First open/close noise budgeting
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<html xmlns="http://www.w3.org/1999/xhtml" lang="en" xml:lang="en">
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<head>
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<!-- 2021-11-30 mar. 15:17 -->
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<!-- 2021-11-30 mar. 17:54 -->
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<meta http-equiv="Content-Type" content="text/html;charset=utf-8" />
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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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@ -39,44 +39,51 @@
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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="#org3fb7374">1. System Kinematics</a>
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<li><a href="#orgbeabbac">1. System Kinematics</a>
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<ul>
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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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<li><a href="#org72840a7">1.1. Bragg Angle</a></li>
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<li><a href="#org330c2ee">1.2. Kinematics (111 Crystal)</a>
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<ul>
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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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<li><a href="#orgcf6be49">1.2.1. Interferometers - 111 Crystal</a></li>
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<li><a href="#org729d4fd">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="#org616bb45">1.3. Save Kinematics</a></li>
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<li><a href="#orgd9564bd">1.3. Save Kinematics</a></li>
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</ul>
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</li>
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<li><a href="#org0000e6d">2. Open Loop System Identification</a>
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<li><a href="#org94165e4">2. Open Loop System Identification</a>
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<ul>
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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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<li><a href="#org49f773d">2.1. Identification</a></li>
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<li><a href="#orga27ac31">2.2. Plant in the frame of the fastjacks</a></li>
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<li><a href="#org1d08b17">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="#org4bda37c">3. Active Damping Plant (Strain gauges)</a>
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<li><a href="#orge94151e">3. Open-Loop Noise Budgeting</a>
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<ul>
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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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<li><a href="#org20060df">3.1. Power Spectral Density of signals</a></li>
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<li><a href="#orgea5b4ad">3.2. Open Loop disturbance and measurement noise</a></li>
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</ul>
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</li>
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<li><a href="#org09dff16">4. Active Damping Plant (Force Sensors)</a>
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<li><a href="#org987b6b1">4. Active Damping Plant (Strain gauges)</a>
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<ul>
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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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<li><a href="#orge45c4d8">4.1. Identification</a></li>
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<li><a href="#org65bc5e9">4.2. Relative Active Damping</a></li>
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<li><a href="#orgc83a897">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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<li><a href="#orgf9782ba">5. Active Damping Plant (Force Sensors)</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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<li><a href="#orge8c03e7">5.1. Identification</a></li>
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<li><a href="#org135d9f2">5.2. Controller - Root Locus</a></li>
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<li><a href="#org4072236">5.3. Damped Plant</a></li>
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</ul>
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</li>
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<li><a href="#org5b57c31">6. HAC-LAC (IFF) architecture</a>
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<ul>
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<li><a href="#orgb4b8d52">6.1. System Identification</a></li>
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<li><a href="#org714ec0f">6.2. High Authority Controller</a></li>
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<li><a href="#org9c9ea48">6.3. Performances</a></li>
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<li><a href="#orgbdf7b28">6.4. Close Loop noise budget</a></li>
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</ul>
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</li>
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</ul>
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@ -94,44 +101,53 @@ 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="#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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<li>Section <a href="#org02dd8b0">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="#orgf62c035">2</a>: the system dynamics is identified in the absence of control.</li>
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<li>Section <a href="#org47df825">3</a>: an open-loop noise budget is performed.</li>
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<li>Section <a href="#org64d9ba7">4</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="#org7fbe16a">5</a>: piezoelectric force sensors are added in series with the piezoelectric actuators and are used to actively damp the plant using the Integral Force Feedback (IFF) control strategy.</li>
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<li>Section <a href="#org226a3fc">6</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-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 id="outline-container-orgbeabbac" class="outline-2">
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<h2 id="orgbeabbac"><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="org14dc352"></a>
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<a id="org02dd8b0"></a>
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</p>
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</div>
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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 id="outline-container-org72840a7" class="outline-3">
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<h3 id="org72840a7"><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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bragg = linspace(5, 80, 1000); <span class="org-comment-delimiter">% </span><span class="org-comment">Bragg angle [deg]</span>
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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>
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</pre>
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</div>
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<p>
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There is a simple relation <a href="eq:bragg_angle_formula">eq:bragg_angle_formula</a> between:
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</p>
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<ul class="org-ul">
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<li>\(d_{\text{off}}\) is the wanted offset between the incident x-ray and the output x-ray</li>
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<li>\(\theta_b\) is the bragg angle</li>
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<li>\(d_z\) is the corresponding distance between the first and second crystals</li>
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</ul>
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<div class="org-src-container">
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<pre class="src src-matlab"><span class="org-matlab-cellbreak">%% Vertical Jack motion as a function of Bragg angle</span>
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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));
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</pre>
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</div>
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\begin{equation} \label{eq:bragg_angle_formula}
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d_z = \frac{d_{\text{off}}}{2 \cos \theta_b}
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\end{equation}
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<p>
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This relation is shown in Figure <a href="#orgd56c0ff">1</a>.
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</p>
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<div id="org6064b2d" class="figure">
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<div id="orgd56c0ff" 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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</div>
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<p>
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The required jack stroke is approximately 25mm.
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</p>
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<div class="org-src-container">
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<pre class="src src-matlab"><span class="org-matlab-cellbreak">%% Required Jack stroke</span>
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<span class="org-matlab-math">ans</span> = 1e3<span class="org-builtin">*</span>(dz(end) <span class="org-builtin">-</span> dz(1))
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@ -144,34 +160,34 @@ 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-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 id="outline-container-org330c2ee" class="outline-3">
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<h3 id="org330c2ee"><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-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 id="outline-container-orgcf6be49" class="outline-4">
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<h4 id="orgcf6be49"><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="#org8f69a58">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="#org7cdbbab">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="org8f69a58" class="figure">
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<div id="org7cdbbab" 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="#org6470cc1">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="#orgea687e1">3</a>):
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</p>
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\begin{equation}
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\begin{bmatrix}
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@ -185,7 +201,7 @@ d_z \\ r_y \\ r_x
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\end{equation}
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<div id="org6470cc1" class="figure">
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<div id="orgea687e1" 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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@ -193,7 +209,7 @@ d_z \\ r_y \\ r_x
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<p>
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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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From the Figure <a href="#org7cdbbab">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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@ -213,7 +229,7 @@ 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="org9ac8ea9" border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
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<table id="org31f833a" 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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@ -245,7 +261,7 @@ 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="#orgacd4853">4</a>).
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The forward kinematics is solved by inverting the Jacobian matrix (see Figure <a href="#org220e3fd">4</a>).
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</p>
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\begin{equation}
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\begin{bmatrix}
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@ -259,13 +275,13 @@ x_1 \\ x_2 \\ x_3
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\end{equation}
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<div id="orgacd4853" class="figure">
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<div id="org220e3fd" 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="org1305abc" border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
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<table id="orga06a454" 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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@ -298,15 +314,15 @@ x_1 \\ x_2 \\ x_3
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</div>
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</div>
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<div id="outline-container-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 id="outline-container-org729d4fd" class="outline-4">
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<h4 id="org729d4fd"><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="#org6ddaa8b">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="#org92f0f11">5</a>.
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</p>
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<div id="org6ddaa8b" class="figure">
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<div id="org92f0f11" 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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@ -327,14 +343,14 @@ d_z \\ r_y \\ r_x
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\end{equation}
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||||
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<div id="orgd947fd1" class="figure">
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<div id="org87ec19c" 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="#org6ddaa8b">5</a>, we obtain:
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Based on the geometry in Figure <a href="#org92f0f11">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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@ -354,7 +370,7 @@ 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="org96d1229" border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
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<table id="orga19c158" 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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@ -400,13 +416,13 @@ d_{u_r} \\ d_{u_h} \\ d_{d}
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\end{equation}
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<div id="orgbdeca35" class="figure">
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<div id="orga0d447c" 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="orgb28ebac" border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
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<table id="orga2ec14c" 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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@ -440,8 +456,8 @@ 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-org616bb45" class="outline-3">
|
||||
<h3 id="org616bb45"><span class="section-number-3">1.3.</span> Save Kinematics</h3>
|
||||
<div id="outline-container-orgd9564bd" class="outline-3">
|
||||
<h3 id="orgd9564bd"><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>)
|
||||
@ -451,15 +467,15 @@ d_{u_r} \\ d_{u_h} \\ d_{d}
|
||||
</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 id="outline-container-org94165e4" class="outline-2">
|
||||
<h2 id="org94165e4"><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>
|
||||
<a id="orgf62c035"></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 id="outline-container-org49f773d" class="outline-3">
|
||||
<h3 id="org49f773d"><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:
|
||||
@ -470,11 +486,11 @@ Let’s considered the system \(\bm{G}(s)\) with:
|
||||
</ul>
|
||||
|
||||
<p>
|
||||
It is schematically shown in Figure <a href="#orga1c2462">8</a>.
|
||||
It is schematically shown in Figure <a href="#org03d9ffb">8</a>.
