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<?xml version="1.0" encoding="utf-8"?>
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<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//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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<!-- 2021-09-01 mer. 10:15 -->
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<meta http-equiv="Content-Type" content="text/html;charset=utf-8" />
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<title>A new method of designing complementary filters for sensor fusion using the $\mathcal{H}_\infty$ synthesis - Tikz Figures</title>
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<meta name="author" content="Thomas Dehaeze" />
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<meta name="generator" content="Org Mode" />
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<link rel="stylesheet" type="text/css" href="https://research.tdehaeze.xyz/css/style.css"/>
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<script type="text/javascript" src="https://research.tdehaeze.xyz/js/script.js"></script>
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<script id="MathJax-script" async
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src="https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-svg.js"></script>
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</head>
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<body>
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<div id="org-div-home-and-up">
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<a accesskey="h" href="../index.html"> UP </a>
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|
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<a accesskey="H" href="../index.html"> HOME </a>
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</div><div id="content" class="content">
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<h1 class="title">A new method of designing complementary filters for sensor fusion using the \(\mathcal{H}_\infty\) synthesis - Tikz Figures</h1>
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<div id="table-of-contents" role="doc-toc">
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<h2>Table of Contents</h2>
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<div id="text-table-of-contents" role="doc-toc">
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<ul>
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<li><a href="#orgaddbc4b">1. Sensor Fusion - Overview</a></li>
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<li><a href="#orgcde34ed">2. Sensor Model</a></li>
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<li><a href="#org881c7cf">3. Sensor Model with calibration</a></li>
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<li><a href="#org52dba0e">4. Sensor Fusion Architecture</a></li>
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<li><a href="#org51c6b92">5. Sensor Model with Uncertainty</a></li>
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<li><a href="#orgd0c3320">6. Sensor Model with Uncertainty - Simplified</a></li>
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<li><a href="#orgd991dd2">7. Sensor fusion architecture with sensor dynamics uncertainty</a></li>
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<li><a href="#orgb45d1ac">8. Uncertainty set of the super sensor dynamics</a></li>
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<li><a href="#orgd194580">9. Generalized plant used for \(\mathcal{H}_\infty\) synthesis of complementary filters</a></li>
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<li><a href="#orgebe7ec5">10. Architecture used for \(\mathcal{H}_\infty\) synthesis of complementary filters</a></li>
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<li><a href="#orgf9a2f39">11. LIGO Sensor Fusion Architecture</a></li>
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<li><a href="#orga2611bb">12. Closed-Loop Complementary Filters</a></li>
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<li><a href="#org5af72e4">13. Closed-Loop Fusion Architecture</a></li>
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<li><a href="#org4e79ee7">14. Feedback Loop Sensor Fusion Architecture</a></li>
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<li><a href="#orgb59c88b">15. Feedback Sensor Fusion - Generalized Plant</a></li>
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<li><a href="#orge593f6d">16. Sensor Fusion - Sequential</a></li>
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<li><a href="#org3b323e8">17. Sensor Fusion - Parallel</a></li>
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<li><a href="#org9baa5b5">18. Architecture for \(\mathcal{H}_\infty\) synthesis of three complementary filters</a></li>
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<li><a href="#org10ffb09">19. Architecture for \(\mathcal{H}_\infty\) synthesis of three complementary filters</a></li>
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</ul>
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</div>
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</div>
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<p>
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Configuration file is accessible <a href="config.html">here</a>.
