Add identification of Newport dynamics and 4QD gain
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index.html
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index.html
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"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
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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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<!-- 2019-09-16 lun. 11:59 -->
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<!-- 2019-09-16 lun. 15:21 -->
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<meta http-equiv="Content-Type" content="text/html;charset=utf-8" />
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<meta name="viewport" content="width=device-width, initial-scale=1" />
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<title>Cercalo Test Bench</title>
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@ -276,271 +276,92 @@ for the JavaScript code in this tag.
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<h2>Table of Contents</h2>
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<div id="text-table-of-contents">
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<ul>
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<li><a href="#org754474f">1. Introduction</a>
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<li><a href="#org56fbace">1. Introduction</a>
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<ul>
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<li><a href="#org5c645da">1.1. Signals</a></li>
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<li><a href="#org0aa0ee3">1.2. Cercalo</a></li>
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<li><a href="#org002d4fa">1.3. Optical Setup</a></li>
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<li><a href="#org39881ce">1.4. 4 quadrant Diode</a></li>
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<li><a href="#orgd6654c5">1.5. ADC/DAC</a></li>
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<li><a href="#org6c0b30a">1.6. Amplifier for the Cercalo</a></li>
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<li><a href="#org4f39adb">1.7. Block Diagram</a></li>
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<li><a href="#orgc71c3a8">1.1. Block Diagram</a></li>
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<li><a href="#org6dd135b">1.2. Amplifier for the Cercalo</a></li>
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<li><a href="#org3f4c55a">1.3. Cercalo</a></li>
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<li><a href="#org8cf1cba">1.4. Optical Setup</a></li>
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<li><a href="#org6d8a72e">1.5. Newport</a></li>
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<li><a href="#orgcf9e532">1.6. 4 quadrant Diode</a></li>
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<li><a href="#orgbfa25c0">1.7. ADC/DAC</a></li>
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</ul>
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</li>
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<li><a href="#org84958b2">2. Identification</a>
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<li><a href="#org549f783">2. Identification</a>
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<ul>
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<li><a href="#orgd0268b2">2.1. Excitation Data</a></li>
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<li><a href="#orgeac50db">2.2. Signals</a></li>
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<li><a href="#org5dc4e7d">2.3. Huddle Test</a></li>
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<li><a href="#org2028261">2.4. Input / Output data</a></li>
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<li><a href="#org15893d8">2.5. Estimation of the Frequency Response Function Matrix</a></li>
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<li><a href="#org48cc1e6">2.6. Coherence</a></li>
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<li><a href="#org3c28674">2.7. Extraction of a transfer function matrix</a></li>
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<li><a href="#orga174838">2.1. Excitation Data</a></li>
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<li><a href="#org4707d35">2.2. Signals</a></li>
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<li><a href="#org96d4dae">2.3. Huddle Test</a></li>
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<li><a href="#org5b29539">2.4. Input / Output data</a></li>
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<li><a href="#org0556947">2.5. Estimation of the Frequency Response Function Matrix</a></li>
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<li><a href="#org6c025b6">2.6. Coherence</a></li>
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<li><a href="#org74eeb16">2.7. Extraction of a transfer function matrix</a></li>
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</ul>
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</li>
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<li><a href="#org14b4ee4">3. Sensor Noise</a></li>
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<li><a href="#org7fd124d">4. Plant Analysis</a>
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<li><a href="#orge157b63">3. Calibration of the 4 Quadrant Diode</a></li>
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<li><a href="#orgc276bb8">4. Plant Scaling</a></li>
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<li><a href="#orgffff224">5. Plant Analysis</a>
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<ul>
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<li><a href="#org0fae2b8">4.1. Load Plant</a></li>
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<li><a href="#orgfeb824b">4.2. RGA-Number</a></li>
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<li><a href="#org1704d2b">4.3. Rotation Matrix</a></li>
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<li><a href="#orge9e5454">5.1. Load Plant</a></li>
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<li><a href="#org425e4e2">5.2. RGA-Number</a></li>
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<li><a href="#orge80bcc0">5.3. Rotation Matrix</a></li>
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</ul>
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</li>
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<li><a href="#org6ec2576">5. Control Objective</a></li>
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<li><a href="#org41cbf8b">6. Plant Scaling</a></li>
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<li><a href="#org0103d04">7. Control Design</a></li>
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<li><a href="#orgfd824b0">6. Control Objective</a></li>
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<li><a href="#orgc7afa66">7. Control Design</a></li>
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<li><a href="#org5791267">8. Measurement of the non-repeatability</a></li>
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</ul>
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</div>
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</div>
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<div id="outline-container-org754474f" class="outline-2">
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<h2 id="org754474f"><span class="section-number-2">1</span> Introduction</h2>
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<div id="outline-container-org56fbace" class="outline-2">
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<h2 id="org56fbace"><span class="section-number-2">1</span> Introduction</h2>
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<div class="outline-text-2" id="text-1">
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</div>
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<div id="outline-container-org5c645da" class="outline-3">
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<h3 id="org5c645da"><span class="section-number-3">1.1</span> Signals</h3>
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<div id="outline-container-orgc71c3a8" class="outline-3">
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<h3 id="orgc71c3a8"><span class="section-number-3">1.1</span> Block Diagram</h3>
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<div class="outline-text-3" id="text-1-1">
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<table border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
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<p>
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The block diagram of the setup to be controlled is shown in Fig. <a href="#orgc07f042">1</a>.
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</p>
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<colgroup>
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<col class="org-left" />
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<div id="orgc07f042" class="figure">
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<p><img src="figs/cercalo_diagram_simplify.png" alt="cercalo_diagram_simplify.png" />
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</p>
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<p><span class="figure-number">Figure 1: </span>Block Diagram of the Experimental Setup</p>
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</div>
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<col class="org-left" />
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<p>
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The transfer functions in the system are:
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</p>
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<ul class="org-ul">
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<li><b>Current Amplifier</b>: from the voltage set by the DAC to the voltage across the Cercalo inductors
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\[ G_i = \begin{bmatrix} G_{i,h} & 0 \\ 0 & G_{i,v} \end{bmatrix} \]</li>
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<li><b>Voltage Amplifier</b>: from the voltage across the Cercalo inductors to the measured voltage
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\[ G_a = \begin{bmatrix} G_{a,h} & 0 \\ 0 & G_{a,v} \end{bmatrix} \]</li>
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<li><b>Cercalo</b>: Transfer function from the Voltage across the cercalo inductors to the 4 quadrant measurement
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\[ G_c = \begin{bmatrix} G_{\frac{V_{p,h}}{\tilde{U}_{c,h}}} & G_{\frac{V_{p,h}}{\tilde{U}_{c,v}}} \\ G_{\frac{V_{p,v}}{\tilde{U}_{c,h}}} & G_{\frac{V_{p,v}}{\tilde{U}_{c,v}}} \end{bmatrix} \]</li>
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<li><b>Newport</b> Transfer function from the command signal of the Newport to the 4 quadrant measurement
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\[ G_n = \begin{bmatrix} G_{\frac{V_{p,h}}{U_{n,h}}} & G_{\frac{V_{p,h}}{U_{n,v}}} \\ G_{\frac{V_{p,v}}{U_{n,h}}} & G_{\frac{V_{n,v}}{U_{n,v}}} \end{bmatrix} \]</li>
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</ul>
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<col class="org-left" />
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<p>
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The block diagram with each transfer function is shown in Fig. <a href="#orgc0fd79e">2</a>.
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</p>
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<col class="org-right" />
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</colgroup>
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<thead>
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<tr>
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<th scope="col" class="org-left">Signal</th>
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<th scope="col" class="org-left">Name</th>
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<th scope="col" class="org-left">Unit</th>
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<th scope="col" class="org-right">Number</th>
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</tr>
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</thead>
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<tbody>
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<tr>
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<td class="org-left">Voltage Sent to Cercalo - Horizontal</td>
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<td class="org-left"><code>Uch</code></td>
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<td class="org-left">[V]</td>
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<td class="org-right">1</td>
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</tr>
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<tr>
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<td class="org-left">Voltage Sent to Cercalo - Vertical</td>
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<td class="org-left"><code>Ucv</code></td>
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<td class="org-left">[V]</td>
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<td class="org-right">2</td>
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</tr>
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<tr>
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<td class="org-left">Voltage Sent to Newport - Horizontal</td>
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<td class="org-left"><code>Unh</code></td>
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<td class="org-left">[V]</td>
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<td class="org-right">3</td>
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</tr>
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<tr>
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<td class="org-left">Voltage Sent to Newport - Vertical</td>
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<td class="org-left"><code>Unv</code></td>
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<td class="org-left">[V]</td>
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<td class="org-right">4</td>
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</tr>
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</tbody>
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<tbody>
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<tr>
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<td class="org-left">4Q Photodiode Measurement - Horizontal</td>
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<td class="org-left"><code>Vph</code></td>
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<td class="org-left">[V]</td>
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<td class="org-right">5</td>
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</tr>
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<tr>
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<td class="org-left">4Q Photodiode Measurement - Vertical</td>
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<td class="org-left"><code>Vpv</code></td>
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<td class="org-left">[V]</td>
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<td class="org-right">6</td>
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</tr>
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<tr>
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<td class="org-left">Measured Voltage across the Inductance - Horizontal</td>
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<td class="org-left"><code>Vch</code></td>
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<td class="org-left">[V]</td>
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<td class="org-right">7</td>
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</tr>
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<tr>
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<td class="org-left">Measured Voltage across the Inductance - Vertical</td>
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<td class="org-left"><code>Vcv</code></td>
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<td class="org-left">[V]</td>
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<td class="org-right">8</td>
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</tr>
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<tr>
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<td class="org-left">Newport Metrology - Horizontal</td>
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<td class="org-left"><code>Vnh</code></td>
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<td class="org-left">[V]</td>
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<td class="org-right">9</td>
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</tr>
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<tr>
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<td class="org-left">Newport Metrology - Vertical</td>
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<td class="org-left"><code>Vnv</code></td>
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<td class="org-left">[V]</td>
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<td class="org-right">10</td>
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</tr>
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</tbody>
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<tbody>
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<tr>
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<td class="org-left">Attocube Measurement</td>
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<td class="org-left"><code>Va</code></td>
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<td class="org-left">[m]</td>
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<td class="org-right">11</td>
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</tr>
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</tbody>
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</table>
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<div id="orgc0fd79e" class="figure">
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<p><img src="figs/cercalo_diagram.png" alt="cercalo_diagram.png" />
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</p>
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<p><span class="figure-number">Figure 2: </span>Block Diagram of the Experimental Setup with detailed dynamics</p>
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</div>
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</div>
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</div>
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<div id="outline-container-org0aa0ee3" class="outline-3">
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<h3 id="org0aa0ee3"><span class="section-number-3">1.2</span> Cercalo</h3>
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<div id="outline-container-org6dd135b" class="outline-3">
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<h3 id="org6dd135b"><span class="section-number-3">1.2</span> Amplifier for the Cercalo</h3>
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<div class="outline-text-3" id="text-1-2">
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<table id="orga3759f1" 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> Cercalo Parameters</caption>
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<colgroup>
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<col class="org-left" />
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<col class="org-right" />
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<col class="org-right" />
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<col class="org-right" />
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<col class="org-right" />
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<col class="org-right" />
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</colgroup>
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<thead>
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<tr>
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<th scope="col" class="org-left"> </th>
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<th scope="col" class="org-right">Maximum Stroke [deg]</th>
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<th scope="col" class="org-right">Resonance Frequency [Hz]</th>
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<th scope="col" class="org-right">DC Gain [mA/deg]</th>
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<th scope="col" class="org-right">Gain at resonance [deg/V]</th>
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<th scope="col" class="org-right">RC Resistance [Ohm]</th>
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</tr>
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</thead>
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<tbody>
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<tr>
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<td class="org-left">AX1</td>
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<td class="org-right">5</td>
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<td class="org-right">411.13</td>
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<td class="org-right">28.4</td>
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<td class="org-right">382.9</td>
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<td class="org-right">9.41</td>
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</tr>
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<tr>
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<td class="org-left">AX2</td>
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<td class="org-right">5</td>
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<td class="org-right">252.5</td>
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<td class="org-right">35.2</td>
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<td class="org-right">350.4</td>
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<td class="org-right"> </td>
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</tr>
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</tbody>
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</table>
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<p>
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L = 0.1mH on both directions
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</p>
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<p>
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Resistance= 41 Ohm
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</p>
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<p>
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Ax2 = 8.3
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Ax1 = 9.3
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</p>
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</div>
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</div>
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<div id="outline-container-org002d4fa" class="outline-3">
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<h3 id="org002d4fa"><span class="section-number-3">1.3</span> Optical Setup</h3>
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</div>
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<div id="outline-container-org39881ce" class="outline-3">
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<h3 id="org39881ce"><span class="section-number-3">1.4</span> 4 quadrant Diode</h3>
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<div class="outline-text-3" id="text-1-4">
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<p>
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Front view of the photodiode.
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</p>
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<div class="figure">
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<p><img src="figs/4qd_naming.png" alt="4qd_naming.png" />
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</p>
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</div>
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<p>
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Signals going to the amplifier on the board. The amplifier is located on the bottom right of the board.
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</p>
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<div class="figure">
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<p><img src="figs/4qd_amplifier.png" alt="4qd_amplifier.png" />
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</p>
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</div>
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</div>
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</div>
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<div id="outline-container-orgd6654c5" class="outline-3">
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<h3 id="orgd6654c5"><span class="section-number-3">1.5</span> ADC/DAC</h3>
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<div class="outline-text-3" id="text-1-5">
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<p>
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||||
Let's compute the theoretical noise of the ADC/DAC.
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</p>
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||||
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\begin{align*}
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||||
\Delta V &= 20 V \\
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||||
n &= 16bits \\
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q &= \Delta V/2^n = 305 \mu V \\
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||||
f_N &= 10kHz \\
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||||
\Gamma_n &= \frac{q^2}{12 f_N} = 7.76 \cdot 10^{-13} \frac{V^2}{Hz}
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\end{align*}
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<p>
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||||
with \(\Delta V\) the total range of the ADC, \(n\) its number of bits, \(q\) the quantization, \(f_N\) the sampling frequency and \(\Gamma_n\) its theoretical Power Spectral Density.
