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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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<!-- 2019-08-14 mer. 12:13 -->
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<!-- 2019-08-21 mer. 13:17 -->
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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>On the Design of Complementary Filters for Control - Computation with Matlab</title>
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@ -279,52 +279,52 @@ 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="#org8101fe3">1. Optimal Sensor Fusion for noise characteristics</a>
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<li><a href="#orga1a1b9d">1. Optimal Sensor Fusion for noise characteristics</a>
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<ul>
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<li><a href="#orgda648ce">1.1. Architecture</a></li>
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<li><a href="#org7737a79">1.2. Noise of the sensors</a></li>
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<li><a href="#orgf85a7f2">1.3. H-Two Synthesis</a></li>
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<li><a href="#org9fcdb98">1.4. Analysis</a></li>
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<li><a href="#orga3597f6">1.1. Architecture</a></li>
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<li><a href="#orgcac1aa3">1.2. Noise of the sensors</a></li>
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<li><a href="#orgdb1ca76">1.3. H-Two Synthesis</a></li>
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<li><a href="#org375cb3a">1.4. Analysis</a></li>
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</ul>
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</li>
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<li><a href="#org7d49713">2. Robustness to sensor dynamics uncertainty</a>
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<li><a href="#org2ebd870">2. Robustness to sensor dynamics uncertainty</a>
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<ul>
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<li><a href="#orgc1b9515">2.1. Unknown sensor dynamics dynamics</a></li>
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<li><a href="#org5df7619">2.2. Design the complementary filters in order to limit the phase and gain uncertainty of the super sensor</a></li>
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<li><a href="#org0328fd7">2.3. First Basic Example with gain mismatch</a></li>
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<li><a href="#org81e622c">2.4. <span class="todo TODO">TODO</span> More Complete example with model uncertainty</a></li>
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<li><a href="#orgf63dc07">2.1. Unknown sensor dynamics dynamics</a></li>
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<li><a href="#org3244858">2.2. Design the complementary filters in order to limit the phase and gain uncertainty of the super sensor</a></li>
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<li><a href="#org20ae980">2.3. First Basic Example with gain mismatch</a></li>
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<li><a href="#org41f2b11">2.4. <span class="todo TODO">TODO</span> More Complete example with model uncertainty</a></li>
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</ul>
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</li>
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<li><a href="#org6d8c8af">3. Complementary filters using analytical formula</a>
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<li><a href="#orgf04a9a4">3. Complementary filters using analytical formula</a>
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<ul>
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<li><a href="#orgb288402">3.1. Analytical 1st order complementary filters</a></li>
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<li><a href="#orgefd1169">3.2. Second Order Complementary Filters</a></li>
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<li><a href="#org32c14e8">3.3. Third Order Complementary Filters</a></li>
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<li><a href="#org7d339a9">3.1. Analytical 1st order complementary filters</a></li>
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<li><a href="#org8edbedb">3.2. Second Order Complementary Filters</a></li>
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<li><a href="#org5fd0c3c">3.3. Third Order Complementary Filters</a></li>
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</ul>
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</li>
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<li><a href="#orgc86d54b">4. H-Infinity synthesis of complementary filters</a>
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<li><a href="#org88004e2">4. H-Infinity synthesis of complementary filters</a>
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<ul>
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<li><a href="#org2288026">4.1. Synthesis Architecture</a></li>
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<li><a href="#orga9b0474">4.2. Weights</a></li>
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<li><a href="#org17ecd06">4.3. H-Infinity Synthesis</a></li>
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<li><a href="#org844c90a">4.4. Obtained Complementary Filters</a></li>
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<li><a href="#orgc51b2bd">4.1. Synthesis Architecture</a></li>
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<li><a href="#orga135818">4.2. Weights</a></li>
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<li><a href="#orgbd3b8d8">4.3. H-Infinity Synthesis</a></li>
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<li><a href="#org0d62ef6">4.4. Obtained Complementary Filters</a></li>
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</ul>
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</li>
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<li><a href="#orgd76dc0a">5. Feedback Control Architecture to generate Complementary Filters</a>
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<li><a href="#orga65dbb6">5. Feedback Control Architecture to generate Complementary Filters</a>
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<ul>
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<li><a href="#org2b0a54f">5.1. Architecture</a></li>
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<li><a href="#org023d478">5.2. Loop Gain Design</a></li>
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<li><a href="#orgef15574">5.3. Complementary Filters Obtained</a></li>
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<li><a href="#orgec7fc24">5.1. Architecture</a></li>
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<li><a href="#org448cd20">5.2. Loop Gain Design</a></li>
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<li><a href="#org1ec0e07">5.3. Complementary Filters Obtained</a></li>
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</ul>
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</li>
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<li><a href="#org9a4e0ed">6. Analytical Formula found in the literature</a>
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<li><a href="#org3b324a8">6. Analytical Formula found in the literature</a>
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<ul>
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<li><a href="#orgfa54829">6.1. Analytical Formula</a></li>
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<li><a href="#org5286b79">6.2. Matlab</a></li>
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<li><a href="#org6092cef">6.3. Discussion</a></li>
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<li><a href="#org8d499d4">6.1. Analytical Formula</a></li>
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<li><a href="#org60e2ce0">6.2. Matlab</a></li>
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<li><a href="#org1038fe9">6.3. Discussion</a></li>
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</ul>
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</li>
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<li><a href="#orgbbfeb6f">7. Comparison of the different methods of synthesis</a></li>
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<li><a href="#org647d1a0">7. Comparison of the different methods of synthesis</a></li>
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</ul>
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</div>
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</div>
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@ -344,31 +344,31 @@ To achieve this, the sensors included in the filter should complement one anothe
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</blockquote>
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<ul class="org-ul">
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<li>in section <a href="#orgc9381ca">1</a>, the optimal design of the complementary filters in order to obtain the lowest resulting "super sensor" noise is studied</li>
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<li>in section <a href="#orge83d075">1</a>, the optimal design of the complementary filters in order to obtain the lowest resulting "super sensor" noise is studied</li>
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</ul>
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<p>
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When blending two sensors using complementary filters with unknown dynamics, phase lag may be introduced that renders the close-loop system unstable.
