.png

      Figure 8: Identification signals when exciting the horizontal direction (png, pdf)

      @@ -870,7 +869,7 @@ uv.t = uv.t - uv.t +

      calib_4qd_v.png

      Figure 9: Identification signals when exciting in the vertical direction (png, pdf)

      @@ -878,8 +877,8 @@ uv.t = uv.t - uv.t -

      2.1.2 Linear Regression to obtain the gain of the 4QD

      +
      +

      2.1.2 Linear Regression to obtain the gain of the 4QD

      We plot the angle of mirror @@ -909,7 +908,7 @@ where:

      -The linear regression is shown in Fig. 10. +The linear regression is shown in Fig. 10.

      @@ -919,17 +918,17 @@ bv = [ones +

      4qd_linear_reg.png

      Figure 10: Linear Regression (png, pdf)

      -Thus, we obtain the "gain of the 4 quadrant photo-diode as shown on table 2. +Thus, we obtain the "gain of the 4 quadrant photo-diode as shown on table 2.

      -
      +
      @@ -973,11 +972,11 @@ We obtain: -
      -

      2.2 Identification of the Cercalo Impedance, Current Amplifier and Voltage Amplifier dynamics

      +
      +

      2.2 Identification of the Cercalo Impedance, Current Amplifier and Voltage Amplifier dynamics

      -We wish here to determine \(G_i\) and \(G_a\) shown in Fig. 1. +We wish here to determine \(G_i\) and \(G_a\) shown in Fig. 1.

      @@ -985,15 +984,15 @@ We ignore the electro-mechanical coupling.

      -
      -

      2.2.1 Electrical Schematic

      +
      +

      2.2.1 Electrical Schematic

      -The schematic of the electrical circuit used to drive the Cercalo is shown in Fig. 11. +The schematic of the electrical circuit used to drive the Cercalo is shown in Fig. 11.

      -
      +

      cercalo_amplifier.png

      Figure 11: Current Amplifier Schematic

      @@ -1080,8 +1079,8 @@ with
      -
      -

      2.2.2 Theoretical Transfer Functions

      +
      +

      2.2.2 Theoretical Transfer Functions

      The values of the components in the current amplifier have been measured. @@ -1111,7 +1110,7 @@ Ga = blkdiag( +

      current_amplifier_tf.png

      Figure 12: Transfer function for the current amplifier (png, pdf)

      @@ -1133,8 +1132,8 @@ Zc = tf(blkdiag -

      2.2.3 Identified Transfer Functions

      +
      +

      2.2.3 Identified Transfer Functions

      Noise is generated using the DAC (\([U_{c,h}\ U_{c,v}]\)) and we measure the output of the voltage amplifier \([V_{c,h}, V_{c,v}]\). @@ -1161,7 +1160,7 @@ We remove the first seconds where the Cercalo is turned on.

      -
      +

      current_amplifier_comp_theory_id.png

      Figure 13: Identified and Theoretical Transfer Function \(G_a G_i\) (png, pdf)

      @@ -1179,7 +1178,7 @@ Gi = tf(blkdiag +

      current_amplifier_comp_theory_id_bis.png

      Figure 14: Identified and Theoretical Transfer Function \(G_a G_i\) (png, pdf)

      @@ -1257,11 +1256,11 @@ Continuous-time zero/pole/gain model.
      -
      -

      2.3 Identification of the Cercalo Dynamics

      +
      +

      2.3 Identification of the Cercalo Dynamics

      -We now wish to identify the dynamics of the Cercalo identified by \(G_c\) on the block diagram in Fig. 1. +We now wish to identify the dynamics of the Cercalo identified by \(G_c\) on the block diagram in Fig. 1.

      @@ -1273,8 +1272,8 @@ The transfer function obtained will be \(G_c G_i\), and because we have already

      -
      -

      2.3.1 Input / Output data

      +
      +

      2.3.1 Input / Output data

      The identification data is loaded @@ -1309,7 +1308,7 @@ uv.t = uv.t - uv.t +

      identification_uh.png

      Figure 15: Identification signals when exciting the horizontal direction (png, pdf)

      @@ -1317,7 +1316,7 @@ uv.t = uv.t - uv.t +

      identification_uv.png

      Figure 16: Identification signals when exciting in the vertical direction (png, pdf)

      @@ -1325,8 +1324,8 @@ uv.t = uv.t - uv.t -

      2.3.2 Coherence

      +
      +

      2.3.2 Coherence

      The window used for the spectral analysis is an hanning windows with temporal size equal to 1 second. @@ -1345,7 +1344,7 @@ The window used for the spectral analysis is an hanning windows wit

      -
      +

      coh_cercalo.png

      Figure 17: Coherence (png, pdf)

      @@ -1353,8 +1352,8 @@ The window used for the spectral analysis is an hanning windows wit
      -
      -

      2.3.3 Estimation of the Frequency Response Function Matrix

      +
      +

      2.3.3 Estimation of the Frequency Response Function Matrix

      We compute an estimate of the transfer functions. @@ -1368,14 +1367,14 @@ We compute an estimate of the transfer functions.

