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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.

-
+

sercalo_diagram_simplify.png

Figure 1: Block Diagram of the Experimental Setup

@@ -447,10 +450,10 @@ 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.

-
+

sercalo_diagram.png

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

@@ -458,14 +461,14 @@ The block diagram with each transfer function is shown in Fig. -

1.2 Sercalo

+
+

1.2 Sercalo

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

- +
@@ -523,11 +526,11 @@ The Inductance and DC resistance of the two axis of the Sercalo have been measur

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

-
+

mech_sercalo.png

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

@@ -566,7 +569,7 @@ 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: \[ \left| \frac{x}{I} \right|(0) = \left| \alpha \frac{x}{F} \right|(0) = 28.4\ \frac{mA}{deg} = 1.63\ \frac{A}{rad} \]

@@ -627,18 +630,18 @@ 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.

@@ -650,14 +653,14 @@ And we have: \end{align*} -

+

newport_doc.png

Figure 4: Documentation of the Newport

-
+

newport_gain.png

Figure 5: Transfer function of the Newport

@@ -665,25 +668,25 @@ 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

@@ -691,8 +694,8 @@ 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. @@ -712,14 +715,14 @@ 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: Sercalo Parameters
@@ -816,8 +819,8 @@ 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. @@ -832,15 +835,15 @@ 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

-
uh = load('mat/data_cal_pd_h.mat', 't', 'Vph', 'Vpv', 'Vnh');
-uv = load('mat/data_cal_pd_v.mat', 't', 'Vph', 'Vpv', 'Vnv');
+
uh = load('mat/data_cal_pd_h.mat', 't', 'Vph', 'Vpv', 'Vnh');
+uv = load('mat/data_cal_pd_v.mat', 't', 'Vph', 'Vpv', 'Vnv');
@@ -867,7 +870,7 @@ uv.t = uv.t - uv.t +

calib_4qd_h.png

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

@@ -875,7 +878,7 @@ uv.t = uv.t - uv.t +

calib_4qd_v.png

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

@@ -883,8 +886,8 @@ 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 @@ -914,7 +917,7 @@ where:

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

@@ -924,17 +927,17 @@ 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.

-
+
@@ -978,11 +981,11 @@ We obtain: -
-

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

+
+

2.2 Identification of the Sercalo 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.

@@ -990,15 +993,15 @@ 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 Sercalo is shown in Fig. 11. +The schematic of the electrical circuit used to drive the Sercalo is shown in Fig. 11.

-
+

sercalo_amplifier.png

Figure 11: Current Amplifier Schematic

@@ -1085,8 +1088,8 @@ 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. @@ -1116,7 +1119,7 @@ Ga = blkdiag( +

current_amplifier_tf.png

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

@@ -1138,8 +1141,8 @@ 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 \([V_{c,h}, V_{c,v}]\). @@ -1150,8 +1153,8 @@ From that, we should be able to identify \(G_a Z_c G_i\). The identification data is loaded.

-
uh = load('mat/data_uch.mat', 't', 'Uch', 'Vch');
-uv = load('mat/data_ucv.mat', 't', 'Ucv', 'Vcv');
+
uh = load('mat/data_uch.mat', 't', 'Uch', 'Vch');
+uv = load('mat/data_ucv.mat', 't', 'Ucv', 'Vcv');
@@ -1166,7 +1169,7 @@ We remove the first seconds where the Sercalo is turned on.
-
+

current_amplifier_comp_theory_id.png

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

@@ -1184,7 +1187,7 @@ Gi = tf(blkdiag +

current_amplifier_comp_theory_id_bis.png

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

@@ -1262,11 +1265,11 @@ Continuous-time zero/pole/gain model.
-
-

2.3 Identification of the Sercalo Dynamics

+
+

2.3 Identification of the Sercalo Dynamics

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

@@ -1278,15 +1281,15 @@ 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

-
uh = load('mat/data_uch.mat', 't', 'Uch', 'Vph', 'Vpv');
-uv = load('mat/data_ucv.mat', 't', 'Ucv', 'Vph', 'Vpv');
+
uh = load('mat/data_uch.mat', 't', 'Uch', 'Vph', 'Vpv');
+uv = load('mat/data_ucv.mat', 't', 'Ucv', 'Vph', 'Vpv');
@@ -1314,7 +1317,7 @@ uv.t = uv.t - uv.t +

identification_uh.png

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

@@ -1322,7 +1325,7 @@ uv.t = uv.t - uv.t +

identification_uv.png

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

@@ -1330,8 +1333,8 @@ 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. @@ -1350,7 +1353,7 @@ The window used for the spectral analysis is an hanning windows wit

-
+

coh_sercalo.png

Figure 17: Coherence (png, pdf)

@@ -1358,8 +1361,8 @@ 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. @@ -1373,14 +1376,14 @@ We compute an estimate of the transfer functions.

-
+

frf_sercalo_gain.png

Figure 18: Frequency Response Matrix (png, pdf)

-
+

frf_sercalo_phase.png

Figure 19: Frequency Response MatrixPhase (png, pdf)

@@ -1388,8 +1391,8 @@ 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. @@ -1420,8 +1423,8 @@ 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. @@ -1449,10 +1452,10 @@ Also, we want more importance where we have a better coherence. Finally, we ignore data above some frequency.

-
weight_Uch_Vph = coh_Uch_Vph';
-weight_Uch_Vpv = coh_Uch_Vpv';
-weight_Ucv_Vph = coh_Ucv_Vph';
-weight_Ucv_Vpv = coh_Ucv_Vpv';
+
weight_Uch_Vph = coh_Uch_Vph';
+weight_Uch_Vpv = coh_Uch_Vpv';
+weight_Ucv_Vph = coh_Ucv_Vph';
+weight_Ucv_Vpv = coh_Ucv_Vpv';
 
 alpha = 0.1;
 
@@ -1471,11 +1474,11 @@ weight_Ucv_Vpv(f
 
 

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

 
 
-

+

 
weights_sercalo.png
 

 
Figure 20: Weights amplitude (png, pdf)

@@ -1527,7 +1530,7 @@ An we run the vectfit3 algorithm.
 