|
||||
</p>
|
||||
|
||||
|
||||
<div id="orga1c2462" class="figure">
|
||||
<div id="org03d9ffb" 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>
|
||||
@ -516,8 +532,8 @@ State-space model with 3 outputs, 3 inputs, and 24 states.
|
||||
</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 id="outline-container-orga27ac31" class="outline-3">
|
||||
<h3 id="orga27ac31"><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>);
|
||||
@ -525,11 +541,11 @@ State-space model with 3 outputs, 3 inputs, and 24 states.
|
||||
</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>).
|
||||
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="#org5f538a8">9</a>).
|
||||
</p>
|
||||
|
||||
|
||||
<div id="org015dc10" class="figure">
|
||||
<div id="org5f538a8" 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>
|
||||
@ -550,7 +566,7 @@ The DC gain of the new system shows that the system is well decoupled at low fre
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
<table id="orgb47db5c" border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
|
||||
<table id="orgdc8a086" 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>
|
||||
@ -582,17 +598,17 @@ 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="#org3a99582">10</a>.
|
||||
The bode plot of \(\bm{G}_{\text{fj}}(s)\) is shown in Figure <a href="#org4618cac">10</a>.
|
||||
</p>
|
||||
|
||||
|
||||
<div id="org3a99582" class="figure">
|
||||
<div id="org4618cac" 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">
|
||||
<div class="important" id="org834d546">
|
||||
<p>
|
||||
Computing the system in the frame of the fastjack gives good decoupling at low frequency (until the first resonance of the system).
|
||||
</p>
|
||||
@ -601,11 +617,11 @@ Computing the system in the frame of the fastjack gives good decoupling at low f
|
||||
</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 id="outline-container-org1d08b17" class="outline-3">
|
||||
<h3 id="org1d08b17"><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">
|
||||
<div id="orgd94c3c6" 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>
|
||||
@ -660,19 +676,102 @@ The main reason is that, as we map forces to the center of the 111 crystal and n
|
||||
</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 id="outline-container-orge94151e" class="outline-2">
|
||||
<h2 id="orge94151e"><span class="section-number-2">3.</span> Open-Loop Noise Budgeting</h2>
|
||||
<div class="outline-text-2" id="text-3">
|
||||
<p>
|
||||
<a id="org80ca2a0"></a>
|
||||
<a id="org47df825"></a>
|
||||
</p>
|
||||
|
||||
<div id="org2ed8210" class="figure">
|
||||
<p><img src="figs/noise_budget_dcm_schematic_fast_jack_frame.png" alt="noise_budget_dcm_schematic_fast_jack_frame.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 12: </span>Schematic representation of the control loop in the frame of one fast jack</p>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org20060df" class="outline-3">
|
||||
<h3 id="org20060df"><span class="section-number-3">3.1.</span> Power Spectral Density of signals</h3>
|
||||
<div class="outline-text-3" id="text-3-1">
|
||||
<p>
|
||||
Interferometer noise:
|
||||
</p>
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab">Wn = 6e<span class="org-builtin">-</span>11<span class="org-builtin">*</span>(1 <span class="org-builtin">+</span> s<span class="org-builtin">/</span>2<span class="org-builtin">/</span><span class="org-matlab-math">pi</span><span class="org-builtin">/</span>200)<span class="org-builtin">/</span>(1 <span class="org-builtin">+</span> s<span class="org-builtin">/</span>2<span class="org-builtin">/</span><span class="org-matlab-math">pi</span><span class="org-builtin">/</span>60); <span class="org-comment-delimiter">% </span><span class="org-comment">m/sqrt(Hz)</span>
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
<pre class="example">
|
||||
Measurement noise: 0.79 [nm,rms]
|
||||
</pre>
|
||||
|
||||
|
||||
<p>
|
||||
DAC noise (amplified by the PI voltage amplifier, and converted to newtons):
|
||||
</p>
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab">Wdac = tf(3e<span class="org-builtin">-</span>8); <span class="org-comment-delimiter">% </span><span class="org-comment">V/sqrt(Hz)</span>
|
||||
Wu = Wdac<span class="org-builtin">*</span>22.5<span class="org-builtin">*</span>10; <span class="org-comment-delimiter">% </span><span class="org-comment">N/sqrt(Hz)</span>
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
<pre class="example">
|
||||
DAC noise: 0.95 [uV,rms]
|
||||
</pre>
|
||||
|
||||
|
||||
<p>
|
||||
Disturbances:
|
||||
</p>
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab">Wd = 5e<span class="org-builtin">-</span>7<span class="org-builtin">/</span>(1 <span class="org-builtin">+</span> s<span class="org-builtin">/</span>2<span class="org-builtin">/</span><span class="org-matlab-math">pi</span>); <span class="org-comment-delimiter">% </span><span class="org-comment">m/sqrt(Hz)</span>
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
<pre class="example">
|
||||
Disturbance motion: 0.61 [um,rms]
|
||||
</pre>
|
||||
|
||||
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab"><span class="org-matlab-cellbreak">%% Save ASD of noise and disturbances</span>
|
||||
save(<span class="org-string">'mat/asd_noises_disturbances.mat'</span>, <span class="org-string">'Wn'</span>, <span class="org-string">'Wu'</span>, <span class="org-string">'Wd'</span>);
|
||||
</pre>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-orgea5b4ad" class="outline-3">
|
||||
<h3 id="orgea5b4ad"><span class="section-number-3">3.2.</span> Open Loop disturbance and measurement noise</h3>
|
||||
<div class="outline-text-3" id="text-3-2">
|
||||
<p>
|
||||
The comparison of the amplitude spectral density of the measurement noise and of the jack parasitic motion is performed in Figure <a href="#orge20b337">13</a>.
|
||||
It confirms that the sensor noise is low enough to measure the motion errors of the crystal.
|
||||
</p>
|
||||
|
||||
|
||||
<div id="orge20b337" class="figure">
|
||||
<p><img src="figs/open_loop_noise_budget_fast_jack.png" alt="open_loop_noise_budget_fast_jack.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 13: </span>Open Loop noise budgeting</p>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org987b6b1" class="outline-2">
|
||||
<h2 id="org987b6b1"><span class="section-number-2">4.</span> Active Damping Plant (Strain gauges)</h2>
|
||||
<div class="outline-text-2" id="text-4">
|
||||
<p>
|
||||
<a id="org64d9ba7"></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 id="outline-container-orge45c4d8" class="outline-3">
|
||||
<h3 id="orge45c4d8"><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;
|
||||
@ -730,15 +829,15 @@ G_sg = linearize(mdl, io);
|
||||
</table>
|
||||
|
||||
|
||||
<div id="org8f04d26" class="figure">
|
||||
<div id="org782b33f" 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>
|
||||
<p><span class="figure-number">Figure 14: </span>Bode Plot of the transfer functions from piezoelectric forces to strain gauges measuremed displacements</p>
|
||||
</div>
|
||||
|
||||
<div class="important" id="orgd585bf5">
|
||||
<div class="important" id="org9ec0d6c">
|
||||
<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.
|
||||
As the distance between the poles and zeros in Figure <a href="#org0e9b8f2">17</a> is very small, little damping can be actively added using the strain gauges.
|
||||
This will be confirmed using a Root Locus plot.
|
||||
</p>
|
||||
|
||||
@ -746,23 +845,23 @@ This will be confirmed using a Root Locus plot.
|
||||
</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 id="outline-container-org65bc5e9" class="outline-3">
|
||||
<h3 id="org65bc5e9"><span class="section-number-3">4.2.</span> Relative Active Damping</h3>
|
||||
<div class="outline-text-3" id="text-4-2">
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab">Krad_g1 = eye(3)<span class="org-builtin">*</span>s<span class="org-builtin">/</span>(s<span class="org-builtin">^</span>2<span class="org-builtin">/</span>(2<span class="org-builtin">*</span><span class="org-matlab-math">pi</span><span class="org-builtin">*</span>500)<span class="org-builtin">^</span>2 <span class="org-builtin">+</span> 2<span class="org-builtin">*</span>s<span class="org-builtin">/</span>(2<span class="org-builtin">*</span><span class="org-matlab-math">pi</span><span class="org-builtin">*</span>500) <span class="org-builtin">+</span> 1);
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
<p>
|
||||
As can be seen in Figure <a href="#orgec235bb">13</a>, very little damping can be added using relative damping strategy using strain gauges.
|
||||
As can be seen in Figure <a href="#orgc3707d5">15</a>, very little damping can be added using relative damping strategy using strain gauges.
|
||||
</p>
|
||||
|
||||
|
||||
<div id="orgec235bb" class="figure">
|
||||
<div id="orgc3707d5" 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>
|
||||
<p><span class="figure-number">Figure 15: </span>Root Locus for the relative damping control</p>
|
||||
</div>
|
||||
|
||||
<div class="org-src-container">
|
||||
@ -772,9 +871,9 @@ As can be seen in Figure <a href="#orgec235bb">13</a>, very little damping can b
|
||||
</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">
|
||||
<div id="outline-container-orgc83a897" class="outline-3">
|
||||
<h3 id="orgc83a897"><span class="section-number-3">4.3.</span> Damped Plant</h3>
|
||||
<div class="outline-text-3" id="text-4-3">
|
||||
<p>
|
||||
The controller is implemented on Simscape, and the damped plant is identified.