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</p>
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<ul class="org-ul">
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<li>in section <a href="#orgc67c206">2</a>, the blending robustness to sensor dynamic uncertainty is studied.</li>
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<li>in section <a href="#orgb2ab616">2</a>, the blending robustness to sensor dynamic uncertainty is studied.</li>
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</ul>
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<p>
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Then, three design methods for generating two complementary filters are proposed:
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</p>
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<ul class="org-ul">
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<li>in section <a href="#org91a8313">3</a>, analytical formulas are proposed</li>
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<li>in section <a href="#org2358787">4</a>, the \(\mathcal{H}_\infty\) synthesis is used</li>
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<li>in section <a href="#org94d0d5f">5</a>, the classical feedback architecture is used</li>
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<li>in section <a href="#orga144d76">6</a>, analytical formulas found in the literature are listed</li>
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<li>in section <a href="#org654b360">3</a>, analytical formulas are proposed</li>
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<li>in section <a href="#org924f36f">4</a>, the \(\mathcal{H}_\infty\) synthesis is used</li>
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<li>in section <a href="#org3e654dc">5</a>, the classical feedback architecture is used</li>
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<li>in section <a href="#org22f6020">6</a>, analytical formulas found in the literature are listed</li>
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</ul>
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<div id="outline-container-org8101fe3" class="outline-2">
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<h2 id="org8101fe3"><span class="section-number-2">1</span> Optimal Sensor Fusion for noise characteristics</h2>
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<div id="outline-container-orga1a1b9d" class="outline-2">
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<h2 id="orga1a1b9d"><span class="section-number-2">1</span> Optimal Sensor Fusion for noise characteristics</h2>
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<div class="outline-text-2" id="text-1">
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<p>
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<a id="orgc9381ca"></a>
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<a id="orge83d075"></a>
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</p>
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<p>
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The idea is to combine sensors that works in different frequency range using complementary filters.
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@ -389,11 +389,11 @@ All the files (data and Matlab scripts) are accessible <a href="data/optimal_com
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</div>
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</div>
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<div id="outline-container-orgda648ce" class="outline-3">
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<h3 id="orgda648ce"><span class="section-number-3">1.1</span> Architecture</h3>
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<div id="outline-container-orga3597f6" class="outline-3">
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<h3 id="orga3597f6"><span class="section-number-3">1.1</span> Architecture</h3>
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<div class="outline-text-3" id="text-1-1">
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<p>
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Let's consider the sensor fusion architecture shown on figure <a href="#org0214c79">1</a> where two sensors 1 and 2 are measuring the same quantity \(x\) with different noise characteristics determined by \(W_1\) and \(W_2\).
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Let's consider the sensor fusion architecture shown on figure <a href="#org80c1b07">1</a> where two sensors 1 and 2 are measuring the same quantity \(x\) with different noise characteristics determined by \(W_1\) and \(W_2\).
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</p>
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<p>
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@ -401,19 +401,20 @@ Let's consider the sensor fusion architecture shown on figure <a href="#org0214c
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</p>
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<div id="org0214c79" class="figure">
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<p><img src="figs/fusion_two_noisy_sensors_with_dyn.png" alt="fusion_two_noisy_sensors_with_dyn.png" />
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<div id="org80c1b07" class="figure">
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<p><object type="image/svg+xml" data="figs-tikz/fusion_two_noisy_sensors_with_dyn.svg" class="org-svg">
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Sorry, your browser does not support SVG.</object>
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</p>
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<p><span class="figure-number">Figure 1: </span>Fusion of two sensors</p>
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</div>
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<p>
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We consider that the two sensor dynamics \(G_1\) and \(G_2\) are ideal (\(G_1 = G_2 = 1\)). We obtain the architecture of figure <a href="#org8edd1ad">2</a>.
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We consider that the two sensor dynamics \(G_1\) and \(G_2\) are ideal (\(G_1 = G_2 = 1\)). We obtain the architecture of figure <a href="#org442f9da">2</a>.
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</p>
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<div id="org8edd1ad" class="figure">
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<p><img src="figs/fusion_two_noisy_sensors.png" alt="fusion_two_noisy_sensors.png" />
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<div id="org442f9da" class="figure">
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<p><img src="figs-tikz/fusion_two_noisy_sensors.png" alt="fusion_two_noisy_sensors.png" />
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</p>
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<p><span class="figure-number">Figure 2: </span>Fusion of two sensors with ideal dynamics</p>
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</div>
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@ -447,8 +448,8 @@ For that, we will use the \(\mathcal{H}_2\) Synthesis.
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</div>
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</div>
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<div id="outline-container-org7737a79" class="outline-3">
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<h3 id="org7737a79"><span class="section-number-3">1.2</span> Noise of the sensors</h3>
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<div id="outline-container-orgcac1aa3" class="outline-3">
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<h3 id="orgcac1aa3"><span class="section-number-3">1.2</span> Noise of the sensors</h3>
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<div class="outline-text-3" id="text-1-2">
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<p>
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Let's define the noise characteristics of the two sensors by choosing \(W_1\) and \(W_2\):
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@ -468,7 +469,7 @@ W2 = <span class="org-rainbow-delimiters-depth-1">(</span><span class="org-rainb
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</div>
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<div id="org535b437" class="figure">
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<div id="org8380e17" class="figure">
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<p><img src="figs/nosie_characteristics_sensors.png" alt="nosie_characteristics_sensors.png" />
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</p>
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<p><span class="figure-number">Figure 3: </span>Noise Characteristics of the two sensors (<a href="./figs/nosie_characteristics_sensors.png">png</a>, <a href="./figs/nosie_characteristics_sensors.pdf">pdf</a>)</p>
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@ -476,16 +477,16 @@ W2 = <span class="org-rainbow-delimiters-depth-1">(</span><span class="org-rainb
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</div>
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</div>
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<div id="outline-container-orgf85a7f2" class="outline-3">
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<h3 id="orgf85a7f2"><span class="section-number-3">1.3</span> H-Two Synthesis</h3>
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<div id="outline-container-orgdb1ca76" class="outline-3">
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<h3 id="orgdb1ca76"><span class="section-number-3">1.3</span> H-Two Synthesis</h3>
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<div class="outline-text-3" id="text-1-3">
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<p>
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We use the generalized plant architecture shown on figure <a href="#org178bd58">4</a>.
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We use the generalized plant architecture shown on figure <a href="#orgfd9fdc8">4</a>.
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</p>
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<div id="org178bd58" class="figure">
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<p><img src="figs/h_infinity_optimal_comp_filters.png" alt="h_infinity_optimal_comp_filters.png" />
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<div id="orgfd9fdc8" class="figure">
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<p><img src="figs-tikz/h_infinity_optimal_comp_filters.png" alt="h_infinity_optimal_comp_filters.png" />
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</p>
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<p><span class="figure-number">Figure 4: </span>\(\mathcal{H}_2\) Synthesis - Generalized plant used for the optimal generation of complementary filters</p>
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</div>
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@ -502,7 +503,7 @@ Thus, if we minimize the \(\mathcal{H}_2\) norm of this transfer function, we mi
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</p>
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<p>
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We define the generalized plant \(P\) on matlab as shown on figure <a href="#org178bd58">4</a>.
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We define the generalized plant \(P\) on matlab as shown on figure <a href="#orgfd9fdc8">4</a>.
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</p>
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<div class="org-src-container">
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<pre class="src src-matlab">P = <span class="org-rainbow-delimiters-depth-1">[</span><span class="org-highlight-numbers-number">0</span> W2 <span class="org-highlight-numbers-number">1</span>;
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@ -532,30 +533,30 @@ Finally, we define \(H_2 = 1 - H_1\).
