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Add noise budget to compare active damping techniques

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Thomas Dehaeze 2019-11-04 17:33:30 +01:00
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@ -3,7 +3,7 @@
"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd"> "http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
<html xmlns="http://www.w3.org/1999/xhtml" lang="en" xml:lang="en"> <html xmlns="http://www.w3.org/1999/xhtml" lang="en" xml:lang="en">
<head> <head>
<!-- 2019-10-28 lun. 17:34 --> <!-- 2019-11-04 lun. 17:33 -->
<meta http-equiv="Content-Type" content="text/html;charset=utf-8" /> <meta http-equiv="Content-Type" content="text/html;charset=utf-8" />
<meta name="viewport" content="width=device-width, initial-scale=1" /> <meta name="viewport" content="width=device-width, initial-scale=1" />
<title>Simscape Uniaxial Model</title> <title>Simscape Uniaxial Model</title>
@ -280,58 +280,60 @@ for the JavaScript code in this tag.
<h2>Table of Contents</h2> <h2>Table of Contents</h2>
<div id="text-table-of-contents"> <div id="text-table-of-contents">
<ul> <ul>
<li><a href="#orgfc3044a">1. Simscape Model</a></li> <li><a href="#orgac9343f">1. Simscape Model</a></li>
<li><a href="#org8da4eb0">2. Undamped System</a> <li><a href="#org3890417">2. Undamped System</a>
<ul> <ul>
<li><a href="#orgcb2b0a1">2.1. Init</a></li> <li><a href="#orgbc01ced">2.1. Init</a></li>
<li><a href="#org7f40bf7">2.2. Identification</a></li> <li><a href="#org699bc2c">2.2. Identification</a></li>
<li><a href="#org7908bab">2.3. Sensitivity to Disturbances</a></li> <li><a href="#org0c59d84">2.3. Sensitivity to Disturbances</a></li>
<li><a href="#org5a57afd">2.4. Plant</a></li> <li><a href="#orgb191841">2.4. Noise Budget</a></li>
<li><a href="#org1b2f77b">2.5. Plant</a></li>
</ul> </ul>
</li> </li>
<li><a href="#org68d1bb0">3. Integral Force Feedback</a> <li><a href="#org27aabe3">3. Integral Force Feedback</a>
<ul> <ul>
<li><a href="#orga5e22eb">3.1. Control Design</a></li> <li><a href="#org04b5ef2">3.1. Control Design</a></li>
<li><a href="#org0fdf2fd">3.2. Identification</a></li> <li><a href="#orgd976043">3.2. Identification</a></li>
<li><a href="#org8b81fd6">3.3. Sensitivity to Disturbance</a></li> <li><a href="#orgb785fd5">3.3. Sensitivity to Disturbance</a></li>
<li><a href="#org80d5d2d">3.4. Damped Plant</a></li> <li><a href="#org7f2e353">3.4. Damped Plant</a></li>
<li><a href="#orga9ed49c">3.5. Conclusion</a></li> <li><a href="#org46695ba">3.5. Conclusion</a></li>
</ul> </ul>
</li> </li>
<li><a href="#org5d0bc94">4. Relative Motion Control</a> <li><a href="#org8f75c3f">4. Relative Motion Control</a>
<ul> <ul>
<li><a href="#org4ffacc7">4.1. Control Design</a></li> <li><a href="#org7291ab1">4.1. Control Design</a></li>
<li><a href="#orgf86862c">4.2. Identification</a></li> <li><a href="#org72b4f0a">4.2. Identification</a></li>
<li><a href="#org0211838">4.3. Sensitivity to Disturbance</a></li> <li><a href="#org7669eee">4.3. Sensitivity to Disturbance</a></li>
<li><a href="#orgefb061f">4.4. Damped Plant</a></li> <li><a href="#orgab2a3d1">4.4. Damped Plant</a></li>
<li><a href="#org467a5d6">4.5. Conclusion</a></li> <li><a href="#orgbc19b27">4.5. Conclusion</a></li>
</ul> </ul>
</li> </li>
<li><a href="#org408eed0">5. Direct Velocity Feedback</a> <li><a href="#org264e55c">5. Direct Velocity Feedback</a>
<ul> <ul>
<li><a href="#org64e7b3f">5.1. Control Design</a></li> <li><a href="#orge1aa5cd">5.1. Control Design</a></li>
<li><a href="#orga75fa6d">5.2. Identification</a></li> <li><a href="#org2a880b1">5.2. Identification</a></li>
<li><a href="#org0d535fa">5.3. Sensitivity to Disturbance</a></li> <li><a href="#orgd7e4638">5.3. Sensitivity to Disturbance</a></li>
<li><a href="#org9643807">5.4. Damped Plant</a></li> <li><a href="#org9524bf4">5.4. Damped Plant</a></li>
<li><a href="#org6e6fd47">5.5. Conclusion</a></li> <li><a href="#org329f7b9">5.5. Conclusion</a></li>
</ul> </ul>
</li> </li>
<li><a href="#orgd792cab">6. With Cedrat Piezo-electric Actuators</a> <li><a href="#org8ffadeb">6. With Cedrat Piezo-electric Actuators</a>
<ul> <ul>
<li><a href="#org7707a0a">6.1. Identification</a></li> <li><a href="#org8f92e1b">6.1. Identification</a></li>
<li><a href="#orgd921ae7">6.2. Control Design</a></li> <li><a href="#org9183f23">6.2. Control Design</a></li>
<li><a href="#org1d5a39c">6.3. Identification</a></li> <li><a href="#orge8484c9">6.3. Identification</a></li>
<li><a href="#orgb163c6c">6.4. Sensitivity to Disturbance</a></li> <li><a href="#org35ee201">6.4. Sensitivity to Disturbance</a></li>
<li><a href="#org552dcab">6.5. Damped Plant</a></li> <li><a href="#orgfdfe26c">6.5. Damped Plant</a></li>
<li><a href="#org5065aae">6.6. Conclusion</a></li> <li><a href="#org7b0ae1d">6.6. Conclusion</a></li>
</ul> </ul>
</li> </li>
<li><a href="#org60dfb12">7. Comparison of Active Damping Techniques</a> <li><a href="#org51051c1">7. Comparison of Active Damping Techniques</a>
<ul> <ul>
<li><a href="#org249a650">7.1. Load the plants</a></li> <li><a href="#orga925887">7.1. Load the plants</a></li>
<li><a href="#org0c1cccb">7.2. Sensitivity to Disturbance</a></li> <li><a href="#orga01b1a9">7.2. Sensitivity to Disturbance</a></li>
<li><a href="#orgb54c9e3">7.3. Damped Plant</a></li> <li><a href="#orga4fdd66">7.3. Noise Budget</a></li>
<li><a href="#org1c67523">7.4. Conclusion</a></li> <li><a href="#orgbb2291d">7.4. Damped Plant</a></li>
<li><a href="#org40f7a4d">7.5. Conclusion</a></li>
</ul> </ul>
</li> </li>
</ul> </ul>
@ -346,11 +348,11 @@ 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. This is done in order to more easily study the system and evaluate control techniques.
</p> </p>
<div id="outline-container-orgfc3044a" class="outline-2"> <div id="outline-container-orgac9343f" class="outline-2">
<h2 id="orgfc3044a"><span class="section-number-2">1</span> Simscape Model</h2> <h2 id="orgac9343f"><span class="section-number-2">1</span> Simscape Model</h2>
<div class="outline-text-2" id="text-1"> <div class="outline-text-2" id="text-1">
<p> <p>
A schematic of the uniaxial model used for simulations is represented in figure <a href="#orgc5e0a56">1</a>. A schematic of the uniaxial model used for simulations is represented in figure <a href="#org7ac1a00">1</a>.
</p> </p>
<p> <p>
@ -394,7 +396,7 @@ The control signal \(u\) is:
</ul> </ul>
<div id="orgc5e0a56" class="figure"> <div id="org7ac1a00" class="figure">
<p><img src="figs/uniaxial-model-nass-flexible.png" alt="uniaxial-model-nass-flexible.png" /> <p><img src="figs/uniaxial-model-nass-flexible.png" alt="uniaxial-model-nass-flexible.png" />
</p> </p>
<p><span class="figure-number">Figure 1: </span>Schematic of the uniaxial model used</p> <p><span class="figure-number">Figure 1: </span>Schematic of the uniaxial model used</p>
@ -403,11 +405,11 @@ The control signal \(u\) is:
<p> <p>
Few active damping techniques will be compared in order to decide which sensor is to be included in the system. 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 <a href="#orgdb9985c">2</a>. Schematics of the active damping techniques are displayed in figure <a href="#orgb379a31">2</a>.
</p> </p>
<div id="orgdb9985c" class="figure"> <div id="orgb379a31" class="figure">
<p><img src="figs/uniaxial-model-nass-flexible-active-damping.png" alt="uniaxial-model-nass-flexible-active-damping.png" /> <p><img src="figs/uniaxial-model-nass-flexible-active-damping.png" alt="uniaxial-model-nass-flexible-active-damping.png" />
</p> </p>
<p><span class="figure-number">Figure 2: </span>Comparison of used active damping techniques</p> <p><span class="figure-number">Figure 2: </span>Comparison of used active damping techniques</p>
@ -415,16 +417,16 @@ Schematics of the active damping techniques are displayed in figure <a href="#or
</div> </div>
</div> </div>
<div id="outline-container-org8da4eb0" class="outline-2"> <div id="outline-container-org3890417" class="outline-2">
<h2 id="org8da4eb0"><span class="section-number-2">2</span> Undamped System</h2> <h2 id="org3890417"><span class="section-number-2">2</span> Undamped System</h2>
<div class="outline-text-2" id="text-2"> <div class="outline-text-2" id="text-2">
<p> <p>
Let's start by study the undamped system. Let's start by study the undamped system.
