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<title>ESRF Double Crystal Monochromator - Dynamical Multi-Body Model</title>
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<h1 class="title">ESRF Double Crystal Monochromator - Dynamical Multi-Body Model</h1>
<div id="table-of-contents" role="doc-toc">
<h2>Table of Contents</h2>
<div id="text-table-of-contents" role="doc-toc">
<ul>
<li><a href="#orge27c493">1. System Kinematics</a>
<ul>
<li><a href="#orge374fa1">1.1. Bragg Angle</a></li>
<li><a href="#orgc14dc65">1.2. Kinematics (311 Crystal)</a>
<ul>
<li><a href="#org6e1caa8">1.2.1. Interferometers - 311 secondary Crystal</a></li>
<li><a href="#orga890b46">1.2.2. Interferometers - 311 primary Crystal</a></li>
<li><a href="#org60df86c">1.2.3. Piezo - 311 Crystal</a></li>
</ul>
</li>
<li><a href="#orgc5862d9">1.3. Kinematics (111 Crystal)</a>
<ul>
<li><a href="#org8a389fe">1.3.1. Interferometers - 111 secondary Crystal</a></li>
<li><a href="#org080d541">1.3.2. Interferometers - 111 primary Crystal</a></li>
<li><a href="#org2b65143">1.3.3. Piezo - 111 Crystal</a></li>
</ul>
</li>
<li><a href="#org8e6b78b">1.4. Kinematics (Metrology Frame)</a></li>
<li><a href="#org74a3774">1.5. Verification</a></li>
<li><a href="#org02c0c35">1.6. Save Kinematics</a></li>
</ul>
</li>
<li><a href="#orgb814544">2. Open Loop System Identification</a>
<ul>
<li><a href="#org588c7cb">2.1. Identification</a></li>
<li><a href="#orgb350e31">2.2. Plant in the frame of the fastjacks</a></li>
<li><a href="#org8c5f00e">2.3. Plant in the frame of the crystal</a></li>
</ul>
</li>
<li><a href="#orga6970e2">3. Open-Loop Noise Budgeting</a>
<ul>
<li><a href="#org540e081">3.1. Power Spectral Density of signals</a></li>
<li><a href="#org28b9340">3.2. Open Loop disturbance and measurement noise</a></li>
</ul>
</li>
<li><a href="#org5d1548f">4. Active Damping Plant (Strain gauges)</a>
<ul>
<li><a href="#org6d76aa1">4.1. Identification</a></li>
<li><a href="#org3f2c1bd">4.2. Relative Active Damping</a></li>
<li><a href="#orgd9be2f7">4.3. Damped Plant</a></li>
</ul>
</li>
<li><a href="#org0f90dfd">5. Active Damping Plant (Force Sensors)</a>
<ul>
<li><a href="#org2b0e4b2">5.1. Identification</a></li>
<li><a href="#org7a5017e">5.2. Controller - Root Locus</a></li>
<li><a href="#org8e7f5c7">5.3. Damped Plant</a></li>
</ul>
</li>
<li><a href="#orgb450951">6. Feedback Control</a></li>
<li><a href="#org0bdc80c">7. HAC-LAC (IFF) architecture</a>
<ul>
<li><a href="#org3d59fea">7.1. System Identification</a></li>
<li><a href="#org66fa33e">7.2. High Authority Controller</a></li>
<li><a href="#orge129b2d">7.3. Performances</a></li>
<li><a href="#org7677800">7.4. Close Loop noise budget</a></li>
</ul>
</li>
</ul>
</div>
</div>
<hr>
<p>This report is also available as a <a href="./dcm-simscape-model.pdf">pdf</a>.</p>
<hr>
<p>
In this document, a Simscape (.e.g. multi-body) model of the ESRF Double Crystal Monochromator (DCM) is presented and used to develop and optimize the control strategy.
</p>
<p>
It is structured as follow:
</p>
<ul class="org-ul">
<li>Section <a href="#orgd6a7599">1</a>: the kinematics of the DCM is presented, and Jacobian matrices which are used to solve the inverse and forward kinematics are computed.</li>
<li>Section <a href="#orgdfb3b70">2</a>: the system dynamics is identified in the absence of control.</li>
<li>Section <a href="#org1f95dda">3</a>: an open-loop noise budget is performed.</li>
<li>Section <a href="#orge98df39">4</a>: it is studied whether if the strain gauges fixed to the piezoelectric actuators can be used to actively damp the plant.</li>
<li>Section <a href="#org8c440d8">5</a>: piezoelectric force sensors are added in series with the piezoelectric actuators and are used to actively damp the plant using the Integral Force Feedback (IFF) control strategy.</li>
<li>Section <a href="#orga62d12a">7</a>: the High Authority Control - Low Authority Control (HAC-LAC) strategy is tested on the Simscape model.</li>
</ul>
<div id="outline-container-orge27c493" class="outline-2">
<h2 id="orge27c493"><span class="section-number-2">1.</span> System Kinematics</h2>
<div class="outline-text-2" id="text-1">
<p>
<a id="orgd6a7599"></a>
</p>
</div>
<div id="outline-container-orge374fa1" class="outline-3">
<h3 id="orge374fa1"><span class="section-number-3">1.1.</span> Bragg Angle</h3>
<div class="outline-text-3" id="text-1-1">
<p>
There is a simple relation <a href="eq:bragg_angle_formula">eq:bragg_angle_formula</a> between:
</p>
<ul class="org-ul">
<li>\(d_{\text{off}}\) is the wanted offset between the incident x-ray and the output x-ray</li>
<li>\(\theta_b\) is the bragg angle</li>
<li>\(d_z\) is the corresponding distance between the first and second crystals</li>
</ul>
\begin{equation} \label{eq:bragg_angle_formula}
d_z = \frac{d_{\text{off}}}{2 \cos \theta_b}
\end{equation}
<p>
This relation is shown in Figure <a href="#org4ff5aad">1</a>.
</p>
<div id="org4ff5aad" class="figure">
<p><img src="figs/jack_motion_bragg_angle.png" alt="jack_motion_bragg_angle.png" />
</p>
<p><span class="figure-number">Figure 1: </span>Jack motion as a function of Bragg angle</p>
</div>
<p>
The required jack stroke is approximately 25mm.
</p>
<div class="org-src-container">
<pre class="src src-matlab"><span class="org-matlab-cellbreak">%% Required Jack stroke</span>
<span class="org-matlab-math">ans</span> = 1e3<span class="org-builtin">*</span>(dz(end) <span class="org-builtin">-</span> dz(1))
</pre>
</div>
<pre class="example">
24.963
</pre>
</div>
</div>
<div id="outline-container-orgc14dc65" class="outline-3">
<h3 id="orgc14dc65"><span class="section-number-3">1.2.</span> Kinematics (311 Crystal)</h3>
<div class="outline-text-3" id="text-1-2">
<p>
The reference frame is taken at the center of the 311 second crystal.
</p>
</div>
<div id="outline-container-org6e1caa8" class="outline-4">
<h4 id="org6e1caa8"><span class="section-number-4">1.2.1.</span> Interferometers - 311 secondary Crystal</h4>
<div class="outline-text-4" id="text-1-2-1">
<p>
Three interferometers are pointed to the bottom surface of the 311 crystal.
</p>
<p>
The position of the measurement points are shown in Figure <a href="#org2dcdb04">2</a> as well as the origin where the motion of the crystal is computed.
</p>
<div id="org2dcdb04" class="figure">
<p><img src="figs/sensor_311_crystal_points.png" alt="sensor_311_crystal_points.png" />
</p>
<p><span class="figure-number">Figure 2: </span>Bottom view of the second crystal 311. Position of the measurement points.</p>
</div>
<p>
The inverse kinematics consisting of deriving the interferometer measurements from the motion of the crystal (see Figure <a href="#org282e108">3</a>):
</p>
\begin{equation}
\begin{bmatrix}
x_1 \\ x_2 \\ x_3
\end{bmatrix}
=
\bm{J}_{s,311}
\begin{bmatrix}
d_z \\ r_y \\ r_x
\end{bmatrix}
\end{equation}
<div id="org282e108" class="figure">
<p><img src="figs/schematic_sensor_jacobian_inverse_kinematics_311.png" alt="schematic_sensor_jacobian_inverse_kinematics_311.png" />
</p>
<p><span class="figure-number">Figure 3: </span>Inverse Kinematics - Interferometers</p>
</div>
<p>
From the Figure <a href="#org2dcdb04">2</a>, the inverse kinematics can be solved as follow (for small motion):
</p>
\begin{equation}
\bm{J}_{s,311}
=
\begin{bmatrix}
1 & 0.07 & -0.015 \\
1 & 0 & 0.015 \\
1 & -0.07 & -0.015
\end{bmatrix}
\end{equation}
<div class="org-src-container">
<pre class="src src-matlab"><span class="org-matlab-cellbreak">%% Sensor Jacobian matrix for 311 crystal</span>
J_s_311 = [1, 0.07, <span class="org-builtin">-</span>0.015
1, 0, 0.015
1, <span class="org-builtin">-</span>0.07, <span class="org-builtin">-</span>0.015];
</pre>
</div>
<table id="org68faf8f" border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
<caption class="t-above"><span class="table-number">Table 1:</span> Sensor Jacobian \(\bm{J}_{s,311}\)</caption>
<colgroup>
<col class="org-right" />
<col class="org-right" />
<col class="org-right" />
</colgroup>
<tbody>
<tr>
<td class="org-right">1.0</td>
<td class="org-right">0.07</td>
<td class="org-right">-0.015</td>
</tr>
<tr>
<td class="org-right">1.0</td>
<td class="org-right">0.0</td>
<td class="org-right">0.015</td>
</tr>
<tr>
<td class="org-right">1.0</td>
<td class="org-right">-0.07</td>
<td class="org-right">-0.015</td>
</tr>
</tbody>
</table>
<p>
The forward kinematics is solved by inverting the Jacobian matrix (see Figure <a href="#orgf0b1992">4</a>).
