Add HAC-IFF strategy and analysis
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<head>
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<!-- 2021-11-30 mar. 11:44 -->
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<!-- 2021-11-30 mar. 15:17 -->
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<meta http-equiv="Content-Type" content="text/html;charset=utf-8" />
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<title>DCM - Dynamical Multi-Body Model</title>
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<title>ESRF Double Crystal Monochromator - Dynamical Multi-Body Model</title>
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<meta name="author" content="Dehaeze Thomas" />
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<meta name="generator" content="Org Mode" />
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<link rel="stylesheet" type="text/css" href="https://research.tdehaeze.xyz/css/style.css"/>
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@ -34,46 +34,51 @@
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<a accesskey="H" href="../index.html"> HOME </a>
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</div><div id="content" class="content">
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<h1 class="title">DCM - Dynamical Multi-Body Model</h1>
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<h1 class="title">ESRF Double Crystal Monochromator - Dynamical Multi-Body Model</h1>
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<div id="table-of-contents" role="doc-toc">
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<h2>Table of Contents</h2>
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<div id="text-table-of-contents" role="doc-toc">
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<ul>
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<li><a href="#orgcb7822a">1. System Kinematics</a>
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<li><a href="#org3fb7374">1. System Kinematics</a>
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<ul>
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<li><a href="#orgc4b429c">1.1. Bragg Angle</a></li>
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<li><a href="#orga30e667">1.2. Kinematics (111 Crystal)</a>
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<li><a href="#org1ef1423">1.1. Bragg Angle</a></li>
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<li><a href="#orgcd8fbe6">1.2. Kinematics (111 Crystal)</a>
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<ul>
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<li><a href="#org390c6c1">1.2.1. Interferometers - 111 Crystal</a></li>
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<li><a href="#org4c00d94">1.2.2. Piezo - 111 Crystal</a></li>
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<li><a href="#org542b06e">1.2.1. Interferometers - 111 Crystal</a></li>
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<li><a href="#org52f68f7">1.2.2. Piezo - 111 Crystal</a></li>
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</ul>
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</li>
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<li><a href="#org82434d5">1.3. Save Kinematics</a></li>
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<li><a href="#org616bb45">1.3. Save Kinematics</a></li>
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</ul>
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</li>
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<li><a href="#orgd95f56d">2. Open Loop System Identification</a>
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<li><a href="#org0000e6d">2. Open Loop System Identification</a>
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<ul>
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<li><a href="#org49ca34a">2.1. Identification</a></li>
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<li><a href="#org170173c">2.2. Plant in the frame of the fastjacks</a></li>
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<li><a href="#orge372095">2.3. Plant in the frame of the crystal</a></li>
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<li><a href="#org16c8552">2.1. Identification</a></li>
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<li><a href="#orgc2236c5">2.2. Plant in the frame of the fastjacks</a></li>
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<li><a href="#orgb0e1668">2.3. Plant in the frame of the crystal</a></li>
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</ul>
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</li>
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<li><a href="#org00855cb">3. Active Damping Plant (Strain gauges)</a>
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<li><a href="#org4bda37c">3. Active Damping Plant (Strain gauges)</a>
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<ul>
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<li><a href="#org6886aa9">3.1. Identification</a></li>
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<li><a href="#orgd1754c8">3.2. Relative Active Damping</a></li>
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<li><a href="#orgf6e5d1c">3.3. Damped Plant</a></li>
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<li><a href="#orga8033f0">3.1. Identification</a></li>
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<li><a href="#org78fe7a9">3.2. Relative Active Damping</a></li>
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<li><a href="#org760bce8">3.3. Damped Plant</a></li>
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</ul>
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</li>
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<li><a href="#org3953e07">4. Active Damping Plant (Force Sensors)</a>
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<li><a href="#org09dff16">4. Active Damping Plant (Force Sensors)</a>
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<ul>
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<li><a href="#org243bdc3">4.1. Identification</a></li>
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<li><a href="#orga2bc3f0">4.2. Controller - Root Locus</a></li>
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<li><a href="#org63d6a74">4.3. Damped Plant</a></li>
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<li><a href="#org8c22e6e">4.4. Save</a></li>
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<li><a href="#orgeb8c92e">4.1. Identification</a></li>
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<li><a href="#orgae5e7fb">4.2. Controller - Root Locus</a></li>
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<li><a href="#orgde5a8cd">4.3. Damped Plant</a></li>
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</ul>
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</li>
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<li><a href="#org27e3538">5. HAC-LAC (IFF) architecture</a>
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<ul>
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<li><a href="#org72519d4">5.1. System Identification</a></li>
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<li><a href="#org6919788">5.2. High Authority Controller</a></li>
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<li><a href="#orgc5ddfb6">5.3. Performances</a></li>
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</ul>
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</li>
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<li><a href="#org053c75c">5. HAC-LAC (IFF) architecture</a></li>
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</ul>
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</div>
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</div>
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@ -89,22 +94,23 @@ In this document, a Simscape (.e.g. multi-body) model of the ESRF Double Crystal
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It is structured as follow:
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</p>
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<ul class="org-ul">
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<li>Section <a href="#org0fd3e9f">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>
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<li>Section <a href="#orgd4eb6dd">2</a>: the system dynamics is identified in the absence of control.</li>
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<li>Section <a href="#orgdb80cfb">3</a>: it is studied whether if the strain gauges fixed to the piezoelectric actuators can be used to actively damp the plant.</li>
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<li>Section <a href="#org8e736b1">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>
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<li>Section <a href="#orga68bafc">5</a>: the High Authority Control - Low Authority Control (HAC-LAC) strategy is tested on the Simscape model.</li>
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<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>
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<li>Section <a href="#orgc1f64db">2</a>: the system dynamics is identified in the absence of control.</li>
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<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>
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<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>
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<li>Section <a href="#orgee34a4d">5</a>: the High Authority Control - Low Authority Control (HAC-LAC) strategy is tested on the Simscape model.</li>
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</ul>
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<div id="outline-container-orgcb7822a" class="outline-2">
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<h2 id="orgcb7822a"><span class="section-number-2">1.</span> System Kinematics</h2>
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<div id="outline-container-org3fb7374" class="outline-2">
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<h2 id="org3fb7374"><span class="section-number-2">1.</span> System Kinematics</h2>
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<div class="outline-text-2" id="text-1">
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<p>
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<a id="org0fd3e9f"></a>
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<a id="org14dc352"></a>
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</p>
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</div>
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<div id="outline-container-orgc4b429c" class="outline-3">
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<h3 id="orgc4b429c"><span class="section-number-3">1.1.</span> Bragg Angle</h3>
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<div id="outline-container-org1ef1423" class="outline-3">
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<h3 id="org1ef1423"><span class="section-number-3">1.1.</span> Bragg Angle</h3>
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<div class="outline-text-3" id="text-1-1">
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<div class="org-src-container">
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<pre class="src src-matlab"><span class="org-matlab-cellbreak">%% Tested bragg angles</span>
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@ -120,7 +126,7 @@ dz = d_off<span class="org-builtin">./</span>(2<span class="org-builtin">*</span
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</div>
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<div id="org122820c" class="figure">
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<div id="org6064b2d" class="figure">
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<p><img src="figs/jack_motion_bragg_angle.png" alt="jack_motion_bragg_angle.png" />
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</p>
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<p><span class="figure-number">Figure 1: </span>Jack motion as a function of Bragg angle</p>
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@ -138,34 +144,34 @@ dz = d_off<span class="org-builtin">./</span>(2<span class="org-builtin">*</span
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</div>
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</div>
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<div id="outline-container-orga30e667" class="outline-3">
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<h3 id="orga30e667"><span class="section-number-3">1.2.</span> Kinematics (111 Crystal)</h3>
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<div id="outline-container-orgcd8fbe6" class="outline-3">
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<h3 id="orgcd8fbe6"><span class="section-number-3">1.2.</span> Kinematics (111 Crystal)</h3>
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<div class="outline-text-3" id="text-1-2">
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<p>
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The reference frame is taken at the center of the 111 second crystal.