      -
      +

      frf_cercalo_gain.png

      Figure 18: Frequency Response Matrix (png, pdf)

      -
      +

      frf_cercalo_phase.png

      Figure 19: Frequency Response MatrixPhase (png, pdf)

      @@ -1383,8 +1382,8 @@ We compute an estimate of the transfer functions.
      -
      -

      2.3.4 Time Delay

      +
      +

      2.3.4 Time Delay

      Now, we would like to remove the time delay included in the FRF prior to the model extraction. @@ -1415,8 +1414,8 @@ tf_Ucv_Vpv = tf_Ucv_Vpv./G_delay_resp;

      -
      -

      2.3.5 Extraction of a transfer function matrix

      +
      +

      2.3.5 Extraction of a transfer function matrix

      First we define the initial guess for the resonance frequencies and the weights associated. @@ -1466,11 +1465,11 @@ weight_Ucv_Vpv(f

      -The weights are shown in Fig. 20. +The weights are shown in Fig. 20.

      -
      +

      weights_cercalo.png

      Figure 20: Weights amplitude (png, pdf)

      @@ -1522,7 +1521,7 @@ An we run the vectfit3 algorithm.
      -
      +

      identification_matrix_fit.png

      Figure 21: Transfer Function Extraction of the FRF matrix (png, pdf)

      @@ -1530,7 +1529,7 @@ An we run the vectfit3 algorithm. -
      +

      identification_matrix_fit_phase.png

      Figure 22: Transfer Function Extraction of the FRF matrix (png, pdf)

      @@ -1553,8 +1552,8 @@ Gc = [G_Uch_Vph, G_Ucv_Vph;
      -
      -

      2.4 Identification of the Newport Dynamics

      +
      +

      2.4 Identification of the Newport Dynamics

      We here identify the transfer function from a reference sent to the Newport \([U_{n,h},\ U_{n,v}]\) to the measurement made by the 4QD \([V_{p,h},\ V_{p,v}]\). @@ -1565,8 +1564,8 @@ To do so, we inject noise to the Newport \([U_{n,h},\ U_{n,v}]\) and we record t

      -
      -

      2.4.1 Input / Output data

      +
      +

      2.4.1 Input / Output data

      The identification data is loaded @@ -1601,14 +1600,14 @@ uv.t = uv.t - uv.t +

      identification_unh.png

      Figure 23: Identification signals when exciting the horizontal direction (png, pdf)

      -
      +

      identification_unv.png

      Figure 24: Identification signals when exciting in the vertical direction (png, pdf)

      @@ -1616,8 +1615,8 @@ uv.t = uv.t - uv.t -

      2.4.2 Coherence

      +
      +

      2.4.2 Coherence

      The window used for the spectral analysis is an hanning windows with temporal size equal to 1 second. @@ -1636,7 +1635,7 @@ The window used for the spectral analysis is an hanning windows wit

      -
      +

      id_newport_coherence.png

      Figure 25: Coherence (png, pdf)

      @@ -1644,8 +1643,8 @@ The window used for the spectral analysis is an hanning windows wit
      -
      -

      2.4.3 Estimation of the Frequency Response Function Matrix

      +
      +

      2.4.3 Estimation of the Frequency Response Function Matrix

      We compute an estimate of the transfer functions. @@ -1659,14 +1658,14 @@ We compute an estimate of the transfer functions.

      -
      +

      frf_newport_gain.png

      Figure 26: Frequency Response Matrix (png, pdf)

      -
      +

      frf_newport_phase.png

      Figure 27: Frequency Response Matrix Phase (png, pdf)

      @@ -1674,8 +1673,8 @@ We compute an estimate of the transfer functions.
      -
      -

      2.4.4 Time Delay

      +
      +

      2.4.4 Time Delay

      Now, we would like to remove the time delay included in the FRF prior to the model extraction. @@ -1697,7 +1696,7 @@ G_delay_resp = squeeze(freqr We then remove the time delay from the frequency response function.

      -
      +

      time_delay_newport.png

      Figure 28: Phase change due to time-delay in the Newport dynamics (png, pdf)

      @@ -1705,11 +1704,11 @@ We then remove the time delay from the frequency response function.
      -
      -

      2.4.5 Extraction of a transfer function matrix

      +
      +

      2.4.5 Extraction of a transfer function matrix

      -From Fig. 26, it seems reasonable to model the Newport dynamics as diagonal and constant. +From Fig. 26, it seems reasonable to model the Newport dynamics as diagonal and constant.

      @@ -1720,8 +1719,8 @@ From Fig. 26, it seems reasonable to model the Newport
      -
      -

      2.5 Full System

      +
      +

      2.5 Full System

      We now have identified: @@ -1783,11 +1782,11 @@ The file mat/plant.mat is accessible here

      -
      -

      3 Huddle Test

      +
      +

      3 Huddle Test

      - +

      The goal is to determine the noise of the photodiodes as well as the noise of the Attocube interferometer. @@ -1797,7 +1796,7 @@ The goal is to determine the noise of the photodiodes as well as the noise of th Multiple measurements are done with different experimental configuration as follow:

      -
      Table 2: Identified Gain of the 4 quadrant diode
      +
      @@ -1855,8 +1854,8 @@ Multiple measurements are done with different experimental configuration as foll
      Table 3: Experimental Configuration for the various Huddle test
      -
      -

      3.1 Load Data

      +
      +

      3.1 Load Data

      ht_1 = load('./mat/data_huddle_test_1.mat', 't', 'Vph', 'Vpv', 'Va');
      @@ -1874,8 +1873,8 @@ ht_4 = load(
      -

      3.2 Pre-processing

      +
      +

      3.2 Pre-processing

      t0 = 1; % [s]
      @@ -1909,18 +1908,18 @@ ht_4 = ht_s{
      -

      3.3 Time domain plots

      +
      +

      3.3 Time domain plots

      -
      +

      huddle_test_Va.png

      Figure 29: Measurement of the Attocube during Huddle Test (png, pdf)