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</p>
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<div id="outline-container-orgaddbc4b" class="outline-2">
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<h2 id="orgaddbc4b"><span class="section-number-2">1.</span> Sensor Fusion - Overview</h2>
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<div class="outline-text-2" id="text-1">
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<div class="org-src-container">
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<pre class="src src-latex"><span class="org-font-latex-sedate">\definecolor</span>{myblue}{rgb}{0, 0.447, 0.741}
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<span class="org-font-latex-sedate">\definecolor</span>{myred}{rgb}{0.8500, 0.325, 0.098}
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<span class="org-font-latex-sedate"><span class="org-keyword">\begin</span></span>{<span class="org-function-name">tikzpicture</span>}
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<span class="org-font-latex-sedate">\node</span>[branch] (x) at (0, 0);
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<span class="org-font-latex-sedate">\node</span>[block, above right=0.3 and 0.5 of x](sensor1){Sensor 1};
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<span class="org-font-latex-sedate">\node</span>[block, below right=0.3 and 0.5 of x](sensor2){Sensor 2};
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<span class="org-font-latex-sedate">\node</span>[block, right=1.1 of sensor1](H1){<span class="org-font-latex-math">$H</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">1(s)$</span>};
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<span class="org-font-latex-sedate">\node</span>[block, right=1.1 of sensor2](H2){<span class="org-font-latex-math">$H</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">2(s)$</span>};
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<span class="org-font-latex-sedate">\node</span>[addb, right=5.0 of x](add){};
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<span class="org-font-latex-sedate">\draw</span>[] (<span class="org-font-latex-math">$(x)+(-0.7, 0)$</span>) node[above right]{<span class="org-font-latex-math">$x$</span>} -- (x.center);
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<span class="org-font-latex-sedate">\draw</span>[->] (x.center) |- (sensor1.west);
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<span class="org-font-latex-sedate">\draw</span>[->] (x.center) |- (sensor2.west);
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<span class="org-font-latex-sedate">\draw</span>[->] (sensor1.east) -- node[midway, above]{<span class="org-font-latex-math">$</span><span class="org-font-latex-sedate"><span class="org-font-latex-math">\hat</span></span><span class="org-font-latex-math">{x}</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">1$</span>} (H1.west);
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<span class="org-font-latex-sedate">\draw</span>[->] (sensor2.east) -- node[midway, above]{<span class="org-font-latex-math">$</span><span class="org-font-latex-sedate"><span class="org-font-latex-math">\hat</span></span><span class="org-font-latex-math">{x}</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">2$</span>} (H2.west);
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<span class="org-font-latex-sedate">\draw</span>[->] (H1) -| (add.north);
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<span class="org-font-latex-sedate">\draw</span>[->] (H2) -| (add.south);
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<span class="org-font-latex-sedate">\draw</span>[->] (add.east) -- ++(0.9, 0) node[above left]{<span class="org-font-latex-math">$</span><span class="org-font-latex-sedate"><span class="org-font-latex-math">\hat</span></span><span class="org-font-latex-math">{x}$</span>};
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<span class="org-font-latex-sedate"><span class="org-keyword">\begin</span></span>{<span class="org-function-name">scope</span>}[on background layer]
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<span class="org-font-latex-sedate">\node</span>[fit={(<span class="org-font-latex-math">$(H2.south-|x) + (0, -0.2)$</span>) (<span class="org-font-latex-math">$(H1.north-|add.east) + (0.2, 0.6)$</span>)}, fill=black!10!white, draw, inner sep=6pt] (supersensor) {};
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<span class="org-font-latex-sedate">\node</span>[below] at (supersensor.north) {Super Sensor};
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<span class="org-font-latex-sedate">\node</span>[fit={(sensor2.south west) (sensor1.north east)}, fill=black!20!white, draw, inner sep=6pt] (sensors) {};
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<span class="org-font-latex-sedate">\node</span>[align=center] at (sensors.center) {{<span class="org-font-latex-sedate"><span class="org-keyword">\tiny</span></span><span class="org-type"> Normalized</span>}<span class="org-font-latex-warning">\\</span>[-0.5em]{<span class="org-font-latex-sedate"><span class="org-keyword">\tiny</span></span><span class="org-type"> Sensors</span>}};
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<span class="org-font-latex-sedate">\node</span>[fit={(H2.south west) (H1.north-|add.east)}, fill=black!20!white, draw, inner sep=6pt] (filters) {};
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<span class="org-font-latex-sedate">\node</span>[align=center] at (<span class="org-font-latex-math">$(filters.center) + (-0.3, 0)$</span>) {{<span class="org-font-latex-sedate"><span class="org-keyword">\tiny</span></span><span class="org-type"> Complementary</span>}<span class="org-font-latex-warning">\\</span>[-0.5em]{<span class="org-font-latex-sedate"><span class="org-keyword">\tiny</span></span><span class="org-type"> Filters</span>}};