 
 
-

+

 
identification_matrix_fit.png
 

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

@@ -1535,7 +1538,7 @@ An we run the vectfit3 algorithm.
 
 
 
-

+

 
identification_matrix_fit_phase.png
 

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

@@ -1558,8 +1561,8 @@ 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 \([V_{p,h},\ V_{p,v}]\).
@@ -1570,15 +1573,15 @@ 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
 

 

-
uh = load('mat/data_unh.mat', 't', 'Unh', 'Vph', 'Vpv');
-uv = load('mat/data_unv.mat', 't', 'Unv', 'Vph', 'Vpv');
+
uh = load('mat/data_unh.mat', 't', 'Unh', 'Vph', 'Vpv');
+uv = load('mat/data_unv.mat', 't', 'Unv', 'Vph', 'Vpv');
 

 

 
@@ -1606,14 +1609,14 @@ 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)

@@ -1621,8 +1624,8 @@ 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.
@@ -1641,7 +1644,7 @@ The window used for the spectral analysis is an hanning windows wit
 

 
 
-

+

 
id_newport_coherence.png
 

 
Figure 25: Coherence (png, pdf)

@@ -1649,8 +1652,8 @@ 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.
@@ -1664,14 +1667,14 @@ 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)

@@ -1679,8 +1682,8 @@ 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.
@@ -1702,7 +1705,7 @@ 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)

@@ -1710,11 +1713,11 @@ 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.
 

 
 

@@ -1725,8 +1728,8 @@ From Fig. 26, it seems reasonable to model the Newport
 

 

 
-

-
2.5 Full System

+

+
2.5 Full System

 

 

 We now have identified:
@@ -1743,23 +1746,23 @@ We now have identified:
 We name the input and output of each transfer function:
 

 

-
Gi.InputName = {'Uch', 'Ucv'};
-Gi.OutputName = {'Ich', 'Icv'};
+
Gi.InputName = {'Uch', 'Ucv'};
+Gi.OutputName = {'Ich', 'Icv'};
 
-Zc.InputName = {'Ich', 'Icv'};
-Zc.OutputName = {'Vtch', 'Vtcv'};
+Zc.InputName = {'Ich', 'Icv'};
+Zc.OutputName = {'Vtch', 'Vtcv'};
 
-Ga.InputName = {'Vtch', 'Vtcv'};
-Ga.OutputName = {'Vch', 'Vcv'};
+Ga.InputName = {'Vtch', 'Vtcv'};
+Ga.OutputName = {'Vch', 'Vcv'};
 
-Gc.InputName = {'Ich', 'Icv'};
-Gc.OutputName = {'Vpch', 'Vpcv'};
+Gc.InputName = {'Ich', 'Icv'};
+Gc.OutputName = {'Vpch', 'Vpcv'};
 
-Gn.InputName = {'Unh', 'Unv'};
-Gn.OutputName = {'Vpnh', 'Vpnv'};
+Gn.InputName = {'Unh', 'Unv'};
+Gn.OutputName = {'Vpnh', 'Vpnv'};
 
-Gd.InputName = {'Rh', 'Rv'};
-Gd.OutputName = {'Vph', 'Vpv'};
+Gd.InputName = {'Rh', 'Rv'};
+Gd.OutputName = {'Vph', 'Vpv'};
 

 

 
@@ -1770,8 +1773,8 @@ Sv = sumblk(
-
inputs  = {'Uch', 'Ucv', 'Unh', 'Unv'};
-outputs = {'Vch', 'Vcv', 'Ich', 'Icv', 'Rh', 'Rv', 'Vph', 'Vpv'};
+
inputs  = {'Uch', 'Ucv', 'Unh', 'Unv'};
+outputs = {'Vch', 'Vcv', 'Ich', 'Icv', 'Rh', 'Rv', 'Vph', 'Vpv'};
 
 sys = connect(Gi, Zc, Ga, Gc, Gn, inv(Gd), Sh, Sv, inputs, outputs);
 

@@ -1781,18 +1784,18 @@ sys = connect(Gi, Zc, Ga, Gc
 The file mat/plant.mat is accessible here.
 

 

-
save('mat/plant.mat', 'sys', 'Gi', 'Zc', 'Ga', 'Gc', 'Gn', 'Gd');
+
save('mat/plant.mat', 'sys', 'Gi', 'Zc', 'Ga', 'Gc', 'Gn', 'Gd');
 

 

 

 

 

 
-

-
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.
@@ -1802,7 +1805,7 @@ 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
+
@@ -1860,14 +1863,14 @@ 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');
-ht_2 = load('./mat/data_huddle_test_2.mat', 't', 'Vph', 'Vpv', 'Va');
-ht_3 = load('./mat/data_huddle_test_3.mat', 't', 'Uch', 'Ucv', 'Vph', 'Vpv', 'Va');
-ht_4 = load('./mat/data_huddle_test_4.mat', 't', 'Vph', 'Vpv', 'Va');
+
ht_1 = load('./mat/data_huddle_test_1.mat', 't', 'Vph', 'Vpv', 'Va');
+ht_2 = load('./mat/data_huddle_test_2.mat', 't', 'Vph', 'Vpv', 'Va');
+ht_3 = load('./mat/data_huddle_test_3.mat', 't', 'Uch', 'Ucv', 'Vph', 'Vpv', 'Va');
+ht_4 = load('./mat/data_huddle_test_4.mat', 't', 'Vph', 'Vpv', 'Va');
 % ht_5 = load('./mat/data_huddle_test_5.mat', 't', 'Uch', 'Ucv', 'Vph', 'Vpv', 'Va');
@@ -1879,8 +1882,8 @@ ht_4 = load( -

3.2 Pre-processing

+
+

3.2 Pre-processing

t0 = 1; % [s]
@@ -1889,7 +1892,7 @@ tend = 100; {ht_1 ht_2 ht_3 ht_4}
 
-for i = 1:length(ht_s)
+for i = 1:length(ht_s)
   ht_s{i}.Vph(ht_s{i}.t<t0) = [];
   ht_s{i}.Vpv(ht_s{i}.t<t0) = [];
   ht_s{i}.Va(ht_s{i}.t<t0)  = [];
@@ -1914,18 +1917,39 @@ ht_4 = ht_s{
-
3.3 Time domain plots

