diff --git a/uniaxial/index.html b/uniaxial/index.html index 4514141..1af0916 100644 --- a/uniaxial/index.html +++ b/uniaxial/index.html @@ -3,7 +3,7 @@ "http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd"> - + Simscape Uniaxial Model @@ -280,72 +280,72 @@ for the JavaScript code in this tag.

Table of Contents

@@ -360,15 +360,15 @@ The idea is to use the same model as the full Simscape Model but to restrict the This is done in order to more easily study the system and evaluate control techniques.

-
-

1 Simscape Model

+
+

1 Simscape Model

- +

-A schematic of the uniaxial model used for simulations is represented in figure 1. +A schematic of the uniaxial model used for simulations is represented in figure 1.

@@ -412,7 +412,7 @@ The control signal \(u\) is: -

+

uniaxial-model-nass-flexible.png

Figure 1: Schematic of the uniaxial model used

@@ -421,11 +421,11 @@ The control signal \(u\) is:

Few active damping techniques will be compared in order to decide which sensor is to be included in the system. -Schematics of the active damping techniques are displayed in figure 2. +Schematics of the active damping techniques are displayed in figure 2.

-
+

uniaxial-model-nass-flexible-active-damping.png

Figure 2: Comparison of used active damping techniques

@@ -433,18 +433,18 @@ Schematics of the active damping techniques are displayed in figure -

2 Undamped System

+
+

2 Undamped System

- +

Let's start by study the undamped system.

-
-

2.1 Init

+
+

2.1 Init

We initialize all the stages with the default parameters. @@ -456,8 +456,8 @@ All the controllers are set to 0 (Open Loop).

-
-

2.2 Identification

+
+

2.2 Identification

We identify the dynamics of the system. @@ -520,19 +520,19 @@ Finally, we save the identified system dynamics for further analysis.

-
-

2.3 Sensitivity to Disturbances

+
+

2.3 Sensitivity to Disturbances

We show several plots representing the sensitivity to disturbances:

    -
  • in figure 3 the transfer functions from ground motion \(D_w\) to the sample position \(D\) and the transfer function from direct force on the sample \(F_s\) to the sample position \(D\) are shown
    • -
  • in figure 4, it is the effect of parasitic forces of the positioning stages (\(F_{ty}\) and \(F_{rz}\)) on the position \(D\) of the sample that are shown
    • +
  • in figure 3 the transfer functions from ground motion \(D_w\) to the sample position \(D\) and the transfer function from direct force on the sample \(F_s\) to the sample position \(D\) are shown
    • +
  • in figure 4, it is the effect of parasitic forces of the positioning stages (\(F_{ty}\) and \(F_{rz}\)) on the position \(D\) of the sample that are shown
-
+

uniaxial-sensitivity-disturbances.png

Figure 3: Sensitivity to disturbances (png, pdf)

@@ -540,7 +540,7 @@ We show several plots representing the sensitivity to disturbances: -
+

uniaxial-sensitivity-force-dist.png

Figure 4: Sensitivity to disturbances (png, pdf)

@@ -548,8 +548,8 @@ We show several plots representing the sensitivity to disturbances:
-
-

2.4 Noise Budget

+
+

2.4 Noise Budget

We first load the measured PSD of the disturbance. @@ -561,12 +561,12 @@ We first load the measured PSD of the disturbance.

The effect of these disturbances on the distance \(D\) is computed below. -The PSD of the obtain distance \(D\) due to each of the perturbation is shown in figure 5 and the Cumulative Amplitude Spectrum is shown in figure 6. +The PSD of the obtain distance \(D\) due to each of the perturbation is shown in figure 5 and the Cumulative Amplitude Spectrum is shown in figure 6.

-The Root Mean Square value of the obtained displacement \(D\) is computed below and can be determined from the figure 6. +The Root Mean Square value of the obtained displacement \(D\) is computed below and can be determined from the figure 6. 