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||||
</p>
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</div>
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</div>
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<div id="outline-container-org6c0b30a" class="outline-3">
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<h3 id="org6c0b30a"><span class="section-number-3">1.6</span> Amplifier for the Cercalo</h3>
|
||||
<div class="outline-text-3" id="text-1-6">
|
||||
|
||||
<div class="figure">
|
||||
<p><img src="figs/cercalo_amplifier.png" alt="cercalo_amplifier.png" />
|
||||
@ -548,13 +369,12 @@ with \(\Delta V\) the total range of the ADC, \(n\) its number of bits, \(q\) th
|
||||
</div>
|
||||
|
||||
<p>
|
||||
The values of the parameters are:
|
||||
The value of the resistor in series with the buffer have been measured for both axis.
|
||||
</p>
|
||||
<ul class="org-ul">
|
||||
<li>\(R_h = 41 \Omega\)</li>
|
||||
<li>\(L_{c,h} = 0.1 mH\)</li>
|
||||
<li>\(R_{c,h} = 9.3 \Omega\)</li>
|
||||
|
||||
<li>\(R_v = 41 \Omega\)</li>
|
||||
<li>\(L_{c,v} = 0.1 mH\)</li>
|
||||
<li>\(R_{c,v} = 8.3 \Omega\)</li>
|
||||
@ -603,6 +423,10 @@ with
|
||||
<li>\(\omega_0 = \frac{R + R_c}{L_c}\)</li>
|
||||
</ul>
|
||||
|
||||
<p>
|
||||
Let's verify that the electrical circuit behaves as a constant current amplifier in the frequency band of interest.
|
||||
</p>
|
||||
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab">Rh = <span class="org-highlight-numbers-number">41</span>; <span class="org-comment">% [Ohm]</span>
|
||||
Lch = <span class="org-highlight-numbers-number">0</span>.<span class="org-highlight-numbers-number">1e</span><span class="org-type">-</span><span class="org-highlight-numbers-number">3</span>; <span class="org-comment">% [H]</span>
|
||||
@ -623,13 +447,6 @@ Gvv = <span class="org-rainbow-delimiters-depth-1">(</span>Rcv <span class="org-
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
<div class="important">
|
||||
<p>
|
||||
The current amplifier has a constant gain over all the frequency band of interest.
|
||||
</p>
|
||||
|
||||
</div>
|
||||
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab">Gih0 = freqresp<span class="org-rainbow-delimiters-depth-1">(</span>Gih, <span class="org-highlight-numbers-number">0</span><span class="org-rainbow-delimiters-depth-1">)</span>;
|
||||
Gvh0 = freqresp<span class="org-rainbow-delimiters-depth-1">(</span>Gvh, <span class="org-highlight-numbers-number">0</span><span class="org-rainbow-delimiters-depth-1">)</span>;
|
||||
@ -637,101 +454,285 @@ Giv0 = freqresp<span class="org-rainbow-delimiters-depth-1">(</span>Giv, <span c
|
||||
Gvv0 = freqresp<span class="org-rainbow-delimiters-depth-1">(</span>Gvv, <span class="org-highlight-numbers-number">0</span><span class="org-rainbow-delimiters-depth-1">)</span>;
|
||||
</pre>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org4f39adb" class="outline-3">
|
||||
<h3 id="org4f39adb"><span class="section-number-3">1.7</span> Block Diagram</h3>
|
||||
<div class="outline-text-3" id="text-1-7">
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-latex"><span class="org-font-latex-sedate"><span class="org-keyword">\begin</span></span>{<span class="org-function-name">tikzpicture</span>}
|
||||
<span class="org-font-latex-sedate">\node</span>[DAC] (dac) at (0, 0) {};
|
||||
<span class="org-font-latex-sedate">\node</span>[block, right=1.2 of dac] (Gi) {<span class="org-font-latex-math">$G</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_i</span></span><span class="org-font-latex-math">$</span>};
|
||||
<span class="org-font-latex-sedate">\node</span>[block, right=1.5 of Gi] (Gc) {<span class="org-font-latex-math">$G</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_c</span></span><span class="org-font-latex-math">$</span>};
|
||||
<span class="org-font-latex-sedate">\node</span>[addb, right=0.5 of Gc] (add) {};
|
||||
<span class="org-font-latex-sedate">\node</span>[block, above= of add] (Gn) {<span class="org-font-latex-math">$G</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_n</span></span><span class="org-font-latex-math">$</span>};
|
||||
<span class="org-font-latex-sedate">\node</span>[ADC, right=1.2 of add] (adc) {};
|
||||
<span class="org-font-latex-sedate">\coordinate</span> (GiGc) at (<span class="org-font-latex-math">$0.8*(Gi.east) + 0.2*(Gc.west)$</span>);
|
||||
<span class="org-font-latex-sedate">\node</span>[block] (Amp) at (GiGc|-Gn) {<span class="org-font-latex-math">$G</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_a</span></span><span class="org-font-latex-math">$</span>};
|
||||
|
||||
<span class="org-font-latex-sedate">\node</span>[above, align=center] at (Gi.north) {Current<span class="org-font-latex-warning">\\</span>Amplifier};
|
||||
<span class="org-font-latex-sedate">\node</span>[above, align=center] at (Gc.north) {Cercalo};
|
||||
<span class="org-font-latex-sedate">\node</span>[left, align=right] at (Gn.west) {Newport};
|
||||
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (<span class="org-font-latex-math">$(dac.west) + (-1, 0)$</span>) --node[midway, sloped]{<span class="org-font-latex-math">$/$</span>} (dac.west);
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (dac.east) -- (Gi.west) node[above left]{<span class="org-font-latex-math">$</span><span class="org-font-latex-sedate"><span class="org-keyword"><span class="org-font-latex-math">\begin</span></span></span><span class="org-font-latex-math">{</span><span class="org-function-name"><span class="org-font-latex-math">bmatrix</span></span><span class="org-font-latex-math">}U</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_{c,h}</span></span><span class="org-font-latex-math"> \\ U</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_{c,v}</span></span><span class="org-font-latex-sedate"><span class="org-keyword"><span class="org-font-latex-math">\end</span></span></span><span class="org-font-latex-math">{</span><span class="org-function-name"><span class="org-font-latex-math">bmatrix</span></span><span class="org-font-latex-math">}$</span>};
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (Gi.east) -- (Gc.west) node[above left]{<span class="org-font-latex-math">$</span><span class="org-font-latex-sedate"><span class="org-keyword"><span class="org-font-latex-math">\begin</span></span></span><span class="org-font-latex-math">{</span><span class="org-function-name"><span class="org-font-latex-math">bmatrix</span></span><span class="org-font-latex-math">}</span><span class="org-font-latex-sedate"><span class="org-font-latex-math">\tilde</span></span><span class="org-font-latex-math">{U}</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_{c,h}</span></span><span class="org-font-latex-math"> \\ </span><span class="org-font-latex-sedate"><span class="org-font-latex-math">\tilde</span></span><span class="org-font-latex-math">{U}</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_{c,v}</span></span><span class="org-font-latex-sedate"><span class="org-keyword"><span class="org-font-latex-math">\end</span></span></span><span class="org-font-latex-math">{</span><span class="org-function-name"><span class="org-font-latex-math">bmatrix</span></span><span class="org-font-latex-math">}$</span>};
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (Gc.east) -- (add.west);
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (Gn.south) -- (add.north);
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (GiGc)node[branch]{} -- (Amp.south);
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (Amp.north) -- ++(0, 1.5) node[below right]{<span class="org-font-latex-math">$</span><span class="org-font-latex-sedate"><span class="org-keyword"><span class="org-font-latex-math">\begin</span></span></span><span class="org-font-latex-math">{</span><span class="org-function-name"><span class="org-font-latex-math">bmatrix</span></span><span class="org-font-latex-math">}V</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_{c,h}</span></span><span class="org-font-latex-math"> \\ V</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_{c,v}</span></span><span class="org-font-latex-sedate"><span class="org-keyword"><span class="org-font-latex-math">\end</span></span></span><span class="org-font-latex-math">{</span><span class="org-function-name"><span class="org-font-latex-math">bmatrix</span></span><span class="org-font-latex-math">}$</span>};
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (<span class="org-font-latex-math">$(Gn.north) + (0, 1.5)$</span>) -- (Gn.north) node[above right]{<span class="org-font-latex-math">$</span><span class="org-font-latex-sedate"><span class="org-keyword"><span class="org-font-latex-math">\begin</span></span></span><span class="org-font-latex-math">{</span><span class="org-function-name"><span class="org-font-latex-math">bmatrix</span></span><span class="org-font-latex-math">}U</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_{n,h}</span></span><span class="org-font-latex-math"> \\ U</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_{n,v}</span></span><span class="org-font-latex-sedate"><span class="org-keyword"><span class="org-font-latex-math">\end</span></span></span><span class="org-font-latex-math">{</span><span class="org-function-name"><span class="org-font-latex-math">bmatrix</span></span><span class="org-font-latex-math">}$</span>};
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (add.east) -- (adc.west) node[above left]{<span class="org-font-latex-math">$</span><span class="org-font-latex-sedate"><span class="org-keyword"><span class="org-font-latex-math">\begin</span></span></span><span class="org-font-latex-math">{</span><span class="org-function-name"><span class="org-font-latex-math">bmatrix</span></span><span class="org-font-latex-math">}V</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_{p,h}</span></span><span class="org-font-latex-math"> \\ V</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_{p,v}</span></span><span class="org-font-latex-sedate"><span class="org-keyword"><span class="org-font-latex-math">\end</span></span></span><span class="org-font-latex-math">{</span><span class="org-function-name"><span class="org-font-latex-math">bmatrix</span></span><span class="org-font-latex-math">}$</span>};
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (adc.east) --node[midway, sloped]{<span class="org-font-latex-math">$/$</span>} ++(1, 0);
|
||||
<span class="org-font-latex-sedate"><span class="org-keyword">\end</span></span>{<span class="org-function-name">tikzpicture</span>}
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
|
||||
<div class="figure">
|
||||
<p><img src="figs/cercalo_diagram_simplify.png" alt="cercalo_diagram_simplify.png" />
|
||||
<div id="orgfec4858" class="figure">
|
||||
<p><img src="figs/current_amplifier_tf.png" alt="current_amplifier_tf.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 4: </span>Transfer function for the current amplifier (<a href="./figs/current_amplifier_tf.png">png</a>, <a href="./figs/current_amplifier_tf.pdf">pdf</a>)</p>
|
||||
</div>
|
||||
|
||||
|
||||
<div class="important">
|
||||
<p>
|
||||
The current amplifier has a constant gain over all the frequency band of interest.
|
||||
\[ G_i(s) = \begin{bmatrix} 0.02 & 0 \\ 0 & 0.02 \end{bmatrix}\quad \left[\frac{A}{V}\right] \]
|
||||
\[ G_a(s) = \begin{bmatrix} 0.185 & 0 \\ 0 & 0.168 \end{bmatrix} \left[\frac{V}{V}\right] \]
|
||||
</p>
|
||||
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org3f4c55a" class="outline-3">
|
||||
<h3 id="org3f4c55a"><span class="section-number-3">1.3</span> Cercalo</h3>
|
||||
<div class="outline-text-3" id="text-1-3">
|
||||
<p>
|
||||
From the Cercalo documentation, we have the parameters shown on table <a href="#orgfb4a003">1</a>.