+

+
3.3 Filter data with low pass filter

 

+

+We filter the data with a first order low pass filter with a crossover frequency of \(\omega_0\).
+

 
-

+

+
w0 = 50; % [Hz]
+
+G_lpf = 1/(1 + s/2/pi/w0);
+
+ht_1.Vaf = lsim(G_lpf, ht_1.Va, ht_1.t);
+ht_2.Vaf = lsim(G_lpf, ht_2.Va, ht_2.t);
+ht_3.Vaf = lsim(G_lpf, ht_3.Va, ht_3.t);
+ht_4.Vaf = lsim(G_lpf, ht_4.Va, ht_4.t);
+

+

+

+

+
+

+
3.4 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)

@@ -1933,9 +1957,9 @@ ht_4 = ht_s{
-
3.4 Power Spectral Density

-

+

+
3.5 Power Spectral Density

+

 

 
win = hanning(ceil(1*fs));
 

@@ -1966,7 +1990,7 @@ ht_4 = ht_s{
+

 
huddle_test_psd_va.png
 

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

@@ -1974,7 +1998,7 @@ ht_4 = ht_s{
+

 
huddle_test_4qd_psd.png
 

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

@@ -1983,9 +2007,9 @@ ht_4 = ht_s{
-
3.5 Conclusion

-

+

+
3.6 Conclusion

+

 

 The Attocube's "Environmental Compensation Unit" does not have a significant effect on the stability of the measurement.
 

@@ -1993,11 +2017,11 @@ The Attocube's "Environmental Compensation Unit" does not have a significant eff
 

 

 
-

-
4 Budget Error

+

+
4 Budget Error

 

 

-
+
 

 

 Goals:
@@ -2033,27 +2057,27 @@ 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 Sercalo mirror on the Attocube measurement
    • 
-
  • in section 4.2:
+
  • in section 4.2:
 The effect of perpendicular motion of the Newport and Sercalo 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 Sercalo is studied.
 Sensor noise, actuator noise and their effects on the control error is discussed.
    • 
 

 

-

-
4.1 Effect of the Sercalo angle error on the measured distance by the Attocube

+

+
4.1 Effect of the Sercalo 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
    • 
@@ -2080,7 +2104,7 @@ L = 0.
+
    
 
    angle_error_schematic_sercalo.png
 
    
 
    Figure 33: Schematic of the geometry used to evaluate the effect of \(\delta \theta_c\) on the measured distance \(\delta L\)
    
@@ -2110,7 +2134,7 @@ We now compute the new path length when there is an error angle \(\delta \theta_
 
    
 
    path_length = zeros(size(thetas_c));
 
-for i = 1:length(thetas_c)
+for i = 1:length(thetas_c)
     theta_c = thetas_c(i);
     Y = [H*tan(2*theta_c), H];
     M = 2*H/(tan(pi/4-theta_c)+1/tan(2*theta_c))*[1, tan(pi/4-theta_c)];
@@ -2122,7 +2146,7 @@ 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 Sercalo angle error (Fig. 34).
+We then compute the distance error and we plot it as a function of the Sercalo angle error (Fig. 34).
 
    
 
    
 
    path_error = path_length - path_nominal;
@@ -2130,14 +2154,14 @@ We then compute the distance error and we plot it as a function of the Sercalo a
 
    
 
 
-
    
+
    
 
    effect_sercalo_angle_distance_meas.png
 
    
 
    Figure 34: Effect of an angle error of the Sercalo 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]
@@ -2154,7 +2178,7 @@ T = [
 
    
 
 
-
    
+
    
 
    simulation_beam_path_high_angle.png
 
    
 
    Figure 35: Simulation of a beam path for high angle error (png, pdf)
    
@@ -2162,7 +2186,7 @@ T = [
 
 
    
 
    
-Based on Fig. 34, we see that an angle error \(\delta\theta_c\) of the Sercalo 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 Sercalo mirror induces a distance error \(\delta L\) measured by the Attocube which is dependent of the square of \(\delta \theta_c\):
 
    
 
 \begin{equation}
@@ -2177,7 +2201,7 @@ with:
 

 
 

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

 
 

@@ -2186,7 +2210,7 @@ The tracking error of the feedback system used to position the Sercalo mirror sh
 
 

 
-
+
@@ -2220,15 +2244,15 @@ The tracking error of the feedback system used to position the Sercalo mirror sh
 
 
-

-
4.2 Unwanted motion of Sercalo/Newport mirrors perpendicular to its surface

+

+
4.2 Unwanted motion of Sercalo/Newport mirrors perpendicular to its surface

 

 

-
+
 

 
 

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

 
 

@@ -2252,14 +2276,14 @@ The error in measured distance by the Attocube will we \(\delta L/2\).
 

 
 
-

+

 
sercalo_perpendicular_motion.png
 

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

 

 
 
-

+

 
newport_perpendicular_motion.png
 

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

@@ -2279,11 +2303,11 @@ 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

 

 

-
+
 

 
 

@@ -2381,8 +2405,8 @@ 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.
@@ -2402,18 +2426,18 @@ Thus, the temperature of the metrology frame should be kept constant to less tha
 

 

 
-

-
4.5 Estimation of the Sercalo angle error due to Noise

+

+
4.5 Estimation of the Sercalo 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 Sercalo to the Beam angle
    • 
@@ -2444,22 +2468,22 @@ 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:
@@ -2494,8 +2518,8 @@ For the ADC used:
 

 

 
-

-
4.5.1.2 DAC Quantization Noise

+

+
4.5.1.2 DAC Quantization Noise

 

 

 The DAC quantization noise is:
@@ -2530,8 +2554,8 @@ 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.
@@ -2581,8 +2605,8 @@ 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:
@@ -2593,7 +2617,7 @@ 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

@@ -2608,8 +2632,8 @@ 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 Sercalo angle error will be equal to the 4 quadrant diode noise.
@@ -2642,11 +2666,11 @@ If we just consider the ADC noise:
 

 

 
-

-
4.5.3 Error due to DAC noise used for the Sercalo

+

+
4.5.3 Error due to DAC noise used for the Sercalo

 

 

-
load('./mat/plant.mat', 'Gi', 'Gc', 'Gd');
+
load('./mat/plant.mat', 'Gi', 'Gc', 'Gd');
 

 

 
@@ -2753,15 +2777,15 @@ The DAC noise use for the Sercalo does not limit the performance of the system.
 

