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matlab/figs-paper
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"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
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"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
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<html xmlns="http://www.w3.org/1999/xhtml" lang="en" xml:lang="en">
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<html xmlns="http://www.w3.org/1999/xhtml" lang="en" xml:lang="en">
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<head>
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<head>
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<!-- 2020-10-05 lun. 11:45 -->
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<!-- 2020-10-05 lun. 15:08 -->
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<meta http-equiv="Content-Type" content="text/html;charset=utf-8" />
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<meta http-equiv="Content-Type" content="text/html;charset=utf-8" />
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<title>Robust and Optimal Sensor Fusion - Matlab Computation</title>
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<title>Robust and Optimal Sensor Fusion - Matlab Computation</title>
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<meta name="generator" content="Org mode" />
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<meta name="generator" content="Org mode" />
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@ -35,50 +35,50 @@
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<h2>Table of Contents</h2>
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<h2>Table of Contents</h2>
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||||||
<div id="text-table-of-contents">
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<div id="text-table-of-contents">
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||||||
<ul>
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<ul>
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||||||
<li><a href="#org27de0ea">1. Sensor Description</a>
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<li><a href="#org0a492b7">1. Sensor Description</a>
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||||||
<ul>
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<ul>
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||||||
<li><a href="#org615e955">1.1. Sensor Dynamics</a></li>
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<li><a href="#org9575585">1.1. Sensor Dynamics</a></li>
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||||||
<li><a href="#org693d3bf">1.2. Sensor Model Uncertainty</a></li>
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<li><a href="#orgec8c81d">1.2. Sensor Model Uncertainty</a></li>
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||||||
<li><a href="#org60da040">1.3. Sensor Noise</a></li>
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<li><a href="#org81d9a34">1.3. Sensor Noise</a></li>
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||||||
<li><a href="#org0b1325d">1.4. Save Model</a></li>
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<li><a href="#org4e23f6c">1.4. Save Model</a></li>
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||||||
</ul>
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</ul>
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||||||
</li>
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</li>
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<li><a href="#org0b389f3">2. Introduction to Sensor Fusion</a>
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<li><a href="#org2cab1a2">2. Introduction to Sensor Fusion</a>
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||||||
<ul>
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<ul>
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||||||
<li><a href="#org1f9e1b3">2.1. Sensor Fusion Architecture</a></li>
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<li><a href="#org0cbc92d">2.1. Sensor Fusion Architecture</a></li>
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<li><a href="#org6ce496e">2.2. Super Sensor Noise</a></li>
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<li><a href="#orge9e0bd4">2.2. Super Sensor Noise</a></li>
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||||||
<li><a href="#org8a5c291">2.3. Super Sensor Dynamical Uncertainty</a></li>
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<li><a href="#orgefb4347">2.3. Super Sensor Dynamical Uncertainty</a></li>
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</ul>
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</ul>
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</li>
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</li>
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<li><a href="#org7cb91ba">3. Optimal Super Sensor Noise: \(\mathcal{H}_2\) Synthesis</a>
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<li><a href="#org5896b60">3. Optimal Super Sensor Noise: \(\mathcal{H}_2\) Synthesis</a>
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||||||
<ul>
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<ul>
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<li><a href="#orga0474b9">3.1. \(\mathcal{H}_2\) Synthesis</a></li>
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<li><a href="#orgf5a8a84">3.1. \(\mathcal{H}_2\) Synthesis</a></li>
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||||||
<li><a href="#org1273dcd">3.2. Super Sensor Noise</a></li>
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<li><a href="#org72159df">3.2. Super Sensor Noise</a></li>
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||||||
<li><a href="#orge9c2d41">3.3. Discrepancy between sensor dynamics and model</a></li>
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<li><a href="#org8ffba19">3.3. Discrepancy between sensor dynamics and model</a></li>
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</ul>
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</ul>
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</li>
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</li>
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<li><a href="#orgef8f365">4. Robust Sensor Fusion: \(\mathcal{H}_\infty\) Synthesis</a>
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<li><a href="#org26ea7b1">4. Robust Sensor Fusion: \(\mathcal{H}_\infty\) Synthesis</a>
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<ul>
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<ul>
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<li><a href="#org6f283c6">4.1. Weighting Function used to bound the super sensor uncertainty</a></li>
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<li><a href="#org8dfd9d2">4.1. Weighting Function used to bound the super sensor uncertainty</a></li>
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<li><a href="#org027886f">4.2. \(\mathcal{H}_\infty\) Synthesis</a></li>
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<li><a href="#org7422ade">4.2. \(\mathcal{H}_\infty\) Synthesis</a></li>
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||||||
<li><a href="#org0ad8fe8">4.3. Super sensor uncertainty</a></li>
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<li><a href="#orga0267c6">4.3. Super sensor uncertainty</a></li>
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<li><a href="#orgd5efe47">4.4. Super sensor noise</a></li>
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<li><a href="#org979fede">4.4. Super sensor noise</a></li>
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<li><a href="#org0355c08">4.5. Conclusion</a></li>
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<li><a href="#orgda33992">4.5. Conclusion</a></li>
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</ul>
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</ul>
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</li>
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</li>
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<li><a href="#org3654cee">5. Optimal and Robust Sensor Fusion: Mixed \(\mathcal{H}_2/\mathcal{H}_\infty\) Synthesis</a>
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<li><a href="#org15afe90">5. Optimal and Robust Sensor Fusion: Mixed \(\mathcal{H}_2/\mathcal{H}_\infty\) Synthesis</a>
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||||||
<ul>
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<ul>
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<li><a href="#org4d41c02">5.1. Mixed \(\mathcal{H}_2\) / \(\mathcal{H}_\infty\) Synthesis</a></li>
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<li><a href="#org0f81a91">5.1. Mixed \(\mathcal{H}_2\) / \(\mathcal{H}_\infty\) Synthesis</a></li>
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<li><a href="#orgc93b489">5.2. Obtained Super Sensor’s noise</a></li>
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<li><a href="#org417aabd">5.2. Obtained Super Sensor’s noise</a></li>
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||||||
<li><a href="#org5adb8ec">5.3. Obtained Super Sensor’s Uncertainty</a></li>
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<li><a href="#org2dce888">5.3. Obtained Super Sensor’s Uncertainty</a></li>
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<li><a href="#org76f00f7">5.4. Conclusion</a></li>
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<li><a href="#org47da78c">5.4. Conclusion</a></li>
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</ul>
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</ul>
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</li>
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</li>
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<li><a href="#orgf68e579">6. Matlab Functions</a>
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<li><a href="#org0afe5ef">6. Matlab Functions</a>
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<ul>
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<ul>
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<li><a href="#orgdec213a">6.1. <code>createWeight</code></a></li>
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<li><a href="#orge81e522">6.1. <code>createWeight</code></a></li>
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<li><a href="#orgad116a6">6.2. <code>plotMagUncertainty</code></a></li>
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<li><a href="#org37ec2b4">6.2. <code>plotMagUncertainty</code></a></li>
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||||||
<li><a href="#orga641eed">6.3. <code>plotPhaseUncertainty</code></a></li>
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<li><a href="#org9f73572">6.3. <code>plotPhaseUncertainty</code></a></li>
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||||||
</ul>
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</ul>
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</li>
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</li>
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</ul>
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</ul>
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@ -89,27 +89,27 @@
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This document is arranged as follows:
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This document is arranged as follows:
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||||||
</p>
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</p>
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<ul class="org-ul">
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<ul class="org-ul">
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||||||
<li>Section <a href="#org740b45e">1</a>: the sensors are described (dynamics, uncertainty, noise)</li>
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<li>Section <a href="#orgee25d07">1</a>: the sensors are described (dynamics, uncertainty, noise)</li>
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||||||
<li>Section <a href="#orge79447b">2</a>: the sensor fusion architecture is described and the super sensor noise and dynamical uncertainty are derived</li>
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<li>Section <a href="#orga64daad">2</a>: the sensor fusion architecture is described and the super sensor noise and dynamical uncertainty are derived</li>
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||||||
<li>Section <a href="#org4d1175a">3</a>: the \(\mathcal{H}_2\) synthesis is used to design complementary filters such that the RMS value of the super sensor’s noise is minimized</li>
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<li>Section <a href="#orgdd6b9ce">3</a>: the \(\mathcal{H}_2\) synthesis is used to design complementary filters such that the RMS value of the super sensor’s noise is minimized</li>
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||||||
<li>Section <a href="#org92f543f">4</a>: the \(\mathcal{H}_\infty\) synthesis is used to design complementary filters such that the super sensor’s uncertainty is bonded to acceptable values</li>
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<li>Section <a href="#org5d93f37">4</a>: the \(\mathcal{H}_\infty\) synthesis is used to design complementary filters such that the super sensor’s uncertainty is bonded to acceptable values</li>
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||||||
<li>Section <a href="#orga0c0443">5</a>: the mixed \(\mathcal{H}_2/\mathcal{H}_\infty\) synthesis is used to both limit the super sensor’s uncertainty and to lower the RMS value of the super sensor’s noise</li>
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<li>Section <a href="#org9f98c16">5</a>: the mixed \(\mathcal{H}_2/\mathcal{H}_\infty\) synthesis is used to both limit the super sensor’s uncertainty and to lower the RMS value of the super sensor’s noise</li>
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||||||
<li>Section <a href="#org4f93e35">6</a>: Matlab functions used for the analysis are described</li>
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<li>Section <a href="#orgf41dc8d">6</a>: Matlab functions used for the analysis are described</li>
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||||||
</ul>
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</ul>
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||||||
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||||||
<div id="outline-container-org27de0ea" class="outline-2">
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<div id="outline-container-org0a492b7" class="outline-2">
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||||||
<h2 id="org27de0ea"><span class="section-number-2">1</span> Sensor Description</h2>
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<h2 id="org0a492b7"><span class="section-number-2">1</span> Sensor Description</h2>
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||||||
<div class="outline-text-2" id="text-1">
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<div class="outline-text-2" id="text-1">
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||||||
<p>
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<p>
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||||||
<a id="org740b45e"></a>
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<a id="orgee25d07"></a>
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||||||
</p>
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</p>
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<p>
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<p>
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||||||
In Figure <a href="#org6428f94">1</a> is shown a schematic of a sensor model that is used in the following study.
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In Figure <a href="#org35e2340">1</a> is shown a schematic of a sensor model that is used in the following study.
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||||||
In this example, the measured quantity \(x\) is the velocity of an object.
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In this example, the measured quantity \(x\) is the velocity of an object.
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||||||
</p>
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</p>
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||||||
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||||||
<table id="orge5eb43f" border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
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<table id="org2fe6194" 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> Description of signals in Figure <a href="#org6428f94">1</a></caption>
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<caption class="t-above"><span class="table-number">Table 1:</span> Description of signals in Figure <a href="#org35e2340">1</a></caption>
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||||||
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||||||
<colgroup>
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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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||||||
@ -176,8 +176,8 @@ In this example, the measured quantity \(x\) is the velocity of an object.
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|||||||
</tbody>
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</tbody>
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||||||
</table>
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</table>
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||||||
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||||||
<table id="org9dd0355" border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
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<table id="org281ecb3" border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
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||||||
<caption class="t-above"><span class="table-number">Table 2:</span> Description of Systems in Figure <a href="#org6428f94">1</a></caption>
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<caption class="t-above"><span class="table-number">Table 2:</span> Description of Systems in Figure <a href="#org35e2340">1</a></caption>
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||||||
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||||||
<colgroup>
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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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||||||
@ -221,18 +221,18 @@ In this example, the measured quantity \(x\) is the velocity of an object.
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|||||||
</table>
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</table>
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||||||
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||||||
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||||||
<div id="org6428f94" class="figure">
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<div id="org35e2340" class="figure">
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||||||
<p><img src="figs-tikz/sensor_model_noise_uncertainty.png" alt="sensor_model_noise_uncertainty.png" />
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<p><img src="figs-paper/sensor_model_noise_uncertainty.png" alt="sensor_model_noise_uncertainty.png" />
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||||||
</p>
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</p>
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||||||
<p><span class="figure-number">Figure 1: </span>Sensor Model</p>
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<p><span class="figure-number">Figure 1: </span>Sensor Model</p>
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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="outline-container-org615e955" class="outline-3">
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<div id="outline-container-org9575585" class="outline-3">
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||||||
<h3 id="org615e955"><span class="section-number-3">1.1</span> Sensor Dynamics</h3>
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<h3 id="org9575585"><span class="section-number-3">1.1</span> Sensor Dynamics</h3>
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<div class="outline-text-3" id="text-1-1">
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<div class="outline-text-3" id="text-1-1">
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||||||
<p>
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<p>
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||||||
<a id="orgc593869"></a>
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<a id="org52415aa"></a>
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||||||
Let’s consider two sensors measuring the velocity of an object.
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Let’s consider two sensors measuring the velocity of an object.
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</p>
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</p>
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||||||
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@ -263,14 +263,14 @@ G2 = g_pos<span class="org-type">/</span>s<span class="org-type">/</span>(1 <spa
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<p>
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<p>
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||||||
These nominal dynamics are also taken as the model of the sensor dynamics.
|
These nominal dynamics are also taken as the model of the sensor dynamics.
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||||||
The true sensor dynamics has some uncertainty associated to it and described in section <a href="#orgba186f9">1.2</a>.
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The true sensor dynamics has some uncertainty associated to it and described in section <a href="#orgf04439d">1.2</a>.
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||||||
</p>
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</p>
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||||||
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||||||
<p>
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<p>
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||||||
Both sensor dynamics in \([\frac{V}{m/s}]\) are shown in Figure <a href="#org8ac4b85">2</a>.
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Both sensor dynamics in \([\frac{V}{m/s}]\) are shown in Figure <a href="#org2d1ef0b">2</a>.
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||||||
</p>
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</p>
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<div id="org8ac4b85" class="figure">
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<div id="org2d1ef0b" class="figure">
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||||||
<p><img src="figs/sensors_nominal_dynamics.png" alt="sensors_nominal_dynamics.png" />
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<p><img src="figs/sensors_nominal_dynamics.png" alt="sensors_nominal_dynamics.png" />
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||||||
</p>
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</p>
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||||||
<p><span class="figure-number">Figure 2: </span>Sensor nominal dynamics from the velocity of the object to the output voltage</p>
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<p><span class="figure-number">Figure 2: </span>Sensor nominal dynamics from the velocity of the object to the output voltage</p>
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||||||
@ -278,12 +278,12 @@ Both sensor dynamics in \([\frac{V}{m/s}]\) are shown in Figure <a href="#org8ac
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</div>
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</div>
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||||||
</div>
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</div>
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||||||
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<div id="outline-container-org693d3bf" class="outline-3">
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<div id="outline-container-orgec8c81d" class="outline-3">
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||||||
<h3 id="org693d3bf"><span class="section-number-3">1.2</span> Sensor Model Uncertainty</h3>
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<h3 id="orgec8c81d"><span class="section-number-3">1.2</span> Sensor Model Uncertainty</h3>
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||||||
<div class="outline-text-3" id="text-1-2">
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<div class="outline-text-3" id="text-1-2">
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||||||
<p>
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<p>
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||||||
<a id="orgba186f9"></a>
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<a id="orgf04439d"></a>
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||||||
The uncertainty on the sensor dynamics is described by multiplicative uncertainty (Figure <a href="#org6428f94">1</a>).
|
The uncertainty on the sensor dynamics is described by multiplicative uncertainty (Figure <a href="#org35e2340">1</a>).