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<span class="org-font-latex-sedate"><span class="org-keyword">\end</span></span>{<span class="org-function-name">scope</span>}
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<span class="org-font-latex-sedate"><span class="org-keyword">\end</span></span>{<span class="org-function-name">tikzpicture</span>}
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</pre>
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</div>
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<div id="orgfdc79fa" class="figure">
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<p><img src="figs/sensor_fusion_overview.png" alt="sensor_fusion_overview.png" />
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</p>
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<p><span class="figure-number">Figure 1: </span>Schematic of a sensor fusion architecture using complementary filters</p>
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</div>
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</div>
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</div>
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<div id="outline-container-orgcde34ed" class="outline-2">
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<h2 id="orgcde34ed"><span class="section-number-2">2.</span> Sensor Model</h2>
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<div class="outline-text-2" id="text-2">
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<div class="org-src-container">
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<pre class="src src-latex"><span class="org-font-latex-sedate"><span class="org-keyword">\begin</span></span>{<span class="org-function-name">tikzpicture</span>}
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<span class="org-font-latex-sedate">\node</span>[addb](add1){};
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<span class="org-font-latex-sedate">\node</span>[block, right=0.8 of add1](G1){<span class="org-font-latex-math">$G</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">i(s)$</span>};
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<span class="org-font-latex-sedate">\draw</span>[->] (<span class="org-font-latex-math">$(add1.west)+(-0.7, 0)$</span>) node[above right]{<span class="org-font-latex-math">$x$</span>} -- (add1.west);
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<span class="org-font-latex-sedate">\draw</span>[<-] (add1.north) -- ++(0, 0.7)node[below right](n1){<span class="org-font-latex-math">$n</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">i$</span>};
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<span class="org-font-latex-sedate">\draw</span>[->] (add1.east) -- (G1.west);
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<span class="org-font-latex-sedate">\draw</span>[->] (G1.east) -- ++(0.7, 0) node[above left]{<span class="org-font-latex-math">$</span><span class="org-font-latex-sedate"><span class="org-font-latex-math">\tilde</span></span><span class="org-font-latex-math">{x}</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">i$</span>};
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<span class="org-font-latex-sedate"><span class="org-keyword">\begin</span></span>{<span class="org-function-name">scope</span>}[on background layer]
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<span class="org-font-latex-sedate">\node</span>[fit={(add1.west |- G1.south) (n1.north -| G1.east)}, fill=black!20!white, draw, inner sep=3pt] (sensor1) {};
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<span class="org-font-latex-sedate">\node</span>[below left] at (sensor1.north east) {Sensor};
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<span class="org-font-latex-sedate"><span class="org-keyword">\end</span></span>{<span class="org-function-name">scope</span>}
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<span class="org-font-latex-sedate"><span class="org-keyword">\end</span></span>{<span class="org-function-name">tikzpicture</span>}
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</pre>
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</div>
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<div id="orga6c4cb8" class="figure">
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<p><img src="figs/sensor_model.png" alt="sensor_model.png" />
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</p>
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<p><span class="figure-number">Figure 2: </span>Basic sensor model consisting of a noise input \(n_i\) and a linear time invariant transfer function \(G_i(s)\)</p>
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</div>
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</div>
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</div>
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<div id="outline-container-org881c7cf" class="outline-2">
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<h2 id="org881c7cf"><span class="section-number-2">3.</span> Sensor Model with calibration</h2>
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<div class="outline-text-2" id="text-3">
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<div class="org-src-container">
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<pre class="src src-latex"><span class="org-font-latex-sedate"><span class="org-keyword">\begin</span></span>{<span class="org-function-name">tikzpicture</span>}
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<span class="org-font-latex-sedate">\node</span>[addb](add1){};
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<span class="org-font-latex-sedate">\node</span>[block, right=0.8 of add1](G1){<span class="org-font-latex-math">$G</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">i(s)$</span>};