|
||||
</p>
|
||||
|
||||
<table id="orgfb4a003" border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
|
||||
<caption class="t-above"><span class="table-number">Table 1:</span> Cercalo Parameters</caption>
|
||||
|
||||
<colgroup>
|
||||
<col class="org-left" />
|
||||
|
||||
<col class="org-right" />
|
||||
|
||||
<col class="org-right" />
|
||||
|
||||
<col class="org-right" />
|
||||
|
||||
<col class="org-right" />
|
||||
|
||||
<col class="org-right" />
|
||||
</colgroup>
|
||||
<thead>
|
||||
<tr>
|
||||
<th scope="col" class="org-left"> </th>
|
||||
<th scope="col" class="org-right">Maximum Stroke [deg]</th>
|
||||
<th scope="col" class="org-right">Resonance Frequency [Hz]</th>
|
||||
<th scope="col" class="org-right">DC Gain [mA/deg]</th>
|
||||
<th scope="col" class="org-right">Gain at resonance [deg/V]</th>
|
||||
<th scope="col" class="org-right">RC Resistance [Ohm]</th>
|
||||
</tr>
|
||||
</thead>
|
||||
<tbody>
|
||||
<tr>
|
||||
<td class="org-left">AX1 (Horizontal)</td>
|
||||
<td class="org-right">5</td>
|
||||
<td class="org-right">411.13</td>
|
||||
<td class="org-right">28.4</td>
|
||||
<td class="org-right">382.9</td>
|
||||
<td class="org-right">9.41</td>
|
||||
</tr>
|
||||
|
||||
<tr>
|
||||
<td class="org-left">AX2 (Vertical)</td>
|
||||
<td class="org-right">5</td>
|
||||
<td class="org-right">252.5</td>
|
||||
<td class="org-right">35.2</td>
|
||||
<td class="org-right">350.4</td>
|
||||
<td class="org-right"> </td>
|
||||
</tr>
|
||||
</tbody>
|
||||
</table>
|
||||
|
||||
<p>
|
||||
The Inductance and DC resistance of the two axis of the Cercalo have been measured:
|
||||
</p>
|
||||
<ul class="org-ul">
|
||||
<li>Transfer function for the <b>Current Amplifier</b>:
|
||||
\[ G_i = \begin{bmatrix} G_{i,h} & 0 \\ 0 & G_{i,v} \end{bmatrix} \]</li>
|
||||
<li>Transfer function for the <b>Voltage Amplifier</b>:
|
||||
\[ G_a = \begin{bmatrix} G_{a,h} & 0 \\ 0 & G_{a,v} \end{bmatrix} \]</li>
|
||||
<li>Transfer function from the Voltage across the <b>Cercalo</b> inductors to the 4 quadrant measurement
|
||||
\[ G_c = \begin{bmatrix} G_{\frac{V_{p,h}}{\tilde{U}_{c,h}}} & G_{\frac{V_{p,h}}{\tilde{U}_{c,v}}} \\ G_{\frac{V_{p,v}}{\tilde{U}_{c,h}}} & G_{\frac{V_{p,v}}{\tilde{U}_{c,v}}} \end{bmatrix} \]</li>
|
||||
<li>Transfer function from the <b>Newport</b> command signal to the 4 quadrant measurement
|
||||
\[ G_n = \begin{bmatrix} G_{\frac{V_{p,h}}{U_{n,h}}} & G_{\frac{V_{p,h}}{U_{n,v}}} \\ G_{\frac{V_{p,v}}{U_{n,h}}} & G_{\frac{V_{n,v}}{U_{n,v}}} \end{bmatrix} \]</li>
|
||||
<li>\(L_{c,h} = 0.1\ \text{mH}\)</li>
|
||||
<li>\(L_{c,v} = 0.1\ \text{mH}\)</li>
|
||||
<li>\(R_{c,h} = 9.3\ \Omega\)</li>
|
||||
<li>\(R_{c,v} = 8.3\ \Omega\)</li>
|
||||
</ul>
|
||||
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-latex"><span class="org-font-latex-sedate"><span class="org-keyword">\begin</span></span>{<span class="org-function-name">tikzpicture</span>}
|
||||
<span class="org-font-latex-sedate">\node</span>[DAC] (dac) at (0, 0) {};
|
||||
<span class="org-font-latex-sedate">\node</span>[block, right=1.5 of dac] (Gi) {<span class="org-font-latex-math">$</span><span class="org-font-latex-sedate"><span class="org-keyword"><span class="org-font-latex-math">\begin</span></span></span><span class="org-font-latex-math">{</span><span class="org-function-name"><span class="org-font-latex-math">bmatrix</span></span><span class="org-font-latex-math">} G</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_{i,h}</span></span><span class="org-font-latex-math"> & 0 \\ 0 & G</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_{i,v}</span></span><span class="org-font-latex-math"> </span><span class="org-font-latex-sedate"><span class="org-keyword"><span class="org-font-latex-math">\end</span></span></span><span class="org-font-latex-math">{</span><span class="org-function-name"><span class="org-font-latex-math">bmatrix</span></span><span class="org-font-latex-math">}$</span>};
|
||||
<span class="org-font-latex-sedate">\node</span>[block, right=1.8 of Gi] (Gc) {<span class="org-font-latex-math">$</span><span class="org-font-latex-sedate"><span class="org-keyword"><span class="org-font-latex-math">\begin</span></span></span><span class="org-font-latex-math">{</span><span class="org-function-name"><span class="org-font-latex-math">bmatrix</span></span><span class="org-font-latex-math">} G</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_{\frac{V_{p,h}}{\tilde{U}_{c,h}}}</span></span><span class="org-font-latex-math"> & G</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_{\frac{V_{p,h}}{\tilde{U}_{c,v}}}</span></span><span class="org-font-latex-math"> \\ G</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_{\frac{V_{p,v}}{\tilde{U}_{c,h}}}</span></span><span class="org-font-latex-math"> & G</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_{\frac{V_{p,v}}{\tilde{U}_{c,v}}}</span></span><span class="org-font-latex-math"> </span><span class="org-font-latex-sedate"><span class="org-keyword"><span class="org-font-latex-math">\end</span></span></span><span class="org-font-latex-math">{</span><span class="org-function-name"><span class="org-font-latex-math">bmatrix</span></span><span class="org-font-latex-math">}$</span>};
|
||||
<span class="org-font-latex-sedate">\node</span>[addb, right= of Gc] (add) {};
|
||||
<span class="org-font-latex-sedate">\node</span>[block, above= of add] (Gn) {<span class="org-font-latex-math">$</span><span class="org-font-latex-sedate"><span class="org-keyword"><span class="org-font-latex-math">\begin</span></span></span><span class="org-font-latex-math">{</span><span class="org-function-name"><span class="org-font-latex-math">bmatrix</span></span><span class="org-font-latex-math">} G</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_{\frac{V_{p,h}}{U_{n,h}}}</span></span><span class="org-font-latex-math"> & G</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_{\frac{V_{p,h}}{U_{n,v}}}</span></span><span class="org-font-latex-math"> \\ G</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_{\frac{V_{p,v}}{U_{n,h}}}</span></span><span class="org-font-latex-math"> & G</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_{\frac{V_{n,v}}{U_{n,v}}}</span></span><span class="org-font-latex-math"> </span><span class="org-font-latex-sedate"><span class="org-keyword"><span class="org-font-latex-math">\end</span></span></span><span class="org-font-latex-math">{</span><span class="org-function-name"><span class="org-font-latex-math">bmatrix</span></span><span class="org-font-latex-math">}$</span>};
|
||||
<span class="org-font-latex-sedate">\node</span>[ADC, right = 1.5 of add] (adc) {};
|
||||
<span class="org-font-latex-sedate">\coordinate</span> (GiGc) at (<span class="org-font-latex-math">$0.7*(Gi.east) + 0.3*(Gc.west)$</span>);
|
||||
<span class="org-font-latex-sedate">\node</span>[block] (Amp) at (GiGc|-Gn) {<span class="org-font-latex-math">$</span><span class="org-font-latex-sedate"><span class="org-keyword"><span class="org-font-latex-math">\begin</span></span></span><span class="org-font-latex-math">{</span><span class="org-function-name"><span class="org-font-latex-math">bmatrix</span></span><span class="org-font-latex-math">} G</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_{a,h}</span></span><span class="org-font-latex-math"> & 0 \\ 0 & G</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_{a,v}</span></span><span class="org-font-latex-math"> </span><span class="org-font-latex-sedate"><span class="org-keyword"><span class="org-font-latex-math">\end</span></span></span><span class="org-font-latex-math">{</span><span class="org-function-name"><span class="org-font-latex-math">bmatrix</span></span><span class="org-font-latex-math">}$</span>};
|
||||
<p>
|
||||
Let's first consider the <b>horizontal direction</b> and we try to model the Cercalo by a spring/mass/damper system (Fig. <a href="#orgd829094">5</a>).
|
||||
</p>
|
||||
|
||||
<span class="org-font-latex-sedate">\node</span>[above, align=center] at (Gi.north) {Current<span class="org-font-latex-warning">\\</span>Amplifier};
|
||||
<span class="org-font-latex-sedate">\node</span>[above, align=center] at (Gc.north) {Cercalo};
|
||||
<span class="org-font-latex-sedate">\node</span>[left, align=right] at (Gn.west) {Newport};
|
||||
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (<span class="org-font-latex-math">$(dac.west) + (-1, 0)$</span>) --node[midway, sloped]{<span class="org-font-latex-math">$/$</span>} (dac.west);
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (dac.east) -- (Gi.west) node[above left]{<span class="org-font-latex-math">$</span><span class="org-font-latex-sedate"><span class="org-keyword"><span class="org-font-latex-math">\begin</span></span></span><span class="org-font-latex-math">{</span><span class="org-function-name"><span class="org-font-latex-math">bmatrix</span></span><span class="org-font-latex-math">}U</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_{c,h}</span></span><span class="org-font-latex-math"> \\ U</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_{c,v}</span></span><span class="org-font-latex-sedate"><span class="org-keyword"><span class="org-font-latex-math">\end</span></span></span><span class="org-font-latex-math">{</span><span class="org-function-name"><span class="org-font-latex-math">bmatrix</span></span><span class="org-font-latex-math">}$</span>};
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (Gi.east) -- (Gc.west) node[above left]{<span class="org-font-latex-math">$</span><span class="org-font-latex-sedate"><span class="org-keyword"><span class="org-font-latex-math">\begin</span></span></span><span class="org-font-latex-math">{</span><span class="org-function-name"><span class="org-font-latex-math">bmatrix</span></span><span class="org-font-latex-math">}</span><span class="org-font-latex-sedate"><span class="org-font-latex-math">\tilde</span></span><span class="org-font-latex-math">{U}</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_{c,h}</span></span><span class="org-font-latex-math"> \\ </span><span class="org-font-latex-sedate"><span class="org-font-latex-math">\tilde</span></span><span class="org-font-latex-math">{U}</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_{c,v}</span></span><span class="org-font-latex-sedate"><span class="org-keyword"><span class="org-font-latex-math">\end</span></span></span><span class="org-font-latex-math">{</span><span class="org-function-name"><span class="org-font-latex-math">bmatrix</span></span><span class="org-font-latex-math">}$</span>};
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (Gc.east) -- (add.west);
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (Gn.south) -- (add.north);
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (GiGc)node[branch]{} -- (Amp.south);
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (Amp.north) -- ++(0, 1.5) node[below right]{<span class="org-font-latex-math">$</span><span class="org-font-latex-sedate"><span class="org-keyword"><span class="org-font-latex-math">\begin</span></span></span><span class="org-font-latex-math">{</span><span class="org-function-name"><span class="org-font-latex-math">bmatrix</span></span><span class="org-font-latex-math">}V</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_{c,h}</span></span><span class="org-font-latex-math"> \\ V</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_{c,v}</span></span><span class="org-font-latex-sedate"><span class="org-keyword"><span class="org-font-latex-math">\end</span></span></span><span class="org-font-latex-math">{</span><span class="org-function-name"><span class="org-font-latex-math">bmatrix</span></span><span class="org-font-latex-math">}$</span>};
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (<span class="org-font-latex-math">$(Gn.north) + (0, 1.5)$</span>) -- (Gn.north) node[above right]{<span class="org-font-latex-math">$</span><span class="org-font-latex-sedate"><span class="org-keyword"><span class="org-font-latex-math">\begin</span></span></span><span class="org-font-latex-math">{</span><span class="org-function-name"><span class="org-font-latex-math">bmatrix</span></span><span class="org-font-latex-math">}U</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_{n,h}</span></span><span class="org-font-latex-math"> \\ U</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_{n,v}</span></span><span class="org-font-latex-sedate"><span class="org-keyword"><span class="org-font-latex-math">\end</span></span></span><span class="org-font-latex-math">{</span><span class="org-function-name"><span class="org-font-latex-math">bmatrix</span></span><span class="org-font-latex-math">}$</span>};
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (add.east) -- (adc.west) node[above left]{<span class="org-font-latex-math">$</span><span class="org-font-latex-sedate"><span class="org-keyword"><span class="org-font-latex-math">\begin</span></span></span><span class="org-font-latex-math">{</span><span class="org-function-name"><span class="org-font-latex-math">bmatrix</span></span><span class="org-font-latex-math">}V</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_{p,h}</span></span><span class="org-font-latex-math"> \\ V</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_{p,v}</span></span><span class="org-font-latex-sedate"><span class="org-keyword"><span class="org-font-latex-math">\end</span></span></span><span class="org-font-latex-math">{</span><span class="org-function-name"><span class="org-font-latex-math">bmatrix</span></span><span class="org-font-latex-math">}$</span>};
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (adc.east) --node[midway, sloped]{<span class="org-font-latex-math">$/$</span>} ++(1, 0);
|
||||
<span class="org-font-latex-sedate"><span class="org-keyword">\end</span></span>{<span class="org-function-name">tikzpicture</span>}
|
||||
</pre>
|
||||
<div id="orgd829094" class="figure">
|
||||
<p><img src="figs/mech_cercalo.png" alt="mech_cercalo.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 5: </span>1 degree-of-freedom model of the Cercalo</p>
|
||||
</div>
|
||||
|
||||
|
||||
<div class="figure">
|
||||
<p><img src="figs/cercalo_diagram.png" alt="cercalo_diagram.png" />
|
||||
<p>
|
||||
The equation of motion is:
|
||||
</p>
|
||||
\begin{align*}
|
||||
\frac{x}{F} &= \frac{1}{k + c s + m s^2} \\
|
||||
&= \frac{G_0}{1 + 2 \xi \frac{s}{\omega_0} + \frac{s^2}{\omega_0^2}}
|
||||
\end{align*}
|
||||
<p>
|
||||
with:
|
||||
</p>
|
||||
<ul class="org-ul">
|
||||
<li>\(G_0 = 1/k\) is the gain at DC in rad/N</li>
|
||||
<li>\(\xi = \frac{c}{2 \sqrt{km}}\) is the damping ratio of the system</li>
|
||||
<li>\(\omega_0 = \sqrt{\frac{k}{m}}\) is the resonance frequency in rad</li>
|
||||
</ul>
|
||||
|
||||
<p>
|
||||
The force \(F\) applied to the mass is proportional to the current \(I\) flowing through the voice coils:
|
||||
\[ \frac{F}{I} = \alpha \]
|
||||
with \(\alpha\) is in \(N/A\) and is to be determined.
|
||||
</p>
|
||||
|
||||
<p>
|
||||
The current \(I\) is also proportional to the voltage at the output of the buffer:
|
||||
</p>
|
||||
\begin{align*}
|
||||
\frac{I_c}{U_c} &= \frac{1}{(R + R_c) + L_c s} \\
|
||||
&\approx 0.02 \left[ \frac{A}{V} \right]
|
||||
\end{align*}
|
||||
|
||||
<p>
|
||||
Let's try to determine the equivalent mass and spring values.
