1653 lines
56 KiB
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1653 lines
56 KiB
HTML
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<!-- 2020-11-24 mar. 13:54 -->
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<title>Test Bench - Amplified Piezoelectric Actuator</title>
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<a accesskey="h" href="../index.html"> UP </a>
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<a accesskey="H" href="../index.html"> HOME </a>
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</div><div id="content">
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<h1 class="title">Test Bench - Amplified Piezoelectric Actuator</h1>
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<div id="table-of-contents">
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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="#org692da4d">1. Experimental Setup</a></li>
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<li><a href="#orgf5c6cd7">2. Estimation of electrical/mechanical relationships</a>
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<ul>
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<li><a href="#orgd8c7c0b">2.1. Estimation from Data-sheet</a></li>
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<li><a href="#org901da31">2.2. Estimation from Piezoelectric parameters</a></li>
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<li><a href="#org616104b">2.3. Estimation from Experiment</a>
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<ul>
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<li><a href="#orgcb3d952">2.3.1. From actuator voltage \(V_a\) to actuator force \(F_a\)</a></li>
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<li><a href="#org138110f">2.3.2. From stack strain \(\Delta h\) to generated voltage \(V_s\)</a></li>
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</ul>
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</li>
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<li><a href="#org7cf58ef">2.4. Conclusion</a></li>
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</ul>
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</li>
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<li><a href="#org36bf85c">3. Simscape model of the test-bench</a>
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<ul>
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<li><a href="#orgca28e7b">3.1. Import Mass Matrix, Stiffness Matrix, and Interface Nodes Coordinates</a></li>
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<li><a href="#orga49ef8e">3.2. Simscape Model</a></li>
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<li><a href="#orgf1f3f75">3.3. Dynamics from Actuator Voltage to Vertical Mass Displacement</a></li>
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<li><a href="#orgd4a750c">3.4. Dynamics from Actuator Voltage to Force Sensor Voltage</a></li>
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<li><a href="#org1ac4d3c">3.5. Save Data for further use</a></li>
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</ul>
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</li>
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<li><a href="#org9de639d">4. Huddle Test</a>
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<ul>
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<li><a href="#orgaf93c0c">4.1. Time Domain Data</a></li>
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<li><a href="#org22ccdcf">4.2. PSD of Measurement Noise</a></li>
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</ul>
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</li>
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<li><a href="#org914a7da">5. Identification of the dynamics from actuator to displacement</a>
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<ul>
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<li><a href="#orga1aace1">5.1. Load Data</a></li>
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<li><a href="#org1284757">5.2. Comparison of the PSD with Huddle Test</a></li>
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<li><a href="#org89453e3">5.3. Compute TF estimate and Coherence</a></li>
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</ul>
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</li>
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<li><a href="#orgfff18d2">6. Identification of the dynamics from actuator to force sensor</a>
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<ul>
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<li><a href="#orgfa2171a">6.1. System Identification</a></li>
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<li><a href="#org633e9f1">6.2. Integral Force Feedback</a></li>
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</ul>
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</li>
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<li><a href="#org1bf58e2">7. Integral Force Feedback</a>
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<ul>
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<li><a href="#org25092d8">7.1. First tests with few gains</a></li>
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<li><a href="#org67fa466">7.2. Second test with many Gains</a></li>
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</ul>
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</li>
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</ul>
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</div>
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</div>
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<ul class="org-ul">
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<li>Section <a href="#org2c6bb95">1</a>:</li>
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<li>Section <a href="#orgb81d61b">3</a>:</li>
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<li>Section [[]]:</li>
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<li>Section [[]]:</li>
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<li>Section [[]]:</li>
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</ul>
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<div id="outline-container-org692da4d" class="outline-2">
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<h2 id="org692da4d"><span class="section-number-2">1</span> Experimental Setup</h2>
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<div class="outline-text-2" id="text-1">
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<p>
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<a id="org2c6bb95"></a>
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</p>
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<p>
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A schematic of the test-bench is shown in Figure <a href="#org57478f1">1</a>.
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</p>
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<p>
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A mass can be vertically moved using the amplified piezoelectric actuator (APA95ML).
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The displacement of the mass (relative to the mechanical frame) is measured by the interferometer.
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</p>
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<p>
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The APA95ML has three stacks that can be used as actuator or as sensors.
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</p>
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<p>
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Pictures of the test bench are shown in Figure <a href="#org4bf2105">2</a> and <a href="#org809b25b">3</a>.
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</p>
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<div id="org57478f1" class="figure">
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<p><img src="figs/test_bench_apa_schematic.png" alt="test_bench_apa_schematic.png" />
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</p>
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<p><span class="figure-number">Figure 1: </span>Schematic of the Setup</p>
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</div>
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<div id="org4bf2105" class="figure">
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<p><img src="figs/setup_picture.png" alt="setup_picture.png" />
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</p>
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<p><span class="figure-number">Figure 2: </span>Picture of the Setup</p>
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</div>
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<div id="org809b25b" class="figure">
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<p><img src="figs/setup_zoom.png" alt="setup_zoom.png" />
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</p>
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<p><span class="figure-number">Figure 3: </span>Zoom on the APA</p>
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</div>
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<div class="note" id="org79ca795">
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<p>
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Here are the equipment used in the test bench:
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</p>
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<ul class="org-ul">
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<li>Attocube interferometer (<a href="doc/IDS3010.pdf">doc</a>)</li>
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<li>Cedrat Amplified Piezoelectric Actuator APA95ML (<a href="doc/APA95ML.pdf">doc</a>)</li>
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<li>Voltage Amplifier LA75B (<a href="doc/LA75B.pdf">doc</a>)</li>
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<li>Speedgoat IO131 with 16bits ADC and DAC (<a href="doc/IO130 IO131 OEM Datasheet.pdf">doc</a>)</li>
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<li>Low Noise Voltage Preamplifier from Ametek (<a href="doc/model_5113.pdf">doc</a>)</li>
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</ul>
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</div>
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</div>
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</div>
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<div id="outline-container-orgf5c6cd7" class="outline-2">
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<h2 id="orgf5c6cd7"><span class="section-number-2">2</span> Estimation of electrical/mechanical relationships</h2>
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<div class="outline-text-2" id="text-2">
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<p>
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In order to correctly model the piezoelectric actuator, we need to determine:
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</p>
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<ol class="org-ol">
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<li>\(g_a\): the ratio of the generated force \(F_a\) to the supply voltage \(V_a\) across the piezoelectric stack</li>
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<li>\(g_s\): the ratio of the generated voltage \(V_s\) across the piezoelectric stack when subject to a strain \(\Delta h\)</li>
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</ol>
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<p>
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We estimate \(g_a\) and \(g_s\) using different approaches:
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</p>
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<ol class="org-ol">
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<li>Section <a href="#org640a15c">2.1</a>: \(g_a\) is estimated from the datasheet of the piezoelectric stack</li>
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<li>Section <a href="#orgeeaae90">2.2</a>: \(g_a\) and \(g_s\) are estimated using the piezoelectric constants</li>
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<li>Section <a href="#org5a92750">2.3</a>: \(g_a\) and \(g_s\) are estimated experimentally</li>
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</ol>
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</div>
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<div id="outline-container-orgd8c7c0b" class="outline-3">
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<h3 id="orgd8c7c0b"><span class="section-number-3">2.1</span> Estimation from Data-sheet</h3>
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<div class="outline-text-3" id="text-2-1">
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<p>
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<a id="org640a15c"></a>
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</p>
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<p>
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The stack parameters taken from the data-sheet are shown in Table <a href="#orgf2c9009">1</a>.
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</p>
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<table id="orgf2c9009" 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> Stack Parameters</caption>
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<colgroup>
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<col class="org-left" />
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<col class="org-left" />
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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">Parameter</th>
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<th scope="col" class="org-left">Unit</th>
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<th scope="col" class="org-right">Value</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">Nominal Stroke</td>
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<td class="org-left">\(\mu m\)</td>
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<td class="org-right">20</td>
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</tr>
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<tr>
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<td class="org-left">Blocked force</td>
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<td class="org-left">\(N\)</td>
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<td class="org-right">4700</td>
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</tr>
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<tr>
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<td class="org-left">Stiffness</td>
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<td class="org-left">\(N/\mu m\)</td>
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<td class="org-right">235</td>
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</tr>
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<tr>
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<td class="org-left">Voltage Range</td>
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<td class="org-left">\(V\)</td>
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<td class="org-right">-20..150</td>
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</tr>
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<tr>
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<td class="org-left">Capacitance</td>
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<td class="org-left">\(\mu F\)</td>
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<td class="org-right">4.4</td>
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</tr>
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<tr>
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<td class="org-left">Length</td>
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<td class="org-left">\(mm\)</td>
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<td class="org-right">20</td>
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</tr>
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<tr>
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<td class="org-left">Stack Area</td>
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<td class="org-left">\(mm^2\)</td>
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<td class="org-right">10x10</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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|
Let’s compute the generated force
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</p>
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<p>
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The stroke is \(L_{\max} = 20\mu m\) for a voltage range of \(V_{\max} = 170 V\).