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<span class="org-font-latex-sedate">\node</span>[block, right=0.8 of G1](G1inv){<span class="org-font-latex-math">$</span><span class="org-font-latex-sedate"><span class="org-font-latex-math">\hat</span></span><span class="org-font-latex-math">{G}</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">i</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">^</span></span><span class="org-font-latex-math">{-1}(s)$</span>};
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<span class="org-font-latex-sedate">\draw</span>[->] (<span class="org-font-latex-math">$(add1.west)+(-0.7, 0)$</span>) node[above right]{<span class="org-font-latex-math">$x$</span>} -- (add1.west);
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<span class="org-font-latex-sedate">\draw</span>[<-] (add1.north) -- ++(0, 0.7)node[below right](n1){<span class="org-font-latex-math">$n</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">i$</span>};
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<span class="org-font-latex-sedate">\draw</span>[->] (add1.east) -- (G1.west);
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<span class="org-font-latex-sedate">\draw</span>[->] (G1.east) -- (G1inv.west) node[above left]{<span class="org-font-latex-math">$</span><span class="org-font-latex-sedate"><span class="org-font-latex-math">\tilde</span></span><span class="org-font-latex-math">{x}</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">i$</span>};
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||||
<span class="org-font-latex-sedate">\draw</span>[->] (G1inv.east) -- ++(0.8, 0) node[above left]{<span class="org-font-latex-math">$</span><span class="org-font-latex-sedate"><span class="org-font-latex-math">\hat</span></span><span class="org-font-latex-math">{x}</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">i$</span>};
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<span class="org-font-latex-sedate"><span class="org-keyword">\begin</span></span>{<span class="org-function-name">scope</span>}[on background layer]
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<span class="org-font-latex-sedate">\node</span>[fit={(add1.west |- G1inv.south) (n1.north -| G1inv.east)}, fill=black!10!white, draw, inner sep=6pt] (sensor1cal) {};
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<span class="org-font-latex-sedate">\node</span>[below left, align=right] at (sensor1cal.north east) {{<span class="org-font-latex-sedate"><span class="org-keyword">\tiny</span></span><span class="org-type"> Normalized</span>}<span class="org-font-latex-warning">\\</span>[-0.5em]{<span class="org-font-latex-sedate"><span class="org-keyword">\tiny</span></span><span class="org-type"> sensor</span>}};
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<span class="org-font-latex-sedate">\node</span>[fit={(add1.west |- G1.south) (n1.north -| G1.east)}, fill=black!20!white, draw, inner sep=3pt] (sensor1) {};
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<span class="org-font-latex-sedate">\node</span>[below left] at (sensor1.north east) {Sensor};
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<span class="org-font-latex-sedate"><span class="org-keyword">\end</span></span>{<span class="org-function-name">scope</span>}
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<span class="org-font-latex-sedate"><span class="org-keyword">\end</span></span>{<span class="org-function-name">tikzpicture</span>}
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</pre>
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</div>
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<div id="orgd38eff7" class="figure">
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<p><img src="figs/sensor_model_calibrated.png" alt="sensor_model_calibrated.png" />
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</p>
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<p><span class="figure-number">Figure 3: </span>Normalized sensors using the inverse of an estimate \(\hat{G}_i(s)\) of the sensor dynamics</p>
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</div>
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</div>
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</div>
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<div id="outline-container-org52dba0e" class="outline-2">
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<h2 id="org52dba0e"><span class="section-number-2">4.</span> Sensor Fusion Architecture</h2>
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<div class="outline-text-2" id="text-4">
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<div class="org-src-container">
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<pre class="src src-latex"><span class="org-font-latex-sedate">\definecolor</span>{myblue}{rgb}{0, 0.447, 0.741}
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<span class="org-font-latex-sedate">\definecolor</span>{myred}{rgb}{0.8500, 0.325, 0.098}
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<span class="org-font-latex-sedate"><span class="org-keyword">\begin</span></span>{<span class="org-function-name">tikzpicture</span>}
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<span class="org-font-latex-sedate">\node</span>[branch] (x) at (0, 0);
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<span class="org-font-latex-sedate">\node</span>[addb, above right=0.8 and 0.5 of x](add1){};
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<span class="org-font-latex-sedate">\node</span>[addb, below right=0.8 and 0.5 of x](add2){};