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</div>
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</div>
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<div id="outline-container-org9fcdb98" class="outline-3">
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<h3 id="org9fcdb98"><span class="section-number-3">1.4</span> Analysis</h3>
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<div id="outline-container-org375cb3a" class="outline-3">
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<h3 id="org375cb3a"><span class="section-number-3">1.4</span> Analysis</h3>
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<div class="outline-text-3" id="text-1-4">
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<p>
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The complementary filters obtained are shown on figure <a href="#org3f036f9">5</a>. The PSD of the <a href="#org272e8f4">6</a>.
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Finally, the RMS value of \(\hat{x}\) is shown on table <a href="#orgf88e62b">1</a>.
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The complementary filters obtained are shown on figure <a href="#org4addc9f">5</a>. The PSD of the <a href="#org52b2e97">6</a>.
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Finally, the RMS value of \(\hat{x}\) is shown on table <a href="#orgccc886e">1</a>.
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The optimal sensor fusion has permitted to reduced the RMS value of the estimation error by a factor 8 compare to when using only one sensor.
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</p>
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<div id="org3f036f9" class="figure">
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<div id="org4addc9f" class="figure">
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<p><img src="figs/htwo_comp_filters.png" alt="htwo_comp_filters.png" />
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</p>
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<p><span class="figure-number">Figure 5: </span>Obtained complementary filters using the \(\mathcal{H}_2\) Synthesis (<a href="./figs/htwo_comp_filters.png">png</a>, <a href="./figs/htwo_comp_filters.pdf">pdf</a>)</p>
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</div>
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<div id="org272e8f4" class="figure">
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<div id="org52b2e97" class="figure">
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<p><img src="figs/psd_sensors_htwo_synthesis.png" alt="psd_sensors_htwo_synthesis.png" />
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</p>
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<p><span class="figure-number">Figure 6: </span>Power Spectral Density of the estimated \(\hat{x}\) using the two sensors alone and using the optimally fused signal (<a href="./figs/psd_sensors_htwo_synthesis.png">png</a>, <a href="./figs/psd_sensors_htwo_synthesis.pdf">pdf</a>)</p>
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</div>
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<table id="orgf88e62b" border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
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<table id="orgccc886e" 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> RMS value of the estimation error when using the sensor individually and when using the two sensor merged using the optimal complementary filters</caption>
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<colgroup>
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@ -589,18 +590,19 @@ The optimal sensor fusion has permitted to reduced the RMS value of the estimati
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</div>
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</div>
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</div>
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<div id="outline-container-org7d49713" class="outline-2">
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<h2 id="org7d49713"><span class="section-number-2">2</span> Robustness to sensor dynamics uncertainty</h2>
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<div id="outline-container-org2ebd870" class="outline-2">
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<h2 id="org2ebd870"><span class="section-number-2">2</span> Robustness to sensor dynamics uncertainty</h2>
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<div class="outline-text-2" id="text-2">
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<p>
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<a id="orgc67c206"></a>
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<a id="orgb2ab616"></a>
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</p>
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<p>
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Let's first consider ideal sensors where \(G_1 = 1\) and \(G_2 = 1\) (figure <a href="#orgd65643e">7</a>).
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Let's first consider ideal sensors where \(G_1 = 1\) and \(G_2 = 1\) (figure <a href="#orgd0780c6">7</a>).
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</p>
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<div id="orgd65643e" class="figure">
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<div id="orgd0780c6" class="figure">
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<p><img src="figs/fusion_two_noisy_sensors_with_dyn_bis.png" alt="fusion_two_noisy_sensors_with_dyn_bis.png" />
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</p>
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<p><span class="figure-number">Figure 7: </span>Fusion of two sensors</p>
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@ -630,17 +632,17 @@ All the files (data and Matlab scripts) are accessible <a href="data/comp_filter
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</div>
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</div>
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<div id="outline-container-orgc1b9515" class="outline-3">
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<h3 id="orgc1b9515"><span class="section-number-3">2.1</span> Unknown sensor dynamics dynamics</h3>
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<div id="outline-container-orgf63dc07" class="outline-3">
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<h3 id="orgf63dc07"><span class="section-number-3">2.1</span> Unknown sensor dynamics dynamics</h3>
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<div class="outline-text-3" id="text-2-1">
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<p>
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||||
In practical systems, the sensor dynamics has always some level of uncertainty.
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Let's represent that with multiplicative input uncertainty as shown on figure <a href="#org0cb23aa">8</a>.
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Let's represent that with multiplicative input uncertainty as shown on figure <a href="#orga6c7451">8</a>.
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</p>
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<div id="org0cb23aa" class="figure">
|
||||
<p><img src="figs/fusion_gain_mismatch.png" alt="fusion_gain_mismatch.png" />
|
||||
<div id="orga6c7451" class="figure">
|
||||
<p><img src="figs-tikz/fusion_gain_mismatch.png" alt="fusion_gain_mismatch.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 8: </span>Fusion of two sensors with input multiplicative uncertainty</p>
|
||||
</div>
|
||||
@ -684,7 +686,7 @@ Which is approximately the same as requiring
|
||||
</p>
|
||||
|
||||
<p>
|
||||
The uncertainty set of the transfer function from \(\hat{x}\) to \(x\) is bounded in the complex plane by a circle centered on 1 and with a radius equal to \(\epsilon\) (figure <a href="#orge9b44f9">9</a>).
|
||||
The uncertainty set of the transfer function from \(\hat{x}\) to \(x\) is bounded in the complex plane by a circle centered on 1 and with a radius equal to \(\epsilon\) (figure <a href="#org5dfc170">9</a>).
|
||||
</p>
|
||||
|
||||
<p>
|
||||
@ -693,8 +695,8 @@ We then have that the angle introduced by the super sensor is bounded by \(\arcs
|
||||
</p>
|
||||
|
||||
|
||||
<div id="orge9b44f9" class="figure">
|
||||
<p><img src="figs/uncertainty_gain_phase_variation.png" alt="uncertainty_gain_phase_variation.png" />
|
||||
<div id="org5dfc170" class="figure">
|
||||
<p><img src="figs-tikz/uncertainty_gain_phase_variation.png" alt="uncertainty_gain_phase_variation.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 9: </span>Maximum phase variation</p>
|
||||
</div>
|
||||
@ -705,8 +707,8 @@ Thus, we choose should choose \(\epsilon\) so that the maximum phase uncertainty
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org5df7619" class="outline-3">
|
||||
<h3 id="org5df7619"><span class="section-number-3">2.2</span> Design the complementary filters in order to limit the phase and gain uncertainty of the super sensor</h3>
|
||||
<div id="outline-container-org3244858" class="outline-3">
|
||||
<h3 id="org3244858"><span class="section-number-3">2.2</span> Design the complementary filters in order to limit the phase and gain uncertainty of the super sensor</h3>
|
||||
<div class="outline-text-3" id="text-2-2">
|
||||
<p>
|
||||
Let's say the two sensors dynamics \(H_1\) and \(H_2\) have been identified with the associated uncertainty weights \(W_1\) and \(W_2\).