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</p>
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</div>
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<div id="outline-container-org390c6c1" class="outline-4">
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<h4 id="org390c6c1"><span class="section-number-4">1.2.1.</span> Interferometers - 111 Crystal</h4>
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<div id="outline-container-org542b06e" class="outline-4">
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<h4 id="org542b06e"><span class="section-number-4">1.2.1.</span> Interferometers - 111 Crystal</h4>
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<div class="outline-text-4" id="text-1-2-1">
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<p>
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Three interferometers are pointed to the bottom surface of the 111 crystal.
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</p>
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<p>
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The position of the measurement points are shown in Figure <a href="#org7147d17">2</a> as well as the origin where the motion of the crystal is computed.
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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.
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</p>
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<div id="org7147d17" class="figure">
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<div id="org8f69a58" class="figure">
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<p><img src="figs/sensor_111_crystal_points.png" alt="sensor_111_crystal_points.png" />
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</p>
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<p><span class="figure-number">Figure 2: </span>Bottom view of the second crystal 111. Position of the measurement points.</p>
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</div>
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<p>
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The inverse kinematics consisting of deriving the interferometer measurements from the motion of the crystal (see Figure <a href="#org8e47ad4">3</a>):
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The inverse kinematics consisting of deriving the interferometer measurements from the motion of the crystal (see Figure <a href="#org6470cc1">3</a>):
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</p>
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\begin{equation}
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\begin{bmatrix}
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@ -179,7 +185,7 @@ d_z \\ r_y \\ r_x
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\end{equation}
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<div id="org8e47ad4" class="figure">
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<div id="org6470cc1" class="figure">
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<p><img src="figs/schematic_sensor_jacobian_inverse_kinematics.png" alt="schematic_sensor_jacobian_inverse_kinematics.png" />
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</p>
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<p><span class="figure-number">Figure 3: </span>Inverse Kinematics - Interferometers</p>
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@ -187,7 +193,7 @@ d_z \\ r_y \\ r_x
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<p>
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From the Figure <a href="#org7147d17">2</a>, the inverse kinematics can be solved as follow (for small motion):
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From the Figure <a href="#org8f69a58">2</a>, the inverse kinematics can be solved as follow (for small motion):
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</p>
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\begin{equation}
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\bm{J}_{s,111}
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@ -207,7 +213,7 @@ J_s_111 = [1, 0.07, <span class="org-builtin">-</span>0.015
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</pre>
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</div>
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<table id="org8c13b9a" border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
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<table id="org9ac8ea9" border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
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<caption class="t-above"><span class="table-number">Table 1:</span> Sensor Jacobian \(\bm{J}_{s,111}\)</caption>
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<colgroup>
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@ -239,7 +245,7 @@ J_s_111 = [1, 0.07, <span class="org-builtin">-</span>0.015
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</table>
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<p>
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The forward kinematics is solved by inverting the Jacobian matrix (see Figure <a href="#org700c4d4">4</a>).
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The forward kinematics is solved by inverting the Jacobian matrix (see Figure <a href="#orgacd4853">4</a>).
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</p>
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\begin{equation}
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\begin{bmatrix}
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@ -253,13 +259,13 @@ x_1 \\ x_2 \\ x_3
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\end{equation}
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<div id="org700c4d4" class="figure">
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<div id="orgacd4853" class="figure">
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<p><img src="figs/schematic_sensor_jacobian_forward_kinematics.png" alt="schematic_sensor_jacobian_forward_kinematics.png" />
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</p>
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<p><span class="figure-number">Figure 4: </span>Forward Kinematics - Interferometers</p>
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</div>
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<table id="org65d7338" border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
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<table id="org1305abc" border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
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<caption class="t-above"><span class="table-number">Table 2:</span> Inverse of the sensor Jacobian \(\bm{J}_{s,111}^{-1}\)</caption>
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<colgroup>
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@ -292,15 +298,15 @@ x_1 \\ x_2 \\ x_3
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</div>
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</div>
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<div id="outline-container-org4c00d94" class="outline-4">
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<h4 id="org4c00d94"><span class="section-number-4">1.2.2.</span> Piezo - 111 Crystal</h4>
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<div id="outline-container-org52f68f7" class="outline-4">
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<h4 id="org52f68f7"><span class="section-number-4">1.2.2.</span> Piezo - 111 Crystal</h4>
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<div class="outline-text-4" id="text-1-2-2">
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<p>
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The location of the actuators with respect with the center of the 111 second crystal are shown in Figure <a href="#orgd329873">5</a>.
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The location of the actuators with respect with the center of the 111 second crystal are shown in Figure <a href="#org6ddaa8b">5</a>.
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</p>
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<div id="orgd329873" class="figure">
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<div id="org6ddaa8b" class="figure">
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<p><img src="figs/actuator_jacobian_111_points.png" alt="actuator_jacobian_111_points.png" />
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</p>
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<p><span class="figure-number">Figure 5: </span>Location of actuators with respect to the center of the 111 second crystal (bottom view)</p>
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@ -321,14 +327,14 @@ d_z \\ r_y \\ r_x
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\end{equation}
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<div id="orga74633d" class="figure">
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<div id="orgd947fd1" class="figure">
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<p><img src="figs/schematic_actuator_jacobian_inverse_kinematics.png" alt="schematic_actuator_jacobian_inverse_kinematics.png" />
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</p>
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<p><span class="figure-number">Figure 6: </span>Inverse Kinematics - Actuators</p>
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</div>
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<p>
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Based on the geometry in Figure <a href="#orgd329873">5</a>, we obtain:
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Based on the geometry in Figure <a href="#org6ddaa8b">5</a>, we obtain:
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</p>
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\begin{equation}
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\bm{J}_{a,111}
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@ -348,7 +354,7 @@ J_a_111 = [1, 0.14, <span class="org-builtin">-</span>0.1525
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</pre>
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</div>
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<table id="org363311b" border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
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<table id="org96d1229" border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
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<caption class="t-above"><span class="table-number">Table 3:</span> Actuator Jacobian \(\bm{J}_{a,111}\)</caption>
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<colgroup>
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@ -394,13 +400,13 @@ d_{u_r} \\ d_{u_h} \\ d_{d}
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\end{equation}
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<div id="orgeeec18d" class="figure">
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<div id="orgbdeca35" class="figure">
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<p><img src="figs/schematic_actuator_jacobian_forward_kinematics.png" alt="schematic_actuator_jacobian_forward_kinematics.png" />
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</p>
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<p><span class="figure-number">Figure 7: </span>Forward Kinematics - Actuators for 111 crystal</p>
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</div>
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<table id="org19c3313" border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
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<table id="orgb28ebac" border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
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<caption class="t-above"><span class="table-number">Table 4:</span> Inverse of the actuator Jacobian \(\bm{J}_{a,111}^{-1}\)</caption>
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<colgroup>
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@ -434,8 +440,8 @@ d_{u_r} \\ d_{u_h} \\ d_{d}
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</div>
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</div>
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<div id="outline-container-org82434d5" class="outline-3">
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<h3 id="org82434d5"><span class="section-number-3">1.3.</span> Save Kinematics</h3>
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<div id="outline-container-org616bb45" class="outline-3">
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<h3 id="org616bb45"><span class="section-number-3">1.3.</span> Save Kinematics</h3>
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<div class="outline-text-3" id="text-1-3">
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<div class="org-src-container">
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<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>)
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@ -445,15 +451,15 @@ d_{u_r} \\ d_{u_h} \\ d_{d}
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</div>
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</div>
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<div id="outline-container-orgd95f56d" class="outline-2">
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<h2 id="orgd95f56d"><span class="section-number-2">2.</span> Open Loop System Identification</h2>
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<div id="outline-container-org0000e6d" class="outline-2">
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<h2 id="org0000e6d"><span class="section-number-2">2.</span> Open Loop System Identification</h2>
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<div class="outline-text-2" id="text-2">
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<p>
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<a id="orgd4eb6dd"></a>
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<a id="orgc1f64db"></a>
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</p>
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</div>
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<div id="outline-container-org49ca34a" class="outline-3">
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<h3 id="org49ca34a"><span class="section-number-3">2.1.</span> Identification</h3>
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<div id="outline-container-org16c8552" class="outline-3">
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<h3 id="org16c8552"><span class="section-number-3">2.1.</span> Identification</h3>
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<div class="outline-text-3" id="text-2-1">
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<p>
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||||
Let’s considered the system \(\bm{G}(s)\) with:
|
||||
@ -464,11 +470,11 @@ Let’s considered the system \(\bm{G}(s)\) with:
|
||||
</ul>
|
||||
|
||||
<p>
|
||||
It is schematically shown in Figure <a href="#org52a4b7c">8</a>.