</p> </p>
</div> </div>
<div id="outline-container-orgcb2b0a1" class="outline-3"> <div id="outline-container-orgbc01ced" class="outline-3">
<h3 id="orgcb2b0a1"><span class="section-number-3">2.1</span> Init</h3> <h3 id="orgbc01ced"><span class="section-number-3">2.1</span> Init</h3>
<div class="outline-text-3" id="text-2-1"> <div class="outline-text-3" id="text-2-1">
<p> <p>
We initialize all the stages with the default parameters. We initialize all the stages with the default parameters.
@ -436,8 +438,8 @@ All the controllers are set to 0 (Open Loop).
</p> </p>
</div> </div>
</div> </div>
<div id="outline-container-org7f40bf7" class="outline-3"> <div id="outline-container-org699bc2c" class="outline-3">
<h3 id="org7f40bf7"><span class="section-number-3">2.2</span> Identification</h3> <h3 id="org699bc2c"><span class="section-number-3">2.2</span> Identification</h3>
<div class="outline-text-3" id="text-2-2"> <div class="outline-text-3" id="text-2-2">
<p> <p>
We identify the dynamics of the system. We identify the dynamics of the system.
@ -500,19 +502,19 @@ Finally, we save the identified system dynamics for further analysis.
</div> </div>
</div> </div>
<div id="outline-container-org7908bab" class="outline-3"> <div id="outline-container-org0c59d84" class="outline-3">
<h3 id="org7908bab"><span class="section-number-3">2.3</span> Sensitivity to Disturbances</h3> <h3 id="org0c59d84"><span class="section-number-3">2.3</span> Sensitivity to Disturbances</h3>
<div class="outline-text-3" id="text-2-3"> <div class="outline-text-3" id="text-2-3">
<p> <p>
We show several plots representing the sensitivity to disturbances: We show several plots representing the sensitivity to disturbances:
</p> </p>
<ul class="org-ul"> <ul class="org-ul">
<li>in figure <a href="#orgd82c2ce">3</a> 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</li> <li>in figure <a href="#orgc6b4646">3</a> 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</li>
<li>in figure <a href="#org72d40e7">4</a>, 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</li> <li>in figure <a href="#orgbcfd91e">4</a>, 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</li>
</ul> </ul>
<div id="orgd82c2ce" class="figure"> <div id="orgc6b4646" class="figure">
<p><img src="figs/uniaxial-sensitivity-disturbances.png" alt="uniaxial-sensitivity-disturbances.png" /> <p><img src="figs/uniaxial-sensitivity-disturbances.png" alt="uniaxial-sensitivity-disturbances.png" />
</p> </p>
<p><span class="figure-number">Figure 3: </span>Sensitivity to disturbances (<a href="./figs/uniaxial-sensitivity-disturbances.png">png</a>, <a href="./figs/uniaxial-sensitivity-disturbances.pdf">pdf</a>)</p> <p><span class="figure-number">Figure 3: </span>Sensitivity to disturbances (<a href="./figs/uniaxial-sensitivity-disturbances.png">png</a>, <a href="./figs/uniaxial-sensitivity-disturbances.pdf">pdf</a>)</p>
@ -520,7 +522,7 @@ We show several plots representing the sensitivity to disturbances:
<div id="org72d40e7" class="figure"> <div id="orgbcfd91e" class="figure">
<p><img src="figs/uniaxial-sensitivity-force-dist.png" alt="uniaxial-sensitivity-force-dist.png" /> <p><img src="figs/uniaxial-sensitivity-force-dist.png" alt="uniaxial-sensitivity-force-dist.png" />
</p> </p>
<p><span class="figure-number">Figure 4: </span>Sensitivity to disturbances (<a href="./figs/uniaxial-sensitivity-force-dist.png">png</a>, <a href="./figs/uniaxial-sensitivity-force-dist.pdf">pdf</a>)</p> <p><span class="figure-number">Figure 4: </span>Sensitivity to disturbances (<a href="./figs/uniaxial-sensitivity-force-dist.png">png</a>, <a href="./figs/uniaxial-sensitivity-force-dist.pdf">pdf</a>)</p>
@ -528,39 +530,81 @@ We show several plots representing the sensitivity to disturbances:
</div> </div>
</div> </div>
<div id="outline-container-org5a57afd" class="outline-3"> <div id="outline-container-orgb191841" class="outline-3">
<h3 id="org5a57afd"><span class="section-number-3">2.4</span> Plant</h3> <h3 id="orgb191841"><span class="section-number-3">2.4</span> Noise Budget</h3>
<div class="outline-text-3" id="text-2-4"> <div class="outline-text-3" id="text-2-4">
<p> <p>
The transfer function from the force \(F\) applied by the nano-hexapod to the position of the sample \(D\) is shown in figure <a href="#org5789c3f">5</a>. We first load the measured PSD of the disturbance.
</p>
<div class="org-src-container">
<pre class="src src-matlab">load<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-string">'./disturbances/mat/dist_psd.mat'</span>, <span class="org-string">'dist_f'</span><span class="org-rainbow-delimiters-depth-1">)</span>;
</pre>
</div>
<p>
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 <a href="#orgb199386">5</a> and the Cumulative Amplitude Spectrum is shown in figure <a href="#org3989b84">6</a>.
</p>
<p>
The Root Mean Square value of the obtained displacement \(D\) is computed below and can be determined from the figure <a href="#org3989b84">6</a>.
</p>
<pre class="example">
3.3793e-06
</pre>
<div id="orgb199386" class="figure">
<p><img src="figs/uniaxial-psd-dist.png" alt="uniaxial-psd-dist.png" />
</p>
<p><span class="figure-number">Figure 5: </span>caption (<a href="./figs/uniaxial-psd-dist.png">png</a>, <a href="./figs/uniaxial-psd-dist.pdf">pdf</a>)</p>
</div>
<div id="org3989b84" class="figure">
<p><img src="figs/uniaxial-cas-dist.png" alt="uniaxial-cas-dist.png" />
</p>
<p><span class="figure-number">Figure 6: </span>caption (<a href="./figs/uniaxial-cas-dist.png">png</a>, <a href="./figs/uniaxial-cas-dist.pdf">pdf</a>)</p>
</div>
</div>
</div>
<div id="outline-container-org1b2f77b" class="outline-3">
<h3 id="org1b2f77b"><span class="section-number-3">2.5</span> Plant</h3>
<div class="outline-text-3" id="text-2-5">
<p>
The transfer function from the force \(F\) applied by the nano-hexapod to the position of the sample \(D\) is shown in figure <a href="#org63b704d">7</a>.
It corresponds to the plant to control. It corresponds to the plant to control.
</p> </p>
<div id="org5789c3f" class="figure"> <div id="org63b704d" class="figure">
<p><img src="figs/uniaxial-plant.png" alt="uniaxial-plant.png" /> <p><img src="figs/uniaxial-plant.png" alt="uniaxial-plant.png" />
</p> </p>
<p><span class="figure-number">Figure 5: </span>Bode plot of the Plant (<a href="./figs/uniaxial-plant.png">png</a>, <a href="./figs/uniaxial-plant.pdf">pdf</a>)</p> <p><span class="figure-number">Figure 7: </span>Bode plot of the Plant (<a href="./figs/uniaxial-plant.png">png</a>, <a href="./figs/uniaxial-plant.pdf">pdf</a>)</p>
</div> </div>
</div> </div>
</div> </div>
</div> </div>
<div id="outline-container-org68d1bb0" class="outline-2"> <div id="outline-container-org27aabe3" class="outline-2">
<h2 id="org68d1bb0"><span class="section-number-2">3</span> Integral Force Feedback</h2> <h2 id="org27aabe3"><span class="section-number-2">3</span> Integral Force Feedback</h2>
<div class="outline-text-2" id="text-3"> <div class="outline-text-2" id="text-3">
<p> <p>
<a id="org36327e7"></a> <a id="orgcfb1bdd"></a>
</p> </p>
<div id="org6ca8a23" class="figure"> <div id="org2faff5f" class="figure">
<p><img src="figs/uniaxial-model-nass-flexible-iff.png" alt="uniaxial-model-nass-flexible-iff.png" /> <p><img src="figs/uniaxial-model-nass-flexible-iff.png" alt="uniaxial-model-nass-flexible-iff.png" />
</p> </p>
<p><span class="figure-number">Figure 6: </span>Uniaxial IFF Control Schematic</p> <p><span class="figure-number">Figure 8: </span>Uniaxial IFF Control Schematic</p>
</div> </div>
</div> </div>
<div id="outline-container-orga5e22eb" class="outline-3"> <div id="outline-container-org04b5ef2" class="outline-3">
<h3 id="orga5e22eb"><span class="section-number-3">3.1</span> Control Design</h3> <h3 id="org04b5ef2"><span class="section-number-3">3.1</span> Control Design</h3>
<div class="outline-text-3" id="text-3-1"> <div class="outline-text-3" id="text-3-1">
<div class="org-src-container"> <div class="org-src-container">
<pre class="src src-matlab">load<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-string">'./uniaxial/mat/plants.mat'</span>, <span class="org-string">'G'</span><span class="org-rainbow-delimiters-depth-1">)</span>; <pre class="src src-matlab">load<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-string">'./uniaxial/mat/plants.mat'</span>, <span class="org-string">'G'</span><span class="org-rainbow-delimiters-depth-1">)</span>;
@ -572,10 +616,10 @@ Let's look at the transfer function from actuator forces in the nano-hexapod to
</p> </p>
<div id="org5063cb4" class="figure"> <div id="org7cefef0" class="figure">
<p><img src="figs/uniaxial_iff_plant.png" alt="uniaxial_iff_plant.png" /> <p><img src="figs/uniaxial_iff_plant.png" alt="uniaxial_iff_plant.png" />
</p> </p>
<p><span class="figure-number">Figure 7: </span>Transfer function from forces applied in the legs to force sensor (<a href="./figs/uniaxial_iff_plant.png">png</a>, <a href="./figs/uniaxial_iff_plant.pdf">pdf</a>)</p> <p><span class="figure-number">Figure 9: </span>Transfer function from forces applied in the legs to force sensor (<a href="./figs/uniaxial_iff_plant.png">png</a>, <a href="./figs/uniaxial_iff_plant.pdf">pdf</a>)</p>
</div> </div>
<p> <p>
@ -587,16 +631,16 @@ The controller for each pair of actuator/sensor is:
</div> </div>
<div id="org495687f" class="figure"> <div id="org2c6c9a0" class="figure">
<p><img src="figs/uniaxial_iff_open_loop.png" alt="uniaxial_iff_open_loop.png" /> <p><img src="figs/uniaxial_iff_open_loop.png" alt="uniaxial_iff_open_loop.png" />
</p> </p>
<p><span class="figure-number">Figure 8: </span>Loop Gain for the Integral Force Feedback (<a href="./figs/uniaxial_iff_open_loop.png">png</a>, <a href="./figs/uniaxial_iff_open_loop.pdf">pdf</a>)</p> <p><span class="figure-number">Figure 10: </span>Loop Gain for the Integral Force Feedback (<a href="./figs/uniaxial_iff_open_loop.png">png</a>, <a href="./figs/uniaxial_iff_open_loop.pdf">pdf</a>)</p>
</div> </div>
</div> </div>
</div> </div>
<div id="outline-container-org0fdf2fd" class="outline-3"> <div id="outline-container-orgd976043" class="outline-3">
<h3 id="org0fdf2fd"><span class="section-number-3">3.2</span> Identification</h3> <h3 id="orgd976043"><span class="section-number-3">3.2</span> Identification</h3>
<div class="outline-text-3" id="text-3-2"> <div class="outline-text-3" id="text-3-2">
<p> <p>
Let's initialize the system prior to identification. Let's initialize the system prior to identification.