|
||||
</p>
|
||||
@ -818,20 +917,20 @@ G_dp.OutputName = {<span class="org-string">'d_ur'</span>, <span class="org-str
|
||||
</div>
|
||||
|
||||
|
||||
<div id="orgca0b154" class="figure">
|
||||
<div id="org5d6f27c" 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>
|
||||
<p><span class="figure-number">Figure 16: </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">
|
||||
<div id="outline-container-orgf9782ba" class="outline-2">
|
||||
<h2 id="orgf9782ba"><span class="section-number-2">5.</span> Active Damping Plant (Force Sensors)</h2>
|
||||
<div class="outline-text-2" id="text-5">
|
||||
<p>
|
||||
<a id="orgb029a8b"></a>
|
||||
<a id="org7fbe16a"></a>
|
||||
</p>
|
||||
<p>
|
||||
Force sensors are added above the piezoelectric actuators.
|
||||
@ -839,9 +938,9 @@ 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 id="outline-container-orge8c03e7" class="outline-3">
|
||||
<h3 id="orge8c03e7"><span class="section-number-3">5.1.</span> Identification</h3>
|
||||
<div class="outline-text-3" id="text-5-1">
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab"><span class="org-matlab-cellbreak">%% Input/Output definition</span>
|
||||
clear io; io_i = 1;
|
||||
@ -863,22 +962,22 @@ G_fs = linearize(mdl, io);
|
||||
</div>
|
||||
|
||||
<p>
|
||||
The Bode plot of the identified dynamics is shown in Figure <a href="#org11a1e17">15</a>.
|
||||
The Bode plot of the identified dynamics is shown in Figure <a href="#org0e9b8f2">17</a>.
|
||||
At high frequency, the diagonal terms are constants while the off-diagonal terms have some roll-off.
|
||||
</p>
|
||||
|
||||
|
||||
<div id="org11a1e17" class="figure">
|
||||
<div id="org0e9b8f2" 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>
|
||||
<p><span class="figure-number">Figure 17: </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">
|
||||
<div id="outline-container-org135d9f2" class="outline-3">
|
||||
<h3 id="org135d9f2"><span class="section-number-3">5.2.</span> Controller - Root Locus</h3>
|
||||
<div class="outline-text-3" id="text-5-2">
|
||||
<p>
|
||||
We want to have integral action around the resonances of the system, but we do not want to integrate at low frequency.
|
||||
Therefore, we can use a low pass filter.
|
||||
@ -891,10 +990,10 @@ Kiff_g1 = eye(3)<span class="org-builtin">*</span>1<span class="org-builtin">/</
|
||||
</div>
|
||||
|
||||
|
||||
<div id="org5b2bab0" class="figure">
|
||||
<div id="org14798b7" 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>
|
||||
<p><span class="figure-number">Figure 18: </span>Root Locus plot for the IFF Control strategy</p>
|
||||
</div>
|
||||
|
||||
<div class="org-src-container">
|
||||
@ -911,38 +1010,38 @@ save(<span class="org-string">'mat/Kiff.mat'</span>, <span class="org-string">'K
|
||||
</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">
|
||||
<div id="outline-container-org4072236" class="outline-3">
|
||||
<h3 id="org4072236"><span class="section-number-3">5.3.</span> Damped Plant</h3>
|
||||
<div class="outline-text-3" id="text-5-3">
|
||||
<p>
|
||||
Both the Open Loop dynamics (see Figure <a href="#org015dc10">9</a>) and the dynamics with IFF (see Figure <a href="#org7a880a7">17</a>) are identified.
|
||||
Both the Open Loop dynamics (see Figure <a href="#org5f538a8">9</a>) and the dynamics with IFF (see Figure <a href="#org8cbe7f9">19</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>.
|
||||
This is schematically represented in Figure <a href="#org8cbe7f9">19</a>.
|
||||
</p>
|
||||
|
||||
|
||||
<div id="org7a880a7" class="figure">
|
||||
<div id="org8cbe7f9" 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>
|
||||
<p><span class="figure-number">Figure 19: </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>.
|
||||
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="#org73485e9">20</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">
|
||||
<div id="org73485e9" 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>
|
||||
<p><span class="figure-number">Figure 20: </span>Bode plot of both the open-loop plant and the damped plant using IFF</p>
|
||||
</div>
|
||||
|
||||
<div class="important" id="org8586fa6">
|
||||
<div class="important" id="org91bfa66">
|
||||
<p>
|
||||
The Integral Force Feedback control strategy is very effective in damping the modes present in the plant.
|
||||
</p>
|
||||
@ -952,42 +1051,42 @@ The Integral Force Feedback control strategy is very effective in damping the mo
|
||||
</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">
|
||||
<div id="outline-container-org5b57c31" class="outline-2">
|
||||
<h2 id="org5b57c31"><span class="section-number-2">6.</span> HAC-LAC (IFF) architecture</h2>
|
||||
<div class="outline-text-2" id="text-6">
|
||||
<p>
|
||||
<a id="orgee34a4d"></a>
|
||||
<a id="org226a3fc"></a>
|
||||
</p>
|
||||
<p>
|
||||
The HAC-LAC architecture is shown in Figure <a href="#orgb03e1da">19</a>.
|
||||
The HAC-LAC architecture is shown in Figure <a href="#org604ef36">21</a>.
|
||||
</p>
|
||||
|
||||
|
||||
<div id="orgb03e1da" class="figure">
|
||||
<div id="org604ef36" 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>
|
||||
<p><span class="figure-number">Figure 21: </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">
|
||||
<div id="outline-container-orgb4b8d52" class="outline-3">
|
||||
<h3 id="orgb4b8d52"><span class="section-number-3">6.1.</span> System Identification</h3>
|
||||
<div class="outline-text-3" id="text-6-1">
|
||||
<p>
|
||||
Let’s identify the damped plant.
|
||||
</p>
|
||||
|
||||
|
||||
<div id="org022508f" class="figure">
|
||||
<div id="orgb9d93cb" 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>
|
||||
<p><span class="figure-number">Figure 22: </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">
|
||||
<div id="outline-container-org714ec0f" class="outline-3">
|
||||
<h3 id="org714ec0f"><span class="section-number-3">6.2.</span> High Authority Controller</h3>
|
||||
<div class="outline-text-3" id="text-6-2">
|
||||
<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.
|
||||
@ -1018,28 +1117,28 @@ L_hac_lac = G_dp <span class="org-builtin">*</span> Khac;
|
||||
</div>
|
||||
|
||||
|
||||
<div id="org1eefea2" class="figure">
|
||||
<div id="orgc55c5d0" 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>
|
||||
<p><span class="figure-number">Figure 23: </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.
|
||||
As shown in the Root Locus plot in Figure <a href="#org54bbcb8">24</a>, the closed loop system should be stable.
|
||||
</p>
|
||||
|
||||
|
||||
<div id="orgc90ee63" class="figure">
|
||||
<div id="org54bbcb8" 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>
|
||||
<p><span class="figure-number">Figure 24: </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">
|
||||
<div id="outline-container-org9c9ea48" class="outline-3">
|
||||
<h3 id="org9c9ea48"><span class="section-number-3">6.3.</span> Performances</h3>
|
||||
<div class="outline-text-3" id="text-6-3">
|
||||
<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>
|
||||
@ -1059,22 +1158,72 @@ It is first verified that the closed-loop system is stable:
|
||||
|
||||
|
||||
<p>
|
||||
And both transmissibilities are compared in Figure <a href="#org152d7e8">23</a>.
|
||||
And both transmissibilities are compared in Figure <a href="#org948d6ba">25</a>.
|
||||
</p>
|
||||
|
||||
|
||||
<div id="org152d7e8" class="figure">
|
||||
<div id="org948d6ba" 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>
|
||||
<p><span class="figure-number">Figure 25: </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">
|
||||
<div class="important" id="orgc81cfdd">
|
||||
<p>
|
||||
The HAC-IFF control strategy can effectively reduce the transmissibility of the motion errors of the stepper motors.
|
||||
This reduction is effective inside the bandwidth of the controller.
|
||||
</p>
|
||||
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-orgbdf7b28" class="outline-3">
|
||||
<h3 id="orgbdf7b28"><span class="section-number-3">6.4.</span> Close Loop noise budget</h3>
|
||||
<div class="outline-text-3" id="text-6-4">
|
||||
<p>
|
||||
Let’s compute the amplitude spectral density of the jack motion errors due to the sensor noise, the actuator noise and disturbances.