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||||||
</p>
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</p>
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||||||
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||||||
<p>
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<p>
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||||||
@ -295,7 +295,7 @@ The true sensor dynamics \(G_i(s)\) is then described by \eqref{eq:sensor_dynami
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|||||||
\end{equation}
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\end{equation}
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||||||
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||||||
<p>
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<p>
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||||||
The weights \(W_i(s)\) representing the dynamical uncertainty are defined below and their magnitude is shown in Figure <a href="#orgd098571">3</a>.
|
The weights \(W_i(s)\) representing the dynamical uncertainty are defined below and their magnitude is shown in Figure <a href="#org557e062">3</a>.
|
||||||
</p>
|
</p>
|
||||||
<div class="org-src-container">
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<div class="org-src-container">
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||||||
<pre class="src src-matlab">W1 = createWeight(<span class="org-string">'n'</span>, 2, <span class="org-string">'w0'</span>, 2<span class="org-type">*</span><span class="org-constant">pi</span><span class="org-type">*</span>3, <span class="org-string">'G0'</span>, 2, <span class="org-string">'G1'</span>, 0.1, <span class="org-string">'Gc'</span>, 1) <span class="org-type">*</span> ...
|
<pre class="src src-matlab">W1 = createWeight(<span class="org-string">'n'</span>, 2, <span class="org-string">'w0'</span>, 2<span class="org-type">*</span><span class="org-constant">pi</span><span class="org-type">*</span>3, <span class="org-string">'G0'</span>, 2, <span class="org-string">'G1'</span>, 0.1, <span class="org-string">'Gc'</span>, 1) <span class="org-type">*</span> ...
|
||||||
@ -306,18 +306,18 @@ W2 = createWeight(<span class="org-string">'n'</span>, 2, <span class="org-strin
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|||||||
</div>
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</div>
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||||||
|
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||||||
<p>
|
<p>
|
||||||
The bode plot of the sensors nominal dynamics as well as their defined dynamical spread are shown in Figure <a href="#org4527c71">4</a>.
|
The bode plot of the sensors nominal dynamics as well as their defined dynamical spread are shown in Figure <a href="#orgc218675">4</a>.
|
||||||
</p>
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</p>
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||||||
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||||||
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||||||
<div id="orgd098571" class="figure">
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<div id="org557e062" class="figure">
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||||||
<p><img src="figs/sensors_uncertainty_weights.png" alt="sensors_uncertainty_weights.png" />
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<p><img src="figs/sensors_uncertainty_weights.png" alt="sensors_uncertainty_weights.png" />
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||||||
</p>
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</p>
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||||||
<p><span class="figure-number">Figure 3: </span>Magnitude of the multiplicative uncertainty weights \(|W_i(j\omega)|\)</p>
|
<p><span class="figure-number">Figure 3: </span>Magnitude of the multiplicative uncertainty weights \(|W_i(j\omega)|\)</p>
|
||||||
</div>
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</div>
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||||||
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||||||
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<div id="org4527c71" class="figure">
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<div id="orgc218675" class="figure">
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||||||
<p><img src="figs/sensors_nominal_dynamics_and_uncertainty.png" alt="sensors_nominal_dynamics_and_uncertainty.png" />
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<p><img src="figs/sensors_nominal_dynamics_and_uncertainty.png" alt="sensors_nominal_dynamics_and_uncertainty.png" />
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||||||
</p>
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</p>
|
||||||
<p><span class="figure-number">Figure 4: </span>Nominal Sensor Dynamics \(\hat{G}_i\) (solid lines) as well as the spread of the dynamical uncertainty (background color)</p>
|
<p><span class="figure-number">Figure 4: </span>Nominal Sensor Dynamics \(\hat{G}_i\) (solid lines) as well as the spread of the dynamical uncertainty (background color)</p>
|
||||||
@ -325,12 +325,12 @@ The bode plot of the sensors nominal dynamics as well as their defined dynamical
|
|||||||
</div>
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</div>
|
||||||
</div>
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</div>
|
||||||
|
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||||||
<div id="outline-container-org60da040" class="outline-3">
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<div id="outline-container-org81d9a34" class="outline-3">
|
||||||
<h3 id="org60da040"><span class="section-number-3">1.3</span> Sensor Noise</h3>
|
<h3 id="org81d9a34"><span class="section-number-3">1.3</span> Sensor Noise</h3>
|
||||||
<div class="outline-text-3" id="text-1-3">
|
<div class="outline-text-3" id="text-1-3">
|
||||||
<p>
|
<p>
|
||||||
<a id="orgb5f9d77"></a>
|
<a id="org71b587e"></a>
|
||||||
The noise of the sensors \(n_i\) are modelled by shaping a white noise with unitary PSD \(\tilde{n}_i\) \eqref{eq:unitary_noise_psd} with a LTI transfer function \(N_i(s)\) (Figure <a href="#org6428f94">1</a>).
|
The noise of the sensors \(n_i\) are modelled by shaping a white noise with unitary PSD \(\tilde{n}_i\) \eqref{eq:unitary_noise_psd} with a LTI transfer function \(N_i(s)\) (Figure <a href="#org35e2340">1</a>).
|
||||||
</p>
|
</p>
|
||||||
\begin{equation}
|
\begin{equation}
|
||||||
\Phi_{\tilde{n}_i}(\omega) = 1 \label{eq:unitary_noise_psd}
|
\Phi_{\tilde{n}_i}(\omega) = 1 \label{eq:unitary_noise_psd}
|
||||||
@ -344,7 +344,7 @@ The Power Spectral Density of the sensor noise \(\Phi_{n_i}(\omega)\) is then co
|
|||||||
\end{equation}
|
\end{equation}
|
||||||
|
|
||||||
<p>
|
<p>
|
||||||
The weights \(N_1\) and \(N_2\) representing the amplitude spectral density of the sensor noises are defined below and shown in Figure <a href="#orgf397a85">5</a>.
|
The weights \(N_1\) and \(N_2\) representing the amplitude spectral density of the sensor noises are defined below and shown in Figure <a href="#org51f9788">5</a>.
|
||||||
</p>
|
</p>
|
||||||
<div class="org-src-container">
|
<div class="org-src-container">
|
||||||
<pre class="src src-matlab">omegac = 0.15<span class="org-type">*</span>2<span class="org-type">*</span><span class="org-constant">pi</span>; G0 = 1e<span class="org-type">-</span>1; Ginf = 1e<span class="org-type">-</span>6;
|
<pre class="src src-matlab">omegac = 0.15<span class="org-type">*</span>2<span class="org-type">*</span><span class="org-constant">pi</span>; G0 = 1e<span class="org-type">-</span>1; Ginf = 1e<span class="org-type">-</span>6;
|
||||||
@ -356,7 +356,7 @@ N2 = (Ginf<span class="org-type">*</span>s<span class="org-type">/</span>omegac
|
|||||||
</div>
|
</div>
|
||||||
|
|
||||||
|
|
||||||
<div id="orgf397a85" class="figure">
|
<div id="org51f9788" class="figure">
|
||||||
<p><img src="figs/sensors_noise.png" alt="sensors_noise.png" />
|
<p><img src="figs/sensors_noise.png" alt="sensors_noise.png" />
|
||||||
</p>
|
</p>
|
||||||
<p><span class="figure-number">Figure 5: </span>Amplitude spectral density of the sensors \(\sqrt{\Phi_{n_i}(\omega)} = |N_i(j\omega)|\)</p>
|
<p><span class="figure-number">Figure 5: </span>Amplitude spectral density of the sensors \(\sqrt{\Phi_{n_i}(\omega)} = |N_i(j\omega)|\)</p>
|
||||||
@ -364,8 +364,8 @@ N2 = (Ginf<span class="org-type">*</span>s<span class="org-type">/</span>omegac
|
|||||||
</div>
|
</div>
|
||||||
</div>
|
</div>
|
||||||
|
|
||||||
<div id="outline-container-org0b1325d" class="outline-3">
|
<div id="outline-container-org4e23f6c" class="outline-3">
|
||||||
<h3 id="org0b1325d"><span class="section-number-3">1.4</span> Save Model</h3>
|
<h3 id="org4e23f6c"><span class="section-number-3">1.4</span> Save Model</h3>
|
||||||
<div class="outline-text-3" id="text-1-4">
|
<div class="outline-text-3" id="text-1-4">
|
||||||
<p>
|
<p>
|
||||||
All the dynamical systems representing the sensors are saved for further use.
|
All the dynamical systems representing the sensors are saved for further use.
|
||||||
@ -379,28 +379,28 @@ All the dynamical systems representing the sensors are saved for further use.
|
|||||||
</div>
|
</div>
|
||||||
</div>
|
</div>
|
||||||
|
|
||||||
<div id="outline-container-org0b389f3" class="outline-2">
|
<div id="outline-container-org2cab1a2" class="outline-2">
|
||||||
<h2 id="org0b389f3"><span class="section-number-2">2</span> Introduction to Sensor Fusion</h2>
|
<h2 id="org2cab1a2"><span class="section-number-2">2</span> Introduction to Sensor Fusion</h2>
|
||||||
<div class="outline-text-2" id="text-2">
|
<div class="outline-text-2" id="text-2">
|
||||||
<p>
|
<p>
|
||||||
<a id="orge79447b"></a>
|
<a id="orga64daad"></a>
|
||||||
</p>
|
</p>
|
||||||
</div>
|
</div>
|
||||||
|
|
||||||
<div id="outline-container-org1f9e1b3" class="outline-3">
|
<div id="outline-container-org0cbc92d" class="outline-3">
|
||||||
<h3 id="org1f9e1b3"><span class="section-number-3">2.1</span> Sensor Fusion Architecture</h3>
|
<h3 id="org0cbc92d"><span class="section-number-3">2.1</span> Sensor Fusion Architecture</h3>
|
||||||
<div class="outline-text-3" id="text-2-1">
|
<div class="outline-text-3" id="text-2-1">
|
||||||
<p>
|
<p>
|
||||||
<a id="org0b744b4"></a>
|
<a id="org31e00a0"></a>
|
||||||
</p>
|
</p>
|
||||||
|
|
||||||
<p>
|
<p>
|
||||||
The two sensors presented in Section <a href="#org740b45e">1</a> are now merged together using complementary filters \(H_1(s)\) and \(H_2(s)\) to form a super sensor (Figure <a href="#org5caaf16">6</a>).
|
The two sensors presented in Section <a href="#orgee25d07">1</a> are now merged together using complementary filters \(H_1(s)\) and \(H_2(s)\) to form a super sensor (Figure <a href="#org48a16fd">6</a>).
|
||||||
</p>
|
</p>
|
||||||
|
|
||||||
|
|
||||||
<div id="org5caaf16" class="figure">
|
<div id="org48a16fd" class="figure">
|
||||||
<p><img src="figs-tikz/sensor_fusion_noise_arch.png" alt="sensor_fusion_noise_arch.png" />
|
<p><img src="figs-paper/sensor_fusion_noise_arch.png" alt="sensor_fusion_noise_arch.png" />
|
||||||
</p>
|
</p>
|
||||||
<p><span class="figure-number">Figure 6: </span>Sensor Fusion Architecture</p>
|
<p><span class="figure-number">Figure 6: </span>Sensor Fusion Architecture</p>
|
||||||
</div>
|
</div>
|
||||||
@ -423,11 +423,11 @@ The super sensor estimate \(\hat{x}\) is given by \eqref{eq:super_sensor_estimat
|
|||||||
</div>
|
</div>
|
||||||
</div>
|
</div>
|
||||||
|
|
||||||
<div id="outline-container-org6ce496e" class="outline-3">
|
<div id="outline-container-orge9e0bd4" class="outline-3">
|
||||||
<h3 id="org6ce496e"><span class="section-number-3">2.2</span> Super Sensor Noise</h3>
|
<h3 id="orge9e0bd4"><span class="section-number-3">2.2</span> Super Sensor Noise</h3>
|
||||||
<div class="outline-text-3" id="text-2-2">
|
<div class="outline-text-3" id="text-2-2">
|
||||||
<p>
|
<p>
|
||||||
<a id="org3cb79e5"></a>
|
<a id="orgff055a3"></a>
|
||||||
</p>
|
</p>
|
||||||
|
|
||||||
<p>
|
<p>
|
||||||
@ -458,15 +458,15 @@ And the Root Mean Square (RMS) value of the super sensor noise \(\sigma_n\) is g
|
|||||||
</div>
|
</div>
|
||||||
</div>
|
</div>
|
||||||
|
|
||||||
<div id="outline-container-org8a5c291" class="outline-3">
|
<div id="outline-container-orgefb4347" class="outline-3">
|
||||||
<h3 id="org8a5c291"><span class="section-number-3">2.3</span> Super Sensor Dynamical Uncertainty</h3>
|
<h3 id="orgefb4347"><span class="section-number-3">2.3</span> Super Sensor Dynamical Uncertainty</h3>
|
||||||
<div class="outline-text-3" id="text-2-3">
|
<div class="outline-text-3" id="text-2-3">
|
||||||
<p>
|
<p>
|
||||||
<a id="org2ff763e"></a>
|
<a id="org859b213"></a>
|
||||||
</p>
|
</p>
|
||||||
|
|
||||||
<p>
|
<p>
|
||||||
If we consider some dynamical uncertainty (the true system dynamics \(G_i\) not being perfectly equal to our model \(\hat{G}_i\)) that we model by the use of multiplicative uncertainty (Figure <a href="#org3a5e8c1">7</a>), the super sensor dynamics is then equals to:
|
If we consider some dynamical uncertainty (the true system dynamics \(G_i\) not being perfectly equal to our model \(\hat{G}_i\)) that we model by the use of multiplicative uncertainty (Figure <a href="#org54538d7">7</a>), the super sensor dynamics is then equals to:
|
||||||
</p>
|
</p>
|
||||||
|
|
||||||
\begin{equation}
|
\begin{equation}
|
||||||
@ -478,19 +478,19 @@ If we consider some dynamical uncertainty (the true system dynamics \(G_i\) not
|
|||||||
\end{equation}
|
\end{equation}
|
||||||
|
|
||||||
|
|
||||||
<div id="org3a5e8c1" class="figure">
|
<div id="org54538d7" class="figure">
|
||||||
<p><img src="figs-tikz/sensor_model_uncertainty.png" alt="sensor_model_uncertainty.png" />
|
<p><img src="figs-paper/sensor_model_uncertainty.png" alt="sensor_model_uncertainty.png" />
|
||||||
</p>
|
</p>
|
||||||
<p><span class="figure-number">Figure 7: </span>Sensor Model including Dynamical Uncertainty</p>
|
<p><span class="figure-number">Figure 7: </span>Sensor Model including Dynamical Uncertainty</p>
|
||||||
</div>
|
</div>
|
||||||
|
|
||||||
<p>
|
<p>
|
||||||
The uncertainty set of the transfer function from \(\hat{x}\) to \(x\) at frequency \(\omega\) is bounded in the complex plane by a circle centered on 1 and with a radius equal to \(|W_1(j\omega) H_1(j\omega)| + |W_2(j\omega) H_2(j\omega)|\) as shown in Figure <a href="#orge995373">8</a>.
|
The uncertainty set of the transfer function from \(\hat{x}\) to \(x\) at frequency \(\omega\) is bounded in the complex plane by a circle centered on 1 and with a radius equal to \(|W_1(j\omega) H_1(j\omega)| + |W_2(j\omega) H_2(j\omega)|\) as shown in Figure <a href="#orgb11ef23">8</a>.
|
||||||
</p>
|
</p>
|
||||||
|
|
||||||
|
|
||||||
<div id="orge995373" class="figure">
|
<div id="orgb11ef23" class="figure">
|
||||||
<p><img src="figs-tikz/uncertainty_set_super_sensor.png" alt="uncertainty_set_super_sensor.png" />
|
<p><img src="figs-paper/uncertainty_set_super_sensor.png" alt="uncertainty_set_super_sensor.png" />
|
||||||
</p>
|
</p>
|
||||||
<p><span class="figure-number">Figure 8: </span>Super Sensor model uncertainty displayed in the complex plane</p>
|
<p><span class="figure-number">Figure 8: </span>Super Sensor model uncertainty displayed in the complex plane</p>
|
||||||
</div>
|
</div>
|
||||||
@ -498,11 +498,11 @@ The uncertainty set of the transfer function from \(\hat{x}\) to \(x\) at freque
|
|||||||
</div>
|
</div>
|
||||||
</div>
|
</div>
|
||||||
|
|
||||||
<div id="outline-container-org7cb91ba" class="outline-2">
|
<div id="outline-container-org5896b60" class="outline-2">
|
||||||
<h2 id="org7cb91ba"><span class="section-number-2">3</span> Optimal Super Sensor Noise: \(\mathcal{H}_2\) Synthesis</h2>
|
<h2 id="org5896b60"><span class="section-number-2">3</span> Optimal Super Sensor Noise: \(\mathcal{H}_2\) Synthesis</h2>
|
||||||
<div class="outline-text-2" id="text-3">
|
<div class="outline-text-2" id="text-3">
|
||||||
<p>
|
<p>
|
||||||
<a id="org4d1175a"></a>
|
<a id="orgdd6b9ce"></a>
|
||||||
</p>
|
</p>
|
||||||
<p>
|
<p>
|
||||||
In this section, the complementary filters \(H_1(s)\) and \(H_2(s)\) are designed in order to minimize the RMS value of super sensor noise \(\sigma_n\).