      -
      +

      huddle_test_4qd.png

      Figure 30: Measurement of the 4QD during the Huddle tests (png, pdf)

      @@ -1928,8 +1927,8 @@ ht_4 = ht_s{ -

      3.4 Power Spectral Density

      +
      +

      3.4 Power Spectral Density

      win = hanning(ceil(1*fs));
      @@ -1961,7 +1960,7 @@ ht_4 = ht_s{
      +

      huddle_test_psd_va.png

      Figure 31: PSD of the Interferometer measurement during Huddle tests (png, pdf)

      @@ -1969,7 +1968,7 @@ ht_4 = ht_s{ +

      huddle_test_4qd_psd.png

      Figure 32: PSD of the 4QD signal during Huddle tests (png, pdf)

      @@ -1978,8 +1977,8 @@ ht_4 = ht_s{ -

      3.5 Conclusion

      +
      +

      3.5 Conclusion

      The Attocube's "Environmental Compensation Unit" does not have a significant effect on the stability of the measurement. @@ -1988,11 +1987,11 @@ The Attocube's "Environmental Compensation Unit" does not have a significant eff

      -
      -

      4 Budget Error

      +
      +

      4 Budget Error

      - +

      Goals: @@ -2028,27 +2027,27 @@ This can be due to change of Temperature, Pressure and Humidity of the air in th Procedure:

        -
      • in section 4.1: +
      • in section 4.1: We estimate the effect of an angle error of the Cercalo mirror on the Attocube measurement
      • -
      • in section 4.2: +
      • in section 4.2: The effect of perpendicular motion of the Newport and Cercalo mirrors on the Attocube measurement is determined.
      • -
      • in section 4.3: +
      • in section 4.3: We estimate the expected change of refractive index of the air in the beam path and the resulting Attocube measurement error
      • -
      • in section 4.5: +
      • in section 4.5: The feedback system using the 4 quadrant diode and the Cercalo is studied. Sensor noise, actuator noise and their effects on the control error is discussed.
      -
      -

      4.1 Effect of the Cercalo angle error on the measured distance by the Attocube

      +
      +

      4.1 Effect of the Cercalo angle error on the measured distance by the Attocube

      - + To simplify, we suppose that the Newport mirror is a flat mirror (instead of a concave one).

      -The geometry of the setup is shown in Fig. 33 where: +The geometry of the setup is shown in Fig. 33 where:

      • \(O\) is the reference surface of the Attocube
      • @@ -2075,7 +2074,7 @@ L = 0. +

        angle_error_schematic_cercalo.png

        Figure 33: Schematic of the geometry used to evaluate the effect of \(\delta \theta_c\) on the measured distance \(\delta L\)

        @@ -2117,7 +2116,7 @@ We now compute the new path length when there is an error angle \(\delta \theta_

        -We then compute the distance error and we plot it as a function of the Cercalo angle error (Fig. 34). +We then compute the distance error and we plot it as a function of the Cercalo angle error (Fig. 34).

        path_error = path_length - path_nominal;
        @@ -2125,14 +2124,14 @@ We then compute the distance error and we plot it as a function of the Cercalo a
         
        -
        +

        effect_cercalo_angle_distance_meas.png

        Figure 34: Effect of an angle error of the Cercalo on the distance error measured by the Attocube (png, pdf)

        -And we plot the beam path using Matlab for an high angle to verify that the code is working (Fig. 35). +And we plot the beam path using Matlab for an high angle to verify that the code is working (Fig. 35).

        theta = 2*2*pi/360; % [rad]
        @@ -2149,7 +2148,7 @@ T = [
         
        -
        +

        simulation_beam_path_high_angle.png

        Figure 35: Simulation of a beam path for high angle error (png, pdf)

        @@ -2157,7 +2156,7 @@ T = [

        -Based on Fig. 34, we see that an angle error \(\delta\theta_c\) of the Cercalo mirror induces a distance error \(\delta L\) measured by the Attocube which is dependent of the square of \(\delta \theta_c\): +Based on Fig. 34, we see that an angle error \(\delta\theta_c\) of the Cercalo mirror induces a distance error \(\delta L\) measured by the Attocube which is dependent of the square of \(\delta \theta_c\):

        \begin{equation} @@ -2172,7 +2171,7 @@ with:

      -Some example are shown in table 4. +Some example are shown in table 4.

      @@ -2181,7 +2180,7 @@ The tracking error of the feedback system used to position the Cercalo mirror sh

      - +
      @@ -2215,15 +2214,15 @@ The tracking error of the feedback system used to position the Cercalo mirror sh -
      -

      4.2 Unwanted motion of Cercalo/Newport mirrors perpendicular to its surface

      +
      +

      4.2 Unwanted motion of Cercalo/Newport mirrors perpendicular to its surface

      - +

      -From Figs 36 and 37, it is clear that perpendicular motions of the Cercalo mirror and of the Newport mirror have an impact on the measured distance by the Attocube interferometer. +From Figs 36 and 37, it is clear that perpendicular motions of the Cercalo mirror and of the Newport mirror have an impact on the measured distance by the Attocube interferometer.

      @@ -2247,14 +2246,14 @@ The error in measured distance by the Attocube will we \(\delta L/2\).