@ -679,39 +723,39 @@ G_iff.OutputName = <span class="org-rainbow-delimiters-depth-1">{</span><span cl
</div> </div>
</div> </div>
<div id="outline-container-org8b81fd6" class="outline-3"> <div id="outline-container-orgb785fd5" class="outline-3">
<h3 id="org8b81fd6"><span class="section-number-3">3.3</span> Sensitivity to Disturbance</h3> <h3 id="orgb785fd5"><span class="section-number-3">3.3</span> Sensitivity to Disturbance</h3>
<div class="outline-text-3" id="text-3-3"> <div class="outline-text-3" id="text-3-3">
<div id="org307c8d8" class="figure"> <div id="org9085a3e" class="figure">
<p><img src="figs/uniaxial_sensitivity_dist_iff.png" alt="uniaxial_sensitivity_dist_iff.png" /> <p><img src="figs/uniaxial_sensitivity_dist_iff.png" alt="uniaxial_sensitivity_dist_iff.png" />
</p> </p>
<p><span class="figure-number">Figure 9: </span>Sensitivity to disturbance once the IFF controller is applied to the system (<a href="./figs/uniaxial_sensitivity_dist_iff.png">png</a>, <a href="./figs/uniaxial_sensitivity_dist_iff.pdf">pdf</a>)</p> <p><span class="figure-number">Figure 11: </span>Sensitivity to disturbance once the IFF controller is applied to the system (<a href="./figs/uniaxial_sensitivity_dist_iff.png">png</a>, <a href="./figs/uniaxial_sensitivity_dist_iff.pdf">pdf</a>)</p>
</div> </div>
<div id="orgabd6245" class="figure"> <div id="org087512a" class="figure">
<p><img src="figs/uniaxial_sensitivity_dist_stages_iff.png" alt="uniaxial_sensitivity_dist_stages_iff.png" /> <p><img src="figs/uniaxial_sensitivity_dist_stages_iff.png" alt="uniaxial_sensitivity_dist_stages_iff.png" />
</p> </p>
<p><span class="figure-number">Figure 10: </span>Sensitivity to force disturbances in various stages when IFF is applied (<a href="./figs/uniaxial_sensitivity_dist_stages_iff.png">png</a>, <a href="./figs/uniaxial_sensitivity_dist_stages_iff.pdf">pdf</a>)</p> <p><span class="figure-number">Figure 12: </span>Sensitivity to force disturbances in various stages when IFF is applied (<a href="./figs/uniaxial_sensitivity_dist_stages_iff.png">png</a>, <a href="./figs/uniaxial_sensitivity_dist_stages_iff.pdf">pdf</a>)</p>
</div> </div>
</div> </div>
</div> </div>
<div id="outline-container-org80d5d2d" class="outline-3"> <div id="outline-container-org7f2e353" class="outline-3">
<h3 id="org80d5d2d"><span class="section-number-3">3.4</span> Damped Plant</h3> <h3 id="org7f2e353"><span class="section-number-3">3.4</span> Damped Plant</h3>
<div class="outline-text-3" id="text-3-4"> <div class="outline-text-3" id="text-3-4">
<div id="org35f8f43" class="figure"> <div id="orge8dfaf6" class="figure">
<p><img src="figs/uniaxial_plant_iff_damped.png" alt="uniaxial_plant_iff_damped.png" /> <p><img src="figs/uniaxial_plant_iff_damped.png" alt="uniaxial_plant_iff_damped.png" />
</p> </p>
<p><span class="figure-number">Figure 11: </span>Damped Plant after IFF is applied (<a href="./figs/uniaxial_plant_iff_damped.png">png</a>, <a href="./figs/uniaxial_plant_iff_damped.pdf">pdf</a>)</p> <p><span class="figure-number">Figure 13: </span>Damped Plant after IFF is applied (<a href="./figs/uniaxial_plant_iff_damped.png">png</a>, <a href="./figs/uniaxial_plant_iff_damped.pdf">pdf</a>)</p>
</div> </div>
</div> </div>
</div> </div>
<div id="outline-container-orga9ed49c" class="outline-3"> <div id="outline-container-org46695ba" class="outline-3">
<h3 id="orga9ed49c"><span class="section-number-3">3.5</span> Conclusion</h3> <h3 id="org46695ba"><span class="section-number-3">3.5</span> Conclusion</h3>
<div class="outline-text-3" id="text-3-5"> <div class="outline-text-3" id="text-3-5">
<div class="important"> <div class="important">
<p> <p>
@ -723,25 +767,25 @@ Integral Force Feedback:
</div> </div>
</div> </div>
<div id="outline-container-org5d0bc94" class="outline-2"> <div id="outline-container-org8f75c3f" class="outline-2">
<h2 id="org5d0bc94"><span class="section-number-2">4</span> Relative Motion Control</h2> <h2 id="org8f75c3f"><span class="section-number-2">4</span> Relative Motion Control</h2>
<div class="outline-text-2" id="text-4"> <div class="outline-text-2" id="text-4">
<p> <p>
<a id="org5737634"></a> <a id="org14dacd3"></a>
</p> </p>
<p> <p>
In the Relative Motion Control (RMC), a derivative feedback is applied between the measured actuator displacement to the actuator force input. In the Relative Motion Control (RMC), a derivative feedback is applied between the measured actuator displacement to the actuator force input.
</p> </p>
<div id="org742e0c1" class="figure"> <div id="orgcb12d53" class="figure">
<p><img src="figs/uniaxial-model-nass-flexible-rmc.png" alt="uniaxial-model-nass-flexible-rmc.png" /> <p><img src="figs/uniaxial-model-nass-flexible-rmc.png" alt="uniaxial-model-nass-flexible-rmc.png" />
</p> </p>
<p><span class="figure-number">Figure 12: </span>Uniaxial RMC Control Schematic</p> <p><span class="figure-number">Figure 14: </span>Uniaxial RMC Control Schematic</p>
</div> </div>
</div> </div>
<div id="outline-container-org4ffacc7" class="outline-3"> <div id="outline-container-org7291ab1" class="outline-3">
<h3 id="org4ffacc7"><span class="section-number-3">4.1</span> Control Design</h3> <h3 id="org7291ab1"><span class="section-number-3">4.1</span> Control Design</h3>
<div class="outline-text-3" id="text-4-1"> <div class="outline-text-3" id="text-4-1">
<div class="org-src-container"> <div class="org-src-container">
<pre class="src src-matlab">load<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-string">'./uniaxial/mat/plants.mat'</span>, <span class="org-string">'G'</span><span class="org-rainbow-delimiters-depth-1">)</span>; <pre class="src src-matlab">load<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-string">'./uniaxial/mat/plants.mat'</span>, <span class="org-string">'G'</span><span class="org-rainbow-delimiters-depth-1">)</span>;
@ -753,10 +797,10 @@ Let's look at the transfer function from actuator forces in the nano-hexapod to
</p> </p>
<div id="org9fd5b87" class="figure"> <div id="org7c9427d" class="figure">
<p><img src="figs/uniaxial_rmc_plant.png" alt="uniaxial_rmc_plant.png" /> <p><img src="figs/uniaxial_rmc_plant.png" alt="uniaxial_rmc_plant.png" />
</p> </p>
<p><span class="figure-number">Figure 13: </span>Transfer function from forces applied in the legs to leg displacement sensor (<a href="./figs/uniaxial_rmc_plant.png">png</a>, <a href="./figs/uniaxial_rmc_plant.pdf">pdf</a>)</p> <p><span class="figure-number">Figure 15: </span>Transfer function from forces applied in the legs to leg displacement sensor (<a href="./figs/uniaxial_rmc_plant.png">png</a>, <a href="./figs/uniaxial_rmc_plant.pdf">pdf</a>)</p>
</div> </div>
<p> <p>
@ -769,16 +813,16 @@ A Low pass Filter is added to make the controller transfer function proper.