|
In this section, the complementary filters \(H_1(s)\) and \(H_2(s)\) are designed in order to minimize the RMS value of super sensor noise \(\sigma_n\).
|
||||||
@ -520,24 +520,24 @@ The RMS value of the super sensor noise is (neglecting the model uncertainty):
|
|||||||
|
|
||||||
<p>
|
<p>
|
||||||
The goal is to design \(H_1(s)\) and \(H_2(s)\) such that \(H_1(s) + H_2(s) = 1\) (complementary property) and such that \(\left\| \begin{matrix} H_1 N_1 \\ H_2 N_2 \end{matrix} \right\|_2\) is minimized (minimized RMS value of the super sensor noise).
|
The goal is to design \(H_1(s)\) and \(H_2(s)\) such that \(H_1(s) + H_2(s) = 1\) (complementary property) and such that \(\left\| \begin{matrix} H_1 N_1 \\ H_2 N_2 \end{matrix} \right\|_2\) is minimized (minimized RMS value of the super sensor noise).
|
||||||
This is done using the \(\mathcal{H}_2\) synthesis in Section <a href="#org15426e5">3.1</a>.
|
This is done using the \(\mathcal{H}_2\) synthesis in Section <a href="#org5bc9386">3.1</a>.
|
||||||
</p>
|
</p>
|
||||||
</div>
|
</div>
|
||||||
|
|
||||||
<div id="outline-container-orga0474b9" class="outline-3">
|
<div id="outline-container-orgf5a8a84" class="outline-3">
|
||||||
<h3 id="orga0474b9"><span class="section-number-3">3.1</span> \(\mathcal{H}_2\) Synthesis</h3>
|
<h3 id="orgf5a8a84"><span class="section-number-3">3.1</span> \(\mathcal{H}_2\) Synthesis</h3>
|
||||||
<div class="outline-text-3" id="text-3-1">
|
<div class="outline-text-3" id="text-3-1">
|
||||||
<p>
|
<p>
|
||||||
<a id="org15426e5"></a>
|
<a id="org5bc9386"></a>
|
||||||
</p>
|
</p>
|
||||||
|
|
||||||
<p>
|
<p>
|
||||||
Consider the generalized plant \(P_{\mathcal{H}_2}\) shown in Figure <a href="#orgb96eccb">9</a> and described by Equation \eqref{eq:H2_generalized_plant}.
|
Consider the generalized plant \(P_{\mathcal{H}_2}\) shown in Figure <a href="#orgd218886">9</a> and described by Equation \eqref{eq:H2_generalized_plant}.
|
||||||
</p>
|
</p>
|
||||||
|
|
||||||
|
|
||||||
<div id="orgb96eccb" class="figure">
|
<div id="orgd218886" class="figure">
|
||||||
<p><img src="figs-tikz/h_two_optimal_fusion.png" alt="h_two_optimal_fusion.png" />
|
<p><img src="figs-paper/h_two_optimal_fusion.png" alt="h_two_optimal_fusion.png" />
|
||||||
</p>
|
</p>
|
||||||
<p><span class="figure-number">Figure 9: </span>Architecture used for \(\mathcal{H}_\infty\) synthesis of complementary filters</p>
|
<p><span class="figure-number">Figure 9: </span>Architecture used for \(\mathcal{H}_\infty\) synthesis of complementary filters</p>
|
||||||
</div>
|
</div>
|
||||||
@ -592,10 +592,10 @@ Finally, \(H_1(s)\) is defined as follows
|
|||||||
</div>
|
</div>
|
||||||
|
|
||||||
<p>
|
<p>
|
||||||
The obtained complementary filters are shown in Figure <a href="#orga2bc39b">10</a>.
|
The obtained complementary filters are shown in Figure <a href="#org8c7ba6b">10</a>.
|
||||||
</p>
|
</p>
|
||||||
|
|
||||||
<div id="orga2bc39b" class="figure">
|
<div id="org8c7ba6b" class="figure">
|
||||||
<p><img src="figs/htwo_comp_filters.png" alt="htwo_comp_filters.png" />
|
<p><img src="figs/htwo_comp_filters.png" alt="htwo_comp_filters.png" />
|
||||||
</p>
|
</p>
|
||||||
<p><span class="figure-number">Figure 10: </span>Obtained complementary filters using the \(\mathcal{H}_2\) Synthesis</p>
|
<p><span class="figure-number">Figure 10: </span>Obtained complementary filters using the \(\mathcal{H}_2\) Synthesis</p>
|
||||||
@ -603,11 +603,11 @@ The obtained complementary filters are shown in Figure <a href="#orga2bc39b">10<
|
|||||||
</div>
|
</div>
|
||||||
</div>
|
</div>
|
||||||
|
|
||||||
<div id="outline-container-org1273dcd" class="outline-3">
|
<div id="outline-container-org72159df" class="outline-3">
|
||||||
<h3 id="org1273dcd"><span class="section-number-3">3.2</span> Super Sensor Noise</h3>
|
<h3 id="org72159df"><span class="section-number-3">3.2</span> Super Sensor Noise</h3>
|
||||||
<div class="outline-text-3" id="text-3-2">
|
<div class="outline-text-3" id="text-3-2">
|
||||||
<p>
|
<p>
|
||||||
<a id="org0a41807"></a>
|
<a id="orgc7cc0a8"></a>
|
||||||
</p>
|
</p>
|
||||||
|
|
||||||
<p>
|
<p>
|
||||||
@ -622,13 +622,13 @@ PSD_H2 = abs(squeeze(freqresp(N1<span class="org-type">*</span>H1, freqs, <span
|
|||||||
</div>
|
</div>
|
||||||
|
|
||||||
<p>
|
<p>
|
||||||
The obtained ASD are shown in Figure <a href="#org9628dea">11</a>.
|
The obtained ASD are shown in Figure <a href="#orge481bcd">11</a>.
|
||||||
</p>
|
</p>
|
||||||
|
|
||||||
<p>
|
<p>
|
||||||
The RMS value of the individual sensors and of the super sensor are listed in Table <a href="#org124b0f8">3</a>.
|
The RMS value of the individual sensors and of the super sensor are listed in Table <a href="#org3918d27">3</a>.
|
||||||
</p>
|
</p>
|
||||||
<table id="org124b0f8" border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
|
<table id="org3918d27" border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
|
||||||
<caption class="t-above"><span class="table-number">Table 3:</span> RMS value of the individual sensor noise and of the super sensor using the \(\mathcal{H}_2\) Synthesis</caption>
|
<caption class="t-above"><span class="table-number">Table 3:</span> RMS value of the individual sensor noise and of the super sensor using the \(\mathcal{H}_2\) Synthesis</caption>
|
||||||
|
|
||||||
<colgroup>
|
<colgroup>
|
||||||
@ -661,7 +661,7 @@ The RMS value of the individual sensors and of the super sensor are listed in Ta
|
|||||||
</table>
|
</table>
|
||||||
|
|
||||||
|
|
||||||
<div id="org9628dea" class="figure">
|
<div id="orge481bcd" class="figure">
|
||||||
<p><img src="figs/psd_sensors_htwo_synthesis.png" alt="psd_sensors_htwo_synthesis.png" />
|
<p><img src="figs/psd_sensors_htwo_synthesis.png" alt="psd_sensors_htwo_synthesis.png" />
|
||||||
</p>
|
</p>
|
||||||
<p><span class="figure-number">Figure 11: </span>Power Spectral Density of the estimated \(\hat{x}\) using the two sensors alone and using the optimally fused signal</p>
|
<p><span class="figure-number">Figure 11: </span>Power Spectral Density of the estimated \(\hat{x}\) using the two sensors alone and using the optimally fused signal</p>
|
||||||
@ -670,19 +670,19 @@ The RMS value of the individual sensors and of the super sensor are listed in Ta
|
|||||||
<p>
|
<p>
|
||||||
A time domain simulation is now performed.
|
A time domain simulation is now performed.
|
||||||
The measured velocity \(x\) is set to be a sweep sine with an amplitude of \(0.1\ [m/s]\).
|
The measured velocity \(x\) is set to be a sweep sine with an amplitude of \(0.1\ [m/s]\).
|
||||||
The velocity estimates from the two sensors and from the super sensors are shown in Figure <a href="#orga63fd84">12</a>.
|
The velocity estimates from the two sensors and from the super sensors are shown in Figure <a href="#org9c2a7e4">12</a>.
|
||||||
The resulting noises are displayed in Figure <a href="#orgf8fd218">13</a>.
|
The resulting noises are displayed in Figure <a href="#org4cc42bf">13</a>.
|
||||||
</p>
|
</p>
|
||||||
|
|
||||||
|
|
||||||
<div id="orga63fd84" class="figure">
|
<div id="org9c2a7e4" class="figure">
|
||||||
<p><img src="figs/super_sensor_time_domain_h2.png" alt="super_sensor_time_domain_h2.png" />
|
<p><img src="figs/super_sensor_time_domain_h2.png" alt="super_sensor_time_domain_h2.png" />
|
||||||
</p>
|
</p>
|
||||||
<p><span class="figure-number">Figure 12: </span>Noise of individual sensors and noise of the super sensor</p>
|
<p><span class="figure-number">Figure 12: </span>Noise of individual sensors and noise of the super sensor</p>
|
||||||
</div>
|
</div>
|
||||||
|
|
||||||
|
|
||||||
<div id="orgf8fd218" class="figure">
|
<div id="org4cc42bf" class="figure">
|
||||||
<p><img src="figs/sensor_noise_H2_time_domain.png" alt="sensor_noise_H2_time_domain.png" />
|
<p><img src="figs/sensor_noise_H2_time_domain.png" alt="sensor_noise_H2_time_domain.png" />
|
||||||
</p>
|
</p>
|
||||||
<p><span class="figure-number">Figure 13: </span>Noise of the two sensors \(n_1, n_2\) and noise of the super sensor \(n\)</p>
|
<p><span class="figure-number">Figure 13: </span>Noise of the two sensors \(n_1, n_2\) and noise of the super sensor \(n\)</p>
|
||||||
@ -690,15 +690,15 @@ The resulting noises are displayed in Figure <a href="#orgf8fd218">13</a>.
|
|||||||
</div>
|
</div>
|
||||||
</div>
|
</div>
|
||||||
|
|
||||||
<div id="outline-container-orge9c2d41" class="outline-3">
|
<div id="outline-container-org8ffba19" class="outline-3">
|
||||||
<h3 id="orge9c2d41"><span class="section-number-3">3.3</span> Discrepancy between sensor dynamics and model</h3>
|
<h3 id="org8ffba19"><span class="section-number-3">3.3</span> Discrepancy between sensor dynamics and model</h3>
|
||||||
<div class="outline-text-3" id="text-3-3">
|
<div class="outline-text-3" id="text-3-3">
|
||||||
<p>
|
<p>
|
||||||
If we consider sensor dynamical uncertainty as explained in Section <a href="#orgba186f9">1.2</a>, we can compute what would be the super sensor dynamical uncertainty when using the complementary filters obtained using the \(\mathcal{H}_2\) Synthesis.
|
If we consider sensor dynamical uncertainty as explained in Section <a href="#orgf04439d">1.2</a>, we can compute what would be the super sensor dynamical uncertainty when using the complementary filters obtained using the \(\mathcal{H}_2\) Synthesis.
|
||||||
</p>
|
</p>
|
||||||
|
|
||||||
<p>
|
<p>
|
||||||
The super sensor dynamical uncertainty is shown in Figure <a href="#orgfce6557">14</a>.
|
The super sensor dynamical uncertainty is shown in Figure <a href="#org865879b">14</a>.
|
||||||
</p>
|
</p>
|
||||||
|
|
||||||
<p>
|
<p>
|
||||||
@ -706,7 +706,7 @@ It is shown that the phase uncertainty is not bounded between 100Hz and 200Hz.
|
|||||||
As a result the super sensor signal can not be used for feedback applications about 100Hz.
|
As a result the super sensor signal can not be used for feedback applications about 100Hz.
|
||||||
</p>
|
</p>
|
||||||
|
|
||||||
<div id="orgfce6557" class="figure">
|
<div id="org865879b" class="figure">
|
||||||
<p><img src="figs/super_sensor_dynamical_uncertainty_H2.png" alt="super_sensor_dynamical_uncertainty_H2.png" />
|
<p><img src="figs/super_sensor_dynamical_uncertainty_H2.png" alt="super_sensor_dynamical_uncertainty_H2.png" />
|
||||||
</p>
|
</p>
|
||||||
<p><span class="figure-number">Figure 14: </span>Super sensor dynamical uncertainty when using the \(\mathcal{H}_2\) Synthesis</p>
|
<p><span class="figure-number">Figure 14: </span>Super sensor dynamical uncertainty when using the \(\mathcal{H}_2\) Synthesis</p>
|
||||||
@ -715,11 +715,11 @@ As a result the super sensor signal can not be used for feedback applications ab
|
|||||||
</div>
|
</div>
|
||||||
</div>
|
</div>
|
||||||
|
|
||||||
<div id="outline-container-orgef8f365" class="outline-2">
|
<div id="outline-container-org26ea7b1" class="outline-2">
|
||||||
<h2 id="orgef8f365"><span class="section-number-2">4</span> Robust Sensor Fusion: \(\mathcal{H}_\infty\) Synthesis</h2>
|
<h2 id="org26ea7b1"><span class="section-number-2">4</span> Robust Sensor Fusion: \(\mathcal{H}_\infty\) Synthesis</h2>
|
||||||
<div class="outline-text-2" id="text-4">
|
<div class="outline-text-2" id="text-4">
|
||||||
<p>
|
<p>
|
||||||
<a id="org92f543f"></a>
|
<a id="org5d93f37"></a>
|
||||||
</p>
|
</p>
|
||||||
<p>
|
<p>
|
||||||
We initially considered perfectly known sensor dynamics so that it can be perfectly inverted.
|
We initially considered perfectly known sensor dynamics so that it can be perfectly inverted.
|
||||||
@ -727,18 +727,18 @@ We initially considered perfectly known sensor dynamics so that it can be perfec
|
|||||||
|
|
||||||
<p>
|
<p>
|
||||||
We now take into account the fact that the sensor dynamics is only partially known.
|
We now take into account the fact that the sensor dynamics is only partially known.
|
||||||
To do so, we model the uncertainty that we have on the sensor dynamics by multiplicative input uncertainty as shown in Figure <a href="#org98d72a2">15</a>.
|
To do so, we model the uncertainty that we have on the sensor dynamics by multiplicative input uncertainty as shown in Figure <a href="#org2b71ca6">15</a>.
|
||||||
</p>
|
</p>
|
||||||
|
|
||||||
|
|
||||||
<div id="org98d72a2" class="figure">
|
<div id="org2b71ca6" class="figure">
|
||||||
<p><img src="figs-tikz/sensor_fusion_arch_uncertainty.png" alt="sensor_fusion_arch_uncertainty.png" />
|
<p><img src="figs-paper/sensor_fusion_arch_uncertainty.png" alt="sensor_fusion_arch_uncertainty.png" />
|
||||||
</p>
|
</p>
|
||||||
<p><span class="figure-number">Figure 15: </span>Sensor fusion architecture with sensor dynamics uncertainty</p>
|
<p><span class="figure-number">Figure 15: </span>Sensor fusion architecture with sensor dynamics uncertainty</p>
|
||||||
</div>
|
</div>
|
||||||
|
|
||||||
<p>
|
<p>
|
||||||
As explained in Section <a href="#orgba186f9">1.2</a>, at each frequency \(\omega\), the dynamical uncertainty of the super sensor can be represented in the complex plane by a circle with a radius equals to \(|H_1(j\omega) W_1(j\omega)| + |H_2(j\omega) W_2(j\omega)|\) and centered on 1.
|
As explained in Section <a href="#orgf04439d">1.2</a>, at each frequency \(\omega\), the dynamical uncertainty of the super sensor can be represented in the complex plane by a circle with a radius equals to \(|H_1(j\omega) W_1(j\omega)| + |H_2(j\omega) W_2(j\omega)|\) and centered on 1.
|
||||||
</p>
|
</p>
|
||||||
|
|
||||||
<p>
|
<p>
|
||||||
@ -760,7 +760,7 @@ In order to specify a wanted upper bound on the dynamical uncertainty, a weight
|
|||||||
\end{align}
|
\end{align}
|
||||||
|
|
||||||
<p>
|
<p>
|
||||||
The choice of \(W_u\) is presented in Section <a href="#org510f718">4.1</a>.
|
The choice of \(W_u\) is presented in Section <a href="#orgefed264">4.1</a>.