|
||||
</p>
|
||||
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab"><span class="org-matlab-cellbreak">%% Computation of ASD of contribution of inputs to the closed-loop motion</span>
|
||||
<span class="org-comment-delimiter">% </span><span class="org-comment">Error due to disturbances</span>
|
||||
asd_d = abs(squeeze(freqresp(Wd<span class="org-builtin">*</span>(1<span class="org-builtin">/</span>(1 <span class="org-builtin">+</span> G_dp(1,1)<span class="org-builtin">*</span>Khac(1,1))), f, <span class="org-string">'Hz'</span>)));
|
||||
<span class="org-comment-delimiter">% </span><span class="org-comment">Error due to actuator noise</span>
|
||||
asd_u = abs(squeeze(freqresp(Wu<span class="org-builtin">*</span>(G_dp(1,1)<span class="org-builtin">/</span>(1 <span class="org-builtin">+</span> G_dp(1,1)<span class="org-builtin">*</span>Khac(1,1))), f, <span class="org-string">'Hz'</span>)));
|
||||
<span class="org-comment-delimiter">% </span><span class="org-comment">Error due to sensor noise</span>
|
||||
asd_n = abs(squeeze(freqresp(Wn<span class="org-builtin">*</span>(G_dp(1,1)<span class="org-builtin">*</span>Khac(1,1)<span class="org-builtin">/</span>(1 <span class="org-builtin">+</span> G_dp(1,1)<span class="org-builtin">*</span>Khac(1,1))), f, <span class="org-string">'Hz'</span>)));
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
<p>
|
||||
The closed-loop ASD is then:
|
||||
</p>
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab"><span class="org-matlab-cellbreak">%% ASD of the closed-loop motion</span>
|
||||
asd_cl = sqrt(asd_d<span class="org-builtin">.^</span>2 <span class="org-builtin">+</span> asd_u<span class="org-builtin">.^</span>2 <span class="org-builtin">+</span> asd_n<span class="org-builtin">.^</span>2);
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
<p>
|
||||
The obtained ASD are shown in Figure <a href="#orgf0027c6">26</a>.
|
||||
</p>
|
||||
|
||||
|
||||
<div id="orgf0027c6" class="figure">
|
||||
<p><img src="figs/close_loop_asd_noise_budget_hac_iff.png" alt="close_loop_asd_noise_budget_hac_iff.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 26: </span>Closed Loop noise budget</p>
|
||||
</div>
|
||||
|
||||
<p>
|
||||
Let’s compare the open-loop and close-loop cases (Figure <a href="#org691e64a">27</a>).
|
||||
</p>
|
||||
|
||||
<div id="org691e64a" class="figure">
|
||||
<p><img src="figs/cps_comp_ol_cl_hac_iff.png" alt="cps_comp_ol_cl_hac_iff.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 27: </span>Cumulative Power Spectrum of the open-loop and closed-loop motion error along one fast-jack</p>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
@ -1082,7 +1231,7 @@ This reduction is effective inside the bandwidth of the controller.
|
||||
</div>
|
||||
<div id="postamble" class="status">
|
||||
<p class="author">Author: Dehaeze Thomas</p>
|
||||
<p class="date">Created: 2021-11-30 mar. 15:17</p>
|
||||
<p class="date">Created: 2021-11-30 mar. 17:54</p>
|
||||
</div>
|
||||
</body>
|
||||
</html>
|
||||
|
||||
249
dcm-simscape.org
249
dcm-simscape.org
@ -56,6 +56,7 @@ In this document, a Simscape (.e.g. multi-body) model of the ESRF Double Crystal
|
||||
It is structured as follow:
|
||||
- Section [[sec:dcm_kinematics]]: the kinematics of the DCM is presented, and Jacobian matrices which are used to solve the inverse and forward kinematics are computed.
|
||||
- Section [[sec:open_loop_identification]]: the system dynamics is identified in the absence of control.
|
||||
- Section [[sec:dcm_noise_budget]]: an open-loop noise budget is performed.
|
||||
- Section [[sec:active_damping_strain_gauges]]: it is studied whether if the strain gauges fixed to the piezoelectric actuators can be used to actively damp the plant.
|
||||
- Section [[sec:active_damping_iff]]: 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.
|
||||
- Section [[sec:hac_iff]]: the High Authority Control - Low Authority Control (HAC-LAC) strategy is tested on the Simscape model.
|
||||
@ -95,17 +96,28 @@ It is structured as follow:
|
||||
#+end_src
|
||||
|
||||
** Bragg Angle
|
||||
#+begin_src matlab
|
||||
There is a simple relation eqref:eq:bragg_angle_formula between:
|
||||
- $d_{\text{off}}$ is the wanted offset between the incident x-ray and the output x-ray
|
||||
- $\theta_b$ is the bragg angle
|
||||
- $d_z$ is the corresponding distance between the first and second crystals
|
||||
|
||||
\begin{equation} \label{eq:bragg_angle_formula}
|
||||
d_z = \frac{d_{\text{off}}}{2 \cos \theta_b}
|
||||
\end{equation}
|
||||
|
||||
#+begin_src matlab :exports none
|
||||
%% Tested bragg angles
|
||||
bragg = linspace(5, 80, 1000); % Bragg angle [deg]
|
||||
d_off = 10.5e-3; % Wanted offset between x-rays [m]
|
||||
#+end_src
|
||||
|
||||
#+begin_src matlab
|
||||
#+begin_src matlab :exports none
|
||||
%% Vertical Jack motion as a function of Bragg angle
|
||||
dz = d_off./(2*cos(bragg*pi/180));
|
||||
#+end_src
|
||||
|
||||
This relation is shown in Figure [[fig:jack_motion_bragg_angle]].
|
||||
|
||||
#+begin_src matlab :exports none
|
||||
%% Jack motion as a function of Bragg angle
|
||||
figure;
|
||||
@ -122,6 +134,8 @@ exportFig('figs/jack_motion_bragg_angle.pdf', 'width', 'wide', 'height', 'normal
|
||||
#+RESULTS:
|
||||
[[file:figs/jack_motion_bragg_angle.png]]
|
||||
|
||||
The required jack stroke is approximately 25mm.
|
||||
|
||||
#+begin_src matlab :results value replace :exports both
|
||||
%% Required Jack stroke
|
||||
ans = 1e3*(dz(end) - dz(1))
|
||||
@ -701,6 +715,148 @@ Here, we map the piezo forces at the center of stiffness.
|
||||
|
||||
Let's first compute the Jacobian:
|
||||
|
||||
* Open-Loop Noise Budgeting
|
||||
:PROPERTIES:
|
||||
:header-args:matlab+: :tangle matlab/dcm_noise_budget.m
|
||||
:END:
|
||||
<<sec:dcm_noise_budget>>
|
||||
|
||||
** Introduction :ignore:
|
||||
|
||||
#+begin_src latex :file noise_budget_dcm_schematic_fast_jack_frame.pdf
|
||||
\begin{tikzpicture}
|
||||
% Blocs
|
||||
\node[block] (G) {$G(s)$};
|
||||
\node[addb, left= of G] (adddu) {};
|
||||
\node[block, left= of adddu] (K) {$K(s)$};
|
||||
\node[addb={+}{}{}{}{-}, left= of K] (subL) {};
|
||||
\node[addb, right= of G] (addd) {};
|
||||
\node[addb, below right=0.8 and 0.6 of addd] (adddn) {};
|
||||
|
||||
% Connections and labels
|
||||
\draw[->] (subL.east) -- (K.west);
|
||||
\draw[->] (K.east) -- (adddu.west);
|
||||
\draw[->] (adddu.east) -- (G.west);
|
||||
\draw[->] (G.east) -- (addd.west);
|
||||
\draw[->] (addd-|adddn)node[branch]{}node[above]{$y_{\text{fj}}$} -- (adddn.north);
|
||||
\draw[->] (adddn.west) -| (subL.south) node[below right]{$y_{\text{fj},m}$};
|
||||
\draw[<-] (adddu.north) -- ++(0, 1) node[below right]{$d_u$};
|
||||