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<span class="org-font-latex-sedate">\node</span>[block, right=0.8 of add1](G1){<span class="org-font-latex-math">$G</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">1(s)$</span>};
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<span class="org-font-latex-sedate">\node</span>[block, right=0.8 of add2](G2){<span class="org-font-latex-math">$G</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">2(s)$</span>};
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<span class="org-font-latex-sedate">\node</span>[block, right=0.8 of G1](G1inv){<span class="org-font-latex-math">$</span><span class="org-font-latex-sedate"><span class="org-font-latex-math">\hat</span></span><span class="org-font-latex-math">{G}</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">1</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">^</span></span><span class="org-font-latex-math">{-1}(s)$</span>};
|
||||
<span class="org-font-latex-sedate">\node</span>[block, right=0.8 of G2](G2inv){<span class="org-font-latex-math">$</span><span class="org-font-latex-sedate"><span class="org-font-latex-math">\hat</span></span><span class="org-font-latex-math">{G}</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">2</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">^</span></span><span class="org-font-latex-math">{-2}(s)$</span>};
|
||||
<span class="org-font-latex-sedate">\node</span>[block, right=0.8 of G1inv](H1){<span class="org-font-latex-math">$H</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">1(s)$</span>};
|
||||
<span class="org-font-latex-sedate">\node</span>[block, right=0.8 of G2inv](H2){<span class="org-font-latex-math">$H</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">2(s)$</span>};
|
||||
<span class="org-font-latex-sedate">\node</span>[addb, right=7 of x](add){};
|
||||
|
||||
<span class="org-font-latex-sedate">\draw</span>[] (<span class="org-font-latex-math">$(x)+(-0.7, 0)$</span>) node[above right]{<span class="org-font-latex-math">$x$</span>} -- (x.center);
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (x.center) |- (add1.west);
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (x.center) |- (add2.west);
|
||||
<span class="org-font-latex-sedate">\draw</span>[<-] (add1.north) -- ++(0, 0.7)node[below right](n1){<span class="org-font-latex-math">$n</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">1$</span>};
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (add1.east) -- (G1.west);
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (G1.east) -- (G1inv.west) node[above left]{<span class="org-font-latex-math">$</span><span class="org-font-latex-sedate"><span class="org-font-latex-math">\tilde</span></span><span class="org-font-latex-math">{x}</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">1$</span>};
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (G1inv.east) -- (H1.west) node[above left]{<span class="org-font-latex-math">$</span><span class="org-font-latex-sedate"><span class="org-font-latex-math">\hat</span></span><span class="org-font-latex-math">{x}</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">1$</span>};
|
||||
<span class="org-font-latex-sedate">\draw</span>[<-] (add2.north) -- ++(0, 0.7)node[below right](n2){<span class="org-font-latex-math">$n</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">2$</span>};
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (add2.east) -- (G2.west);
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (G2.east) -- (G2inv.west) node[above left]{<span class="org-font-latex-math">$</span><span class="org-font-latex-sedate"><span class="org-font-latex-math">\tilde</span></span><span class="org-font-latex-math">{x}</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">2$</span>};
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (G2inv.east) -- (H2.west) node[above left]{<span class="org-font-latex-math">$</span><span class="org-font-latex-sedate"><span class="org-font-latex-math">\hat</span></span><span class="org-font-latex-math">{x}</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">2$</span>};
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (H1) -| (add.north);
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (H2) -| (add.south);
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (add.east) -- ++(0.7, 0) node[above left]{<span class="org-font-latex-math">$</span><span class="org-font-latex-sedate"><span class="org-font-latex-math">\hat</span></span><span class="org-font-latex-math">{x}$</span>};
|
||||
|
||||
<span class="org-font-latex-sedate"><span class="org-keyword">\begin</span></span>{<span class="org-function-name">scope</span>}[on background layer]
|
||||
<span class="org-font-latex-sedate">\node</span>[fit={(G2.south-|x) (n1.north-|add.east)}, fill=black!10!white, draw, inner sep=9pt] (supersensor) {};
|
||||
<span class="org-font-latex-sedate">\node</span>[below left] at (supersensor.north east) {Super Sensor};
|
||||
|
||||
<span class="org-font-latex-sedate">\node</span>[fit={(add1.west |- G1inv.south) (n1.north -| G1inv.east)}, fill=myblue!20!white, draw, inner sep=6pt] (sensor1cal) {};
|
||||
<span class="org-font-latex-sedate">\node</span>[below left, align=right] at (sensor1cal.north east) {{<span class="org-font-latex-sedate"><span class="org-keyword">\tiny</span></span><span class="org-type"> Normalized</span>}<span class="org-font-latex-warning">\\</span>[-0.5em]{<span class="org-font-latex-sedate"><span class="org-keyword">\tiny</span></span><span class="org-type"> sensor</span>}};
|
||||
<span class="org-font-latex-sedate">\node</span>[fit={(add1.west |- G1.south) (n1.north -| G1.east)}, fill=myblue!30!white, draw, inner sep=3pt] (sensor1) {};
|
||||
<span class="org-font-latex-sedate">\node</span>[below left] at (sensor1.north east) {Sensor 1};
|
||||
|
||||
<span class="org-font-latex-sedate">\node</span>[fit={(add2.west |- G2inv.south) (n2.north -| G2inv.east)}, fill=myred!20!white, draw, inner sep=6pt] (sensor2cal) {};