      -
      +

      cercalo_perpendicular_motion.png

      Figure 36: Effect of a Perpendicular motion of the Cercalo Mirror

      -
      +

      newport_perpendicular_motion.png

      Figure 37: Effect of a Perpendicular motion of the Newport Mirror

      @@ -2274,11 +2273,11 @@ However, the non repeatability of this motion should be less than few nano-meter
      -
      -

      4.3 Change in refractive index of the air in the beam path

      +
      +

      4.3 Change in refractive index of the air in the beam path

      - +

      @@ -2376,8 +2375,8 @@ The beam path should be protected using aluminum to minimize the change in refra

      -
      -

      4.4 Thermal Expansion of the Metrology Frame

      +
      +

      4.4 Thermal Expansion of the Metrology Frame

      The material used for the metrology frame is Aluminum. @@ -2397,18 +2396,18 @@ Thus, the temperature of the metrology frame should be kept constant to less tha

      -
      -

      4.5 Estimation of the Cercalo angle error due to Noise

      +
      +

      4.5 Estimation of the Cercalo angle error due to Noise

      - +

      In this section, we seek to estimate the angle error \(\delta \theta\)

      -Consider the block diagram in Fig. 38 with: +Consider the block diagram in Fig. 38 with:

      • \(G\): represents the transfer function from a voltage applied by the Speedgoat DAC used for the Cercalo to the Beam angle
      • @@ -2439,22 +2438,22 @@ It includes:
      -
      +

      feedback_diagram.png

      Figure 38: Block Diagram of the Feedback system

      -
      -

      4.5.1 Estimation of sources of noise and disturbances

      +
      +

      4.5.1 Estimation of sources of noise and disturbances

      Let's estimate the values of \(d_u\), \(d\) and \(n_\theta\).

      -
      -
      4.5.1.1 ADC Quantization Noise
      +
      +
      4.5.1.1 ADC Quantization Noise

      The ADC quantization noise is: @@ -2489,8 +2488,8 @@ For the ADC used:

      -
      -
      4.5.1.2 DAC Quantization Noise
      +
      +
      4.5.1.2 DAC Quantization Noise

      The DAC quantization noise is: @@ -2525,8 +2524,8 @@ For the DAC used:

      -
      -
      4.5.1.3 Noise of the Newport Mirror angle
      +
      +
      4.5.1.3 Noise of the Newport Mirror angle

      Plus, we estimate the effect of DAC quantization noise on the angle error on the Newport mirror. @@ -2576,8 +2575,8 @@ If we suppose a white noise, the power spectral density of the beam angle due to

      -
      -
      4.5.1.4 Disturbances due the Newport Mirror Rotation
      +
      +
      4.5.1.4 Disturbances due the Newport Mirror Rotation

      We will rotate the Newport mirror in order to simulate a displacement of the Sample: @@ -2588,7 +2587,7 @@ We will rotate the Newport mirror in order to simulate a displacement of the Sam -

      +

      newport_angle_concave_mirror.png

      Figure 39: Rotation of the (concave) Newport mirror

      @@ -2603,8 +2602,8 @@ where \(\alpha\) is the rotation of the Newport mirror.
      -
      -

      4.5.2 Perfect Control

      +
      +

      4.5.2 Perfect Control

      If the feedback is perfect, the Cercalo angle error will be equal to the 4 quadrant diode noise. @@ -2637,8 +2636,8 @@ If we just consider the ADC noise:

      -
      -

      4.5.3 Error due to DAC noise used for the Cercalo

      +
      +

      4.5.3 Error due to DAC noise used for the Cercalo

      load('./mat/plant.mat', 'Gi', 'Gc', 'Gd');
      @@ -2748,18 +2747,18 @@ The DAC noise use for the Cercalo does not limit the performance of the system.
       
      -
      -

      5 Plant Scaling

      +
      +

      5 Plant Scaling

      - +

      The goal is the scale the plant prior to control synthesis. This will simplify the choice of weighting functions and will yield useful insight on the controllability of the plant.

      -
      Table 4: Effect of an angle error \(\delta \theta_c\) of the Cercalo's mirror on the measurement error \(\delta L\) by the Attocube
      +
      @@ -2810,8 +2809,8 @@ This will simplify the choice of weighting functions and will yield useful insig
      Table 5: Maximum wanted values for various signals
      -
      -

      5.1 Control Objective

      +
      +

      5.1 Control Objective

      The maximum expected stroke is \(y_\text{max} = 3mm \approx 5e^{-2} rad\) at \(1Hz\). @@ -2833,15 +2832,15 @@ In terms of loop gain, this is equivalent to:

      -
      -

      5.2 General Configuration

      +
      +

      5.2 General Configuration

      -The plant is put in a general configuration as shown in Fig. 40. +The plant is put in a general configuration as shown in Fig. 40.

      -
      +

      general_control_names.png

      Figure 40: General Control Configuration

      @@ -2850,15 +2849,15 @@ The plant is put in a general configuration as shown in Fig. -

      6 Plant Analysis

      +
      +

      6 Plant Analysis

      - +

      -
      -

      6.1 Load Plant

      +
      +

      6.1 Load Plant

      load('mat/plant.mat', 'G');
      @@ -2867,8 +2866,8 @@ The plant is put in a general configuration as shown in Fig. 
      -

      6.2 RGA-Number

      +
      +

      6.2 RGA-Number

      freqs = logspace(2, 4, 1000);
      @@ -2907,8 +2906,8 @@ V = zeros(
      -

      6.3 Rotation Matrix

      +
      +

      6.3 Rotation Matrix

      G0 = freqresp(G, 0);
      @@ -2918,15 +2917,15 @@ V = zeros(
      -

      7 Active Damping

      +
      +

      7 Active Damping

      -
      -

      7.1 Load Plant

      +
      +

      7.1 Load Plant

      load('mat/plant.mat', 'sys', 'Gi', 'Zc', 'Ga', 'Gc', 'Gn', 'Gd');
      @@ -2935,8 +2934,8 @@ V = zeros(
      -

      7.2 Integral Force Feedback

      +
      +

      7.2 Integral Force Feedback

      bode(sys({'Vch', 'Vcv'}, {'Uch', 'Ucv'}));
      @@ -2963,8 +2962,8 @@ sys_cl = connect(sys, Kppf,
       
      -
      -

      7.3 Conclusion

      +
      +

      7.3 Conclusion

      Active damping does not seems to be applicable here. @@ -2973,11 +2972,11 @@ Active damping does not seems to be applicable here.