\draw[<-] (addd.north) -- ++(0, 1) node[below right]{$d_{\text{fj}}$};
|
||||
\draw[<-] (adddn.east) -- ++(1, 0)coordinate(dn) node[above left]{$n_{\text{fj}}$};
|
||||
\draw[->] (addd.east) -- (addd-|dn);
|
||||
\end{tikzpicture}
|
||||
#+end_src
|
||||
|
||||
#+name: fig:noise_budget_dcm_schematic_fast_jack_frame
|
||||
#+caption: Schematic representation of the control loop in the frame of one fast jack
|
||||
#+RESULTS:
|
||||
[[file:figs/noise_budget_dcm_schematic_fast_jack_frame.png]]
|
||||
|
||||
** Matlab Init :noexport:ignore:
|
||||
#+begin_src matlab
|
||||
%% dcm_noise_budget.m
|
||||
% Basic uniaxial noise budgeting
|
||||
#+end_src
|
||||
|
||||
#+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name)
|
||||
<<matlab-dir>>
|
||||
#+end_src
|
||||
|
||||
#+begin_src matlab :exports none :results silent :noweb yes
|
||||
<<matlab-init>>
|
||||
#+end_src
|
||||
|
||||
#+begin_src matlab :tangle no :noweb yes
|
||||
<<m-init-path>>
|
||||
#+end_src
|
||||
|
||||
#+begin_src matlab :eval no :noweb yes
|
||||
<<m-init-path-tangle>>
|
||||
#+end_src
|
||||
|
||||
#+begin_src matlab :noweb yes
|
||||
<<m-init-other>>
|
||||
|
||||
%% Frequency vector for noise budget [Hz]
|
||||
f = logspace(-1, 3, 1000);
|
||||
#+end_src
|
||||
|
||||
** Power Spectral Density of signals
|
||||
|
||||
Interferometer noise:
|
||||
#+begin_src matlab
|
||||
Wn = 6e-11*(1 + s/2/pi/200)/(1 + s/2/pi/60); % m/sqrt(Hz)
|
||||
#+end_src
|
||||
|
||||
#+begin_src matlab :results value replace :exports results :tangle no
|
||||
sprintf('Measurement noise: %.2f [nm,rms]', 1e9*sqrt(trapz(f, abs(squeeze(freqresp(Wn, f, 'Hz'))).^2)));
|
||||
#+end_src
|
||||
|
||||
#+RESULTS:
|
||||
: Measurement noise: 0.79 [nm,rms]
|
||||
|
||||
DAC noise (amplified by the PI voltage amplifier, and converted to newtons):
|
||||
#+begin_src matlab
|
||||
Wdac = tf(3e-8); % V/sqrt(Hz)
|
||||
Wu = Wdac*22.5*10; % N/sqrt(Hz)
|
||||
#+end_src
|
||||
|
||||
#+begin_src matlab :results value replace :exports results :tangle no
|
||||
sprintf('DAC noise: %.2f [uV,rms]', 1e6*sqrt(trapz(f, abs(squeeze(freqresp(Wdac, f, 'Hz'))).^2)));
|
||||
#+end_src
|
||||
|
||||
#+RESULTS:
|
||||
: DAC noise: 0.95 [uV,rms]
|
||||
|
||||
Disturbances:
|
||||
#+begin_src matlab
|
||||
Wd = 5e-7/(1 + s/2/pi); % m/sqrt(Hz)
|
||||
#+end_src
|
||||
|
||||
#+begin_src matlab :results value replace :exports results :tangle no
|
||||
sprintf('Disturbance motion: %.2f [um,rms]', 1e6*sqrt(trapz(f, abs(squeeze(freqresp(Wd, f, 'Hz'))).^2)));
|
||||
#+end_src
|
||||
|
||||
#+RESULTS:
|
||||
: Disturbance motion: 0.61 [um,rms]
|
||||
|
||||
#+begin_src matlab :exports none :tangle no
|
||||
save('matlab/mat/asd_noises_disturbances.mat', 'Wn', 'Wu', 'Wd');
|
||||
#+end_src
|
||||
|
||||
#+begin_src matlab :eval no
|
||||
%% Save ASD of noise and disturbances
|
||||
save('mat/asd_noises_disturbances.mat', 'Wn', 'Wu', 'Wd');
|
||||
#+end_src
|
||||
|
||||
** Open Loop disturbance and measurement noise
|
||||
The comparison of the amplitude spectral density of the measurement noise and of the jack parasitic motion is performed in Figure [[fig:open_loop_noise_budget_fast_jack]].
|
||||
It confirms that the sensor noise is low enough to measure the motion errors of the crystal.
|
||||
|
||||
#+begin_src matlab :exports none
|
||||
%% Bode plot for the plant (strain gauge output)
|
||||
figure;
|
||||
hold on;
|
||||
plot(f, abs(squeeze(freqresp(Wn, f, 'Hz'))), ...
|
||||
'DisplayName', 'n');
|
||||
plot(f, abs(squeeze(freqresp(Wd, f, 'Hz'))), ...
|
||||
'DisplayName', 'd');
|
||||
hold off;
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
||||
xlabel('Frequency [Hz]'); ylabel('ASD [$m/\sqrt{Hz}$]');
|
||||
legend('location', 'northeast', 'FontSize', 8, 'NumColumns', 2);
|
||||
xlim([f(1), f(end)]);
|
||||
#+end_src
|
||||
|
||||
#+begin_src matlab :tangle no :exports results :results file replace
|
||||
exportFig('figs/open_loop_noise_budget_fast_jack.pdf', 'width', 'wide', 'height', 'normal');
|
||||
#+end_src
|
||||
|
||||
#+name: fig:open_loop_noise_budget_fast_jack
|
||||
#+caption: Open Loop noise budgeting
|
||||
#+RESULTS:
|
||||
[[file:figs/open_loop_noise_budget_fast_jack.png]]
|
||||
|
||||
* Active Damping Plant (Strain gauges)
|
||||
:PROPERTIES:
|
||||
:header-args:matlab+: :tangle matlab/dcm_active_damping_strain_gauges.m
|
||||
@ -1673,6 +1829,95 @@ The HAC-IFF control strategy can effectively reduce the transmissibility of the
|
||||
This reduction is effective inside the bandwidth of the controller.
|
||||
#+end_important
|
||||
|
||||
** Close Loop noise budget
|
||||
#+begin_src matlab :exports none
|
||||
%% Load disturbances
|
||||
load('asd_noises_disturbances.mat');
|
||||
#+end_src
|
||||
|
||||
Let's compute the amplitude spectral density of the jack motion errors due to the sensor noise, the actuator noise and disturbances.
|
||||
|
||||
#+begin_src matlab
|
||||
%% Computation of ASD of contribution of inputs to the closed-loop motion
|
||||
% Error due to disturbances
|
||||
asd_d = abs(squeeze(freqresp(Wd*(1/(1 + G_dp(1,1)*Khac(1,1))), f, 'Hz')));
|
||||
% Error due to actuator noise
|
||||
asd_u = abs(squeeze(freqresp(Wu*(G_dp(1,1)/(1 + G_dp(1,1)*Khac(1,1))), f, 'Hz')));
|
||||
% Error due to sensor noise
|
||||
asd_n = abs(squeeze(freqresp(Wn*(G_dp(1,1)*Khac(1,1)/(1 + G_dp(1,1)*Khac(1,1))), f, 'Hz')));
|
||||
#+end_src
|
||||
|
||||
The closed-loop ASD is then:
|
||||
#+begin_src matlab
|
||||
%% ASD of the closed-loop motion
|
||||
asd_cl = sqrt(asd_d.^2 + asd_u.^2 + asd_n.^2);
|
||||
#+end_src
|
||||
|
||||
The obtained ASD are shown in Figure [[fig:close_loop_asd_noise_budget_hac_iff]].
|
||||
|
||||
#+begin_src matlab :exports none
|
||||
%% Noise Budget (ASD)
|
||||
f = logspace(-1, 3, 1000);
|
||||
|
||||
figure;
|
||||
hold on;
|
||||
plot(f, asd_n, 'DisplayName', '$n$');
|
||||
plot(f, asd_u, 'DisplayName', '$d_u$');
|
||||
plot(f, asd_d, 'DisplayName', '$d$');
|
||||
plot(f, asd_cl, 'k--', 'DisplayName', '$y$');
|
||||
hold off;
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
||||
xlabel('Frequency [Hz]'); ylabel('ASD [$m/\sqrt{Hz}$]');
|
||||
legend('location', 'southeast', 'FontSize', 8, 'NumColumns', 2);
|
||||
xlim([f(1), f(end)]);
|
||||
ylim([1e-16, 1e-8]);
|
||||
#+end_src
|
||||
|
||||
#+begin_src matlab :tangle no :exports results :results file replace
|
||||
exportFig('figs/close_loop_asd_noise_budget_hac_iff.pdf', 'width', 'wide', 'height', 'tall');
|
||||
#+end_src
|
||||
|
||||
#+name: fig:close_loop_asd_noise_budget_hac_iff
|
||||
#+caption: Closed Loop noise budget
|
||||
#+RESULTS:
|
||||
[[file:figs/close_loop_asd_noise_budget_hac_iff.png]]
|
||||
|
||||
Let's compare the open-loop and close-loop cases (Figure [[fig:cps_comp_ol_cl_hac_iff]]).