|
||||
@ -739,8 +741,8 @@ This is of primary importance in order to ensure the stability of the feedback l
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org0328fd7" class="outline-3">
|
||||
<h3 id="org0328fd7"><span class="section-number-3">2.3</span> First Basic Example with gain mismatch</h3>
|
||||
<div id="outline-container-org20ae980" class="outline-3">
|
||||
<h3 id="org20ae980"><span class="section-number-3">2.3</span> First Basic Example with gain mismatch</h3>
|
||||
<div class="outline-text-3" id="text-2-3">
|
||||
<p>
|
||||
Let's consider two ideal sensors except one sensor has not an expected gain of one but a gain of \(0.6\).
|
||||
@ -752,19 +754,19 @@ G2 = <span class="org-highlight-numbers-number">0</span>.<span class="org-highli
|
||||
</div>
|
||||
|
||||
<p>
|
||||
Let's design two complementary filters as shown on figure <a href="#orgba6e31e">10</a>.
|
||||
Let's design two complementary filters as shown on figure <a href="#orgc3e7049">10</a>.
|
||||
The complementary filters shown in blue does not present a bump as the red ones but provides less sensor separation at high and low frequencies.
|
||||
</p>
|
||||
|
||||
|
||||
<div id="orgba6e31e" class="figure">
|
||||
<div id="orgc3e7049" class="figure">
|
||||
<p><img src="figs/comp_filters_robustness_test.png" alt="comp_filters_robustness_test.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 10: </span>The two complementary filters designed for the robustness test (<a href="./figs/comp_filters_robustness_test.png">png</a>, <a href="./figs/comp_filters_robustness_test.pdf">pdf</a>)</p>
|
||||
</div>
|
||||
|
||||
<p>
|
||||
We then compute the bode plot of the super sensor transfer function \(H_1*G_1 + H_2*G_2\) for both complementary filters pair (figure <a href="#orgd412a5a">11</a>).
|
||||
We then compute the bode plot of the super sensor transfer function \(H_1*G_1 + H_2*G_2\) for both complementary filters pair (figure <a href="#org124e3b5">11</a>).
|
||||
</p>
|
||||
|
||||
<p>
|
||||
@ -772,7 +774,7 @@ We see that the blue complementary filters with a lower maximum norm permits to
|
||||
</p>
|
||||
|
||||
|
||||
<div id="orgd412a5a" class="figure">
|
||||
<div id="org124e3b5" class="figure">
|
||||
<p><img src="figs/tf_super_sensor_comp.png" alt="tf_super_sensor_comp.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 11: </span>Comparison of the obtained super sensor transfer functions (<a href="./figs/tf_super_sensor_comp.png">png</a>, <a href="./figs/tf_super_sensor_comp.pdf">pdf</a>)</p>
|
||||
@ -780,15 +782,15 @@ We see that the blue complementary filters with a lower maximum norm permits to
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org81e622c" class="outline-3">
|
||||
<h3 id="org81e622c"><span class="section-number-3">2.4</span> <span class="todo TODO">TODO</span> More Complete example with model uncertainty</h3>
|
||||
<div id="outline-container-org41f2b11" class="outline-3">
|
||||
<h3 id="org41f2b11"><span class="section-number-3">2.4</span> <span class="todo TODO">TODO</span> More Complete example with model uncertainty</h3>
|
||||
</div>
|
||||
</div>
|
||||
<div id="outline-container-org6d8c8af" class="outline-2">
|
||||
<h2 id="org6d8c8af"><span class="section-number-2">3</span> Complementary filters using analytical formula</h2>
|
||||
<div id="outline-container-orgf04a9a4" class="outline-2">
|
||||
<h2 id="orgf04a9a4"><span class="section-number-2">3</span> Complementary filters using analytical formula</h2>
|
||||
<div class="outline-text-2" id="text-3">
|
||||
<p>
|
||||
<a id="org91a8313"></a>
|
||||
<a id="org654b360"></a>
|
||||
</p>
|
||||
<div class="note">
|
||||
<p>
|
||||
@ -798,8 +800,8 @@ All the files (data and Matlab scripts) are accessible <a href="data/comp_filter
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-orgb288402" class="outline-3">
|
||||
<h3 id="orgb288402"><span class="section-number-3">3.1</span> Analytical 1st order complementary filters</h3>
|
||||
<div id="outline-container-org7d339a9" class="outline-3">
|
||||
<h3 id="org7d339a9"><span class="section-number-3">3.1</span> Analytical 1st order complementary filters</h3>
|
||||
<div class="outline-text-3" id="text-3-1">
|
||||
<p>
|
||||
First order complementary filters are defined with following equations:
|
||||
@ -810,7 +812,7 @@ First order complementary filters are defined with following equations:
|
||||
\end{align}
|
||||
|
||||
<p>
|
||||
Their bode plot is shown figure <a href="#orga970e88">12</a>.
|
||||
Their bode plot is shown figure <a href="#orgd35e336">12</a>.
|
||||
</p>
|
||||
|
||||
<div class="org-src-container">
|
||||
@ -822,7 +824,7 @@ Hl1 = <span class="org-highlight-numbers-number">1</span><span class="org-type">
|
||||
</div>
|
||||
|
||||
|
||||
<div id="orga970e88" class="figure">
|
||||
<div id="orgd35e336" class="figure">
|
||||
<p><img src="figs/comp_filter_1st_order.png" alt="comp_filter_1st_order.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 12: </span>Bode plot of first order complementary filter (<a href="./figs/comp_filter_1st_order.png">png</a>, <a href="./figs/comp_filter_1st_order.pdf">pdf</a>)</p>
|
||||
@ -830,8 +832,8 @@ Hl1 = <span class="org-highlight-numbers-number">1</span><span class="org-type">
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-orgefd1169" class="outline-3">
|
||||
<h3 id="orgefd1169"><span class="section-number-3">3.2</span> Second Order Complementary Filters</h3>
|
||||
<div id="outline-container-org8edbedb" class="outline-3">
|
||||
<h3 id="org8edbedb"><span class="section-number-3">3.2</span> Second Order Complementary Filters</h3>
|
||||
<div class="outline-text-3" id="text-3-2">
|
||||
<p>
|
||||
We here use analytical formula for the complementary filters \(H_L\) and \(H_H\).
|
||||
@ -857,12 +859,12 @@ where:
|
||||
</ul>
|
||||
|
||||
<p>
|
||||
This is illustrated on figure <a href="#orga420ceb">13</a>.
|
||||
This is illustrated on figure <a href="#org551e9a2">13</a>.