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Furthermore, the stiffness is \(k_a = 235 \cdot 10^6 N/m\).
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</p>
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<p>
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The relation between the applied voltage and the generated force can be estimated as follows:
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</p>
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\begin{equation}
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g_a = k_a \frac{L_{\max}}{V_{\max}}
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\end{equation}
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<div class="org-src-container">
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<pre class="src src-matlab">ka = 235e6; <span class="org-comment">% [N/m]</span>
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Lmax = 20e<span class="org-type">-</span>6; <span class="org-comment">% [m]</span>
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Vmax = 170; <span class="org-comment">% [V]</span>
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</pre>
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</div>
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<div class="org-src-container">
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<pre class="src src-matlab">ka<span class="org-type">*</span>Lmax<span class="org-type">/</span>Vmax <span class="org-comment">% [N/V]</span>
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</pre>
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</div>
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<pre class="example">
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27.647
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</pre>
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<p>
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From the parameters of the stack, it seems not possible to estimate the relation between the strain and the generated voltage.
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</p>
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</div>
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</div>
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<div id="outline-container-org901da31" class="outline-3">
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<h3 id="org901da31"><span class="section-number-3">2.2</span> Estimation from Piezoelectric parameters</h3>
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<div class="outline-text-3" id="text-2-2">
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<p>
|
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<a id="orgeeaae90"></a>
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</p>
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<p>
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In order to make the conversion from applied voltage to generated force or from the strain to the generated voltage, we need to defined some parameters corresponding to the piezoelectric material:
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</p>
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<div class="org-src-container">
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<pre class="src src-matlab">d33 = 300e<span class="org-type">-</span>12; <span class="org-comment">% Strain constant [m/V]</span>
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n = 80; <span class="org-comment">% Number of layers per stack</span>
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ka = 235e6; <span class="org-comment">% Stack stiffness [N/m]</span>
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</pre>
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</div>
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<p>
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The ratio of the developed force to applied voltage is:
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</p>
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\begin{equation}
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\label{org8300de8}
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F_a = g_a V_a, \quad g_a = d_{33} n k_a
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\end{equation}
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<p>
|
|
where:
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</p>
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<ul class="org-ul">
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<li>\(F_a\): developed force in [N]</li>
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<li>\(n\): number of layers of the actuator stack</li>
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<li>\(d_{33}\): strain constant in [m/V]</li>
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<li>\(k_a\): actuator stack stiffness in [N/m]</li>
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<li>\(V_a\): applied voltage in [V]</li>
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</ul>
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<p>
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|
If we take the numerical values, we obtain:
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</p>
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<div class="org-src-container">
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<pre class="src src-matlab">ga = d33<span class="org-type">*</span>n<span class="org-type">*</span>ka; <span class="org-comment">% [N/V]</span>
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</pre>
|
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</div>
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<pre class="example">
|
|
ga = 5.6 [N/V]
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</pre>
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<p>
|
|
From (<a href="#citeproc_bib_item_1">Fleming and Leang 2014</a>) (page 123), the relation between relative displacement of the sensor stack and generated voltage is:
|
|
</p>
|
|
\begin{equation}
|
|
\label{org312083f}
|
|
V_s = \frac{d_{33}}{\epsilon^T s^D n} \Delta h
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\end{equation}
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<p>
|
|
where:
|
|
</p>
|
|
<ul class="org-ul">
|
|
<li>\(V_s\): measured voltage in [V]</li>
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|
<li>\(d_{33}\): strain constant in [m/V]</li>
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|
<li>\(\epsilon^T\): permittivity under constant stress in [F/m]</li>
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|
<li>\(s^D\): elastic compliance under constant electric displacement in [m^2/N]</li>
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|
<li>\(n\): number of layers of the sensor stack</li>
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|
<li>\(\Delta h\): relative displacement in [m]</li>
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</ul>
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|
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|
<p>
|
|
If we take the numerical values, we obtain:
|
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</p>
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">d33 = 300e<span class="org-type">-</span>12; <span class="org-comment">% Strain constant [m/V]</span>
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|
n = 80; <span class="org-comment">% Number of layers per stack</span>
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|
eT = 5.3e<span class="org-type">-</span>9; <span class="org-comment">% Permittivity under constant stress [F/m]</span>
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|
sD = 2e<span class="org-type">-</span>11; <span class="org-comment">% Compliance under constant electric displacement [m2/N]</span>
|
|
</pre>
|
|
</div>
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<div class="org-src-container">
|
|
<pre class="src src-matlab">gs = d33<span class="org-type">/</span>(eT<span class="org-type">*</span>sD<span class="org-type">*</span>n); <span class="org-comment">% [V/m]</span>
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|
</pre>
|
|
</div>
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|
|
<pre class="example">
|
|
gs = 35.4 [V/um]
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|
</pre>
|
|
</div>
|
|
</div>
|
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|
|
<div id="outline-container-org616104b" class="outline-3">
|
|
<h3 id="org616104b"><span class="section-number-3">2.3</span> Estimation from Experiment</h3>
|
|
<div class="outline-text-3" id="text-2-3">
|
|
<p>
|
|
<a id="org5a92750"></a>
|
|
</p>
|
|
<p>
|
|
The idea here is to obtain the parameters \(g_a\) and \(g_s\) from the comparison of an experimental identification and the identification using Simscape.
|
|
</p>
|
|
|
|
<p>
|
|
Using the experimental identification, we can easily obtain the gain from the applied voltage to the generated displacement, but not to the generated force.
|
|
However, from the Simscape model, we can easily have the link from the generated force to the displacement, them we can computed \(g_a\).
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</p>
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|
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<p>
|
|
Similarly, it is fairly easy to experimentally obtain the gain from the stack displacement to the generated voltage across the stack.
|
|
To link that to the strain of the sensor stack, the simscape model is used.
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|
</p>
|
|
</div>
|
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<div id="outline-container-orgcb3d952" class="outline-4">
|
|
<h4 id="orgcb3d952"><span class="section-number-4">2.3.1</span> From actuator voltage \(V_a\) to actuator force \(F_a\)</h4>
|
|
<div class="outline-text-4" id="text-2-3-1">
|
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<p>
|
|
The data from the identification test is loaded.
|
|
</p>
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">load(<span class="org-string">'apa95ml_5kg_Amp_E505.mat'</span>, <span class="org-string">'t'</span>, <span class="org-string">'um'</span>, <span class="org-string">'y'</span>);
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<span class="org-comment">% Any offset value is removed:</span>
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um = detrend(um, 0); <span class="org-comment">% Amplifier Input Voltage [V]</span>
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y = detrend(y , 0); <span class="org-comment">% Mass displacement [m]</span>
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</pre>
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</div>
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<p>
|
|
Now we add a factor 10 to take into account the gain of the voltage amplifier and thus obtain the voltage across the piezoelectric stack.
|
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</p>
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<div class="org-src-container">
|
|
<pre class="src src-matlab">um = 10<span class="org-type">*</span>um; <span class="org-comment">% Stack Actuator Input Voltage [V]</span>
|
|
</pre>
|
|
</div>
|
|
|
|
<p>
|
|
Then, the transfer function from the stack voltage to the vertical displacement is computed.
|
|
</p>
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">Ts = t(end)<span class="org-type">/</span>(length(t)<span class="org-type">-</span>1);
|
|
Fs = 1<span class="org-type">/</span>Ts;
|
|
|
|
win = hanning(ceil(1<span class="org-type">*</span>Fs));
|
|
|
|
[tf_est, f] = tfestimate(um, y, win, [], [], 1<span class="org-type">/</span>Ts);
|
|
</pre>
|
|
</div>
|
|
|
|
<p>
|
|
The gain from input voltage of the stack to the vertical displacement is determined:
|
|
</p>
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">g_d_Va = 4e<span class="org-type">-</span>7; <span class="org-comment">% [m/V]</span>
|
|
</pre>
|
|
</div>
|
|
|
|
|
|
<div id="orgf3201ac" class="figure">
|
|
<p><img src="figs/gain_Va_to_d.png" alt="gain_Va_to_d.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 4: </span>Transfer function from actuator stack voltage \(V_a\) to vertical displacement of the mass \(d\)</p>
|
|
</div>
|
|
|
|
<p>
|
|
Then, the transfer function from forces applied by the stack actuator to the vertical displacement of the mass is identified from the Simscape model.