      -
      -

      8 Decentralized Control of the Cercalo

      +
      +

      8 Decentralized Control of the Cercalo

      - +

      In this section, we try to implement a simple decentralized controller. @@ -2989,8 +2988,8 @@ All the files (data and Matlab scripts) are accessible -

      8.1 Load Plant

      +
      +

      8.1 Load Plant

      load('mat/plant.mat', 'sys', 'Gi', 'Zc', 'Ga', 'Gc', 'Gn', 'Gd');
      @@ -2999,12 +2998,12 @@ All the files (data and Matlab scripts) are accessible 
      -

      8.2 Diagonal Controller

      +
      +

      8.2 Diagonal Controller

      Using SISOTOOL, a diagonal controller is designed. -The two SISO loop gains are shown in Fig. 41. +The two SISO loop gains are shown in Fig. 41.

      Kh = -0.25598*(s+112)*(s^2 + 15.93*s + 6.686e06)/((s^2*(s+352.5)*(1+s/2/pi/2000)));
      @@ -3017,14 +3016,14 @@ K.OutputName = {
       
       
      -
      +

      diag_contr_loop_gain.png

      Figure 41: Loop Gain using the Decentralized Diagonal Controller (png, pdf)

      -We then close the loop and we look at the transfer function from the Newport rotation signal to the beam angle (Fig. 42). +We then close the loop and we look at the transfer function from the Newport rotation signal to the beam angle (Fig. 42).

      inputs  = {'Uch', 'Ucv', 'Unh', 'Unv'};
      @@ -3035,7 +3034,7 @@ sys_cl = connect(sys, 
       
       
      -
      +

      diag_contr_effect_newport.png

      Figure 42: Effect of the Newport rotation on the beam position when the loop is closed using the Decentralized Diagonal Controller (png, pdf)

      @@ -3043,8 +3042,8 @@ sys_cl = connect(sys,
      -
      -

      8.3 Save the Controller

      +
      +

      8.3 Save the Controller

      Kd = c2d(K, 1e-4, 'tustin');
      @@ -3062,11 +3061,11 @@ The diagonal controller is accessible here.
       
      -
      -

      9 Newport Control

      +
      +

      9 Newport Control

      - +

      In this section, we try to implement a simple decentralized controller for the Newport. @@ -3079,8 +3078,8 @@ This can be used to align the 4QD:

    1. finally, we are sure to be aligned when the command signal of the Newport is 0
    2. -
      -

      9.1 Load Plant

      +
      +

      9.1 Load Plant

      load('mat/plant.mat', 'Gn', 'Gd');
      @@ -3089,8 +3088,8 @@ This can be used to align the 4QD:
       
      -
      -

      9.2 Analysis

      +
      +

      9.2 Analysis

      The plant is basically a constant until frequencies up to the required bandwidth. @@ -3117,7 +3116,7 @@ Knv = 1

      -
      +

      loop_gain_newport.png

      Figure 43: Diagonal Loop Gain for the Newport (png, pdf)

      @@ -3125,8 +3124,8 @@ Knv = 1
      -
      -

      9.3 Save

      +
      +

      9.3 Save

      Kn = blkdiag(Knh, Knv);
      @@ -3147,18 +3146,22 @@ The controllers can be downloaded here.
       
      -
      -

      10 Measurement of the non-repeatability

      +
      +

      10 Measurement of the non-repeatability

      - + +

      +

      +The goal here is the measure the non-repeatability of the setup. +

      + +

      +All sources of error (detailed in the budget error in Section 4) will contribute to the non-repeatability of the system.

      -
        -
      • Explanation of the procedure
      • -
      -
      -

      10.1 Data Load

      +
      +

      10.1 Data Load and pre-processing

      uh = load('mat/data_rep_h.mat', ...
      @@ -3221,9 +3224,21 @@ uv.t = uv.t - uv.t
      -

      10.2 Verify Tracking Angle Error

      +
      +

      10.2 Some Time domain plots

      + +
      +

      repeat_time_signals.png +

      +

      Figure 44: Time domain Signals for the repeatability measurement (png, pdf)

      +
      +
      +
      + +
      +

      10.3 Verify Tracking Angle Error

      +

      Let's verify that the positioning error of the beam is small and what could be the effect on the distance measured by the intereferometer.