|
||||||
</p>
|
</p>
|
||||||
|
|
||||||
|
|
||||||
@ -778,15 +778,15 @@ The objective is to design \(H_1(s)\) and \(H_2(s)\) such that \(H_1(s) + H_2(s)
|
|||||||
</p>
|
</p>
|
||||||
|
|
||||||
<p>
|
<p>
|
||||||
This is done using the \(\mathcal{H}_\infty\) synthesis in Section <a href="#org48c47d7">4.2</a>.
|
This is done using the \(\mathcal{H}_\infty\) synthesis in Section <a href="#org2c990ce">4.2</a>.
|
||||||
</p>
|
</p>
|
||||||
</div>
|
</div>
|
||||||
|
|
||||||
<div id="outline-container-org6f283c6" class="outline-3">
|
<div id="outline-container-org8dfd9d2" class="outline-3">
|
||||||
<h3 id="org6f283c6"><span class="section-number-3">4.1</span> Weighting Function used to bound the super sensor uncertainty</h3>
|
<h3 id="org8dfd9d2"><span class="section-number-3">4.1</span> Weighting Function used to bound the super sensor uncertainty</h3>
|
||||||
<div class="outline-text-3" id="text-4-1">
|
<div class="outline-text-3" id="text-4-1">
|
||||||
<p>
|
<p>
|
||||||
<a id="org510f718"></a>
|
<a id="orgefed264"></a>
|
||||||
</p>
|
</p>
|
||||||
|
|
||||||
<p>
|
<p>
|
||||||
@ -799,7 +799,7 @@ This is done using the \(\mathcal{H}_\infty\) synthesis in Section <a href="#org
|
|||||||
\end{align}
|
\end{align}
|
||||||
|
|
||||||
<p>
|
<p>
|
||||||
The uncertainty bounds of the two individual sensor as well as the wanted maximum uncertainty bounds of the super sensor are shown in Figure <a href="#org3a8c93b">16</a>.
|
The uncertainty bounds of the two individual sensor as well as the wanted maximum uncertainty bounds of the super sensor are shown in Figure <a href="#orgdffee80">16</a>.
|
||||||
</p>
|
</p>
|
||||||
|
|
||||||
<div class="org-src-container">
|
<div class="org-src-container">
|
||||||
@ -810,7 +810,7 @@ Wu = createWeight(<span class="org-string">'n'</span>, 2, <span class="org-strin
|
|||||||
</div>
|
</div>
|
||||||
|
|
||||||
|
|
||||||
<div id="org3a8c93b" class="figure">
|
<div id="orgdffee80" class="figure">
|
||||||
<p><img src="figs/weight_uncertainty_bounds_Wu.png" alt="weight_uncertainty_bounds_Wu.png" />
|
<p><img src="figs/weight_uncertainty_bounds_Wu.png" alt="weight_uncertainty_bounds_Wu.png" />
|
||||||
</p>
|
</p>
|
||||||
<p><span class="figure-number">Figure 16: </span>Uncertainty region of the two sensors as well as the wanted maximum uncertainty of the super sensor (dashed lines)</p>
|
<p><span class="figure-number">Figure 16: </span>Uncertainty region of the two sensors as well as the wanted maximum uncertainty of the super sensor (dashed lines)</p>
|
||||||
@ -818,20 +818,20 @@ Wu = createWeight(<span class="org-string">'n'</span>, 2, <span class="org-strin
|
|||||||
</div>
|
</div>
|
||||||
</div>
|
</div>
|
||||||
|
|
||||||
<div id="outline-container-org027886f" class="outline-3">
|
<div id="outline-container-org7422ade" class="outline-3">
|
||||||
<h3 id="org027886f"><span class="section-number-3">4.2</span> \(\mathcal{H}_\infty\) Synthesis</h3>
|
<h3 id="org7422ade"><span class="section-number-3">4.2</span> \(\mathcal{H}_\infty\) Synthesis</h3>
|
||||||
<div class="outline-text-3" id="text-4-2">
|
<div class="outline-text-3" id="text-4-2">
|
||||||
<p>
|
<p>
|
||||||
<a id="org48c47d7"></a>
|
<a id="org2c990ce"></a>
|
||||||
</p>
|
</p>
|
||||||
|
|
||||||
<p>
|
<p>
|
||||||
The generalized plant \(P_{\mathcal{H}_\infty}\) used for the \(\mathcal{H}_\infty\) Synthesis of the complementary filters is shown in Figure <a href="#orgaac3e7e">17</a> and is described by Equation \eqref{eq:Hinf_generalized_plant}.
|
The generalized plant \(P_{\mathcal{H}_\infty}\) used for the \(\mathcal{H}_\infty\) Synthesis of the complementary filters is shown in Figure <a href="#org59777f5">17</a> and is described by Equation \eqref{eq:Hinf_generalized_plant}.
|
||||||
</p>
|
</p>
|
||||||
|
|
||||||
|
|
||||||
<div id="orgaac3e7e" class="figure">
|
<div id="org59777f5" class="figure">
|
||||||
<p><img src="figs-tikz/h_infinity_robust_fusion.png" alt="h_infinity_robust_fusion.png" />
|
<p><img src="figs-paper/h_infinity_robust_fusion.png" alt="h_infinity_robust_fusion.png" />
|
||||||
</p>
|
</p>
|
||||||
<p><span class="figure-number">Figure 17: </span>Architecture used for \(\mathcal{H}_\infty\) synthesis of complementary filters</p>
|
<p><span class="figure-number">Figure 17: </span>Architecture used for \(\mathcal{H}_\infty\) synthesis of complementary filters</p>
|
||||||
</div>
|
</div>
|
||||||
@ -897,11 +897,11 @@ The \(\mathcal{H}_\infty\) is successful as the \(\mathcal{H}_\infty\) norm of t
|
|||||||
</div>
|
</div>
|
||||||
|
|
||||||
<p>
|
<p>
|
||||||
The obtained complementary filters as well as the wanted upper bounds are shown in Figure <a href="#org4673b81">18</a>.
|
The obtained complementary filters as well as the wanted upper bounds are shown in Figure <a href="#orga1806e3">18</a>.
|
||||||
</p>
|
</p>
|
||||||
|
|
||||||
|
|
||||||
<div id="org4673b81" class="figure">
|
<div id="orga1806e3" class="figure">
|
||||||
<p><img src="figs/hinf_comp_filters.png" alt="hinf_comp_filters.png" />
|
<p><img src="figs/hinf_comp_filters.png" alt="hinf_comp_filters.png" />
|
||||||
</p>
|
</p>
|
||||||
<p><span class="figure-number">Figure 18: </span>Obtained complementary filters using the \(\mathcal{H}_\infty\) Synthesis</p>
|
<p><span class="figure-number">Figure 18: </span>Obtained complementary filters using the \(\mathcal{H}_\infty\) Synthesis</p>
|
||||||
@ -909,11 +909,11 @@ The obtained complementary filters as well as the wanted upper bounds are shown
|
|||||||
</div>
|
</div>
|
||||||
</div>
|
</div>
|
||||||
|
|
||||||
<div id="outline-container-org0ad8fe8" class="outline-3">
|
<div id="outline-container-orga0267c6" class="outline-3">
|
||||||
<h3 id="org0ad8fe8"><span class="section-number-3">4.3</span> Super sensor uncertainty</h3>
|
<h3 id="orga0267c6"><span class="section-number-3">4.3</span> Super sensor uncertainty</h3>
|
||||||
<div class="outline-text-3" id="text-4-3">
|
<div class="outline-text-3" id="text-4-3">
|
||||||
<p>
|
<p>
|
||||||
The super sensor dynamical uncertainty is displayed in Figure <a href="#org9f35650">19</a>.
|
The super sensor dynamical uncertainty is displayed in Figure <a href="#orge75f5ef">19</a>.
|
||||||
It is confirmed that the super sensor dynamical uncertainty is less than the maximum allowed uncertainty defined by the norm of \(W_u(s)\).
|
It is confirmed that the super sensor dynamical uncertainty is less than the maximum allowed uncertainty defined by the norm of \(W_u(s)\).
|
||||||
</p>
|
</p>
|
||||||
|
|
||||||
@ -922,7 +922,7 @@ The \(\mathcal{H}_\infty\) synthesis thus allows to design filters such that the
|
|||||||
</p>
|
</p>
|
||||||
|
|
||||||
|
|
||||||
<div id="org9f35650" class="figure">
|
<div id="orge75f5ef" class="figure">
|
||||||
<p><img src="figs/super_sensor_dynamical_uncertainty_Hinf.png" alt="super_sensor_dynamical_uncertainty_Hinf.png" />
|
<p><img src="figs/super_sensor_dynamical_uncertainty_Hinf.png" alt="super_sensor_dynamical_uncertainty_Hinf.png" />
|
||||||
</p>
|
</p>
|
||||||
<p><span class="figure-number">Figure 19: </span>Super sensor dynamical uncertainty (solid curve) when using the \(\mathcal{H}_\infty\) Synthesis</p>
|
<p><span class="figure-number">Figure 19: </span>Super sensor dynamical uncertainty (solid curve) when using the \(\mathcal{H}_\infty\) Synthesis</p>
|
||||||
@ -930,12 +930,12 @@ The \(\mathcal{H}_\infty\) synthesis thus allows to design filters such that the
|
|||||||
</div>
|
</div>
|
||||||
</div>
|
</div>
|
||||||
|
|
||||||
<div id="outline-container-orgd5efe47" class="outline-3">
|
<div id="outline-container-org979fede" class="outline-3">
|
||||||
<h3 id="orgd5efe47"><span class="section-number-3">4.4</span> Super sensor noise</h3>
|
<h3 id="org979fede"><span class="section-number-3">4.4</span> Super sensor noise</h3>
|
||||||
<div class="outline-text-3" id="text-4-4">
|
<div class="outline-text-3" id="text-4-4">
|
||||||
<p>
|
<p>
|
||||||
We now compute the obtain Power Spectral Density of the super sensor’s noise.
|
We now compute the obtain Power Spectral Density of the super sensor’s noise.
|
||||||
The Amplitude Spectral Densities are shown in Figure <a href="#orgf375b8c">20</a>.
|
The Amplitude Spectral Densities are shown in Figure <a href="#org5aac73f">20</a>.
|
||||||
</p>
|
</p>
|
||||||
|
|
||||||
<div class="org-src-container">
|
<div class="org-src-container">
|
||||||
@ -947,18 +947,18 @@ PSD_Hinf = abs(squeeze(freqresp(N1<span class="org-type">*</span>H1, freqs, <spa
|
|||||||
</div>
|
</div>
|
||||||
|
|
||||||
<p>
|
<p>
|
||||||
The obtained RMS of the super sensor noise in the \(\mathcal{H}_2\) and \(\mathcal{H}_\infty\) case are shown in Table <a href="#org2c81207">4</a>.
|
The obtained RMS of the super sensor noise in the \(\mathcal{H}_2\) and \(\mathcal{H}_\infty\) case are shown in Table <a href="#org3e08452">4</a>.
|
||||||
As expected, the super sensor obtained from the \(\mathcal{H}_\infty\) synthesis is much noisier than the super sensor obtained from the \(\mathcal{H}_2\) synthesis.
|
As expected, the super sensor obtained from the \(\mathcal{H}_\infty\) synthesis is much noisier than the super sensor obtained from the \(\mathcal{H}_2\) synthesis.
|
||||||
</p>
|
</p>
|
||||||
|
|
||||||
|
|
||||||
<div id="orgf375b8c" class="figure">
|
<div id="org5aac73f" class="figure">
|
||||||
<p><img src="figs/psd_sensors_hinf_synthesis.png" alt="psd_sensors_hinf_synthesis.png" />
|
<p><img src="figs/psd_sensors_hinf_synthesis.png" alt="psd_sensors_hinf_synthesis.png" />
|
||||||
</p>
|
</p>
|
||||||
<p><span class="figure-number">Figure 20: </span>Power Spectral Density of the estimated \(\hat{x}\) using the two sensors alone and using the</p>
|
<p><span class="figure-number">Figure 20: </span>Power Spectral Density of the estimated \(\hat{x}\) using the two sensors alone and using the</p>
|
||||||
</div>
|
</div>
|
||||||
|
|
||||||
<table id="org2c81207" border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
|
<table id="org3e08452" border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
|
||||||
<caption class="t-above"><span class="table-number">Table 4:</span> Comparison of the obtained RMS noise of the super sensor</caption>
|
<caption class="t-above"><span class="table-number">Table 4:</span> Comparison of the obtained RMS noise of the super sensor</caption>
|
||||||
|
|
||||||
<colgroup>
|
<colgroup>
|
||||||
@ -987,8 +987,8 @@ As expected, the super sensor obtained from the \(\mathcal{H}_\infty\) synthesis
|
|||||||
</div>
|
</div>
|
||||||
</div>
|
</div>
|
||||||
|
|
||||||
<div id="outline-container-org0355c08" class="outline-3">
|
<div id="outline-container-orgda33992" class="outline-3">
|
||||||
<h3 id="org0355c08"><span class="section-number-3">4.5</span> Conclusion</h3>
|
<h3 id="orgda33992"><span class="section-number-3">4.5</span> Conclusion</h3>
|
||||||
<div class="outline-text-3" id="text-4-5">
|
<div class="outline-text-3" id="text-4-5">
|
||||||
<p>
|
<p>
|
||||||
Using the \(\mathcal{H}_\infty\) synthesis, the dynamical uncertainty of the super sensor can be bounded to acceptable values.
|
Using the \(\mathcal{H}_\infty\) synthesis, the dynamical uncertainty of the super sensor can be bounded to acceptable values.
|
||||||
@ -1001,23 +1001,23 @@ However, the RMS of the super sensor noise is not optimized as it was the case w
|
|||||||
</div>
|
</div>
|
||||||
</div>
|
</div>
|
||||||
|
|
||||||
<div id="outline-container-org3654cee" class="outline-2">
|
<div id="outline-container-org15afe90" class="outline-2">
|
||||||
<h2 id="org3654cee"><span class="section-number-2">5</span> Optimal and Robust Sensor Fusion: Mixed \(\mathcal{H}_2/\mathcal{H}_\infty\) Synthesis</h2>
|
<h2 id="org15afe90"><span class="section-number-2">5</span> Optimal and Robust Sensor Fusion: Mixed \(\mathcal{H}_2/\mathcal{H}_\infty\) Synthesis</h2>
|
||||||
<div class="outline-text-2" id="text-5">
|
<div class="outline-text-2" id="text-5">
|
||||||
<p>
|
<p>
|
||||||
<a id="orga0c0443"></a>
|
<a id="org9f98c16"></a>
|
||||||
</p>
|
</p>
|
||||||
<p>
|
<p>
|
||||||
The (optima) \(\mathcal{H}_2\) synthesis and the (robust) \(\mathcal{H}_\infty\) synthesis are now combined to form an Optimal and Robust synthesis of complementary filters for sensor fusion.
|
The (optima) \(\mathcal{H}_2\) synthesis and the (robust) \(\mathcal{H}_\infty\) synthesis are now combined to form an Optimal and Robust synthesis of complementary filters for sensor fusion.
|
||||||
</p>
|
</p>
|
||||||
|
|
||||||
<p>
|
<p>
|
||||||
The sensor fusion architecture is shown in Figure <a href="#org6b7e130">21</a> (\(\hat{G}_i\) are omitted for space reasons).
|
The sensor fusion architecture is shown in Figure <a href="#org3cc874e">21</a> (\(\hat{G}_i\) are omitted for space reasons).
|
||||||
</p>
|
</p>
|
||||||
|
|
||||||
|
|
||||||
<div id="org6b7e130" class="figure">
|
<div id="org3cc874e" class="figure">
|
||||||
<p><img src="figs-tikz/sensor_fusion_arch_full.png" alt="sensor_fusion_arch_full.png" />
|
<p><img src="figs-paper/sensor_fusion_arch_full.png" alt="sensor_fusion_arch_full.png" />
|
||||||
</p>
|
</p>
|
||||||
<p><span class="figure-number">Figure 21: </span>Sensor fusion architecture with sensor dynamics uncertainty</p>
|
<p><span class="figure-number">Figure 21: </span>Sensor fusion architecture with sensor dynamics uncertainty</p>
|
||||||
</div>
|
</div>
|
||||||
@ -1031,18 +1031,18 @@ The goal is to design complementary filters such that:
|
|||||||
</ul>
|
</ul>
|
||||||
|
|
||||||
<p>
|
<p>
|
||||||
To do so, we can use the Mixed \(\mathcal{H}_2/\mathcal{H}_\infty\) Synthesis presented in Section <a href="#org7eb3cad">5.1</a>.