</p>
\begin{equation}
\begin{bmatrix}
d_z \\ r_y \\ r_x
\end{bmatrix}
=
\bm{J}_{s,311}^{-1}
\begin{bmatrix}
x_1 \\ x_2 \\ x_3
\end{bmatrix}
\end{equation}
<div id="orgf0b1992" class="figure">
<p><img src="figs/schematic_sensor_jacobian_forward_kinematics_311.png" alt="schematic_sensor_jacobian_forward_kinematics_311.png" />
</p>
<p><span class="figure-number">Figure 4: </span>Forward Kinematics - Interferometers</p>
</div>
<table id="org25a9e49" border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
<caption class="t-above"><span class="table-number">Table 2:</span> Inverse of the sensor Jacobian \(\bm{J}_{s,311}^{-1}\)</caption>
<colgroup>
<col class="org-right" />
<col class="org-right" />
<col class="org-right" />
</colgroup>
<tbody>
<tr>
<td class="org-right">0.25</td>
<td class="org-right">0.5</td>
<td class="org-right">0.25</td>
</tr>
<tr>
<td class="org-right">7.14</td>
<td class="org-right">0.0</td>
<td class="org-right">-7.14</td>
</tr>
<tr>
<td class="org-right">-16.67</td>
<td class="org-right">33.33</td>
<td class="org-right">-16.67</td>
</tr>
</tbody>
</table>
</div>
</div>
<div id="outline-container-orga890b46" class="outline-4">
<h4 id="orga890b46"><span class="section-number-4">1.2.2.</span> Interferometers - 311 primary Crystal</h4>
<div class="outline-text-4" id="text-1-2-2">
<p>
Three interferometers are pointed to the bottom surface of the 311 crystal.
</p>
<p>
The position of the measurement points are shown in Figure <a href="#org016732d">5</a> as well as the origin where the motion of the crystal is computed.
</p>
<div id="org016732d" class="figure">
<p><img src="figs/sensor_311_crystal_points_primary.png" alt="sensor_311_crystal_points_primary.png" />
</p>
<p><span class="figure-number">Figure 5: </span>Top view of the primary crystal 311. Position of the measurement points.</p>
</div>
<p>
The inverse kinematics consisting of deriving the interferometer measurements from the motion of the crystal (see Figure <a href="#org282e108">3</a>):
</p>
\begin{equation}
\begin{bmatrix}
x_1 \\ x_2 \\ x_3
\end{bmatrix}
=
\bm{J}_{s,311}
\begin{bmatrix}
d_z \\ r_y \\ r_x
\end{bmatrix}
\end{equation}
<div id="org8235e80" class="figure">
<p><img src="figs/schematic_sensor_jacobian_inverse_kinematics_311.png" alt="schematic_sensor_jacobian_inverse_kinematics_311.png" />
</p>
<p><span class="figure-number">Figure 6: </span>Inverse Kinematics - Interferometers</p>
</div>
<p>
From the Figure <a href="#org2dcdb04">2</a>, the inverse kinematics can be solved as follow (for small motion):
</p>
\begin{equation}
\bm{J}_{s,311}
=
\begin{bmatrix}
-1 & -0.07 & 0.015 \\
-1 & 0 & -0.015 \\
-1 & 0.07 & 0.015
\end{bmatrix}
\end{equation}
<div class="org-src-container">
<pre class="src src-matlab"><span class="org-matlab-cellbreak">%% Sensor Jacobian matrix for 311 crystal</span>
J_s_311_1 = [<span class="org-builtin">-</span>1, 0.07, <span class="org-builtin">-</span>0.015
<span class="org-builtin">-</span>1, 0, 0.015
<span class="org-builtin">-</span>1, <span class="org-builtin">-</span>0.07, <span class="org-builtin">-</span>0.015];
</pre>
</div>
<table id="org193992d" border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
<caption class="t-above"><span class="table-number">Table 3:</span> Sensor Jacobian - Primary Crystal - \(\bm{J}_{s,311}\)</caption>
<colgroup>
<col class="org-right" />
<col class="org-right" />
<col class="org-right" />
</colgroup>
<tbody>
<tr>
<td class="org-right">-1.0</td>
<td class="org-right">0.07</td>
<td class="org-right">-0.015</td>
</tr>
<tr>
<td class="org-right">-1.0</td>
<td class="org-right">0.0</td>
<td class="org-right">0.015</td>
</tr>
<tr>
<td class="org-right">-1.0</td>
<td class="org-right">-0.07</td>
<td class="org-right">-0.015</td>
</tr>
</tbody>
</table>
<p>
The forward kinematics is solved by inverting the Jacobian matrix (see Figure <a href="#orgf0b1992">4</a>).
</p>
\begin{equation}
\begin{bmatrix}
d_z \\ r_y \\ r_x
\end{bmatrix}
=
\bm{J}_{s,311}^{-1}
\begin{bmatrix}
x_1 \\ x_2 \\ x_3
\end{bmatrix}
\end{equation}
<div id="org5eae2a7" class="figure">
<p><img src="figs/schematic_sensor_jacobian_forward_kinematics_311.png" alt="schematic_sensor_jacobian_forward_kinematics_311.png" />
</p>
<p><span class="figure-number">Figure 7: </span>Forward Kinematics - Interferometers</p>
</div>
<table id="org5c458f6" border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
<caption class="t-above"><span class="table-number">Table 4:</span> Inverse of the sensor Jacobian \(\bm{J}_{s,311}^{-1}\)</caption>
<colgroup>
<col class="org-right" />
<col class="org-right" />
<col class="org-right" />
</colgroup>
<tbody>
<tr>
<td class="org-right">-0.25</td>
<td class="org-right">-0.5</td>
<td class="org-right">-0.25</td>
</tr>
<tr>
<td class="org-right">7.14</td>
<td class="org-right">0.0</td>
<td class="org-right">-7.14</td>
</tr>
<tr>
<td class="org-right">-16.67</td>
<td class="org-right">33.33</td>
<td class="org-right">-16.67</td>
</tr>
</tbody>
</table>
</div>
</div>
<div id="outline-container-org60df86c" class="outline-4">
<h4 id="org60df86c"><span class="section-number-4">1.2.3.</span> Piezo - 311 Crystal</h4>
<div class="outline-text-4" id="text-1-2-3">
<p>
The location of the actuators with respect with the center of the 311 second crystal are shown in Figure <a href="#org2bdd89a">8</a>.
</p>
<div id="org2bdd89a" class="figure">
<p><img src="figs/actuator_jacobian_311_points.png" alt="actuator_jacobian_311_points.png" />
</p>
<p><span class="figure-number">Figure 8: </span>Location of actuators with respect to the center of the 311 second crystal (bottom view)</p>
</div>
<p>
Inverse Kinematics consist of deriving the axial (z) motion of the 3 actuators from the motion of the crystal&rsquo;s center.
</p>
\begin{equation}
\begin{bmatrix}
d_{u_r} \\ d_{u_h} \\ d_{d}
\end{bmatrix}
=
\bm{J}_{a,311}
\begin{bmatrix}
d_z \\ r_y \\ r_x
\end{bmatrix}
\end{equation}
<div id="orgd4ccae6" class="figure">
<p><img src="figs/schematic_actuator_jacobian_inverse_kinematics_311.png" alt="schematic_actuator_jacobian_inverse_kinematics_311.png" />
</p>
<p><span class="figure-number">Figure 9: </span>Inverse Kinematics - Actuators</p>
</div>
<p>
Based on the geometry in Figure <a href="#org2bdd89a">8</a>, we obtain:
</p>
\begin{equation}
\bm{J}_{a,311}
=
\begin{bmatrix}
1 & 0.14 & -0.1525 \\
1 & 0.14 & 0.0675 \\
1 & -0.14 & -0.0425
\end{bmatrix}
\end{equation}
<div class="org-src-container">
<pre class="src src-matlab"><span class="org-matlab-cellbreak">%% Actuator Jacobian - 311 crystal</span>
J_a_311 = [1, 0.14, <span class="org-builtin">-</span>0.1525
1, 0.14, 0.0675
1, <span class="org-builtin">-</span>0.14, <span class="org-builtin">-</span>0.0425];
</pre>
</div>
<table id="org27e50a6" border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
<caption class="t-above"><span class="table-number">Table 5:</span> Actuator Jacobian \(\bm{J}_{a,311}\)</caption>
<colgroup>
<col class="org-right" />
<col class="org-right" />
<col class="org-right" />
</colgroup>
<tbody>
<tr>
<td class="org-right">1.0</td>
<td class="org-right">0.14</td>
<td class="org-right">-0.1525</td>
</tr>
<tr>
<td class="org-right">1.0</td>
<td class="org-right">0.14</td>
<td class="org-right">0.0675</td>
</tr>
<tr>
<td class="org-right">1.0</td>
<td class="org-right">-0.14</td>
<td class="org-right">-0.0425</td>
</tr>
</tbody>
</table>
<p>
The forward Kinematics is solved by inverting the Jacobian matrix:
</p>
\begin{equation}
\begin{bmatrix}
d_z \\ r_y \\ r_x
\end{bmatrix}
=
\bm{J}_{a,311}^{-1}
\begin{bmatrix}
d_{u_r} \\ d_{u_h} \\ d_{d}
\end{bmatrix}
\end{equation}
<div id="org0308f1b" class="figure">
<p><img src="figs/schematic_actuator_jacobian_forward_kinematics_311.png" alt="schematic_actuator_jacobian_forward_kinematics_311.png" />
</p>
<p><span class="figure-number">Figure 10: </span>Forward Kinematics - Actuators for 311 crystal</p>
</div>
<table id="org13d8bc5" border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
<caption class="t-above"><span class="table-number">Table 6:</span> Inverse of the actuator Jacobian \(\bm{J}_{a,311}^{-1}\)</caption>
<colgroup>
<col class="org-right" />
<col class="org-right" />
<col class="org-right" />
</colgroup>
<tbody>
<tr>
<td class="org-right">0.0568</td>
<td class="org-right">0.4432</td>
<td class="org-right">0.5</td>
</tr>
<tr>
<td class="org-right">1.7857</td>
<td class="org-right">1.7857</td>
<td class="org-right">-3.5714</td>
</tr>
<tr>
<td class="org-right">-4.5455</td>
<td class="org-right">4.5455</td>
<td class="org-right">0.0</td>
</tr>
</tbody>
</table>
</div>
</div>
</div>
<div id="outline-container-orgc5862d9" class="outline-3">
<h3 id="orgc5862d9"><span class="section-number-3">1.3.</span> Kinematics (111 Crystal)</h3>
<div class="outline-text-3" id="text-1-3">
<p>
The reference frame is taken at the center of the 111 second crystal.