 

 
-

-
5 Plant Uncertainty

+

+
5 Plant Uncertainty

 

 

-

-
5.1 Coprime Factorization

+

+
5.1 Coprime Factorization

 

 

-
load('mat/plant.mat', 'sys', 'Gi', 'Zc', 'Ga', 'Gc', 'Gn', 'Gd');
+
load('mat/plant.mat', 'sys', 'Gi', 'Zc', 'Ga', 'Gc', 'Gn', 'Gd');
 

 

 
@@ -2773,18 +2797,18 @@ The DAC noise use for the Sercalo does not limit the performance of the system.
 

 

 
-

-
6 Plant Scaling

+

+
6 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 Sercalo's mirror on the measurement error \(\delta L\) by the Attocube
 
 

 

+
@@ -2835,8 +2859,8 @@ This will simplify the choice of weighting functions and will yield useful insig
 

 Table 5: Maximum wanted values for various signals
 
 

 

 

-

-
6.1 Control Objective

+

+
6.1 Control Objective

 

 

 The maximum expected stroke is \(y_\text{max} = 3mm \approx 5e^{-2} rad\) at \(1Hz\).
@@ -2858,15 +2882,15 @@ In terms of loop gain, this is equivalent to:
 

 

 
-

-
6.2 General Configuration

+

+
6.2 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

@@ -2875,25 +2899,25 @@ The plant is put in a general configuration as shown in Fig. 
-
7 Plant Analysis

+

+
7 Plant Analysis

 

 

-
+
 

 

-

-
7.1 Load Plant

+

+
7.1 Load Plant

 

 

-
load('mat/plant.mat', 'G');
+
load('mat/plant.mat', 'G');
 

 

 

 

 
-

-
7.2 RGA-Number

+

+
7.2 RGA-Number

 

 

 
freqs = logspace(2, 4, 1000);
@@ -2901,8 +2925,8 @@ G_resp = freqresp(G, freqs,
 A = zeros(size(G_resp));
 RGAnum = zeros(1, length(freqs));
 
-for i = 1:length(freqs)
-  A(:, :, i) = G_resp(:, :, i).*inv(G_resp(:, :, i))';
+for i = 1:length(freqs)
+  A(:, :, i) = G_resp(:, :, i).*inv(G_resp(:, :, i))';
   RGAnum(i) = sum(sum(abs(A(:, :, i)-eye(2))));
 end
 % RGA = G0.*inv(G0)';
@@ -2912,7 +2936,7 @@ RGAnum = zeros(
 
figure;
 plot(freqs, RGAnum);
-set(gca, 'xscale', 'log');
+set(gca, 'xscale', 'log');
 

 

 
@@ -2921,7 +2945,7 @@ plot(freqs, RGAnum(2, 2, length(freqs))
 V = zeros(2, 2, length(freqs));
 
-for i = 1:length(freqs)
+for i = 1:length(freqs)
   [Ui, Si, Vi] = svd(G_resp(:, :, i));
   U(:, :, i) = Ui;
   S(:, :, i) = Si;
@@ -2932,8 +2956,8 @@ V = zeros(
-
7.3 Rotation Matrix

+

+
7.3 Rotation Matrix

 

 

 
G0 = freqresp(G, 0);
@@ -2943,53 +2967,53 @@ V = zeros(
-
8 Active Damping

+

+
8 Active Damping

 

 

-
+
 

 

-

-
8.1 Load Plant

+

+
8.1 Load Plant

 

 

-
load('mat/plant.mat', 'sys', 'Gi', 'Zc', 'Ga', 'Gc', 'Gn', 'Gd');
+
load('mat/plant.mat', 'sys', 'Gi', 'Zc', 'Ga', 'Gc', 'Gn', 'Gd');
 

 

 

 

 
-

-
8.2 Integral Force Feedback

+

+
8.2 Integral Force Feedback

 

 

-
bode(sys({'Vch', 'Vcv'}, {'Uch', 'Ucv'}));
+
bode(sys({'Vch', 'Vcv'}, {'Uch', 'Ucv'}));
 

 

 
 

 
Kppf = blkdiag(-10000/s, tf(0));
 
-Kppf.InputName  = {'Vch', 'Vcv'};
-Kppf.OutputName = {'Uch', 'Ucv'};
+Kppf.InputName  = {'Vch', 'Vcv'};
+Kppf.OutputName = {'Uch', 'Ucv'};
 

 

 
 

-
inputs  = {'Uch', 'Ucv', 'Unh', 'Unv'};
-outputs = {'Ich', 'Icv', 'Rh', 'Rv', 'Vph', 'Vpv'};
+
inputs  = {'Uch', 'Ucv', 'Unh', 'Unv'};
+outputs = {'Ich', 'Icv', 'Rh', 'Rv', 'Vph', 'Vpv'};
 
 sys_cl = connect(sys, Kppf, inputs, outputs);
 
-figure; bode(sys_cl({'Vph', 'Vpv'}, {'Uch', 'Ucv'}), sys({'Vph', 'Vpv'}, {'Uch', 'Ucv'}))
+figure; bode(sys_cl({'Vph', 'Vpv'}, {'Uch', 'Ucv'}), sys({'Vph', 'Vpv'}, {'Uch', 'Ucv'}))
 

 

 

 

 
-

-
8.3 Conclusion

+

+
8.3 Conclusion

 

 

 Active damping does not seems to be applicable here.
@@ -2998,11 +3022,11 @@ Active damping does not seems to be applicable here.
 