|
To do so, we can use the Mixed \(\mathcal{H}_2/\mathcal{H}_\infty\) Synthesis presented in Section <a href="#orgbbc8594">5.1</a>.
|
||||||
</p>
|
</p>
|
||||||
</div>
|
</div>
|
||||||
<div id="outline-container-org4d41c02" class="outline-3">
|
<div id="outline-container-org0f81a91" class="outline-3">
|
||||||
<h3 id="org4d41c02"><span class="section-number-3">5.1</span> Mixed \(\mathcal{H}_2\) / \(\mathcal{H}_\infty\) Synthesis</h3>
|
<h3 id="org0f81a91"><span class="section-number-3">5.1</span> Mixed \(\mathcal{H}_2\) / \(\mathcal{H}_\infty\) Synthesis</h3>
|
||||||
<div class="outline-text-3" id="text-5-1">
|
<div class="outline-text-3" id="text-5-1">
|
||||||
<p>
|
<p>
|
||||||
<a id="org7eb3cad"></a>
|
<a id="orgbbc8594"></a>
|
||||||
</p>
|
</p>
|
||||||
|
|
||||||
<p>
|
<p>
|
||||||
The synthesis architecture that is used here is shown in Figure <a href="#orgb136ff8">22</a>.
|
The synthesis architecture that is used here is shown in Figure <a href="#orga971cdb">22</a>.
|
||||||
</p>
|
</p>
|
||||||
|
|
||||||
<p>
|
<p>
|
||||||
@ -1054,8 +1054,8 @@ The filter \(H_2(s)\) is synthesized such that it:
|
|||||||
</ul>
|
</ul>
|
||||||
|
|
||||||
|
|
||||||
<div id="orgb136ff8" class="figure">
|
<div id="orga971cdb" class="figure">
|
||||||
<p><img src="figs-tikz/mixed_h2_hinf_synthesis.png" alt="mixed_h2_hinf_synthesis.png" />
|
<p><img src="figs-paper/mixed_h2_hinf_synthesis.png" alt="mixed_h2_hinf_synthesis.png" />
|
||||||
</p>
|
</p>
|
||||||
<p><span class="figure-number">Figure 22: </span>Mixed \(\mathcal{H}_2/\mathcal{H}_\infty\) Synthesis</p>
|
<p><span class="figure-number">Figure 22: </span>Mixed \(\mathcal{H}_2/\mathcal{H}_\infty\) Synthesis</p>
|
||||||
</div>
|
</div>
|
||||||
@ -1098,11 +1098,11 @@ And the mixed \(\mathcal{H}_2/\mathcal{H}_\infty\) synthesis is performed.
|
|||||||
</div>
|
</div>
|
||||||
|
|
||||||
<p>
|
<p>
|
||||||
The obtained complementary filters are shown in Figure <a href="#org1bf2ce4">23</a>.
|
The obtained complementary filters are shown in Figure <a href="#org30e8a3f">23</a>.
|
||||||
</p>
|
</p>
|
||||||
|
|
||||||
|
|
||||||
<div id="org1bf2ce4" class="figure">
|
<div id="org30e8a3f" class="figure">
|
||||||
<p><img src="figs/htwo_hinf_comp_filters.png" alt="htwo_hinf_comp_filters.png" />
|
<p><img src="figs/htwo_hinf_comp_filters.png" alt="htwo_hinf_comp_filters.png" />
|
||||||
</p>
|
</p>
|
||||||
<p><span class="figure-number">Figure 23: </span>Obtained complementary filters after mixed \(\mathcal{H}_2/\mathcal{H}_\infty\) synthesis</p>
|
<p><span class="figure-number">Figure 23: </span>Obtained complementary filters after mixed \(\mathcal{H}_2/\mathcal{H}_\infty\) synthesis</p>
|
||||||
@ -1110,19 +1110,19 @@ The obtained complementary filters are shown in Figure <a href="#org1bf2ce4">23<
|
|||||||
</div>
|
</div>
|
||||||
</div>
|
</div>
|
||||||
|
|
||||||
<div id="outline-container-orgc93b489" class="outline-3">
|
<div id="outline-container-org417aabd" class="outline-3">
|
||||||
<h3 id="orgc93b489"><span class="section-number-3">5.2</span> Obtained Super Sensor’s noise</h3>
|
<h3 id="org417aabd"><span class="section-number-3">5.2</span> Obtained Super Sensor’s noise</h3>
|
||||||
<div class="outline-text-3" id="text-5-2">
|
<div class="outline-text-3" id="text-5-2">
|
||||||
<p>
|
<p>
|
||||||
The Amplitude Spectral Density of the super sensor’s noise is shown in Figure <a href="#org6c93dd1">24</a>.
|
The Amplitude Spectral Density of the super sensor’s noise is shown in Figure <a href="#orgfdfbf76">24</a>.
|
||||||
</p>
|
</p>
|
||||||
|
|
||||||
<p>
|
<p>
|
||||||
A time domain simulation is shown in Figure <a href="#orgb1e4b20">25</a>.
|
A time domain simulation is shown in Figure <a href="#orgdf5905f">25</a>.
|
||||||
</p>
|
</p>
|
||||||
|
|
||||||
<p>
|
<p>
|
||||||
The RMS values of the super sensor noise for the presented three synthesis are listed in Table <a href="#orga287858">5</a>.
|
The RMS values of the super sensor noise for the presented three synthesis are listed in Table <a href="#org3b7df55">5</a>.
|
||||||
</p>
|
</p>
|
||||||
|
|
||||||
<div class="org-src-container">
|
<div class="org-src-container">
|
||||||
@ -1134,20 +1134,20 @@ PSD_H2Hinf = abs(squeeze(freqresp(N1<span class="org-type">*</span>H1, freqs, <s
|
|||||||
</div>
|
</div>
|
||||||
|
|
||||||
|
|
||||||
<div id="org6c93dd1" class="figure">
|
<div id="orgfdfbf76" class="figure">
|
||||||
<p><img src="figs/psd_sensors_htwo_hinf_synthesis.png" alt="psd_sensors_htwo_hinf_synthesis.png" />
|
<p><img src="figs/psd_sensors_htwo_hinf_synthesis.png" alt="psd_sensors_htwo_hinf_synthesis.png" />
|
||||||
</p>
|
</p>
|
||||||
<p><span class="figure-number">Figure 24: </span>Power Spectral Density of the Super Sensor obtained with the mixed \(\mathcal{H}_2/\mathcal{H}_\infty\) synthesis</p>
|
<p><span class="figure-number">Figure 24: </span>Power Spectral Density of the Super Sensor obtained with the mixed \(\mathcal{H}_2/\mathcal{H}_\infty\) synthesis</p>
|
||||||
</div>
|
</div>
|
||||||
|
|
||||||
|
|
||||||
<div id="orgb1e4b20" class="figure">
|
<div id="orgdf5905f" class="figure">
|
||||||
<p><img src="figs/super_sensor_time_domain_h2_hinf.png" alt="super_sensor_time_domain_h2_hinf.png" />
|
<p><img src="figs/super_sensor_time_domain_h2_hinf.png" alt="super_sensor_time_domain_h2_hinf.png" />
|
||||||
</p>
|
</p>
|
||||||
<p><span class="figure-number">Figure 25: </span>Noise of individual sensors and noise of the super sensor</p>
|
<p><span class="figure-number">Figure 25: </span>Noise of individual sensors and noise of the super sensor</p>
|
||||||
</div>
|
</div>
|
||||||
|
|
||||||
<table id="orga287858" border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
|
<table id="org3b7df55" border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
|
||||||
<caption class="t-above"><span class="table-number">Table 5:</span> Comparison of the obtained RMS noise of the super sensor</caption>
|
<caption class="t-above"><span class="table-number">Table 5:</span> Comparison of the obtained RMS noise of the super sensor</caption>
|
||||||
|
|
||||||
<colgroup>
|
<colgroup>
|
||||||
@ -1181,15 +1181,15 @@ PSD_H2Hinf = abs(squeeze(freqresp(N1<span class="org-type">*</span>H1, freqs, <s
|
|||||||
</div>
|
</div>
|
||||||
</div>
|
</div>
|
||||||
|
|
||||||
<div id="outline-container-org5adb8ec" class="outline-3">
|
<div id="outline-container-org2dce888" class="outline-3">
|
||||||
<h3 id="org5adb8ec"><span class="section-number-3">5.3</span> Obtained Super Sensor’s Uncertainty</h3>
|
<h3 id="org2dce888"><span class="section-number-3">5.3</span> Obtained Super Sensor’s Uncertainty</h3>
|
||||||
<div class="outline-text-3" id="text-5-3">
|
<div class="outline-text-3" id="text-5-3">
|
||||||
<p>
|
<p>
|
||||||
The uncertainty on the super sensor’s dynamics is shown in Figure <a href="#org82b5806">26</a>.
|
The uncertainty on the super sensor’s dynamics is shown in Figure <a href="#orgb2d28c5">26</a>.
|
||||||
</p>
|
</p>
|
||||||
|
|
||||||
|
|
||||||
<div id="org82b5806" class="figure">
|
<div id="orgb2d28c5" class="figure">
|
||||||
<p><img src="figs/super_sensor_dynamical_uncertainty_Htwo_Hinf.png" alt="super_sensor_dynamical_uncertainty_Htwo_Hinf.png" />
|
<p><img src="figs/super_sensor_dynamical_uncertainty_Htwo_Hinf.png" alt="super_sensor_dynamical_uncertainty_Htwo_Hinf.png" />
|
||||||
</p>
|
</p>
|
||||||
<p><span class="figure-number">Figure 26: </span>Super sensor dynamical uncertainty (solid curve) when using the mixed \(\mathcal{H}_2/\mathcal{H}_\infty\) Synthesis</p>
|
<p><span class="figure-number">Figure 26: </span>Super sensor dynamical uncertainty (solid curve) when using the mixed \(\mathcal{H}_2/\mathcal{H}_\infty\) Synthesis</p>
|
||||||
@ -1197,8 +1197,8 @@ The uncertainty on the super sensor’s dynamics is shown in Figure <a href=
|
|||||||
</div>
|
</div>
|
||||||
</div>
|
</div>
|
||||||
|
|
||||||
<div id="outline-container-org76f00f7" class="outline-3">
|
<div id="outline-container-org47da78c" class="outline-3">
|
||||||
<h3 id="org76f00f7"><span class="section-number-3">5.4</span> Conclusion</h3>
|
<h3 id="org47da78c"><span class="section-number-3">5.4</span> Conclusion</h3>
|
||||||
<div class="outline-text-3" id="text-5-4">
|
<div class="outline-text-3" id="text-5-4">
|
||||||
<p>
|
<p>
|
||||||
The mixed \(\mathcal{H}_2/\mathcal{H}_\infty\) synthesis of the complementary filters allows to:
|
The mixed \(\mathcal{H}_2/\mathcal{H}_\infty\) synthesis of the complementary filters allows to:
|
||||||
@ -1211,18 +1211,18 @@ The mixed \(\mathcal{H}_2/\mathcal{H}_\infty\) synthesis of the complementary fi
|
|||||||
</div>
|
</div>
|
||||||
</div>
|
</div>
|
||||||
|
|
||||||
<div id="outline-container-orgf68e579" class="outline-2">
|
<div id="outline-container-org0afe5ef" class="outline-2">
|
||||||
<h2 id="orgf68e579"><span class="section-number-2">6</span> Matlab Functions</h2>
|
<h2 id="org0afe5ef"><span class="section-number-2">6</span> Matlab Functions</h2>
|
||||||
<div class="outline-text-2" id="text-6">
|
<div class="outline-text-2" id="text-6">
|
||||||
<p>
|
<p>
|
||||||
<a id="org4f93e35"></a>
|
<a id="orgf41dc8d"></a>
|
||||||
</p>
|
</p>
|
||||||
</div>
|
</div>
|
||||||
<div id="outline-container-orgdec213a" class="outline-3">
|
<div id="outline-container-orge81e522" class="outline-3">
|
||||||
<h3 id="orgdec213a"><span class="section-number-3">6.1</span> <code>createWeight</code></h3>
|
<h3 id="orge81e522"><span class="section-number-3">6.1</span> <code>createWeight</code></h3>
|
||||||
<div class="outline-text-3" id="text-6-1">
|
<div class="outline-text-3" id="text-6-1">
|
||||||
<p>
|
<p>
|
||||||
<a id="orgb6d8184"></a>
|
<a id="org89bad6d"></a>
|
||||||
</p>
|
</p>
|
||||||
|
|
||||||
<p>
|
<p>
|
||||||
@ -1274,11 +1274,11 @@ This Matlab function is accessible <a href="src/createWeight.m">here</a>.
|
|||||||
</div>
|
</div>
|
||||||
</div>
|
</div>
|
||||||
|
|
||||||
<div id="outline-container-orgad116a6" class="outline-3">
|
<div id="outline-container-org37ec2b4" class="outline-3">
|
||||||
<h3 id="orgad116a6"><span class="section-number-3">6.2</span> <code>plotMagUncertainty</code></h3>
|
<h3 id="org37ec2b4"><span class="section-number-3">6.2</span> <code>plotMagUncertainty</code></h3>
|
||||||
<div class="outline-text-3" id="text-6-2">
|
<div class="outline-text-3" id="text-6-2">
|
||||||
<p>
|
<p>
|
||||||
<a id="orgd8e37bd"></a>
|
<a id="org8739875"></a>
|
||||||
</p>
|
</p>
|
||||||
|
|
||||||
<p>
|
<p>
|
||||||
@ -1329,11 +1329,11 @@ p.FaceAlpha = args.opacity;
|
|||||||
</div>
|
</div>
|
||||||
</div>
|
</div>
|
||||||
|
|
||||||
<div id="outline-container-orga641eed" class="outline-3">
|
<div id="outline-container-org9f73572" class="outline-3">
|
||||||
<h3 id="orga641eed"><span class="section-number-3">6.3</span> <code>plotPhaseUncertainty</code></h3>
|
<h3 id="org9f73572"><span class="section-number-3">6.3</span> <code>plotPhaseUncertainty</code></h3>
|
||||||
<div class="outline-text-3" id="text-6-3">
|
<div class="outline-text-3" id="text-6-3">
|
||||||
<p>
|
<p>
|
||||||
<a id="org5f016a4"></a>
|
<a id="org49e21eb"></a>
|
||||||
</p>
|
</p>
|
||||||
|
|
||||||
<p>
|
<p>
|
||||||
@ -1397,7 +1397,7 @@ p.FaceAlpha = args.opacity;
|
|||||||
</div>
|
</div>
|
||||||
<div id="postamble" class="status">
|
<div id="postamble" class="status">
|
||||||
<p class="author">Author: Thomas Dehaeze</p>
|
<p class="author">Author: Thomas Dehaeze</p>
|
||||||
<p class="date">Created: 2020-10-05 lun. 11:45</p>
|
<p class="date">Created: 2020-10-05 lun. 15:08</p>
|
||||||
</div>
|
</div>
|
||||||
</body>
|
</body>
|
||||||
</html>
|
</html>
|
||||||
|
|||||||
@ -100,7 +100,7 @@ In this example, the measured quantity $x$ is the velocity of an object.
|
|||||||
#+name: fig:sensor_model_noise_uncertainty
|
#+name: fig:sensor_model_noise_uncertainty
|
||||||
#+caption: Sensor Model
|
#+caption: Sensor Model
|
||||||
#+RESULTS:
|
#+RESULTS:
|
||||||
[[file:figs-tikz/sensor_model_noise_uncertainty.png]]
|
[[file:figs-paper/sensor_model_noise_uncertainty.png]]
|
||||||
|
|
||||||
** Matlab Init :noexport:ignore:
|
** Matlab Init :noexport:ignore:
|
||||||
#+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name)
|
#+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name)
|
||||||
@ -325,7 +325,7 @@ The two sensors presented in Section [[sec:sensor_description]] are now merged t
|
|||||||
|
|
||||||
#+name: fig:sensor_fusion_noise_arch
|
#+name: fig:sensor_fusion_noise_arch
|
||||||
#+caption: Sensor Fusion Architecture
|
#+caption: Sensor Fusion Architecture
|
||||||
[[file:figs-tikz/sensor_fusion_noise_arch.png]]
|
[[file:figs-paper/sensor_fusion_noise_arch.png]]
|
||||||
|
|
||||||
The complementary property of $H_1(s)$ and $H_2(s)$ means that the sum of their transfer function is equal to $1$ eqref:eq:complementary_property.