</p>
</div>
<div id="outline-container-org8a389fe" class="outline-4">
<h4 id="org8a389fe"><span class="section-number-4">1.3.1.</span> Interferometers - 111 secondary Crystal</h4>
<div class="outline-text-4" id="text-1-3-1">
<p>
Three interferometers are pointed to the bottom surface of the 111 crystal.
</p>
<p>
The position of the measurement points are shown in Figure <a href="#orgca4ae97">11</a> as well as the origin where the motion of the crystal is computed.
</p>
<div id="orgca4ae97" class="figure">
<p><img src="figs/sensor_111_crystal_points.png" alt="sensor_111_crystal_points.png" />
</p>
<p><span class="figure-number">Figure 11: </span>Bottom view of the second crystal 111. Position of the measurement points.</p>
</div>
<p>
The inverse kinematics consisting of deriving the interferometer measurements from the motion of the crystal (see Figure <a href="#org72de77d">12</a>):
</p>
\begin{equation}
\begin{bmatrix}
x_1 \\ x_2 \\ x_3
\end{bmatrix}
=
\bm{J}_{s,111}
\begin{bmatrix}
d_z \\ r_y \\ r_x
\end{bmatrix}
\end{equation}
<div id="org72de77d" class="figure">
<p><img src="figs/schematic_sensor_jacobian_inverse_kinematics_111.png" alt="schematic_sensor_jacobian_inverse_kinematics_111.png" />
</p>
<p><span class="figure-number">Figure 12: </span>Inverse Kinematics - Interferometers</p>
</div>
<p>
From the Figure <a href="#orgca4ae97">11</a>, the inverse kinematics can be solved as follow (for small motion):
</p>
\begin{equation}
\bm{J}_{s,111}
=
\begin{bmatrix}
1 & 0.07 & 0.015 \\
1 & 0 & -0.015 \\
1 & -0.07 & 0.015
\end{bmatrix}
\end{equation}
<div class="org-src-container">
<pre class="src src-matlab"><span class="org-matlab-cellbreak">%% Sensor Jacobian matrix for 111 crystal</span>
J_s_111 = [1, 0.07, 0.015
1, 0, <span class="org-builtin">-</span>0.015
1, <span class="org-builtin">-</span>0.07, 0.015];
</pre>
</div>
<table id="org4bcd25b" border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
<caption class="t-above"><span class="table-number">Table 7:</span> Sensor Jacobian \(\bm{J}_{s,111}\)</caption>
<colgroup>
<col class="org-right" />
<col class="org-right" />
<col class="org-right" />
</colgroup>
<tbody>
<tr>
<td class="org-right">1.0</td>
<td class="org-right">0.07</td>
<td class="org-right">0.015</td>
</tr>
<tr>
<td class="org-right">1.0</td>
<td class="org-right">0.0</td>
<td class="org-right">-0.015</td>
</tr>
<tr>
<td class="org-right">1.0</td>
<td class="org-right">-0.07</td>
<td class="org-right">0.015</td>
</tr>
</tbody>
</table>
<p>
The forward kinematics is solved by inverting the Jacobian matrix (see Figure <a href="#org8d6627b">13</a>).
</p>
\begin{equation}
\begin{bmatrix}
d_z \\ r_y \\ r_x
\end{bmatrix}
=
\bm{J}_{s,111}^{-1}
\begin{bmatrix}
x_1 \\ x_2 \\ x_3
\end{bmatrix}
\end{equation}
<div id="org8d6627b" class="figure">
<p><img src="figs/schematic_sensor_jacobian_forward_kinematics_111.png" alt="schematic_sensor_jacobian_forward_kinematics_111.png" />
</p>
<p><span class="figure-number">Figure 13: </span>Forward Kinematics - Interferometers</p>
</div>
<table id="orgcffa8c9" border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
<caption class="t-above"><span class="table-number">Table 8:</span> Inverse of the sensor Jacobian \(\bm{J}_{s,111}^{-1}\)</caption>
<colgroup>
<col class="org-right" />
<col class="org-right" />
<col class="org-right" />
</colgroup>
<tbody>
<tr>
<td class="org-right">0.25</td>
<td class="org-right">0.5</td>
<td class="org-right">0.25</td>
</tr>
<tr>
<td class="org-right">7.14</td>
<td class="org-right">0.0</td>
<td class="org-right">-7.14</td>
</tr>
<tr>
<td class="org-right">16.67</td>
<td class="org-right">-33.33</td>
<td class="org-right">16.67</td>
</tr>
</tbody>
</table>
</div>
</div>
<div id="outline-container-org080d541" class="outline-4">
<h4 id="org080d541"><span class="section-number-4">1.3.2.</span> Interferometers - 111 primary Crystal</h4>
<div class="outline-text-4" id="text-1-3-2">
<p>
Three interferometers are pointed to the bottom surface of the 111 crystal.
</p>
<p>
The position of the measurement points are shown in Figure <a href="#org347923f">14</a> as well as the origin where the motion of the crystal is computed.
</p>
<div id="org347923f" class="figure">
<p><img src="figs/sensor_111_crystal_points_primary.png" alt="sensor_111_crystal_points_primary.png" />
</p>
<p><span class="figure-number">Figure 14: </span>Top view of the primary crystal 111. Position of the measurement points.</p>
</div>
<p>
The inverse kinematics consisting of deriving the interferometer measurements from the motion of the crystal (see Figure <a href="#org72de77d">12</a>):
</p>
\begin{equation}
\begin{bmatrix}
x_1 \\ x_2 \\ x_3
\end{bmatrix}
=
\bm{J}_{s,111}
\begin{bmatrix}
d_z \\ r_y \\ r_x
\end{bmatrix}
\end{equation}
<div id="org9e24dcb" class="figure">
<p><img src="figs/schematic_sensor_jacobian_inverse_kinematics_111.png" alt="schematic_sensor_jacobian_inverse_kinematics_111.png" />
</p>
<p><span class="figure-number">Figure 15: </span>Inverse Kinematics - Interferometers</p>
</div>
<p>
From the Figure <a href="#orgca4ae97">11</a>, the inverse kinematics can be solved as follow (for small motion):
</p>
\begin{equation}
\bm{J}_{s,111}
=
\begin{bmatrix}
1 & -0.036 & -0.015 \\
1 & 0 & 0.015 \\
1 & 0.036 & -0.015
\end{bmatrix}
\end{equation}
<div class="org-src-container">
<pre class="src src-matlab"><span class="org-matlab-cellbreak">%% Sensor Jacobian matrix for 111 crystal</span>
J_s_111_1 = [<span class="org-builtin">-</span>1, <span class="org-builtin">-</span>0.036, <span class="org-builtin">-</span>0.015
<span class="org-builtin">-</span>1, 0, 0.015
<span class="org-builtin">-</span>1, 0.036, <span class="org-builtin">-</span>0.015];
</pre>
</div>
<table id="org94a1144" border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
<caption class="t-above"><span class="table-number">Table 9:</span> Sensor Jacobian \(\bm{J}_{s,111}\)</caption>
<colgroup>
<col class="org-right" />
<col class="org-right" />
<col class="org-right" />
</colgroup>
<tbody>
<tr>
<td class="org-right">-1.0</td>
<td class="org-right">-0.036</td>
<td class="org-right">-0.015</td>
</tr>
<tr>
<td class="org-right">-1.0</td>
<td class="org-right">0.0</td>
<td class="org-right">0.015</td>
</tr>
<tr>
<td class="org-right">-1.0</td>
<td class="org-right">0.036</td>
<td class="org-right">-0.015</td>
</tr>
</tbody>
</table>
<p>
The forward kinematics is solved by inverting the Jacobian matrix (see Figure <a href="#org8d6627b">13</a>).
</p>
\begin{equation}
\begin{bmatrix}
d_z \\ r_y \\ r_x
\end{bmatrix}
=
\bm{J}_{s,111}^{-1}
\begin{bmatrix}
x_1 \\ x_2 \\ x_3
\end{bmatrix}
\end{equation}
<div id="org80212a3" class="figure">
<p><img src="figs/schematic_sensor_jacobian_forward_kinematics_111.png" alt="schematic_sensor_jacobian_forward_kinematics_111.png" />
</p>
<p><span class="figure-number">Figure 16: </span>Forward Kinematics - Interferometers</p>
</div>
<table id="org2d30850" border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
<caption class="t-above"><span class="table-number">Table 10:</span> Inverse of the sensor Jacobian \(\bm{J}_{s,111}^{-1}\)</caption>
<colgroup>
<col class="org-right" />
<col class="org-right" />
<col class="org-right" />
</colgroup>
<tbody>
<tr>
<td class="org-right">-0.25</td>
<td class="org-right">-0.5</td>
<td class="org-right">-0.25</td>
</tr>
<tr>
<td class="org-right">-13.89</td>
<td class="org-right">0.0</td>
<td class="org-right">13.89</td>
</tr>
<tr>
<td class="org-right">-16.67</td>
<td class="org-right">33.33</td>
<td class="org-right">-16.67</td>
</tr>
</tbody>
</table>
</div>
</div>
<div id="outline-container-org2b65143" class="outline-4">
<h4 id="org2b65143"><span class="section-number-4">1.3.3.</span> Piezo - 111 Crystal</h4>
<div class="outline-text-4" id="text-1-3-3">
<p>
The location of the actuators with respect with the center of the 111 second crystal are shown in Figure <a href="#org702dec8">17</a>.