|
|
</p>
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">m = 5.5;
|
|
|
|
<span class="org-matlab-cellbreak"><span class="org-comment">%% Name of the Simulink File</span></span>
|
|
mdl = <span class="org-string">'piezo_amplified_3d'</span>;
|
|
|
|
<span class="org-matlab-cellbreak"><span class="org-comment">%% Input/Output definition</span></span>
|
|
clear io; io_i = 1;
|
|
io(io_i) = linio([mdl, <span class="org-string">'/Fa'</span>], 1, <span class="org-string">'openinput'</span>); io_i = io_i <span class="org-type">+</span> 1; <span class="org-comment">% Actuator Force [N]</span>
|
|
io(io_i) = linio([mdl, <span class="org-string">'/y'</span>], 1, <span class="org-string">'openoutput'</span>); io_i = io_i <span class="org-type">+</span> 1; <span class="org-comment">% Vertical Displacement [m]</span>
|
|
|
|
Gd = linearize(mdl, io);
|
|
</pre>
|
|
</div>
|
|
|
|
<p>
|
|
The DC gain the the identified dynamics
|
|
</p>
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">g_d_Fa = abs(dcgain(Gd)); <span class="org-comment">% [m/N]</span>
|
|
</pre>
|
|
</div>
|
|
|
|
<pre class="example">
|
|
G_d_Fa = 1.2e-08 [m/N]
|
|
</pre>
|
|
|
|
|
|
|
|
<p>
|
|
And finally, the gain \(g_a\) from the the actuator voltage \(V_a\) to the generated force \(F_a\) can be computed:
|
|
</p>
|
|
\begin{equation}
|
|
g_a = \frac{F_a}{V_a} = \frac{F_a}{d} \frac{d}{V_a}
|
|
\end{equation}
|
|
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">ga = g_d_Va<span class="org-type">/</span>g_d_Fa;
|
|
</pre>
|
|
</div>
|
|
|
|
<pre class="example">
|
|
ga = 33.7 [N/V]
|
|
</pre>
|
|
|
|
|
|
<p>
|
|
The obtained comparison between the Simscape model and the identified dynamics is shown in Figure <a href="#orgc727518">5</a>.
|
|
</p>
|
|
|
|
|
|
<div id="orgc727518" class="figure">
|
|
<p><img src="figs/compare_Gd_id_simscape.png" alt="compare_Gd_id_simscape.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 5: </span>Comparison of the identified transfer function between actuator voltage \(V_a\) and vertical mass displacement \(d\)</p>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-org138110f" class="outline-4">
|
|
<h4 id="org138110f"><span class="section-number-4">2.3.2</span> From stack strain \(\Delta h\) to generated voltage \(V_s\)</h4>
|
|
<div class="outline-text-4" id="text-2-3-2">
|
|
<p>
|
|
Now, the gain from the stack strain \(\Delta h\) to the generated voltage \(V_s\) is estimated.
|
|
</p>
|
|
|
|
<p>
|
|
We can determine the gain from actuator voltage \(V_a\) to sensor voltage \(V_s\) thanks to the identification.
|
|
Using the simscape model, we can have the transfer function from the actuator voltage \(V_a\) (using the previously estimated gain \(g_a\)) to the sensor stack strain \(\Delta h\).
|
|
</p>
|
|
|
|
<p>
|
|
Finally, using these two values, we can compute the gain \(g_s\) from the stack strain \(\Delta h\) to the generated Voltage \(V_s\).
|
|
</p>
|
|
|
|
<p>
|
|
Identification data is loaded.
|
|
</p>
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">load(<span class="org-string">'apa95ml_5kg_2a_1s.mat'</span>, <span class="org-string">'t'</span>, <span class="org-string">'u'</span>, <span class="org-string">'v'</span>);
|
|
|
|
u = detrend(u, 0); <span class="org-comment">% Input Voltage of the Amplifier [V]</span>
|
|
v = detrend(v, 0); <span class="org-comment">% Voltage accross the stack sensor [V]</span>
|
|
</pre>
|
|
</div>
|
|
|
|
<p>
|
|
Here, an amplifier with a gain of 20 is used.
|
|
</p>
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">u = 20<span class="org-type">*</span>u; <span class="org-comment">% Input Voltage of the Amplifier [V]</span>
|
|
</pre>
|
|
</div>
|
|
|
|
<p>
|
|
Then, the transfer function from \(V_a\) to \(V_s\) is identified and its DC gain is estimated (Figure <a href="#org6245a69">6</a>).
|
|
</p>
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">Ts = t(end)<span class="org-type">/</span>(length(t)<span class="org-type">-</span>1);
|
|
Fs = 1<span class="org-type">/</span>Ts;
|
|
|
|
win = hann(ceil(10<span class="org-type">/</span>Ts));
|
|
|
|
[tf_est, f] = tfestimate(u, v, win, [], [], 1<span class="org-type">/</span>Ts);
|
|
</pre>
|
|
</div>
|
|
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">g_Vs_Va = 0.022; <span class="org-comment">% [V/V]</span>
|
|
</pre>
|
|
</div>
|
|
|
|
|
|
<div id="org6245a69" class="figure">
|
|
<p><img src="figs/gain_Va_to_Vs.png" alt="gain_Va_to_Vs.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 6: </span>Transfer function from actuator stack voltage \(V_a\) to sensor stack voltage \(V_s\)</p>
|
|
</div>
|
|
|
|
|
|
<p>
|
|
Now the transfer function from the actuator stack voltage to the sensor stack strain is estimated using the Simscape model.
|
|
</p>
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">m = 5.5;
|
|
|
|
<span class="org-matlab-cellbreak"><span class="org-comment">%% Name of the Simulink File</span></span>
|
|
mdl = <span class="org-string">'piezo_amplified_3d'</span>;
|
|
|
|
<span class="org-matlab-cellbreak"><span class="org-comment">%% Input/Output definition</span></span>
|
|
clear io; io_i = 1;
|
|
io(io_i) = linio([mdl, <span class="org-string">'/Va'</span>], 1, <span class="org-string">'openinput'</span>); io_i = io_i <span class="org-type">+</span> 1; <span class="org-comment">% Actuator Voltage [V]</span>
|
|
io(io_i) = linio([mdl, <span class="org-string">'/dL'</span>], 1, <span class="org-string">'openoutput'</span>); io_i = io_i <span class="org-type">+</span> 1; <span class="org-comment">% Sensor Stack displacement [m]</span>
|
|
|
|
Gf = linearize(mdl, io);
|
|
</pre>
|
|
</div>
|
|
|
|
<p>
|
|
The gain from the actuator stack voltage to the sensor stack strain is estimated below.
|
|
</p>
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">G_dh_Va = abs(dcgain(Gf));
|
|
</pre>
|
|
</div>
|
|
|
|
<pre class="example">
|
|
G_dh_Va = 6.2e-09 [m/V]
|
|
</pre>
|
|
|
|
|
|
<p>
|
|
And finally, the gain \(g_s\) from the sensor stack strain to the generated voltage can be estimated:
|
|
</p>
|
|
\begin{equation}
|
|
g_s = \frac{V_s}{\Delta h} = \frac{V_s}{V_a} \frac{V_a}{\Delta h}
|
|
\end{equation}
|
|
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">gs = g_Vs_Va<span class="org-type">/</span>G_dh_Va; <span class="org-comment">% [V/m]</span>
|
|
</pre>
|
|
</div>
|
|
|
|
<pre class="example">
|
|
gs = 3.5 [V/um]
|
|
</pre>
|
|
|
|
|
|
|
|
<div id="org6b6b451" class="figure">
|
|
<p><img src="figs/compare_Gf_id_simscape.png" alt="compare_Gf_id_simscape.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 7: </span>Comparison of the identified transfer function between actuator voltage \(V_a\) and sensor stack voltage</p>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-org7cf58ef" class="outline-3">
|
|
<h3 id="org7cf58ef"><span class="section-number-3">2.4</span> Conclusion</h3>
|
|
<div class="outline-text-3" id="text-2-4">
|
|
<p>
|
|
The obtained parameters \(g_a\) and \(g_s\) are not consistent between the different methods.
|
|
</p>
|
|
|
|
<p>
|
|
The one using the experimental data are saved and further used.