|
The complementary property of $H_1(s)$ and $H_2(s)$ means that the sum of their transfer function is equal to $1$ eqref:eq:complementary_property.
|
||||||
|
|
||||||
@ -377,13 +377,13 @@ If we consider some dynamical uncertainty (the true system dynamics $G_i$ not be
|
|||||||
|
|
||||||
#+name: fig:sensor_model_uncertainty
|
#+name: fig:sensor_model_uncertainty
|
||||||
#+caption: Sensor Model including Dynamical Uncertainty
|
#+caption: Sensor Model including Dynamical Uncertainty
|
||||||
[[file:figs-tikz/sensor_model_uncertainty.png]]
|
[[file:figs-paper/sensor_model_uncertainty.png]]
|
||||||
|
|
||||||
The uncertainty set of the transfer function from $\hat{x}$ to $x$ at frequency $\omega$ is bounded in the complex plane by a circle centered on 1 and with a radius equal to $|W_1(j\omega) H_1(j\omega)| + |W_2(j\omega) H_2(j\omega)|$ as shown in Figure [[fig:uncertainty_set_super_sensor]].
|
The uncertainty set of the transfer function from $\hat{x}$ to $x$ at frequency $\omega$ is bounded in the complex plane by a circle centered on 1 and with a radius equal to $|W_1(j\omega) H_1(j\omega)| + |W_2(j\omega) H_2(j\omega)|$ as shown in Figure [[fig:uncertainty_set_super_sensor]].
|
||||||
|
|
||||||
#+name: fig:uncertainty_set_super_sensor
|
#+name: fig:uncertainty_set_super_sensor
|
||||||
#+caption: Super Sensor model uncertainty displayed in the complex plane
|
#+caption: Super Sensor model uncertainty displayed in the complex plane
|
||||||
[[file:figs-tikz/uncertainty_set_super_sensor.png]]
|
[[file:figs-paper/uncertainty_set_super_sensor.png]]
|
||||||
|
|
||||||
* Optimal Super Sensor Noise: $\mathcal{H}_2$ Synthesis
|
* Optimal Super Sensor Noise: $\mathcal{H}_2$ Synthesis
|
||||||
:PROPERTIES:
|
:PROPERTIES:
|
||||||
@ -427,7 +427,7 @@ Consider the generalized plant $P_{\mathcal{H}_2}$ shown in Figure [[fig:h_two_o
|
|||||||
|
|
||||||
#+name: fig:h_two_optimal_fusion
|
#+name: fig:h_two_optimal_fusion
|
||||||
#+caption: Architecture used for $\mathcal{H}_\infty$ synthesis of complementary filters
|
#+caption: Architecture used for $\mathcal{H}_\infty$ synthesis of complementary filters
|
||||||
[[file:figs-tikz/h_two_optimal_fusion.png]]
|
[[file:figs-paper/h_two_optimal_fusion.png]]
|
||||||
|
|
||||||
\begin{equation} \label{eq:H2_generalized_plant}
|
\begin{equation} \label{eq:H2_generalized_plant}
|
||||||
\begin{pmatrix}
|
\begin{pmatrix}
|
||||||
@ -696,7 +696,7 @@ To do so, we model the uncertainty that we have on the sensor dynamics by multip
|
|||||||
|
|
||||||
#+name: fig:sensor_fusion_arch_uncertainty
|
#+name: fig:sensor_fusion_arch_uncertainty
|
||||||
#+caption: Sensor fusion architecture with sensor dynamics uncertainty
|
#+caption: Sensor fusion architecture with sensor dynamics uncertainty
|
||||||
[[file:figs-tikz/sensor_fusion_arch_uncertainty.png]]
|
[[file:figs-paper/sensor_fusion_arch_uncertainty.png]]
|
||||||
|
|
||||||
As explained in Section [[sec:sensor_uncertainty]], at each frequency $\omega$, the dynamical uncertainty of the super sensor can be represented in the complex plane by a circle with a radius equals to $|H_1(j\omega) W_1(j\omega)| + |H_2(j\omega) W_2(j\omega)|$ and centered on 1.
|
As explained in Section [[sec:sensor_uncertainty]], at each frequency $\omega$, the dynamical uncertainty of the super sensor can be represented in the complex plane by a circle with a radius equals to $|H_1(j\omega) W_1(j\omega)| + |H_2(j\omega) W_2(j\omega)|$ and centered on 1.
|
||||||
|
|
||||||
@ -818,7 +818,7 @@ The generalized plant $P_{\mathcal{H}_\infty}$ used for the $\mathcal{H}_\infty$
|
|||||||
|
|
||||||
#+name: fig:h_infinity_robust_fusion
|
#+name: fig:h_infinity_robust_fusion
|
||||||
#+caption: Architecture used for $\mathcal{H}_\infty$ synthesis of complementary filters
|
#+caption: Architecture used for $\mathcal{H}_\infty$ synthesis of complementary filters
|
||||||
[[file:figs-tikz/h_infinity_robust_fusion.png]]
|
[[file:figs-paper/h_infinity_robust_fusion.png]]
|
||||||
|
|
||||||
\begin{equation} \label{eq:Hinf_generalized_plant}
|
\begin{equation} \label{eq:Hinf_generalized_plant}
|
||||||
\begin{pmatrix}
|
\begin{pmatrix}
|
||||||
@ -1017,7 +1017,7 @@ As expected, the super sensor obtained from the $\mathcal{H}_\infty$ synthesis i
|
|||||||
#+end_src
|
#+end_src
|
||||||
|
|
||||||
#+name: fig:psd_sensors_hinf_synthesis
|
#+name: fig:psd_sensors_hinf_synthesis
|
||||||
#+caption: Power Spectral Density of the estimated $\hat{x}$ using the two sensors alone and using the
|
#+caption: Power Spectral Density of the estimated $\hat{x}$ using the two sensors alone and using the $\mathcal{H}_\infty$ synthesis
|
||||||
#+RESULTS:
|
#+RESULTS:
|
||||||
[[file:figs/psd_sensors_hinf_synthesis.png]]
|
[[file:figs/psd_sensors_hinf_synthesis.png]]
|
||||||
|
|
||||||
@ -1054,7 +1054,7 @@ The sensor fusion architecture is shown in Figure [[fig:sensor_fusion_arch_full]
|
|||||||
|
|
||||||
#+name: fig:sensor_fusion_arch_full
|
#+name: fig:sensor_fusion_arch_full
|
||||||
#+caption: Sensor fusion architecture with sensor dynamics uncertainty
|
#+caption: Sensor fusion architecture with sensor dynamics uncertainty
|
||||||
[[file:figs-tikz/sensor_fusion_arch_full.png]]
|
[[file:figs-paper/sensor_fusion_arch_full.png]]
|
||||||
|
|
||||||
The goal is to design complementary filters such that:
|
The goal is to design complementary filters such that:
|
||||||
- the maximum uncertainty of the super sensor is bounded to acceptable values (defined by $W_u(s)$)
|
- the maximum uncertainty of the super sensor is bounded to acceptable values (defined by $W_u(s)$)
|
||||||
@ -1087,7 +1087,7 @@ The filter $H_2(s)$ is synthesized such that it:
|
|||||||
|
|
||||||
#+name: fig:mixed_h2_hinf_synthesis
|
#+name: fig:mixed_h2_hinf_synthesis
|
||||||
#+caption: Mixed $\mathcal{H}_2/\mathcal{H}_\infty$ Synthesis
|
#+caption: Mixed $\mathcal{H}_2/\mathcal{H}_\infty$ Synthesis
|
||||||
[[file:figs-tikz/mixed_h2_hinf_synthesis.png]]
|
[[file:figs-paper/mixed_h2_hinf_synthesis.png]]
|
||||||
|
|
||||||
Let's see that
|
Let's see that
|
||||||
with $H_1(s)= 1 - H_2(s)$
|
with $H_1(s)= 1 - H_2(s)$
|
||||||
@ -1213,7 +1213,7 @@ The RMS values of the super sensor noise for the presented three synthesis are l
|
|||||||
#+end_src
|
#+end_src
|
||||||
|
|
||||||
#+name: fig:psd_sensors_htwo_hinf_synthesis
|
#+name: fig:psd_sensors_htwo_hinf_synthesis
|
||||||
#+CAPTION: Power Spectral Density of the Super Sensor obtained with the mixed $\mathcal{H}_2/\mathcal{H}_\infty$ synthesis
|
#+caption: Power Spectral Density of the Super Sensor obtained with the mixed $\mathcal{H}_2/\mathcal{H}_\infty$ synthesis
|
||||||
#+RESULTS:
|
#+RESULTS:
|
||||||
[[file:figs/psd_sensors_htwo_hinf_synthesis.png]]
|
[[file:figs/psd_sensors_htwo_hinf_synthesis.png]]
|
||||||
|
|
||||||
|
|||||||
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@ -221,13 +221,46 @@ We then have that the $\mathcal{H}_2$ synthesis applied on $P_{\mathcal{H}_2}$ g
|
|||||||
|
|
||||||
** Example
|
** Example
|
||||||
|
|
||||||
#+name: fig:figure_name
|
#+name: fig:sensors_nominal_dynamics
|
||||||
#+caption: Figure caption
|
#+caption: Sensor nominal dynamics from the velocity of the object to the output voltage
|
||||||
#+attr_latex: :scale 1
|
#+attr_latex: :scale 1
|
||||||
[[file:figs/sensors_nominal_dynamics.pdf]]
|
[[file:figs/sensors_nominal_dynamics.pdf]]
|
||||||
|
|
||||||
|
#+name: fig:sensors_noise
|
||||||
|
#+caption: Amplitude spectral density of the sensors $\sqrt{\Phi_{n_i}(\omega)} = |N_i(j\omega)|$
|
||||||
|
#+attr_latex: :scale 1
|
||||||
|
[[file:figs/sensors_noise.pdf]]
|
||||||
|
|
||||||
|
|
||||||
|
#+name: fig:htwo_comp_filters
|
||||||
|
#+caption: Obtained complementary filters using the $\mathcal{H}_2$ Synthesis
|
||||||
|
#+attr_latex: :scale 1
|
||||||
|
[[file:figs/htwo_comp_filters.pdf]]
|
||||||
|
|
||||||
|
|
||||||
|
#+name: fig:psd_sensors_htwo_synthesis
|
||||||
|
#+caption: Power Spectral Density of the estimated $\hat{x}$ using the two sensors alone and using the optimally fused signal
|
||||||
|
#+attr_latex: :scale 1
|
||||||
|
[[file:figs/psd_sensors_htwo_synthesis.pdf]]
|
||||||
|
|
||||||
|
|
||||||
|
#+name: fig:super_sensor_time_domain_h2
|
||||||
|
#+caption: Noise of individual sensors and noise of the super sensor
|
||||||
|
#+attr_latex: :scale 1
|
||||||
|
[[file:figs/super_sensor_time_domain_h2.pdf]]
|
||||||
|
|
||||||
** Robustness Problem
|
** Robustness Problem
|
||||||
|
|
||||||
|
#+name: fig:sensors_nominal_dynamics_and_uncertainty
|
||||||
|
#+caption: Nominal Sensor Dynamics $\hat{G}_i$ (solid lines) as well as the spread of the dynamical uncertainty (background color)
|
||||||
|
#+attr_latex: :scale 1
|
||||||
|
[[file:figs/sensors_nominal_dynamics_and_uncertainty.pdf]]
|
||||||
|
|
||||||
|
#+name: fig:super_sensor_dynamical_uncertainty_H2
|
||||||
|
#+caption: Super sensor dynamical uncertainty when using the $\mathcal{H}_2$ Synthesis
|
||||||
|
#+attr_latex: :scale 1
|
||||||
|
[[file:figs/super_sensor_dynamical_uncertainty_H2.pdf]]
|
||||||
|
|
||||||
* Robust Sensor Fusion: $\mathcal{H}_\infty$ Synthesis
|
* Robust Sensor Fusion: $\mathcal{H}_\infty$ Synthesis
|
||||||
<<sec:robust_fusion>>
|
<<sec:robust_fusion>>
|
||||||
|
|
||||||
@ -334,6 +367,33 @@ The $\mathcal{H}_\infty$ norm of Eq. eqref:eq:Hinf_norm is equals to $\sigma_n$
|
|||||||
|
|
||||||
** Example
|
** Example
|
||||||
|
|
||||||
|
#+name: fig:sensors_uncertainty_weights
|
||||||
|
#+caption: Magnitude of the multiplicative uncertainty weights $|W_i(j\omega)|$
|
||||||
|
#+attr_latex: :scale 1
|
||||||
|
[[file:figs/sensors_uncertainty_weights.pdf]]
|
||||||
|
|
||||||
|
|
||||||
|
#+name: fig:weight_uncertainty_bounds_Wu
|
||||||
|
#+caption: Uncertainty region of the two sensors as well as the wanted maximum uncertainty of the super sensor (dashed lines)
|
||||||
|
#+attr_latex: :scale 1
|
||||||
|
[[file:figs/weight_uncertainty_bounds_Wu.pdf]]
|
||||||
|
|
||||||
|
#+name: fig:hinf_comp_filters
|
||||||
|
#+caption: Obtained complementary filters using the $\mathcal{H}_\infty$ Synthesis
|
||||||
|
#+attr_latex: :scale 1
|
||||||
|
[[file:figs/hinf_comp_filters.pdf]]
|
||||||
|
|
||||||
|
#+name: fig:super_sensor_dynamical_uncertainty_Hinf
|
||||||
|
#+caption: Super sensor dynamical uncertainty (solid curve) when using the $\mathcal{H}_\infty$ Synthesis
|
||||||
|
#+attr_latex: :scale 1
|
||||||
|
[[file:figs/super_sensor_dynamical_uncertainty_Hinf.pdf]]
|
||||||
|
|
||||||
|
#+name: fig:psd_sensors_hinf_synthesis
|
||||||
|
#+caption: Power Spectral Density of the estimated $\hat{x}$ using the two sensors alone and using the $\mathcal{H}_\infty$ synthesis
|
||||||
|
#+attr_latex: :scale 1
|
||||||
|
[[file:figs/psd_sensors_hinf_synthesis.pdf]]
|
||||||
|
|
||||||
|
|
||||||
* Optimal and Robust Sensor Fusion: Mixed $\mathcal{H}_2/\mathcal{H}_\infty$ Synthesis
|
* Optimal and Robust Sensor Fusion: Mixed $\mathcal{H}_2/\mathcal{H}_\infty$ Synthesis
|
||||||
<<sec:optimal_robust_fusion>>
|
<<sec:optimal_robust_fusion>>
|
||||||
|
|
||||||
@ -414,6 +474,26 @@ The synthesis objective is to:
|
|||||||
|
|
||||||
** Example
|
** Example
|
||||||
|
|
||||||
|
#+name: fig:htwo_hinf_comp_filters
|
||||||
|
#+caption: Obtained complementary filters after mixed $\mathcal{H}_2/\mathcal{H}_\infty$ synthesis
|
||||||
|
#+attr_latex: :scale 1
|
||||||
|
[[file:figs/htwo_hinf_comp_filters.pdf]]
|
||||||
|
|
||||||
|
#+name: fig:psd_sensors_htwo_hinf_synthesis
|
||||||
|
#+caption: Power Spectral Density of the Super Sensor obtained with the mixed $\mathcal{H}_2/\mathcal{H}_\infty$ synthesis
|
||||||
|
#+attr_latex: :scale 1
|
||||||
|
[[file:figs/psd_sensors_htwo_hinf_synthesis.pdf]]
|
||||||
|
|
||||||
|
#+name: fig:super_sensor_time_domain_h2_hinf
|
||||||
|
#+caption: Noise of individual sensors and noise of the super sensor
|
||||||
|
#+attr_latex: :scale 1
|
||||||
|
[[file:figs/super_sensor_time_domain_h2_hinf.pdf]]
|
||||||
|
|
||||||
|
#+name: fig:super_sensor_dynamical_uncertainty_Htwo_Hinf
|
||||||
|
#+caption: Super sensor dynamical uncertainty (solid curve) when using the mixed $\mathcal{H}_2/\mathcal{H}_\infty$ Synthesis
|
||||||
|
#+attr_latex: :scale 1
|
||||||
|
[[file:figs/super_sensor_dynamical_uncertainty_Htwo_Hinf.pdf]]
|
||||||
|
|
||||||
* Experimental Validation
|
* Experimental Validation
|
||||||
<<sec:experimental_validation>>
|
<<sec:experimental_validation>>
|
||||||
|
|
||||||
|
|||||||
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@ -1,4 +1,4 @@
|
|||||||
% Created 2020-09-23 mer. 14:15
|
% Created 2020-10-05 lun. 15:33
|
||||||
% Intended LaTeX compiler: pdflatex
|
% Intended LaTeX compiler: pdflatex
|
||||||
\documentclass[conference]{IEEEtran}
|
\documentclass[conference]{IEEEtran}
|
||||||
\usepackage[utf8]{inputenc}
|
\usepackage[utf8]{inputenc}
|
||||||
@ -35,7 +35,7 @@
|
|||||||
\def\BibTeX{{\rm B\kern-.05em{\sc i\kern-.025em b}\kern-.08em T\kern-.1667em\lower.7ex\hbox{E}\kern-.125emX}}
|
\def\BibTeX{{\rm B\kern-.05em{\sc i\kern-.025em b}\kern-.08em T\kern-.1667em\lower.7ex\hbox{E}\kern-.125emX}}
|
||||||
\usepackage{showframe}
|
\usepackage{showframe}
|
||||||
\author{\IEEEauthorblockN{Dehaeze Thomas} \IEEEauthorblockA{\textit{European Synchrotron Radiation Facility} \\ Grenoble, France\\ \textit{Precision Mechatronics Laboratory} \\ \textit{University of Liege}, Belgium \\ thomas.dehaeze@esrf.fr }\and \IEEEauthorblockN{Collette Christophe} \IEEEauthorblockA{\textit{BEAMS Department}\\ \textit{Free University of Brussels}, Belgium\\ \textit{Precision Mechatronics Laboratory} \\ \textit{University of Liege}, Belgium \\ ccollett@ulb.ac.be }}
|
\author{\IEEEauthorblockN{Dehaeze Thomas} \IEEEauthorblockA{\textit{European Synchrotron Radiation Facility} \\ Grenoble, France\\ \textit{Precision Mechatronics Laboratory} \\ \textit{University of Liege}, Belgium \\ thomas.dehaeze@esrf.fr }\and \IEEEauthorblockN{Collette Christophe} \IEEEauthorblockA{\textit{BEAMS Department}\\ \textit{Free University of Brussels}, Belgium\\ \textit{Precision Mechatronics Laboratory} \\ \textit{University of Liege}, Belgium \\ ccollett@ulb.ac.be }}
|
||||||
\date{2020-09-23}
|
\date{2020-10-05}
|
||||||
\title{Optimal and Robust Sensor Fusion}
|
\title{Optimal and Robust Sensor Fusion}
|
||||||
\begin{document}
|
\begin{document}
|
||||||
|
|
||||||
@ -50,7 +50,7 @@ Complementary Filters, Sensor Fusion, H-Infinity Synthesis
|
|||||||
\end{IEEEkeywords}
|
\end{IEEEkeywords}
|
||||||
|
|
||||||
\section{Introduction}
|
\section{Introduction}
|
||||||
\label{sec:org88afd51}
|
\label{sec:org26a7400}
|
||||||
\label{sec:introduction}
|
\label{sec:introduction}
|
||||||
|
|
||||||
\begin{itemize}
|
\begin{itemize}
|
||||||
@ -61,12 +61,11 @@ Complementary Filters, Sensor Fusion, H-Infinity Synthesis
|
|||||||
\end{itemize}
|
\end{itemize}
|
||||||
|
|
||||||
\section{Optimal Super Sensor Noise: \(\mathcal{H}_2\) Synthesis}
|
\section{Optimal Super Sensor Noise: \(\mathcal{H}_2\) Synthesis}
|
||||||
\label{sec:org5853545}
|
\label{sec:org49e80fd}
|
||||||
\label{sec:optimal_fusion}
|
\label{sec:optimal_fusion}
|
||||||
|
|
||||||
\subsection{Sensor Model}
|
\subsection{Sensor Model}
|
||||||
\label{sec:org565ea86}
|
\label{sec:org9555932}
|
||||||
|
|
||||||
Let's consider a sensor measuring a physical quantity \(x\) (Figure \ref{fig:sensor_model_noise}).