|
||||
From table <a href="#orgfb4a003">1</a>, for the horizontal direction:
|
||||
\[ \left| \frac{x}{I} \right|(0) = \left| \alpha \frac{x}{F} \right|(0) = 28.4\ \frac{mA}{deg} = 1.63\ \frac{A}{rad} \]
|
||||
</p>
|
||||
|
||||
<p>
|
||||
So:
|
||||
\[ \alpha \frac{1}{k} = 1.63 \Longleftrightarrow k = \frac{\alpha}{1.63} \left[\frac{N}{rad}\right] \]
|
||||
</p>
|
||||
|
||||
<p>
|
||||
We also know the resonance frequency:
|
||||
\[ \omega_0 = 411.1\ \text{Hz} = 2583\ \frac{rad}{s} \]
|
||||
</p>
|
||||
|
||||
<p>
|
||||
And the gain at resonance:
|
||||
</p>
|
||||
\begin{align*}
|
||||
\left| \frac{x}{U_c} \right|(j\omega_0) &= \left| 0.02 \frac{x}{I_c} \right| (j\omega_0) \\
|
||||
&= \left| 0.02 \alpha \frac{x}{F} \right| (j\omega_0) \\
|
||||
&= 0.02 \alpha \frac{1/k}{2\xi} \\
|
||||
&= 282.9\ \left[\frac{deg}{V}\right] \\
|
||||
&= 4.938\ \left[\frac{rad}{V}\right]
|
||||
\end{align*}
|
||||
|
||||
<p>
|
||||
Thus:
|
||||
</p>
|
||||
\begin{align*}
|
||||
& \frac{\alpha}{2 \xi k} = 245 \\
|
||||
\Leftrightarrow & \frac{1.63}{2 \xi} = 245 \\
|
||||
\Leftrightarrow & \xi = 0.0033 \\
|
||||
\Leftrightarrow & \xi = 0.33 \%
|
||||
\end{align*}
|
||||
|
||||
<div class="important">
|
||||
\begin{align*}
|
||||
G_0 &= \frac{1.63}{\alpha}\ \frac{rad}{N} \\
|
||||
\xi &= 0.0033 \\
|
||||
\omega_0 &= 2583\ \frac{rad}{s}
|
||||
\end{align*}
|
||||
<p>
|
||||
and in terms of the physical properties:
|
||||
</p>
|
||||
\begin{align*}
|
||||
k &= \frac{\alpha}{1.63}\ \frac{N}{rad} \\
|
||||
\xi &= 0.0033 \\
|
||||
m &= \frac{\alpha}{1.1 \cdot 10^7}\ \frac{kg}{m^2}
|
||||
\end{align*}
|
||||
|
||||
<p>
|
||||
Thus, we have to determine \(\alpha\).
|
||||
This can be done experimentally by determining the gain at DC or at resonance of the system.
|
||||
For that, we need to know the angle of the mirror, thus we need to <b>calibrate</b> the photo-diodes.
|
||||
This will be done using the Newport.
|
||||
</p>
|
||||
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org8cf1cba" class="outline-3">
|
||||
<h3 id="org8cf1cba"><span class="section-number-3">1.4</span> Optical Setup</h3>
|
||||
</div>
|
||||
<div id="outline-container-org6d8a72e" class="outline-3">
|
||||
<h3 id="org6d8a72e"><span class="section-number-3">1.5</span> Newport</h3>
|
||||
<div class="outline-text-3" id="text-1-5">
|
||||
<p>
|
||||
Parameters of the Newport are shown in Fig. <a href="#org893d45e">6</a>.
|
||||
</p>
|
||||
|
||||
<p>
|
||||
It's dynamics for small angle excitation is shown in Fig. <a href="#org4e41b71">7</a>.
|
||||
</p>
|
||||
|
||||
<p>
|
||||
And we have:
|
||||
</p>
|
||||
\begin{align*}
|
||||
G_{n, h}(0) &= 2.62 \cdot 10^{-3}\ \frac{rad}{V} \\
|
||||
G_{n, v}(0) &= 2.62 \cdot 10^{-3}\ \frac{rad}{V}
|
||||
\end{align*}
|
||||
|
||||
|
||||
<div id="org893d45e" class="figure">
|
||||
<p><img src="figs/newport_doc.png" alt="newport_doc.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 6: </span>Documentation of the Newport</p>
|
||||
</div>
|
||||
|
||||
|
||||
<div id="org4e41b71" class="figure">
|
||||
<p><img src="figs/newport_gain.png" alt="newport_gain.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 7: </span>Transfer function of the Newport</p>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-orgcf9e532" class="outline-3">
|
||||
<h3 id="orgcf9e532"><span class="section-number-3">1.6</span> 4 quadrant Diode</h3>
|
||||
<div class="outline-text-3" id="text-1-6">
|
||||
<p>
|
||||
The front view of the 4 quadrant photo-diode is shown in Fig. <a href="#org279009d">8</a>.
|
||||
</p>
|
||||
|
||||
|
||||
<div id="org279009d" class="figure">
|
||||
<p><img src="figs/4qd_naming.png" alt="4qd_naming.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 8: </span>Front view of the 4QD</p>
|
||||
</div>
|
||||
|
||||
<p>
|
||||
Each of the photo-diode is amplified using a 4-channel amplifier as shown in Fig. <a href="#org7f931ef">9</a>.
|
||||
</p>
|
||||
|
||||
<div id="org7f931ef" class="figure">
|
||||
<p><img src="figs/4qd_amplifier.png" alt="4qd_amplifier.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 9: </span>Wiring of the amplifier. The amplifier is located on the bottom right of the board</p>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-orgbfa25c0" class="outline-3">
|
||||
<h3 id="orgbfa25c0"><span class="section-number-3">1.7</span> ADC/DAC</h3>
|
||||
<div class="outline-text-3" id="text-1-7">
|
||||
<p>
|
||||
Let's compute the theoretical noise of the ADC/DAC.
|
||||
</p>
|
||||
|
||||
\begin{align*}
|
||||
\Delta V &= 20 V \\
|
||||
n &= 16bits \\
|
||||
q &= \Delta V/2^n = 305 \mu V \\
|
||||
f_N &= 10kHz \\
|
||||
\Gamma_n &= \frac{q^2}{12 f_N} = 7.76 \cdot 10^{-13} \frac{V^2}{Hz}
|
||||
\end{align*}
|
||||
<p>
|
||||
with \(\Delta V\) the total range of the ADC, \(n\) its number of bits, \(q\) the quantization, \(f_N\) the sampling frequency and \(\Gamma_n\) its theoretical Power Spectral Density.
|
||||
</p>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org84958b2" class="outline-2">
|
||||
<h2 id="org84958b2"><span class="section-number-2">2</span> Identification</h2>
|
||||
<div id="outline-container-org549f783" class="outline-2">
|
||||
<h2 id="org549f783"><span class="section-number-2">2</span> Identification</h2>
|
||||
<div class="outline-text-2" id="text-2">
|
||||
<p>
|
||||
<a id="orgf9e9ac3"></a>
|
||||
<a id="orgaf96e37"></a>
|
||||
</p>
|
||||
<div class="note">
|
||||
<p>
|
||||
@ -740,8 +741,8 @@ All the files (data and Matlab scripts) are accessible <a href="data/plant_ident
|
||||
|
||||
</div>
|
||||
</div>
|
||||
<div id="outline-container-orgd0268b2" class="outline-3">
|
||||
<h3 id="orgd0268b2"><span class="section-number-3">2.1</span> Excitation Data</h3>
|
||||
<div id="outline-container-orga174838" class="outline-3">
|
||||
<h3 id="orga174838"><span class="section-number-3">2.1</span> Excitation Data</h3>
|
||||
<div class="outline-text-3" id="text-2-1">
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab">fs = <span class="org-highlight-numbers-number">1e4</span>;
|
||||
@ -778,8 +779,8 @@ Discrete-time zero/pole/gain model.
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-orgeac50db" class="outline-3">
|
||||
<h3 id="orgeac50db"><span class="section-number-3">2.2</span> Signals</h3>
|
||||
<div id="outline-container-org4707d35" class="outline-3">
|
||||
<h3 id="org4707d35"><span class="section-number-3">2.2</span> Signals</h3>
|
||||
<div class="outline-text-3" id="text-2-2">
|
||||
<table border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
|
||||
|
||||
@ -871,8 +872,8 @@ Discrete-time zero/pole/gain model.
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org5dc4e7d" class="outline-3">
|
||||
<h3 id="org5dc4e7d"><span class="section-number-3">2.3</span> Huddle Test</h3>
|
||||
<div id="outline-container-org96d4dae" class="outline-3">
|
||||
<h3 id="org96d4dae"><span class="section-number-3">2.3</span> Huddle Test</h3>
|
||||
<div class="outline-text-3" id="text-2-3">
|
||||
<p>
|
||||
We load the data taken during the Huddle Test.
|
||||
@ -957,8 +958,8 @@ xlim<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-rainbo
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org2028261" class="outline-3">
|
||||
<h3 id="org2028261"><span class="section-number-3">2.4</span> Input / Output data</h3>
|
||||
<div id="outline-container-org5b29539" class="outline-3">
|
||||
<h3 id="org5b29539"><span class="section-number-3">2.4</span> Input / Output data</h3>
|
||||
<div class="outline-text-3" id="text-2-4">
|
||||
<p>
|
||||
The identification data is loaded
|
||||
@ -1023,24 +1024,24 @@ uv.t = uv.t <span class="org-type">-</span> uv.t<span class="org-rainbow-delimit
|
||||
</div>
|
||||
|
||||
|
||||
<div id="orgeb45e59" class="figure">
|
||||
<div id="orga6bc5e6" class="figure">
|
||||
<p><img src="figs/identification_uh.png" alt="identification_uh.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 6: </span>Identification signals when exciting the horizontal direction (<a href="./figs/identification_uh.png">png</a>, <a href="./figs/identification_uh.pdf">pdf</a>)</p>
|
||||
<p><span class="figure-number">Figure 10: </span>Identification signals when exciting the horizontal direction (<a href="./figs/identification_uh.png">png</a>, <a href="./figs/identification_uh.pdf">pdf</a>)</p>
|
||||
</div>
|
||||
|
||||
|
||||
|
||||
<div id="org37217b5" class="figure">
|
||||
<div id="org82bc95e" class="figure">
|
||||
<p><img src="figs/identification_uv.png" alt="identification_uv.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 7: </span>Identification signals when exciting in the vertical direction (<a href="./figs/identification_uv.png">png</a>, <a href="./figs/identification_uv.pdf">pdf</a>)</p>
|
||||
<p><span class="figure-number">Figure 11: </span>Identification signals when exciting in the vertical direction (<a href="./figs/identification_uv.png">png</a>, <a href="./figs/identification_uv.pdf">pdf</a>)</p>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org15893d8" class="outline-3">
|
||||
<h3 id="org15893d8"><span class="section-number-3">2.5</span> Estimation of the Frequency Response Function Matrix</h3>
|
||||
<div id="outline-container-org0556947" class="outline-3">
|
||||
<h3 id="org0556947"><span class="section-number-3">2.5</span> Estimation of the Frequency Response Function Matrix</h3>
|
||||
<div class="outline-text-3" id="text-2-5">
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab">win = hanning<span class="org-rainbow-delimiters-depth-1">(</span>ceil<span class="org-rainbow-delimiters-depth-2">(</span><span class="org-highlight-numbers-number">1</span><span class="org-type">*</span>fs<span class="org-rainbow-delimiters-depth-2">)</span><span class="org-rainbow-delimiters-depth-1">)</span>;
|
||||
@ -1059,16 +1060,16 @@ We compute an estimate of the transfer functions.
|
||||
</div>
|
||||
|
||||
|
||||
<div id="org6ee4e7c" class="figure">
|
||||
<div id="org171e666" class="figure">
|
||||
<p><img src="figs/frequency_response_matrix.png" alt="frequency_response_matrix.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 8: </span>Frequency Response Matrix (<a href="./figs/frequency_response_matrix.png">png</a>, <a href="./figs/frequency_response_matrix.pdf">pdf</a>)</p>
|
||||
<p><span class="figure-number">Figure 12: </span>Frequency Response Matrix (<a href="./figs/frequency_response_matrix.png">png</a>, <a href="./figs/frequency_response_matrix.pdf">pdf</a>)</p>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org48cc1e6" class="outline-3">
|
||||
<h3 id="org48cc1e6"><span class="section-number-3">2.6</span> Coherence</h3>
|
||||
<div id="outline-container-org6c025b6" class="outline-3">
|
||||
<h3 id="org6c025b6"><span class="section-number-3">2.6</span> Coherence</h3>
|
||||
<div class="outline-text-3" id="text-2-6">
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab"><span class="org-rainbow-delimiters-depth-1">[</span>coh_Uch_Vph, f<span class="org-rainbow-delimiters-depth-1">]</span> = mscohere<span class="org-rainbow-delimiters-depth-1">(</span>uh.Uch, uh.Vph, win, <span class="org-rainbow-delimiters-depth-2">[]</span>, <span class="org-rainbow-delimiters-depth-2">[]</span>, fs<span class="org-rainbow-delimiters-depth-1">)</span>;
|
||||
@ -1079,17 +1080,17 @@ We compute an estimate of the transfer functions.
|
||||
</div>
|
||||
|
||||
|
||||
<div id="org3ec05cb" class="figure">
|
||||
<div id="orgd207281" class="figure">
|
||||
<p><img src="figs/identification_coherence.png" alt="identification_coherence.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 9: </span>Coherence (<a href="./figs/identification_coherence.png">png</a>, <a href="./figs/identification_coherence.pdf">pdf</a>)</p>
|
||||
<p><span class="figure-number">Figure 13: </span>Coherence (<a href="./figs/identification_coherence.png">png</a>, <a href="./figs/identification_coherence.pdf">pdf</a>)</p>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
|
||||
<div id="outline-container-org3c28674" class="outline-3">
|
||||
<h3 id="org3c28674"><span class="section-number-3">2.7</span> Extraction of a transfer function matrix</h3>
|
||||
<div id="outline-container-org74eeb16" class="outline-3">
|
||||
<h3 id="org74eeb16"><span class="section-number-3">2.7</span> Extraction of a transfer function matrix</h3>
|
||||
<div class="outline-text-3" id="text-2-7">
|
||||
<p>
|
||||
First we define the initial guess for the resonance frequencies and the weights associated.
|
||||
@ -1152,10 +1153,10 @@ weight_Ucv_Vpv<span class="org-rainbow-delimiters-depth-1">(</span>f<span class=
|
||||
</div>
|
||||
|
||||
|
||||
<div id="org60f2a97" class="figure">
|
||||
<div id="org9848359" class="figure">
|
||||
<p><img src="figs/weights.png" alt="weights.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 10: </span>Weights amplitude (<a href="./figs/weights.png">png</a>, <a href="./figs/weights.pdf">pdf</a>)</p>
|
||||
<p><span class="figure-number">Figure 14: </span>Weights amplitude (<a href="./figs/weights.png">png</a>, <a href="./figs/weights.pdf">pdf</a>)</p>
|
||||
</div>
|
||||
|
||||
<p>
|
||||
@ -1204,18 +1205,18 @@ An we run the <code>vectfit3</code> algorithm.