|
|
</p>
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">save(<span class="org-string">'./matlab/mat/apa95ml_params.mat'</span>, <span class="org-string">'ga'</span>, <span class="org-string">'gs'</span>);
|
|
</pre>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-org36bf85c" class="outline-2">
|
|
<h2 id="org36bf85c"><span class="section-number-2">3</span> Simscape model of the test-bench</h2>
|
|
<div class="outline-text-2" id="text-3">
|
|
<p>
|
|
<a id="orgb81d61b"></a>
|
|
</p>
|
|
<p>
|
|
The idea here is to model the test-bench using Simscape.
|
|
</p>
|
|
|
|
<p>
|
|
Whereas the suspended mass and metrology frame can be considered as rigid bodies in the frequency range of interest, the Amplified Piezoelectric Actuator (APA) is flexible.
|
|
</p>
|
|
|
|
<p>
|
|
To model the APA, a Finite Element Model (FEM) is used (Figure <a href="#orgbdafa32">8</a>) and imported into Simscape.
|
|
</p>
|
|
|
|
|
|
<div id="orgbdafa32" class="figure">
|
|
<p><img src="figs/APA95ML_FEM.png" alt="APA95ML_FEM.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 8: </span>Finite Element Model of the APA95ML</p>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-orgca28e7b" class="outline-3">
|
|
<h3 id="orgca28e7b"><span class="section-number-3">3.1</span> Import Mass Matrix, Stiffness Matrix, and Interface Nodes Coordinates</h3>
|
|
<div class="outline-text-3" id="text-3-1">
|
|
<p>
|
|
We first extract the stiffness and mass matrices.
|
|
</p>
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">K = readmatrix(<span class="org-string">'APA95ML_K.CSV'</span>);
|
|
M = readmatrix(<span class="org-string">'APA95ML_M.CSV'</span>);
|
|
</pre>
|
|
</div>
|
|
|
|
<table border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
|
|
<caption class="t-above"><span class="table-number">Table 2:</span> First 10x10 elements of the Stiffness matrix</caption>
|
|
|
|
<colgroup>
|
|
<col class="org-right" />
|
|
|
|
<col class="org-right" />
|
|
|
|
<col class="org-right" />
|
|
|
|
<col class="org-right" />
|
|
|
|
<col class="org-right" />
|
|
|
|
<col class="org-right" />
|
|
|
|
<col class="org-right" />
|
|
|
|
<col class="org-right" />
|
|
|
|
<col class="org-right" />
|
|
|
|
<col class="org-right" />
|
|
</colgroup>
|
|
<tbody>
|
|
<tr>
|
|
<td class="org-right">300000000.0</td>
|
|
<td class="org-right">-1000.0</td>
|
|
<td class="org-right">-30000.0</td>
|
|
<td class="org-right">-40.0</td>
|
|
<td class="org-right">70000.0</td>
|
|
<td class="org-right">300.0</td>
|
|
<td class="org-right">20000000.0</td>
|
|
<td class="org-right">-30.0</td>
|
|
<td class="org-right">-5000.0</td>
|
|
<td class="org-right">5</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-right">-1000.0</td>
|
|
<td class="org-right">50000000.0</td>
|
|
<td class="org-right">-7000.0</td>
|
|
<td class="org-right">800000.0</td>
|
|
<td class="org-right">-20.0</td>
|
|
<td class="org-right">300.0</td>
|
|
<td class="org-right">3000.0</td>
|
|
<td class="org-right">5000000.0</td>
|
|
<td class="org-right">400.0</td>
|
|
<td class="org-right">-40000.0</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-right">-30000.0</td>
|
|
<td class="org-right">-7000.0</td>
|
|
<td class="org-right">100000000.0</td>
|
|
<td class="org-right">-200.0</td>
|
|
<td class="org-right">-60.0</td>
|
|
<td class="org-right">70.0</td>
|
|
<td class="org-right">3000.0</td>
|
|
<td class="org-right">3000.0</td>
|
|
<td class="org-right">-8000000.0</td>
|
|
<td class="org-right">-30.0</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-right">-40.0</td>
|
|
<td class="org-right">800000.0</td>
|
|
<td class="org-right">-200.0</td>
|
|
<td class="org-right">20000.0</td>
|
|
<td class="org-right">-0.4</td>
|
|
<td class="org-right">4</td>
|
|
<td class="org-right">30.0</td>
|
|
<td class="org-right">40000.0</td>
|
|
<td class="org-right">7</td>
|
|
<td class="org-right">-300.0</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-right">70000.0</td>
|
|
<td class="org-right">-20.0</td>
|
|
<td class="org-right">-60.0</td>
|
|
<td class="org-right">-0.4</td>
|
|
<td class="org-right">3000.0</td>
|
|
<td class="org-right">1</td>
|
|
<td class="org-right">-6000.0</td>
|
|
<td class="org-right">10.0</td>
|
|
<td class="org-right">8</td>
|
|
<td class="org-right">-0.1</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-right">300.0</td>
|
|
<td class="org-right">300.0</td>
|
|
<td class="org-right">70.0</td>
|
|
<td class="org-right">4</td>
|
|
<td class="org-right">1</td>
|
|
<td class="org-right">40000.0</td>
|
|
<td class="org-right">-10.0</td>
|
|
<td class="org-right">-10.0</td>
|
|
<td class="org-right">30.0</td>
|
|
<td class="org-right">0.1</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-right">20000000.0</td>
|
|
<td class="org-right">3000.0</td>
|
|
<td class="org-right">3000.0</td>
|
|
<td class="org-right">30.0</td>
|
|
<td class="org-right">-6000.0</td>
|
|
<td class="org-right">-10.0</td>
|
|
<td class="org-right">300000000.0</td>
|
|
<td class="org-right">2000.0</td>
|
|
<td class="org-right">9000.0</td>
|
|
<td class="org-right">-100.0</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-right">-30.0</td>
|
|
<td class="org-right">5000000.0</td>
|
|
<td class="org-right">3000.0</td>
|
|
<td class="org-right">40000.0</td>
|
|
<td class="org-right">10.0</td>
|
|
<td class="org-right">-10.0</td>
|
|
<td class="org-right">2000.0</td>
|
|
<td class="org-right">50000000.0</td>
|
|
<td class="org-right">-3000.0</td>
|
|
<td class="org-right">-800000.0</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-right">-5000.0</td>
|
|
<td class="org-right">400.0</td>
|
|
<td class="org-right">-8000000.0</td>
|
|
<td class="org-right">7</td>
|
|
<td class="org-right">8</td>
|
|
<td class="org-right">30.0</td>
|
|
<td class="org-right">9000.0</td>
|
|
<td class="org-right">-3000.0</td>
|
|
<td class="org-right">100000000.0</td>
|
|
<td class="org-right">100.0</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-right">5</td>
|
|
<td class="org-right">-40000.0</td>
|
|
<td class="org-right">-30.0</td>
|
|
<td class="org-right">-300.0</td>
|
|
<td class="org-right">-0.1</td>
|
|
<td class="org-right">0.1</td>
|
|
<td class="org-right">-100.0</td>
|
|
<td class="org-right">-800000.0</td>
|
|
<td class="org-right">100.0</td>
|
|
<td class="org-right">20000.0</td>
|
|
</tr>
|
|
</tbody>
|
|
</table>
|
|
|
|
|
|
<table border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
|
|
<caption class="t-above"><span class="table-number">Table 3:</span> First 10x10 elements of the Mass matrix</caption>
|
|
|
|
<colgroup>
|
|
<col class="org-right" />
|
|
|
|
<col class="org-right" />
|
|
|
|
<col class="org-right" />
|
|
|
|
<col class="org-right" />
|
|
|
|
<col class="org-right" />
|
|
|
|
<col class="org-right" />
|
|
|
|
<col class="org-right" />
|
|
|
|
<col class="org-right" />
|
|
|
|
<col class="org-right" />
|
|
|
|
<col class="org-right" />
|
|
</colgroup>
|
|
<tbody>
|
|
<tr>
|
|
<td class="org-right">0.03</td>
|
|
<td class="org-right">7e-08</td>
|
|
<td class="org-right">2e-06</td>
|
|
<td class="org-right">-3e-09</td>
|
|
<td class="org-right">-0.0002</td>
|
|
<td class="org-right">-6e-08</td>
|
|
<td class="org-right">-0.001</td>
|
|
<td class="org-right">8e-07</td>
|
|
<td class="org-right">6e-07</td>
|
|
<td class="org-right">-8e-09</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-right">7e-08</td>
|
|
<td class="org-right">0.02</td>
|
|
<td class="org-right">-1e-06</td>
|
|
<td class="org-right">9e-05</td>
|
|
<td class="org-right">-3e-09</td>
|
|
<td class="org-right">-4e-09</td>