|
||||
The slope of those filters at high and low frequencies is \(-2\) and \(2\) respectively for \(H_L\) and \(H_H\).
|
||||
</p>
|
||||
|
||||
|
||||
<div id="orga420ceb" class="figure">
|
||||
<div id="org551e9a2" class="figure">
|
||||
<p><img src="figs/comp_filters_param_alpha.png" alt="comp_filters_param_alpha.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 13: </span>Effect of the parameter \(\alpha\) on the shape of the generated second order complementary filters (<a href="./figs/comp_filters_param_alpha.png">png</a>, <a href="./figs/comp_filters_param_alpha.pdf">pdf</a>)</p>
|
||||
@ -880,7 +882,7 @@ xlabel<span class="org-rainbow-delimiters-depth-1">(</span>'$<span class="org-ty
|
||||
</div>
|
||||
|
||||
|
||||
<div id="orgf07fa17" class="figure">
|
||||
<div id="org9c950bd" class="figure">
|
||||
<p><img src="figs/param_alpha_hinf_norm.png" alt="param_alpha_hinf_norm.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 14: </span>Evolution of the H-Infinity norm of the complementary filters with the parameter \(\alpha\) (<a href="./figs/param_alpha_hinf_norm.png">png</a>, <a href="./figs/param_alpha_hinf_norm.pdf">pdf</a>)</p>
|
||||
@ -888,8 +890,8 @@ xlabel<span class="org-rainbow-delimiters-depth-1">(</span>'$<span class="org-ty
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org32c14e8" class="outline-3">
|
||||
<h3 id="org32c14e8"><span class="section-number-3">3.3</span> Third Order Complementary Filters</h3>
|
||||
<div id="outline-container-org5fd0c3c" class="outline-3">
|
||||
<h3 id="org5fd0c3c"><span class="section-number-3">3.3</span> Third Order Complementary Filters</h3>
|
||||
<div class="outline-text-3" id="text-3-3">
|
||||
<p>
|
||||
The following formula gives complementary filters with slopes of \(-3\) and \(3\):
|
||||
@ -908,7 +910,7 @@ The parameters are:
|
||||
</ul>
|
||||
|
||||
<p>
|
||||
The filters are defined below and the result is shown on figure <a href="#orgbe15178">15</a>.
|
||||
The filters are defined below and the result is shown on figure <a href="#org8791569">15</a>.
|
||||
</p>
|
||||
|
||||
<div class="org-src-container">
|
||||
@ -922,7 +924,7 @@ Hl3_ana = <span class="org-rainbow-delimiters-depth-1">(</span><span class="org-
|
||||
</div>
|
||||
|
||||
|
||||
<div id="orgbe15178" class="figure">
|
||||
<div id="org8791569" class="figure">
|
||||
<p><img src="figs/complementary_filters_third_order.png" alt="complementary_filters_third_order.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 15: </span>Third order complementary filters using the analytical formula (<a href="./figs/complementary_filters_third_order.png">png</a>, <a href="./figs/complementary_filters_third_order.pdf">pdf</a>)</p>
|
||||
@ -931,11 +933,11 @@ Hl3_ana = <span class="org-rainbow-delimiters-depth-1">(</span><span class="org-
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-orgc86d54b" class="outline-2">
|
||||
<h2 id="orgc86d54b"><span class="section-number-2">4</span> H-Infinity synthesis of complementary filters</h2>
|
||||
<div id="outline-container-org88004e2" class="outline-2">
|
||||
<h2 id="org88004e2"><span class="section-number-2">4</span> H-Infinity synthesis of complementary filters</h2>
|
||||
<div class="outline-text-2" id="text-4">
|
||||
<p>
|
||||
<a id="org2358787"></a>
|
||||
<a id="org924f36f"></a>
|
||||
</p>
|
||||
<div class="note">
|
||||
<p>
|
||||
@ -945,8 +947,8 @@ All the files (data and Matlab scripts) are accessible <a href="data/h_inf_synth
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org2288026" class="outline-3">
|
||||
<h3 id="org2288026"><span class="section-number-3">4.1</span> Synthesis Architecture</h3>
|
||||
<div id="outline-container-orgc51b2bd" class="outline-3">
|
||||
<h3 id="orgc51b2bd"><span class="section-number-3">4.1</span> Synthesis Architecture</h3>
|
||||
<div class="outline-text-3" id="text-4-1">
|
||||
<p>
|
||||
We here synthesize the complementary filters using the \(\mathcal{H}_\infty\) synthesis.
|
||||
@ -954,18 +956,18 @@ The goal is to specify upper bounds on the norms of \(H_L\) and \(H_H\) while en
|
||||
</p>
|
||||
|
||||
<p>
|
||||
In order to do so, we use the generalized plant shown on figure <a href="#org4764d16">16</a> where \(w_L\) and \(w_H\) weighting transfer functions that will be used to shape \(H_L\) and \(H_H\) respectively.
|
||||
In order to do so, we use the generalized plant shown on figure <a href="#orge2545fd">16</a> where \(w_L\) and \(w_H\) weighting transfer functions that will be used to shape \(H_L\) and \(H_H\) respectively.