</p>
<div id="org702dec8" class="figure">
<p><img src="figs/actuator_jacobian_111_points.png" alt="actuator_jacobian_111_points.png" />
</p>
<p><span class="figure-number">Figure 17: </span>Location of actuators with respect to the center of the 111 second crystal (bottom view)</p>
</div>
<p>
Inverse Kinematics consist of deriving the axial (z) motion of the 3 actuators from the motion of the crystal&rsquo;s center.
</p>
\begin{equation}
\begin{bmatrix}
d_{u_r} \\ d_{u_h} \\ d_{d}
\end{bmatrix}
=
\bm{J}_{a,111}
\begin{bmatrix}
d_z \\ r_y \\ r_x
\end{bmatrix}
\end{equation}
<div id="orgb5c4db1" class="figure">
<p><img src="figs/schematic_actuator_jacobian_inverse_kinematics_111.png" alt="schematic_actuator_jacobian_inverse_kinematics_111.png" />
</p>
<p><span class="figure-number">Figure 18: </span>Inverse Kinematics - Actuators</p>
</div>
<p>
Based on the geometry in Figure <a href="#org702dec8">17</a>, we obtain:
</p>
\begin{equation}
\bm{J}_{a,111}
=
\begin{bmatrix}
1 & 0.14 & -0.0675 \\
1 & 0.14 & 0.1525 \\
1 & -0.14 & 0.0425
\end{bmatrix}
\end{equation}
<div class="org-src-container">
<pre class="src src-matlab"><span class="org-matlab-cellbreak">%% Actuator Jacobian - 111 crystal</span>
J_a_111 = [1, 0.14, <span class="org-builtin">-</span>0.0675
1, 0.14, 0.1525
1, <span class="org-builtin">-</span>0.14, 0.0425];
</pre>
</div>
<table id="orge172358" border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
<caption class="t-above"><span class="table-number">Table 11:</span> Actuator Jacobian \(\bm{J}_{a,111}\)</caption>
<colgroup>
<col class="org-right" />
<col class="org-right" />
<col class="org-right" />
</colgroup>
<tbody>
<tr>
<td class="org-right">1.0</td>
<td class="org-right">0.14</td>
<td class="org-right">-0.1525</td>
</tr>
<tr>
<td class="org-right">1.0</td>
<td class="org-right">0.14</td>
<td class="org-right">0.0675</td>
</tr>
<tr>
<td class="org-right">1.0</td>
<td class="org-right">-0.14</td>
<td class="org-right">0.0425</td>
</tr>
</tbody>
</table>
<p>
The forward Kinematics is solved by inverting the Jacobian matrix:
</p>
\begin{equation}
\begin{bmatrix}
d_z \\ r_y \\ r_x
\end{bmatrix}
=
\bm{J}_{a,111}^{-1}
\begin{bmatrix}
d_{u_r} \\ d_{u_h} \\ d_{d}
\end{bmatrix}
\end{equation}
<div id="orgd7de153" class="figure">
<p><img src="figs/schematic_actuator_jacobian_forward_kinematics_111.png" alt="schematic_actuator_jacobian_forward_kinematics_111.png" />
</p>
<p><span class="figure-number">Figure 19: </span>Forward Kinematics - Actuators for 111 crystal</p>
</div>
<table id="org0ced99d" border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
<caption class="t-above"><span class="table-number">Table 12:</span> Inverse of the actuator Jacobian \(\bm{J}_{a,111}^{-1}\)</caption>
<colgroup>
<col class="org-right" />
<col class="org-right" />
<col class="org-right" />
</colgroup>
<tbody>
<tr>
<td class="org-right">0.25</td>
<td class="org-right">0.25</td>
<td class="org-right">0.5</td>
</tr>
<tr>
<td class="org-right">0.4058</td>
<td class="org-right">3.1656</td>
<td class="org-right">-3.5714</td>
</tr>
<tr>
<td class="org-right">-4.5455</td>
<td class="org-right">4.5455</td>
<td class="org-right">0.0</td>
</tr>
</tbody>
</table>
</div>
</div>
</div>
<div id="outline-container-org8e6b78b" class="outline-3">
<h3 id="org8e6b78b"><span class="section-number-3">1.4.</span> Kinematics (Metrology Frame)</h3>
<div class="outline-text-3" id="text-1-4">
<div id="org910184c" class="figure">
<p><img src="figs/jacobian_metrology_frame.png" alt="jacobian_metrology_frame.png" />
</p>
<p><span class="figure-number">Figure 20: </span>Top View - Top metrology frame</p>
</div>
<div class="org-src-container">
<pre class="src src-matlab"><span class="org-matlab-cellbreak">%% Sensor Jacobian matrix for 111 crystal</span>
J_m = [1, 0.102, 0
1, <span class="org-builtin">-</span>0.088, 0.1275
1, <span class="org-builtin">-</span>0.088, <span class="org-builtin">-</span>0.1275];
</pre>
</div>
<table border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
<colgroup>
<col class="org-right" />
<col class="org-right" />
<col class="org-right" />
</colgroup>
<tbody>
<tr>
<td class="org-right">1.0</td>
<td class="org-right">0.102</td>
<td class="org-right">0.0</td>
</tr>
<tr>
<td class="org-right">1.0</td>
<td class="org-right">-0.088</td>
<td class="org-right">0.1275</td>
</tr>
<tr>
<td class="org-right">1.0</td>
<td class="org-right">-0.088</td>
<td class="org-right">-0.1275</td>
</tr>
</tbody>
</table>
<table border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
<colgroup>
<col class="org-right" />
<col class="org-right" />
<col class="org-right" />
</colgroup>
<tbody>
<tr>
<td class="org-right">0.463</td>
<td class="org-right">0.268</td>
<td class="org-right">0.268</td>
</tr>
<tr>
<td class="org-right">5.263</td>
<td class="org-right">-2.632</td>
<td class="org-right">-2.632</td>
</tr>
<tr>
<td class="org-right">0.0</td>
<td class="org-right">3.922</td>
<td class="org-right">-3.922</td>
</tr>
</tbody>
</table>
</div>
</div>
<div id="outline-container-org74a3774" class="outline-3">
<h3 id="org74a3774"><span class="section-number-3">1.5.</span> Verification</h3>
<div class="outline-text-3" id="text-1-5">
<div class="org-src-container">
<pre class="src src-matlab">mdl = <span class="org-string">'coordinate_transform'</span>;
<span class="org-matlab-cellbreak">%% Input/Output definition</span>
clear io; io_i = 1;
<span class="org-matlab-cellbreak">%% Inputs</span>
io(io_i) = linio([mdl, <span class="org-string">'/interferometers'</span>], 1, <span class="org-string">'openinput'</span>); io_i = io_i <span class="org-builtin">+</span> 1;
<span class="org-matlab-cellbreak">%% Outputs</span>
io(io_i) = linio([mdl, <span class="org-string">'/xtal_111'</span>], 1, <span class="org-string">'openoutput'</span>); io_i = io_i <span class="org-builtin">+</span> 1;
</pre>
</div>
<div class="org-src-container">
<pre class="src src-matlab"><span class="org-matlab-cellbreak">%% Extraction of the dynamics</span>
G_311 = linearize(mdl, io);
G_311 = G_311(<span class="org-builtin">:</span>,[1 2 3 7 8 9 13 14 15]);
</pre>
</div>
<div class="org-src-container">
<pre class="src src-matlab">mdl = <span class="org-string">'coordinate_transform'</span>;
<span class="org-matlab-cellbreak">%% Input/Output definition</span>
clear io; io_i = 1;
<span class="org-matlab-cellbreak">%% Inputs</span>
io(io_i) = linio([mdl, <span class="org-string">'/interferometers'</span>], 1, <span class="org-string">'openinput'</span>); io_i = io_i <span class="org-builtin">+</span> 1;
<span class="org-matlab-cellbreak">%% Outputs</span>
io(io_i) = linio([mdl, <span class="org-string">'/xtal_111'</span>], 1, <span class="org-string">'openoutput'</span>); io_i = io_i <span class="org-builtin">+</span> 1;
</pre>
</div>
<div class="org-src-container">
<pre class="src src-matlab">G_111 = linearize(mdl, io);
G_111 = G_111(<span class="org-builtin">:</span>,[4 5 6 10 11 12 13 14 15]);
</pre>
</div>
<div class="org-src-container">
<pre class="src src-matlab"><span class="org-matlab-cellbreak">%% Sensor Jacobian matrix for 1st 111 crystal</span>
J_s_111_1 = [<span class="org-builtin">-</span>1, <span class="org-builtin">-</span>0.036, <span class="org-builtin">-</span>0.015
<span class="org-builtin">-</span>1, 0, 0.015
<span class="org-builtin">-</span>1, 0.036, <span class="org-builtin">-</span>0.015];
</pre>
</div>
<div class="org-src-container">
<pre class="src src-matlab"><span class="org-matlab-cellbreak">%% Sensor Jacobian matrix for 2nd 111 crystal</span>
J_s_111_2 = [1, 0.07, 0.015
1, 0, <span class="org-builtin">-</span>0.015
1, <span class="org-builtin">-</span>0.07, 0.015];
</pre>
</div>
<div class="org-src-container">
<pre class="src src-matlab"><span class="org-matlab-cellbreak">%% Sensor Jacobian matrix for 111 crystal</span>
J_m = [1, 0.102, 0
1, <span class="org-builtin">-</span>0.088, 0.1275
1, <span class="org-builtin">-</span>0.088, <span class="org-builtin">-</span>0.1275];
</pre>
</div>
<div class="org-src-container">
<pre class="src src-matlab">G_111_t = [<span class="org-builtin">-</span>inv(J_s_111_1), inv(J_s_111_2), <span class="org-builtin">-</span>inv(J_m)]