|
||||
</div>
|
||||
|
||||
|
||||
<div id="org74c7619" class="figure">
|
||||
<div id="org61a8901" class="figure">
|
||||
<p><img src="figs/identification_matrix_fit.png" alt="identification_matrix_fit.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 11: </span>Transfer Function Extraction of the FRF matrix (<a href="./figs/identification_matrix_fit.png">png</a>, <a href="./figs/identification_matrix_fit.pdf">pdf</a>)</p>
|
||||
<p><span class="figure-number">Figure 15: </span>Transfer Function Extraction of the FRF matrix (<a href="./figs/identification_matrix_fit.png">png</a>, <a href="./figs/identification_matrix_fit.pdf">pdf</a>)</p>
|
||||
</div>
|
||||
|
||||
|
||||
|
||||
<div id="org82ac5fe" class="figure">
|
||||
<div id="org05ccf64" class="figure">
|
||||
<p><img src="figs/identification_matrix_fit_phase.png" alt="identification_matrix_fit_phase.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 12: </span>Transfer Function Extraction of the FRF matrix (<a href="./figs/identification_matrix_fit_phase.png">png</a>, <a href="./figs/identification_matrix_fit_phase.pdf">pdf</a>)</p>
|
||||
<p><span class="figure-number">Figure 16: </span>Transfer Function Extraction of the FRF matrix (<a href="./figs/identification_matrix_fit_phase.png">png</a>, <a href="./figs/identification_matrix_fit_phase.pdf">pdf</a>)</p>
|
||||
</div>
|
||||
|
||||
<p>
|
||||
@ -1240,16 +1241,28 @@ G = <span class="org-rainbow-delimiters-depth-1">[</span>G_uh_xh, G_uv_xh;
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org14b4ee4" class="outline-2">
|
||||
<h2 id="org14b4ee4"><span class="section-number-2">3</span> Sensor Noise</h2>
|
||||
<div id="outline-container-orge157b63" class="outline-2">
|
||||
<h2 id="orge157b63"><span class="section-number-2">3</span> Calibration of the 4 Quadrant Diode</h2>
|
||||
</div>
|
||||
<div id="outline-container-org7fd124d" class="outline-2">
|
||||
<h2 id="org7fd124d"><span class="section-number-2">4</span> Plant Analysis</h2>
|
||||
<div id="outline-container-orgc276bb8" class="outline-2">
|
||||
<h2 id="orgc276bb8"><span class="section-number-2">4</span> Plant Scaling</h2>
|
||||
<div class="outline-text-2" id="text-4">
|
||||
<ul class="org-ul">
|
||||
<li>measured noise</li>
|
||||
<li>expected perturbations</li>
|
||||
<li>maximum input usage</li>
|
||||
<li>maximum wanted error</li>
|
||||
</ul>
|
||||
</div>
|
||||
<div id="outline-container-org0fae2b8" class="outline-3">
|
||||
<h3 id="org0fae2b8"><span class="section-number-3">4.1</span> Load Plant</h3>
|
||||
<div class="outline-text-3" id="text-4-1">
|
||||
</div>
|
||||
|
||||
<div id="outline-container-orgffff224" class="outline-2">
|
||||
<h2 id="orgffff224"><span class="section-number-2">5</span> Plant Analysis</h2>
|
||||
<div class="outline-text-2" id="text-5">
|
||||
</div>
|
||||
<div id="outline-container-orge9e5454" class="outline-3">
|
||||
<h3 id="orge9e5454"><span class="section-number-3">5.1</span> Load Plant</h3>
|
||||
<div class="outline-text-3" id="text-5-1">
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab">load<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-string">'mat/plant.mat', 'G'</span><span class="org-rainbow-delimiters-depth-1">)</span>;
|
||||
</pre>
|
||||
@ -1257,9 +1270,9 @@ G = <span class="org-rainbow-delimiters-depth-1">[</span>G_uh_xh, G_uv_xh;
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-orgfeb824b" class="outline-3">
|
||||
<h3 id="orgfeb824b"><span class="section-number-3">4.2</span> RGA-Number</h3>
|
||||
<div class="outline-text-3" id="text-4-2">
|
||||
<div id="outline-container-org425e4e2" class="outline-3">
|
||||
<h3 id="org425e4e2"><span class="section-number-3">5.2</span> RGA-Number</h3>
|
||||
<div class="outline-text-3" id="text-5-2">
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab">freqs = logspace<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-highlight-numbers-number">2</span>, <span class="org-highlight-numbers-number">4</span>, <span class="org-highlight-numbers-number">1000</span><span class="org-rainbow-delimiters-depth-1">)</span>;
|
||||
G_resp = freqresp<span class="org-rainbow-delimiters-depth-1">(</span>G, freqs, <span class="org-string">'Hz'</span><span class="org-rainbow-delimiters-depth-1">)</span>;
|
||||
@ -1297,9 +1310,9 @@ V = zeros<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-h
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org1704d2b" class="outline-3">
|
||||
<h3 id="org1704d2b"><span class="section-number-3">4.3</span> Rotation Matrix</h3>
|
||||
<div class="outline-text-3" id="text-4-3">
|
||||
<div id="outline-container-orge80bcc0" class="outline-3">
|
||||
<h3 id="orge80bcc0"><span class="section-number-3">5.3</span> Rotation Matrix</h3>
|
||||
<div class="outline-text-3" id="text-5-3">
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab">G0 = freqresp<span class="org-rainbow-delimiters-depth-1">(</span>G, <span class="org-highlight-numbers-number">0</span><span class="org-rainbow-delimiters-depth-1">)</span>;
|
||||
</pre>
|
||||
@ -1308,9 +1321,9 @@ V = zeros<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-h
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org6ec2576" class="outline-2">
|
||||
<h2 id="org6ec2576"><span class="section-number-2">5</span> Control Objective</h2>
|
||||
<div class="outline-text-2" id="text-5">
|
||||
<div id="outline-container-orgfd824b0" class="outline-2">
|
||||
<h2 id="orgfd824b0"><span class="section-number-2">6</span> Control Objective</h2>
|
||||
<div class="outline-text-2" id="text-6">
|
||||
<p>
|
||||
The maximum expected stroke is \(y_\text{max} = 3mm \approx 5e^{-2} rad\) at \(1Hz\).
|
||||
The maximum wanted error is \(e_\text{max} = 10 \mu rad\).
|
||||
@ -1331,25 +1344,16 @@ In terms of loop gain, this is equivalent to:
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org41cbf8b" class="outline-2">
|
||||
<h2 id="org41cbf8b"><span class="section-number-2">6</span> Plant Scaling</h2>
|
||||
<div class="outline-text-2" id="text-6">
|
||||
<ul class="org-ul">
|
||||
<li>measured noise</li>
|
||||
<li>expected perturbations</li>
|
||||
<li>maximum input usage</li>
|
||||
<li>maximum wanted error</li>
|
||||
</ul>
|
||||
<div id="outline-container-orgc7afa66" class="outline-2">
|
||||
<h2 id="orgc7afa66"><span class="section-number-2">7</span> Control Design</h2>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org0103d04" class="outline-2">
|
||||
<h2 id="org0103d04"><span class="section-number-2">7</span> Control Design</h2>
|
||||
<div id="outline-container-org5791267" class="outline-2">
|
||||
<h2 id="org5791267"><span class="section-number-2">8</span> Measurement of the non-repeatability</h2>
|
||||
</div>
|
||||
</div>
|
||||
<div id="postamble" class="status">
|
||||
<p class="author">Author: Dehaeze Thomas</p>
|
||||
<p class="date">Created: 2019-09-16 lun. 11:59</p>
|
||||
<p class="date">Created: 2019-09-16 lun. 15:21</p>
|
||||
<p class="validation"><a href="http://validator.w3.org/check?uri=referer">Validate</a></p>
|
||||
</div>
|
||||
</body>
|
||||
|
||||
795
index.org
795
index.org
@ -358,6 +358,25 @@ This will be done using the Newport.
|
||||
#+end_important
|
||||
|
||||
** Optical Setup
|
||||
** Newport
|
||||
Parameters of the Newport are shown in Fig. [[fig:Newport_doc]].
|
||||
|
||||
It's dynamics for small angle excitation is shown in Fig. [[fig:Newport_gain]].
|
||||
|
||||
And we have:
|
||||
\begin{align*}
|
||||
G_{n, h}(0) &= 2.62 \cdot 10^{-3}\ \frac{rad}{V} \\
|
||||
G_{n, v}(0) &= 2.62 \cdot 10^{-3}\ \frac{rad}{V}
|
||||
\end{align*}
|
||||
|
||||
#+name: fig:Newport_doc
|
||||
#+caption: Documentation of the Newport
|
||||
[[file:figs/Newport_doc.png]]
|
||||
|
||||
#+name: fig:Newport_gain
|
||||
#+caption: Transfer function of the Newport
|
||||
[[file:figs/Newport_gain.png]]
|
||||
|
||||
** 4 quadrant Diode
|
||||
The front view of the 4 quadrant photo-diode is shown in Fig. [[fig:4qd_naming]].
|
||||
|
||||
@ -406,7 +425,7 @@ Let's compute the theoretical noise of the ADC/DAC.
|
||||
\end{align*}
|
||||
with $\Delta V$ the total range of the ADC, $n$ its number of bits, $q$ the quantization, $f_N$ the sampling frequency and $\Gamma_n$ its theoretical Power Spectral Density.
|
||||
|
||||
* Identification
|
||||
* Identification of the Cercalo Dynamics
|
||||
:PROPERTIES:
|
||||
:header-args:matlab+: :tangle matlab/plant_identification.m
|
||||
:header-args:matlab+: :comments org :mkdirp yes
|
||||
@ -439,12 +458,13 @@ with $\Delta V$ the total range of the ADC, $n$ its number of bits, $q$ the quan
|
||||
<<matlab-init>>
|
||||
#+end_src
|
||||
|
||||
** Excitation Data
|
||||
#+begin_src matlab
|
||||
fs = 1e4;
|
||||
Ts = 1/fs;
|
||||
freqs = logspace(1, 3, 1000);
|
||||
#+end_src
|
||||
|
||||
** Excitation Data
|
||||
We generate white noise with the "random number" simulink block, and we filter that noise.
|
||||
|
||||
#+begin_src matlab
|
||||
@ -489,7 +509,7 @@ Discrete-time zero/pole/gain model.
|
||||
** Huddle Test
|
||||
We load the data taken during the Huddle Test.
|
||||
#+begin_src matlab
|
||||
load('mat/data_test.mat', ...
|
||||
load('mat/data_huddle_test.mat', ...
|
||||
't', 'Uch', 'Ucv', ...
|
||||
'Unh', 'Unv', ...
|
||||
'Vph', 'Vpv', ...
|
||||
@ -581,7 +601,7 @@ We compute the Power Spectral Density of the voltage across the inductance used
|
||||
** Input / Output data
|
||||
The identification data is loaded
|
||||
#+begin_src matlab
|
||||
uh = load('mat/data_uh.mat', ...
|
||||
uh = load('mat/data_uch.mat', ...
|
||||
't', 'Uch', 'Ucv', ...
|
||||
'Unh', 'Unv', ...
|
||||
'Vph', 'Vpv', ...
|
||||
@ -589,7 +609,7 @@ The identification data is loaded
|
||||
'Vnh', 'Vnv', ...
|
||||
'Va');
|
||||
|
||||
uv = load('mat/data_uh.mat', ...
|
||||
uv = load('mat/data_ucv.mat', ...
|
||||
't', 'Uch', 'Ucv', ...
|
||||
'Unh', 'Unv', ...
|
||||
'Vph', 'Vpv', ...
|
||||
@ -602,35 +622,35 @@ We remove the first seconds where the Cercalo is turned on.
|
||||
#+begin_src matlab
|
||||
t0 = 1;
|
||||
|
||||
uh.Uch(t<t0) = [];
|
||||
uh.Ucv(t<t0) = [];
|
||||
uh.Unh(t<t0) = [];
|
||||
uh.Unv(t<t0) = [];
|
||||
uh.Vph(t<t0) = [];
|
||||
uh.Vpv(t<t0) = [];
|
||||
uh.Vch(t<t0) = [];
|
||||
uh.Vcv(t<t0) = [];
|
||||
uh.Vnh(t<t0) = [];
|
||||
uh.Vnv(t<t0) = [];
|
||||
uh.Va(t<t0) = [];
|
||||
uh.t(t<t0) = [];
|
||||
uh.Uch(uh.t<t0) = [];
|
||||
uh.Ucv(uh.t<t0) = [];
|
||||
uh.Unh(uh.t<t0) = [];
|
||||
uh.Unv(uh.t<t0) = [];
|
||||
uh.Vph(uh.t<t0) = [];
|
||||
uh.Vpv(uh.t<t0) = [];
|
||||
uh.Vch(uh.t<t0) = [];
|
||||
uh.Vcv(uh.t<t0) = [];
|
||||
uh.Vnh(uh.t<t0) = [];
|
||||
uh.Vnv(uh.t<t0) = [];
|
||||
uh.Va(uh.t<t0) = [];
|
||||
uh.t(uh.t<t0) = [];
|
||||
|
||||
uh.t = uh.t - uh.t(1); % We start at t=0
|
||||
|
||||
t0 = 1;
|
||||
|
||||
uv.Uch(t<t0) = [];
|
||||
uv.Ucv(t<t0) = [];
|
||||
uv.Unh(t<t0) = [];
|
||||
uv.Unv(t<t0) = [];
|
||||
uv.Vph(t<t0) = [];
|
||||
uv.Vpv(t<t0) = [];
|
||||
uv.Vch(t<t0) = [];
|
||||
uv.Vcv(t<t0) = [];
|
||||
uv.Vnh(t<t0) = [];
|
||||
uv.Vnv(t<t0) = [];
|
||||
uv.Va(t<t0) = [];
|
||||
uv.t(t<t0) = [];
|
||||
uv.Uch(uv.t<t0) = [];
|
||||
uv.Ucv(uv.t<t0) = [];
|
||||
uv.Unh(uv.t<t0) = [];
|
||||
uv.Unv(uv.t<t0) = [];
|
||||
uv.Vph(uv.t<t0) = [];
|
||||
uv.Vpv(uv.t<t0) = [];
|
||||
uv.Vch(uv.t<t0) = [];
|
||||
uv.Vcv(uv.t<t0) = [];
|
||||
uv.Vnh(uv.t<t0) = [];
|
||||
uv.Vnv(uv.t<t0) = [];
|
||||
uv.Va(uv.t<t0) = [];
|
||||
uv.t(uv.t<t0) = [];
|
||||
|
||||
uv.t = uv.t - uv.t(1); % We start at t=0
|
||||
#+end_src
|
||||
@ -695,11 +715,67 @@ We remove the first seconds where the Cercalo is turned on.