|
||||
<span class="org-font-latex-sedate">\node</span>[below left, align=right] at (sensor2cal.north east) {{<span class="org-font-latex-sedate"><span class="org-keyword">\tiny</span></span><span class="org-type"> Normalized</span>}<span class="org-font-latex-warning">\\</span>[-0.5em]{<span class="org-font-latex-sedate"><span class="org-keyword">\tiny</span></span><span class="org-type"> sensor</span>}};
|
||||
<span class="org-font-latex-sedate">\node</span>[fit={(add2.west |- G2.south) (n2.north -| G2.east)}, fill=myred!30!white, draw, inner sep=3pt] (sensor2) {};
|
||||
<span class="org-font-latex-sedate">\node</span>[below left] at (sensor2.north east) {Sensor 2};
|
||||
<span class="org-font-latex-sedate"><span class="org-keyword">\end</span></span>{<span class="org-function-name">scope</span>}
|
||||
<span class="org-font-latex-sedate"><span class="org-keyword">\end</span></span>{<span class="org-function-name">tikzpicture</span>}
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
|
||||
<div id="org63acf85" class="figure">
|
||||
<p><img src="figs/fusion_super_sensor.png" alt="fusion_super_sensor.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 4: </span>Sensor fusion architecture with two normalized sensors</p>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org51c6b92" class="outline-2">
|
||||
<h2 id="org51c6b92"><span class="section-number-2">5.</span> Sensor Model with Uncertainty</h2>
|
||||
<div class="outline-text-2" id="text-5">
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-latex"><span class="org-font-latex-sedate"><span class="org-keyword">\begin</span></span>{<span class="org-function-name">tikzpicture</span>}
|
||||
<span class="org-font-latex-sedate">\node</span>[branch] (input) at (0,0) {};
|
||||
<span class="org-font-latex-sedate">\node</span>[block, above right= 0.4 and 0.4 of input](W1){<span class="org-font-latex-math">$w</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">1(s)$</span>};
|
||||
<span class="org-font-latex-sedate">\node</span>[block, right=0.4 of W1](delta1){<span class="org-font-latex-math">$</span><span class="org-font-latex-sedate"><span class="org-font-latex-math">\Delta</span></span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">1(s)$</span>};
|
||||
<span class="org-font-latex-sedate">\node</span>[addb] (addu) at (<span class="org-font-latex-math">$(delta1.east|-input) + (0.4, 0)$</span>) {};
|
||||
<span class="org-font-latex-sedate">\node</span>[addb, right=0.4 of addu] (addn) {};
|
||||
<span class="org-font-latex-sedate">\node</span>[block, right=0.4 of addn] (G1) {<span class="org-font-latex-math">$</span><span class="org-font-latex-sedate"><span class="org-font-latex-math">\hat</span></span><span class="org-font-latex-math">{G}</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">1(s)$</span>};
|
||||
<span class="org-font-latex-sedate">\node</span>[block, right=0.8 of G1](G1inv){<span class="org-font-latex-math">$</span><span class="org-font-latex-sedate"><span class="org-font-latex-math">\hat</span></span><span class="org-font-latex-math">{G}</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">1</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">^</span></span><span class="org-font-latex-math">{-1}(s)$</span>};
|
||||
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (<span class="org-font-latex-math">$(input)+(-0.7, 0)$</span>) node[above right]{<span class="org-font-latex-math">$x$</span>} -- (addu);
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (input.center) |- (W1.west);
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (W1.east) -- (delta1.west);
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (delta1.east) -| (addu.north);
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (addu.east) -- (addn.west);
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (addn.east) -- (G1.west);
|
||||
<span class="org-font-latex-sedate">\draw</span>[<-] (addn.north) -- ++(0, 0.7)node[below right](n1){<span class="org-font-latex-math">$n</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">1$</span>};
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (G1.east) -- (G1inv.west) node[above left]{<span class="org-font-latex-math">$</span><span class="org-font-latex-sedate"><span class="org-font-latex-math">\tilde</span></span><span class="org-font-latex-math">{x}</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">1$</span>};
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (G1inv.east) -- ++(0.8, 0) node[above left]{<span class="org-font-latex-math">$</span><span class="org-font-latex-sedate"><span class="org-font-latex-math">\hat</span></span><span class="org-font-latex-math">{x}</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">1$</span>};
|
||||
|
||||
<span class="org-font-latex-sedate"><span class="org-keyword">\begin</span></span>{<span class="org-function-name">scope</span>}[on background layer]
|
||||
<span class="org-font-latex-sedate">\node</span>[fit={(input.west |- G1inv.south) (delta1.north -| G1inv.east)}, fill=black!10!white, draw, inner sep=6pt] (sensor1cal) {};
|
||||
<span class="org-font-latex-sedate">\node</span>[below left, align=right] at (sensor1cal.north east) {{<span class="org-font-latex-sedate"><span class="org-keyword">\tiny</span></span><span class="org-type"> Normalized</span>}<span class="org-font-latex-warning">\\</span>[-0.5em]{<span class="org-font-latex-sedate"><span class="org-keyword">\tiny</span></span><span class="org-type"> sensor</span>}};
|
||||
|
||||
<span class="org-font-latex-sedate">\node</span>[fit={(input.west |- G1.south) (delta1.north -| G1.east)}, fill=black!20!white, draw, inner sep=3pt] (sensor1) {};
|
||||
<span class="org-font-latex-sedate">\node</span>[below left] at (sensor1.north east) {Sensor};