G_111_t(1,<span class="org-builtin">:</span>) = <span class="org-builtin">-</span>G_111_t(1,<span class="org-builtin">:</span>);
</pre>
</div>
<table border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
<colgroup>
<col class="org-right" />
<col class="org-right" />
<col class="org-right" />
<col class="org-right" />
<col class="org-right" />
<col class="org-right" />
<col class="org-right" />
<col class="org-right" />
<col class="org-right" />
</colgroup>
<tbody>
<tr>
<td class="org-right">-0.25</td>
<td class="org-right">-0.5</td>
<td class="org-right">-0.25</td>
<td class="org-right">-0.25</td>
<td class="org-right">-0.5</td>
<td class="org-right">-0.25</td>
<td class="org-right">0.463</td>
<td class="org-right">0.268</td>
<td class="org-right">0.268</td>
</tr>
<tr>
<td class="org-right">13.889</td>
<td class="org-right">0.0</td>
<td class="org-right">-13.889</td>
<td class="org-right">7.143</td>
<td class="org-right">0.0</td>
<td class="org-right">-7.143</td>
<td class="org-right">-5.263</td>
<td class="org-right">2.632</td>
<td class="org-right">2.632</td>
</tr>
<tr>
<td class="org-right">16.667</td>
<td class="org-right">-33.333</td>
<td class="org-right">16.667</td>
<td class="org-right">16.667</td>
<td class="org-right">-33.333</td>
<td class="org-right">16.667</td>
<td class="org-right">0.0</td>
<td class="org-right">-3.922</td>
<td class="org-right">3.922</td>
</tr>
</tbody>
</table>
<table border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
<colgroup>
<col class="org-right" />
<col class="org-right" />
<col class="org-right" />
<col class="org-right" />
<col class="org-right" />
<col class="org-right" />
<col class="org-right" />
<col class="org-right" />
<col class="org-right" />
</colgroup>
<tbody>
<tr>
<td class="org-right">-0.25</td>
<td class="org-right">-0.5</td>
<td class="org-right">-0.25</td>
<td class="org-right">-0.25</td>
<td class="org-right">-0.5</td>
<td class="org-right">-0.25</td>
<td class="org-right">0.333</td>
<td class="org-right">0.333</td>
<td class="org-right">0.333</td>
</tr>
<tr>
<td class="org-right">13.889</td>
<td class="org-right">0.0</td>
<td class="org-right">-13.889</td>
<td class="org-right">7.143</td>
<td class="org-right">0.0</td>
<td class="org-right">-7.143</td>
<td class="org-right">-5.263</td>
<td class="org-right">2.632</td>
<td class="org-right">2.632</td>
</tr>
<tr>
<td class="org-right">16.667</td>
<td class="org-right">-33.333</td>
<td class="org-right">16.667</td>
<td class="org-right">16.667</td>
<td class="org-right">-33.333</td>
<td class="org-right">16.667</td>
<td class="org-right">0.0</td>
<td class="org-right">-3.922</td>
<td class="org-right">3.922</td>
</tr>
</tbody>
</table>
</div>
</div>
<div id="outline-container-org02c0c35" class="outline-3">
<h3 id="org02c0c35"><span class="section-number-3">1.6.</span> Save Kinematics</h3>
<div class="outline-text-3" id="text-1-6">
<div class="org-src-container">
<pre class="src src-matlab">save(<span class="org-string">'mat/dcm_kinematics.mat'</span>, <span class="org-string">'J_a_311'</span>, <span class="org-string">'J_s_311'</span>, <span class="org-string">'J_a_111'</span>, <span class="org-string">'J_s_111'</span>)
</pre>
</div>
</div>
</div>
</div>
<div id="outline-container-orgb814544" class="outline-2">
<h2 id="orgb814544"><span class="section-number-2">2.</span> Open Loop System Identification</h2>
<div class="outline-text-2" id="text-2">
<p>
<a id="orgdfb3b70"></a>
</p>
</div>
<div id="outline-container-org588c7cb" class="outline-3">
<h3 id="org588c7cb"><span class="section-number-3">2.1.</span> Identification</h3>
<div class="outline-text-3" id="text-2-1">
<p>
Let&rsquo;s considered the system \(\bm{G}(s)\) with:
</p>
<ul class="org-ul">
<li>3 inputs: force applied to the 3 fast jacks</li>
<li>3 outputs: measured displacement by the 3 interferometers pointing at the 111 second crystal</li>
</ul>
<p>
It is schematically shown in Figure <a href="#org0473910">21</a>.
</p>
<div id="org0473910" class="figure">
<p><img src="figs/schematic_system_inputs_outputs.png" alt="schematic_system_inputs_outputs.png" />
</p>
<p><span class="figure-number">Figure 21: </span>Dynamical system with inputs and outputs</p>
</div>
<p>
The system is identified from the Simscape model.
</p>
<div class="org-src-container">
<pre class="src src-matlab"><span class="org-matlab-cellbreak">%% Input/Output definition</span>
clear io; io_i = 1;
<span class="org-matlab-cellbreak">%% Inputs</span>
<span class="org-comment-delimiter">% </span><span class="org-comment">Control Input {3x1} [N]</span>
io(io_i) = linio([mdl, <span class="org-string">'/control_system'</span>], 1, <span class="org-string">'openinput'</span>); io_i = io_i <span class="org-builtin">+</span> 1;
<span class="org-matlab-cellbreak">%% Outputs</span>
<span class="org-comment-delimiter">% </span><span class="org-comment">Interferometers {3x1} [m]</span>
io(io_i) = linio([mdl, <span class="org-string">'/DCM'</span>], 1, <span class="org-string">'openoutput'</span>); io_i = io_i <span class="org-builtin">+</span> 1;
</pre>
</div>
<div class="org-src-container">
<pre class="src src-matlab"><span class="org-matlab-cellbreak">%% Extraction of the dynamics</span>
G = linearize(mdl, io);
</pre>
</div>
<div class="org-src-container">
<pre class="src src-matlab">size(G)
</pre>
</div>
<pre class="example">
size(G)
State-space model with 3 outputs, 3 inputs, and 24 states.
</pre>
</div>
</div>
<div id="outline-container-orgb350e31" class="outline-3">
<h3 id="orgb350e31"><span class="section-number-3">2.2.</span> Plant in the frame of the fastjacks</h3>
<div class="outline-text-3" id="text-2-2">
<div class="org-src-container">
<pre class="src src-matlab">load(<span class="org-string">'dcm_kinematics.mat'</span>);
</pre>
</div>
<p>
Using the forward and inverse kinematics, we can computed the dynamics from piezo forces to axial motion of the 3 fastjacks (see Figure <a href="#orgd6e3126">22</a>).
</p>
<div id="orgd6e3126" class="figure">
<p><img src="figs/schematic_jacobian_frame_fastjack.png" alt="schematic_jacobian_frame_fastjack.png" />
</p>
<p><span class="figure-number">Figure 22: </span>Use of Jacobian matrices to obtain the system in the frame of the fastjacks</p>
</div>
<div class="org-src-container">
<pre class="src src-matlab"><span class="org-matlab-cellbreak">%% Compute the system in the frame of the fastjacks</span>
G_pz = J_a_311<span class="org-builtin">*</span>inv(J_s_311)<span class="org-builtin">*</span>G;
</pre>
</div>
<p>
The DC gain of the new system shows that the system is well decoupled at low frequency.
</p>
<div class="org-src-container">
<pre class="src src-matlab">dcgain(G_pz)
</pre>
</div>
<table id="org87317dc" border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
<caption class="t-above"><span class="table-number">Table 13:</span> DC gain of the plant in the frame of the fast jacks \(\bm{G}_{\text{fj}}\)</caption>
<colgroup>
<col class="org-right" />
<col class="org-right" />
<col class="org-right" />
</colgroup>
<tbody>
<tr>
<td class="org-right">4.4407e-09</td>
<td class="org-right">2.7656e-12</td>
<td class="org-right">1.0132e-12</td>
</tr>
<tr>
<td class="org-right">2.7656e-12</td>
<td class="org-right">4.4407e-09</td>
<td class="org-right">1.0132e-12</td>
</tr>
<tr>
<td class="org-right">1.0109e-12</td>
<td class="org-right">1.0109e-12</td>
<td class="org-right">4.4424e-09</td>
</tr>
</tbody>
</table>
<p>
The bode plot of \(\bm{G}_{\text{fj}}(s)\) is shown in Figure <a href="#org9f81aff">23</a>.
</p>
<div class="org-src-container">
<pre class="src src-matlab">G_pz = diag(1<span class="org-builtin">./</span>diag(dcgain(G_pz)))<span class="org-builtin">*</span>G_pz;
</pre>
</div>
<div id="org9f81aff" class="figure">
<p><img src="figs/bode_plot_plant_fj.png" alt="bode_plot_plant_fj.png" />
</p>
<p><span class="figure-number">Figure 23: </span>Bode plot of the diagonal and off-diagonal elements of the plant in the frame of the fast jacks</p>
</div>
<div class="important" id="orge732c0b">
<p>
Computing the system in the frame of the fastjack gives good decoupling at low frequency (until the first resonance of the system).