      @@ -3233,28 +3248,11 @@ Let's verify that the positioning error of the beam is small and what could be t
      -
      -
      figure;
      -ax1 = subplot(1, 2, 1);
      -hold on;
      -plot(uh.t(1:2*fs), 1e6*uh.Vph(1:2*fs)/freqresp(Gd(1,1), 0), 'DisplayName', '$\theta_{h}$');
      -plot(uh.t(1:2*fs), 1e6*uh.Vpv(1:2*fs)/freqresp(Gd(2,2), 0), 'DisplayName', '$\theta_{v}$');
      -hold off;
      -xlabel('Time [s]'); ylabel('$\theta$ [$\mu$ rad]');
      -title('Newport Tilt - Horizontal Direction');
      -legend();
       
      -ax2 = subplot(1, 2, 2);
      -hold on;
      -plot(uv.t(1:2*fs), 1e6*uv.Vph(1:2*fs)/freqresp(Gd(1,1), 0), 'DisplayName', '$\theta_{h}$');
      -plot(uv.t(1:2*fs), 1e6*uv.Vpv(1:2*fs)/freqresp(Gd(2,2), 0), 'DisplayName', '$\theta_{v}$');
      -hold off;
      -xlabel('Time [s]'); ylabel('$\theta$ [$\mu$ rad]');
      -title('Newport Tilt - Vertical Direction');
      -legend();
      -
      -linkaxes([ax1,ax2],'xy');
      -
      +
      +

      repeat_tracking_errors.png +

      +

      Figure 45: Tracking errors during the repeatability measurement (png, pdf)

      @@ -3269,6 +3267,14 @@ Let's compute the PSD of the error to see the frequency content.

      + +
      +

      psd_tracking_error_rad.png +

      +

      Figure 46: Power Spectral Density of the tracking errors (png, pdf)

      +
      + +

      Let's convert that to errors in distance

      @@ -3297,19 +3303,33 @@ with

      -Now, compare that with the PSD of the measured distance by the interferometer. +Now, compare that with the PSD of the measured distance by the interferometer (Fig. 47).

      [psd_Lh, f] = pwelch(uh.Va, hanning(ceil(1*fs)), [], [], fs);
       [psd_Lv, ~] = pwelch(uv.Va, hanning(ceil(1*fs)), [], [], fs);
       
      + + +
      +

      compare_tracking_error_attocube_meas.png +

      +

      Figure 47: Comparison of the effect of tracking error on the measured distance and the measured distance by the Attocube (png, pdf)

      +
      + +
      +

      +The tracking errors are a limiting factor. +

      + +
      -
      -

      10.3 Processing

      -
      +
      +

      10.4 Processing

      +

      First, we get the mean value as measured by the interferometer for each value of the Newport angle.

      @@ -3322,42 +3342,19 @@ Unvm = mean(reshape
      -

      -We then subtract + +

      +

      repeat_plot_raw.png

      -
      -
      figure;
      -hold on;
      -plot(uh.Unh, 1e9*(uh.Va - repmat(Vam, length(uh.t)/length(Vam),1)))
      -hold off;
      -xlabel('$V_{n,h}$ [V]'); ylabel('$V_a$ [nm]');
      -
      -
      -
      +

      Figure 48: Repeatability of the measurement (png, pdf)

      -
      -

      10.4 Some Plots

      -
      -
      -
      figure;
      -hold on;
      -plot(uh.Unh, uh.Va);
      -plot(Unhm, Vam)
      -hold off;
      -xlabel('$V_{n,h}$ [V]'); ylabel('$V_a$ [m]');
      -
      -
      -
      -
      -
      -

      10.5 Repeatability

      -
      -
      -
      bh = [ones(size(Vnh)) Vnh]\Vph;
      -bv = [ones(size(Vnv)) Vnv]\Vpv;
      -
      + +
      +

      repeat_plot_subtract_mean.png +

      +

      Figure 49: Repeatability of the measurement after subtracting the mean value (png, pdf)