|
||||
#+begin_src matlab :exports none
|
||||
% Amplitude spectral density of the open loop motion errors [m/sqrt(Hz)]
|
||||
asd_ol = abs(squeeze(freqresp(Wd, f, 'Hz')));
|
||||
#+end_src
|
||||
|
||||
#+begin_src matlab :exports none
|
||||
% CPS of open-loop motion [m^2]
|
||||
cps_ol = flip(-cumtrapz(flip(f), flip(asd_ol.^2)));
|
||||
% CPS of closed-loop motion [m^2]
|
||||
cps_cl = flip(-cumtrapz(flip(f), flip(asd_cl.^2)));
|
||||
#+end_src
|
||||
|
||||
#+begin_src matlab :exports none
|
||||
%% Cumulative Power Spectrum - Motion error of fast jack
|
||||
figure;
|
||||
hold on;
|
||||
plot(f, cps_ol, 'DisplayName', sprintf('OL, $\\epsilon_d = %.0f$ [nm,rms]', 1e9*sqrt(cps_ol(1))));
|
||||
plot(f, cps_cl, 'DisplayName', sprintf('CL, $\\epsilon_d = %.0f$ [nm,rms]', 1e9*sqrt(cps_cl(1))));
|
||||
hold off;
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
||||
xlabel('Frequency [Hz]'); ylabel('CPS [$m^2$]');
|
||||
legend('location', 'southwest', 'FontSize', 8);
|
||||
xlim([f(1), f(end)]);
|
||||
% ylim([1e-16, 1e-8]);
|
||||
#+end_src
|
||||
|
||||
#+begin_src matlab :tangle no :exports results :results file replace
|
||||
exportFig('figs/cps_comp_ol_cl_hac_iff.pdf', 'width', 'wide', 'height', 'normal');
|
||||
#+end_src
|
||||
|
||||
#+name: fig:cps_comp_ol_cl_hac_iff
|
||||
#+caption: Cumulative Power Spectrum of the open-loop and closed-loop motion error along one fast-jack
|
||||
#+RESULTS:
|
||||
[[file:figs/cps_comp_ol_cl_hac_iff.png]]
|
||||
|
||||
* Helping Functions :noexport:
|
||||
** Initialize Path
|
||||
#+NAME: m-init-path
|
||||
|
||||
BIN
dcm-simscape.pdf
BIN
dcm-simscape.pdf
Binary file not shown.
BIN
figs/close_loop_asd_noise_budget_hac_iff.pdf
Normal file
BIN
figs/close_loop_asd_noise_budget_hac_iff.pdf
Normal file
Binary file not shown.
BIN
figs/close_loop_asd_noise_budget_hac_iff.png
Normal file
BIN
figs/close_loop_asd_noise_budget_hac_iff.png
Normal file
Binary file not shown.
|
After Width: | Height: | Size: 107 KiB |
1241
figs/cps_comp_ol_cl_hac_iff.pdf
Normal file
1241
figs/cps_comp_ol_cl_hac_iff.pdf
Normal file
File diff suppressed because it is too large
Load Diff
BIN
figs/cps_comp_ol_cl_hac_iff.png
Normal file
BIN
figs/cps_comp_ol_cl_hac_iff.png
Normal file
Binary file not shown.
|
After Width: | Height: | Size: 70 KiB |
BIN
figs/noise_budget_dcm_schematic_fast_jack_frame.pdf
Normal file
BIN
figs/noise_budget_dcm_schematic_fast_jack_frame.pdf
Normal file
Binary file not shown.
BIN
figs/noise_budget_dcm_schematic_fast_jack_frame.png
Normal file
BIN
figs/noise_budget_dcm_schematic_fast_jack_frame.png
Normal file
Binary file not shown.
|
After Width: | Height: | Size: 12 KiB |
211
figs/noise_budget_dcm_schematic_fast_jack_frame.svg
Normal file
211
figs/noise_budget_dcm_schematic_fast_jack_frame.svg
Normal file
@ -0,0 +1,211 @@
|
||||
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||||
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<g style="fill:rgb(0%,0%,0%);fill-opacity:1;">
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<use xlink:href="#glyph4-1" x="292.628256" y="70.122517"/>
|
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<use xlink:href="#glyph4-2" x="295.071288" y="70.122517"/>
|
||||
</g>
|
||||
<path style="fill:none;stroke-width:0.99628;stroke-linecap:butt;stroke-linejoin:miter;stroke:rgb(0%,0%,0%);stroke-opacity:1;stroke-miterlimit:10;" d="M 65.778592 0.00197369 L 120.590593 0.00197369 " transform="matrix(0.989034,0,0,-0.989034,177.934917,38.876952)"/>
|
||||
<path style=" stroke:none;fill-rule:nonzero;fill:rgb(0%,0%,0%);fill-opacity:1;" d="M 300.382812 38.875 L 295.984375 37.210938 L 297.449219 38.875 L 295.984375 40.542969 Z M 300.382812 38.875 "/>
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<g clip-path="url(#clip6)" clip-rule="nonzero">
|
||||
<path style="fill:none;stroke-width:0.99628;stroke-linecap:butt;stroke-linejoin:miter;stroke:rgb(0%,0%,0%);stroke-opacity:1;stroke-miterlimit:10;" d="M 6.054536 0.00197369 L 1.60733 1.684486 L 3.088416 0.00197369 L 1.60733 -1.684489 Z M 6.054536 0.00197369 " transform="matrix(0.989034,0,0,-0.989034,294.39467,38.876952)"/>
|
||||
</g>
|
||||
</g>
|
||||
</svg>
|
||||
|
After Width: | Height: | Size: 33 KiB |
BIN
figs/open_loop_noise_budget_fast_jack.pdf
Normal file
BIN
figs/open_loop_noise_budget_fast_jack.pdf
Normal file
Binary file not shown.
BIN
figs/open_loop_noise_budget_fast_jack.png
Normal file
BIN
figs/open_loop_noise_budget_fast_jack.png
Normal file
Binary file not shown.
|
After Width: | Height: | Size: 63 KiB |
@ -54,10 +54,8 @@ G_fs.OutputName = {'fs_ur', 'fs_uh', 'fs_d'};
|
||||
|
||||
|
||||
|
||||
% #+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.
|
||||
|
||||
|
||||
%% Bode plot for the plant
|
||||
@ -82,7 +80,7 @@ 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]);
|
||||
|
||||
@ -102,7 +100,11 @@ linkaxes([ax1,ax2],'x');
|
||||
xlim([freqs(1), freqs(end)]);
|
||||
|
||||
% 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.
|
||||
|
||||
|
||||
%% Integral Force Feedback Controller
|
||||
Kiff_g1 = eye(3)*1/(1 + s/2/pi/20);
|
||||
|
||||
%% Root Locus for IFF
|
||||
@ -141,10 +143,19 @@ legend('location', 'northwest');
|
||||
% [[file:figs/iff_root_locus.png]]
|
||||
|
||||
|
||||
%% Integral Force Feedback Controller
|
||||
%% Integral Force Feedback Controller with optimal gain
|
||||
Kiff = g*Kiff_g1;
|
||||
|
||||
% Damped Plant
|
||||
%% Save the IFF controller
|
||||
save('mat/Kiff.mat', 'Kiff');
|
||||
|
||||
|
||||
|
||||
% #+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]]
|
||||
|
||||
|
||||
%% Input/Output definition
|
||||
clear io; io_i = 1;
|
||||
@ -157,8 +168,8 @@ io(io_i) = linio([mdl, '/control_system'], 1, 'input'); io_i = io_i + 1;
|
||||
% Force Sensor {3x1} [m]
|
||||
io(io_i) = linio([mdl, '/DCM'], 1, 'openoutput'); io_i = io_i + 1;
|
||||
|
||||
%% DCM Kinematics
|
||||
load('mat/dcm_kinematics.mat');
|
||||
%% Load DCM Kinematics
|
||||
load('dcm_kinematics.mat');
|
||||
|
||||
%% Identification of the Open Loop plant
|
||||
controller.type = 0; % Open Loop
|
||||
@ -172,6 +183,12 @@ 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'};
|
||||
|
||||
|
||||
|
||||
% 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.