|
||||
It is schematically shown in Figure <a href="#orga1c2462">8</a>.
|
||||
</p>
|
||||
|
||||
|
||||
<div id="org52a4b7c" class="figure">
|
||||
<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>
|
||||
@ -510,20 +516,20 @@ State-space model with 3 outputs, 3 inputs, and 24 states.
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org170173c" class="outline-3">
|
||||
<h3 id="org170173c"><span class="section-number-3">2.2.</span> Plant in the frame of the fastjacks</h3>
|
||||
<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">'mat/dcm_kinematics.mat'</span>);
|
||||
<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="#orgf9b4903">9</a>).
|
||||
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="orgf9b4903" class="figure">
|
||||
<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>
|
||||
@ -544,7 +550,7 @@ The DC gain of the new system shows that the system is well decoupled at low fre
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
<table id="org19e4a7e" border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
|
||||
<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>
|
||||
@ -576,17 +582,17 @@ The DC gain of the new system shows that the system is well decoupled at low fre
|
||||
</table>
|
||||
|
||||
<p>
|
||||
The bode plot of \(\bm{G}_{\text{fj}}(s)\) is shown in Figure <a href="#org6777e66">10</a>.
|
||||
The bode plot of \(\bm{G}_{\text{fj}}(s)\) is shown in Figure <a href="#org3a99582">10</a>.
|
||||
</p>
|
||||
|
||||
|
||||
<div id="org6777e66" class="figure">
|
||||
<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="orgcdf2cc1">
|
||||
<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>
|
||||
@ -595,11 +601,11 @@ Computing the system in the frame of the fastjack gives good decoupling at low f
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-orge372095" class="outline-3">
|
||||
<h3 id="orge372095"><span class="section-number-3">2.3.</span> Plant in the frame of the crystal</h3>
|
||||
<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="org4377a93" class="figure">
|
||||
<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>
|
||||
@ -654,18 +660,18 @@ The main reason is that, as we map forces to the center of the 111 crystal and n
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org00855cb" class="outline-2">
|
||||
<h2 id="org00855cb"><span class="section-number-2">3.</span> Active Damping Plant (Strain gauges)</h2>
|
||||
<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="orgdb80cfb"></a>
|
||||
<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-org6886aa9" class="outline-3">
|
||||
<h3 id="org6886aa9"><span class="section-number-3">3.1.</span> Identification</h3>
|
||||
<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>
|
||||
@ -724,15 +730,15 @@ G_sg = linearize(mdl, io);
|
||||
</table>
|
||||
|
||||
|
||||
<div id="org72b48ee" class="figure">
|
||||
<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="org070666c">
|
||||
<div class="important" id="orgd585bf5">
|
||||
<p>
|
||||
As the distance between the poles and zeros in Figure <a href="#org16f0105">15</a> is very small, little damping can be actively added using the strain gauges.
|
||||
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>
|
||||
|
||||
@ -740,8 +746,8 @@ This will be confirmed using a Root Locus plot.
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-orgd1754c8" class="outline-3">
|
||||
<h3 id="orgd1754c8"><span class="section-number-3">3.2.</span> Relative Active Damping</h3>
|
||||
<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);
|
||||
@ -749,11 +755,11 @@ This will be confirmed using a Root Locus plot.
|
||||
</div>
|
||||
|
||||
<p>
|
||||
As can be seen in Figure <a href="#org16a4fc0">13</a>, very little damping can be added using relative damping strategy using strain gauges.
|
||||
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="org16a4fc0" class="figure">
|
||||
<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>
|
||||
@ -766,8 +772,8 @@ As can be seen in Figure <a href="#org16a4fc0">13</a>, very little damping can b
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-orgf6e5d1c" class="outline-3">
|
||||
<h3 id="orgf6e5d1c"><span class="section-number-3">3.3.</span> Damped Plant</h3>
|
||||
<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.
|
||||
@ -789,7 +795,7 @@ io(io_i) = linio([mdl, <span class="org-string">'/DCM'</span>], 1, <span class="
|
||||
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab"><span class="org-matlab-cellbreak">%% DCM Kinematics</span>
|
||||
load(<span class="org-string">'mat/dcm_kinematics.mat'</span>);
|
||||
load(<span class="org-string">'dcm_kinematics.mat'</span>);
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
@ -812,7 +818,7 @@ G_dp.OutputName = {<span class="org-string">'d_ur'</span>, <span class="org-str
|
||||
</div>
|
||||
|
||||
|
||||
<div id="org908c24c" class="figure">
|
||||
<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>
|
||||
@ -821,11 +827,11 @@ G_dp.OutputName = {<span class="org-string">'d_ur'</span>, <span class="org-str
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org3953e07" class="outline-2">
|
||||
<h2 id="org3953e07"><span class="section-number-2">4.</span> Active Damping Plant (Force Sensors)</h2>
|
||||
<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="org8e736b1"></a>
|
||||
<a id="orgb029a8b"></a>
|
||||
</p>
|
||||
<p>
|
||||
Force sensors are added above the piezoelectric actuators.