|
|
<td class="org-right">-1e-06</td>
|
|
<td class="org-right">-0.0006</td>
|
|
<td class="org-right">-4e-08</td>
|
|
<td class="org-right">5e-06</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-right">2e-06</td>
|
|
<td class="org-right">-1e-06</td>
|
|
<td class="org-right">0.02</td>
|
|
<td class="org-right">-3e-08</td>
|
|
<td class="org-right">-4e-08</td>
|
|
<td class="org-right">1e-08</td>
|
|
<td class="org-right">1e-07</td>
|
|
<td class="org-right">-2e-07</td>
|
|
<td class="org-right">0.0003</td>
|
|
<td class="org-right">1e-09</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-right">-3e-09</td>
|
|
<td class="org-right">9e-05</td>
|
|
<td class="org-right">-3e-08</td>
|
|
<td class="org-right">1e-06</td>
|
|
<td class="org-right">-3e-11</td>
|
|
<td class="org-right">-3e-13</td>
|
|
<td class="org-right">-7e-09</td>
|
|
<td class="org-right">-5e-06</td>
|
|
<td class="org-right">-3e-10</td>
|
|
<td class="org-right">3e-08</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-right">-0.0002</td>
|
|
<td class="org-right">-3e-09</td>
|
|
<td class="org-right">-4e-08</td>
|
|
<td class="org-right">-3e-11</td>
|
|
<td class="org-right">2e-06</td>
|
|
<td class="org-right">6e-10</td>
|
|
<td class="org-right">2e-06</td>
|
|
<td class="org-right">-7e-09</td>
|
|
<td class="org-right">-2e-09</td>
|
|
<td class="org-right">7e-11</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-right">-6e-08</td>
|
|
<td class="org-right">-4e-09</td>
|
|
<td class="org-right">1e-08</td>
|
|
<td class="org-right">-3e-13</td>
|
|
<td class="org-right">6e-10</td>
|
|
<td class="org-right">1e-06</td>
|
|
<td class="org-right">1e-08</td>
|
|
<td class="org-right">3e-09</td>
|
|
<td class="org-right">-2e-09</td>
|
|
<td class="org-right">2e-13</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-right">-0.001</td>
|
|
<td class="org-right">-1e-06</td>
|
|
<td class="org-right">1e-07</td>
|
|
<td class="org-right">-7e-09</td>
|
|
<td class="org-right">2e-06</td>
|
|
<td class="org-right">1e-08</td>
|
|
<td class="org-right">0.03</td>
|
|
<td class="org-right">4e-08</td>
|
|
<td class="org-right">-2e-06</td>
|
|
<td class="org-right">8e-09</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-right">8e-07</td>
|
|
<td class="org-right">-0.0006</td>
|
|
<td class="org-right">-2e-07</td>
|
|
<td class="org-right">-5e-06</td>
|
|
<td class="org-right">-7e-09</td>
|
|
<td class="org-right">3e-09</td>
|
|
<td class="org-right">4e-08</td>
|
|
<td class="org-right">0.02</td>
|
|
<td class="org-right">-9e-07</td>
|
|
<td class="org-right">-9e-05</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-right">6e-07</td>
|
|
<td class="org-right">-4e-08</td>
|
|
<td class="org-right">0.0003</td>
|
|
<td class="org-right">-3e-10</td>
|
|
<td class="org-right">-2e-09</td>
|
|
<td class="org-right">-2e-09</td>
|
|
<td class="org-right">-2e-06</td>
|
|
<td class="org-right">-9e-07</td>
|
|
<td class="org-right">0.02</td>
|
|
<td class="org-right">2e-08</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-right">-8e-09</td>
|
|
<td class="org-right">5e-06</td>
|
|
<td class="org-right">1e-09</td>
|
|
<td class="org-right">3e-08</td>
|
|
<td class="org-right">7e-11</td>
|
|
<td class="org-right">2e-13</td>
|
|
<td class="org-right">8e-09</td>
|
|
<td class="org-right">-9e-05</td>
|
|
<td class="org-right">2e-08</td>
|
|
<td class="org-right">1e-06</td>
|
|
</tr>
|
|
</tbody>
|
|
</table>
|
|
|
|
|
|
<p>
|
|
Then, we extract the coordinates of the interface nodes.
|
|
</p>
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">[int_xyz, int_i, n_xyz, n_i, nodes] = extractNodes(<span class="org-string">'APA95ML_out_nodes_3D.txt'</span>);
|
|
</pre>
|
|
</div>
|
|
|
|
<p>
|
|
The interface nodes are shown in Figure <a href="#org7dd0e69">9</a> and their coordinates are listed in Table <a href="#orgc8e88d6">4</a>.
|
|
</p>
|
|
|
|
<table border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
|
|
|
|
|
|
<colgroup>
|
|
<col class="org-left" />
|
|
|
|
<col class="org-right" />
|
|
</colgroup>
|
|
<tbody>
|
|
<tr>
|
|
<td class="org-left">Total number of Nodes</td>
|
|
<td class="org-right">7</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-left">Number of interface Nodes</td>
|
|
<td class="org-right">7</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-left">Number of Modes</td>
|
|
<td class="org-right">6</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-left">Size of M and K matrices</td>
|
|
<td class="org-right">48</td>
|
|
</tr>
|
|
</tbody>
|
|
</table>
|
|
|
|
<table id="orgc8e88d6" border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
|
|
<caption class="t-above"><span class="table-number">Table 4:</span> Coordinates of the interface nodes</caption>
|
|
|
|
<colgroup>
|
|
<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-right">Node i</th>
|
|
<th scope="col" class="org-right">Node Number</th>
|
|
<th scope="col" class="org-right">x [m]</th>
|
|
<th scope="col" class="org-right">y [m]</th>
|
|
<th scope="col" class="org-right">z [m]</th>
|
|
</tr>
|
|
</thead>
|
|
<tbody>
|
|
<tr>
|
|
<td class="org-right">1.0</td>
|
|
<td class="org-right">40467.0</td>
|
|
<td class="org-right">0.0</td>
|
|
<td class="org-right">0.0</td>
|
|
<td class="org-right">0.029997</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-right">2.0</td>
|
|
<td class="org-right">40469.0</td>
|
|
<td class="org-right">0.0</td>
|
|
<td class="org-right">0.0</td>
|
|
<td class="org-right">-0.029997</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-right">3.0</td>
|
|
<td class="org-right">40470.0</td>
|
|
<td class="org-right">-0.035</td>
|
|
<td class="org-right">0.0</td>
|
|
<td class="org-right">0.0</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-right">4.0</td>
|
|
<td class="org-right">40475.0</td>
|
|
<td class="org-right">-0.015</td>
|
|
<td class="org-right">0.0</td>
|
|
<td class="org-right">0.0</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-right">5.0</td>
|
|
<td class="org-right">40476.0</td>
|
|
<td class="org-right">-0.005</td>
|
|
<td class="org-right">0.0</td>
|
|
<td class="org-right">0.0</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-right">6.0</td>
|
|
<td class="org-right">40477.0</td>
|
|
<td class="org-right">0.015</td>
|
|
<td class="org-right">0.0</td>
|
|
<td class="org-right">0.0</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td class="org-right">7.0</td>
|
|
<td class="org-right">40478.0</td>
|
|
<td class="org-right">0.035</td>
|
|
<td class="org-right">0.0</td>
|
|
<td class="org-right">0.0</td>
|
|
</tr>
|
|
</tbody>
|
|
</table>
|
|
|
|
|
|
<div id="org7dd0e69" class="figure">
|
|
<p><img src="figs/APA95ML_nodes_1.png" alt="APA95ML_nodes_1.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 9: </span>Interface Nodes chosen for the APA95ML</p>
|
|
</div>
|
|
|
|
<p>
|
|
Using <code>K</code>, <code>M</code> and <code>int_xyz</code>, we can use the <code>Reduced Order Flexible Solid</code> simscape block.
|
|
</p>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-orga49ef8e" class="outline-3">
|
|
<h3 id="orga49ef8e"><span class="section-number-3">3.2</span> Simscape Model</h3>
|
|
<div class="outline-text-3" id="text-3-2">
|
|
<p>
|
|
The flexible element is imported using the <code>Reduced Order Flexible Solid</code> Simscape block.
|
|
</p>
|
|
|
|
<p>
|
|
To model the actuator, an <code>Internal Force</code> block is added between the nodes 3 and 12.
|
|
A <code>Relative Motion Sensor</code> block is added between the nodes 1 and 2 to measure the displacement and the amplified piezo.