 

 
-

-
9 Decentralized Control of the Sercalo

+

+
9 Decentralized Control of the Sercalo

 

 

-
+
 

 

 In this section, we try to implement a simple decentralized controller.
@@ -3014,53 +3038,53 @@ All the files (data and Matlab scripts) are accessible 
-
9.1 Load Plant

+

+
9.1 Load Plant

 

 

-
load('mat/plant.mat', 'sys', 'Gi', 'Zc', 'Ga', 'Gc', 'Gn', 'Gd');
+
load('mat/plant.mat', 'sys', 'Gi', 'Zc', 'Ga', 'Gc', 'Gn', 'Gd');
 

 

 

 

 
-

-
9.2 Diagonal Controller

+

+
9.2 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)));
 Kv = 10207*(s+55.15)*(s^2 + 17.45*s + 2.491e06)/(s^2*(s+491.2)*(s+7695));
 
 K = blkdiag(Kh, Kv);
-K.InputName  = {'Rh', 'Rv'};
-K.OutputName = {'Uch', 'Ucv'};
+K.InputName  = {'Rh', 'Rv'};
+K.OutputName = {'Uch', 'Ucv'};
 

 

 
 
-

+

 
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'};
-outputs = {'Vch', 'Vcv', 'Ich', 'Icv', 'Rh', 'Rv', 'Vph', 'Vpv'};
+
inputs  = {'Uch', 'Ucv', 'Unh', 'Unv'};
+outputs = {'Vch', 'Vcv', 'Ich', 'Icv', 'Rh', 'Rv', 'Vph', 'Vpv'};
 
 sys_cl = connect(sys, -K, inputs, outputs);
 

 

 
 
-

+

 
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)

@@ -3068,8 +3092,8 @@ sys_cl = connect(sys, 
 

 
-

-
9.3 Save the Controller

+

+
9.3 Save the Controller

 

 

 
Kd = c2d(K, 1e-4, 'tustin');
@@ -3080,18 +3104,18 @@ sys_cl = connect(sys, here.
 

 

-
save('mat/K_diag.mat', 'K', 'Kd');
+
save('mat/K_diag.mat', 'K', 'Kd');
 

 

 

 

 

 
-

-
10 Newport Control

+

+
10 Newport Control

 

 

-
+
 

 

 In this section, we try to implement a simple decentralized controller for the Newport.
@@ -3104,18 +3128,18 @@ This can be used to align the 4QD:
 
  • finally, we are sure to be aligned when the command signal of the Newport is 0
    • 
 
 
    
-
    
-
    10.1 Load Plant
    
+
    
+
    10.1 Load Plant
    
 
    
 
    
-
    load('mat/plant.mat', 'Gn', 'Gd');
+
    load('mat/plant.mat', 'Gn', 'Gd');
 
    
 
    
 
    
 
    
 
-
    
-
    10.2 Analysis
    
+
    
+
    10.2 Analysis
    
 
    
 
    
 The plant is basically a constant until frequencies up to the required bandwidth.
@@ -3142,7 +3166,7 @@ Knv = 1
 
    
 
 
-
    
+
    
 
    loop_gain_newport.png
 
    
 
    Figure 43: Diagonal Loop Gain for the Newport (png, pdf)
    
@@ -3150,8 +3174,8 @@ Knv = 1
 
    
 
    
 
-
    
-
    10.3 Save
    
+
    
+
    10.3 Save
    
 
    
 
    
 
    Kn = blkdiag(Knh, Knv);
@@ -3165,43 +3189,43 @@ The controllers can be downloaded here.
 
    
 
 
    
-
    save('mat/K_newport.mat', 'Kn', 'Knd');
+
    save('mat/K_newport.mat', 'Kn', 'Knd');
 
    
 
    
 
    
 
    
 
    
 
-
    
-
    11 Measurement of the non-repeatability
    
+
    
+
    11 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.
+All sources of error (detailed in the budget error in Section 4) will contribute to the non-repeatability of the system.
 
    
 
    
-
    
-
    11.1 Data Load and pre-processing
    
+
    
+
    11.1 Data Load and pre-processing
    
 
    
 
    
-
    uh = load('mat/data_rep_h.mat', ...
-     't', 'Uch', 'Ucv', ...
-     'Unh', 'Unv', ...
-     'Vph', 'Vpv', ...
-     'Vnh', 'Vnv', ...
+
    uh = load('mat/data_rep_h.mat', ...
+     't', 'Uch', 'Ucv', ...
+     'Unh', 'Unv', ...
+     'Vph', 'Vpv', ...
+     'Vnh', 'Vnv', ...
      'Va');
 
-uv = load('mat/data_rep_v.mat', ...
-     't', 'Uch', 'Ucv', ...
-     'Unh', 'Unv', ...
-     'Vph', 'Vpv', ...
-     'Vnh', 'Vnv', ...
+uv = load('mat/data_rep_v.mat', ...
+     't', 'Uch', 'Ucv', ...
+     'Unh', 'Unv', ...
+     'Vph', 'Vpv', ...
+     'Vnh', 'Vnv', ...
      'Va');
 
    
 
    
@@ -3250,11 +3274,11 @@ uv.t = uv.t - uv.t
-
    11.2 Some Time domain plots
    
+
    
+
    11.2 Some Time domain plots
    
 
    
 
-
    
+
    
 
    repeat_time_signals.png
 
    
 
    Figure 44: Time domain Signals for the repeatability measurement (png, pdf)
    
@@ -3262,20 +3286,20 @@ uv.t = uv.t - uv.t
-
    11.3 Verify Tracking Angle Error
    
+
    
+
    11.3 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.
 
    
 
 
    
-
    load('./mat/plant.mat', 'Gd');
+
    load('./mat/plant.mat', 'Gd');
 
    
 
    
 
 
-
    
+
    
 
    repeat_tracking_errors.png
 
    
 
    Figure 45: Tracking errors during the repeatability measurement (png, pdf)
    
@@ -3294,7 +3318,7 @@ 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)
    
@@ -3329,7 +3353,7 @@ with
 
    
 
 
    
-Now, compare that with the PSD of the measured distance by the interferometer (Fig. 47).
+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);
@@ -3338,7 +3362,7 @@ Now, compare that with the PSD of the measured distance by the interferometer (F
 
    
 
 
-
    
+
    
 
    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)
    
@@ -3353,8 +3377,8 @@ The tracking errors are a limiting factor.
 