      @@ -3365,7 +3362,7 @@ bv = [ones

      Author: Dehaeze Thomas

      -

      Created: 2019-09-20 ven. 09:25

      +

      Created: 2019-09-20 ven. 11:34

      Validate

      diff --git a/index.org b/index.org index 4962201..e94485e 100644 --- a/index.org +++ b/index.org @@ -2022,7 +2022,7 @@ Multiple measurements are done with different experimental configuration as foll hold off; set(gca, 'xscale', 'log'); set(gca, 'yscale', 'log'); xlabel('Frequency [Hz]'); - ylabel('ASD $\left[\frac{m}{\sqrt{Hz}}\right]$'); + ylabel('ASD $\left[\frac{1}{\sqrt{Hz}}\right]$'); legend('location', 'northeast'); xlim([1, 1000]); #+end_src @@ -3011,7 +3011,9 @@ The controllers can be downloaded [[./mat/K_newport.mat][here]]. * Measurement of the non-repeatability <> ** Introduction :ignore: -- Explanation of the procedure +The goal here is the measure the non-repeatability of the setup. + +All sources of error (detailed in the budget error in Section [[sec:budget_error]]) will contribute to the non-repeatability of the system. ** Matlab Init :noexport:ignore: #+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name) @@ -3026,7 +3028,7 @@ The controllers can be downloaded [[./mat/K_newport.mat][here]]. fs = 1e4; #+end_src -** Data Load +** Data Load and pre-processing #+begin_src matlab uh = load('mat/data_rep_h.mat', ... 't', 'Uch', 'Ucv', ... @@ -3083,6 +3085,53 @@ The controllers can be downloaded [[./mat/K_newport.mat][here]]. uv.t = uv.t - uv.t(1); #+end_src +** Some Time domain plots +#+begin_src matlab :exports none + tend = 5; % [s] + + figure; + ax1 = subplot(2, 2, 1); + hold on; + plot(uh.t(1:tend*fs), uh.Unh(1:tend*fs)); + hold off; + xlabel('Time [s]'); ylabel('Voltage [V]'); + title('Newport Tilt - Horizontal Direction'); + + ax3 = subplot(2, 2, 3); + hold on; + plot(uh.t(1:tend*fs), 1e9*uh.Va(1:tend*fs)); + hold off; + xlabel('Time [s]'); ylabel('Distance [nm]'); + title('Attocube - Horizontal Direction'); + + ax2 = subplot(2, 2, 2); + hold on; + plot(uv.t(1:tend*fs), uv.Unv(1:tend*fs)); + hold off; + xlabel('Time [s]'); ylabel('Voltage [V]'); + title('Newport Tilt - Vertical Direction'); + + ax4 = subplot(2, 2, 4); + hold on; + plot(uv.t(1:tend*fs), 1e9*uv.Va(1:tend*fs)); + hold off; + xlabel('Time [s]'); ylabel('Distance [nm]'); + title('Attocube - Vertical Direction'); + + linkaxes([ax1,ax2,ax3,ax4],'x'); + linkaxes([ax1,ax2],'xy'); + linkaxes([ax3,ax4],'xy'); +#+end_src + +#+HEADER: :tangle no :exports results :results none :noweb yes +#+begin_src matlab :var filepath="figs/repeat_time_signals.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png") + <> +#+end_src + +#+NAME: fig:repeat_time_signals +#+CAPTION: Time domain Signals for the repeatability measurement ([[./figs/repeat_time_signals.png][png]], [[./figs/repeat_time_signals.pdf][pdf]]) +[[file:figs/repeat_time_signals.png]] + ** Verify Tracking Angle Error Let's verify that the positioning error of the beam is small and what could be the effect on the distance measured by the intereferometer. @@ -3090,7 +3139,7 @@ Let's verify that the positioning error of the beam is small and what could be t load('./mat/plant.mat', 'Gd'); #+end_src -#+begin_src matlab +#+begin_src matlab :exports none figure; ax1 = subplot(1, 2, 1); hold on; @@ -3113,6 +3162,15 @@ Let's verify that the positioning error of the beam is small and what could be t linkaxes([ax1,ax2],'xy'); #+end_src +#+HEADER: :tangle no :exports results :results none :noweb yes +#+begin_src matlab :var filepath="figs/repeat_tracking_errors.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png") + <> +#+end_src + +#+NAME: fig:repeat_tracking_errors +#+CAPTION: Tracking errors during the repeatability measurement ([[./figs/repeat_tracking_errors.png][png]], [[./figs/repeat_tracking_errors.pdf][pdf]]) +[[file:figs/repeat_tracking_errors.png]] + Let's compute the PSD of the error to see the frequency content. #+begin_src matlab @@ -3148,6 +3206,16 @@ Let's compute the PSD of the error to see the frequency content. xlim([1, 1000]); #+end_src +#+HEADER: :tangle no :exports results :results none :noweb yes +#+begin_src matlab :var filepath="figs/psd_tracking_error_rad.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png") + <> +#+end_src + +#+NAME: fig:psd_tracking_error_rad +#+CAPTION: Power Spectral Density of the tracking errors ([[./figs/psd_tracking_error_rad.png][png]], [[./figs/psd_tracking_error_rad.pdf][pdf]]) +[[file:figs/psd_tracking_error_rad.png]] + + Let's convert that to errors in distance \[ \Delta L = L^\prime - L = \frac{L}{\cos(\alpha)} - L \approx \frac{L \alpha^2}{2} \] @@ -3167,33 +3235,7 @@ with [psd_UvLv, ~] = pwelch(0.5*L*(uv.Vpv/freqresp(Gd(2,2), 0)).