|
||||
#+CAPTION: Identification signals when exciting in the vertical direction ([[./figs/identification_uv.png][png]], [[./figs/identification_uv.pdf][pdf]])
|
||||
[[file:figs/identification_uv.png]]
|
||||
|
||||
** Estimation of the Frequency Response Function Matrix
|
||||
** Coherence
|
||||
The window used for the spectral analysis is an =hanning= windows with temporal size equal to 1 second.
|
||||
#+begin_src matlab
|
||||
win = hanning(ceil(1*fs));
|
||||
#+end_src
|
||||
|
||||
#+begin_src matlab
|
||||
[coh_Uch_Vph, f] = mscohere(uh.Uch, uh.Vph, win, [], [], fs);
|
||||
[coh_Uch_Vpv, ~] = mscohere(uh.Uch, uh.Vpv, win, [], [], fs);
|
||||
[coh_Ucv_Vph, ~] = mscohere(uv.Ucv, uv.Vph, win, [], [], fs);
|
||||
[coh_Ucv_Vpv, ~] = mscohere(uv.Ucv, uv.Vpv, win, [], [], fs);
|
||||
#+end_src
|
||||
|
||||
#+begin_src matlab :exports none
|
||||
figure;
|
||||
ax11 = subplot(2, 2, 1);
|
||||
hold on;
|
||||
plot(f, coh_Uch_Vph)
|
||||
set(gca, 'Xscale', 'log');
|
||||
title('Coherence $\frac{Vp_h}{Uc_h}$')
|
||||
ylabel('Coherence')
|
||||
hold off;
|
||||
|
||||
ax12 = subplot(2, 2, 2);
|
||||
hold on;
|
||||
plot(f, coh_Ucv_Vph)
|
||||
set(gca, 'Xscale', 'log');
|
||||
title('Coherence $\frac{Vp_h}{Uc_v}$')
|
||||
hold off;
|
||||
|
||||
ax21 = subplot(2, 2, 3);
|
||||
hold on;
|
||||
plot(f, coh_Uch_Vpv)
|
||||
set(gca, 'Xscale', 'log');
|
||||
title('Coherence $\frac{Vp_v}{Uc_h}$')
|
||||
ylabel('Coherence')
|
||||
xlabel('Frequency [Hz]')
|
||||
hold off;
|
||||
|
||||
ax22 = subplot(2, 2, 4);
|
||||
hold on;
|
||||
plot(f, coh_Ucv_Vpv)
|
||||
set(gca, 'Xscale', 'log');
|
||||
title('Coherence $\frac{Vp_v}{Uc_v}$')
|
||||
xlabel('Frequency [Hz]')
|
||||
hold off;
|
||||
|
||||
linkaxes([ax11,ax12,ax21,ax22],'xy');
|
||||
xlim([10, 1000]); ylim([0, 1]);
|
||||
#+end_src
|
||||
|
||||
#+HEADER: :tangle no :exports results :results none :noweb yes
|
||||
#+begin_src matlab :var filepath="figs/identification_coherence.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png")
|
||||
<<plt-matlab>>
|
||||
#+end_src
|
||||
|
||||
#+NAME: fig:identification_coherence
|
||||
#+CAPTION: Coherence ([[./figs/identification_coherence.png][png]], [[./figs/identification_coherence.pdf][pdf]])
|
||||
[[file:figs/identification_coherence.png]]
|
||||
|
||||
** Estimation of the Frequency Response Function Matrix
|
||||
We compute an estimate of the transfer functions.
|
||||
#+begin_src matlab
|
||||
[tf_Uch_Vph, f] = tfestimate(uh.Uch, uh.Vph, win, [], [], fs);
|
||||
@ -755,61 +831,76 @@ We compute an estimate of the transfer functions.
|
||||
#+CAPTION: Frequency Response Matrix ([[./figs/frequency_response_matrix.png][png]], [[./figs/frequency_response_matrix.pdf][pdf]])
|
||||
[[file:figs/frequency_response_matrix.png]]
|
||||
|
||||
** Coherence
|
||||
#+begin_src matlab
|
||||
[coh_Uch_Vph, f] = mscohere(uh.Uch, uh.Vph, win, [], [], fs);
|
||||
[coh_Uch_Vpv, ~] = mscohere(uh.Uch, uh.Vpv, win, [], [], fs);
|
||||
[coh_Ucv_Vph, ~] = mscohere(uv.Ucv, uv.Vph, win, [], [], fs);
|
||||
[coh_Ucv_Vpv, ~] = mscohere(uv.Ucv, uv.Vpv, win, [], [], fs);
|
||||
#+end_src
|
||||
|
||||
#+begin_src matlab :exports none
|
||||
figure;
|
||||
ax11 = subplot(2, 2, 1);
|
||||
hold on;
|
||||
plot(f, coh_Uch_Vph)
|
||||
plot(f, 180/pi*unwrap(angle(tf_Uch_Vph)))
|
||||
title('Frequency Response Function $\frac{Vp_h}{Uc_h}$')
|
||||
set(gca, 'Xscale', 'log');
|
||||
title('Coherence $\frac{Vp_h}{Uc_h}$')
|
||||
ylabel('Coherence')
|
||||
ylabel('Phase [deg]')
|
||||
yticks([-180:90:180]);
|
||||
hold off;
|
||||
|
||||
ax12 = subplot(2, 2, 2);
|
||||
hold on;
|
||||
plot(f, coh_Ucv_Vph)
|
||||
plot(f, 180/pi*unwrap(angle(tf_Ucv_Vph)))
|
||||
title('Frequency Response Function $\frac{Vp_h}{Uc_v}$')
|
||||
set(gca, 'Xscale', 'log');
|
||||
title('Coherence $\frac{Vp_h}{Uc_v}$')
|
||||
yticks([-180:90:180]);
|
||||
hold off;
|
||||
|
||||
ax21 = subplot(2, 2, 3);
|
||||
hold on;
|
||||
plot(f, coh_Uch_Vpv)
|
||||
plot(f, 180/pi*unwrap(angle(tf_Uch_Vpv)))
|
||||
title('Frequency Response Function $\frac{Vp_v}{Uc_h}$')
|
||||
set(gca, 'Xscale', 'log');
|
||||
title('Coherence $\frac{Vp_v}{Uc_h}$')
|
||||
ylabel('Coherence')
|
||||
ylabel('Phase [deg]')
|
||||
xlabel('Frequency [Hz]')
|
||||
yticks([-180:90:180]);
|
||||
hold off;
|
||||
|
||||
ax22 = subplot(2, 2, 4);
|
||||
hold on;
|
||||
plot(f, coh_Ucv_Vpv)
|
||||
plot(f, 180/pi*unwrap(angle(tf_Ucv_Vpv)))
|
||||
title('Frequency Response Function $\frac{Vp_v}{Uc_v}$')
|
||||
set(gca, 'Xscale', 'log');
|
||||
title('Coherence $\frac{Vp_v}{Uc_v}$')
|
||||
xlabel('Frequency [Hz]')
|
||||
yticks([-180:90:180]);
|
||||
hold off;
|
||||
|
||||
linkaxes([ax11,ax12,ax21,ax22],'xy');
|
||||
xlim([10, 1000]); ylim([0, 1]);
|
||||
xlim([10, 1000]); ylim([-200, 200]);
|
||||
#+end_src
|
||||
|
||||
#+HEADER: :tangle no :exports results :results none :noweb yes
|
||||
#+begin_src matlab :var filepath="figs/identification_coherence.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png")
|
||||
#+begin_src matlab :var filepath="figs/frequency_response_matrix_phase.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png")
|
||||
<<plt-matlab>>
|
||||
#+end_src
|
||||
|
||||
#+NAME: fig:identification_coherence
|
||||
#+CAPTION: Coherence ([[./figs/identification_coherence.png][png]], [[./figs/identification_coherence.pdf][pdf]])
|
||||
[[file:figs/identification_coherence.png]]
|
||||
#+NAME: fig:frequency_response_matrix_phase
|
||||
#+CAPTION: Frequency Response Matrix_Phase ([[./figs/frequency_response_matrix_phase.png][png]], [[./figs/frequency_response_matrix_phase.pdf][pdf]])
|
||||
[[file:figs/frequency_response_matrix_phase.png]]
|
||||
|
||||
** Time Delay
|
||||
Now, we would like to remove the time delay included in the FRF prior to the model extraction.
|
||||
|
||||
Estimation of the time delay:
|
||||
#+begin_src matlab
|
||||
Ts_delay = Ts; % [s]
|
||||
|
||||
G_delay = tf(1, 1, 'InputDelay', Ts_delay);
|
||||
|
||||
G_delay_resp = squeeze(freqresp(G_delay, f, 'Hz'));
|
||||
#+end_src
|
||||
|
||||
We then remove the time delay from the frequency response function.
|
||||
#+begin_src matlab
|
||||
tf_Uch_Vph = tf_Uch_Vph./G_delay_resp;
|
||||
tf_Uch_Vpv = tf_Uch_Vpv./G_delay_resp;
|
||||
tf_Ucv_Vph = tf_Ucv_Vph./G_delay_resp;
|
||||
tf_Ucv_Vpv = tf_Ucv_Vpv./G_delay_resp;
|
||||
#+end_src
|
||||
|
||||
** Extraction of a transfer function matrix
|
||||
First we define the initial guess for the resonance frequencies and the weights associated.
|
||||
@ -1025,19 +1116,603 @@ An we run the =vectfit3= algorithm.
|
||||
|
||||
And finally, we create the identified state space model:
|
||||
#+begin_src matlab
|
||||
G_uh_xh = minreal(ss(full(SER_uh_xh.A),SER_uh_xh.B,SER_uh_xh.C,SER_uh_xh.D));
|
||||
G_uv_xh = minreal(ss(full(SER_uv_xh.A),SER_uv_xh.B,SER_uv_xh.C,SER_uv_xh.D));
|
||||
G_uh_xv = minreal(ss(full(SER_uh_xv.A),SER_uh_xv.B,SER_uh_xv.C,SER_uh_xv.D));
|
||||
G_uv_xv = minreal(ss(full(SER_uv_xv.A),SER_uv_xv.B,SER_uv_xv.C,SER_uv_xv.D));
|
||||
G_Uch_Vph = minreal(ss(full(SER_Uch_Vph.A),SER_Uch_Vph.B,SER_Uch_Vph.C,SER_Uch_Vph.D));
|
||||
G_Ucv_Vph = minreal(ss(full(SER_Ucv_Vph.A),SER_Ucv_Vph.B,SER_Ucv_Vph.C,SER_Ucv_Vph.D));
|
||||
G_Uch_Vpv = minreal(ss(full(SER_Uch_Vpv.A),SER_Uch_Vpv.B,SER_Uch_Vpv.C,SER_Uch_Vpv.D));
|
||||
G_Ucv_Vpv = minreal(ss(full(SER_Ucv_Vpv.A),SER_Ucv_Vpv.B,SER_Ucv_Vpv.C,SER_Ucv_Vpv.D));
|
||||
|
||||
G = [G_uh_xh, G_uv_xh;
|
||||
G_uh_xv, G_uv_xv];
|
||||
Gc = [G_Uch_Vph, G_Ucv_Vph;
|
||||
G_Uch_Vpv, G_Ucv_Vpv];
|
||||
#+end_src
|
||||
|
||||
#+begin_src matlab
|
||||
save('mat/plant.mat', 'G');
|
||||
save('mat/plant.mat', 'Gc');
|
||||
#+end_src
|
||||
|
||||
** Parameter extraction
|
||||
#+begin_src matlab
|
||||
[Gz, Gp, Gk] = zpkdata(G_Uch_Vph);
|
||||
G_Uch_Vph = zpk([], Gp, -prod(Gz{:})*Gk);
|
||||
[Gz, Gp, Gk] = zpkdata(G_Ucv_Vph);
|
||||
G_Ucv_Vph = zpk([], Gp, -prod(Gz{:})*Gk);
|
||||
[Gz, Gp, Gk] = zpkdata(G_Uch_Vpv);
|
||||
G_Uch_Vpv = zpk([], Gp, -prod(Gz{:})*Gk);
|
||||
[Gz, Gp, Gk] = zpkdata(G_Ucv_Vpv);
|
||||
G_Ucv_Vpv = zpk([], Gp, -prod(Gz{:})*Gk);
|
||||
|
||||
% Plant without the Zero
|
||||
Gb = [G_Uch_Vph, G_Ucv_Vph;
|
||||
G_Uch_Vpv, G_Ucv_Vpv];
|
||||
#+end_src
|
||||
|
||||
* Identification of the Newport Dynamics
|
||||
** Matlab Init :noexport:ignore:
|
||||
#+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
|
||||
fs = 1e4;
|
||||
Ts = 1/fs;
|
||||
freqs = logspace(1, 3, 1000);
|
||||
#+end_src
|
||||
|
||||
** Input / Output data
|
||||
The identification data is loaded
|
||||
#+begin_src matlab
|
||||
uh = load('mat/data_unh.mat', ...
|
||||
't', 'Uch', 'Ucv', ...
|
||||
'Unh', 'Unv', ...
|
||||
'Vph', 'Vpv', ...
|
||||
'Vch', 'Vcv', ...
|
||||
'Vnh', 'Vnv', ...
|
||||
'Va');
|
||||
|
||||
uv = load('mat/data_unv.mat', ...
|
||||
't', 'Uch', 'Ucv', ...
|
||||
'Unh', 'Unv', ...
|
||||
'Vph', 'Vpv', ...
|
||||
'Vch', 'Vcv', ...