</p>
</div>
</div>
</div>
<div id="outline-container-org8c5f00e" class="outline-3">
<h3 id="org8c5f00e"><span class="section-number-3">2.3.</span> Plant in the frame of the crystal</h3>
<div class="outline-text-3" id="text-2-3">
<div id="org1c276c1" class="figure">
<p><img src="figs/schematic_jacobian_frame_crystal.png" alt="schematic_jacobian_frame_crystal.png" />
</p>
<p><span class="figure-number">Figure 24: </span>Use of Jacobian matrices to obtain the system in the frame of the crystal</p>
</div>
<div class="org-src-container">
<pre class="src src-matlab">G_mr = inv(J_s_111)<span class="org-builtin">*</span>G<span class="org-builtin">*</span>inv(J_a_111<span class="org-builtin">'</span>);
</pre>
</div>
<div class="org-src-container">
<pre class="src src-matlab">dcgain(G_mr)
</pre>
</div>
<table border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
<colgroup>
<col class="org-right" />
<col class="org-right" />
<col class="org-right" />
</colgroup>
<tbody>
<tr>
<td class="org-right">1.9978e-09</td>
<td class="org-right">3.9657e-09</td>
<td class="org-right">7.7944e-09</td>
</tr>
<tr>
<td class="org-right">3.9656e-09</td>
<td class="org-right">8.4979e-08</td>
<td class="org-right">-1.5135e-17</td>
</tr>
<tr>
<td class="org-right">7.7944e-09</td>
<td class="org-right">-3.9252e-17</td>
<td class="org-right">1.834e-07</td>
</tr>
</tbody>
</table>
<p>
This results in a coupled system.
The main reason is that, as we map forces to the center of the 111 crystal and not at the center of mass/stiffness of the moving part, vertical forces will induce rotation and torques will induce vertical motion.
</p>
</div>
</div>
</div>
<div id="outline-container-orga6970e2" class="outline-2">
<h2 id="orga6970e2"><span class="section-number-2">3.</span> Open-Loop Noise Budgeting</h2>
<div class="outline-text-2" id="text-3">
<p>
<a id="org1f95dda"></a>
</p>
<div id="org61f57fd" class="figure">
<p><img src="figs/noise_budget_dcm_schematic_fast_jack_frame.png" alt="noise_budget_dcm_schematic_fast_jack_frame.png" />
</p>
<p><span class="figure-number">Figure 25: </span>Schematic representation of the control loop in the frame of one fast jack</p>
</div>
</div>
<div id="outline-container-org540e081" class="outline-3">
<h3 id="org540e081"><span class="section-number-3">3.1.</span> Power Spectral Density of signals</h3>
<div class="outline-text-3" id="text-3-1">
<p>
Interferometer noise:
</p>
<div class="org-src-container">
<pre class="src src-matlab">Wn = 6e<span class="org-builtin">-</span>11<span class="org-builtin">*</span>(1 <span class="org-builtin">+</span> s<span class="org-builtin">/</span>2<span class="org-builtin">/</span><span class="org-matlab-math">pi</span><span class="org-builtin">/</span>200)<span class="org-builtin">/</span>(1 <span class="org-builtin">+</span> s<span class="org-builtin">/</span>2<span class="org-builtin">/</span><span class="org-matlab-math">pi</span><span class="org-builtin">/</span>60); <span class="org-comment-delimiter">% </span><span class="org-comment">m/sqrt(Hz)</span>
</pre>
</div>
<pre class="example">
Measurement noise: 0.79 [nm,rms]
</pre>
<p>
DAC noise (amplified by the PI voltage amplifier, and converted to newtons):
</p>
<div class="org-src-container">
<pre class="src src-matlab">Wdac = tf(3e<span class="org-builtin">-</span>8); <span class="org-comment-delimiter">% </span><span class="org-comment">V/sqrt(Hz)</span>
Wu = Wdac<span class="org-builtin">*</span>22.5<span class="org-builtin">*</span>10; <span class="org-comment-delimiter">% </span><span class="org-comment">N/sqrt(Hz)</span>
</pre>
</div>
<pre class="example">
DAC noise: 0.95 [uV,rms]
</pre>
<p>
Disturbances:
</p>
<div class="org-src-container">
<pre class="src src-matlab">Wd = 5e<span class="org-builtin">-</span>7<span class="org-builtin">/</span>(1 <span class="org-builtin">+</span> s<span class="org-builtin">/</span>2<span class="org-builtin">/</span><span class="org-matlab-math">pi</span>); <span class="org-comment-delimiter">% </span><span class="org-comment">m/sqrt(Hz)</span>
</pre>
</div>
<pre class="example">
Disturbance motion: 0.61 [um,rms]
</pre>
<div class="org-src-container">
<pre class="src src-matlab"><span class="org-matlab-cellbreak">%% Save ASD of noise and disturbances</span>
save(<span class="org-string">'mat/asd_noises_disturbances.mat'</span>, <span class="org-string">'Wn'</span>, <span class="org-string">'Wu'</span>, <span class="org-string">'Wd'</span>);
</pre>
</div>
</div>
</div>
<div id="outline-container-org28b9340" class="outline-3">
<h3 id="org28b9340"><span class="section-number-3">3.2.</span> Open Loop disturbance and measurement noise</h3>
<div class="outline-text-3" id="text-3-2">
<p>
The comparison of the amplitude spectral density of the measurement noise and of the jack parasitic motion is performed in Figure <a href="#org16088d5">26</a>.
It confirms that the sensor noise is low enough to measure the motion errors of the crystal.
</p>
<div id="org16088d5" class="figure">
<p><img src="figs/open_loop_noise_budget_fast_jack.png" alt="open_loop_noise_budget_fast_jack.png" />
</p>
<p><span class="figure-number">Figure 26: </span>Open Loop noise budgeting</p>
</div>
</div>
</div>
</div>
<div id="outline-container-org5d1548f" class="outline-2">
<h2 id="org5d1548f"><span class="section-number-2">4.</span> Active Damping Plant (Strain gauges)</h2>
<div class="outline-text-2" id="text-4">
<p>
<a id="orge98df39"></a>
</p>
<p>
In this section, we wish to see whether if strain gauges fixed to the piezoelectric actuator can be used for active damping.
</p>
</div>
<div id="outline-container-org6d76aa1" class="outline-3">
<h3 id="org6d76aa1"><span class="section-number-3">4.1.</span> Identification</h3>
<div class="outline-text-3" id="text-4-1">
<div class="org-src-container">
<pre class="src src-matlab"><span class="org-matlab-cellbreak">%% Input/Output definition</span>
clear io; io_i = 1;
<span class="org-matlab-cellbreak">%% Inputs</span>
<span class="org-comment-delimiter">% </span><span class="org-comment">Control Input {3x1} [N]</span>
io(io_i) = linio([mdl, <span class="org-string">'/control_system'</span>], 1, <span class="org-string">'openinput'</span>); io_i = io_i <span class="org-builtin">+</span> 1;
<span class="org-matlab-cellbreak">%% Outputs</span>
<span class="org-comment-delimiter">% </span><span class="org-comment">Strain Gauges {3x1} [m]</span>
io(io_i) = linio([mdl, <span class="org-string">'/DCM'</span>], 2, <span class="org-string">'openoutput'</span>); io_i = io_i <span class="org-builtin">+</span> 1;
</pre>
</div>
<div class="org-src-container">
<pre class="src src-matlab"><span class="org-matlab-cellbreak">%% Extraction of the dynamics</span>
G_sg = linearize(mdl, io);
</pre>
</div>
<div class="org-src-container">
<pre class="src src-matlab">dcgain(G_sg)
</pre>
</div>
<table border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
<colgroup>
<col class="org-right" />
<col class="org-right" />
<col class="org-right" />
</colgroup>
<tbody>
<tr>
<td class="org-right">4.4443e-09</td>
<td class="org-right">1.0339e-13</td>
<td class="org-right">3.774e-14</td>
</tr>
<tr>
<td class="org-right">1.0339e-13</td>
<td class="org-right">4.4443e-09</td>
<td class="org-right">3.774e-14</td>
</tr>
<tr>
<td class="org-right">3.7792e-14</td>
<td class="org-right">3.7792e-14</td>
<td class="org-right">4.4444e-09</td>
</tr>
</tbody>
</table>
<div id="org6b673e4" class="figure">
<p><img src="figs/strain_gauge_plant_bode_plot.png" alt="strain_gauge_plant_bode_plot.png" />
</p>
<p><span class="figure-number">Figure 27: </span>Bode Plot of the transfer functions from piezoelectric forces to strain gauges measuremed displacements</p>
</div>
<div class="important" id="org0af437c">
<p>
As the distance between the poles and zeros in Figure <a href="#orge45f47e">30</a> is very small, little damping can be actively added using the strain gauges.
This will be confirmed using a Root Locus plot.
</p>
</div>
</div>
</div>
<div id="outline-container-org3f2c1bd" class="outline-3">
<h3 id="org3f2c1bd"><span class="section-number-3">4.2.</span> Relative Active Damping</h3>
<div class="outline-text-3" id="text-4-2">
<div class="org-src-container">
<pre class="src src-matlab">Krad_g1 = eye(3)<span class="org-builtin">*</span>s<span class="org-builtin">/</span>(s<span class="org-builtin">^</span>2<span class="org-builtin">/</span>(2<span class="org-builtin">*</span><span class="org-matlab-math">pi</span><span class="org-builtin">*</span>500)<span class="org-builtin">^</span>2 <span class="org-builtin">+</span> 2<span class="org-builtin">*</span>s<span class="org-builtin">/</span>(2<span class="org-builtin">*</span><span class="org-matlab-math">pi</span><span class="org-builtin">*</span>500) <span class="org-builtin">+</span> 1);
</pre>
</div>
<p>
As can be seen in Figure <a href="#org64577e5">28</a>, very little damping can be added using relative damping strategy using strain gauges.