|
||||
|
||||
|
||||
%% Comparison of the damped and undamped plant
|
||||
figure;
|
||||
tiledlayout(3, 1, 'TileSpacing', 'Compact', 'Padding', 'None');
|
||||
@ -221,5 +238,3 @@ ylim([-180, 0]);
|
||||
|
||||
linkaxes([ax1,ax2],'x');
|
||||
xlim([freqs(1), freqs(end)]);
|
||||
|
||||
save('mat/Kiff.mat', 'Kiff');
|
||||
|
||||
@ -40,13 +40,11 @@ clear io; io_i = 1;
|
||||
|
||||
%% Inputs
|
||||
% Control Input {3x1} [N]
|
||||
io(io_i) = linio([mdl, '/u'], 1, 'openinput'); io_i = io_i + 1;
|
||||
% % Stepper Displacement {3x1} [m]
|
||||
% io(io_i) = linio([mdl, '/d'], 1, 'openinput'); io_i = io_i + 1;
|
||||
io(io_i) = linio([mdl, '/control_system'], 1, 'openinput'); io_i = io_i + 1;
|
||||
|
||||
%% Outputs
|
||||
% Strain Gauges {3x1} [m]
|
||||
io(io_i) = linio([mdl, '/sg'], 1, 'openoutput'); io_i = io_i + 1;
|
||||
io(io_i) = linio([mdl, '/DCM'], 2, 'openoutput'); io_i = io_i + 1;
|
||||
|
||||
%% Extraction of the dynamics
|
||||
G_sg = linearize(mdl, io);
|
||||
@ -57,12 +55,12 @@ G_sg.OutputName = {'sg_ur', 'sg_uh', 'sg_d'};
|
||||
|
||||
|
||||
% #+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 |
|
||||
% | 4.4443e-09 | 1.0339e-13 | 3.774e-14 |
|
||||
% | 1.0339e-13 | 4.4443e-09 | 3.774e-14 |
|
||||
% | 3.7792e-14 | 3.7792e-14 | 4.4444e-09 |
|
||||
|
||||
|
||||
%% Bode plot for the plant
|
||||
%% Bode plot for the plant (strain gauge output)
|
||||
figure;
|
||||
tiledlayout(3, 1, 'TileSpacing', 'Compact', 'Padding', 'None');
|
||||
|
||||
@ -83,7 +81,8 @@ end
|
||||
hold off;
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
||||
ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]);
|
||||
legend('location', 'southwest', 'FontSize', 8, 'NumColumns', 2);
|
||||
legend('location', 'southeast', 'FontSize', 8, 'NumColumns', 2);
|
||||
ylim([1e-14, 1e-7]);
|
||||
|
||||
ax2 = nexttile;
|
||||
hold on;
|
||||
@ -95,7 +94,133 @@ set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
|
||||
xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
|
||||
hold off;
|
||||
yticks(-360:90:360);
|
||||
ylim([-180, 180]);
|
||||
ylim([-180, 0]);
|
||||
|
||||
linkaxes([ax1,ax2],'x');
|
||||
xlim([freqs(1), freqs(end)]);
|
||||
|
||||
% Relative Active Damping
|
||||
|
||||
Krad_g1 = eye(3)*s/(s^2/(2*pi*500)^2 + 2*s/(2*pi*500) + 1);
|
||||
|
||||
|
||||
|
||||
% As can be seen in Figure [[fig:relative_damping_root_locus]], very little damping can be added using relative damping strategy using strain gauges.
|
||||
|
||||
|
||||
%% Root Locus for IFF
|
||||
gains = logspace(3, 8, 200);
|
||||
|
||||
figure;
|
||||
|
||||
hold on;
|
||||
plot(real(pole(G_sg)), imag(pole(G_sg)), 'x', 'color', colors(1,:), ...
|
||||
'DisplayName', '$g = 0$');
|
||||
plot(real(tzero(G_sg)), imag(tzero(G_sg)), 'o', 'color', colors(1,:), ...
|
||||
'HandleVisibility', 'off');
|
||||
|
||||
for g = gains
|
||||
clpoles = pole(feedback(G_sg, g*Krad_g1, -1));
|
||||
plot(real(clpoles), imag(clpoles), '.', 'color', colors(1,:), ...
|
||||
'HandleVisibility', 'off');
|
||||
end
|
||||
|
||||
% Optimal gain
|
||||
g = 2e5;
|
||||
clpoles = pole(feedback(G_sg, g*Krad_g1, -1));
|
||||
plot(real(clpoles), imag(clpoles), 'x', 'color', colors(2,:), ...
|
||||
'DisplayName', sprintf('$g=%.0e$', g));
|
||||
hold off;
|
||||
xlim([-6, 0]); ylim([0, 2700]);
|
||||
xlabel('Real Part'); ylabel('Imaginary Part');
|
||||
legend('location', 'northwest');
|
||||
|
||||
|
||||
|
||||
% #+name: fig:relative_damping_root_locus
|
||||
% #+caption: Root Locus for the relative damping control
|
||||
% #+RESULTS:
|
||||
% [[file:figs/relative_damping_root_locus.png]]
|
||||
|
||||
|
||||
Krad = -g*Krad_g1;
|
||||
|
||||
% Damped Plant
|
||||
% The controller is implemented on Simscape, and the damped plant is identified.
|
||||
|
||||
|
||||
%% 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;
|
||||
|
||||
%% DCM Kinematics
|
||||
load('dcm_kinematics.mat');
|
||||
|
||||
%% Identification of the Open Loop plant
|
||||
controller.type = 0; % Open Loop
|
||||
G_ol = J_a_111*inv(J_s_111)*linearize(mdl, io);
|
||||
G_ol.InputName = {'u_ur', 'u_uh', 'u_d'};
|
||||
G_ol.OutputName = {'d_ur', 'd_uh', 'd_d'};
|
||||
|
||||
%% Identification of the damped plant with Relative Active Damping
|
||||
controller.type = 2; % RAD
|
||||
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'};
|
||||
|
||||
%% 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_ol(1,1), freqs, 'Hz'))), ...
|
||||
'DisplayName', 'd - OL');
|
||||
plot(freqs, abs(squeeze(freqresp(G_ol(2,2), freqs, 'Hz'))), ...
|
||||
'DisplayName', 'uh - OL');
|
||||
plot(freqs, abs(squeeze(freqresp(G_ol(3,3), freqs, 'Hz'))), ...
|
||||
'DisplayName', 'ur - OL');
|
||||
set(gca,'ColorOrderIndex',1)
|
||||
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-6]);
|
||||
|
||||
ax2 = nexttile;
|
||||
hold on;
|
||||
plot(freqs, 180/pi*angle(squeeze(freqresp(G_ol(1,1), freqs, 'Hz'))));
|
||||
plot(freqs, 180/pi*angle(squeeze(freqresp(G_ol(2,2), freqs, 'Hz'))));
|
||||
plot(freqs, 180/pi*angle(squeeze(freqresp(G_ol(3,3), freqs, 'Hz'))));
|
||||
set(gca,'ColorOrderIndex',1)
|
||||
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)]);
|
||||
|
||||
@ -32,3 +32,308 @@ colors = colororder;
|
||||
|
||||
%% Frequency Vector
|
||||
freqs = logspace(1, 3, 1000);
|
||||
|
||||
% System Identification
|
||||
% Let's identify the damped plant.
|
||||
|
||||
|
||||
%% 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;
|
||||
|
||||
%% Load DCM Kinematics and IFF controller
|
||||
load('dcm_kinematics.mat');
|
||||
load('Kiff.mat');
|
||||
|
||||
%% 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'};
|
||||
|
||||
%% 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)]);
|
||||
|
||||
% 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.
|
||||
|
||||
|
||||
%% 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
|
||||
|
||||
%% Save the HAC controller
|
||||
save('mat/Khac_iff.mat', 'Khac');
|
||||
|
||||
%% Loop Gain
|
||||
L_hac_lac = G_dp * Khac;
|
||||
|
||||
%% 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)]);
|
||||
|
||||
|
||||
|
||||
% #+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]]
|
||||
|
||||
|
||||
%% 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
|
||||
|
||||
|
||||
|
||||
% As shown in the Root Locus plot in Figure [[fig:loci_hac_iff_fast_jack]], the closed loop system should be stable.
|
||||
|
||||
|
||||
%% Plot of the eigenvalues of L in the complex plane
|
||||
figure;
|
||||
hold on;
|
||||
% Angle used to draw the circles
|
||||
theta = linspace(0, 2*pi, 100);
|
||||
% Unit circle
|
||||
plot(cos(theta), sin(theta), '--');
|
||||
% Circle for module margin
|
||||
plot(-1 + min(min(abs(Ldet + 1)))*cos(theta), min(min(abs(Ldet + 1)))*sin(theta), '--');
|
||||
|
||||
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
|
||||
% Unstable Point
|
||||
plot(-1, 0, 'kx', 'HandleVisibility', 'off');
|
||||
hold off;
|
||||
set(gca, 'XScale', 'lin'); set(gca, 'YScale', 'lin');
|
||||
xlabel('Real'); ylabel('Imag');
|
||||
axis square;
|
||||
xlim([-3, 1]); ylim([-2, 2]);
|
||||
|
||||
% 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.
|
||||
|
||||
|
||||
%% 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;
|
||||
|
||||
%% 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'};
|
||||
|
||||
%% Load DCM Kinematics and IFF controller
|
||||
load('dcm_kinematics.mat');
|
||||
load('Kiff.mat');
|
||||
|
||||
%% 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'};
|
||||
|
||||
|
||||
|
||||
% #+RESULTS:
|
||||
% : 1
|
||||
|
||||
% And both transmissibilities are compared in Figure [[fig:stepper_transmissibility_comp_ol_hac_iff]].
|
||||
|
||||
|
||||
%% 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)]);
|
||||
|
||||
% Close Loop noise budget
|
||||
|
||||
%% Load disturbances
|
||||
load('asd_noises_disturbances.mat');
|
||||
|
||||
|
||||
|
||||
% Let's compute the amplitude spectral density of the jack motion errors due to the sensor noise, the actuator noise and disturbances.