    
 
    
 
-
    
-
    11.4 Processing
    
+
    
+
    11.4 Processing
    
 
    
 
    
 First, we get the mean value as measured by the interferometer for each value of the Newport angle.
@@ -3394,15 +3418,14 @@ And we can compute the RMS value of the non-repeatable part:
 
 
 
-
    
+
    
 
    repeat_plot_raw.png
 
    
 
    Figure 48: Repeatability of the measurement (png, pdf)
    
 
    
 
 
-
-
    
+
    
 
    repeat_plot_subtract_mean.png
 
    
 
    Figure 49: Repeatability of the measurement after subtracting the mean value (png, pdf)
    
@@ -3410,8 +3433,8 @@ And we can compute the RMS value of the non-repeatable part:
 
    
 
    
 
-
    
-
    11.5 Analysis of the non-repeatable contributions
    
+
    
+
    11.5 Analysis of the non-repeatable contributions
    
 
    
 
    
 Let's know try to determine where does the non-repeatability comes from.
@@ -3430,22 +3453,96 @@ We also plot the displacement measured during the huddle test.
 
    
 
 
    
-All the signals are shown on Fig. 50.
+All the signals are shown on Fig. 50.
 
    
 
 
-
    
+
    
 
    non-repeatability-parts.png
 
    
 
    Figure 50: Non repeatabilities (png, pdf)
    
 
    
 
    
 
    
+
    
+
    11.6 Results with a low pass filter
    
+
    
+
    
+We filter the data with a first order low pass filter with a crossover frequency of \(\omega_0\).
+
    
+
+
    
+
    w0 = 10; % [Hz]
+
+G_lpf = 1/(1 + s/2/pi/w0);
+
+uh.Vaf = lsim(G_lpf, uh.Va, uh.t);
+uv.Vaf = lsim(G_lpf, uv.Va, uv.t);
+
    
+
    
+
    
+
    
+
+
    
+
    11.7 Processing
    
+
    
+
    
+First, we get the mean value as measured by the interferometer for each value of the Newport angle.
+
    
+
    
+
    Vahm = mean(reshape(uh.Vaf, [fs floor(length(uh.t)/fs)]),2);
+Unhm = mean(reshape(uh.Unh, [fs floor(length(uh.t)/fs)]),2);
+
+Vavm = mean(reshape(uv.Vaf, [fs floor(length(uv.t)/fs)]),2);
+Unvm = mean(reshape(uv.Unv, [fs floor(length(uv.t)/fs)]),2);
+
    
+
    
+
+
    
+And we can compute the RMS value of the non-repeatable part:
+
    
+
+
+
+++
++
+
+
+
+
+
+
+
+
+
+
+
+
+
    



    Va - Horizontal [nm rms]Va - Vertical [nm rms]
    22.913.9
    
+
+
+
    
+
    repeat_plot_lpf.png
+
    
+
    Figure 51: Repeatability of the measurement (png, pdf)
    
+
    
+
+
+
    
+
    repeat_plot_subtract_mean_lpf.png
+
    
+
    Figure 52: Repeatability of the measurement after subtracting the mean value (png, pdf)
    
+
    
+
    
+
    
 
    
 
    
 
    
 
    Author: Dehaeze Thomas
    
-
    Created: 2019-10-08 mar. 16:18
    
+
    Created: 2019-10-15 mar. 11:01
    
 
    Validate
    
 
    
 
diff --git a/index.org b/index.org
index 556f5ef..123b8f7 100644
--- a/index.org
+++ b/index.org
@@ -18,7 +18,7 @@
 #+LATEX_CLASS_OPTIONS: []
 #+LATEX_HEADER: \usepackage{minted}
 
-#+PROPERTY: header-args:latex  :headers '("\\usepackage{tikz}" "\\usepackage{import}" "\\import{/home/thomas/Cloud/These/LaTeX/}{config.tex}")
+#+PROPERTY: header-args:latex  :headers '("\\usepackage{tikz}" "\\usepackage{import}" "\\import{/home/thomas/Cloud/thesis/latex/}{config.tex}")
 #+PROPERTY: header-args:latex+ :imagemagick t :fit yes
 #+PROPERTY: header-args:latex+ :iminoptions -scale 100% -density 150
 #+PROPERTY: header-args:latex+ :imoutoptions -quality 100
@@ -1868,36 +1868,54 @@ Multiple measurements are done with different experimental configuration as foll
   ht_4 = ht_s{4};
 #+end_src
 
+** Filter data with low pass filter
+We filter the data with a first order low pass filter with a crossover frequency of $\omega_0$.
+
+#+begin_src matlab
+  w0 = 50; % [Hz]
+
+  G_lpf = 1/(1 + s/2/pi/w0);
+
+  ht_1.Vaf = lsim(G_lpf, ht_1.Va, ht_1.t);
+  ht_2.Vaf = lsim(G_lpf, ht_2.Va, ht_2.t);
+  ht_3.Vaf = lsim(G_lpf, ht_3.Va, ht_3.t);
+  ht_4.Vaf = lsim(G_lpf, ht_4.Va, ht_4.t);
+#+end_src
+
 ** Time domain plots
 #+begin_src matlab :exports none
   figure;
-  ax1 = subaxis(2, 2, 1)
+  ax1 = subplot(2, 2, 1);
   hold on;
   plot(ht_1.t, 1e9*ht_1.Va);
+  plot(ht_1.t, 1e9*ht_1.Vaf);
   hold off;
   ylabel('Displacement [nm]');
   set(gca, 'XTickLabel',[]);
   title('OL');
 