|
|
</p>
|
|
|
|
<p>
|
|
One mass is fixed at one end of the piezo-electric stack actuator, the other end is fixed to the world frame.
|
|
</p>
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">m = 5;
|
|
</pre>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-orgf1f3f75" class="outline-3">
|
|
<h3 id="orgf1f3f75"><span class="section-number-3">3.3</span> Dynamics from Actuator Voltage to Vertical Mass Displacement</h3>
|
|
<div class="outline-text-3" id="text-3-3">
|
|
<p>
|
|
The identified dynamics is shown in Figure <a href="#orgd2ca2ca">10</a>.
|
|
</p>
|
|
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab"><span class="org-matlab-cellbreak"><span class="org-comment">%% Name of the Simulink File</span></span>
|
|
mdl = <span class="org-string">'piezo_amplified_3d'</span>;
|
|
|
|
<span class="org-matlab-cellbreak"><span class="org-comment">%% Input/Output definition</span></span>
|
|
clear io; io_i = 1;
|
|
io(io_i) = linio([mdl, <span class="org-string">'/Va'</span>], 1, <span class="org-string">'openinput'</span>); io_i = io_i <span class="org-type">+</span> 1; <span class="org-comment">% Actuator Voltage [V]</span>
|
|
io(io_i) = linio([mdl, <span class="org-string">'/y'</span>], 1, <span class="org-string">'openoutput'</span>); io_i = io_i <span class="org-type">+</span> 1; <span class="org-comment">% Vertical Displacement [m]</span>
|
|
|
|
Ghm = <span class="org-type">-</span>linearize(mdl, io);
|
|
</pre>
|
|
</div>
|
|
|
|
|
|
<div id="orgd2ca2ca" class="figure">
|
|
<p><img src="figs/dynamics_act_disp_comp_mass.png" alt="dynamics_act_disp_comp_mass.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 10: </span>Dynamics from \(F\) to \(d\) without a payload and with a 5kg payload</p>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-orgd4a750c" class="outline-3">
|
|
<h3 id="orgd4a750c"><span class="section-number-3">3.4</span> Dynamics from Actuator Voltage to Force Sensor Voltage</h3>
|
|
<div class="outline-text-3" id="text-3-4">
|
|
<p>
|
|
The obtained dynamics is shown in Figure <a href="#orgb135de4">11</a>.
|
|
</p>
|
|
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab"><span class="org-matlab-cellbreak"><span class="org-comment">%% Name of the Simulink File</span></span>
|
|
mdl = <span class="org-string">'piezo_amplified_3d'</span>;
|
|
|
|
<span class="org-matlab-cellbreak"><span class="org-comment">%% Input/Output definition</span></span>
|
|
clear io; io_i = 1;
|
|
io(io_i) = linio([mdl, <span class="org-string">'/Va'</span>], 1, <span class="org-string">'openinput'</span>); io_i = io_i <span class="org-type">+</span> 1; <span class="org-comment">% Voltage Actuator [V]</span>
|
|
io(io_i) = linio([mdl, <span class="org-string">'/Vs'</span>], 1, <span class="org-string">'openoutput'</span>); io_i = io_i <span class="org-type">+</span> 1; <span class="org-comment">% Sensor Voltage [V]</span>
|
|
|
|
Gfm = linearize(mdl, io);
|
|
</pre>
|
|
</div>
|
|
|
|
|
|
<div id="orgb135de4" class="figure">
|
|
<p><img src="figs/dynamics_force_force_sensor_comp_mass.png" alt="dynamics_force_force_sensor_comp_mass.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 11: </span>Dynamics from \(F\) to \(F_m\) for \(m=0\) and \(m = 10kg\)</p>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-org1ac4d3c" class="outline-3">
|
|
<h3 id="org1ac4d3c"><span class="section-number-3">3.5</span> Save Data for further use</h3>
|
|
<div class="outline-text-3" id="text-3-5">
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">save(<span class="org-string">'matlab/mat/fem_simscape_models.mat'</span>, <span class="org-string">'Ghm'</span>, <span class="org-string">'Gfm'</span>)
|
|
</pre>
|
|
</div>
|
|
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">save(<span class="org-string">'mat/fem_simscape_models.mat'</span>, <span class="org-string">'Ghm'</span>, <span class="org-string">'Gfm'</span>)
|
|
</pre>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-org9de639d" class="outline-2">
|
|
<h2 id="org9de639d"><span class="section-number-2">4</span> Huddle Test</h2>
|
|
<div class="outline-text-2" id="text-4">
|
|
<p>
|
|
<a id="org9acdf87"></a>
|
|
</p>
|
|
</div>
|
|
<div id="outline-container-orgaf93c0c" class="outline-3">
|
|
<h3 id="orgaf93c0c"><span class="section-number-3">4.1</span> Time Domain Data</h3>
|
|
<div class="outline-text-3" id="text-4-1">
|
|
|
|
<div id="orgd048822" class="figure">
|
|
<p><img src="figs/huddle_test_time_domain.png" alt="huddle_test_time_domain.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 12: </span>Measurement of the Mass displacement during Huddle Test</p>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-org22ccdcf" class="outline-3">
|
|
<h3 id="org22ccdcf"><span class="section-number-3">4.2</span> PSD of Measurement Noise</h3>
|
|
<div class="outline-text-3" id="text-4-2">
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">Ts = t(end)<span class="org-type">/</span>(length(t)<span class="org-type">-</span>1);
|
|
Fs = 1<span class="org-type">/</span>Ts;
|
|
|
|
win = hanning(ceil(1<span class="org-type">*</span>Fs));
|
|
</pre>
|
|
</div>
|
|
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">[pxx, f] = pwelch(y(1000<span class="org-type">:</span>end), win, [], [], Fs);
|
|
</pre>
|
|
</div>
|
|
|
|
|
|
<div id="orgd309d3e" class="figure">
|
|
<p><img src="figs/huddle_test_pdf.png" alt="huddle_test_pdf.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 13: </span>Amplitude Spectral Density of the Displacement during Huddle Test</p>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-org914a7da" class="outline-2">
|
|
<h2 id="org914a7da"><span class="section-number-2">5</span> Identification of the dynamics from actuator to displacement</h2>
|
|
<div class="outline-text-2" id="text-5">
|
|
<p>
|
|
<a id="org8cacbcf"></a>
|
|
</p>
|
|
<ul class="org-ul">
|
|
<li class="off"><code>[ ]</code> List of equipment</li>
|
|
<li class="off"><code>[ ]</code> Schematic</li>
|
|
<li class="off"><code>[ ]</code> Problem of matching between the models? (there is a factor 10)</li>
|
|
</ul>
|
|
|
|
<p>
|
|
E505 with gain of 10.
|
|
</p>
|
|
</div>
|
|
<div id="outline-container-orga1aace1" class="outline-3">
|
|
<h3 id="orga1aace1"><span class="section-number-3">5.1</span> Load Data</h3>
|
|
<div class="outline-text-3" id="text-5-1">
|
|
<p>
|
|
The data from the “noise test” and the identification test are loaded.
|
|
</p>
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">ht = load(<span class="org-string">'huddle_test.mat'</span>, <span class="org-string">'t'</span>, <span class="org-string">'u'</span>, <span class="org-string">'y'</span>);
|
|
load(<span class="org-string">'apa95ml_5kg_Amp_E505.mat'</span>, <span class="org-string">'t'</span>, <span class="org-string">'um'</span>, <span class="org-string">'y'</span>);
|
|
</pre>
|
|
</div>
|
|
|
|
<p>
|
|
Any offset value is removed:
|
|
</p>
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">um = detrend(um, 0); <span class="org-comment">% Input Voltage [V]</span>
|
|
y = detrend(y , 0); <span class="org-comment">% Mass displacement [m]</span>
|
|
|
|
ht.u = detrend(ht.u, 0);
|
|
ht.y = detrend(ht.y, 0);
|
|
</pre>
|
|
</div>
|
|
|
|
<p>
|
|
Now we add a factor 10 to take into account the gain of the voltage amplifier.