 3.3793e-06
@@ -574,32 +574,32 @@ The Root Mean Square value of the obtained displacement \(D\) is computed below
 
 
 
-

+

 
uniaxial-psd-dist.png
 

-
Figure 5: caption (png, pdf)

+
Figure 5: PSD of the effect of disturbances on \(D\) (png, pdf)

 

 
 
 
-

+

 
uniaxial-cas-dist.png
 

-
Figure 6: caption (png, pdf)

+
Figure 6: CAS of the effect of disturbances on \(D\) (png, pdf)

 

 

 

 
-

-
2.5 Plant

+

+
2.5 Plant

 

 

-The transfer function from the force \(F\) applied by the nano-hexapod to the position of the sample \(D\) is shown in figure 7.
+The transfer function from the force \(F\) applied by the nano-hexapod to the position of the sample \(D\) is shown in figure 7.
 It corresponds to the plant to control.
 

 
 
-

+

 
uniaxial-plant.png
 

 
Figure 7: Bode plot of the Plant (png, pdf)

@@ -608,21 +608,21 @@ It corresponds to the plant to control.
 

 

 
-

-
3 Integral Force Feedback

+

+
3 Integral Force Feedback

 

 

-
+
 

 
-

+

 
uniaxial-model-nass-flexible-iff.png
 

 
Figure 8: Uniaxial IFF Control Schematic

 

 

-

-
3.1 Control Design

+

+
3.1 Control Design

 

 

 
load('./uniaxial/mat/plants.mat', 'G');
@@ -634,7 +634,7 @@ Let's look at the transfer function from actuator forces in the nano-hexapod to
 

 
 
-

+

 
uniaxial_iff_plant.png
 

 
Figure 9: Transfer function from forces applied in the legs to force sensor (png, pdf)

@@ -649,7 +649,7 @@ The controller for each pair of actuator/sensor is:
 

 
 
-

+

 
uniaxial_iff_open_loop.png
 

 
Figure 10: Loop Gain for the Integral Force Feedback (png, pdf)

@@ -657,8 +657,8 @@ The controller for each pair of actuator/sensor is:
 

 

 
-

-
3.2 Identification

+

+
3.2 Identification

 

 

 Let's initialize the system prior to identification.
@@ -741,18 +741,18 @@ G_iff.OutputName = {
 

 
-

-
3.3 Sensitivity to Disturbance

+

+
3.3 Sensitivity to Disturbance

 

 
-

+

 
uniaxial_sensitivity_dist_iff.png
 

 
Figure 11: Sensitivity to disturbance once the IFF controller is applied to the system (png, pdf)

 

 
 
-

+

 
uniaxial_sensitivity_dist_stages_iff.png
 

 
Figure 12: Sensitivity to force disturbances in various stages when IFF is applied (png, pdf)

@@ -760,11 +760,11 @@ G_iff.OutputName = {
 

 
-

-
3.4 Damped Plant

+

+
3.4 Damped Plant

 

 
-

+

 
uniaxial_plant_iff_damped.png
 

 
Figure 13: Damped Plant after IFF is applied (png, pdf)

@@ -772,8 +772,8 @@ G_iff.OutputName = {
 

 
-

-
3.5 Conclusion

+

+
3.5 Conclusion

 

 

 

@@ -785,25 +785,25 @@ Integral Force Feedback:
 

 

 
-

-
4 Relative Motion Control

+

+
4 Relative Motion Control

 

 

-
+
 

 

 In the Relative Motion Control (RMC), a derivative feedback is applied between the measured actuator displacement to the actuator force input.
 

 
 
-

+

 
uniaxial-model-nass-flexible-rmc.png
 

 
Figure 14: Uniaxial RMC Control Schematic

 

 

-

-
4.1 Control Design

+

+
4.1 Control Design

 

 

 
load('./uniaxial/mat/plants.mat', 'G');
@@ -815,7 +815,7 @@ Let's look at the transfer function from actuator forces in the nano-hexapod to
 

 
 
-

+

 
uniaxial_rmc_plant.png
 

 
Figure 15: Transfer function from forces applied in the legs to leg displacement sensor (png, pdf)

@@ -831,7 +831,7 @@ A Low pass Filter is added to make the controller transfer function proper.
 

 
 
-

+

 
uniaxial_rmc_open_loop.png
 

 
Figure 16: Loop Gain for the Integral Force Feedback (png, pdf)

@@ -839,8 +839,8 @@ A Low pass Filter is added to make the controller transfer function proper.
 