</div> </div>
<div id="org7d6a1ae" class="figure"> <div id="org782296b" class="figure">
<p><img src="figs/uniaxial_rmc_open_loop.png" alt="uniaxial_rmc_open_loop.png" /> <p><img src="figs/uniaxial_rmc_open_loop.png" alt="uniaxial_rmc_open_loop.png" />
</p> </p>
<p><span class="figure-number">Figure 14: </span>Loop Gain for the Integral Force Feedback (<a href="./figs/uniaxial_rmc_open_loop.png">png</a>, <a href="./figs/uniaxial_rmc_open_loop.pdf">pdf</a>)</p> <p><span class="figure-number">Figure 16: </span>Loop Gain for the Integral Force Feedback (<a href="./figs/uniaxial_rmc_open_loop.png">png</a>, <a href="./figs/uniaxial_rmc_open_loop.pdf">pdf</a>)</p>
</div> </div>
</div> </div>
</div> </div>
<div id="outline-container-orgf86862c" class="outline-3"> <div id="outline-container-org72b4f0a" class="outline-3">
<h3 id="orgf86862c"><span class="section-number-3">4.2</span> Identification</h3> <h3 id="org72b4f0a"><span class="section-number-3">4.2</span> Identification</h3>
<div class="outline-text-3" id="text-4-2"> <div class="outline-text-3" id="text-4-2">
<p> <p>
Let's initialize the system prior to identification. Let's initialize the system prior to identification.
@ -862,39 +906,39 @@ G_rmc.OutputName = <span class="org-rainbow-delimiters-depth-1">{</span><span cl
</div> </div>
<div id="outline-container-org0211838" class="outline-3"> <div id="outline-container-org7669eee" class="outline-3">
<h3 id="org0211838"><span class="section-number-3">4.3</span> Sensitivity to Disturbance</h3> <h3 id="org7669eee"><span class="section-number-3">4.3</span> Sensitivity to Disturbance</h3>
<div class="outline-text-3" id="text-4-3"> <div class="outline-text-3" id="text-4-3">
<div id="org00d0d6e" class="figure"> <div id="orga53e45b" class="figure">
<p><img src="figs/uniaxial_sensitivity_dist_rmc.png" alt="uniaxial_sensitivity_dist_rmc.png" /> <p><img src="figs/uniaxial_sensitivity_dist_rmc.png" alt="uniaxial_sensitivity_dist_rmc.png" />
</p> </p>
<p><span class="figure-number">Figure 15: </span>Sensitivity to disturbance once the RMC controller is applied to the system (<a href="./figs/uniaxial_sensitivity_dist_rmc.png">png</a>, <a href="./figs/uniaxial_sensitivity_dist_rmc.pdf">pdf</a>)</p> <p><span class="figure-number">Figure 17: </span>Sensitivity to disturbance once the RMC controller is applied to the system (<a href="./figs/uniaxial_sensitivity_dist_rmc.png">png</a>, <a href="./figs/uniaxial_sensitivity_dist_rmc.pdf">pdf</a>)</p>
</div> </div>
<div id="org0006b16" class="figure"> <div id="orgb21d169" class="figure">
<p><img src="figs/uniaxial_sensitivity_dist_stages_rmc.png" alt="uniaxial_sensitivity_dist_stages_rmc.png" /> <p><img src="figs/uniaxial_sensitivity_dist_stages_rmc.png" alt="uniaxial_sensitivity_dist_stages_rmc.png" />
</p> </p>
<p><span class="figure-number">Figure 16: </span>Sensitivity to force disturbances in various stages when RMC is applied (<a href="./figs/uniaxial_sensitivity_dist_stages_rmc.png">png</a>, <a href="./figs/uniaxial_sensitivity_dist_stages_rmc.pdf">pdf</a>)</p> <p><span class="figure-number">Figure 18: </span>Sensitivity to force disturbances in various stages when RMC is applied (<a href="./figs/uniaxial_sensitivity_dist_stages_rmc.png">png</a>, <a href="./figs/uniaxial_sensitivity_dist_stages_rmc.pdf">pdf</a>)</p>
</div> </div>
</div> </div>
</div> </div>
<div id="outline-container-orgefb061f" class="outline-3"> <div id="outline-container-orgab2a3d1" class="outline-3">
<h3 id="orgefb061f"><span class="section-number-3">4.4</span> Damped Plant</h3> <h3 id="orgab2a3d1"><span class="section-number-3">4.4</span> Damped Plant</h3>
<div class="outline-text-3" id="text-4-4"> <div class="outline-text-3" id="text-4-4">
<div id="org2092a67" class="figure"> <div id="orgc8a382a" class="figure">
<p><img src="figs/uniaxial_plant_rmc_damped.png" alt="uniaxial_plant_rmc_damped.png" /> <p><img src="figs/uniaxial_plant_rmc_damped.png" alt="uniaxial_plant_rmc_damped.png" />
</p> </p>
<p><span class="figure-number">Figure 17: </span>Damped Plant after RMC is applied (<a href="./figs/uniaxial_plant_rmc_damped.png">png</a>, <a href="./figs/uniaxial_plant_rmc_damped.pdf">pdf</a>)</p> <p><span class="figure-number">Figure 19: </span>Damped Plant after RMC is applied (<a href="./figs/uniaxial_plant_rmc_damped.png">png</a>, <a href="./figs/uniaxial_plant_rmc_damped.pdf">pdf</a>)</p>
</div> </div>
</div> </div>
</div> </div>
<div id="outline-container-org467a5d6" class="outline-3"> <div id="outline-container-orgbc19b27" class="outline-3">
<h3 id="org467a5d6"><span class="section-number-3">4.5</span> Conclusion</h3> <h3 id="orgbc19b27"><span class="section-number-3">4.5</span> Conclusion</h3>
<div class="outline-text-3" id="text-4-5"> <div class="outline-text-3" id="text-4-5">
<div class="important"> <div class="important">
<p> <p>
@ -906,25 +950,25 @@ Relative Motion Control:
</div> </div>
</div> </div>
<div id="outline-container-org408eed0" class="outline-2"> <div id="outline-container-org264e55c" class="outline-2">
<h2 id="org408eed0"><span class="section-number-2">5</span> Direct Velocity Feedback</h2> <h2 id="org264e55c"><span class="section-number-2">5</span> Direct Velocity Feedback</h2>
<div class="outline-text-2" id="text-5"> <div class="outline-text-2" id="text-5">
<p> <p>
<a id="orgfc1ffa1"></a> <a id="org96d3a12"></a>
</p> </p>
<p> <p>
In the Relative Motion Control (RMC), a feedback is applied between the measured velocity of the platform to the actuator force input. In the Relative Motion Control (RMC), a feedback is applied between the measured velocity of the platform to the actuator force input.
</p> </p>
<div id="org070b73d" class="figure"> <div id="org0856bca" class="figure">
<p><img src="figs/uniaxial-model-nass-flexible-dvf.png" alt="uniaxial-model-nass-flexible-dvf.png" /> <p><img src="figs/uniaxial-model-nass-flexible-dvf.png" alt="uniaxial-model-nass-flexible-dvf.png" />
</p> </p>
<p><span class="figure-number">Figure 18: </span>Uniaxial DVF Control Schematic</p> <p><span class="figure-number">Figure 20: </span>Uniaxial DVF Control Schematic</p>
</div> </div>
</div> </div>
<div id="outline-container-org64e7b3f" class="outline-3"> <div id="outline-container-orge1aa5cd" class="outline-3">
<h3 id="org64e7b3f"><span class="section-number-3">5.1</span> Control Design</h3> <h3 id="orge1aa5cd"><span class="section-number-3">5.1</span> Control Design</h3>
<div class="outline-text-3" id="text-5-1"> <div class="outline-text-3" id="text-5-1">
<div class="org-src-container"> <div class="org-src-container">
<pre class="src src-matlab">load<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-string">'./uniaxial/mat/plants.mat'</span>, <span class="org-string">'G'</span><span class="org-rainbow-delimiters-depth-1">)</span>; <pre class="src src-matlab">load<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-string">'./uniaxial/mat/plants.mat'</span>, <span class="org-string">'G'</span><span class="org-rainbow-delimiters-depth-1">)</span>;
@ -932,10 +976,10 @@ In the Relative Motion Control (RMC), a feedback is applied between the measured
</div> </div>
<div id="org56e8509" class="figure"> <div id="orga963d24" class="figure">
<p><img src="figs/uniaxial_dvf_plant.png" alt="uniaxial_dvf_plant.png" /> <p><img src="figs/uniaxial_dvf_plant.png" alt="uniaxial_dvf_plant.png" />
</p> </p>
<p><span class="figure-number">Figure 19: </span>Transfer function from forces applied in the legs to leg velocity sensor (<a href="./figs/uniaxial_dvf_plant.png">png</a>, <a href="./figs/uniaxial_dvf_plant.pdf">pdf</a>)</p> <p><span class="figure-number">Figure 21: </span>Transfer function from forces applied in the legs to leg velocity sensor (<a href="./figs/uniaxial_dvf_plant.png">png</a>, <a href="./figs/uniaxial_dvf_plant.pdf">pdf</a>)</p>
</div> </div>
<div class="org-src-container"> <div class="org-src-container">
@ -944,16 +988,16 @@ In the Relative Motion Control (RMC), a feedback is applied between the measured
</div> </div>
<div id="orgc80a1c2" class="figure"> <div id="org84510c2" class="figure">
<p><img src="figs/uniaxial_dvf_loop_gain.png" alt="uniaxial_dvf_loop_gain.png" /> <p><img src="figs/uniaxial_dvf_loop_gain.png" alt="uniaxial_dvf_loop_gain.png" />
</p> </p>
<p><span class="figure-number">Figure 20: </span>Transfer function from forces applied in the legs to leg velocity sensor (<a href="./figs/uniaxial_dvf_loop_gain.png">png</a>, <a href="./figs/uniaxial_dvf_loop_gain.pdf">pdf</a>)</p> <p><span class="figure-number">Figure 22: </span>Transfer function from forces applied in the legs to leg velocity sensor (<a href="./figs/uniaxial_dvf_loop_gain.png">png</a>, <a href="./figs/uniaxial_dvf_loop_gain.pdf">pdf</a>)</p>
</div> </div>
</div> </div>
</div> </div>
<div id="outline-container-orga75fa6d" class="outline-3"> <div id="outline-container-org2a880b1" class="outline-3">
<h3 id="orga75fa6d"><span class="section-number-3">5.2</span> Identification</h3> <h3 id="org2a880b1"><span class="section-number-3">5.2</span> Identification</h3>
<div class="outline-text-3" id="text-5-2"> <div class="outline-text-3" id="text-5-2">
<p> <p>
Let's initialize the system prior to identification. Let's initialize the system prior to identification.