|
||||
@ -833,8 +839,8 @@ 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-org243bdc3" class="outline-3">
|
||||
<h3 id="org243bdc3"><span class="section-number-3">4.1.</span> Identification</h3>
|
||||
<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>
|
||||
@ -856,44 +862,13 @@ G_fs = linearize(mdl, io);
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab">dcgain(G_fs)
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
<table border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
|
||||
<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>
|
||||
|
||||
|
||||
<colgroup>
|
||||
<col class="org-right" />
|
||||
|
||||
<col class="org-right" />
|
||||
|
||||
<col class="org-right" />
|
||||
</colgroup>
|
||||
<tbody>
|
||||
<tr>
|
||||
<td class="org-right">-1.4113e-13</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">-1.4113e-13</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">-7.5585e-14</td>
|
||||
</tr>
|
||||
</tbody>
|
||||
</table>
|
||||
|
||||
|
||||
<div id="org16f0105" class="figure">
|
||||
<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>
|
||||
@ -901,109 +876,213 @@ G_fs = linearize(mdl, io);
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-orga2bc3f0" class="outline-3">
|
||||
<h3 id="orga2bc3f0"><span class="section-number-3">4.2.</span> Controller - Root Locus</h3>
|
||||
<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">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 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="org4041645" class="figure">
|
||||
<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</span>
|
||||
<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-org63d6a74" class="outline-3">
|
||||
<h3 id="org63d6a74"><span class="section-number-3">4.3.</span> Damped Plant</h3>
|
||||
<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">
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab"><span class="org-matlab-cellbreak">%% Input/Output definition</span>
|
||||
clear io; io_i = 1;
|
||||
<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>
|
||||
|
||||
<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;
|
||||
<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>
|
||||
|
||||
<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 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>
|
||||
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab"><span class="org-matlab-cellbreak">%% DCM Kinematics</span>
|
||||
load(<span class="org-string">'mat/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 IFF</span>
|
||||
controller.type = 1; <span class="org-comment-delimiter">% </span><span class="org-comment">IFF</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>
|
||||
<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="org8455151" class="figure">
|
||||
<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 17: </span>Bode plot of both the open-loop plant and the damped plant using IFF</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="orgebe4b09">
|
||||
<div class="important" id="org8586fa6">
|
||||
<p>
|
||||
The Integral Force Feedback control strategy is very effective in damping the suspension modes of the DCM.
|
||||
The Integral Force Feedback control strategy is very effective in damping the modes present in the plant.
|
||||
</p>
|
||||
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org8c22e6e" class="outline-3">
|
||||
<h3 id="org8c22e6e"><span class="section-number-3">4.4.</span> Save</h3>
|
||||
<div class="outline-text-3" id="text-4-4">
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab">save(<span class="org-string">'mat/Kiff.mat'</span>, <span class="org-string">'Kiff'</span>);
|
||||
</pre>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org053c75c" class="outline-2">
|
||||
<h2 id="org053c75c"><span class="section-number-2">5.</span> HAC-LAC (IFF) architecture</h2>
|
||||
<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="orga68bafc"></a>
|
||||
<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. 11:44</p>
|
||||
<p class="date">Created: 2021-11-30 mar. 15:17</p>
|
||||
</div>
|
||||
</body>
|
||||
</html>
|
||||
|
||||
444
dcm-simscape.org
444
dcm-simscape.org
@ -1,4 +1,4 @@
|
||||
#+TITLE: DCM - Dynamical Multi-Body Model
|
||||
#+TITLE: ESRF Double Crystal Monochromator - Dynamical Multi-Body Model
|
||||
:DRAWER:
|
||||
#+LANGUAGE: en
|
||||
#+EMAIL: dehaeze.thomas@gmail.com
|
||||
@ -65,6 +65,7 @@ It is structured as follow:
|
||||
:header-args:matlab+: :tangle matlab/dcm_kinematics.m
|
||||
:END:
|
||||
<<sec:dcm_kinematics>>
|
||||
|
||||
** Introduction :ignore:
|
||||
|
||||
** Matlab Init :noexport:ignore:
|
||||
@ -378,6 +379,13 @@ data2orgtable(inv(J_a_111), {}, {}, ' %.4f ');
|
||||
| 1.7857 | 1.7857 | -3.5714 |
|
||||
| -4.5455 | 4.5455 | 0.0 |
|
||||
|
||||
** TODO Inputs and Outputs :noexport:
|
||||
|
||||
Disturbances:
|
||||
- Motion errors of the stepper motor
|
||||
- Vibrations from the outside
|
||||
- Vibrations from the cooling system directly applied on the crystals
|
||||
|
||||
** Save Kinematics
|
||||
#+begin_src matlab :exports none :tangle no
|
||||
save('matlab/mat/dcm_kinematics.mat', 'J_a_111', 'J_s_111')
|
||||
@ -482,7 +490,7 @@ size(G)
|
||||
|
||||
** Plant in the frame of the fastjacks
|
||||
#+begin_src matlab
|
||||
load('mat/dcm_kinematics.mat');
|
||||
load('dcm_kinematics.mat');
|
||||
#+end_src
|
||||
|
||||
Using the forward and inverse kinematics, we can computed the dynamics from piezo forces to axial motion of the 3 fastjacks (see Figure [[fig:schematic_jacobian_frame_fastjack]]).
|
||||
@ -886,7 +894,7 @@ io(io_i) = linio([mdl, '/DCM'], 1, 'openoutput'); io_i = io_i + 1;
|
||||
|
||||
#+begin_src matlab
|
||||
%% DCM Kinematics
|
||||
load('mat/dcm_kinematics.mat');
|
||||
load('dcm_kinematics.mat');
|
||||
#+end_src
|
||||
|
||||
#+begin_src matlab
|
||||
@ -1030,14 +1038,8 @@ G_fs.InputName = {'u_ur', 'u_uh', 'u_d'};
|
||||
G_fs.OutputName = {'fs_ur', 'fs_uh', 'fs_d'};
|
||||
#+end_src
|
||||
|
||||
#+begin_src matlab :results value replace :exports both :tangle no
|
||||
dcgain(G_fs)
|
||||
#+end_src
|
||||
|
||||
#+RESULTS:
|
||||
| -1.4113e-13 | 1.0339e-13 | 3.774e-14 |
|
||||
| 1.0339e-13 | -1.4113e-13 | 3.774e-14 |
|
||||
| 3.7792e-14 | 3.7792e-14 | -7.5585e-14 |
|
||||
The Bode plot of the identified dynamics is shown in Figure [[fig:iff_plant_bode_plot]].
|
||||
At high frequency, the diagonal terms are constants while the off-diagonal terms have some roll-off.
|
||||
|
||||
#+begin_src matlab :exports none
|
||||
%% Bode plot for the plant
|
||||
@ -1062,7 +1064,7 @@ for i = 1:2
|
||||
end
|
||||
hold off;
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
||||
ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]);
|
||||
ylabel('Amplitude'); set(gca, 'XTickLabel',[]);
|
||||
legend('location', 'northwest', 'FontSize', 8, 'NumColumns', 2);
|
||||
ylim([1e-13, 1e-7]);
|
||||
|
||||
@ -1092,7 +1094,11 @@ exportFig('figs/iff_plant_bode_plot.pdf', 'width', 'wide', 'height', 'tall');
|
||||
[[file:figs/iff_plant_bode_plot.png]]
|
||||
|
||||
** Controller - Root Locus
|
||||
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.