|
||||
'Vnh', 'Vnv', ...
|
||||
'Va');
|
||||
#+end_src
|
||||
|
||||
We remove the first seconds where the Cercalo is turned on.
|
||||
#+begin_src matlab
|
||||
t0 = 3;
|
||||
|
||||
uh.Uch(uh.t<t0) = [];
|
||||
uh.Ucv(uh.t<t0) = [];
|
||||
uh.Unh(uh.t<t0) = [];
|
||||
uh.Unv(uh.t<t0) = [];
|
||||
uh.Vph(uh.t<t0) = [];
|
||||
uh.Vpv(uh.t<t0) = [];
|
||||
uh.Vch(uh.t<t0) = [];
|
||||
uh.Vcv(uh.t<t0) = [];
|
||||
uh.Vnh(uh.t<t0) = [];
|
||||
uh.Vnv(uh.t<t0) = [];
|
||||
uh.Va(uh.t<t0) = [];
|
||||
uh.t(uh.t<t0) = [];
|
||||
|
||||
uh.t = uh.t - uh.t(1); % We start at t=0
|
||||
|
||||
t0 = 1.5;
|
||||
|
||||
uv.Uch(uv.t<t0) = [];
|
||||
uv.Ucv(uv.t<t0) = [];
|
||||
uv.Unh(uv.t<t0) = [];
|
||||
uv.Unv(uv.t<t0) = [];
|
||||
uv.Vph(uv.t<t0) = [];
|
||||
uv.Vpv(uv.t<t0) = [];
|
||||
uv.Vch(uv.t<t0) = [];
|
||||
uv.Vcv(uv.t<t0) = [];
|
||||
uv.Vnh(uv.t<t0) = [];
|
||||
uv.Vnv(uv.t<t0) = [];
|
||||
uv.Va(uv.t<t0) = [];
|
||||
uv.t(uv.t<t0) = [];
|
||||
|
||||
uv.t = uv.t - uv.t(1); % We start at t=0
|
||||
#+end_src
|
||||
|
||||
#+begin_src matlab :exports none
|
||||
figure;
|
||||
ax1 = subplot(1, 2, 1);
|
||||
plot(uh.t, uh.Unh, 'DisplayName', '$Un_h$');
|
||||
xlabel('Time [s]');
|
||||
ylabel('Amplitude [V]');
|
||||
legend();
|
||||
|
||||
ax2 = subplot(1, 2, 2);
|
||||
hold on;
|
||||
plot(uh.t, uh.Vph, 'DisplayName', '$Vp_h$');
|
||||
plot(uh.t, uh.Vpv, 'DisplayName', '$Vp_v$');
|
||||
hold off;
|
||||
xlabel('Time [s]');
|
||||
ylabel('Amplitude [V]');
|
||||
legend();
|
||||
|
||||
linkaxes([ax1,ax2],'x');
|
||||
xlim([uh.t(1), uh.t(end)]);
|
||||
#+end_src
|
||||
|
||||
#+HEADER: :tangle no :exports results :results none :noweb yes
|
||||
#+begin_src matlab :var filepath="figs/identification_unh.pdf" :var figsize="full-normal" :post pdf2svg(file=*this*, ext="png")
|
||||
<<plt-matlab>>
|
||||
#+end_src
|
||||
|
||||
#+NAME: fig:identification_unh
|
||||
#+CAPTION: Identification signals when exciting the horizontal direction ([[./figs/identification_unh.png][png]], [[./figs/identification_unh.pdf][pdf]])
|
||||
[[file:figs/identification_unh.png]]
|
||||
|
||||
|
||||
#+begin_src matlab :exports none
|
||||
figure;
|
||||
ax1 = subplot(1, 2, 1);
|
||||
plot(uv.t, uv.Unv, 'DisplayName', '$Uc_v$');
|
||||
xlabel('Time [s]');
|
||||
ylabel('Amplitude [V]');
|
||||
|
||||
ax2 = subplot(1, 2, 2);
|
||||
hold on;
|
||||
plot(uv.t, uv.Vpv, 'DisplayName', '$Vp_v$');
|
||||
plot(uv.t, uv.Vph, 'DisplayName', '$Vp_h$');
|
||||
hold off;
|
||||
xlabel('Time [s]');
|
||||
ylabel('Amplitude [V]');
|
||||
legend();
|
||||
|
||||
linkaxes([ax1,ax2],'x');
|
||||
xlim([uv.t(1), uv.t(end)]);
|
||||
#+end_src
|
||||
|
||||
#+HEADER: :tangle no :exports results :results none :noweb yes
|
||||
#+begin_src matlab :var filepath="figs/identification_unv.pdf" :var figsize="full-normal" :post pdf2svg(file=*this*, ext="png")
|
||||
<<plt-matlab>>
|
||||
#+end_src
|
||||
|
||||
#+NAME: fig:identification_unv
|
||||
#+CAPTION: Identification signals when exciting in the vertical direction ([[./figs/identification_unv.png][png]], [[./figs/identification_unv.pdf][pdf]])
|
||||
[[file:figs/identification_unv.png]]
|
||||
|
||||
|
||||
** Coherence
|
||||
The window used for the spectral analysis is an =hanning= windows with temporal size equal to 1 second.
|
||||
#+begin_src matlab
|
||||
win = hanning(ceil(1*fs));
|
||||
#+end_src
|
||||
|
||||
#+begin_src matlab
|
||||
[coh_Unh_Vph, f] = mscohere(uh.Unh, uh.Vph, win, [], [], fs);
|
||||
[coh_Unh_Vpv, ~] = mscohere(uh.Unh, uh.Vpv, win, [], [], fs);
|
||||
[coh_Unv_Vph, ~] = mscohere(uv.Unv, uv.Vph, win, [], [], fs);
|
||||
[coh_Unv_Vpv, ~] = mscohere(uv.Unv, uv.Vpv, win, [], [], fs);
|
||||
#+end_src
|
||||
|
||||
#+begin_src matlab :exports none
|
||||
figure;
|
||||
ax11 = subplot(2, 2, 1);
|
||||
hold on;
|
||||
plot(f, coh_Unh_Vph)
|
||||
set(gca, 'Xscale', 'log');
|
||||
title('Coherence $\frac{Vp_h}{Un_h}$')
|
||||
ylabel('Coherence')
|
||||
hold off;
|
||||
|
||||
ax12 = subplot(2, 2, 2);
|
||||
hold on;
|
||||
plot(f, coh_Unv_Vph)
|
||||
set(gca, 'Xscale', 'log');
|
||||
title('Coherence $\frac{Vp_h}{Un_v}$')
|
||||
hold off;
|
||||
|
||||
ax21 = subplot(2, 2, 3);
|
||||
hold on;
|
||||
plot(f, coh_Unh_Vpv)
|
||||
set(gca, 'Xscale', 'log');
|
||||
title('Coherence $\frac{Vp_v}{Un_h}$')
|
||||
ylabel('Coherence')
|
||||
xlabel('Frequency [Hz]')
|
||||
hold off;
|
||||
|
||||
ax22 = subplot(2, 2, 4);
|
||||
hold on;
|
||||
plot(f, coh_Unv_Vpv)
|
||||
set(gca, 'Xscale', 'log');
|
||||
title('Coherence $\frac{Vp_v}{Un_v}$')
|
||||
xlabel('Frequency [Hz]')
|
||||
hold off;
|
||||
|
||||
linkaxes([ax11,ax12,ax21,ax22],'xy');
|
||||
xlim([10, 1000]); ylim([0, 1]);
|
||||
#+end_src
|
||||
|
||||
#+HEADER: :tangle no :exports results :results none :noweb yes
|
||||
#+begin_src matlab :var filepath="figs/id_Newport_coherence.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png")
|
||||
<<plt-matlab>>
|
||||
#+end_src
|
||||
|
||||
#+NAME: fig:id_Newport_coherence
|
||||
#+CAPTION: Coherence ([[./figs/id_Newport_coherence.png][png]], [[./figs/id_Newport_coherence.pdf][pdf]])
|
||||
[[file:figs/id_Newport_coherence.png]]
|
||||
|
||||
** Estimation of the Frequency Response Function Matrix
|
||||
We compute an estimate of the transfer functions.
|
||||
#+begin_src matlab
|
||||
[tf_Unh_Vph, f] = tfestimate(uh.Unh, uh.Vph, win, [], [], fs);
|
||||
[tf_Unh_Vpv, ~] = tfestimate(uh.Unh, uh.Vpv, win, [], [], fs);
|
||||
[tf_Unv_Vph, ~] = tfestimate(uv.Unv, uv.Vph, win, [], [], fs);
|
||||
[tf_Unv_Vpv, ~] = tfestimate(uv.Unv, uv.Vpv, win, [], [], fs);
|
||||
#+end_src
|
||||
|
||||
#+begin_src matlab :exports none
|
||||
figure;
|
||||
ax11 = subplot(2, 2, 1);
|
||||
hold on;
|
||||
plot(f, abs(tf_Unh_Vph))
|
||||
title('Frequency Response Function $\frac{Vp_h}{Un_h}$')
|
||||
set(gca, 'Xscale', 'log'); set(gca, 'Yscale', 'log');
|
||||
ylabel('Amplitude')
|
||||
hold off;
|
||||
|
||||
ax12 = subplot(2, 2, 2);
|
||||
hold on;
|
||||
plot(f, abs(tf_Unv_Vph))
|
||||
title('Frequency Response Function $\frac{Vp_h}{Un_v}$')
|
||||
set(gca, 'Xscale', 'log'); set(gca, 'Yscale', 'log');
|
||||
hold off;
|
||||
|
||||
ax21 = subplot(2, 2, 3);
|
||||
hold on;
|
||||
plot(f, abs(tf_Unh_Vpv))
|
||||
title('Frequency Response Function $\frac{Vp_v}{Un_h}$')
|
||||
set(gca, 'Xscale', 'log'); set(gca, 'Yscale', 'log');
|
||||
ylabel('Amplitude')
|
||||
xlabel('Frequency [Hz]')
|
||||
hold off;
|
||||
|
||||
ax22 = subplot(2, 2, 4);
|
||||
hold on;
|
||||
plot(f, abs(tf_Unv_Vpv))
|
||||
title('Frequency Response Function $\frac{Vp_v}{Un_v}$')
|
||||
set(gca, 'Xscale', 'log'); set(gca, 'Yscale', 'log');
|
||||
xlabel('Frequency [Hz]')
|
||||
hold off;
|
||||
|
||||
linkaxes([ax11,ax12,ax21,ax22],'xy');
|
||||
xlim([10, 1000]); ylim([1e-4, 1e1]);
|
||||
#+end_src
|
||||
|
||||
#+HEADER: :tangle no :exports results :results none :noweb yes
|
||||
#+begin_src matlab :var filepath="figs/frequency_response_matrix.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png")
|
||||
<<plt-matlab>>
|
||||
#+end_src
|
||||
|
||||
#+NAME: fig:frequency_response_matrix
|
||||
#+CAPTION: Frequency Response Matrix ([[./figs/frequency_response_matrix.png][png]], [[./figs/frequency_response_matrix.pdf][pdf]])
|
||||
[[file:figs/frequency_response_matrix.png]]
|
||||
|
||||
#+begin_src matlab :exports none
|
||||
figure;
|
||||
ax11 = subplot(2, 2, 1);
|
||||
hold on;
|
||||
plot(f, 180/pi*unwrap(angle(tf_Unh_Vph)))
|
||||
title('Frequency Response Function $\frac{Vp_h}{Un_h}$')
|
||||
set(gca, 'Xscale', 'log');
|
||||
ylabel('Phase [deg]')
|
||||
yticks([-180:90:180]);
|
||||
hold off;
|
||||
|
||||
ax12 = subplot(2, 2, 2);
|
||||
hold on;
|
||||
plot(f, 180/pi*unwrap(angle(tf_Unv_Vph)))
|
||||
title('Frequency Response Function $\frac{Vp_h}{Un_v}$')
|
||||
set(gca, 'Xscale', 'log');
|
||||
yticks([-180:90:180]);
|
||||
hold off;
|
||||
|
||||
ax21 = subplot(2, 2, 3);
|
||||
hold on;
|
||||
plot(f, 180/pi*unwrap(angle(tf_Unh_Vpv)))
|
||||
title('Frequency Response Function $\frac{Vp_v}{Un_h}$')
|
||||
set(gca, 'Xscale', 'log');
|
||||
ylabel('Phase [deg]')
|
||||
xlabel('Frequency [Hz]')
|
||||
yticks([-180:90:180]);
|
||||
hold off;
|
||||
|
||||
ax22 = subplot(2, 2, 4);
|
||||
hold on;
|
||||
plot(f, 180/pi*unwrap(angle(tf_Unv_Vpv)))
|
||||
title('Frequency Response Function $\frac{Vp_v}{Un_v}$')
|
||||
set(gca, 'Xscale', 'log');
|
||||
xlabel('Frequency [Hz]')
|
||||
yticks([-180:90:180]);
|
||||
hold off;
|
||||
|
||||
linkaxes([ax11,ax12,ax21,ax22],'xy');
|
||||
xlim([10, 1000]); ylim([-200, 200]);
|
||||
#+end_src
|
||||
|
||||
#+HEADER: :tangle no :exports results :results none :noweb yes
|
||||
#+begin_src matlab :var filepath="figs/id_Newport_phase.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png")
|
||||
<<plt-matlab>>
|
||||
#+end_src
|
||||
|
||||
#+NAME: fig:id_Newport_phase
|
||||
#+CAPTION: Frequency Response Matrix_Phase ([[./figs/id_Newport_phase.png][png]], [[./figs/id_Newport_phase.pdf][pdf]])
|
||||
[[file:figs/id_Newport_phase.png]]
|
||||
|
||||
** Time Delay
|
||||
Now, we would like to remove the time delay included in the FRF prior to the model extraction.
|
||||
|
||||
Estimation of the time delay:
|
||||
#+begin_src matlab
|
||||
Ts_delay = 0.0005; % [s]
|
||||
|
||||
G_delay = tf(1, 1, 'InputDelay', Ts_delay);
|
||||
|
||||
G_delay_resp = squeeze(freqresp(G_delay, f, 'Hz'));
|
||||
#+end_src
|
||||
|
||||
We then remove the time delay from the frequency response function.