@ -1036,39 +1080,39 @@ G_dvf.OutputName = <span class="org-rainbow-delimiters-depth-1">{</span><span cl
</div> </div>
</div> </div>
<div id="outline-container-org0d535fa" class="outline-3"> <div id="outline-container-orgd7e4638" class="outline-3">
<h3 id="org0d535fa"><span class="section-number-3">5.3</span> Sensitivity to Disturbance</h3> <h3 id="orgd7e4638"><span class="section-number-3">5.3</span> Sensitivity to Disturbance</h3>
<div class="outline-text-3" id="text-5-3"> <div class="outline-text-3" id="text-5-3">
<div id="org30e1316" class="figure"> <div id="org5a094e1" class="figure">
<p><img src="figs/uniaxial_sensitivity_dist_dvf.png" alt="uniaxial_sensitivity_dist_dvf.png" /> <p><img src="figs/uniaxial_sensitivity_dist_dvf.png" alt="uniaxial_sensitivity_dist_dvf.png" />
</p> </p>
<p><span class="figure-number">Figure 21: </span>Sensitivity to disturbance once the DVF controller is applied to the system (<a href="./figs/uniaxial_sensitivity_dist_dvf.png">png</a>, <a href="./figs/uniaxial_sensitivity_dist_dvf.pdf">pdf</a>)</p> <p><span class="figure-number">Figure 23: </span>Sensitivity to disturbance once the DVF controller is applied to the system (<a href="./figs/uniaxial_sensitivity_dist_dvf.png">png</a>, <a href="./figs/uniaxial_sensitivity_dist_dvf.pdf">pdf</a>)</p>
</div> </div>
<div id="orge40e605" class="figure"> <div id="orga67a694" class="figure">
<p><img src="figs/uniaxial_sensitivity_dist_stages_dvf.png" alt="uniaxial_sensitivity_dist_stages_dvf.png" /> <p><img src="figs/uniaxial_sensitivity_dist_stages_dvf.png" alt="uniaxial_sensitivity_dist_stages_dvf.png" />
</p> </p>
<p><span class="figure-number">Figure 22: </span>Sensitivity to force disturbances in various stages when DVF is applied (<a href="./figs/uniaxial_sensitivity_dist_stages_dvf.png">png</a>, <a href="./figs/uniaxial_sensitivity_dist_stages_dvf.pdf">pdf</a>)</p> <p><span class="figure-number">Figure 24: </span>Sensitivity to force disturbances in various stages when DVF is applied (<a href="./figs/uniaxial_sensitivity_dist_stages_dvf.png">png</a>, <a href="./figs/uniaxial_sensitivity_dist_stages_dvf.pdf">pdf</a>)</p>
</div> </div>
</div> </div>
</div> </div>
<div id="outline-container-org9643807" class="outline-3"> <div id="outline-container-org9524bf4" class="outline-3">
<h3 id="org9643807"><span class="section-number-3">5.4</span> Damped Plant</h3> <h3 id="org9524bf4"><span class="section-number-3">5.4</span> Damped Plant</h3>
<div class="outline-text-3" id="text-5-4"> <div class="outline-text-3" id="text-5-4">
<div id="org48982d0" class="figure"> <div id="org6b78506" class="figure">
<p><img src="figs/uniaxial_plant_dvf_damped.png" alt="uniaxial_plant_dvf_damped.png" /> <p><img src="figs/uniaxial_plant_dvf_damped.png" alt="uniaxial_plant_dvf_damped.png" />
</p> </p>
<p><span class="figure-number">Figure 23: </span>Damped Plant after DVF is applied (<a href="./figs/uniaxial_plant_dvf_damped.png">png</a>, <a href="./figs/uniaxial_plant_dvf_damped.pdf">pdf</a>)</p> <p><span class="figure-number">Figure 25: </span>Damped Plant after DVF is applied (<a href="./figs/uniaxial_plant_dvf_damped.png">png</a>, <a href="./figs/uniaxial_plant_dvf_damped.pdf">pdf</a>)</p>
</div> </div>
</div> </div>
</div> </div>
<div id="outline-container-org6e6fd47" class="outline-3"> <div id="outline-container-org329f7b9" class="outline-3">
<h3 id="org6e6fd47"><span class="section-number-3">5.5</span> Conclusion</h3> <h3 id="org329f7b9"><span class="section-number-3">5.5</span> Conclusion</h3>
<div class="outline-text-3" id="text-5-5"> <div class="outline-text-3" id="text-5-5">
<div class="important"> <div class="important">
<p> <p>
@ -1079,12 +1123,12 @@ Direct Velocity Feedback:
</div> </div>
</div> </div>
</div> </div>
<div id="outline-container-orgd792cab" class="outline-2"> <div id="outline-container-org8ffadeb" class="outline-2">
<h2 id="orgd792cab"><span class="section-number-2">6</span> With Cedrat Piezo-electric Actuators</h2> <h2 id="org8ffadeb"><span class="section-number-2">6</span> With Cedrat Piezo-electric Actuators</h2>
<div class="outline-text-2" id="text-6"> <div class="outline-text-2" id="text-6">
</div> </div>
<div id="outline-container-org7707a0a" class="outline-3"> <div id="outline-container-org8f92e1b" class="outline-3">
<h3 id="org7707a0a"><span class="section-number-3">6.1</span> Identification</h3> <h3 id="org8f92e1b"><span class="section-number-3">6.1</span> Identification</h3>
<div class="outline-text-3" id="text-6-1"> <div class="outline-text-3" id="text-6-1">
<p> <p>
We identify the dynamics of the system. We identify the dynamics of the system.
@ -1139,18 +1183,18 @@ G.OutputName = <span class="org-rainbow-delimiters-depth-1">{</span><span class=
</div> </div>
</div> </div>
<div id="outline-container-orgd921ae7" class="outline-3"> <div id="outline-container-org9183f23" class="outline-3">
<h3 id="orgd921ae7"><span class="section-number-3">6.2</span> Control Design</h3> <h3 id="org9183f23"><span class="section-number-3">6.2</span> Control Design</h3>
<div class="outline-text-3" id="text-6-2"> <div class="outline-text-3" id="text-6-2">
<p> <p>
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. 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.
</p> </p>
<div id="org8bae400" class="figure"> <div id="org8d3fc99" class="figure">
<p><img src="figs/uniaxial_cedrat_plant.png" alt="uniaxial_cedrat_plant.png" /> <p><img src="figs/uniaxial_cedrat_plant.png" alt="uniaxial_cedrat_plant.png" />
</p> </p>
<p><span class="figure-number">Figure 24: </span>Transfer function from forces applied in the legs to force sensor (<a href="./figs/uniaxial_cedrat_plant.png">png</a>, <a href="./figs/uniaxial_cedrat_plant.pdf">pdf</a>)</p> <p><span class="figure-number">Figure 26: </span>Transfer function from forces applied in the legs to force sensor (<a href="./figs/uniaxial_cedrat_plant.png">png</a>, <a href="./figs/uniaxial_cedrat_plant.pdf">pdf</a>)</p>
</div> </div>
<p> <p>
@ -1162,16 +1206,16 @@ The controller for each pair of actuator/sensor is:
</div> </div>
<div id="org0e53970" class="figure"> <div id="orgc5810ab" class="figure">
<p><img src="figs/uniaxial_cedrat_open_loop.png" alt="uniaxial_cedrat_open_loop.png" /> <p><img src="figs/uniaxial_cedrat_open_loop.png" alt="uniaxial_cedrat_open_loop.png" />
</p> </p>
<p><span class="figure-number">Figure 25: </span>Loop Gain for the Integral Force Feedback (<a href="./figs/uniaxial_cedrat_open_loop.png">png</a>, <a href="./figs/uniaxial_cedrat_open_loop.pdf">pdf</a>)</p> <p><span class="figure-number">Figure 27: </span>Loop Gain for the Integral Force Feedback (<a href="./figs/uniaxial_cedrat_open_loop.png">png</a>, <a href="./figs/uniaxial_cedrat_open_loop.pdf">pdf</a>)</p>
</div> </div>
</div> </div>
</div> </div>
<div id="outline-container-org1d5a39c" class="outline-3"> <div id="outline-container-orge8484c9" class="outline-3">
<h3 id="org1d5a39c"><span class="section-number-3">6.3</span> Identification</h3> <h3 id="orge8484c9"><span class="section-number-3">6.3</span> Identification</h3>
<div class="outline-text-3" id="text-6-3"> <div class="outline-text-3" id="text-6-3">
<p> <p>
Let's initialize the system prior to identification. Let's initialize the system prior to identification.