|
||||
|
||||
#+begin_src matlab
|
||||
%% Integral Force Feedback Controller
|
||||
Kiff_g1 = eye(3)*1/(1 + s/2/pi/20);
|
||||
#+end_src
|
||||
|
||||
@ -1136,12 +1142,61 @@ exportFig('figs/iff_root_locus.pdf', 'width', 'wide', 'height', 'tall');
|
||||
[[file:figs/iff_root_locus.png]]
|
||||
|
||||
#+begin_src matlab
|
||||
%% Integral Force Feedback Controller
|
||||
%% Integral Force Feedback Controller with optimal gain
|
||||
Kiff = g*Kiff_g1;
|
||||
#+end_src
|
||||
|
||||
#+begin_src matlab :exports none :tangle no
|
||||
save('matlab/mat/Kiff.mat', 'Kiff');
|
||||
#+end_src
|
||||
|
||||
#+begin_src matlab :eval no
|
||||
%% Save the IFF controller
|
||||
save('mat/Kiff.mat', 'Kiff');
|
||||
#+end_src
|
||||
|
||||
** Damped Plant
|
||||
#+begin_src matlab
|
||||
Both the Open Loop dynamics (see Figure [[fig:schematic_jacobian_frame_fastjack]]) and the dynamics with IFF (see Figure [[fig:schematic_jacobian_frame_fastjack_iff]]) are identified.
|
||||
|
||||
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 [[fig:schematic_jacobian_frame_fastjack_iff]].
|
||||
|
||||
#+begin_src latex :file schematic_jacobian_frame_fastjack_iff.pdf
|
||||
\begin{tikzpicture}
|
||||
% Blocs
|
||||
\node[block={4.0cm}{3.0cm}] (G) {$\bm{G}(s)$};
|
||||
\coordinate[] (inputF) at ($(G.south west)!0.5!(G.north west)$);
|
||||
\coordinate[] (outputF) at ($(G.south east)!0.8!(G.north east)$);
|
||||
\coordinate[] (outputL) at ($(G.south east)!0.2!(G.north east)$);
|
||||
|
||||
\node[block, right=1.5 of outputL] (Js) {$\bm{J}_{s}^{-1}$};
|
||||
\node[block, right=1.5 of Js] (Ja) {$\bm{J}_{a}$};
|
||||
\node[addb, left= of G] (addF) {};
|
||||
\node[block, above=0.5 of G] (Kiff) {$\bm{K}_{\text{IFF}}(s)$};
|
||||
|
||||
% Connections and labels
|
||||
\draw[->] ($(addF.west)+(-1.5,0)$) node[above right]{$\begin{bmatrix} u_{u_r}^\prime \\ u_{u_h}^\prime \\ u_d^\prime \end{bmatrix}$} -- (addF.west);
|
||||
\draw[->] (addF.east) -- node[miday, above]{$\begin{bmatrix} u_{u_r} \\ u_{u_h} \\ u_d \end{bmatrix}$} (inputF);
|
||||
\draw[->] (outputF) -- ++(2.0, 0) node[above left]{$\begin{bmatrix} \tau_{u_r} \\ \tau_{u_h} \\ \tau_d \end{bmatrix}$};
|
||||
\draw[->] ($(outputF) + (0.6, 0)$)node[branch]{} |- (Kiff.east);
|
||||
\draw[->] (Kiff.west) -| (addF.north);
|
||||
\draw[->] (outputL) -- node[midway, above]{$\begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix}$} (Js.west);
|
||||
\draw[->] (Js.east) -- node[midway, above]{$\begin{bmatrix} d_z \\ r_y \\ r_x \end{bmatrix}$} (Ja.west);
|
||||
\draw[->] (Ja.east) -- ++(1.5, 0) node[above left]{$\begin{bmatrix} d_{u_r} \\ d_{u_h} \\ d_{d} \end{bmatrix}$};
|
||||
|
||||
\begin{scope}[on background layer]
|
||||
\node[fit={(G.south -| addF.west) (Ja.east |- Kiff.north)}, fill=black!20!white, draw, inner sep=6pt] (system) {};
|
||||
\node[above] at (system.north) {$\bm{G}_{\text{fj,IFF}}(s)$};
|
||||
\end{scope}
|
||||
\end{tikzpicture}
|
||||
#+end_src
|
||||
|
||||
#+name: fig:schematic_jacobian_frame_fastjack_iff
|
||||
#+caption: Use of Jacobian matrices to obtain the system in the frame of the fastjacks
|
||||
#+RESULTS:
|
||||
[[file:figs/schematic_jacobian_frame_fastjack_iff.png]]
|
||||
|
||||
#+begin_src matlab :exports none
|
||||
%% Input/Output definition
|
||||
clear io; io_i = 1;
|
||||
|
||||
@ -1154,12 +1209,12 @@ io(io_i) = linio([mdl, '/control_system'], 1, 'input'); io_i = io_i + 1;
|
||||
io(io_i) = linio([mdl, '/DCM'], 1, 'openoutput'); io_i = io_i + 1;
|
||||
#+end_src
|
||||
|
||||
#+begin_src matlab
|
||||
%% DCM Kinematics
|
||||
load('mat/dcm_kinematics.mat');
|
||||
#+begin_src matlab :exports none
|
||||
%% Load DCM Kinematics
|
||||
load('dcm_kinematics.mat');
|
||||
#+end_src
|
||||
|
||||
#+begin_src matlab
|
||||
#+begin_src matlab :exports none
|
||||
%% Identification of the Open Loop plant
|
||||
controller.type = 0; % Open Loop
|
||||
G_ol = J_a_111*inv(J_s_111)*linearize(mdl, io);
|
||||
@ -1167,7 +1222,7 @@ G_ol.InputName = {'u_ur', 'u_uh', 'u_d'};
|
||||
G_ol.OutputName = {'d_ur', 'd_uh', 'd_d'};
|
||||
#+end_src
|
||||
|
||||
#+begin_src matlab
|
||||
#+begin_src matlab :exports none
|
||||
%% Identification of the damped plant with IFF
|
||||
controller.type = 1; % IFF
|
||||
G_dp = J_a_111*inv(J_s_111)*linearize(mdl, io);
|
||||
@ -1175,6 +1230,9 @@ G_dp.InputName = {'u_ur', 'u_uh', 'u_d'};
|
||||
G_dp.OutputName = {'d_ur', 'd_uh', 'd_d'};
|
||||
#+end_src
|
||||
|
||||
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 [[fig:comp_damped_undamped_plant_iff_bode_plot]].
|
||||
It is clear that the Integral Force Feedback control strategy is very effective in damping the resonances of the plant.
|
||||
|
||||
#+begin_src matlab :exports none
|
||||
%% Comparison of the damped and undamped plant
|
||||
figure;
|
||||
@ -1237,24 +1295,67 @@ exportFig('figs/comp_damped_undamped_plant_iff_bode_plot.pdf', 'width', 'wide',
|
||||
[[file:figs/comp_damped_undamped_plant_iff_bode_plot.png]]
|
||||
|
||||
#+begin_important
|
||||
The Integral Force Feedback control strategy is very effective in damping the suspension modes of the DCM.
|
||||
The Integral Force Feedback control strategy is very effective in damping the modes present in the plant.
|
||||
#+end_important
|
||||
|
||||
** Save
|
||||
#+begin_src matlab :exports none :tangle no
|
||||
save('matlab/mat/Kiff.mat', 'Kiff');
|
||||
#+end_src
|
||||
|
||||
#+begin_src matlab :eval no
|
||||
save('mat/Kiff.mat', 'Kiff');
|
||||
#+end_src
|
||||
|
||||
* HAC-LAC (IFF) architecture
|
||||
:PROPERTIES:
|
||||
:header-args:matlab+: :tangle matlab/dcm_hac_iff.m
|
||||
:END:
|
||||
<<sec:hac_iff>>
|
||||
** Introduction :ignore:
|
||||
|
||||
The HAC-LAC architecture is shown in Figure [[fig:schematic_jacobian_frame_fastjack_hac_iff]].