|
||||
#+begin_src matlab :exports none
|
||||
figure;
|
||||
ax11 = subplot(2, 2, 1);
|
||||
hold on;
|
||||
plot(f, 180/pi*unwrap(angle(tf_Unh_Vph)))
|
||||
plot(f, 180/pi*unwrap(angle(tf_Unh_Vph./G_delay_resp)))
|
||||
title('Frequency Response Function $\frac{Vp_h}{Un_h}$')
|
||||
set(gca, 'Xscale', 'log');
|
||||
ylabel('Phase [deg]')
|
||||
yticks([-180:90:180]);
|
||||
hold off;
|
||||
|
||||
ax12 = subplot(2, 2, 2);
|
||||
hold on;
|
||||
plot(f, 180/pi*unwrap(angle(tf_Unv_Vph)))
|
||||
plot(f, 180/pi*unwrap(angle(tf_Unv_Vph./G_delay_resp)))
|
||||
title('Frequency Response Function $\frac{Vp_h}{Un_v}$')
|
||||
set(gca, 'Xscale', 'log');
|
||||
yticks([-180:90:180]);
|
||||
hold off;
|
||||
|
||||
ax21 = subplot(2, 2, 3);
|
||||
hold on;
|
||||
plot(f, 180/pi*unwrap(angle(tf_Unh_Vpv)))
|
||||
plot(f, 180/pi*unwrap(angle(tf_Unh_Vpv./G_delay_resp)))
|
||||
title('Frequency Response Function $\frac{Vp_v}{Un_h}$')
|
||||
set(gca, 'Xscale', 'log');
|
||||
ylabel('Phase [deg]')
|
||||
xlabel('Frequency [Hz]')
|
||||
yticks([-180:90:180]);
|
||||
hold off;
|
||||
|
||||
ax22 = subplot(2, 2, 4);
|
||||
hold on;
|
||||
plot(f, 180/pi*unwrap(angle(tf_Unv_Vpv)))
|
||||
plot(f, 180/pi*unwrap(angle(tf_Unv_Vpv./G_delay_resp)))
|
||||
title('Frequency Response Function $\frac{Vp_v}{Un_v}$')
|
||||
set(gca, 'Xscale', 'log');
|
||||
xlabel('Frequency [Hz]')
|
||||
yticks([-180:90:180]);
|
||||
hold off;
|
||||
|
||||
linkaxes([ax11,ax12,ax21,ax22],'xy');
|
||||
xlim([10, 1000]); ylim([-200, 200]);
|
||||
#+end_src
|
||||
|
||||
#+HEADER: :tangle no :exports results :results none :noweb yes
|
||||
#+begin_src matlab :var filepath="figs/time_delay_Newport.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png")
|
||||
<<plt-matlab>>
|
||||
#+end_src
|
||||
|
||||
#+NAME: fig:time_delay_Newport
|
||||
#+CAPTION: Phase change due to time-delay in the Newport dynamics ([[./figs/time_delay_Newport.png][png]], [[./figs/time_delay_Newport.pdf][pdf]])
|
||||
[[file:figs/time_delay_Newport.png]]
|
||||
|
||||
** Extraction of a transfer function matrix
|
||||
From Fig. [[fig:frequency_response_matrix]], it seems reasonable to model the Newport dynamics as a constant.
|
||||
|
||||
#+begin_src matlab
|
||||
Gnh0 = tf(mean(abs(tf_Unh_Vph(f>10 & f<100))));
|
||||
Gnv0 = tf(mean(abs(tf_Unv_Vpv(f>10 & f<100))));
|
||||
|
||||
Gn = [Gnh0, 0;
|
||||
0, Gnv0];
|
||||
#+end_src
|
||||
|
||||
* Calibration of the 4 Quadrant Diode
|
||||
** Introduction :ignore:
|
||||
|
||||
We know precisely the Newport angle thanks to its metrology included.
|
||||
Thus, we can choose precisely the angle of the Newport mirror and see what is the value measured by the 4 Quadrant Diode.
|
||||
The goal is to obtain the "gain" of the 4QD in [V/rad].
|
||||
|
||||
** Input / Output data
|
||||
The identification data is loaded
|
||||
#+begin_src matlab
|
||||
uh = load('mat/data_cal_pd_h.mat', ...
|
||||
't', 'Uch', 'Ucv', ...
|
||||
'Unh', 'Unv', ...
|
||||
'Vph', 'Vpv', ...
|
||||
'Vch', 'Vcv', ...
|
||||
'Vnh', 'Vnv', ...
|
||||
'Va');
|
||||
|
||||
uv = load('mat/data_cal_pd_v.mat', ...
|
||||
't', 'Uch', 'Ucv', ...
|
||||
'Unh', 'Unv', ...
|
||||
'Vph', 'Vpv', ...
|
||||
'Vch', 'Vcv', ...
|
||||
'Vnh', 'Vnv', ...
|
||||
'Va');
|
||||
#+end_src
|
||||
|
||||
We remove the first seconds where the Cercalo is turned on.
|
||||
#+begin_src matlab
|
||||
t0 = 1;
|
||||
|
||||
uh.Uch(uh.t<t0) = [];
|
||||
uh.Ucv(uh.t<t0) = [];
|
||||
uh.Unh(uh.t<t0) = [];
|
||||
uh.Unv(uh.t<t0) = [];
|
||||
uh.Vph(uh.t<t0) = [];
|
||||
uh.Vpv(uh.t<t0) = [];
|
||||
uh.Vch(uh.t<t0) = [];
|
||||
uh.Vcv(uh.t<t0) = [];
|
||||
uh.Vnh(uh.t<t0) = [];
|
||||
uh.Vnv(uh.t<t0) = [];
|
||||
uh.Va(uh.t<t0) = [];
|
||||
uh.t(uh.t<t0) = [];
|
||||
|
||||
uh.t = uh.t - uh.t(1); % We start at t=0
|
||||
|
||||
t0 = 1;
|
||||
|
||||
uv.Uch(uv.t<t0) = [];
|
||||
uv.Ucv(uv.t<t0) = [];
|
||||
uv.Unh(uv.t<t0) = [];
|
||||
uv.Unv(uv.t<t0) = [];
|
||||
uv.Vph(uv.t<t0) = [];
|
||||
uv.Vpv(uv.t<t0) = [];
|
||||
uv.Vch(uv.t<t0) = [];
|
||||
uv.Vcv(uv.t<t0) = [];
|
||||
uv.Vnh(uv.t<t0) = [];
|
||||
uv.Vnv(uv.t<t0) = [];
|
||||
uv.Va(uv.t<t0) = [];
|
||||
uv.t(uv.t<t0) = [];
|
||||
|
||||
uv.t = uv.t - uv.t(1); % We start at t=0
|
||||
#+end_src
|
||||
|
||||
#+begin_src matlab :exports none
|
||||
figure;
|
||||
ax1 = subplot(1, 2, 1);
|
||||
plot(uh.t, uh.Unh, 'DisplayName', '$Un_h$');
|
||||
xlabel('Time [s]');
|
||||
ylabel('Amplitude [V]');
|
||||
legend();
|
||||
|
||||
ax2 = subplot(1, 2, 2);
|
||||
hold on;
|
||||
plot(uh.t, uh.Vph, 'DisplayName', '$Vp_h$');
|
||||
plot(uh.t, uh.Vpv, 'DisplayName', '$Vp_v$');
|
||||
hold off;
|
||||
xlabel('Time [s]');
|
||||
ylabel('Amplitude [V]');
|
||||
legend();
|
||||
|
||||
linkaxes([ax1,ax2],'x');
|
||||
xlim([uh.t(1), uh.t(end)]);
|
||||
#+end_src
|
||||
|
||||
#+HEADER: :tangle no :exports results :results none :noweb yes
|
||||
#+begin_src matlab :var filepath="figs/calib_4qd_h.pdf" :var figsize="full-normal" :post pdf2svg(file=*this*, ext="png")
|
||||
<<plt-matlab>>
|
||||
#+end_src
|
||||
|
||||
#+NAME: fig:calib_4qd_h
|
||||
#+CAPTION: Identification signals when exciting the horizontal direction ([[./figs/calib_4qd_h.png][png]], [[./figs/calib_4qd_h.pdf][pdf]])
|
||||
[[file:figs/calib_4qd_h.png]]
|
||||
|
||||
|
||||
#+begin_src matlab :exports none
|
||||
figure;
|
||||
ax1 = subplot(1, 2, 1);
|
||||
plot(uv.t, uv.Unv, 'DisplayName', '$Uc_v$');
|
||||
xlabel('Time [s]');
|
||||
ylabel('Amplitude [V]');
|
||||
|
||||
ax2 = subplot(1, 2, 2);
|
||||
hold on;
|
||||
plot(uv.t, uv.Vpv, 'DisplayName', '$Vp_v$');
|
||||
plot(uv.t, uv.Vph, 'DisplayName', '$Vp_h$');
|
||||
hold off;
|
||||
xlabel('Time [s]');
|
||||
ylabel('Amplitude [V]');
|
||||
legend();
|
||||
|
||||
linkaxes([ax1,ax2],'x');
|
||||
xlim([uv.t(1), uv.t(end)]);
|
||||
#+end_src
|
||||
|
||||
#+HEADER: :tangle no :exports results :results none :noweb yes
|
||||
#+begin_src matlab :var filepath="figs/calib_4qd_v.pdf" :var figsize="full-normal" :post pdf2svg(file=*this*, ext="png")
|
||||
<<plt-matlab>>
|
||||
#+end_src
|
||||
|
||||
#+NAME: fig:calib_4qd_v
|
||||
#+CAPTION: Identification signals when exciting in the vertical direction ([[./figs/calib_4qd_v.png][png]], [[./figs/calib_4qd_v.pdf][pdf]])
|
||||
[[file:figs/calib_4qd_v.png]]
|
||||
|
||||
** Linear Regression
|
||||
#+begin_src matlab
|
||||
bh = [ones(size(uh.Unh)) uh.Unh]\uh.Vph;
|
||||
bv = [ones(size(uv.Unv)) uv.Unv]\uv.Vpv;
|
||||
#+end_src
|
||||
|
||||
#+begin_src matlab :exports none
|
||||
figure;
|
||||
ax1 = subplot(1, 2, 1);
|
||||
hold on;
|
||||
plot(uh.Unh, uh.Vph, 'o', 'DisplayName', 'Exp. data');
|
||||
plot([min(uh.Unh) max(uh.Unh)], [min(uh.Unh) max(uh.Unh)].*bh(2) + bh(1), 'k--', 'DisplayName', sprintf('%.1e x + %.1e', bh(2), bh(1)))
|
||||
hold off;
|
||||
xlabel('$Un_h$ [V]'); ylabel('$Vp_h$ [V]');
|
||||
legend();
|
||||
|
||||
ax2 = subplot(1, 2, 2);
|
||||
hold on;
|
||||
plot(uv.Unv, uv.Vpv, 'o', 'DisplayName', 'Exp. data');
|
||||
plot([min(uv.Unv) max(uv.Unv)], [min(uv.Unv) max(uv.Unv)].*bv(2) + bv(1), 'k--', 'DisplayName', sprintf('%.1e x + %.1e', bv(2), bv(1)))
|
||||
hold off;
|
||||
xlabel('$Un_v$ [V]');
|
||||
ylabel('$Vp_v$ [V]');
|
||||
legend();
|
||||
#+end_src
|
||||
|
||||
#+HEADER: :tangle no :exports results :results none :noweb yes
|
||||
#+begin_src matlab :var filepath="figs/4qd_linear_reg.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png")
|
||||
<<plt-matlab>>
|
||||
#+end_src
|
||||
|
||||
#+NAME: fig:4qd_linear_reg
|
||||
#+CAPTION: Linear Regression ([[./figs/4qd_linear_reg.png][png]], [[./figs/4qd_linear_reg.pdf][pdf]])
|
||||
[[file:figs/4qd_linear_reg.png]]
|
||||
|
||||
** Gain of the photodiodes
|
||||
If we note:
|
||||
- $\alpha_o$ the angular displacement of the beam
|
||||
- $\alpha_m$ the angular displacement of the mirror
|
||||
- $V_{qd}$ the voltage measured by the 4QD
|
||||
- $V_n$ the voltage input of the Newport
|
||||
|
||||
The angular displacement of the beam is twice the mechanical angular displacement of the Newport mirror (supposing here that the mirror is flat and not concave):
|
||||
\[ \alpha_o = 2 \alpha_m \]
|
||||
|
||||
We want to determine $\frac{V_{qd}}{\alpha_o}$:
|
||||
\begin{align*}
|
||||
\frac{V_{qd}}{\alpha_o} &= \frac{1}{2} \frac{V_{qd}}{\frac{1}{2} \alpha_o} \\
|
||||
&= \frac{1}{2} \frac{V_{qd}}{V_n} \frac{V_n}{\alpha_m}
|
||||
\end{align*}
|
||||
with:
|
||||
- $\frac{V_n}{\alpha_m} = \frac{1}{2.62 \cdot 10^{-3}}\ \left[\frac{V}{rad}\right]$ is the inverse of the gain of the Newport
|
||||
- $\frac{V_{qd}}{V_n} = 0.185\ \left[\frac{V}{V}\right]$ is the identified DC gain of the transfer function from a voltage applied to the Newport to the 4QD measurement
|
||||
|
||||
Thus:
|
||||
#+begin_src matlab
|
||||
Gqd = 0.5 * mean(diag(dcgain(Gn))) * 1/(2.62*10^-3);
|
||||
#+end_src
|
||||
|
||||
We obtain:
|
||||
\[ \frac{\alpha_0}{V_{qd}} \approx 35.2\ \left[ \frac{V}{rad} \right] \]
|
||||
|
||||
\[ \frac{\alpha_0}{V_{qd}} \approx 0.35\ \left[ \frac{mV}{\mu rad} \right] \]
|
||||
|
||||
* Plant Scaling
|
||||
- measured noise
|
||||
- expected perturbations
|
||||
|
||||
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mat/data_uv.mat
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mat/data_uv.mat
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mat/plant.mat
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mat/plant.mat
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Reference in New Issue
Block a user