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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="#org3fb7374">1. System Kinematics</a>
<ul>
<li><a href="#org1ef1423">1.1. Bragg Angle</a></li>
<li><a href="#orgcd8fbe6">1.2. Kinematics (111 Crystal)</a>
<ul>
<li><a href="#org542b06e">1.2.1. Interferometers - 111 Crystal</a></li>
<li><a href="#org52f68f7">1.2.2. Piezo - 111 Crystal</a></li>
</ul>
</li>
<li><a href="#org616bb45">1.3. Save Kinematics</a></li>
</ul>
</li>
<li><a href="#org0000e6d">2. Open Loop System Identification</a>
<ul>
<li><a href="#org16c8552">2.1. Identification</a></li>
<li><a href="#orgc2236c5">2.2. Plant in the frame of the fastjacks</a></li>
<li><a href="#orgb0e1668">2.3. Plant in the frame of the crystal</a></li>
</ul>
</li>
<li><a href="#org4bda37c">3. Active Damping Plant (Strain gauges)</a>
<ul>
<li><a href="#orga8033f0">3.1. Identification</a></li>
<li><a href="#org78fe7a9">3.2. Relative Active Damping</a></li>
<li><a href="#org760bce8">3.3. Damped Plant</a></li>
</ul>
</li>
<li><a href="#org09dff16">4. Active Damping Plant (Force Sensors)</a>
<ul>
<li><a href="#orgeb8c92e">4.1. Identification</a></li>
<li><a href="#orgae5e7fb">4.2. Controller - Root Locus</a></li>
<li><a href="#orgde5a8cd">4.3. Damped Plant</a></li>
</ul>
</li>
<li><a href="#org27e3538">5. HAC-LAC (IFF) architecture</a>
<ul>
<li><a href="#org72519d4">5.1. System Identification</a></li>
<li><a href="#org6919788">5.2. High Authority Controller</a></li>
<li><a href="#orgc5ddfb6">5.3. Performances</a></li>
</ul>
</li>
</ul>
</div>
</div>
<hr>
<p>This report is also available as a <a href="./dcm-simscape.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="#org14dc352">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="#orgc1f64db">2</a>: the system dynamics is identified in the absence of control.</li>
<li>Section <a href="#org80ca2a0">3</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="#orgb029a8b">4</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="#orgee34a4d">5</a>: the High Authority Control - Low Authority Control (HAC-LAC) strategy is tested on the Simscape model.</li>
</ul>
<div id="outline-container-org3fb7374" class="outline-2">
<h2 id="org3fb7374"><span class="section-number-2">1.</span> System Kinematics</h2>
<div class="outline-text-2" id="text-1">
<p>
<a id="org14dc352"></a>
</p>
</div>
<div id="outline-container-org1ef1423" class="outline-3">
<h3 id="org1ef1423"><span class="section-number-3">1.1.</span> Bragg Angle</h3>
<div class="outline-text-3" id="text-1-1">
<div class="org-src-container">
<pre class="src src-matlab"><span class="org-matlab-cellbreak">%% Tested bragg angles</span>
bragg = linspace(5, 80, 1000); <span class="org-comment-delimiter">% </span><span class="org-comment">Bragg angle [deg]</span>
d_off = 10.5e<span class="org-builtin">-</span>3; <span class="org-comment-delimiter">% </span><span class="org-comment">Wanted offset between x-rays [m]</span>
</pre>
</div>
<div class="org-src-container">
<pre class="src src-matlab"><span class="org-matlab-cellbreak">%% Vertical Jack motion as a function of Bragg angle</span>
dz = d_off<span class="org-builtin">./</span>(2<span class="org-builtin">*</span>cos(bragg<span class="org-builtin">*</span><span class="org-matlab-math">pi</span><span class="org-builtin">/</span>180));
</pre>
</div>
<div id="org6064b2d" 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>
<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-orgcd8fbe6" class="outline-3">
<h3 id="orgcd8fbe6"><span class="section-number-3">1.2.</span> Kinematics (111 Crystal)</h3>
<div class="outline-text-3" id="text-1-2">
<p>
The reference frame is taken at the center of the 111 second crystal.
</p>
</div>
<div id="outline-container-org542b06e" class="outline-4">
<h4 id="org542b06e"><span class="section-number-4">1.2.1.</span> Interferometers - 111 Crystal</h4>
<div class="outline-text-4" id="text-1-2-1">
<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="#org8f69a58">2</a> as well as the origin where the motion of the crystal is computed.