|
||||
<span class="org-font-latex-sedate"><span class="org-keyword">\end</span></span>{<span class="org-function-name">scope</span>}
|
||||
<span class="org-font-latex-sedate"><span class="org-keyword">\end</span></span>{<span class="org-function-name">tikzpicture</span>}
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
|
||||
<div id="org937c435" class="figure">
|
||||
<p><img src="figs/sensor_model_uncertainty.png" alt="sensor_model_uncertainty.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 5: </span>Sensor with multiplicative input uncertainty</p>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-orgd0c3320" class="outline-2">
|
||||
<h2 id="orgd0c3320"><span class="section-number-2">6.</span> Sensor Model with Uncertainty - Simplified</h2>
|
||||
<div class="outline-text-2" id="text-6">
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-latex"><span class="org-font-latex-sedate"><span class="org-keyword">\begin</span></span>{<span class="org-function-name">tikzpicture</span>}
|
||||
<span class="org-font-latex-sedate">\node</span>[branch] (input) at (0,0) {};
|
||||
<span class="org-font-latex-sedate">\node</span>[block, above right= 0.4 and 0.4 of input](W1){<span class="org-font-latex-math">$w</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">1(s)$</span>};
|
||||
<span class="org-font-latex-sedate">\node</span>[block, right=0.4 of W1](delta1){<span class="org-font-latex-math">$</span><span class="org-font-latex-sedate"><span class="org-font-latex-math">\Delta</span></span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">1(s)$</span>};
|
||||
<span class="org-font-latex-sedate">\node</span>[addb] (addu) at (<span class="org-font-latex-math">$(delta1.east|-input) + (0.4, 0)$</span>) {};
|
||||
<span class="org-font-latex-sedate">\node</span>[addb, right=0.4 of addu] (addn) {};
|
||||
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (<span class="org-font-latex-math">$(input)+(-0.8, 0)$</span>) node[above right]{<span class="org-font-latex-math">$x$</span>} -- (addu);
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (input.center) |- (W1.west);
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (W1.east) -- (delta1.west);
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (delta1.east) -| (addu.north);
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (addu.east) -- (addn.west);
|
||||
<span class="org-font-latex-sedate">\draw</span>[<-] (addn.north) -- ++(0, 0.6)node[below right](n1){<span class="org-font-latex-math">$n</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">1$</span>};
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (addn.east) -- ++(0.9, 0) node[above left]{<span class="org-font-latex-math">$</span><span class="org-font-latex-sedate"><span class="org-font-latex-math">\hat</span></span><span class="org-font-latex-math">{x}</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">1$</span>};
|
||||
|
||||
<span class="org-font-latex-sedate"><span class="org-keyword">\begin</span></span>{<span class="org-function-name">scope</span>}[on background layer]
|
||||
<span class="org-font-latex-sedate">\node</span>[fit={(input.west |- addu.south) (<span class="org-font-latex-math">$(delta1.north -| addn.east) + (0.1, 0)$</span>)}, fill=black!10!white, draw, inner sep=6pt] (sensor1cal) {};
|
||||
<span class="org-font-latex-sedate">\node</span>[below left, align=right] at (sensor1cal.north east) {{<span class="org-font-latex-sedate"><span class="org-keyword">\tiny</span></span><span class="org-type"> Normalized</span>}<span class="org-font-latex-warning">\\</span>[-0.5em]{<span class="org-font-latex-sedate"><span class="org-keyword">\tiny</span></span><span class="org-type"> sensor</span>}};
|
||||
<span class="org-font-latex-sedate"><span class="org-keyword">\end</span></span>{<span class="org-function-name">scope</span>}
|
||||
<span class="org-font-latex-sedate"><span class="org-keyword">\end</span></span>{<span class="org-function-name">tikzpicture</span>}
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
|
||||
<div id="orgce8343e" class="figure">
|
||||
<p><img src="figs/sensor_model_uncertainty_simplified.png" alt="sensor_model_uncertainty_simplified.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 6: </span>Simplified sensor model</p>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-orgd991dd2" class="outline-2">
|
||||
<h2 id="orgd991dd2"><span class="section-number-2">7.</span> Sensor fusion architecture with sensor dynamics uncertainty</h2>
|
||||
<div class="outline-text-2" id="text-7">
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-latex"><span class="org-font-latex-sedate">\definecolor</span>{myblue}{rgb}{0, 0.447, 0.741}
|
||||
<span class="org-font-latex-sedate">\definecolor</span>{myred}{rgb}{0.8500, 0.325, 0.098}
|
||||
|
||||
<span class="org-font-latex-sedate"><span class="org-keyword">\begin</span></span>{<span class="org-function-name">tikzpicture</span>}
|
||||
<span class="org-font-latex-sedate">\node</span>[branch] (x) at (0, 0);
|
||||
|
||||
<span class="org-font-latex-sedate">\node</span>[branch, above right=1.0 and 0.3 of x] (input1) {};
|
||||
<span class="org-font-latex-sedate">\node</span>[branch, below right=1.0 and 0.3 of x] (input2) {};
|
||||
<span class="org-font-latex-sedate">\node</span>[block, above right= 0.4 and 0.3 of input1](W1){<span class="org-font-latex-math">$w</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">1(s)$</span>};
|
||||
<span class="org-font-latex-sedate">\node</span>[block, above right= 0.4 and 0.3 of input2](W2){<span class="org-font-latex-math">$w</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">2(s)$</span>};
|
||||
<span class="org-font-latex-sedate">\node</span>[block, right=0.4 of W1](delta1){<span class="org-font-latex-math">$</span><span class="org-font-latex-sedate"><span class="org-font-latex-math">\Delta</span></span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">1(s)$</span>};