@ -1254,39 +1298,39 @@ G_cedrat.OutputName = <span class="org-rainbow-delimiters-depth-1">{</span><span
</div> </div>
</div> </div>
<div id="outline-container-orgb163c6c" class="outline-3"> <div id="outline-container-org35ee201" class="outline-3">
<h3 id="orgb163c6c"><span class="section-number-3">6.4</span> Sensitivity to Disturbance</h3> <h3 id="org35ee201"><span class="section-number-3">6.4</span> Sensitivity to Disturbance</h3>
<div class="outline-text-3" id="text-6-4"> <div class="outline-text-3" id="text-6-4">
<div id="org6c93b19" class="figure"> <div id="org25c9462" class="figure">
<p><img src="figs/uniaxial_sensitivity_dist_cedrat.png" alt="uniaxial_sensitivity_dist_cedrat.png" /> <p><img src="figs/uniaxial_sensitivity_dist_cedrat.png" alt="uniaxial_sensitivity_dist_cedrat.png" />
</p> </p>
<p><span class="figure-number">Figure 26: </span>Sensitivity to disturbance once the CEDRAT controller is applied to the system (<a href="./figs/uniaxial_sensitivity_dist_cedrat.png">png</a>, <a href="./figs/uniaxial_sensitivity_dist_cedrat.pdf">pdf</a>)</p> <p><span class="figure-number">Figure 28: </span>Sensitivity to disturbance once the CEDRAT controller is applied to the system (<a href="./figs/uniaxial_sensitivity_dist_cedrat.png">png</a>, <a href="./figs/uniaxial_sensitivity_dist_cedrat.pdf">pdf</a>)</p>
</div> </div>
<div id="org1b2d2df" class="figure"> <div id="org401a0e9" class="figure">
<p><img src="figs/uniaxial_sensitivity_dist_stages_cedrat.png" alt="uniaxial_sensitivity_dist_stages_cedrat.png" /> <p><img src="figs/uniaxial_sensitivity_dist_stages_cedrat.png" alt="uniaxial_sensitivity_dist_stages_cedrat.png" />
</p> </p>
<p><span class="figure-number">Figure 27: </span>Sensitivity to force disturbances in various stages when CEDRAT is applied (<a href="./figs/uniaxial_sensitivity_dist_stages_cedrat.png">png</a>, <a href="./figs/uniaxial_sensitivity_dist_stages_cedrat.pdf">pdf</a>)</p> <p><span class="figure-number">Figure 29: </span>Sensitivity to force disturbances in various stages when CEDRAT is applied (<a href="./figs/uniaxial_sensitivity_dist_stages_cedrat.png">png</a>, <a href="./figs/uniaxial_sensitivity_dist_stages_cedrat.pdf">pdf</a>)</p>
</div> </div>
</div> </div>
</div> </div>
<div id="outline-container-org552dcab" class="outline-3"> <div id="outline-container-orgfdfe26c" class="outline-3">
<h3 id="org552dcab"><span class="section-number-3">6.5</span> Damped Plant</h3> <h3 id="orgfdfe26c"><span class="section-number-3">6.5</span> Damped Plant</h3>
<div class="outline-text-3" id="text-6-5"> <div class="outline-text-3" id="text-6-5">
<div id="orge59303f" class="figure"> <div id="org96e840c" class="figure">
<p><img src="figs/uniaxial_plant_cedrat_damped.png" alt="uniaxial_plant_cedrat_damped.png" /> <p><img src="figs/uniaxial_plant_cedrat_damped.png" alt="uniaxial_plant_cedrat_damped.png" />
</p> </p>
<p><span class="figure-number">Figure 28: </span>Damped Plant after CEDRAT is applied (<a href="./figs/uniaxial_plant_cedrat_damped.png">png</a>, <a href="./figs/uniaxial_plant_cedrat_damped.pdf">pdf</a>)</p> <p><span class="figure-number">Figure 30: </span>Damped Plant after CEDRAT is applied (<a href="./figs/uniaxial_plant_cedrat_damped.png">png</a>, <a href="./figs/uniaxial_plant_cedrat_damped.pdf">pdf</a>)</p>
</div> </div>
</div> </div>
</div> </div>
<div id="outline-container-org5065aae" class="outline-3"> <div id="outline-container-org7b0ae1d" class="outline-3">
<h3 id="org5065aae"><span class="section-number-3">6.6</span> Conclusion</h3> <h3 id="org7b0ae1d"><span class="section-number-3">6.6</span> Conclusion</h3>
<div class="outline-text-3" id="text-6-6"> <div class="outline-text-3" id="text-6-6">
<div class="important"> <div class="important">
<p> <p>
@ -1298,15 +1342,15 @@ This gives similar results than with a classical force sensor.
</div> </div>
</div> </div>
<div id="outline-container-org60dfb12" class="outline-2"> <div id="outline-container-org51051c1" class="outline-2">
<h2 id="org60dfb12"><span class="section-number-2">7</span> Comparison of Active Damping Techniques</h2> <h2 id="org51051c1"><span class="section-number-2">7</span> Comparison of Active Damping Techniques</h2>
<div class="outline-text-2" id="text-7"> <div class="outline-text-2" id="text-7">
<p> <p>
<a id="orgc7002a8"></a> <a id="orgf321de8"></a>
</p> </p>
</div> </div>
<div id="outline-container-org249a650" class="outline-3"> <div id="outline-container-orga925887" class="outline-3">
<h3 id="org249a650"><span class="section-number-3">7.1</span> Load the plants</h3> <h3 id="orga925887"><span class="section-number-3">7.1</span> Load the plants</h3>
<div class="outline-text-3" id="text-7-1"> <div class="outline-text-3" id="text-7-1">
<div class="org-src-container"> <div class="org-src-container">
<pre class="src src-matlab">load<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-string">'./uniaxial/mat/plants.mat'</span>, <span class="org-string">'G'</span>, <span class="org-string">'G_iff'</span>, <span class="org-string">'G_rmc'</span>, <span class="org-string">'G_dvf'</span><span class="org-rainbow-delimiters-depth-1">)</span>; <pre class="src src-matlab">load<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-string">'./uniaxial/mat/plants.mat'</span>, <span class="org-string">'G'</span>, <span class="org-string">'G_iff'</span>, <span class="org-string">'G_rmc'</span>, <span class="org-string">'G_dvf'</span><span class="org-rainbow-delimiters-depth-1">)</span>;
@ -1315,60 +1359,125 @@ This gives similar results than with a classical force sensor.
</div> </div>
</div> </div>
<div id="outline-container-org0c1cccb" class="outline-3"> <div id="outline-container-orga01b1a9" class="outline-3">
<h3 id="org0c1cccb"><span class="section-number-3">7.2</span> Sensitivity to Disturbance</h3> <h3 id="orga01b1a9"><span class="section-number-3">7.2</span> Sensitivity to Disturbance</h3>
<div class="outline-text-3" id="text-7-2"> <div class="outline-text-3" id="text-7-2">
<div id="org9992967" class="figure"> <div id="orgafd9b97" class="figure">
<p><img src="figs/uniaxial_sensitivity_ground_motion.png" alt="uniaxial_sensitivity_ground_motion.png" /> <p><img src="figs/uniaxial_sensitivity_ground_motion.png" alt="uniaxial_sensitivity_ground_motion.png" />
</p> </p>
<p><span class="figure-number">Figure 29: </span>Sensitivity to Ground Motion - Comparison (<a href="./figs/uniaxial_sensitivity_ground_motion.png">png</a>, <a href="./figs/uniaxial_sensitivity_ground_motion.pdf">pdf</a>)</p> <p><span class="figure-number">Figure 31: </span>Sensitivity to Ground Motion - Comparison (<a href="./figs/uniaxial_sensitivity_ground_motion.png">png</a>, <a href="./figs/uniaxial_sensitivity_ground_motion.pdf">pdf</a>)</p>
</div> </div>
<div id="orgc7a133b" class="figure"> <div id="org3efc30e" class="figure">
<p><img src="figs/uniaxial_sensitivity_direct_force.png" alt="uniaxial_sensitivity_direct_force.png" /> <p><img src="figs/uniaxial_sensitivity_direct_force.png" alt="uniaxial_sensitivity_direct_force.png" />
</p> </p>
<p><span class="figure-number">Figure 30: </span>Sensitivity to disturbance - Comparison (<a href="./figs/uniaxial_sensitivity_direct_force.png">png</a>, <a href="./figs/uniaxial_sensitivity_direct_force.pdf">pdf</a>)</p> <p><span class="figure-number">Figure 32: </span>Sensitivity to disturbance - Comparison (<a href="./figs/uniaxial_sensitivity_direct_force.png">png</a>, <a href="./figs/uniaxial_sensitivity_direct_force.pdf">pdf</a>)</p>
</div> </div>
<div id="org8eaf52e" class="figure"> <div id="orgb329c3d" class="figure">
<p><img src="figs/uniaxial_sensitivity_fty.png" alt="uniaxial_sensitivity_fty.png" /> <p><img src="figs/uniaxial_sensitivity_fty.png" alt="uniaxial_sensitivity_fty.png" />
</p> </p>
<p><span class="figure-number">Figure 31: </span>Sensitivity to force disturbances - Comparison (<a href="./figs/uniaxial_sensitivity_fty.png">png</a>, <a href="./figs/uniaxial_sensitivity_fty.pdf">pdf</a>)</p> <p><span class="figure-number">Figure 33: </span>Sensitivity to force disturbances - Comparison (<a href="./figs/uniaxial_sensitivity_fty.png">png</a>, <a href="./figs/uniaxial_sensitivity_fty.pdf">pdf</a>)</p>
</div> </div>
<div id="orgb554437" class="figure"> <div id="orgc3f8a25" class="figure">
<p><img src="figs/uniaxial_sensitivity_frz.png" alt="uniaxial_sensitivity_frz.png" /> <p><img src="figs/uniaxial_sensitivity_frz.png" alt="uniaxial_sensitivity_frz.png" />
</p> </p>
<p><span class="figure-number">Figure 32: </span>Sensitivity to force disturbances - Comparison (<a href="./figs/uniaxial_sensitivity_frz.png">png</a>, <a href="./figs/uniaxial_sensitivity_frz.pdf">pdf</a>)</p> <p><span class="figure-number">Figure 34: </span>Sensitivity to force disturbances - Comparison (<a href="./figs/uniaxial_sensitivity_frz.png">png</a>, <a href="./figs/uniaxial_sensitivity_frz.pdf">pdf</a>)</p>
</div> </div>
</div> </div>
</div> </div>
<div id="outline-container-orgb54c9e3" class="outline-3"> <div id="outline-container-orga4fdd66" class="outline-3">
<h3 id="orgb54c9e3"><span class="section-number-3">7.3</span> Damped Plant</h3> <h3 id="orga4fdd66"><span class="section-number-3">7.3</span> Noise Budget</h3>
<div class="outline-text-3" id="text-7-3"> <div class="outline-text-3" id="text-7-3">
<p>
We first load the measured PSD of the disturbance.