|
||||
|
||||
#+begin_src latex :file schematic_jacobian_frame_fastjack_hac_iff.pdf
|
||||
\begin{tikzpicture}
|
||||
% Blocs
|
||||
\node[block={4.0cm}{3.0cm}] (G) {$\bm{G}(s)$};
|
||||
\coordinate[] (inputF) at ($(G.south west)!0.5!(G.north west)$);
|
||||
\coordinate[] (outputF) at ($(G.south east)!0.8!(G.north east)$);
|
||||
\coordinate[] (outputL) at ($(G.south east)!0.2!(G.north east)$);
|
||||
|
||||
\node[block, right=1.3 of outputL] (Js) {$\bm{J}_{s}^{-1}$};
|
||||
|
||||
\node[addb, left= of G] (addF) {};
|
||||
\node[block, above=0.5 of G] (Kiff) {$\bm{K}_{\text{IFF}}(s)$};
|
||||
|
||||
\node[block, left=1.3 of addF] (Khac) {$\bm{K}_{\text{HAC}}(s)$};
|
||||
\node[block, left=1.3 of Khac] (Ja) {$\bm{J}_{a}$};
|
||||
\node[addb={+}{}{}{}{-}, left=1.3 of Ja] (subL) {};
|
||||
\coordinate[left=1.3 of subL] (mux);
|
||||
\node[block] (bragg2d) at ($(mux) + (-1.2, 0.8)$) {$\frac{d_{\text{off}}}{2 \cos \theta_b}$};
|
||||
\coordinate (rx) at ($(mux) + (0, -0.8)$);
|
||||
|
||||
|
||||
% Connections and labels
|
||||
\draw[->] (Khac.east) -- node[midway, above]{$\begin{bmatrix} u_{u_r}^\prime \\ u_{u_h}^\prime \\ u_d^\prime \end{bmatrix}$} (addF.west);
|
||||
\draw[->] (addF.east) -- node[midway, above]{$\begin{bmatrix} u_{u_r} \\ u_{u_h} \\ u_d \end{bmatrix}$} (inputF);
|
||||
\draw[->] (outputF) -- ++(2.0, 0) node[above left]{$\begin{bmatrix} \tau_{u_r} \\ \tau_{u_h} \\ \tau_d \end{bmatrix}$};
|
||||
\draw[->] ($(outputF) + (0.6, 0)$)node[branch]{} |- (Kiff.east);
|
||||
\draw[->] (Kiff.west) -| (addF.north);
|
||||
\draw[->] (outputL) -- node[midway, above]{$\begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix}$} (Js.west);
|
||||
\draw[->] (Js.east) -- ++(1.0, 0);
|
||||
|
||||
|
||||
\draw[line width=4pt] ($(mux) + (0, -1.0)$) -- ($(mux) + (0, 1.0)$);
|
||||
\draw[->] (mux) -- node[midway, above]{$\begin{bmatrix} r_{d_z} \\ r_{r_y} \\ r_{r_x} \end{bmatrix}$} (subL.west);
|
||||
\draw[->] (subL.east) -- node[midway, above]{$\begin{bmatrix} \epsilon_{d_z} \\ \epsilon_{r_y} \\ \epsilon_{r_x} \end{bmatrix}$} (Ja.west);
|
||||
\draw[->] (Ja.east) -- node[midway, above]{$\begin{bmatrix} \epsilon_{d_{u_r}} \\ \epsilon_{d_{u_h}} \\ \epsilon_{d_d} \end{bmatrix}$} (Khac.west);
|
||||
|
||||
\draw[->] ($(Js.east) + (0.6, 0)$)node[branch]{}node[above]{$\begin{bmatrix} d_z \\ r_y \\ r_x \end{bmatrix}$} -- ++(0, -1.0) -| (subL.south);
|
||||
\draw[->] ($(bragg2d.west) + (-0.6, 0)$)coordinate(ref_input)node[above right]{$\theta_b$} -- (bragg2d.west);
|
||||
\draw[->] (bragg2d.east) --node[midway, above]{$r_{d_z}$} (bragg2d-|mux);
|
||||
\draw[->] (ref_input|-mux)node[above right]{$r_{r_y}$} -- (mux);
|
||||
\draw[->] (ref_input|-rx) node[above right]{$r_{r_x}$} -- (rx-|mux);
|
||||
\end{tikzpicture}
|
||||
#+end_src
|
||||
|
||||
#+name: fig:schematic_jacobian_frame_fastjack_hac_iff
|
||||
#+caption: HAC-LAC architecture
|
||||
#+RESULTS:
|
||||
[[file:figs/schematic_jacobian_frame_fastjack_hac_iff.png]]
|
||||
|
||||
** Matlab Init :noexport:ignore:
|
||||
#+begin_src matlab
|
||||
%% dcm_hac_iff.m
|
||||
@ -1285,6 +1386,291 @@ save('mat/Kiff.mat', 'Kiff');
|
||||
<<m-init-other>>
|
||||
#+end_src
|
||||
|
||||
** System Identification
|
||||
Let's identify the damped plant.
|
||||
|
||||
#+begin_src matlab :exports none
|
||||
%% Input/Output definition
|
||||
clear io; io_i = 1;
|
||||
|
||||
%% Inputs
|
||||
% Control Input {3x1} [N]
|
||||
io(io_i) = linio([mdl, '/control_system'], 1, 'input'); io_i = io_i + 1;
|
||||
|
||||
%% Outputs
|
||||
% Force Sensor {3x1} [m]
|
||||
io(io_i) = linio([mdl, '/DCM'], 1, 'openoutput'); io_i = io_i + 1;
|
||||
#+end_src
|
||||
|
||||
#+begin_src matlab :exports none
|
||||
%% Load DCM Kinematics and IFF controller
|
||||
load('dcm_kinematics.mat');
|
||||
load('Kiff.mat');
|
||||
#+end_src
|
||||
|
||||
#+begin_src matlab :exports none
|
||||
%% Identification of the damped plant with IFF
|
||||
controller.type = 1; % IFF
|
||||
G_dp = J_a_111*inv(J_s_111)*linearize(mdl, io);
|
||||
G_dp.InputName = {'u_ur', 'u_uh', 'u_d'};
|
||||
G_dp.OutputName = {'d_ur', 'd_uh', 'd_d'};
|
||||
#+end_src
|
||||
|
||||
#+begin_src matlab :exports none
|
||||
%% Comparison of the damped and undamped plant
|
||||
figure;
|
||||
tiledlayout(3, 1, 'TileSpacing', 'Compact', 'Padding', 'None');
|
||||
|
||||
ax1 = nexttile([2,1]);
|
||||
hold on;
|
||||
plot(freqs, abs(squeeze(freqresp(G_dp(1,1), freqs, 'Hz'))), '-', ...
|
||||
'DisplayName', 'd - IFF');
|
||||
plot(freqs, abs(squeeze(freqresp(G_dp(2,2), freqs, 'Hz'))), '-', ...
|
||||
'DisplayName', 'uh - IFF');
|
||||
plot(freqs, abs(squeeze(freqresp(G_dp(3,3), freqs, 'Hz'))), '-', ...
|
||||
'DisplayName', 'ur - IFF');
|
||||
for i = 1:2
|
||||
for j = i+1:3
|
||||
plot(freqs, abs(squeeze(freqresp(G_dp(i,j), freqs, 'Hz'))), 'color', [0, 0, 0, 0.2], ...