 

 
-

-
4.2 Identification

+

+
4.2 Identification

 

 

 Let's initialize the system prior to identification.
@@ -924,18 +924,18 @@ G_rmc.OutputName = {
 
 
-

-
4.3 Sensitivity to Disturbance

+

+
4.3 Sensitivity to Disturbance

 

 
-

+

 
uniaxial_sensitivity_dist_rmc.png
 

 
Figure 17: Sensitivity to disturbance once the RMC controller is applied to the system (png, pdf)

 

 
 
-

+

 
uniaxial_sensitivity_dist_stages_rmc.png
 

 
Figure 18: Sensitivity to force disturbances in various stages when RMC is applied (png, pdf)

@@ -943,11 +943,11 @@ G_rmc.OutputName = {
 

 
-

-
4.4 Damped Plant

+

+
4.4 Damped Plant

 

 
-

+

 
uniaxial_plant_rmc_damped.png
 

 
Figure 19: Damped Plant after RMC is applied (png, pdf)

@@ -955,8 +955,8 @@ G_rmc.OutputName = {
 

 
-

-
4.5 Conclusion

+

+
4.5 Conclusion

 

 

 

@@ -968,25 +968,25 @@ Relative Motion Control:
 

 

 
-

-
5 Direct Velocity Feedback

+

+
5 Direct Velocity Feedback

 

 

-
+
 

 

 In the Relative Motion Control (RMC), a feedback is applied between the measured velocity of the platform to the actuator force input.
 

 
 
-

+

 
uniaxial-model-nass-flexible-dvf.png
 

 
Figure 20: Uniaxial DVF Control Schematic

 

 

-

-
5.1 Control Design

+

+
5.1 Control Design

 

 

 
load('./uniaxial/mat/plants.mat', 'G');
@@ -994,7 +994,7 @@ In the Relative Motion Control (RMC), a feedback is applied between the measured
 

 
 
-

+

 
uniaxial_dvf_plant.png
 

 
Figure 21: Transfer function from forces applied in the legs to leg velocity sensor (png, pdf)

@@ -1006,7 +1006,7 @@ In the Relative Motion Control (RMC), a feedback is applied between the measured
 

 
 
-

+

 
uniaxial_dvf_loop_gain.png
 

 
Figure 22: Transfer function from forces applied in the legs to leg velocity sensor (png, pdf)

@@ -1014,8 +1014,8 @@ In the Relative Motion Control (RMC), a feedback is applied between the measured
 

 

 
-

-
5.2 Identification

+

+
5.2 Identification

 

 

 Let's initialize the system prior to identification.
@@ -1098,18 +1098,18 @@ G_dvf.OutputName = {
 

 
-

-
5.3 Sensitivity to Disturbance

+

+
5.3 Sensitivity to Disturbance

 

 
-

+

 
uniaxial_sensitivity_dist_dvf.png
 

 
Figure 23: Sensitivity to disturbance once the DVF controller is applied to the system (png, pdf)

 

 
 
-

+

 
uniaxial_sensitivity_dist_stages_dvf.png
 

 
Figure 24: Sensitivity to force disturbances in various stages when DVF is applied (png, pdf)

@@ -1117,11 +1117,11 @@ G_dvf.OutputName = {
 

 
-

-
5.4 Damped Plant

+

+
5.4 Damped Plant

 

 
-

+

 
uniaxial_plant_dvf_damped.png
 

 
Figure 25: Damped Plant after DVF is applied (png, pdf)

@@ -1129,8 +1129,8 @@ G_dvf.OutputName = {
 

 
-

-
5.5 Conclusion

+

+
5.5 Conclusion

 

 

 

@@ -1141,25 +1141,25 @@ Direct Velocity Feedback:
 

 

 

-

-
6 With Cedrat Piezo-electric Actuators

+

+
6 With Cedrat Piezo-electric Actuators

 

 

-
+
 

 

-The model used for the Cedrat actuator is shown in figure 26.
+The model used for the Cedrat actuator is shown in figure 26.
 