</p>
<div id="org64577e5" class="figure">
<p><img src="figs/relative_damping_root_locus.png" alt="relative_damping_root_locus.png" />
</p>
<p><span class="figure-number">Figure 28: </span>Root Locus for the relative damping control</p>
</div>
<div class="org-src-container">
<pre class="src src-matlab">Krad = <span class="org-builtin">-</span>g<span class="org-builtin">*</span>Krad_g1;
</pre>
</div>
</div>
</div>
<div id="outline-container-orgd9be2f7" class="outline-3">
<h3 id="orgd9be2f7"><span class="section-number-3">4.3.</span> Damped Plant</h3>
<div class="outline-text-3" id="text-4-3">
<p>
The controller is implemented on Simscape, and the damped plant is identified.
</p>
<div class="org-src-container">
<pre class="src src-matlab"><span class="org-matlab-cellbreak">%% Input/Output definition</span>
clear io; io_i = 1;
<span class="org-matlab-cellbreak">%% Inputs</span>
<span class="org-comment-delimiter">% </span><span class="org-comment">Control Input {3x1} [N]</span>
io(io_i) = linio([mdl, <span class="org-string">'/control_system'</span>], 1, <span class="org-string">'input'</span>); io_i = io_i <span class="org-builtin">+</span> 1;
<span class="org-matlab-cellbreak">%% Outputs</span>
<span class="org-comment-delimiter">% </span><span class="org-comment">Force Sensor {3x1} [m]</span>
io(io_i) = linio([mdl, <span class="org-string">'/DCM'</span>], 1, <span class="org-string">'openoutput'</span>); io_i = io_i <span class="org-builtin">+</span> 1;
</pre>
</div>
<div class="org-src-container">
<pre class="src src-matlab"><span class="org-matlab-cellbreak">%% DCM Kinematics</span>
load(<span class="org-string">'dcm_kinematics.mat'</span>);
</pre>
</div>
<div class="org-src-container">
<pre class="src src-matlab"><span class="org-matlab-cellbreak">%% Identification of the Open Loop plant</span>
controller.type = 0; <span class="org-comment-delimiter">% </span><span class="org-comment">Open Loop</span>
G_ol = J_a_111<span class="org-builtin">*</span>inv(J_s_111)<span class="org-builtin">*</span>linearize(mdl, io);
G_ol.InputName = {<span class="org-string">'u_ur'</span>, <span class="org-string">'u_uh'</span>, <span class="org-string">'u_d'</span>};
G_ol.OutputName = {<span class="org-string">'d_ur'</span>, <span class="org-string">'d_uh'</span>, <span class="org-string">'d_d'</span>};
</pre>
</div>
<div class="org-src-container">
<pre class="src src-matlab"><span class="org-matlab-cellbreak">%% Identification of the damped plant with Relative Active Damping</span>
controller.type = 2; <span class="org-comment-delimiter">% </span><span class="org-comment">RAD</span>
G_dp = J_a_111<span class="org-builtin">*</span>inv(J_s_111)<span class="org-builtin">*</span>linearize(mdl, io);
G_dp.InputName = {<span class="org-string">'u_ur'</span>, <span class="org-string">'u_uh'</span>, <span class="org-string">'u_d'</span>};
G_dp.OutputName = {<span class="org-string">'d_ur'</span>, <span class="org-string">'d_uh'</span>, <span class="org-string">'d_d'</span>};
</pre>
</div>
<div id="orgaf3b6fc" class="figure">
<p><img src="figs/comp_damp_undamped_plant_rad_bode_plot.png" alt="comp_damp_undamped_plant_rad_bode_plot.png" />
</p>
<p><span class="figure-number">Figure 29: </span>Bode plot of both the open-loop plant and the damped plant using relative active damping</p>
</div>
</div>
</div>
</div>
<div id="outline-container-org0f90dfd" class="outline-2">
<h2 id="org0f90dfd"><span class="section-number-2">5.</span> Active Damping Plant (Force Sensors)</h2>
<div class="outline-text-2" id="text-5">
<p>
<a id="org8c440d8"></a>
</p>
<p>
Force sensors are added above the piezoelectric actuators.
They can consists of a simple piezoelectric ceramic stack.
See for instance <a href="fleming10_integ_strain_force_feedb_high">fleming10_integ_strain_force_feedb_high</a>.
</p>
</div>
<div id="outline-container-org2b0e4b2" class="outline-3">
<h3 id="org2b0e4b2"><span class="section-number-3">5.1.</span> Identification</h3>
<div class="outline-text-3" id="text-5-1">
<div class="org-src-container">
<pre class="src src-matlab"><span class="org-matlab-cellbreak">%% Input/Output definition</span>
clear io; io_i = 1;
<span class="org-matlab-cellbreak">%% Inputs</span>
<span class="org-comment-delimiter">% </span><span class="org-comment">Control Input {3x1} [N]</span>
io(io_i) = linio([mdl, <span class="org-string">'/control_system'</span>], 1, <span class="org-string">'openinput'</span>); io_i = io_i <span class="org-builtin">+</span> 1;
<span class="org-matlab-cellbreak">%% Outputs</span>
<span class="org-comment-delimiter">% </span><span class="org-comment">Force Sensor {3x1} [m]</span>
io(io_i) = linio([mdl, <span class="org-string">'/DCM'</span>], 3, <span class="org-string">'openoutput'</span>); io_i = io_i <span class="org-builtin">+</span> 1;
</pre>
</div>
<div class="org-src-container">
<pre class="src src-matlab"><span class="org-matlab-cellbreak">%% Extraction of the dynamics</span>
G_fs = linearize(mdl, io);
</pre>
</div>
<p>
The Bode plot of the identified dynamics is shown in Figure <a href="#orge45f47e">30</a>.
At high frequency, the diagonal terms are constants while the off-diagonal terms have some roll-off.
</p>
<div id="orge45f47e" class="figure">
<p><img src="figs/iff_plant_bode_plot.png" alt="iff_plant_bode_plot.png" />
</p>
<p><span class="figure-number">Figure 30: </span>Bode plot of IFF Plant</p>
</div>
</div>
</div>
<div id="outline-container-org7a5017e" class="outline-3">
<h3 id="org7a5017e"><span class="section-number-3">5.2.</span> Controller - Root Locus</h3>
<div class="outline-text-3" id="text-5-2">
<p>
We want to have integral action around the resonances of the system, but we do not want to integrate at low frequency.
Therefore, we can use a low pass filter.
</p>
<div class="org-src-container">
<pre class="src src-matlab"><span class="org-matlab-cellbreak">%% Integral Force Feedback Controller</span>
Kiff_g1 = eye(3)<span class="org-builtin">*</span>1<span class="org-builtin">/</span>(1 <span class="org-builtin">+</span> s<span class="org-builtin">/</span>2<span class="org-builtin">/</span><span class="org-matlab-math">pi</span><span class="org-builtin">/</span>20);
</pre>
</div>
<div id="org1559bdf" class="figure">
<p><img src="figs/iff_root_locus.png" alt="iff_root_locus.png" />
</p>
<p><span class="figure-number">Figure 31: </span>Root Locus plot for the IFF Control strategy</p>
</div>
<div class="org-src-container">
<pre class="src src-matlab"><span class="org-matlab-cellbreak">%% Integral Force Feedback Controller with optimal gain</span>
Kiff = g<span class="org-builtin">*</span>Kiff_g1;
</pre>
</div>
<div class="org-src-container">
<pre class="src src-matlab"><span class="org-matlab-cellbreak">%% Save the IFF controller</span>
save(<span class="org-string">'mat/Kiff.mat'</span>, <span class="org-string">'Kiff'</span>);
</pre>
</div>
</div>
</div>
<div id="outline-container-org8e7f5c7" class="outline-3">
<h3 id="org8e7f5c7"><span class="section-number-3">5.3.</span> Damped Plant</h3>
<div class="outline-text-3" id="text-5-3">
<p>
Both the Open Loop dynamics (see Figure <a href="#orgd6e3126">22</a>) and the dynamics with IFF (see Figure <a href="#org2cde26d">32</a>) are identified.
</p>
<p>
We are here interested in the dynamics from \(\bm{u}^\prime = [u_{u_r}^\prime,\ u_{u_h}^\prime,\ u_d^\prime]\) (input of the damped plant) to \(\bm{d}_{\text{fj}} = [d_{u_r},\ d_{u_h},\ d_d]\) (motion of the crystal expressed in the frame of the fast-jacks).
This is schematically represented in Figure <a href="#org2cde26d">32</a>.
</p>
<div id="org2cde26d" class="figure">
<p><img src="figs/schematic_jacobian_frame_fastjack_iff.png" alt="schematic_jacobian_frame_fastjack_iff.png" />
</p>
<p><span class="figure-number">Figure 32: </span>Use of Jacobian matrices to obtain the system in the frame of the fastjacks</p>
</div>
<p>
The dynamics from \(\bm{u}\) to \(\bm{d}_{\text{fj}}\) (open-loop dynamics) and from \(\bm{u}^\prime\) to \(\bm{d}_{\text{fs}}\) are compared in Figure <a href="#orge8469c1">33</a>.
It is clear that the Integral Force Feedback control strategy is very effective in damping the resonances of the plant.
</p>
<div id="orge8469c1" class="figure">
<p><img src="figs/comp_damped_undamped_plant_iff_bode_plot.png" alt="comp_damped_undamped_plant_iff_bode_plot.png" />
</p>
<p><span class="figure-number">Figure 33: </span>Bode plot of both the open-loop plant and the damped plant using IFF</p>
</div>
<div class="important" id="org3b156c7">
<p>
The Integral Force Feedback control strategy is very effective in damping the modes present in the plant.