|
||||
'HandleVisibility', 'off');
|
||||
end
|
||||
end
|
||||
hold off;
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
||||
ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]);
|
||||
legend('location', 'southeast', 'FontSize', 8, 'NumColumns', 2);
|
||||
ylim([1e-12, 1e-8]);
|
||||
|
||||
ax2 = nexttile;
|
||||
hold on;
|
||||
plot(freqs, 180/pi*angle(squeeze(freqresp(G_dp(1,1), freqs, 'Hz'))), '-');
|
||||
plot(freqs, 180/pi*angle(squeeze(freqresp(G_dp(2,2), freqs, 'Hz'))), '-');
|
||||
plot(freqs, 180/pi*angle(squeeze(freqresp(G_dp(3,3), freqs, 'Hz'))), '-');
|
||||
hold off;
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
|
||||
xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
|
||||
hold off;
|
||||
yticks(-360:90:360);
|
||||
ylim([-180, 0]);
|
||||
|
||||
linkaxes([ax1,ax2],'x');
|
||||
xlim([freqs(1), freqs(end)]);
|
||||
#+end_src
|
||||
|
||||
#+begin_src matlab :tangle no :exports results :results file replace
|
||||
exportFig('figs/bode_plot_hac_iff_plant.pdf', 'width', 'wide', 'height', 'tall');
|
||||
#+end_src
|
||||
|
||||
#+name: fig:bode_plot_hac_iff_plant
|
||||
#+caption: Bode Plot of the plant for the High Authority Controller (transfer function from $\bm{u}^\prime$ to $\bm{\epsilon}_d$)
|
||||
#+RESULTS:
|
||||
[[file:figs/bode_plot_hac_iff_plant.png]]
|
||||
|
||||
** High Authority Controller
|
||||
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.
|
||||
|
||||
#+begin_src matlab
|
||||
%% Controller design
|
||||
wc = 2*pi*100; % Wanted crossover frequency [rad/s]
|
||||
a = 2; % Lead parameter
|
||||
|
||||
Khac = diag(1./diag(abs(evalfr(G_dp, 1j*wc)))) * ... % Diagonal controller
|
||||
wc/s * ... % Integrator
|
||||
1/(sqrt(a))*(1 + s/(wc/sqrt(a)))/(1 + s/(wc*sqrt(a))) * ... % Lead
|
||||
1/(s^2/(4*wc)^2 + 2*s/(4*wc) + 1); % Low pass filter
|
||||
#+end_src
|
||||
|
||||
#+begin_src matlab :exports none :tangle no
|
||||
save('matlab/mat/Khac_iff.mat', 'Khac');
|
||||
#+end_src
|
||||
|
||||
#+begin_src matlab :eval no
|
||||
%% Save the HAC controller
|
||||
save('mat/Khac_iff.mat', 'Khac');
|
||||
#+end_src
|
||||
|
||||
#+begin_src matlab
|
||||
%% Loop Gain
|
||||
L_hac_lac = G_dp * Khac;
|
||||
#+end_src
|
||||
|
||||
#+begin_src matlab :exports none
|
||||
%% Bode Plot of the Loop Gain
|
||||
figure;
|
||||
tiledlayout(3, 1, 'TileSpacing', 'Compact', 'Padding', 'None');
|
||||
|
||||
ax1 = nexttile([2,1]);
|
||||
hold on;
|
||||
plot(freqs, abs(squeeze(freqresp(L_hac_lac(1,1), freqs, 'Hz'))), '-', ...
|
||||
'DisplayName', 'd');
|
||||
plot(freqs, abs(squeeze(freqresp(L_hac_lac(2,2), freqs, 'Hz'))), '-', ...
|
||||
'DisplayName', 'uh');
|
||||
plot(freqs, abs(squeeze(freqresp(L_hac_lac(3,3), freqs, 'Hz'))), '-', ...
|
||||
'DisplayName', 'ur');
|
||||
for i = 1:2
|
||||
for j = i+1:3
|
||||
plot(freqs, abs(squeeze(freqresp(L_hac_lac(i,j), freqs, 'Hz'))), 'color', [0, 0, 0, 0.2], ...
|
||||
'HandleVisibility', 'off');
|
||||
end
|
||||
end
|
||||
hold off;
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
||||
ylabel('Loop Gain'); set(gca, 'XTickLabel',[]);
|
||||
legend('location', 'northeast', 'FontSize', 8, 'NumColumns', 2);
|
||||
ylim([1e-2, 1e1]);
|
||||
|
||||
ax2 = nexttile;
|
||||
hold on;
|
||||
plot(freqs, 180/pi*angle(squeeze(freqresp(L_hac_lac(1,1), freqs, 'Hz'))), '-');
|
||||
plot(freqs, 180/pi*angle(squeeze(freqresp(L_hac_lac(2,2), freqs, 'Hz'))), '-');
|
||||
plot(freqs, 180/pi*angle(squeeze(freqresp(L_hac_lac(3,3), freqs, 'Hz'))), '-');
|
||||
hold off;
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
|
||||
xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
|
||||
hold off;
|
||||
yticks(-360:90:360);
|
||||
ylim([-180, 0]);
|
||||
|
||||
linkaxes([ax1,ax2],'x');
|
||||
xlim([freqs(1), freqs(end)]);
|
||||
#+end_src
|
||||
|
||||
#+begin_src matlab :tangle no :exports results :results file replace
|
||||
exportFig('figs/hac_iff_loop_gain_bode_plot.pdf', 'width', 'wide', 'height', 'tall');
|
||||
#+end_src
|
||||
|
||||
#+name: fig:hac_iff_loop_gain_bode_plot
|
||||
#+caption: Bode Plot of the Loop gain for the High Authority Controller
|
||||
#+RESULTS:
|
||||
[[file:figs/hac_iff_loop_gain_bode_plot.png]]
|
||||
|
||||
#+begin_src matlab :exports none
|
||||
%% Compute the Eigenvalues of the loop gain
|
||||
Ldet = zeros(3, length(freqs));
|
||||
|
||||
Lmimo = squeeze(freqresp(L_hac_lac, freqs, 'Hz'));
|
||||
for i_f = 1:length(freqs)
|
||||
Ldet(:, i_f) = eig(squeeze(Lmimo(:,:,i_f)));
|
||||
end
|
||||
#+end_src
|
||||
|
||||
As shown in the Root Locus plot in Figure [[fig:loci_hac_iff_fast_jack]], the closed loop system should be stable.
|
||||
|
||||
#+begin_src matlab :exports none
|
||||
%% Plot of the eigenvalues of L in the complex plane
|
||||
figure;
|
||||
hold on;
|
||||
for i = 1:3
|
||||
plot(real(squeeze(Ldet(i,:))), imag(squeeze(Ldet(i,:))), 'k.');
|
||||
plot(real(squeeze(Ldet(i,:))), -imag(squeeze(Ldet(i,:))), 'k.');
|
||||
end
|
||||
plot(-1, 0, 'kx', 'HandleVisibility', 'off');
|
||||
hold off;
|
||||
set(gca, 'XScale', 'lin'); set(gca, 'YScale', 'lin');
|
||||
xlabel('Real'); ylabel('Imag');
|
||||
xlim([-3, 1]); ylim([-2, 2]);
|
||||
#+end_src
|
||||
|
||||
#+begin_src matlab :tangle no :exports results :results file replace
|
||||
exportFig('figs/loci_hac_iff_fast_jack.pdf', 'width', 'wide', 'height', 'tall');
|
||||
#+end_src
|
||||
|
||||
#+name: fig:loci_hac_iff_fast_jack
|
||||
#+caption: Root Locus for the High Authority Controller
|
||||
#+RESULTS:
|
||||
[[file:figs/loci_hac_iff_fast_jack.png]]
|
||||
|
||||
** Performances
|
||||
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.