|
||||
</p>
|
||||
|
||||
|
||||
<div id="org4764d16" class="figure">
|
||||
<p><img src="figs/sf_hinf_filters_plant_b.png" alt="sf_hinf_filters_plant_b.png" />
|
||||
<div id="orge2545fd" class="figure">
|
||||
<p><img src="figs-tikz/sf_hinf_filters_plant_b.png" alt="sf_hinf_filters_plant_b.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 16: </span>Generalized plant used for the \(\mathcal{H}_\infty\) synthesis of the complementary filters</p>
|
||||
</div>
|
||||
|
||||
<p>
|
||||
The \(\mathcal{H}_\infty\) synthesis applied on this generalized plant will give a transfer function \(H_L\) (figure <a href="#orgd665b69">17</a>) such that the \(\mathcal{H}_\infty\) norm of the transfer function from \(w\) to \([z_H,\ z_L]\) is less than one:
|
||||
The \(\mathcal{H}_\infty\) synthesis applied on this generalized plant will give a transfer function \(H_L\) (figure <a href="#org130d2c0">17</a>) such that the \(\mathcal{H}_\infty\) norm of the transfer function from \(w\) to \([z_H,\ z_L]\) is less than one:
|
||||
\[ \left\| \begin{array}{c} H_L w_L \\ (1 - H_L) w_H \end{array} \right\|_\infty < 1 \]
|
||||
</p>
|
||||
|
||||
@ -984,16 +986,16 @@ We then see that \(w_L\) and \(w_H\) can be used to shape both \(H_L\) and \(H_H
|
||||
</p>
|
||||
|
||||
|
||||
<div id="orgd665b69" class="figure">
|
||||
<p><img src="figs/sf_hinf_filters_b.png" alt="sf_hinf_filters_b.png" />
|
||||
<div id="org130d2c0" class="figure">
|
||||
<p><img src="figs-tikz/sf_hinf_filters_b.png" alt="sf_hinf_filters_b.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 17: </span>\(\mathcal{H}_\infty\) synthesis of the complementary filters</p>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-orga9b0474" class="outline-3">
|
||||
<h3 id="orga9b0474"><span class="section-number-3">4.2</span> Weights</h3>
|
||||
<div id="outline-container-orga135818" class="outline-3">
|
||||
<h3 id="orga135818"><span class="section-number-3">4.2</span> Weights</h3>
|
||||
<div class="outline-text-3" id="text-4-2">
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab">omegab = <span class="org-highlight-numbers-number">2</span><span class="org-type">*</span><span class="org-constant">pi</span><span class="org-type">*</span><span class="org-highlight-numbers-number">9</span>;
|
||||
@ -1004,7 +1006,7 @@ wL = <span class="org-rainbow-delimiters-depth-1">(</span>s <span class="org-typ
|
||||
</div>
|
||||
|
||||
|
||||
<div id="org9dc1a51" class="figure">
|
||||
<div id="org3cb0b70" class="figure">
|
||||
<p><img src="figs/weights_wl_wh.png" alt="weights_wl_wh.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 18: </span>Weights on the complementary filters \(w_L\) and \(w_H\) and the associated performance weights (<a href="./figs/weights_wl_wh.png">png</a>, <a href="./figs/weights_wl_wh.pdf">pdf</a>)</p>
|
||||
@ -1012,8 +1014,8 @@ wL = <span class="org-rainbow-delimiters-depth-1">(</span>s <span class="org-typ
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org17ecd06" class="outline-3">
|
||||
<h3 id="org17ecd06"><span class="section-number-3">4.3</span> H-Infinity Synthesis</h3>
|
||||
<div id="outline-container-orgbd3b8d8" class="outline-3">
|
||||
<h3 id="orgbd3b8d8"><span class="section-number-3">4.3</span> H-Infinity Synthesis</h3>
|
||||
<div class="outline-text-3" id="text-4-3">
|
||||
<p>
|
||||
We define the generalized plant \(P\) on matlab.
|
||||
@ -1055,7 +1057,7 @@ Test bounds: 0.0000 < gamma <= 1.7285
|
||||
</pre>
|
||||
|
||||
<p>
|
||||
We then define the high pass filter \(H_H = 1 - H_L\). The bode plot of both \(H_L\) and \(H_H\) is shown on figure <a href="#orge5933cc">19</a>.
|
||||
We then define the high pass filter \(H_H = 1 - H_L\). The bode plot of both \(H_L\) and \(H_H\) is shown on figure <a href="#orgc1ec9ab">19</a>.
|
||||
</p>
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab">Hh_hinf = <span class="org-highlight-numbers-number">1</span> <span class="org-type">-</span> Hl_hinf;
|
||||
@ -1064,15 +1066,15 @@ We then define the high pass filter \(H_H = 1 - H_L\). The bode plot of both \(H
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org844c90a" class="outline-3">
|
||||
<h3 id="org844c90a"><span class="section-number-3">4.4</span> Obtained Complementary Filters</h3>
|
||||
<div id="outline-container-org0d62ef6" class="outline-3">
|
||||
<h3 id="org0d62ef6"><span class="section-number-3">4.4</span> Obtained Complementary Filters</h3>
|
||||
<div class="outline-text-3" id="text-4-4">
|
||||
<p>
|
||||
The obtained complementary filters are shown on figure <a href="#orge5933cc">19</a>.
|
||||
The obtained complementary filters are shown on figure <a href="#orgc1ec9ab">19</a>.
|
||||
</p>
|
||||
|
||||
|
||||
<div id="orge5933cc" class="figure">
|
||||
<div id="orgc1ec9ab" class="figure">
|
||||
<p><img src="figs/hinf_filters_results.png" alt="hinf_filters_results.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 19: </span>Obtained complementary filters using \(\mathcal{H}_\infty\) synthesis (<a href="./figs/hinf_filters_results.png">png</a>, <a href="./figs/hinf_filters_results.pdf">pdf</a>)</p>
|
||||
@ -1081,11 +1083,11 @@ The obtained complementary filters are shown on figure <a href="#orge5933cc">19<
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-orgd76dc0a" class="outline-2">
|
||||
<h2 id="orgd76dc0a"><span class="section-number-2">5</span> Feedback Control Architecture to generate Complementary Filters</h2>
|
||||
<div id="outline-container-orga65dbb6" class="outline-2">
|
||||
<h2 id="orga65dbb6"><span class="section-number-2">5</span> Feedback Control Architecture to generate Complementary Filters</h2>
|
||||
<div class="outline-text-2" id="text-5">
|
||||
<p>
|
||||
<a id="org94d0d5f"></a>
|
||||
<a id="org3e654dc"></a>
|
||||
</p>
|
||||
<p>
|
||||
The idea is here to use the fact that in a classical feedback architecture, \(S + T = 1\), in order to design complementary filters.
|
||||
@ -1102,12 +1104,12 @@ All the files (data and Matlab scripts) are accessible <a href="data/feedback_ge
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org2b0a54f" class="outline-3">
|
||||
<h3 id="org2b0a54f"><span class="section-number-3">5.1</span> Architecture</h3>
|
||||
<div id="outline-container-orgec7fc24" class="outline-3">
|
||||
<h3 id="orgec7fc24"><span class="section-number-3">5.1</span> Architecture</h3>
|
||||
<div class="outline-text-3" id="text-5-1">
|
||||
|
||||
<div id="orgb7d4dd2" class="figure">
|
||||
<p><img src="figs/complementary_filters_feedback_architecture.png" alt="complementary_filters_feedback_architecture.png" />
|
||||
<div id="orgf3202e6" class="figure">
|
||||
<p><img src="figs-tikz/complementary_filters_feedback_architecture.png" alt="complementary_filters_feedback_architecture.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 20: </span>Architecture used to generate the complementary filters</p>
|
||||
</div>
|
||||
@ -1133,8 +1135,8 @@ Which contains two integrator and a lead. \(\omega_c\) is used to tune the cross
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org023d478" class="outline-3">
|
||||
<h3 id="org023d478"><span class="section-number-3">5.2</span> Loop Gain Design</h3>
|
||||
<div id="outline-container-org448cd20" class="outline-3">
|
||||
<h3 id="org448cd20"><span class="section-number-3">5.2</span> Loop Gain Design</h3>
|
||||
<div class="outline-text-3" id="text-5-2">
|
||||
<p>
|
||||
Let's first define the loop gain \(L\).