^2, hanning(ceil(1*fs)), [], [], fs); #+end_src -#+begin_src matlab :exports none - figure; - ax1 = subplot(1, 2, 1); - hold on; - plot(f, sqrt(psd_UhLh), 'DisplayName', '$\Gamma_{L_h}$'); - plot(f, sqrt(psd_UhLv), 'DisplayName', '$\Gamma_{L_v}$'); - hold off; - set(gca, 'xscale', 'log'); set(gca, 'yscale', 'log'); - xlabel('Frequency [Hz]'); ylabel('ASD $\left[\frac{m}{\sqrt{Hz}}\right]$') - legend('Location', 'southwest'); - title('Newport Tilt - Horizontal Direction'); - - ax2 = subplot(1, 2, 2); - hold on; - plot(f, sqrt(psd_UvLh), 'DisplayName', '$\Gamma_{L_h}$'); - plot(f, sqrt(psd_UvLv), 'DisplayName', '$\Gamma_{L_v}$'); - hold off; - set(gca, 'xscale', 'log'); set(gca, 'yscale', 'log'); - xlabel('Frequency [Hz]'); ylabel('ASD $\left[\frac{m}{\sqrt{Hz}}\right]$') - legend('Location', 'southwest'); - title('Newport Tilt - Vertical Direction'); - - linkaxes([ax1,ax2],'xy'); - xlim([1, 1000]); -#+end_src - -Now, compare that with the PSD of the measured distance by the interferometer. +Now, compare that with the PSD of the measured distance by the interferometer (Fig. [[fig:compare_tracking_error_attocube_meas]]). #+begin_src matlab [psd_Lh, f] = pwelch(uh.Va, hanning(ceil(1*fs)), [], [], fs); [psd_Lv, ~] = pwelch(uv.Va, hanning(ceil(1*fs)), [], [], fs); @@ -3227,6 +3269,19 @@ Now, compare that with the PSD of the measured distance by the interferometer. xlim([1, 1000]); #+end_src +#+HEADER: :tangle no :exports results :results none :noweb yes +#+begin_src matlab :var filepath="figs/compare_tracking_error_attocube_meas.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png") + <> +#+end_src + +#+NAME: fig:compare_tracking_error_attocube_meas +#+CAPTION: Comparison of the effect of tracking error on the measured distance and the measured distance by the Attocube ([[./figs/compare_tracking_error_attocube_meas.png][png]], [[./figs/compare_tracking_error_attocube_meas.pdf][pdf]]) +[[file:figs/compare_tracking_error_attocube_meas.png]] + +#+begin_important +The tracking errors are a limiting factor. +#+end_important + ** Processing First, we get the mean value as measured by the interferometer for each value of the Newport angle. #+begin_src matlab @@ -3252,8 +3307,20 @@ First, we get the mean value as measured by the interferometer for each value of plot(Unvm, Vavm) hold off; xlabel('$V_{n,v}$ [V]'); ylabel('$V_a$ [m]'); + + linkaxes([ax1,ax2],'xy'); #+end_src +#+HEADER: :tangle no :exports results :results none :noweb yes +#+begin_src matlab :var filepath="figs/repeat_plot_raw.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png") + <> +#+end_src + +#+NAME: fig:repeat_plot_raw +#+CAPTION: Repeatability of the measurement ([[./figs/repeat_plot_raw.png][png]], [[./figs/repeat_plot_raw.pdf][pdf]]) +[[file:figs/repeat_plot_raw.png]] + + #+begin_src matlab :exports none figure; ax1 = subplot(1, 2, 1); @@ -3272,93 +3339,11 @@ First, we get the mean value as measured by the interferometer for each value of ylim([-100 100]); #+end_src -We then subtract -#+begin_src matlab - figure; - hold on; - plot(uh.Unh, 1e9*(uh.Va - repmat(Vam, length(uh.t)/length(Vam),1))) - hold off; - xlabel('$V_{n,h}$ [V]'); ylabel('$V_a$ [nm]'); +#+HEADER: :tangle no :exports results :results none :noweb yes +#+begin_src matlab :var filepath="figs/repeat_plot_subtract_mean.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png") + <> #+end_src -** Some Plots -#+begin_src matlab - figure; - hold on; - plot(uh.Unh, uh.Va); - plot(Unhm, Vam) - hold off; - xlabel('$V_{n,h}$ [V]'); ylabel('$V_a$ [m]'); -#+end_src - -#+begin_src matlab :exports none - figure; - ax1 = subplot(1, 2, 1); - hold on; - plot(uh.Vnh(1:fs/2), uh.Va(1:fs/2)); - hold off; - xlabel('$V_{n,h}$ [V]'); ylabel('$V_a$ [m]'); - - ax2 = subplot(1, 2, 2); - hold on; - plot(uv.Vnv, uv.Va); - hold off; - xlabel('$V_{n,v}$ [V]'); ylabel('$V_a$ [m]'); -#+end_src - - -#+begin_src matlab :exports none - figure; - ax1 = subplot(1, 2, 1); - hold on; - plot(uh.Vnh, uh.Va); - hold off; - xlabel('$V_{n,h}$ [V]'); ylabel('$V_a$ [m]'); - - ax2 = subplot(1, 2, 2); - hold on; - plot(uv.Vnv, uv.Va); - hold off; - xlabel('$V_{n,v}$ [V]'); ylabel('$V_a$ [m]'); -#+end_src - -** Repeatability -#+begin_src matlab :exports none - figure; - ax1 = subplot(1, 2, 1); - hold on; - plot(Vnh, Va); - hold off; - xlabel('$V_{n,h}$ [V]'); ylabel('$V_a$ [m]'); - - ax2 = subplot(1, 2, 2); - hold on; - plot(Vnv, Va); - hold off; - xlabel('$V_{n,v}$ [V]'); ylabel('$V_a$ [m]'); -#+end_src - - -#+begin_src matlab - bh = [ones(size(Vnh)) Vnh]\Vph; - bv = [ones(size(Vnv)) Vnv]\Vpv; -#+end_src - -#+begin_src matlab :exports none - figure; - ax1 = subplot(1, 2, 1); - hold on; - plot(2*gn0*uh.Vnh, uh.Vph, 'o', 'DisplayName', 'Exp. data'); - plot(2*gn0*[min(uh.Vnh) max(uh.Vnh)], 2*gn0*[min(uh.Vnh) max(uh.Vnh)].*bh(2) + bh(1), 'k--', 'DisplayName', sprintf('%.1e x + %.1e', bh(2), bh(1))) - hold off; - xlabel('$\alpha_{0,h}$ [rad]'); ylabel('$Vp_h$ [V]'); - legend(); - - ax2 = subplot(1, 2, 2); - hold on; - plot(2*gn0*uv.Vnv, uv.Vpv, 'o', 'DisplayName', 'Exp. data'); - plot(2*gn0*[min(uv.Vnv) max(uv.Vnv)], 2*gn0*[min(uv.Vnv) max(uv.Vnv)].*bv(2) + bv(1), 'k--', 'DisplayName', sprintf('%.1e x + %.1e', bv(2), bv(1))) - hold off; - xlabel('$\alpha_{0,v}$ [rad]'); ylabel('$Vp_v$ [V]'); - legend(); -#+end_src +#+NAME: fig:repeat_plot_subtract_mean +#+CAPTION: Repeatability of the measurement after subtracting the mean value ([[./figs/repeat_plot_subtract_mean.png][png]], [[./figs/repeat_plot_subtract_mean.pdf][pdf]]) +[[file:figs/repeat_plot_subtract_mean.png]]

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