|
|
</p>
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">um = 10<span class="org-type">*</span>um;
|
|
ht.u = 10<span class="org-type">*</span>ht.u;
|
|
</pre>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-org1284757" class="outline-3">
|
|
<h3 id="org1284757"><span class="section-number-3">5.2</span> Comparison of the PSD with Huddle Test</h3>
|
|
<div class="outline-text-3" id="text-5-2">
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">Ts = t(end)<span class="org-type">/</span>(length(t)<span class="org-type">-</span>1);
|
|
Fs = 1<span class="org-type">/</span>Ts;
|
|
|
|
win = hanning(ceil(1<span class="org-type">*</span>Fs));
|
|
</pre>
|
|
</div>
|
|
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">[pxx, f] = pwelch(y, win, [], [], Fs);
|
|
[pht, <span class="org-type">~</span>] = pwelch(ht.y, win, [], [], Fs);
|
|
</pre>
|
|
</div>
|
|
|
|
|
|
<div id="org87d1b81" class="figure">
|
|
<p><img src="figs/apa95ml_5kg_PI_pdf_comp_huddle.png" alt="apa95ml_5kg_PI_pdf_comp_huddle.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 14: </span>Comparison of the ASD for the identification test and the huddle test</p>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-org89453e3" class="outline-3">
|
|
<h3 id="org89453e3"><span class="section-number-3">5.3</span> Compute TF estimate and Coherence</h3>
|
|
<div class="outline-text-3" id="text-5-3">
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">[tf_est, f] = tfestimate(um, <span class="org-type">-</span>y, win, [], [], 1<span class="org-type">/</span>Ts);
|
|
[co_est, <span class="org-type">~</span>] = mscohere( um, <span class="org-type">-</span>y, win, [], [], 1<span class="org-type">/</span>Ts);
|
|
</pre>
|
|
</div>
|
|
|
|
|
|
<div id="org69f52ac" class="figure">
|
|
<p><img src="figs/apa95ml_5kg_PI_coh.png" alt="apa95ml_5kg_PI_coh.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 15: </span>Coherence</p>
|
|
</div>
|
|
|
|
<p>
|
|
Comparison with the FEM model
|
|
</p>
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">load(<span class="org-string">'mat/fem_simscape_models.mat'</span>, <span class="org-string">'Ghm'</span>);
|
|
</pre>
|
|
</div>
|
|
|
|
|
|
<div id="orgac63fe3" class="figure">
|
|
<p><img src="figs/apa95ml_5kg_pi_comp_fem.png" alt="apa95ml_5kg_pi_comp_fem.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 16: </span>Comparison of the identified transfer function and the one estimated from the FE model</p>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-orgfff18d2" class="outline-2">
|
|
<h2 id="orgfff18d2"><span class="section-number-2">6</span> Identification of the dynamics from actuator to force sensor</h2>
|
|
<div class="outline-text-2" id="text-6">
|
|
<p>
|
|
<a id="org2eda6f2"></a>
|
|
</p>
|
|
<p>
|
|
Two measurements are performed:
|
|
</p>
|
|
<ul class="org-ul">
|
|
<li>Speedgoat DAC => Voltage Amplifier (x20) => 1 Piezo Stack => … => 2 Stacks as Force Sensor (parallel) => Speedgoat ADC</li>
|
|
<li>Speedgoat DAC => Voltage Amplifier (x20) => 2 Piezo Stacks (parallel) => … => 1 Stack as Force Sensor => Speedgoat ADC</li>
|
|
</ul>
|
|
|
|
<p>
|
|
The obtained dynamics from force actuator to force sensor are compare with the FEM model.
|
|
</p>
|
|
<p>
|
|
The data are loaded:
|
|
</p>
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">load(<span class="org-string">'apa95ml_5kg_2a_1s.mat'</span>, <span class="org-string">'t'</span>, <span class="org-string">'u'</span>, <span class="org-string">'v'</span>);
|
|
</pre>
|
|
</div>
|
|
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">u = detrend(u, 0);
|
|
v = detrend(v, 0);
|
|
</pre>
|
|
</div>
|
|
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">u = 20<span class="org-type">*</span>u;
|
|
</pre>
|
|
</div>
|
|
<p>
|
|
Let’s use the amplifier gain to obtain the true voltage applied to the actuator stack(s)
|
|
</p>
|
|
|
|
<p>
|
|
The parameters of the piezoelectric stacks are defined below:
|
|
</p>
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">d33 = 3e<span class="org-type">-</span>10; <span class="org-comment">% Strain constant [m/V]</span>
|
|
n = 80; <span class="org-comment">% Number of layers per stack</span>
|
|
eT = 1.6e<span class="org-type">-</span>8; <span class="org-comment">% Permittivity under constant stress [F/m]</span>
|
|
sD = 2e<span class="org-type">-</span>11; <span class="org-comment">% Elastic compliance under constant electric displacement [m2/N]</span>
|
|
ka = 235e6; <span class="org-comment">% Stack stiffness [N/m]</span>
|
|
</pre>
|
|
</div>
|
|
|
|
<p>
|
|
From the FEM, we construct the transfer function from DAC voltage to ADC voltage.
|
|
</p>
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">Gfem_aa_s = exp(<span class="org-type">-</span>s<span class="org-type">/</span>1e4)<span class="org-type">*</span>20<span class="org-type">*</span>(2<span class="org-type">*</span>d33<span class="org-type">*</span>n<span class="org-type">*</span>ka)<span class="org-type">*</span>(G(3,1)<span class="org-type">+</span>G(3,2))<span class="org-type">*</span>d33<span class="org-type">/</span>(eT<span class="org-type">*</span>sD<span class="org-type">*</span>n);
|
|
Gfem_a_ss = exp(<span class="org-type">-</span>s<span class="org-type">/</span>1e4)<span class="org-type">*</span>20<span class="org-type">*</span>( d33<span class="org-type">*</span>n<span class="org-type">*</span>ka)<span class="org-type">*</span>(G(3,1)<span class="org-type">+</span>G(2,1))<span class="org-type">*</span>d33<span class="org-type">/</span>(eT<span class="org-type">*</span>sD<span class="org-type">*</span>n);
|
|
</pre>
|
|
</div>
|
|
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">Gfem_aa_s = exp(<span class="org-type">-</span>s<span class="org-type">/</span>1e4)<span class="org-type">*</span>20<span class="org-type">*</span>(2<span class="org-type">*</span>d33<span class="org-type">*</span>n<span class="org-type">*</span>ka)<span class="org-type">*</span>Gfm<span class="org-type">*</span>d33<span class="org-type">/</span>(eT<span class="org-type">*</span>sD<span class="org-type">*</span>n);
|
|
Gfem_a_ss = exp(<span class="org-type">-</span>s<span class="org-type">/</span>1e4)<span class="org-type">*</span>20<span class="org-type">*</span>( d33<span class="org-type">*</span>n<span class="org-type">*</span>ka)<span class="org-type">*</span>Gfm<span class="org-type">*</span>d33<span class="org-type">/</span>(eT<span class="org-type">*</span>sD<span class="org-type">*</span>n);
|
|
</pre>
|
|
</div>
|
|
<p>
|
|
The transfer function from input voltage to output voltage are computed and shown in Figure <a href="#org157bd9b">17</a>.