</p>
</div>
</div>
</div>
</div>
<div id="outline-container-orgb450951" class="outline-2">
<h2 id="orgb450951"><span class="section-number-2">6.</span> Feedback Control</h2>
<div class="outline-text-2" id="text-6">
<div id="org14c0807" class="figure">
<p><img src="figs/schematic_jacobian_frame_fastjack_feedback.png" alt="schematic_jacobian_frame_fastjack_feedback.png" />
</p>
</div>
</div>
</div>
<div id="outline-container-org0bdc80c" class="outline-2">
<h2 id="org0bdc80c"><span class="section-number-2">7.</span> HAC-LAC (IFF) architecture</h2>
<div class="outline-text-2" id="text-7">
<p>
<a id="orga62d12a"></a>
</p>
<p>
The HAC-LAC architecture is shown in Figure <a href="#orgb3c374a">34</a>.
</p>
<div id="orgb3c374a" class="figure">
<p><img src="figs/schematic_jacobian_frame_fastjack_hac_iff.png" alt="schematic_jacobian_frame_fastjack_hac_iff.png" />
</p>
<p><span class="figure-number">Figure 34: </span>HAC-LAC architecture</p>
</div>
</div>
<div id="outline-container-org3d59fea" class="outline-3">
<h3 id="org3d59fea"><span class="section-number-3">7.1.</span> System Identification</h3>
<div class="outline-text-3" id="text-7-1">
<p>
Let&rsquo;s identify the damped plant.
</p>
<div id="org6ce2805" class="figure">
<p><img src="figs/bode_plot_hac_iff_plant.png" alt="bode_plot_hac_iff_plant.png" />
</p>
<p><span class="figure-number">Figure 35: </span>Bode Plot of the plant for the High Authority Controller (transfer function from \(\bm{u}^\prime\) to \(\bm{\epsilon}_d\))</p>
</div>
</div>
</div>
<div id="outline-container-org66fa33e" class="outline-3">
<h3 id="org66fa33e"><span class="section-number-3">7.2.</span> High Authority Controller</h3>
<div class="outline-text-3" id="text-7-2">
<p>
Let&rsquo;s design a controller with a bandwidth of 100Hz.
As the plant is well decoupled and well approximated by a constant at low frequency, the high authority controller can easily be designed with SISO loop shaping.
</p>
<div class="org-src-container">
<pre class="src src-matlab"><span class="org-matlab-cellbreak">%% Controller design</span>
wc = 2<span class="org-builtin">*</span><span class="org-matlab-math">pi</span><span class="org-builtin">*</span>100; <span class="org-comment-delimiter">% </span><span class="org-comment">Wanted crossover frequency [rad/s]</span>
a = 2; <span class="org-comment-delimiter">% </span><span class="org-comment">Lead parameter</span>
Khac = diag(1<span class="org-builtin">./</span>diag(abs(evalfr(G_dp, 1<span class="org-constant">j</span><span class="org-builtin">*</span>wc)))) <span class="org-builtin">*</span> <span class="org-comment-delimiter">.</span><span class="org-comment">.. % Diagonal controller</span>
wc<span class="org-builtin">/</span>s <span class="org-builtin">*</span> <span class="org-comment-delimiter">.</span><span class="org-comment">.. % Integrator</span>
1<span class="org-builtin">/</span>(sqrt(a))<span class="org-builtin">*</span>(1 <span class="org-builtin">+</span> s<span class="org-builtin">/</span>(wc<span class="org-builtin">/</span>sqrt(a)))<span class="org-builtin">/</span>(1 <span class="org-builtin">+</span> s<span class="org-builtin">/</span>(wc<span class="org-builtin">*</span>sqrt(a))) <span class="org-builtin">*</span> <span class="org-comment-delimiter">.</span><span class="org-comment">.. % Lead</span>
1<span class="org-builtin">/</span>(s<span class="org-builtin">^</span>2<span class="org-builtin">/</span>(4<span class="org-builtin">*</span>wc)<span class="org-builtin">^</span>2 <span class="org-builtin">+</span> 2<span class="org-builtin">*</span>s<span class="org-builtin">/</span>(4<span class="org-builtin">*</span>wc) <span class="org-builtin">+</span> 1); <span class="org-comment-delimiter">% </span><span class="org-comment">Low pass filter</span>
</pre>
</div>
<div class="org-src-container">
<pre class="src src-matlab"><span class="org-matlab-cellbreak">%% Save the HAC controller</span>
save(<span class="org-string">'mat/Khac_iff.mat'</span>, <span class="org-string">'Khac'</span>);
</pre>
</div>
<div class="org-src-container">
<pre class="src src-matlab"><span class="org-matlab-cellbreak">%% Loop Gain</span>
L_hac_lac = G_dp <span class="org-builtin">*</span> Khac;
</pre>
</div>
<div id="orgd67e07a" class="figure">
<p><img src="figs/hac_iff_loop_gain_bode_plot.png" alt="hac_iff_loop_gain_bode_plot.png" />
</p>
<p><span class="figure-number">Figure 36: </span>Bode Plot of the Loop gain for the High Authority Controller</p>
</div>
<p>
As shown in the Root Locus plot in Figure <a href="#orgc5d8a2d">37</a>, the closed loop system should be stable.
</p>
<div id="orgc5d8a2d" class="figure">
<p><img src="figs/loci_hac_iff_fast_jack.png" alt="loci_hac_iff_fast_jack.png" />
</p>
<p><span class="figure-number">Figure 37: </span>Root Locus for the High Authority Controller</p>
</div>
</div>
</div>
<div id="outline-container-orge129b2d" class="outline-3">
<h3 id="orge129b2d"><span class="section-number-3">7.3.</span> Performances</h3>
<div class="outline-text-3" id="text-7-3">
<p>
In order to estimate the performances of the HAC-IFF control strategy, the transfer function from motion errors of the stepper motors to the motion error of the crystal is identified both in open loop and with the HAC-IFF strategy.
</p>
<p>
It is first verified that the closed-loop system is stable:
</p>
<div class="org-src-container">
<pre class="src src-matlab">isstable(T_hl)
</pre>
</div>
<pre class="example">
1
</pre>
<p>
And both transmissibilities are compared in Figure <a href="#org62d5403">38</a>.
</p>
<div id="org62d5403" class="figure">
<p><img src="figs/stepper_transmissibility_comp_ol_hac_iff.png" alt="stepper_transmissibility_comp_ol_hac_iff.png" />
</p>
<p><span class="figure-number">Figure 38: </span>Comparison of the transmissibility of errors from vibrations of the stepper motor between the open-loop case and the hac-iff case.</p>
</div>
<div class="important" id="orgbb7f9da">
<p>
The HAC-IFF control strategy can effectively reduce the transmissibility of the motion errors of the stepper motors.
This reduction is effective inside the bandwidth of the controller.
</p>
</div>
</div>
</div>
<div id="outline-container-org7677800" class="outline-3">
<h3 id="org7677800"><span class="section-number-3">7.4.</span> Close Loop noise budget</h3>
<div class="outline-text-3" id="text-7-4">
<p>
Let&rsquo;s compute the amplitude spectral density of the jack motion errors due to the sensor noise, the actuator noise and disturbances.
</p>
<div class="org-src-container">
<pre class="src src-matlab"><span class="org-matlab-cellbreak">%% Computation of ASD of contribution of inputs to the closed-loop motion</span>
<span class="org-comment-delimiter">% </span><span class="org-comment">Error due to disturbances</span>
asd_d = abs(squeeze(freqresp(Wd<span class="org-builtin">*</span>(1<span class="org-builtin">/</span>(1 <span class="org-builtin">+</span> G_dp(1,1)<span class="org-builtin">*</span>Khac(1,1))), f, <span class="org-string">'Hz'</span>)));
<span class="org-comment-delimiter">% </span><span class="org-comment">Error due to actuator noise</span>
asd_u = abs(squeeze(freqresp(Wu<span class="org-builtin">*</span>(G_dp(1,1)<span class="org-builtin">/</span>(1 <span class="org-builtin">+</span> G_dp(1,1)<span class="org-builtin">*</span>Khac(1,1))), f, <span class="org-string">'Hz'</span>)));
<span class="org-comment-delimiter">% </span><span class="org-comment">Error due to sensor noise</span>
asd_n = abs(squeeze(freqresp(Wn<span class="org-builtin">*</span>(G_dp(1,1)<span class="org-builtin">*</span>Khac(1,1)<span class="org-builtin">/</span>(1 <span class="org-builtin">+</span> G_dp(1,1)<span class="org-builtin">*</span>Khac(1,1))), f, <span class="org-string">'Hz'</span>)));
</pre>
</div>
<p>
The closed-loop ASD is then:
</p>
<div class="org-src-container">
<pre class="src src-matlab"><span class="org-matlab-cellbreak">%% ASD of the closed-loop motion</span>
asd_cl = sqrt(asd_d<span class="org-builtin">.^</span>2 <span class="org-builtin">+</span> asd_u<span class="org-builtin">.^</span>2 <span class="org-builtin">+</span> asd_n<span class="org-builtin">.^</span>2);
</pre>
</div>
<p>
The obtained ASD are shown in Figure <a href="#org2c14cd0">39</a>.
</p>
<div id="org2c14cd0" class="figure">
<p><img src="figs/close_loop_asd_noise_budget_hac_iff.png" alt="close_loop_asd_noise_budget_hac_iff.png" />
</p>
<p><span class="figure-number">Figure 39: </span>Closed Loop noise budget</p>
</div>
<p>
Let&rsquo;s compare the open-loop and close-loop cases (Figure <a href="#org6177f9e">40</a>).
</p>
<div id="org6177f9e" class="figure">
<p><img src="figs/cps_comp_ol_cl_hac_iff.png" alt="cps_comp_ol_cl_hac_iff.png" />
</p>
<p><span class="figure-number">Figure 40: </span>Cumulative Power Spectrum of the open-loop and closed-loop motion error along one fast-jack</p>
</div>
</div>
</div>
</div>
</div>
<div id="postamble" class="status">
<p class="author">Author: Dehaeze Thomas</p>
<p class="date">Created: 2022-02-15 mar. 14:18</p>
</div>
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