|
Let's consider a sensor measuring a physical quantity \(x\) (Figure \ref{fig:sensor_model_noise}).
|
||||||
The sensor has an internal dynamics which is here modelled with a Linear Time Invariant (LTI) system transfer function \(G_i(s)\).
|
The sensor has an internal dynamics which is here modelled with a Linear Time Invariant (LTI) system transfer function \(G_i(s)\).
|
||||||
|
|
||||||
@ -102,8 +101,7 @@ In order to obtain an estimate \(\hat{x}_i\) of \(x\), a model \(\hat{G}_i\) of
|
|||||||
\end{figure}
|
\end{figure}
|
||||||
|
|
||||||
\subsection{Sensor Fusion Architecture}
|
\subsection{Sensor Fusion Architecture}
|
||||||
\label{sec:org1ae73e8}
|
\label{sec:orga12ae12}
|
||||||
|
|
||||||
Let's now consider two sensors measuring the same physical quantity \(x\) but with different dynamics \((G_1, G_2)\) and noise characteristics \((N_1, N_2)\) (Figure \ref{fig:sensor_fusion_noise_arch}).
|
Let's now consider two sensors measuring the same physical quantity \(x\) but with different dynamics \((G_1, G_2)\) and noise characteristics \((N_1, N_2)\) (Figure \ref{fig:sensor_fusion_noise_arch}).
|
||||||
|
|
||||||
The noise sources \(\tilde{n}_1\) and \(\tilde{n}_2\) are considered to be uncorrelated.
|
The noise sources \(\tilde{n}_1\) and \(\tilde{n}_2\) are considered to be uncorrelated.
|
||||||
@ -140,7 +138,7 @@ In such case, the super sensor estimate \(\hat{x}\) is equal to \(x\) plus the n
|
|||||||
\end{equation}
|
\end{equation}
|
||||||
|
|
||||||
\subsection{Super Sensor Noise}
|
\subsection{Super Sensor Noise}
|
||||||
\label{sec:orgb2e8dd6}
|
\label{sec:org924b750}
|
||||||
Let's note \(n\) the super sensor noise.
|
Let's note \(n\) the super sensor noise.
|
||||||
\begin{equation}
|
\begin{equation}
|
||||||
n = \left( H_1 N_1 \right) \tilde{n}_1 + \left( H_2 N_2 \right) \tilde{n}_2
|
n = \left( H_1 N_1 \right) \tilde{n}_1 + \left( H_2 N_2 \right) \tilde{n}_2
|
||||||
@ -154,7 +152,7 @@ As the noise of both sensors are considered to be uncorrelated, the PSD of the s
|
|||||||
It is clear that the PSD of the super sensor depends on the norm of the complementary filters.
|
It is clear that the PSD of the super sensor depends on the norm of the complementary filters.
|
||||||
|
|
||||||
\subsection{\(\mathcal{H}_2\) Synthesis of Complementary Filters}
|
\subsection{\(\mathcal{H}_2\) Synthesis of Complementary Filters}
|
||||||
\label{sec:orga4cf5f1}
|
\label{sec:org042a601}
|
||||||
The goal is to design \(H_1(s)\) and \(H_2(s)\) such that the effect of the noise sources \(\tilde{n}_1\) and \(\tilde{n}_2\) has the smallest possible effect on the noise \(n\) of the estimation \(\hat{x}\).
|
The goal is to design \(H_1(s)\) and \(H_2(s)\) such that the effect of the noise sources \(\tilde{n}_1\) and \(\tilde{n}_2\) has the smallest possible effect on the noise \(n\) of the estimation \(\hat{x}\).
|
||||||
|
|
||||||
And the goal is the minimize the Root Mean Square (RMS) value of \(n\):
|
And the goal is the minimize the Root Mean Square (RMS) value of \(n\):
|
||||||
@ -170,8 +168,8 @@ This can be cast into an \(\mathcal{H}_2\) synthesis problem by considering the
|
|||||||
\begin{pmatrix}
|
\begin{pmatrix}
|
||||||
z_1 \\ z_2 \\ v
|
z_1 \\ z_2 \\ v
|
||||||
\end{pmatrix} = \underbrace{\begin{bmatrix}
|
\end{pmatrix} = \underbrace{\begin{bmatrix}
|
||||||
N_1 & N_1 \\
|
N_1 & -N_1 \\
|
||||||
0 & N_2 \\
|
0 & N_2 \\
|
||||||
1 & 0
|
1 & 0
|
||||||
\end{bmatrix}}_{P_{\mathcal{H}_2}} \begin{pmatrix}
|
\end{bmatrix}}_{P_{\mathcal{H}_2}} \begin{pmatrix}
|
||||||
w \\ u
|
w \\ u
|
||||||
@ -198,17 +196,62 @@ We then have that the \(\mathcal{H}_2\) synthesis applied on \(P_{\mathcal{H}_2}
|
|||||||
\end{figure}
|
\end{figure}
|
||||||
|
|
||||||
\subsection{Example}
|
\subsection{Example}
|
||||||
\label{sec:org74634c9}
|
\label{sec:org98c54c2}
|
||||||
|
|
||||||
|
\begin{figure}[htbp]
|
||||||
|
\centering
|
||||||
|
\includegraphics[scale=1]{figs/sensors_nominal_dynamics.pdf}
|
||||||
|
\caption{\label{fig:sensors_nominal_dynamics}Sensor nominal dynamics from the velocity of the object to the output voltage}
|
||||||
|
\end{figure}
|
||||||
|
|
||||||
|
\begin{figure}[htbp]
|
||||||
|
\centering
|
||||||
|
\includegraphics[scale=1]{figs/sensors_noise.pdf}
|
||||||
|
\caption{\label{fig:sensors_noise}Amplitude spectral density of the sensors \(\sqrt{\Phi_{n_i}(\omega)} = |N_i(j\omega)|\)}
|
||||||
|
\end{figure}
|
||||||
|
|
||||||
|
|
||||||
|
\begin{figure}[htbp]
|
||||||
|
\centering
|
||||||
|
\includegraphics[scale=1]{figs/htwo_comp_filters.pdf}
|
||||||
|
\caption{\label{fig:htwo_comp_filters}Obtained complementary filters using the \(\mathcal{H}_2\) Synthesis}
|
||||||
|
\end{figure}
|
||||||
|
|
||||||
|
|
||||||
|
\begin{figure}[htbp]
|
||||||
|
\centering
|
||||||
|
\includegraphics[scale=1]{figs/psd_sensors_htwo_synthesis.pdf}
|
||||||
|
\caption{\label{fig:psd_sensors_htwo_synthesis}Power Spectral Density of the estimated \(\hat{x}\) using the two sensors alone and using the optimally fused signal}
|
||||||
|
\end{figure}
|
||||||
|
|
||||||
|
|
||||||
|
\begin{figure}[htbp]
|
||||||
|
\centering
|
||||||
|
\includegraphics[scale=1]{figs/super_sensor_time_domain_h2.pdf}
|
||||||
|
\caption{\label{fig:super_sensor_time_domain_h2}Noise of individual sensors and noise of the super sensor}
|
||||||
|
\end{figure}
|
||||||
|
|
||||||
\subsection{Robustness Problem}
|
\subsection{Robustness Problem}
|
||||||
\label{sec:org5fda5c1}
|
\label{sec:org81a0772}
|
||||||
|
|
||||||
|
\begin{figure}[htbp]
|
||||||
|
\centering
|
||||||
|
\includegraphics[scale=1]{figs/sensors_nominal_dynamics_and_uncertainty.pdf}
|
||||||
|
\caption{\label{fig:sensors_nominal_dynamics_and_uncertainty}Nominal Sensor Dynamics \(\hat{G}_i\) (solid lines) as well as the spread of the dynamical uncertainty (background color)}
|
||||||
|
\end{figure}
|
||||||
|
|
||||||
|
\begin{figure}[htbp]
|
||||||
|
\centering
|
||||||
|
\includegraphics[scale=1]{figs/super_sensor_dynamical_uncertainty_H2.pdf}
|
||||||
|
\caption{\label{fig:super_sensor_dynamical_uncertainty_H2}Super sensor dynamical uncertainty when using the \(\mathcal{H}_2\) Synthesis}
|
||||||
|
\end{figure}
|
||||||
|
|
||||||
\section{Robust Sensor Fusion: \(\mathcal{H}_\infty\) Synthesis}
|
\section{Robust Sensor Fusion: \(\mathcal{H}_\infty\) Synthesis}
|
||||||
\label{sec:orgc88050f}
|
\label{sec:org78ced60}
|
||||||
\label{sec:robust_fusion}
|
\label{sec:robust_fusion}
|
||||||
|
|
||||||
\subsection{Representation of Sensor Dynamical Uncertainty}
|
\subsection{Representation of Sensor Dynamical Uncertainty}
|
||||||
\label{sec:orgb09aa5a}
|
\label{sec:org9df3b01}
|
||||||
|
|
||||||
In Section \ref{sec:optimal_fusion}, the model \(\hat{G}_i(s)\) of the sensor was considered to be perfect.
|
In Section \ref{sec:optimal_fusion}, the model \(\hat{G}_i(s)\) of the sensor was considered to be perfect.
|
||||||
In reality, there are always uncertainty (neglected dynamics) associated with the estimation of the sensor dynamics.
|
In reality, there are always uncertainty (neglected dynamics) associated with the estimation of the sensor dynamics.
|
||||||
@ -228,7 +271,7 @@ The sensor can then be represented as shown in Figure \ref{fig:sensor_model_unce
|
|||||||
\end{figure}
|
\end{figure}
|
||||||
|
|
||||||
\subsection{Sensor Fusion Architecture}
|
\subsection{Sensor Fusion Architecture}
|
||||||
\label{sec:org1d92a74}
|
\label{sec:orgf4531ff}
|
||||||
Let's consider the sensor fusion architecture shown in Figure \ref{fig:sensor_fusion_arch_uncertainty} where the dynamical uncertainties of both sensors are included.
|
Let's consider the sensor fusion architecture shown in Figure \ref{fig:sensor_fusion_arch_uncertainty} where the dynamical uncertainties of both sensors are included.
|
||||||
|
|
||||||
The super sensor estimate is then:
|
The super sensor estimate is then:
|
||||||
@ -253,8 +296,7 @@ As \(H_1\) and \(H_2\) are complementary filters, we finally have:
|
|||||||
\end{figure}
|
\end{figure}
|
||||||
|
|
||||||
\subsection{Super Sensor Dynamical Uncertainty}
|
\subsection{Super Sensor Dynamical Uncertainty}
|
||||||
\label{sec:org81db1d8}
|
\label{sec:orgf5bb33e}
|
||||||
|
|
||||||
The uncertainty set of the transfer function from \(\hat{x}\) to \(x\) at frequency \(\omega\) is bounded in the complex plane by a circle centered on 1 and with a radius equal to \(|W_1(j\omega) H_1(j\omega)| + |W_2(j\omega) H_2(j\omega)|\) as shown in Figure \ref{fig:uncertainty_set_super_sensor}.
|
The uncertainty set of the transfer function from \(\hat{x}\) to \(x\) at frequency \(\omega\) is bounded in the complex plane by a circle centered on 1 and with a radius equal to \(|W_1(j\omega) H_1(j\omega)| + |W_2(j\omega) H_2(j\omega)|\) as shown in Figure \ref{fig:uncertainty_set_super_sensor}.
|
||||||
|
|
||||||
|
|
||||||
@ -269,7 +311,7 @@ And we can see that the dynamical uncertainty of the super sensor is equal to th
|
|||||||
At frequencies where \(\left|W_i(j\omega)\right| > 1\) the uncertainty exceeds \(100\%\) and sensor fusion is impossible.
|
At frequencies where \(\left|W_i(j\omega)\right| > 1\) the uncertainty exceeds \(100\%\) and sensor fusion is impossible.
|
||||||
|
|
||||||
\subsection{\(\mathcal{H_\infty}\) Synthesis of Complementary Filters}
|
\subsection{\(\mathcal{H_\infty}\) Synthesis of Complementary Filters}
|
||||||
\label{sec:org0e2a7a8}
|
\label{sec:orgf07efa7}
|
||||||
In order for the fusion to be ``robust'', meaning no phase drop will be induced in the super sensor dynamics,
|
In order for the fusion to be ``robust'', meaning no phase drop will be induced in the super sensor dynamics,
|
||||||
|
|
||||||
The goal is to design two complementary filters \(H_1(s)\) and \(H_2(s)\) such that the super sensor noise uncertainty is kept reasonably small.