 
 
-

+

 
cedrat-uniaxial-actuator.png
 

 
Figure 26: Schematic of the model used for the Cedrat Actuator

 

 

-

-
6.1 Identification

+

+
6.1 Identification

 

 

 Let's initialize the system prior to identification.
@@ -1247,15 +1247,15 @@ G.OutputName = {
 

 
-

-
6.2 Control Design

+

+
6.2 Control Design

 

 

 Let's look at the transfer function from actuator forces in the nano-hexapod to the force sensor in the nano-hexapod legs for all 6 pairs of actuator/sensor.
 

 
 
-

+

 
uniaxial_cedrat_plant.png
 

 
Figure 27: Transfer function from forces applied in the legs to force sensor (png, pdf)

@@ -1270,7 +1270,7 @@ The controller for each pair of actuator/sensor is:
 

 
 
-

+

 
uniaxial_cedrat_open_loop.png
 

 
Figure 28: Loop Gain for the Integral Force Feedback (png, pdf)

@@ -1278,8 +1278,8 @@ The controller for each pair of actuator/sensor is:
 

 

 
-

-
6.3 Identification

+

+
6.3 Identification

 

 

 Let's initialize the system prior to identification.
@@ -1363,18 +1363,18 @@ G_cedrat.OutputName = {
 

 
-

-
6.4 Sensitivity to Disturbance

+

+
6.4 Sensitivity to Disturbance

 

 
-

+

 
uniaxial_sensitivity_dist_cedrat.png
 

 
Figure 29: Sensitivity to disturbance once the CEDRAT controller is applied to the system (png, pdf)

 

 
 
-

+

 
uniaxial_sensitivity_dist_stages_cedrat.png
 

 
Figure 30: Sensitivity to force disturbances in various stages when CEDRAT is applied (png, pdf)

@@ -1382,11 +1382,11 @@ G_cedrat.OutputName = {
 

 
-

-
6.5 Damped Plant

+

+
6.5 Damped Plant

 

 
-

+

 
uniaxial_plant_cedrat_damped.png
 

 
Figure 31: Damped Plant after CEDRAT is applied (png, pdf)

@@ -1394,8 +1394,8 @@ G_cedrat.OutputName = {
 

 
-

-
6.6 Conclusion

+

+
6.6 Conclusion

 

 

 

@@ -1407,15 +1407,15 @@ This gives similar results than with a classical force sensor.
 

 

 
-

-
7 Comparison of Active Damping Techniques

+

+
7 Comparison of Active Damping Techniques

 

 

-
+
 

 

-

-
7.1 Load the plants

+

+
7.1 Load the plants

 

 

 
load('./uniaxial/mat/plants.mat', 'G', 'G_iff', 'G_rmc', 'G_dvf');
@@ -1424,11 +1424,11 @@ This gives similar results than with a classical force sensor.
 

 

 
-

-
7.2 Sensitivity to Disturbance

+

+
7.2 Sensitivity to Disturbance

 

 
-

+

 
uniaxial_sensitivity_ground_motion.png
 

 
Figure 32: Sensitivity to Ground Motion - Comparison (png, pdf)

@@ -1436,21 +1436,21 @@ This gives similar results than with a classical force sensor.
 
 
 
-

+

 
uniaxial_sensitivity_direct_force.png
 

 
Figure 33: Sensitivity to disturbance - Comparison (png, pdf)

 

 
 
-

+

 
uniaxial_sensitivity_fty.png
 

 
Figure 34: Sensitivity to force disturbances - Comparison (png, pdf)

 

 
 
-

+

 
uniaxial_sensitivity_frz.png
 

 
Figure 35: Sensitivity to force disturbances - Comparison (png, pdf)

@@ -1458,8 +1458,8 @@ This gives similar results than with a classical force sensor.
 

 

 
-

-
7.3 Noise Budget

+

+
7.3 Noise Budget

 

 

 We first load the measured PSD of the disturbance.
@@ -1471,10 +1471,10 @@ We first load the measured PSD of the disturbance.
 
 

 The effect of these disturbances on the distance \(D\) is computed for all active damping techniques.
-We then compute the Cumulative Amplitude Spectrum (figure 36).
+We then compute the Cumulative Amplitude Spectrum (figure 36).
 