|
|
</p>
|
|
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">Ts = t(end)<span class="org-type">/</span>(length(t)<span class="org-type">-</span>1);
|
|
Fs = 1<span class="org-type">/</span>Ts;
|
|
|
|
win = hann(ceil(10<span class="org-type">/</span>Ts));
|
|
|
|
[tf_est, f] = tfestimate(u, v, win, [], [], 1<span class="org-type">/</span>Ts);
|
|
[coh, <span class="org-type">~</span>] = mscohere( u, v, win, [], [], 1<span class="org-type">/</span>Ts);
|
|
</pre>
|
|
</div>
|
|
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">load(<span class="org-string">'mat/fem_simscape_models.mat'</span>, <span class="org-string">'Gfm'</span>);
|
|
</pre>
|
|
</div>
|
|
|
|
|
|
<div id="org157bd9b" class="figure">
|
|
<p><img src="figs/bode_plot_force_sensor_voltage_comp_fem.png" alt="bode_plot_force_sensor_voltage_comp_fem.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 17: </span>Comparison of the identified dynamics from voltage output to voltage input and the FEM</p>
|
|
</div>
|
|
</div>
|
|
<div id="outline-container-orgfa2171a" class="outline-3">
|
|
<h3 id="orgfa2171a"><span class="section-number-3">6.1</span> System Identification</h3>
|
|
<div class="outline-text-3" id="text-6-1">
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">w_z = 2<span class="org-type">*</span><span class="org-constant">pi</span><span class="org-type">*</span>111; <span class="org-comment">% Zeros frequency [rad/s]</span>
|
|
w_p = 2<span class="org-type">*</span><span class="org-constant">pi</span><span class="org-type">*</span>255; <span class="org-comment">% Pole frequency [rad/s]</span>
|
|
xi_z = 0.05;
|
|
xi_p = 0.015;
|
|
G_inf = 0.1;
|
|
|
|
Gi = G_inf<span class="org-type">*</span>(s<span class="org-type">^</span>2 <span class="org-type">-</span> 2<span class="org-type">*</span>xi_z<span class="org-type">*</span>w_z<span class="org-type">*</span>s <span class="org-type">+</span> w_z<span class="org-type">^</span>2)<span class="org-type">/</span>(s<span class="org-type">^</span>2 <span class="org-type">+</span> 2<span class="org-type">*</span>xi_p<span class="org-type">*</span>w_p<span class="org-type">*</span>s <span class="org-type">+</span> w_p<span class="org-type">^</span>2);
|
|
</pre>
|
|
</div>
|
|
|
|
|
|
<div id="orgf552545" class="figure">
|
|
<p><img src="figs/iff_plant_identification_apa95ml.png" alt="iff_plant_identification_apa95ml.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 18: </span>Identification of the IFF plant</p>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
|
|
<div id="outline-container-org633e9f1" class="outline-3">
|
|
<h3 id="org633e9f1"><span class="section-number-3">6.2</span> Integral Force Feedback</h3>
|
|
<div class="outline-text-3" id="text-6-2">
|
|
|
|
<div id="orgca6756f" class="figure">
|
|
<p><img src="figs/root_locus_iff_apa95ml_identification.png" alt="root_locus_iff_apa95ml_identification.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 19: </span>Root Locus for IFF</p>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-org1bf58e2" class="outline-2">
|
|
<h2 id="org1bf58e2"><span class="section-number-2">7</span> Integral Force Feedback</h2>
|
|
<div class="outline-text-2" id="text-7">
|
|
<p>
|
|
<a id="org6db5225"></a>
|
|
</p>
|
|
|
|
<div id="org86a6667" class="figure">
|
|
<p><img src="figs/test_bench_apa_schematic_iff.png" alt="test_bench_apa_schematic_iff.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 20: </span>Schematic of the test bench using IFF</p>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-org25092d8" class="outline-3">
|
|
<h3 id="org25092d8"><span class="section-number-3">7.1</span> First tests with few gains</h3>
|
|
<div class="outline-text-3" id="text-7-1">
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">iff_g10 = load(<span class="org-string">'apa95ml_iff_g10_res.mat'</span>, <span class="org-string">'u'</span>, <span class="org-string">'t'</span>, <span class="org-string">'y'</span>, <span class="org-string">'v'</span>);
|
|
iff_g100 = load(<span class="org-string">'apa95ml_iff_g100_res.mat'</span>, <span class="org-string">'u'</span>, <span class="org-string">'t'</span>, <span class="org-string">'y'</span>, <span class="org-string">'v'</span>);
|
|
iff_of = load(<span class="org-string">'apa95ml_iff_off_res.mat'</span>, <span class="org-string">'u'</span>, <span class="org-string">'t'</span>, <span class="org-string">'y'</span>, <span class="org-string">'v'</span>);
|
|
</pre>
|
|
</div>
|
|
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">Ts = 1e<span class="org-type">-</span>4;
|
|
win = hann(ceil(10<span class="org-type">/</span>Ts));
|
|
|
|
[tf_iff_g10, f] = tfestimate(iff_g10.u, iff_g10.y, win, [], [], 1<span class="org-type">/</span>Ts);
|
|
[co_iff_g10, <span class="org-type">~</span>] = mscohere(iff_g10.u, iff_g10.y, win, [], [], 1<span class="org-type">/</span>Ts);
|
|
|
|
[tf_iff_g100, <span class="org-type">~</span>] = tfestimate(iff_g100.u, iff_g100.y, win, [], [], 1<span class="org-type">/</span>Ts);
|
|
[co_iff_g100, <span class="org-type">~</span>] = mscohere(iff_g100.u, iff_g100.y, win, [], [], 1<span class="org-type">/</span>Ts);
|
|
|
|
[tf_iff_of, <span class="org-type">~</span>] = tfestimate(iff_of.u, iff_of.y, win, [], [], 1<span class="org-type">/</span>Ts);
|
|
[co_iff_of, <span class="org-type">~</span>] = mscohere(iff_of.u, iff_of.y, win, [], [], 1<span class="org-type">/</span>Ts);
|
|
</pre>
|
|
</div>
|
|
|
|
|
|
<div id="org2da6c81" class="figure">
|
|
<p><img src="figs/iff_first_test_coherence.png" alt="iff_first_test_coherence.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 21: </span>Coherence</p>
|
|
</div>
|
|
|
|
|
|
|
|
<div id="org407e7b6" class="figure">
|
|
<p><img src="figs/iff_first_test_bode_plot.png" alt="iff_first_test_bode_plot.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 22: </span>Bode plot for different values of IFF gain</p>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-org67fa466" class="outline-3">
|
|
<h3 id="org67fa466"><span class="section-number-3">7.2</span> Second test with many Gains</h3>
|
|
<div class="outline-text-3" id="text-7-2">
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">load(<span class="org-string">'apa95ml_iff_test.mat'</span>, <span class="org-string">'results'</span>);
|
|
</pre>
|
|
</div>
|
|
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">Ts = 1e<span class="org-type">-</span>4;
|
|
win = hann(ceil(10<span class="org-type">/</span>Ts));
|
|
</pre>
|
|
</div>
|
|
|
|
<div class="org-src-container">
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<pre class="src src-matlab">tf_iff = {zeros(1, length(results))};
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co_iff = {zeros(1, length(results))};
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g_iff = [0, 1, 5, 10, 50, 100];
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<span class="org-keyword">for</span> <span class="org-variable-name"><span class="org-constant">i</span></span>=<span class="org-constant">1:length(results)</span>
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[tf_est, f] = tfestimate(results{<span class="org-constant">i</span>}.u, results{<span class="org-constant">i</span>}.y, win, [], [], 1<span class="org-type">/</span>Ts);
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[co_est, <span class="org-type">~</span>] = mscohere(results{<span class="org-constant">i</span>}.u, results{<span class="org-constant">i</span>}.y, win, [], [], 1<span class="org-type">/</span>Ts);
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tf_iff(<span class="org-constant">i</span>) = {tf_est};
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co_iff(<span class="org-constant">i</span>) = {co_est};
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<span class="org-keyword">end</span>
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</pre>
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</div>
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<div id="orgfa8bc3c" class="figure">
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<p><img src="figs/iff_results_bode_plots.png" alt="iff_results_bode_plots.png" />
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</p>
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</div>
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<div class="org-src-container">
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<pre class="src src-matlab">G_id = {zeros(1,length(results))};
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f_start = 70; <span class="org-comment">% [Hz]</span>
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f_end = 500; <span class="org-comment">% [Hz]</span>
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<span class="org-keyword">for</span> <span class="org-variable-name"><span class="org-constant">i</span></span> = <span class="org-constant">1:length(results)</span>
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tf_id = tf_iff{<span class="org-constant">i</span>}(sum(f<span class="org-type"><</span>f_start)<span class="org-type">:</span>length(f)<span class="org-type">-</span>sum(f<span class="org-type">></span>f_end));
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f_id = f(sum(f<span class="org-type"><</span>f_start)<span class="org-type">:</span>length(f)<span class="org-type">-</span>sum(f<span class="org-type">></span>f_end));
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gfr = idfrd(tf_id, 2<span class="org-type">*</span><span class="org-constant">pi</span><span class="org-type">*</span>f_id, Ts);
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G_id(<span class="org-constant">i</span>) = {procest(gfr,<span class="org-string">'P2UDZ'</span>)};
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<span class="org-keyword">end</span>
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</pre>
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</div>
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<div id="org62b4b70" class="figure">
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<p><img src="figs/iff_results_bode_plots_identification.png" alt="iff_results_bode_plots_identification.png" />
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</p>
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</div>
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<div id="org49a571f" class="figure">
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<p><img src="figs/iff_results_root_locus.png" alt="iff_results_root_locus.png" />
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</p>
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</div>
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<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><h2 class='citeproc-org-bib-h2'>Bibliography</h2>
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<div class="csl-bib-body">
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<div class="csl-entry"><a name="citeproc_bib_item_1"></a>Fleming, Andrew J., and Kam K. Leang. 2014. <i>Design, Modeling and Control of Nanopositioning Systems</i>. Advances in Industrial Control. Springer International Publishing. <a href="https://doi.org/10.1007/978-3-319-06617-2">https://doi.org/10.1007/978-3-319-06617-2</a>.</div>
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</div>
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</div>
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</div>
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</div>
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</div>
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|
<div id="postamble" class="status">
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<p class="author">Author: Dehaeze Thomas</p>
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<p class="date">Created: 2020-11-24 mar. 13:54</p>
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</div>
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</body>
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</html>
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