</p>
<div class="org-src-container">
<pre class="src src-matlab">load<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-string">'./disturbances/mat/dist_psd.mat'</span>, <span class="org-string">'dist_f'</span><span class="org-rainbow-delimiters-depth-1">)</span>;
</pre>
</div>
<div id="org9375b1e" class="figure"> <p>
The effect of these disturbances on the distance \(D\) is computed for all active damping techniques.
We then compute the Cumulative Amplitude Spectrum (figure <a href="#orge43e5e4">35</a>).
</p>
<div id="orge43e5e4" class="figure">
<p><img src="figs/uniaxial-comp-cas-dist.png" alt="uniaxial-comp-cas-dist.png" />
</p>
<p><span class="figure-number">Figure 35: </span>Comparison of the Cumulative Amplitude Spectrum of \(D\) for different active damping techniques (<a href="./figs/uniaxial-comp-cas-dist.png">png</a>, <a href="./figs/uniaxial-comp-cas-dist.pdf">pdf</a>)</p>
</div>
<p>
The obtained Root Mean Square Value for each active damping technique is shown below.
</p>
<table id="org3b74f43" border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
<caption class="t-above"><span class="table-number">Table 1:</span> Obtain Root Mean Square value of \(D\) for each Active Damping Technique applied</caption>
<colgroup>
<col class="org-left" />
<col class="org-right" />
</colgroup>
<thead>
<tr>
<th scope="col" class="org-left">&#xa0;</th>
<th scope="col" class="org-right">D [m rms]</th>
</tr>
</thead>
<tbody>
<tr>
<td class="org-left">OL</td>
<td class="org-right">3.38e-06</td>
</tr>
<tr>
<td class="org-left">IFF</td>
<td class="org-right">3.40e-06</td>
</tr>
<tr>
<td class="org-left">RMC</td>
<td class="org-right">3.37e-06</td>
</tr>
<tr>
<td class="org-left">DVF</td>
<td class="org-right">3.38e-06</td>
</tr>
</tbody>
</table>
<p>
It is important to note that the effect of direct forces applied to the sample are not taken into account here.
</p>
</div>
</div>
<div id="outline-container-orgbb2291d" class="outline-3">
<h3 id="orgbb2291d"><span class="section-number-3">7.4</span> Damped Plant</h3>
<div class="outline-text-3" id="text-7-4">
<div id="org3213591" class="figure">
<p><img src="figs/uniaxial_plant_damped_comp.png" alt="uniaxial_plant_damped_comp.png" /> <p><img src="figs/uniaxial_plant_damped_comp.png" alt="uniaxial_plant_damped_comp.png" />
</p> </p>
<p><span class="figure-number">Figure 33: </span>Damped Plant - Comparison (<a href="./figs/uniaxial_plant_damped_comp.png">png</a>, <a href="./figs/uniaxial_plant_damped_comp.pdf">pdf</a>)</p> <p><span class="figure-number">Figure 36: </span>Damped Plant - Comparison (<a href="./figs/uniaxial_plant_damped_comp.png">png</a>, <a href="./figs/uniaxial_plant_damped_comp.pdf">pdf</a>)</p>
</div> </div>
</div> </div>
</div> </div>
<div id="outline-container-org1c67523" class="outline-3"> <div id="outline-container-org40f7a4d" class="outline-3">
<h3 id="org1c67523"><span class="section-number-3">7.4</span> Conclusion</h3> <h3 id="org40f7a4d"><span class="section-number-3">7.5</span> Conclusion</h3>
<div class="outline-text-3" id="text-7-4"> <div class="outline-text-3" id="text-7-5">
<p> <table id="org5ad7ed4" border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
#name: tab:active<sub>damping</sub><sub>comparison</sub> <caption class="t-above"><span class="table-number">Table 2:</span> Comparison of proposed active damping techniques</caption>
</p>
<table border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
<caption class="t-above"><span class="table-number">Table 1:</span> Comparison of proposed active damping techniques</caption>
<colgroup> <colgroup>
<col class="org-left" /> <col class="org-left" />
@ -1422,22 +1531,22 @@ This gives similar results than with a classical force sensor.
<td class="org-left">-</td> <td class="org-left">-</td>
<td class="org-left">+</td> <td class="org-left">+</td>
</tr> </tr>
<tr>
<td class="org-left">Overall RMS of \(D\)</td>
<td class="org-left">=</td>
<td class="org-left">=</td>
<td class="org-left">=</td>
</tr>
</tbody> </tbody>
</table> </table>
<div class="important">
<p>
The next step is to take into account the power spectral density of each disturbance.
</p>
</div>
</div> </div>
</div> </div>
</div> </div>
</div> </div>
<div id="postamble" class="status"> <div id="postamble" class="status">
<p class="author">Author: Dehaeze Thomas</p> <p class="author">Author: Dehaeze Thomas</p>
<p class="date">Created: 2019-10-28 lun. 17:34</p> <p class="date">Created: 2019-11-04 lun. 17:33</p>
<p class="validation"><a href="http://validator.w3.org/check?uri=referer">Validate</a></p> <p class="validation"><a href="http://validator.w3.org/check?uri=referer">Validate</a></p>
</div> </div>
</body> </body>

160
uniaxial/index.org
View File

@ -765,12 +765,14 @@ We show several plots representing the sensitivity to disturbances:
subplot(2, 1, 1); subplot(2, 1, 1);
title('$F_{ty}$ to $D$'); title('$F_{ty}$ to $D$');
hold on;
plot(freqs, abs(squeeze(freqresp(G('D', 'Fty'), freqs, 'Hz'))), 'k-'); plot(freqs, abs(squeeze(freqresp(G('D', 'Fty'), freqs, 'Hz'))), 'k-');
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Amplitude [m/N]'); xlabel('Frequency [Hz]'); ylabel('Amplitude [m/N]'); xlabel('Frequency [Hz]');
subplot(2, 1, 2); subplot(2, 1, 2);
title('$F_{rz}$ to $D$'); title('$F_{rz}$ to $D$');
hold on;
plot(freqs, abs(squeeze(freqresp(G('D', 'Frz'), freqs, 'Hz'))), 'k-'); plot(freqs, abs(squeeze(freqresp(G('D', 'Frz'), freqs, 'Hz'))), 'k-');
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Amplitude [m/N]'); xlabel('Frequency [Hz]'); ylabel('Amplitude [m/N]'); xlabel('Frequency [Hz]');
@ -785,6 +787,81 @@ We show several plots representing the sensitivity to disturbances:
#+CAPTION: Sensitivity to disturbances ([[./figs/uniaxial-sensitivity-force-dist.png][png]], [[./figs/uniaxial-sensitivity-force-dist.pdf][pdf]]) #+CAPTION: Sensitivity to disturbances ([[./figs/uniaxial-sensitivity-force-dist.png][png]], [[./figs/uniaxial-sensitivity-force-dist.pdf][pdf]])
[[file:figs/uniaxial-sensitivity-force-dist.png]] [[file:figs/uniaxial-sensitivity-force-dist.png]]
** Noise Budget
We first load the measured PSD of the disturbance.
#+begin_src matlab
load('./disturbances/mat/dist_psd.mat', 'dist_f');
#+end_src
The effect of these disturbances on the distance $D$ is computed below.
#+begin_src matlab :exports none
% Power Spectral Density of the relative Displacement [m^2/Hz]
psd_gm_d = dist_f.psd_gm.*abs(squeeze(freqresp(G('D', 'Dw'), dist_f.f, 'Hz'))).^2;
psd_ty_d = dist_f.psd_ty.*abs(squeeze(freqresp(G('D', 'Fty'), dist_f.f, 'Hz'))).^2;
psd_rz_d = dist_f.psd_rz.*abs(squeeze(freqresp(G('D', 'Frz'), dist_f.f, 'Hz'))).^2;
#+end_src
The PSD of the obtain distance $D$ due to each of the perturbation is shown in figure [[fig:uniaxial-psd-dist]] and the Cumulative Amplitude Spectrum is shown in figure [[fig:uniaxial-cas-dist]].
The Root Mean Square value of the obtained displacement $D$ is computed below and can be determined from the figure [[fig:uniaxial-cas-dist]].