 
-

+

 
uniaxial-comp-cas-dist.png
 

 
Figure 36: Comparison of the Cumulative Amplitude Spectrum of \(D\) for different active damping techniques (png, pdf)

@@ -1483,7 +1483,7 @@ We then compute the Cumulative Amplitude Spectrum (figure 
 

 The obtained Root Mean Square Value for each active damping technique is shown below.
 

-
+
@@ -1526,11 +1526,11 @@ It is important to note that the effect of direct forces applied to the sample a
 
 
-

-
7.4 Damped Plant

+

+
7.4 Damped Plant

 

 
-

+

 
uniaxial_plant_damped_comp.png
 

 
Figure 37: Damped Plant - Comparison (png, pdf)

@@ -1538,10 +1538,10 @@ It is important to note that the effect of direct forces applied to the sample a
 

 

 
-

-
7.5 Conclusion

+

+
7.5 Conclusion

 

-

 Table 1: Obtain Root Mean Square value of \(D\) for each Active Damping Technique applied
 
 

 

+
@@ -1608,15 +1608,15 @@ It is important to note that the effect of direct forces applied to the sample a
 
-

-
8 Voice Coil

+

+
8 Voice Coil

 

 

-
+
 

 

-

-
8.1 Init

+

+
8.1 Init

 

 

 We initialize all the stages with the default parameters.
@@ -1628,8 +1628,8 @@ All the controllers are set to 0 (Open Loop).
 

 

 

-

-
8.2 Identification

+

+
8.2 Identification

 

 

 We identify the dynamics of the system.
@@ -1692,8 +1692,8 @@ Finally, we save the identified system dynamics for further analysis.
 

 

 
-

-
8.3 Sensitivity to Disturbances

+

+
8.3 Sensitivity to Disturbances

 

 

 We load the dynamics when using a piezo-electric nano hexapod to compare the results.
@@ -1707,19 +1707,19 @@ We load the dynamics when using a piezo-electric nano hexapod to compare the res
 We show several plots representing the sensitivity to disturbances:
 

 
    
-
  • in figure 38 the transfer functions from ground motion \(D_w\) to the sample position \(D\) and the transfer function from direct force on the sample \(F_s\) to the sample position \(D\) are shown
    • 
-
  • in figure 39, it is the effect of parasitic forces of the positioning stages (\(F_{ty}\) and \(F_{rz}\)) on the position \(D\) of the sample that are shown
    • 
+
  • in figure 38 the transfer functions from ground motion \(D_w\) to the sample position \(D\) and the transfer function from direct force on the sample \(F_s\) to the sample position \(D\) are shown
    • 
+
  • in figure 39, it is the effect of parasitic forces of the positioning stages (\(F_{ty}\) and \(F_{rz}\)) on the position \(D\) of the sample that are shown
    • 
 

 
 
-

+

 
uniaxial-sensitivity-vc-disturbances.png
 

 
Figure 38: Sensitivity to disturbances (png, pdf)

 

 
 
-

+

 
uniaxial-sensitivity-vc-force-dist.png
 

 
Figure 39: Sensitivity to disturbances (png, pdf)

@@ -1727,8 +1727,8 @@ We show several plots representing the sensitivity to disturbances:
 

 

 
-

-
8.4 Noise Budget

+

+
8.4 Noise Budget

 

 

 We first load the measured PSD of the disturbance.
@@ -1740,11 +1740,11 @@ We first load the measured PSD of the disturbance.
 
 

 The effect of these disturbances on the distance \(D\) is computed below.
-The PSD of the obtain distance \(D\) due to each of the perturbation is shown in figure 40 and the Cumulative Amplitude Spectrum is shown in figure 41.
+The PSD of the obtain distance \(D\) due to each of the perturbation is shown in figure 40 and the Cumulative Amplitude Spectrum is shown in figure 41.
 

 
 

-The Root Mean Square value of the obtained displacement \(D\) is computed below and can be determined from the figure 41.
+The Root Mean Square value of the obtained displacement \(D\) is computed below and can be determined from the figure 41.
 