-  ax2 = subaxis(2, 2, 2)
+  ax2 = subplot(2, 2, 2);
   hold on;
   plot(ht_2.t, 1e9*ht_2.Va);
+  plot(ht_2.t, 1e9*ht_2.Vaf);
   hold off;
   set(gca, 'XTickLabel',[]);
   set(gca, 'YTickLabel',[]);
   title('OL + CU');
 
-  ax3 = subaxis(2, 2, 3)
+  ax3 = subplot(2, 2, 3);
   hold on;
   plot(ht_3.t, 1e9*ht_3.Va);
+  plot(ht_3.t, 1e9*ht_3.Vaf);
   hold off;
   xlabel('Time [s]');
   ylabel('Displacement [nm]');
   title('CL + CU');
 
-  ax4 = subaxis(2, 2, 4)
+  ax4 = subplot(2, 2, 4);
   hold on;
   plot(ht_4.t, 1e9*ht_4.Va);
+  plot(ht_4.t, 1e9*ht_4.Vaf);
   hold off;
   xlabel('Time [s]');
   set(gca, 'YTickLabel',[]);
@@ -1917,7 +1935,7 @@ Multiple measurements are done with different experimental configuration as foll
 
 #+begin_src matlab :exports none
   figure;
-  ax1 = subaxis(2, 2, 1)
+  ax1 = subplot(2, 2, 1)
   hold on;
   plot(ht_1.t, ht_1.Vph);
   plot(ht_1.t, ht_1.Vpv);
@@ -1926,7 +1944,7 @@ Multiple measurements are done with different experimental configuration as foll
   set(gca, 'XTickLabel',[]);
   title('OL');
 
-  ax2 = subaxis(2, 2, 2)
+  ax2 = subplot(2, 2, 2)
   hold on;
   plot(ht_2.t, ht_2.Vph);
   plot(ht_2.t, ht_2.Vpv);
@@ -1935,7 +1953,7 @@ Multiple measurements are done with different experimental configuration as foll
   set(gca, 'YTickLabel',[]);
   title('OL + CU');
 
-  ax3 = subaxis(2, 2, 3)
+  ax3 = subplot(2, 2, 3)
   hold on;
   plot(ht_3.t, ht_3.Vph);
   plot(ht_3.t, ht_3.Vpv);
@@ -1944,7 +1962,7 @@ Multiple measurements are done with different experimental configuration as foll
   ylabel('Voltage [V]');
   title('CL + CU');
 
-  ax4 = subaxis(2, 2, 4)
+  ax4 = subplot(2, 2, 4)
   hold on;
   plot(ht_4.t, ht_4.Vph);
   plot(ht_4.t, ht_4.Vpv);
@@ -2784,7 +2802,7 @@ The plant is put in a general configuration as shown in Fig. [[fig:general_contr
 #+begin_src matlab :exports none
   figure;
   % Magnitude
-  ax1 = subaxis(2,1,1);
+  ax1 = subplot(2,1,1);
   hold on;
   plot(freqs, abs(squeeze(freqresp(G, freqs, 'Hz'))), 'k-');
   set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
@@ -2793,7 +2811,7 @@ The plant is put in a general configuration as shown in Fig. [[fig:general_contr
   hold off;
 
   % Phase
-  ax2 = subaxis(2,1,2);
+  ax2 = subplot(2,1,2);
   hold on;
   plot(freqs, 180/pi*angle(squeeze(freqresp(G, freqs, 'Hz'))), 'k-');
   set(gca,'xscale','log');
@@ -2875,7 +2893,7 @@ The two SISO loop gains are shown in Fig. [[fig:diag_contr_loop_gain]].
 #+begin_src matlab :exports none
   figure;
   % Magnitude
-  ax1 = subaxis(2,1,1);
+  ax1 = subplot(2,1,1);
   hold on;
   plot(freqs, abs(squeeze(freqresp(Kh*sys('Rh', 'Uch'), freqs, 'Hz'))), 'DisplayName', '$L_h = K_h G_{d,h}^{-1} G_{\frac{V_{p,h}}{\tilde{U}_{c,h}}} G_{i,h} $');
   plot(freqs, abs(squeeze(freqresp(Kv*sys('Rv', 'Ucv'), freqs, 'Hz'))), 'DisplayName', '$L_v = K_v G_{d,v}^{-1} G_{\frac{V_{p,v}}{\tilde{U}_{c,v}}} G_{i,v} $');
@@ -2886,7 +2904,7 @@ The two SISO loop gains are shown in Fig. [[fig:diag_contr_loop_gain]].
   legend('location', 'northeast');
 
   % Phase
-  ax2 = subaxis(2,1,2);
+  ax2 = subplot(2,1,2);
   hold on;
   plot(freqs, 180/pi*angle(squeeze(freqresp(Kh*sys('Rh', 'Uch'), freqs, 'Hz'))));
   plot(freqs, 180/pi*angle(squeeze(freqresp(Kv*sys('Rv', 'Ucv'), freqs, 'Hz'))));
@@ -3329,14 +3347,14 @@ And we can compute the RMS value of the non-repeatable part:
   figure;
   ax1 = subplot(1, 2, 1);
   hold on;
-  plot(uh.Unh, uh.Va);
+  plot(uh.Unh, uh.Vaf);
   plot(Unhm, Vahm)
   hold off;
   xlabel('$V_{n,h}$ [V]'); ylabel('$V_a$ [m]');
 
   ax2 = subplot(1, 2, 2);
   hold on;
-  plot(uv.Unv, uv.Va);
+  plot(uv.Unv, uv.Vaf);
   plot(Unvm, Vavm)
   hold off;
   xlabel('$V_{n,v}$ [V]'); ylabel('$V_a$ [m]');
@@ -3353,18 +3371,17 @@ And we can compute the RMS value of the non-repeatable part:
 #+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);
   hold on;
-  plot(uh.Unh, 1e9*(uh.Va - repmat(Vahm, length(uh.t)/length(Vahm),1)));
+  plot(uh.Unh, 1e9*(uh.Vaf - repmat(Vahm, length(uh.t)/length(Vahm),1)));
   hold off;
   xlabel('$V_{n,h}$ [V]'); ylabel('$V_a$ [nm]');
 
   ax2 = subplot(1, 2, 2);
   hold on;
-  plot(uv.Unv, 1e9*(uv.Va - repmat(Vavm, length(uv.t)/length(Vavm),1)));
+  plot(uv.Unv, 1e9*(uv.Vaf - repmat(Vavm, length(uv.t)/length(Vavm),1)));
   hold off;
   xlabel('$V_{n,v}$ [V]'); ylabel('$V_a$ [nm]');
 