</p>
<div id="org8f69a58" class="figure">
<p><img src="figs/sensor_111_crystal_points.png" alt="sensor_111_crystal_points.png" />
</p>
<p><span class="figure-number">Figure 2: </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="#org6470cc1">3</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="org6470cc1" class="figure">
<p><img src="figs/schematic_sensor_jacobian_inverse_kinematics.png" alt="schematic_sensor_jacobian_inverse_kinematics.png" />
</p>
<p><span class="figure-number">Figure 3: </span>Inverse Kinematics - Interferometers</p>
</div>
<p>
From the Figure <a href="#org8f69a58">2</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, <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="org9ac8ea9" border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
<caption class="t-above"><span class="table-number">Table 1:</span> Sensor Jacobian \(\bm{J}_{s,111}\)</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="#orgacd4853">4</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="orgacd4853" class="figure">
<p><img src="figs/schematic_sensor_jacobian_forward_kinematics.png" alt="schematic_sensor_jacobian_forward_kinematics.png" />
</p>
<p><span class="figure-number">Figure 4: </span>Forward Kinematics - Interferometers</p>
</div>
<table id="org1305abc" 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 \(\bm{J}_{s,111}^{-1}\)</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-org52f68f7" class="outline-4">
<h4 id="org52f68f7"><span class="section-number-4">1.2.2.</span> Piezo - 111 Crystal</h4>
<div class="outline-text-4" id="text-1-2-2">
<p>
The location of the actuators with respect with the center of the 111 second crystal are shown in Figure <a href="#org6ddaa8b">5</a>.
</p>
<div id="org6ddaa8b" class="figure">
<p><img src="figs/actuator_jacobian_111_points.png" alt="actuator_jacobian_111_points.png" />
</p>
<p><span class="figure-number">Figure 5: </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’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="orgd947fd1" class="figure">
<p><img src="figs/schematic_actuator_jacobian_inverse_kinematics.png" alt="schematic_actuator_jacobian_inverse_kinematics.png" />
</p>
<p><span class="figure-number">Figure 6: </span>Inverse Kinematics - Actuators</p>
</div>
<p>
Based on the geometry in Figure <a href="#org6ddaa8b">5</a>, we obtain:
</p>
\begin{equation}
\bm{J}_{a,111}
=
\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 - 111 crystal</span>
J_a_111 = [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="org96d1229" border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
<caption class="t-above"><span class="table-number">Table 3:</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="orgbdeca35" class="figure">
<p><img src="figs/schematic_actuator_jacobian_forward_kinematics.png" alt="schematic_actuator_jacobian_forward_kinematics.png" />
</p>
<p><span class="figure-number">Figure 7: </span>Forward Kinematics - Actuators for 111 crystal</p>
</div>
<table id="orgb28ebac" border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
<caption class="t-above"><span class="table-number">Table 4:</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">0.0568</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-org616bb45" class="outline-3">
<h3 id="org616bb45"><span class="section-number-3">1.3.</span> Save Kinematics</h3>
<div class="outline-text-3" id="text-1-3">
<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_111'</span>, <span class="org-string">'J_s_111'</span>)
</pre>
</div>
</div>
</div>
</div>
<div id="outline-container-org0000e6d" class="outline-2">
<h2 id="org0000e6d"><span class="section-number-2">2.</span> Open Loop System Identification</h2>
<div class="outline-text-2" id="text-2">
<p>
<a id="orgc1f64db"></a>
</p>
</div>
<div id="outline-container-org16c8552" class="outline-3">
<h3 id="org16c8552"><span class="section-number-3">2.1.</span> Identification</h3>
<div class="outline-text-3" id="text-2-1">
<p>
Let’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="#orga1c2462">8</a>.
</p>
<div id="orga1c2462" class="figure">
<p><img src="figs/schematic_system_inputs_outputs.png" alt="schematic_system_inputs_outputs.png" />
</p>
<p><span class="figure-number">Figure 8: </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-orgc2236c5" class="outline-3">
<h3 id="orgc2236c5"><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="#org015dc10">9</a>).
</p>
<div id="org015dc10" class="figure">
<p><img src="figs/schematic_jacobian_frame_fastjack.png" alt="schematic_jacobian_frame_fastjack.png" />
</p>
<p><span class="figure-number">Figure 9: </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_111<span class="org-builtin">*</span>inv(J_s_111)<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="orgb47db5c" border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
<caption class="t-above"><span class="table-number">Table 5:</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="#org3a99582">10</a>.
</p>
<div id="org3a99582" class="figure">
<p><img src="figs/bode_plot_plant_fj.png" alt="bode_plot_plant_fj.png" />
</p>
<p><span class="figure-number">Figure 10: </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="orge3e331d">
<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-orgb0e1668" class="outline-3">
<h3 id="orgb0e1668"><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="orge8c1108" class="figure">
<p><img src="figs/schematic_jacobian_frame_crystal.png" alt="schematic_jacobian_frame_crystal.png" />
</p>
<p><span class="figure-number">Figure 11: </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-org4bda37c" class="outline-2">
<h2 id="org4bda37c"><span class="section-number-2">3.</span> Active Damping Plant (Strain gauges)</h2>
<div class="outline-text-2" id="text-3">
<p>
<a id="org80ca2a0"></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-orga8033f0" class="outline-3">
<h3 id="orga8033f0"><span class="section-number-3">3.1.</span> Identification</h3>
<div class="outline-text-3" id="text-3-1">
<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="org8f04d26" 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 12: </span>Bode Plot of the transfer functions from piezoelectric forces to strain gauges measuremed displacements</p>
</div>
<div class="important" id="orgd585bf5">
<p>
As the distance between the poles and zeros in Figure <a href="#org11a1e17">15</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-org78fe7a9" class="outline-3">
<h3 id="org78fe7a9"><span class="section-number-3">3.2.</span> Relative Active Damping</h3>
<div class="outline-text-3" id="text-3-2">
<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="#orgec235bb">13</a>, very little damping can be added using relative damping strategy using strain gauges.