 
 4.8793e-06
@@ -1752,14 +1752,14 @@ The Root Mean Square value of the obtained displacement \(D\) is computed below
 
 
 
-

+

 
uniaxial-vc-psd-dist.png
 

 
Figure 40: PSD of the displacement \(D\) due to disturbances (png, pdf)

 

 
 
-

+

 
uniaxial-vc-cas-dist.png
 

 
Figure 41: CAS of the displacement \(D\) due the disturbances (png, pdf)

@@ -1777,8 +1777,8 @@ Thus, it may be desirable to use voice coil actuators.
 

 

 

-

-
8.5 Integral Force Feedback

+

+
8.5 Integral Force Feedback

 

 

 
K_iff = -20/s;
@@ -1786,7 +1786,7 @@ Thus, it may be desirable to use voice coil actuators.
 

 
 
-

+

 
uniaxial_iff_vc_open_loop.png
 

 
Figure 42: Open Loop Transfer Function for IFF control when using a voice coil actuator (png, pdf)

@@ -1794,8 +1794,8 @@ Thus, it may be desirable to use voice coil actuators.
 

 

 
-

-
8.6 Identification of the Damped Plant

+

+
8.6 Identification of the Damped Plant

 

 

 Let's initialize the system prior to identification.
@@ -1873,14 +1873,14 @@ G_vc_iff.OutputName = {
 

 
-

-
8.7 Noise Budget

+

+
8.7 Noise Budget

 

 

 We compute the obtain PSD of the displacement \(D\) when using IFF.
 

 
-

+

 
uniaxial-cas-iff-vc.png
 

 
Figure 43: CAS of the displacement \(D\) (png, pdf)

@@ -1888,8 +1888,8 @@ We compute the obtain PSD of the displacement \(D\) when using IFF.
 

 

 
-

-
8.8 Conclusion

+

+
8.8 Conclusion

 

 

 

@@ -1907,7 +1907,7 @@ Similarly, it would require much lower bandwidth to attain the same level of dis
 

 

 
Author: Dehaeze Thomas

-
Created: 2019-11-05 mar. 11:27

+
Created: 2019-11-11 lun. 14:50

 
Validate

 

 
diff --git a/uniaxial/index.org b/uniaxial/index.org
index 48c6226..074d1c3 100644
--- a/uniaxial/index.org
+++ b/uniaxial/index.org
@@ -825,7 +825,7 @@ The Root Mean Square value of the obtained displacement $D$ is computed below an
   plot(dist_f.f, psd_rz_d, 'DisplayName', '$R_z$');
   hold off;
   set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
-  ylabel('CAS of the effect of disturbances on $D$ $\left[\frac{m^2}{Hz}\right]$');  xlabel('Frequency [Hz]');
+  ylabel('PSD of the effect of disturbances on $D$ $\left[\frac{m^2}{Hz}\right]$');  xlabel('Frequency [Hz]');
   legend('location', 'northeast')
   xlim([0.5, 500]);
 #+end_src
@@ -836,7 +836,7 @@ The Root Mean Square value of the obtained displacement $D$ is computed below an
 #+end_src
 
 #+NAME: fig:uniaxial-psd-dist
-#+CAPTION: caption ([[./figs/uniaxial-psd-dist.png][png]], [[./figs/uniaxial-psd-dist.pdf][pdf]])
+#+CAPTION: PSD of the effect of disturbances on $D$ ([[./figs/uniaxial-psd-dist.png][png]], [[./figs/uniaxial-psd-dist.pdf][pdf]])
 [[file:figs/uniaxial-psd-dist.png]]
 
 
@@ -862,7 +862,7 @@ The Root Mean Square value of the obtained displacement $D$ is computed below an
 #+end_src
 
 #+NAME: fig:uniaxial-cas-dist
-#+CAPTION: caption ([[./figs/uniaxial-cas-dist.png][png]], [[./figs/uniaxial-cas-dist.pdf][pdf]])
+#+CAPTION: CAS of the effect of disturbances on $D$ ([[./figs/uniaxial-cas-dist.png][png]], [[./figs/uniaxial-cas-dist.pdf][pdf]])
 [[file:figs/uniaxial-cas-dist.png]]
 
 ** Plant


 Table 2: Comparison of proposed active damping techniques