|
||||
@ -1148,7 +1150,7 @@ L = <span class="org-rainbow-delimiters-depth-1">(</span>wc<span class="org-type
|
||||
</div>
|
||||
|
||||
|
||||
<div id="org5563034" class="figure">
|
||||
<div id="orgc090be3" class="figure">
|
||||
<p><img src="figs/loop_gain_bode_plot.png" alt="loop_gain_bode_plot.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 21: </span>Bode plot of the loop gain \(L\) (<a href="./figs/loop_gain_bode_plot.png">png</a>, <a href="./figs/loop_gain_bode_plot.pdf">pdf</a>)</p>
|
||||
@ -1156,8 +1158,8 @@ L = <span class="org-rainbow-delimiters-depth-1">(</span>wc<span class="org-type
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-orgef15574" class="outline-3">
|
||||
<h3 id="orgef15574"><span class="section-number-3">5.3</span> Complementary Filters Obtained</h3>
|
||||
<div id="outline-container-org1ec0e07" class="outline-3">
|
||||
<h3 id="org1ec0e07"><span class="section-number-3">5.3</span> Complementary Filters Obtained</h3>
|
||||
<div class="outline-text-3" id="text-5-3">
|
||||
<p>
|
||||
We then compute the resulting low pass and high pass filters.
|
||||
@ -1169,7 +1171,7 @@ Hh = <span class="org-highlight-numbers-number">1</span><span class="org-type">/
|
||||
</div>
|
||||
|
||||
|
||||
<div id="org7a3c2de" class="figure">
|
||||
<div id="org84c20e0" class="figure">
|
||||
<p><img src="figs/low_pass_high_pass_filters.png" alt="low_pass_high_pass_filters.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 22: </span>Low pass and High pass filters \(H_L\) and \(H_H\) for different values of \(\alpha\) (<a href="./figs/low_pass_high_pass_filters.png">png</a>, <a href="./figs/low_pass_high_pass_filters.pdf">pdf</a>)</p>
|
||||
@ -1178,16 +1180,16 @@ Hh = <span class="org-highlight-numbers-number">1</span><span class="org-type">/
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org9a4e0ed" class="outline-2">
|
||||
<h2 id="org9a4e0ed"><span class="section-number-2">6</span> Analytical Formula found in the literature</h2>
|
||||
<div id="outline-container-org3b324a8" class="outline-2">
|
||||
<h2 id="org3b324a8"><span class="section-number-2">6</span> Analytical Formula found in the literature</h2>
|
||||
<div class="outline-text-2" id="text-6">
|
||||
<p>
|
||||
<a id="orga144d76"></a>
|
||||
<a id="org22f6020"></a>
|
||||
</p>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-orgfa54829" class="outline-3">
|
||||
<h3 id="orgfa54829"><span class="section-number-3">6.1</span> Analytical Formula</h3>
|
||||
<div id="outline-container-org8d499d4" class="outline-3">
|
||||
<h3 id="org8d499d4"><span class="section-number-3">6.1</span> Analytical Formula</h3>
|
||||
<div class="outline-text-3" id="text-6-1">
|
||||
<p>
|
||||
<a class='org-ref-reference' href="#min15_compl_filter_desig_angle_estim">min15_compl_filter_desig_angle_estim</a>
|
||||
@ -1236,8 +1238,8 @@ Hh = <span class="org-highlight-numbers-number">1</span><span class="org-type">/
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org5286b79" class="outline-3">
|
||||
<h3 id="org5286b79"><span class="section-number-3">6.2</span> Matlab</h3>
|
||||
<div id="outline-container-org60e2ce0" class="outline-3">
|
||||
<h3 id="org60e2ce0"><span class="section-number-3">6.2</span> Matlab</h3>
|
||||
<div class="outline-text-3" id="text-6-2">
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab">omega0 = <span class="org-highlight-numbers-number">1</span><span class="org-type">*</span><span class="org-highlight-numbers-number">2</span><span class="org-type">*</span><span class="org-constant">pi</span>; <span class="org-comment">% [rad/s]</span>
|
||||
@ -1255,7 +1257,7 @@ HL3 = <span class="org-rainbow-delimiters-depth-1">(</span><span class="org-high
|
||||
</div>
|
||||
|
||||
|
||||
<div id="orge38d4ce" class="figure">
|
||||
<div id="org5dd1179" class="figure">
|
||||
<p><img src="figs/comp_filters_literature.png" alt="comp_filters_literature.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 23: </span>Comparison of some complementary filters found in the literature (<a href="./figs/comp_filters_literature.png">png</a>, <a href="./figs/comp_filters_literature.pdf">pdf</a>)</p>
|
||||
@ -1263,8 +1265,8 @@ HL3 = <span class="org-rainbow-delimiters-depth-1">(</span><span class="org-high
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org6092cef" class="outline-3">
|
||||
<h3 id="org6092cef"><span class="section-number-3">6.3</span> Discussion</h3>
|
||||
<div id="outline-container-org1038fe9" class="outline-3">
|
||||
<h3 id="org1038fe9"><span class="section-number-3">6.3</span> Discussion</h3>
|
||||
<div class="outline-text-3" id="text-6-3">
|
||||
<p>
|
||||
Analytical Formula found in the literature provides either no parameter for tuning the robustness / performance trade-off.
|
||||
@ -1273,11 +1275,11 @@ Analytical Formula found in the literature provides either no parameter for tuni
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-orgbbfeb6f" class="outline-2">
|
||||
<h2 id="orgbbfeb6f"><span class="section-number-2">7</span> Comparison of the different methods of synthesis</h2>
|
||||
<div id="outline-container-org647d1a0" class="outline-2">
|
||||
<h2 id="org647d1a0"><span class="section-number-2">7</span> Comparison of the different methods of synthesis</h2>
|
||||
<div class="outline-text-2" id="text-7">
|
||||
<p>
|
||||
<a id="orgfee2d57"></a>
|
||||
<a id="orgcbb346f"></a>
|
||||
The generated complementary filters using \(\mathcal{H}_\infty\) and the analytical formulas are very close to each other. However there is some difference to note here:
|
||||
</p>
|
||||
<ul class="org-ul">
|
||||
@ -1286,6 +1288,7 @@ The generated complementary filters using \(\mathcal{H}_\infty\) and the analyti
|
||||
</ul>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<p>
|
||||
|
||||
<h1 class='org-ref-bib-h1'>Bibliography</h1>
|
||||
@ -1300,7 +1303,7 @@ The generated complementary filters using \(\mathcal{H}_\infty\) and the analyti
|
||||
</div>
|
||||
<div id="postamble" class="status">
|
||||
<p class="author">Author: Thomas Dehaeze</p>
|
||||
<p class="date">Created: 2019-08-14 mer. 12:13</p>
|
||||
<p class="date">Created: 2019-08-21 mer. 13:17</p>
|
||||
<p class="validation"><a href="http://validator.w3.org/check?uri=referer">Validate</a></p>
|
||||
</div>
|
||||
</body>
|
||||
|
||||
@ -1,4 +1,4 @@
|
||||
#+TITLE: On the Design of Complementary Filters for Control - Computation with Matlab
|
||||
#+TITLE: Robust and Optimal Sensor Fusion - Matlab Computation
|
||||
:DRAWER:
|
||||
#+HTML_LINK_HOME: ../index.html
|
||||
#+HTML_LINK_UP: ../index.html
|
||||
@ -92,13 +92,13 @@ $n_1$ and $n_2$ are white noise (constant power spectral density over all freque
|
||||
|
||||
#+name: fig:fusion_two_noisy_sensors_with_dyn
|
||||
#+caption: Fusion of two sensors
|
||||
[[file:figs/fusion_two_noisy_sensors_with_dyn.png]]
|
||||
[[file:figs-tikz/fusion_two_noisy_sensors_with_dyn.png]]
|
||||
|
||||
We consider that the two sensor dynamics $G_1$ and $G_2$ are ideal ($G_1 = G_2 = 1$). We obtain the architecture of figure [[fig:fusion_two_noisy_sensors]].