</p>
<div id="orgec235bb" class="figure">
<p><img src="figs/relative_damping_root_locus.png" alt="relative_damping_root_locus.png" />
</p>
<p><span class="figure-number">Figure 13: </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-org760bce8" class="outline-3">
<h3 id="org760bce8"><span class="section-number-3">3.3.</span> Damped Plant</h3>
<div class="outline-text-3" id="text-3-3">
<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="orgca0b154" 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 14: </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-org09dff16" class="outline-2">
<h2 id="org09dff16"><span class="section-number-2">4.</span> Active Damping Plant (Force Sensors)</h2>
<div class="outline-text-2" id="text-4">
<p>
<a id="orgb029a8b"></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-orgeb8c92e" class="outline-3">
<h3 id="orgeb8c92e"><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">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="#org11a1e17">15</a>.
At high frequency, the diagonal terms are constants while the off-diagonal terms have some roll-off.
</p>
<div id="org11a1e17" class="figure">
<p><img src="figs/iff_plant_bode_plot.png" alt="iff_plant_bode_plot.png" />
</p>
<p><span class="figure-number">Figure 15: </span>Bode plot of IFF Plant</p>
</div>
</div>
</div>
<div id="outline-container-orgae5e7fb" class="outline-3">
<h3 id="orgae5e7fb"><span class="section-number-3">4.2.</span> Controller - Root Locus</h3>
<div class="outline-text-3" id="text-4-2">
<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="org5b2bab0" class="figure">
<p><img src="figs/iff_root_locus.png" alt="iff_root_locus.png" />
</p>
<p><span class="figure-number">Figure 16: </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-orgde5a8cd" class="outline-3">
<h3 id="orgde5a8cd"><span class="section-number-3">4.3.</span> Damped Plant</h3>
<div class="outline-text-3" id="text-4-3">
<p>
Both the Open Loop dynamics (see Figure <a href="#org015dc10">9</a>) and the dynamics with IFF (see Figure <a href="#org7a880a7">17</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="#org7a880a7">17</a>.
</p>
<div id="org7a880a7" 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 17: </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="#org9f5d048">18</a>.
It is clear that the Integral Force Feedback control strategy is very effective in damping the resonances of the plant.
</p>
<div id="org9f5d048" 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 18: </span>Bode plot of both the open-loop plant and the damped plant using IFF</p>
</div>
<div class="important" id="org8586fa6">
<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-org27e3538" class="outline-2">
<h2 id="org27e3538"><span class="section-number-2">5.</span> HAC-LAC (IFF) architecture</h2>
<div class="outline-text-2" id="text-5">
<p>
<a id="orgee34a4d"></a>
</p>
<p>
The HAC-LAC architecture is shown in Figure <a href="#orgb03e1da">19</a>.
</p>
<div id="orgb03e1da" 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 19: </span>HAC-LAC architecture</p>
</div>
</div>
<div id="outline-container-org72519d4" class="outline-3">
<h3 id="org72519d4"><span class="section-number-3">5.1.</span> System Identification</h3>
<div class="outline-text-3" id="text-5-1">
<p>
Let’s identify the damped plant.
</p>
<div id="org022508f" 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 20: </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-org6919788" class="outline-3">
<h3 id="org6919788"><span class="section-number-3">5.2.</span> High Authority Controller</h3>
<div class="outline-text-3" id="text-5-2">
<p>
Let’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="org1eefea2" 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 21: </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="#orgc90ee63">22</a>, the closed loop system should be stable.
</p>
<div id="orgc90ee63" 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 22: </span>Root Locus for the High Authority Controller</p>
</div>
</div>
</div>
<div id="outline-container-orgc5ddfb6" class="outline-3">
<h3 id="orgc5ddfb6"><span class="section-number-3">5.3.</span> Performances</h3>
<div class="outline-text-3" id="text-5-3">
<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="#org152d7e8">23</a>.
</p>
<div id="org152d7e8" 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 23: </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="org755e221">
<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>
</div>
<div id="postamble" class="status">
<p class="author">Author: Dehaeze Thomas</p>
<p class="date">Created: 2021-11-30 mar. 15:17</p>
</div>
</body>
</html>