|
The goal is to design two complementary filters \(H_1(s)\) and \(H_2(s)\) such that the super sensor noise uncertainty is kept reasonably small.
|
||||||
@ -279,7 +321,7 @@ To define what by ``small'' we mean, we use a weighting filter \(W_u(s)\) such t
|
|||||||
\left| W_1(j\omega)H_1(j\omega) \right| + \left| W_2(j\omega)H_2(j\omega) \right| < \frac{1}{\left| W_u(j\omega) \right|}, \quad \forall \omega
|
\left| W_1(j\omega)H_1(j\omega) \right| + \left| W_2(j\omega)H_2(j\omega) \right| < \frac{1}{\left| W_u(j\omega) \right|}, \quad \forall \omega
|
||||||
\end{equation}
|
\end{equation}
|
||||||
|
|
||||||
This is actually almost equivalent (to within a factor \(\sqrt{2}\)) equivalent as to have:
|
This is actually almost equivalent as to have (within a factor \(\sqrt{2}\)):
|
||||||
\begin{equation}
|
\begin{equation}
|
||||||
\left\| \begin{matrix} W_u W_1 H_1 \\ W_u W_2 H_2 \end{matrix} \right\|_\infty < 1
|
\left\| \begin{matrix} W_u W_1 H_1 \\ W_u W_2 H_2 \end{matrix} \right\|_\infty < 1
|
||||||
\end{equation}
|
\end{equation}
|
||||||
@ -289,9 +331,9 @@ This problem can thus be dealt with an \(\mathcal{H}_\infty\) synthesis problem
|
|||||||
\begin{pmatrix}
|
\begin{pmatrix}
|
||||||
z_1 \\ z_2 \\ v
|
z_1 \\ z_2 \\ v
|
||||||
\end{pmatrix} = \underbrace{\begin{bmatrix}
|
\end{pmatrix} = \underbrace{\begin{bmatrix}
|
||||||
W_u W_1 & W_u W_1 \\
|
W_u W_1 & -W_u W_1 \\
|
||||||
0 & W_u W_2 \\
|
0 & W_u W_2 \\
|
||||||
1 & 0
|
1 & 0
|
||||||
\end{bmatrix}}_{P_{\mathcal{H}_\infty}} \begin{pmatrix}
|
\end{bmatrix}}_{P_{\mathcal{H}_\infty}} \begin{pmatrix}
|
||||||
w \\ u
|
w \\ u
|
||||||
\end{pmatrix}
|
\end{pmatrix}
|
||||||
@ -315,15 +357,58 @@ The \(\mathcal{H}_\infty\) norm of Eq. \eqref{eq:Hinf_norm} is equals to \(\sigm
|
|||||||
\end{figure}
|
\end{figure}
|
||||||
|
|
||||||
\subsection{Example}
|
\subsection{Example}
|
||||||
\label{sec:org0122000}
|
\label{sec:org0ca6ef9}
|
||||||
|
|
||||||
|
\begin{figure}[htbp]
|
||||||
|
\centering
|
||||||
|
\includegraphics[scale=1]{figs/sensors_uncertainty_weights.pdf}
|
||||||
|
\caption{\label{fig:sensors_uncertainty_weights}Magnitude of the multiplicative uncertainty weights \(|W_i(j\omega)|\)}
|
||||||
|
\end{figure}
|
||||||
|
|
||||||
|
|
||||||
|
\begin{figure}[htbp]
|
||||||
|
\centering
|
||||||
|
\includegraphics[scale=1]{figs/weight_uncertainty_bounds_Wu.pdf}
|
||||||
|
\caption{\label{fig:weight_uncertainty_bounds_Wu}Uncertainty region of the two sensors as well as the wanted maximum uncertainty of the super sensor (dashed lines)}
|
||||||
|
\end{figure}
|
||||||
|
|
||||||
|
\begin{figure}[htbp]
|
||||||
|
\centering
|
||||||
|
\includegraphics[scale=1]{figs/hinf_comp_filters.pdf}
|
||||||
|
\caption{\label{fig:hinf_comp_filters}Obtained complementary filters using the \(\mathcal{H}_\infty\) Synthesis}
|
||||||
|
\end{figure}
|
||||||
|
|
||||||
|
\begin{figure}[htbp]
|
||||||
|
\centering
|
||||||
|
\includegraphics[scale=1]{figs/super_sensor_dynamical_uncertainty_Hinf.pdf}
|
||||||
|
\caption{\label{fig:super_sensor_dynamical_uncertainty_Hinf}Super sensor dynamical uncertainty (solid curve) when using the \(\mathcal{H}_\infty\) Synthesis}
|
||||||
|
\end{figure}
|
||||||
|
|
||||||
|
\begin{figure}[htbp]
|
||||||
|
\centering
|
||||||
|
\includegraphics[scale=1]{figs/psd_sensors_hinf_synthesis.pdf}
|
||||||
|
\caption{\label{fig:psd_sensors_hinf_synthesis}Power Spectral Density of the estimated \(\hat{x}\) using the two sensors alone and using the \(\mathcal{H}_\infty\) synthesis}
|
||||||
|
\end{figure}
|
||||||
|
|
||||||
|
|
||||||
\section{Optimal and Robust Sensor Fusion: Mixed \(\mathcal{H}_2/\mathcal{H}_\infty\) Synthesis}
|
\section{Optimal and Robust Sensor Fusion: Mixed \(\mathcal{H}_2/\mathcal{H}_\infty\) Synthesis}
|
||||||
\label{sec:orgdf5a196}
|
\label{sec:orgf642e73}
|
||||||
\label{sec:optimal_robust_fusion}
|
\label{sec:optimal_robust_fusion}
|
||||||
|
|
||||||
\subsection{Sensor Fusion Architecture}
|
\subsection{Sensor with noise and model uncertainty}
|
||||||
\label{sec:orge16b510}
|
\label{sec:org8949812}
|
||||||
|
We wish now to combine the two previous synthesis, that is to say
|
||||||
|
|
||||||
|
The sensors are now modelled by a white noise with unitary PSD \(\tilde{n}_i\) shaped by a LTI transfer function \(N_i(s)\).
|
||||||
|
The dynamical uncertainty of the sensor is modelled using multiplicative uncertainty
|
||||||
|
\begin{equation}
|
||||||
|
v_i = \hat{G}_i (1 + W_i \Delta_i) x + \hat{G_i} (1 + W_i \Delta_i) N_i \tilde{n}_i
|
||||||
|
\end{equation}
|
||||||
|
|
||||||
|
Multiplying by the inverse of the nominal model of the sensor dynamics gives an estimate \(\hat{x}_i\) of \(x\):
|
||||||
|
\begin{equation}
|
||||||
|
\hat{x} = (1 + W_i \Delta_i) x + (1 + W_i \Delta_i) N_i \tilde{n}_i
|
||||||
|
\end{equation}
|
||||||
|
|
||||||
\begin{figure}[htbp]
|
\begin{figure}[htbp]
|
||||||
\centering
|
\centering
|
||||||
@ -331,6 +416,26 @@ The \(\mathcal{H}_\infty\) norm of Eq. \eqref{eq:Hinf_norm} is equals to \(\sigm
|
|||||||
\caption{\label{fig:sensor_model_noise_uncertainty}Sensor Model including Noise and Dynamical Uncertainty}
|
\caption{\label{fig:sensor_model_noise_uncertainty}Sensor Model including Noise and Dynamical Uncertainty}
|
||||||
\end{figure}
|
\end{figure}
|
||||||
|
|
||||||
|
\subsection{Sensor Fusion Architecture}
|
||||||
|
\label{sec:orgcbc3d54}
|
||||||
|
|
||||||
|
For reason of space, the blocks \(\hat{G}_i\) and \(\hat{G}_i^{-1}\) are omitted.
|
||||||
|
|
||||||
|
\begin{equation}
|
||||||
|
\begin{aligned}
|
||||||
|
\hat{x} = &\Big( H_1 (1 + W_1 \Delta_1) + H_2 (1 + W_2 \Delta_2) \Big) x \\
|
||||||
|
&+ \Big( H_1 (1 + W_1 \Delta_1) N_1 \Big) \tilde{n}_1 + \Big( H_2 (1 + W_2 \Delta_2) N_2 \Big) \tilde{n}_2
|
||||||
|
\end{aligned}
|
||||||
|
\end{equation}
|
||||||
|
|
||||||
|
\begin{equation}
|
||||||
|
\begin{aligned}
|
||||||
|
\hat{x} = &\Big( 1 + H_1 W_1 \Delta_1 + H_2 W_2 \Delta_2 \Big) x \\
|
||||||
|
&+ \Big( H_1 (1 + W_1 \Delta_1) N_1 \Big) \tilde{n}_1 + \Big( H_2 (1 + W_2 \Delta_2) N_2 \Big) \tilde{n}_2
|
||||||
|
\end{aligned}
|
||||||
|
\end{equation}
|
||||||
|
|
||||||
|
The estimate \(\hat{x}\) of \(x\)
|
||||||
|
|
||||||
\begin{figure}[htbp]
|
\begin{figure}[htbp]
|
||||||
\centering
|
\centering
|
||||||
@ -338,11 +443,34 @@ The \(\mathcal{H}_\infty\) norm of Eq. \eqref{eq:Hinf_norm} is equals to \(\sigm
|
|||||||
\caption{\label{fig:sensor_fusion_arch_full}Super Sensor Fusion with both sensor noise and sensor model uncertainty}
|
\caption{\label{fig:sensor_fusion_arch_full}Super Sensor Fusion with both sensor noise and sensor model uncertainty}
|
||||||
\end{figure}
|
\end{figure}
|
||||||
|
|
||||||
\subsection{Synthesis Objective}
|
|
||||||
\label{sec:orgb4b43b3}
|
|
||||||
|
|
||||||
\subsection{Mixed \(\mathcal{H}_2/\mathcal{H}_\infty\) Synthesis}
|
\subsection{Mixed \(\mathcal{H}_2/\mathcal{H}_\infty\) Synthesis}
|
||||||
\label{sec:orgb9b52ad}
|
\label{sec:org9d3f160}
|
||||||
|
|
||||||
|
The synthesis objective is to generate two complementary filters \(H_1(s)\) and \(H_2(s)\) such that the uncertainty associated with the super sensor is kept reasonably small and such that the RMS value of super sensors noise is minimized.
|
||||||
|
|
||||||
|
To specify how small we want the super sensor dynamic spread, we use a weighting filter \(W_u(s)\) as was done in Section \ref{sec:robust_fusion}.
|
||||||
|
|
||||||
|
|
||||||
|
This synthesis problem can be solved using the mixed \(\mathcal{H}_2/\mathcal{H}_\infty\) synthesis on the following generalized plant:
|
||||||
|
\begin{equation}
|
||||||
|
\begin{pmatrix}
|
||||||
|
z_{\infty, 1} \\ z_{\infty, 2} \\ z_{2, 1} \\ z_{2, 2} \\ v
|
||||||
|
\end{pmatrix} = \underbrace{\begin{bmatrix}
|
||||||
|
W_u W_1 & W_u W_1 \\
|
||||||
|
0 & W_u W_2 \\
|
||||||
|
N_1 & N_1 \\
|
||||||
|
0 & N_2 \\
|
||||||
|
1 & 0
|
||||||
|
\end{bmatrix}}_{P_{\mathcal{H}_2/\mathcal{H}_\infty}} \begin{pmatrix}
|
||||||
|
w \\ u
|
||||||
|
\end{pmatrix}
|
||||||
|
\end{equation}
|
||||||
|
|
||||||
|
The synthesis objective is to:
|
||||||
|
\begin{itemize}
|
||||||
|
\item Keep the \(\mathcal{H}_\infty\) norm from \(w\) to \((z_{\infty,1}, z_{\infty,2})\) below \(1\)
|
||||||
|
\item Minimize the \(\mathcal{H}_2\) norm from \(w\) to \((z_{2,1}, z_{2,2})\)
|
||||||
|
\end{itemize}
|
||||||
|
|
||||||
\begin{figure}[htbp]
|
\begin{figure}[htbp]
|
||||||
\centering
|
\centering
|
||||||
@ -351,30 +479,54 @@ The \(\mathcal{H}_\infty\) norm of Eq. \eqref{eq:Hinf_norm} is equals to \(\sigm
|
|||||||
\end{figure}
|
\end{figure}
|
||||||
|
|
||||||
\subsection{Example}
|
\subsection{Example}
|
||||||
\label{sec:orgc881f20}
|
\label{sec:org85f304b}
|
||||||
|
|
||||||
|
\begin{figure}[htbp]
|
||||||
|
\centering
|
||||||
|
\includegraphics[scale=1]{figs/htwo_hinf_comp_filters.pdf}
|
||||||
|
\caption{\label{fig:htwo_hinf_comp_filters}Obtained complementary filters after mixed \(\mathcal{H}_2/\mathcal{H}_\infty\) synthesis}
|
||||||
|
\end{figure}
|
||||||
|
|
||||||
|
\begin{figure}[htbp]
|
||||||
|
\centering
|
||||||
|
\includegraphics[scale=1]{figs/psd_sensors_htwo_hinf_synthesis.pdf}
|
||||||
|
\caption{\label{fig:psd_sensors_htwo_hinf_synthesis}Power Spectral Density of the Super Sensor obtained with the mixed \(\mathcal{H}_2/\mathcal{H}_\infty\) synthesis}
|
||||||
|
\end{figure}
|
||||||
|
|
||||||
|
\begin{figure}[htbp]
|
||||||
|
\centering
|
||||||
|
\includegraphics[scale=1]{figs/super_sensor_time_domain_h2_hinf.pdf}
|
||||||
|
\caption{\label{fig:super_sensor_time_domain_h2_hinf}Noise of individual sensors and noise of the super sensor}
|
||||||
|
\end{figure}
|
||||||
|
|
||||||
|
\begin{figure}[htbp]
|
||||||
|
\centering
|
||||||
|
\includegraphics[scale=1]{figs/super_sensor_dynamical_uncertainty_Htwo_Hinf.pdf}
|
||||||
|
\caption{\label{fig:super_sensor_dynamical_uncertainty_Htwo_Hinf}Super sensor dynamical uncertainty (solid curve) when using the mixed \(\mathcal{H}_2/\mathcal{H}_\infty\) Synthesis}
|
||||||
|
\end{figure}
|
||||||
|
|
||||||
\section{Experimental Validation}
|
\section{Experimental Validation}
|
||||||
\label{sec:org05b79a0}
|
\label{sec:org49bf34a}
|
||||||
\label{sec:experimental_validation}
|
\label{sec:experimental_validation}
|
||||||
|
|
||||||
\subsection{Experimental Setup}
|
\subsection{Experimental Setup}
|
||||||
\label{sec:orgc3daf35}
|
\label{sec:orgdd8fce6}
|
||||||
|
|
||||||
\subsection{Sensor Noise and Dynamical Uncertainty}
|
\subsection{Sensor Noise and Dynamical Uncertainty}
|
||||||
\label{sec:org26fedf6}
|
\label{sec:org21add72}
|
||||||
|
|
||||||
\subsection{Mixed \(\mathcal{H}_2/\mathcal{H}_\infty\) Synthesis}
|
\subsection{Mixed \(\mathcal{H}_2/\mathcal{H}_\infty\) Synthesis}
|
||||||
\label{sec:org72f2969}
|
\label{sec:org30521a3}
|
||||||
|
|
||||||
\subsection{Super Sensor Noise and Dynamical Uncertainty}
|
\subsection{Super Sensor Noise and Dynamical Uncertainty}
|
||||||
\label{sec:orgf66f78b}
|
\label{sec:org86cde79}
|
||||||
|
|
||||||
\section{Conclusion}
|
\section{Conclusion}
|
||||||
\label{sec:orge0f0a43}
|
\label{sec:org16245b7}
|
||||||
\label{sec:conclusion}
|
\label{sec:conclusion}
|
||||||
|
|
||||||
\section{Acknowledgment}
|
\section{Acknowledgment}
|
||||||
\label{sec:orgb16559e}
|
\label{sec:orgd992049}
|
||||||
|
|
||||||
\bibliography{ref}
|
\bibliography{ref}
|
||||||
\end{document}
|
\end{document}
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||||||
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Before Width: | Height: | Size: 30 KiB |
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Reference in New Issue
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