|
||||
|
||||
|
||||
%% Computation of ASD of contribution of inputs to the closed-loop motion
|
||||
% Error due to disturbances
|
||||
asd_d = abs(squeeze(freqresp(Wd*(1/(1 + G_dp(1,1)*Khac(1,1))), f, 'Hz')));
|
||||
% Error due to actuator noise
|
||||
asd_u = abs(squeeze(freqresp(Wu*(G_dp(1,1)/(1 + G_dp(1,1)*Khac(1,1))), f, 'Hz')));
|
||||
% Error due to sensor noise
|
||||
asd_n = abs(squeeze(freqresp(Wn*(G_dp(1,1)*Khac(1,1)/(1 + G_dp(1,1)*Khac(1,1))), f, 'Hz')));
|
||||
|
||||
|
||||
|
||||
% The closed-loop ASD is then:
|
||||
|
||||
%% ASD of the closed-loop motion
|
||||
asd_cl = sqrt(asd_d.^2 + asd_u.^2 + asd_n.^2);
|
||||
|
||||
|
||||
|
||||
% The obtained ASD are shown in Figure [[fig:close_loop_asd_noise_budget_hac_iff]].
|
||||
|
||||
|
||||
%% Noise Budget (ASD)
|
||||
f = logspace(-1, 3, 1000);
|
||||
|
||||
figure;
|
||||
hold on;
|
||||
plot(f, asd_n, 'DisplayName', '$n$');
|
||||
plot(f, asd_u, 'DisplayName', '$d_u$');
|
||||
plot(f, asd_d, 'DisplayName', '$d$');
|
||||
plot(f, asd_cl, 'k--', 'DisplayName', '$y$');
|
||||
hold off;
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
||||
xlabel('Frequency [Hz]'); ylabel('ASD [$m/\sqrt{Hz}$]');
|
||||
legend('location', 'southeast', 'FontSize', 8, 'NumColumns', 2);
|
||||
xlim([f(1), f(end)]);
|
||||
ylim([1e-16, 1e-8]);
|
||||
|
||||
|
||||
|
||||
% #+name: fig:close_loop_asd_noise_budget_hac_iff
|
||||
% #+caption: Closed Loop noise budget
|
||||
% #+RESULTS:
|
||||
% [[file:figs/close_loop_asd_noise_budget_hac_iff.png]]
|
||||
|
||||
% Let's compare the open-loop and close-loop cases (Figure [[fig:cps_comp_ol_cl_hac_iff]]).
|
||||
|
||||
% Amplitude spectral density of the open loop motion errors [m/sqrt(Hz)]
|
||||
asd_ol = abs(squeeze(freqresp(Wd, f, 'Hz')));
|
||||
|
||||
% CPS of open-loop motion [m^2]
|
||||
cps_ol = flip(-cumtrapz(flip(f), flip(asd_ol.^2)));
|
||||
% CPS of closed-loop motion [m^2]
|
||||
cps_cl = flip(-cumtrapz(flip(f), flip(asd_cl.^2)));
|
||||
|
||||
%% Cumulative Power Spectrum - Motion error of fast jack
|
||||
figure;
|
||||
hold on;
|
||||
plot(f, cps_ol, 'DisplayName', sprintf('OL, $\\epsilon_d = %.0f$ [nm,rms]', 1e9*sqrt(cps_ol(1))));
|
||||
plot(f, cps_cl, 'DisplayName', sprintf('CL, $\\epsilon_d = %.0f$ [nm,rms]', 1e9*sqrt(cps_cl(1))));
|
||||
hold off;
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
||||
xlabel('Frequency [Hz]'); ylabel('CPS [$m^2$]');
|
||||
legend('location', 'southwest', 'FontSize', 8);
|
||||
xlim([f(1), f(end)]);
|
||||
% ylim([1e-16, 1e-8]);
|
||||
|
||||
@ -63,7 +63,7 @@ G.OutputName = {'int_111_1', 'int_111_2', 'int_111_3'};
|
||||
|
||||
% Plant in the frame of the fastjacks
|
||||
|
||||
load('mat/dcm_kinematics.mat');
|
||||
load('dcm_kinematics.mat');
|
||||
|
||||
|
||||
|
||||
|
||||
@ -22,6 +22,15 @@ colors = colororder;
|
||||
freqs = logspace(1, 3, 1000);
|
||||
|
||||
% Bragg Angle
|
||||
% There is a simple relation eqref:eq:bragg_angle_formula between:
|
||||
% - $d_{\text{off}}$ is the wanted offset between the incident x-ray and the output x-ray
|
||||
% - $\theta_b$ is the bragg angle
|
||||
% - $d_z$ is the corresponding distance between the first and second crystals
|
||||
|
||||
% \begin{equation} \label{eq:bragg_angle_formula}
|
||||
% d_z = \frac{d_{\text{off}}}{2 \cos \theta_b}
|
||||
% \end{equation}
|
||||
|
||||
|
||||
%% Tested bragg angles
|
||||
bragg = linspace(5, 80, 1000); % Bragg angle [deg]
|
||||
@ -30,6 +39,11 @@ d_off = 10.5e-3; % Wanted offset between x-rays [m]
|
||||
%% Vertical Jack motion as a function of Bragg angle
|
||||
dz = d_off./(2*cos(bragg*pi/180));
|
||||
|
||||
|
||||
|
||||
% This relation is shown in Figure [[fig:jack_motion_bragg_angle]].
|
||||
|
||||
|
||||
%% Jack motion as a function of Bragg angle
|
||||
figure;
|
||||
plot(bragg, 1e3*dz)
|
||||
@ -42,6 +56,8 @@ xlabel('Bragg angle [deg]'); ylabel('Jack Motion [mm]');
|
||||
% #+RESULTS:
|
||||
% [[file:figs/jack_motion_bragg_angle.png]]
|
||||
|
||||
% The required jack stroke is approximately 25mm.
|
||||
|
||||
|
||||
%% Required Jack stroke
|
||||
ans = 1e3*(dz(end) - dz(1))
|
||||
|
||||
71
matlab/dcm_noise_budget.m
Normal file
71
matlab/dcm_noise_budget.m
Normal file
@ -0,0 +1,71 @@
|
||||
% Matlab Init :noexport:ignore:
|
||||
|
||||
%% dcm_noise_budget.m
|
||||
% Basic uniaxial noise budgeting
|
||||
|
||||
%% Clear Workspace and Close figures
|
||||
clear; close all; clc;
|
||||
|
||||
%% Intialize Laplace variable
|
||||
s = zpk('s');
|
||||
|
||||
%% Path for functions, data and scripts
|
||||
addpath('./mat/'); % Path for data
|
||||
|
||||
%% Simscape Model - Nano Hexapod
|
||||
addpath('./STEPS/')
|
||||
|
||||
%% Colors for the figures
|
||||
colors = colororder;
|
||||
|
||||
%% Frequency Vector
|
||||
freqs = logspace(1, 3, 1000);
|
||||
|
||||
%% Frequency vector for noise budget [Hz]
|
||||
f = logspace(-1, 3, 1000);
|
||||
|
||||
% Power Spectral Density of signals
|
||||
|
||||
% Interferometer noise:
|
||||
|
||||
Wn = 6e-11*(1 + s/2/pi/200)/(1 + s/2/pi/60); % m/sqrt(Hz)
|
||||
|
||||
|
||||
|
||||
% #+RESULTS:
|
||||
% : Measurement noise: 0.79 [nm,rms]
|
||||
|
||||
% DAC noise (amplified by the PI voltage amplifier, and converted to newtons):
|
||||
|
||||
Wdac = tf(3e-8); % V/sqrt(Hz)
|
||||
Wu = Wdac*22.5*10; % N/sqrt(Hz)
|
||||
|
||||
|
||||
|
||||
% #+RESULTS:
|
||||
% : DAC noise: 0.95 [uV,rms]
|
||||
|
||||
% Disturbances:
|
||||
|
||||
Wd = 5e-7/(1 + s/2/pi); % m/sqrt(Hz)
|
||||
|
||||
%% Save ASD of noise and disturbances
|
||||
save('mat/asd_noises_disturbances.mat', 'Wn', 'Wu', 'Wd');
|
||||
|
||||
% Open Loop disturbance and measurement noise
|
||||
% The comparison of the amplitude spectral density of the measurement noise and of the jack parasitic motion is performed in Figure [[fig:open_loop_noise_budget_fast_jack]].
|
||||
% It confirms that the sensor noise is low enough to measure the motion errors of the crystal.
|
||||
|
||||
|
||||
%% Bode plot for the plant (strain gauge output)
|
||||
figure;
|
||||
hold on;
|
||||
plot(f, abs(squeeze(freqresp(Wn, f, 'Hz'))), ...
|
||||
'DisplayName', 'n');
|
||||
plot(f, abs(squeeze(freqresp(Wd, f, 'Hz'))), ...
|
||||
'DisplayName', 'd');
|
||||
hold off;
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
||||
xlabel('Frequency [Hz]'); ylabel('ASD [$m/\sqrt{Hz}$]');
|
||||
legend('location', 'northeast', 'FontSize', 8, 'NumColumns', 2);
|
||||
xlim([f(1), f(end)]);
|
||||
BIN
matlab/mat/Khac_iff.mat
Normal file
BIN
matlab/mat/Khac_iff.mat
Normal file
Binary file not shown.
BIN
matlab/mat/asd_noises_disturbances.mat
Normal file
BIN
matlab/mat/asd_noises_disturbances.mat
Normal file
Binary file not shown.
Loading…
Reference in New Issue
Block a user