#+begin_src matlab :results value replace :exports results
cas_tot_d = sqrt(cumtrapz(dist_f.f, psd_rz_d+psd_ty_d+psd_gm_d)); cas_tot_d(end)
#+end_src
#+RESULTS:
: 3.3793e-06
#+begin_src matlab :exports none
freqs = logspace(0, 3, 1000);
figure;
hold on;
plot(dist_f.f, psd_gm_d, 'DisplayName', '$D_w$');
plot(dist_f.f, psd_ty_d, 'DisplayName', '$T_y$');
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]');
legend('location', 'northeast')
xlim([0.5, 500]);
#+end_src
#+HEADER: :tangle no :exports results :results none :noweb yes
#+begin_src matlab :var filepath="figs/uniaxial-psd-dist.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png")
<<plt-matlab>>
#+end_src
#+NAME: fig:uniaxial-psd-dist
#+CAPTION: caption ([[./figs/uniaxial-psd-dist.png][png]], [[./figs/uniaxial-psd-dist.pdf][pdf]])
[[file:figs/uniaxial-psd-dist.png]]
#+begin_src matlab :exports none
freqs = logspace(0, 3, 1000);
figure;
hold on;
plot(dist_f.f, flip(sqrt(-cumtrapz(flip(dist_f.f), flip(psd_gm_d)))), 'DisplayName', '$D_w$');
plot(dist_f.f, flip(sqrt(-cumtrapz(flip(dist_f.f), flip(psd_ty_d)))), 'DisplayName', '$T_y$');
plot(dist_f.f, flip(sqrt(-cumtrapz(flip(dist_f.f), flip(psd_rz_d)))), 'DisplayName', '$R_z$');
plot(dist_f.f, flip(sqrt(-cumtrapz(flip(dist_f.f), flip(psd_gm_d+psd_ty_d+psd_rz_d)))), 'k-', 'DisplayName', 'tot');
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('CAS of the effect of disturbances on $D$ [m]'); xlabel('Frequency [Hz]');
legend('location', 'northeast')
xlim([0.5, 500]); ylim([1e-12, 1e-6]);
#+end_src
#+HEADER: :tangle no :exports results :results none :noweb yes
#+begin_src matlab :var filepath="figs/uniaxial-cas-dist.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png")
<<plt-matlab>>
#+end_src
#+NAME: fig:uniaxial-cas-dist
#+CAPTION: caption ([[./figs/uniaxial-cas-dist.png][png]], [[./figs/uniaxial-cas-dist.pdf][pdf]])
[[file:figs/uniaxial-cas-dist.png]]
** Plant ** Plant
The transfer function from the force $F$ applied by the nano-hexapod to the position of the sample $D$ is shown in figure [[fig:uniaxial-plant]]. The transfer function from the force $F$ applied by the nano-hexapod to the position of the sample $D$ is shown in figure [[fig:uniaxial-plant]].
It corresponds to the plant to control. It corresponds to the plant to control.
@ -2543,6 +2620,82 @@ This gives similar results than with a classical force sensor.
#+CAPTION: Sensitivity to force disturbances - Comparison ([[./figs/uniaxial_sensitivity_frz.png][png]], [[./figs/uniaxial_sensitivity_frz.pdf][pdf]]) #+CAPTION: Sensitivity to force disturbances - Comparison ([[./figs/uniaxial_sensitivity_frz.png][png]], [[./figs/uniaxial_sensitivity_frz.pdf][pdf]])
[[file:figs/uniaxial_sensitivity_frz.png]] [[file:figs/uniaxial_sensitivity_frz.png]]
** Noise Budget
We first load the measured PSD of the disturbance.
#+begin_src matlab
load('./disturbances/mat/dist_psd.mat', 'dist_f');
#+end_src
The effect of these disturbances on the distance $D$ is computed for all active damping techniques.
#+begin_src matlab :exports none
% Power Spectral Density of the relative Displacement [m^2/Hz]
psd_ol_gm_d = dist_f.psd_gm.*abs(squeeze(freqresp(G('D', 'Dw'), dist_f.f, 'Hz'))).^2;
psd_ol_ty_d = dist_f.psd_ty.*abs(squeeze(freqresp(G('D', 'Fty'), dist_f.f, 'Hz'))).^2;
psd_ol_rz_d = dist_f.psd_rz.*abs(squeeze(freqresp(G('D', 'Frz'), dist_f.f, 'Hz'))).^2;
psd_iff_gm_d = dist_f.psd_gm.*abs(squeeze(freqresp(G_iff('D', 'Dw'), dist_f.f, 'Hz'))).^2;
psd_iff_ty_d = dist_f.psd_ty.*abs(squeeze(freqresp(G_iff('D', 'Fty'), dist_f.f, 'Hz'))).^2;
psd_iff_rz_d = dist_f.psd_rz.*abs(squeeze(freqresp(G_iff('D', 'Frz'), dist_f.f, 'Hz'))).^2;
psd_rmc_gm_d = dist_f.psd_gm.*abs(squeeze(freqresp(G_rmc('D', 'Dw'), dist_f.f, 'Hz'))).^2;
psd_rmc_ty_d = dist_f.psd_ty.*abs(squeeze(freqresp(G_rmc('D', 'Fty'), dist_f.f, 'Hz'))).^2;
psd_rmc_rz_d = dist_f.psd_rz.*abs(squeeze(freqresp(G_rmc('D', 'Frz'), dist_f.f, 'Hz'))).^2;
psd_dvf_gm_d = dist_f.psd_gm.*abs(squeeze(freqresp(G_dvf('D', 'Dw'), dist_f.f, 'Hz'))).^2;
psd_dvf_ty_d = dist_f.psd_ty.*abs(squeeze(freqresp(G_dvf('D', 'Fty'), dist_f.f, 'Hz'))).^2;
psd_dvf_rz_d = dist_f.psd_rz.*abs(squeeze(freqresp(G_dvf('D', 'Frz'), dist_f.f, 'Hz'))).^2;
#+end_src
We then compute the Cumulative Amplitude Spectrum (figure [[fig:uniaxial-comp-cas-dist]]).
#+begin_src matlab :exports none
cas_ol_tot_d = flip(sqrt(-cumtrapz(flip(dist_f.f), flip(psd_ol_rz_d+psd_ol_ty_d+psd_ol_gm_d))));
cas_iff_tot_d = flip(sqrt(-cumtrapz(flip(dist_f.f), flip(psd_iff_rz_d+psd_iff_ty_d+psd_iff_gm_d))));
cas_rmc_tot_d = flip(sqrt(-cumtrapz(flip(dist_f.f), flip(psd_rmc_rz_d+psd_rmc_ty_d+psd_rmc_gm_d))));
cas_dvf_tot_d = flip(sqrt(-cumtrapz(flip(dist_f.f), flip(psd_dvf_rz_d+psd_dvf_ty_d+psd_dvf_gm_d))));
#+end_src
#+begin_src matlab :exports none
freqs = logspace(0, 3, 1000);
figure;
hold on;
plot(dist_f.f, cas_ol_tot_d, 'DisplayName', 'OL');
plot(dist_f.f, cas_iff_tot_d, 'DisplayName', 'IFF');
plot(dist_f.f, cas_rmc_tot_d, 'DisplayName', 'RMC');
plot(dist_f.f, cas_dvf_tot_d, 'DisplayName', 'DVF');
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('CAS of the effect of disturbances on $D$ [m]'); xlabel('Frequency [Hz]');
legend('location', 'northeast')
xlim([0.5, 500]); ylim([1e-11, 1e-6]);
#+end_src
#+HEADER: :tangle no :exports results :results none :noweb yes
#+begin_src matlab :var filepath="figs/uniaxial-comp-cas-dist.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png")
<<plt-matlab>>
#+end_src
#+NAME: fig:uniaxial-comp-cas-dist
#+CAPTION: Comparison of the Cumulative Amplitude Spectrum of $D$ for different active damping techniques ([[./figs/uniaxial-comp-cas-dist.png][png]], [[./figs/uniaxial-comp-cas-dist.pdf][pdf]])
[[file:figs/uniaxial-comp-cas-dist.png]]
The obtained Root Mean Square Value for each active damping technique is shown below.
#+begin_src matlab :exports results :results value table replace :tangle no :post addhdr(*this*)
data2orgtable([cas_ol_tot_d(1); cas_iff_tot_d(1); cas_rmc_tot_d(1); cas_dvf_tot_d(1)], {'OL', 'IFF', 'RMC', 'DVF'}, {'D [m rms]'}, ' %.2e ');
#+end_src
#+name: tab:rms_d_comp
#+caption: Obtain Root Mean Square value of $D$ for each Active Damping Technique applied
#+RESULTS:
| | D [m rms] |
|-----+-----------|
| OL | 3.38e-06 |
| IFF | 3.40e-06 |
| RMC | 3.37e-06 |
| DVF | 3.38e-06 |
It is important to note that the effect of direct forces applied to the sample are not taken into account here.
** Damped Plant ** Damped Plant
#+begin_src matlab :exports none #+begin_src matlab :exports none
freqs = logspace(0, 3, 1000); freqs = logspace(0, 3, 1000);
@ -2586,7 +2739,7 @@ This gives similar results than with a classical force sensor.
** Conclusion ** Conclusion
#name: tab:active_damping_comparison #+name: tab:active_damping_comparison
#+caption: Comparison of proposed active damping techniques #+caption: Comparison of proposed active damping techniques
| | IFF | RMC | DVF | | | IFF | RMC | DVF |
|---------------------------+-----------------+-----------------+----------| |---------------------------+-----------------+-----------------+----------|
@ -2595,7 +2748,4 @@ This gives similar results than with a classical force sensor.
| Sensitivity ($D_w$) | - | + | - | | Sensitivity ($D_w$) | - | + | - |
| Sensitivity ($F_s$) | - (at low freq) | + | + | | Sensitivity ($F_s$) | - (at low freq) | + | + |
| Sensitivity ($F_{ty,rz}$) | + | - | + | | Sensitivity ($F_{ty,rz}$) | + | - | + |
| Overall RMS of $D$ | = | = | = |
#+begin_important
The next step is to take into account the power spectral density of each disturbance.
#+end_important

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