@@ -3418,6 +3435,24 @@ All the signals are shown on Fig. [[fig:non-repeatability-parts]].
   htm = 1e9*ht.Va(1:length(Vaheq)) - repmat(mean(1e9*ht.Va(1:length(Vaheq))), length(uh.t)/length(Vaheq),1);
 #+end_src
 
+#+begin_src matlab :exports none
+  figure;
+  hold on;
+  plot(uh.Unh, 1e9*(uh.Va - repmat(Vahm, length(uh.t)/length(Vahm),1)), 'DisplayName', 'Measured Non-Repeatability');
+  plot(uh.Unh, 1e9*ht.Va(1:length(Vaheq))-mean(1e9*ht.Va(1:length(Vaheq))), 'DisplayName', 'Huddle Test');
+  plot(uh.Unh, 1e9*Vaheq, 'DisplayName', 'Due to Sercalo Angle Error');
+  hold off;
+  xlabel('$V_{n,h}$ [V]'); ylabel('$V_a$ [nm]');
+  ylim([-100 100]);
+  legend();
+#+end_src
+
+#+HEADER: :tangle no :exports results :results none :noweb yes
+#+begin_src matlab :var filepath="figs/non-repeatability-parts-half.pdf" :var figsize="normal-tall" :post pdf2svg(file=*this*, ext="png")
+  <>
+#+end_src
+
+
 #+begin_src matlab :exports none
   figure;
   ax1 = subplot(1, 2, 1);
@@ -3449,3 +3484,89 @@ All the signals are shown on Fig. [[fig:non-repeatability-parts]].
 #+NAME: fig:non-repeatability-parts
 #+CAPTION: Non repeatabilities ([[./figs/non-repeatability-parts.png][png]], [[./figs/non-repeatability-parts.pdf][pdf]])
 [[file:figs/non-repeatability-parts.png]]
+** Results with a low pass filter
+We filter the data with a first order low pass filter with a crossover frequency of $\omega_0$.
+
+#+begin_src matlab
+  w0 = 10; % [Hz]
+
+  G_lpf = 1/(1 + s/2/pi/w0);
+
+  uh.Vaf = lsim(G_lpf, uh.Va, uh.t);
+  uv.Vaf = lsim(G_lpf, uv.Va, uv.t);
+#+end_src
+
+** Processing
+First, we get the mean value as measured by the interferometer for each value of the Newport angle.
+#+begin_src matlab
+  Vahm = mean(reshape(uh.Vaf, [fs floor(length(uh.t)/fs)]),2);
+  Unhm = mean(reshape(uh.Unh, [fs floor(length(uh.t)/fs)]),2);
+
+  Vavm = mean(reshape(uv.Vaf, [fs floor(length(uv.t)/fs)]),2);
+  Unvm = mean(reshape(uv.Unv, [fs floor(length(uv.t)/fs)]),2);
+#+end_src
+
+And we can compute the RMS value of the non-repeatable part:
+#+begin_src matlab :exports results :results value table replace :tangle no :post addhdr(*this*)
+  data2orgtable([rms(1e9*(uh.Vaf - repmat(Vahm, length(uh.t)/length(Vahm),1))), rms(1e9*(uv.Vaf - repmat(Vavm, length(uv.t)/length(Vavm),1)))], {}, {'Va - Horizontal [nm rms]', 'Va - Vertical [nm rms]'}, ' %.1f ');
+#+end_src
+
+#+RESULTS:
+| Va - Horizontal [nm rms] | Va - Vertical [nm rms] |
+|--------------------------+------------------------|
+|                     22.9 |                   13.9 |
+
+#+begin_src matlab :exports none
+  figure;
+  ax1 = subplot(1, 2, 1);
+  hold on;
+  plot(uh.Unh, uh.Vaf);
+  plot(Unhm, Vahm)
+  hold off;
+  xlabel('$V_{n,h}$ [V]'); ylabel('$V_a$ [m]');
+
+  ax2 = subplot(1, 2, 2);
+  hold on;
+  plot(uv.Unv, uv.Vaf);
+  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_lpf.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_lpf.png][png]], [[./figs/repeat_plot_lpf.pdf][pdf]])
+[[file:figs/repeat_plot_lpf.png]]
+
+#+begin_src matlab :exports none
+  figure;
+  ax1 = subplot(1, 2, 1);
+  hold on;
+  plot(uh.Unh, 1e9*(uh.Vaf - repmat(Vahm, length(uh.t)/length(Vahm),1)));
+  hold off;
+  xlabel('$V_{n,h}$ [V]'); ylabel('$V_a$ [nm]');
+
+  ax2 = subplot(1, 2, 2);
+  hold on;
+  plot(uv.Unv, 1e9*(uv.Vaf - repmat(Vavm, length(uv.t)/length(Vavm),1)));
+  hold off;
+  xlabel('$V_{n,v}$ [V]'); ylabel('$V_a$ [nm]');
+
+  linkaxes([ax1,ax2],'xy');
+  ylim([-60 60]);
+#+end_src
+
+#+HEADER: :tangle no :exports results :results none :noweb yes
+#+begin_src matlab :var filepath="figs/repeat_plot_subtract_mean_lpf.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png")
+  <>
+#+end_src
+
+#+NAME: fig:repeat_plot_subtract_mean_lpf
+#+CAPTION: Repeatability of the measurement after subtracting the mean value ([[./figs/repeat_plot_subtract_mean_lpf.png][png]], [[./figs/repeat_plot_subtract_mean_lpf.pdf][pdf]])
+[[file:figs/repeat_plot_subtract_mean_lpf.png]]