|
||||
|
||||
#+begin_src matlab :exports none
|
||||
%% Input/Output definition
|
||||
clear io; io_i = 1;
|
||||
|
||||
%% Inputs
|
||||
% Jack Motion Erros {3x1} [m]
|
||||
io(io_i) = linio([mdl, '/stepper_errors'], 1, 'input'); io_i = io_i + 1;
|
||||
|
||||
%% Outputs
|
||||
% Interferometer Output {3x1} [m]
|
||||
io(io_i) = linio([mdl, '/DCM'], 1, 'output'); io_i = io_i + 1;
|
||||
#+end_src
|
||||
|
||||
#+begin_src matlab :exports none
|
||||
%% Identification of the transmissibility of errors in open-loop
|
||||
controller.type = 0; % Open Loop
|
||||
T_ol = inv(J_s_111)*linearize(mdl, io)*J_a_111;
|
||||
T_ol.InputName = {'e_dz', 'e_ry', 'e_rx'};
|
||||
T_ol.OutputName = {'dx', 'ry', 'rx'};
|
||||
#+end_src
|
||||
|
||||
#+begin_src matlab :exports none
|
||||
%% Load DCM Kinematics and IFF controller
|
||||
load('dcm_kinematics.mat');
|
||||
load('Kiff.mat');
|
||||
#+end_src
|
||||
|
||||
#+begin_src matlab :exports none
|
||||
%% Identification of the transmissibility of errors with HAC-IFF
|
||||
controller.type = 3; % IFF
|
||||
T_hl = inv(J_s_111)*linearize(mdl, io)*J_a_111;
|
||||
T_hl.InputName = {'e_dz', 'e_ry', 'e_rx'};
|
||||
T_hl.OutputName = {'dx', 'ry', 'rx'};
|
||||
#+end_src
|
||||
|
||||
It is first verified that the closed-loop system is stable:
|
||||
|
||||
#+begin_src matlab :results value replace :exports both :tangle no
|
||||
isstable(T_hl)
|
||||
#+end_src
|
||||
|
||||
#+RESULTS:
|
||||
: 1
|
||||
|
||||
And both transmissibilities are compared in Figure [[fig:stepper_transmissibility_comp_ol_hac_iff]].
|
||||
|
||||
#+begin_src matlab :exports none
|
||||
%% Transmissibility of stepper errors
|
||||
f = logspace(0, 3, 1000);
|
||||
|
||||
figure;
|
||||
hold on;
|
||||
plot(f, abs(squeeze(freqresp(T_ol(1,1), f, 'Hz'))), '-', ...
|
||||
'DisplayName', '$d_z$ - OL');
|
||||
plot(f, abs(squeeze(freqresp(T_ol(2,2), f, 'Hz'))), '-', ...
|
||||
'DisplayName', '$r_y$ - OL');
|
||||
plot(f, abs(squeeze(freqresp(T_ol(3,3), f, 'Hz'))), '-', ...
|
||||
'DisplayName', '$r_x$ - OL');
|
||||
set(gca,'ColorOrderIndex',1)
|
||||
plot(f, abs(squeeze(freqresp(T_hl(1,1), f, 'Hz'))), '--', ...
|
||||
'DisplayName', '$d_z$ - HAC-IFF');
|
||||
plot(f, abs(squeeze(freqresp(T_hl(2,2), f, 'Hz'))), '--', ...
|
||||
'DisplayName', '$r_y$ - HAC-IFF');
|
||||
plot(f, abs(squeeze(freqresp(T_hl(3,3), f, 'Hz'))), '--', ...
|
||||
'DisplayName', '$r_x$ - HAC-IFF');
|
||||
hold off;
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
||||
xlabel('Frequency [Hz]'); ylabel('Stepper transmissibility');
|
||||
legend('location', 'southeast', 'FontSize', 8, 'NumColumns', 2);
|
||||
ylim([1e-2, 1e2]);
|
||||
xlim([f(1), f(end)]);
|
||||
#+end_src
|
||||
|
||||
#+begin_src matlab :tangle no :exports results :results file replace
|
||||
exportFig('figs/stepper_transmissibility_comp_ol_hac_iff.pdf', 'width', 'wide', 'height', 'tall');
|
||||
#+end_src
|
||||
|
||||
#+name: fig:stepper_transmissibility_comp_ol_hac_iff
|
||||
#+caption: Comparison of the transmissibility of errors from vibrations of the stepper motor between the open-loop case and the hac-iff case.
|
||||
#+RESULTS:
|
||||
[[file:figs/stepper_transmissibility_comp_ol_hac_iff.png]]
|
||||
|
||||
#+begin_important
|
||||
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.
|
||||
#+end_important
|
||||
|
||||
* Helping Functions :noexport:
|
||||
** Initialize Path
|
||||
#+NAME: m-init-path
|
||||
|
||||
BIN
dcm-simscape.pdf
BIN
dcm-simscape.pdf
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BIN
figs/bode_plot_hac_iff_plant.pdf
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BIN
figs/bode_plot_hac_iff_plant.pdf
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figs/bode_plot_hac_iff_plant.png
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figs/bode_plot_hac_iff_plant.png
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|
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1288
figs/bode_plot_iff_plant.pdf
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1288
figs/bode_plot_iff_plant.pdf
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Load Diff
BIN
figs/bode_plot_iff_plant.png
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BIN
figs/bode_plot_iff_plant.png
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|
After Width: | Height: | Size: 100 KiB |
BIN
figs/hac_iff_loop_gain_bode_plot.pdf
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BIN
figs/hac_iff_loop_gain_bode_plot.pdf
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BIN
figs/hac_iff_loop_gain_bode_plot.png
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BIN
figs/hac_iff_loop_gain_bode_plot.png
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|
After Width: | Height: | Size: 96 KiB |
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|
Before Width: | Height: | Size: 117 KiB After Width: | Height: | Size: 116 KiB |
BIN
figs/loci_hac_iff_fast_jack.pdf
Normal file
BIN
figs/loci_hac_iff_fast_jack.pdf
Normal file
Binary file not shown.
BIN
figs/loci_hac_iff_fast_jack.png
Normal file
BIN
figs/loci_hac_iff_fast_jack.png
Normal file
Binary file not shown.
|
After Width: | Height: | Size: 34 KiB |
BIN
figs/schematic_jacobian_frame_fastjack_hac_iff.pdf
Normal file
BIN
figs/schematic_jacobian_frame_fastjack_hac_iff.pdf
Normal file
Binary file not shown.
BIN
figs/schematic_jacobian_frame_fastjack_hac_iff.png
Normal file
BIN
figs/schematic_jacobian_frame_fastjack_hac_iff.png
Normal file
Binary file not shown.
|
After Width: | Height: | Size: 26 KiB |
696
figs/schematic_jacobian_frame_fastjack_hac_iff.svg
Normal file
696
figs/schematic_jacobian_frame_fastjack_hac_iff.svg
Normal file
@ -0,0 +1,696 @@
|
||||
<?xml version="1.0" encoding="UTF-8"?>
|
||||
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<defs>
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<g>
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<symbol overflow="visible" id="glyph0-0">
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Reference in New Issue
Block a user