|
||||
|
||||
#+name: fig:fusion_two_noisy_sensors
|
||||
#+caption: Fusion of two sensors with ideal dynamics
|
||||
[[file:figs/fusion_two_noisy_sensors.png]]
|
||||
[[file:figs-tikz/fusion_two_noisy_sensors.png]]
|
||||
|
||||
$H_1$ and $H_2$ are complementary filters ($H_1 + H_2 = 1$). The goal is to design $H_1$ and $H_2$ such that the effect of the noise sources $n_1$ and $n_2$ has the smallest possible effect on the estimation $\hat{x}$.
|
||||
|
||||
@ -162,7 +162,7 @@ We use the generalized plant architecture shown on figure [[fig:h_infinity_optim
|
||||
|
||||
#+name: fig:h_infinity_optimal_comp_filters
|
||||
#+caption: $\mathcal{H}_2$ Synthesis - Generalized plant used for the optimal generation of complementary filters
|
||||
[[file:figs/h_infinity_optimal_comp_filters.png]]
|
||||
[[file:figs-tikz/h_infinity_optimal_comp_filters.png]]
|
||||
|
||||
The transfer function from $[n_1, n_2]$ to $\hat{x}$ is:
|
||||
\[ \begin{bmatrix} W_1 H_1 \\ W_2 (1 - H_1) \end{bmatrix} \]
|
||||
@ -249,6 +249,7 @@ The optimal sensor fusion has permitted to reduced the RMS value of the estimati
|
||||
| Sensor 1 | 1.1e-02 |
|
||||
| Sensor 2 | 1.3e-03 |
|
||||
| Optimal Sensor Fusion | 1.5e-04 |
|
||||
|
||||
* Robustness to sensor dynamics uncertainty
|
||||
:PROPERTIES:
|
||||
:header-args:matlab+: :tangle matlab/comp_filter_robustness.m
|
||||
@ -327,7 +328,7 @@ Let's represent that with multiplicative input uncertainty as shown on figure [[
|
||||
|
||||
#+name: fig:fusion_gain_mismatch
|
||||
#+caption: Fusion of two sensors with input multiplicative uncertainty
|
||||
[[file:figs/fusion_gain_mismatch.png]]
|
||||
[[file:figs-tikz/fusion_gain_mismatch.png]]
|
||||
|
||||
We have:
|
||||
\begin{align*}
|
||||
@ -360,7 +361,7 @@ We then have that the angle introduced by the super sensor is bounded by $\arcsi
|
||||
|
||||
#+name: fig:uncertainty_gain_phase_variation
|
||||
#+caption: Maximum phase variation
|
||||
[[file:figs/uncertainty_gain_phase_variation.png]]
|
||||
[[file:figs-tikz/uncertainty_gain_phase_variation.png]]
|
||||
|
||||
Thus, we choose should choose $\epsilon$ so that the maximum phase uncertainty introduced by the sensors is of an acceptable value.
|
||||
|
||||
@ -765,7 +766,7 @@ In order to do so, we use the generalized plant shown on figure [[fig:sf_hinf_fi
|
||||
|
||||
#+name: fig:sf_hinf_filters_plant_b
|
||||
#+caption: Generalized plant used for the $\mathcal{H}_\infty$ synthesis of the complementary filters
|
||||
[[file:figs/sf_hinf_filters_plant_b.png]]
|
||||
[[file:figs-tikz/sf_hinf_filters_plant_b.png]]
|
||||
|
||||
The $\mathcal{H}_\infty$ synthesis applied on this generalized plant will give a transfer function $H_L$ (figure [[fig:sf_hinf_filters_b]]) such that the $\mathcal{H}_\infty$ norm of the transfer function from $w$ to $[z_H,\ z_L]$ is less than one:
|
||||
\[ \left\| \begin{array}{c} H_L w_L \\ (1 - H_L) w_H \end{array} \right\|_\infty < 1 \]
|
||||
@ -782,7 +783,7 @@ We then see that $w_L$ and $w_H$ can be used to shape both $H_L$ and $H_H$ while
|
||||
|
||||
#+name: fig:sf_hinf_filters_b
|
||||
#+caption: $\mathcal{H}_\infty$ synthesis of the complementary filters
|
||||
[[file:figs/sf_hinf_filters_b.png]]
|
||||
[[file:figs-tikz/sf_hinf_filters_b.png]]
|
||||
|
||||
** Weights
|
||||
|
||||
@ -934,7 +935,7 @@ Thus, all the tools that has been developed for classical feedback control can b
|
||||
** Architecture
|
||||
#+name: fig:complementary_filters_feedback_architecture
|
||||
#+caption: Architecture used to generate the complementary filters
|
||||
[[file:figs/complementary_filters_feedback_architecture.png]]
|
||||
[[file:figs-tikz/complementary_filters_feedback_architecture.png]]
|
||||
|
||||
We have:
|
||||
\[ y = \underbrace{\frac{L}{L + 1}}_{H_L} y_1 + \underbrace{\frac{1}{L + 1}}_{H_H} y_2 \]
|
||||
@ -1138,6 +1139,7 @@ Analytical Formula found in the literature provides either no parameter for tuni
|
||||
The generated complementary filters using $\mathcal{H}_\infty$ and the analytical formulas are very close to each other. However there is some difference to note here:
|
||||
- the analytical formula provides a very simple way to generate the complementary filters (and thus the controller), they could even be used to tune the controller online using the parameters $\alpha$ and $\omega_0$. However, these formula have the property that $|H_H|$ and $|H_L|$ are symmetrical with the frequency $\omega_0$ which may not be desirable.
|
||||
- while the $\mathcal{H}_\infty$ synthesis of the complementary filters is not as straightforward as using the analytical formula, it provides a more optimized procedure to obtain the complementary filters
|
||||
|
||||
* Bibliography :ignore:
|
||||
bibliographystyle:unsrt
|
||||
bibliography:ref.bib
|
||||
|
||||
Loading…
Reference in New Issue
Block a user