|
||||
<span class="org-font-latex-sedate">\node</span>[block, right=0.4 of W2](delta2){<span class="org-font-latex-math">$</span><span class="org-font-latex-sedate"><span class="org-font-latex-math">\Delta</span></span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">2(s)$</span>};
|
||||
<span class="org-font-latex-sedate">\node</span>[addb] (addu1) at (<span class="org-font-latex-math">$(delta1.east|-input1) + (0.4, 0)$</span>) {};
|
||||
<span class="org-font-latex-sedate">\node</span>[addb] (addu2) at (<span class="org-font-latex-math">$(delta2.east|-input2) + (0.4, 0)$</span>) {};
|
||||
<span class="org-font-latex-sedate">\node</span>[addb, right=0.4 of addu1] (addn1) {};
|
||||
<span class="org-font-latex-sedate">\node</span>[addb, right=0.4 of addu2] (addn2) {};
|
||||
<span class="org-font-latex-sedate">\node</span>[block, right=0.9 of addn1](H1){<span class="org-font-latex-math">$H</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">1(s)$</span>};
|
||||
<span class="org-font-latex-sedate">\node</span>[block, right=0.9 of addn2](H2){<span class="org-font-latex-math">$H</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">2(s)$</span>};
|
||||
|
||||
<span class="org-font-latex-sedate">\node</span>[addb, right=7 of x](add){};
|
||||
|
||||
|
||||
<span class="org-font-latex-sedate">\draw</span>[] (<span class="org-font-latex-math">$(x)+(-0.7, 0)$</span>) node[above right]{<span class="org-font-latex-math">$x$</span>} -- (x.center);
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (x.center) |- (addu1.west);
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (x.center) |- (addu2.west);
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (input1.center) |- (W1.west);
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (W1.east) -- (delta1.west);
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (delta1.east) -| (addu1.north);
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (addu1.east) -- (addn1.west);
|
||||
<span class="org-font-latex-sedate">\draw</span>[<-] (addn1.north) -- ++(0, 0.6)node[below right](n1){<span class="org-font-latex-math">$n</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">1$</span>};
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (input2.center) |- (W2.west);
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (W2.east) -- (delta2.west);
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (delta2.east) -| (addu2.north);
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (addu2.east) -- (addn2.west);
|
||||
<span class="org-font-latex-sedate">\draw</span>[<-] (addn2.north) -- ++(0, 0.6)node[below right](n2){<span class="org-font-latex-math">$n</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">2$</span>};
|
||||
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (addn1.east) -- (H1.west) node[above left]{<span class="org-font-latex-math">$</span><span class="org-font-latex-sedate"><span class="org-font-latex-math">\hat</span></span><span class="org-font-latex-math">{x}</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">1$</span>};
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (addn2.east) -- (H2.west) node[above left]{<span class="org-font-latex-math">$</span><span class="org-font-latex-sedate"><span class="org-font-latex-math">\hat</span></span><span class="org-font-latex-math">{x}</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">2$</span>};
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (H1) -| (add.north);
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (H2) -| (add.south);
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (add.east) -- ++(0.7, 0) node[above left]{<span class="org-font-latex-math">$</span><span class="org-font-latex-sedate"><span class="org-font-latex-math">\hat</span></span><span class="org-font-latex-math">{x}$</span>};
|
||||
|
||||
<span class="org-font-latex-sedate"><span class="org-keyword">\begin</span></span>{<span class="org-function-name">scope</span>}[on background layer]
|
||||
<span class="org-font-latex-sedate">\node</span>[fit={(addn2.south-|x) (delta1.north-|add.east)}, fill=black!10!white, draw, inner sep=9pt] (supersensor) {};
|
||||
<span class="org-font-latex-sedate">\node</span>[below left] at (supersensor.north east) {Super Sensor};
|
||||
|
||||
<span class="org-font-latex-sedate">\node</span>[fit={(input1.west |- addu1.south) (<span class="org-font-latex-math">$(delta1.north -| addn1.east) + (0.1, 0.0)$</span>)}, fill=myblue!20!white, draw, inner sep=6pt] (sensor1cal) {};
|
||||
<span class="org-font-latex-sedate">\node</span>[below left, align=right] at (sensor1cal.north east) {{<span class="org-font-latex-sedate"><span class="org-keyword">\tiny</span></span><span class="org-type"> Normalized</span>}<span class="org-font-latex-warning">\\</span>[-0.5em]{<span class="org-font-latex-sedate"><span class="org-keyword">\tiny</span></span><span class="org-type"> sensor 1</span>}};
|
||||
|
||||
<span class="org-font-latex-sedate">\node</span>[fit={(input2.west |- addu2.south) (<span class="org-font-latex-math">$(delta2.north -| addn1.east) + (0.1, 0.0)$</span>)}, fill=myred!20!white, draw, inner sep=6pt] (sensor2cal) {};
|
||||
<span class="org-font-latex-sedate">\node</span>[below left, align=right] at (sensor2cal.north east) {{<span class="org-font-latex-sedate"><span class="org-keyword">\tiny</span></span><span class="org-type"> Normalized</span>}<span class="org-font-latex-warning">\\</span>[-0.5em]{<span class="org-font-latex-sedate"><span class="org-keyword">\tiny</span></span><span class="org-type"> sensor 2</span>}};
|
||||
<span class="org-font-latex-sedate"><span class="org-keyword">\end</span></span>{<span class="org-function-name">scope</span>}
|
||||
<span class="org-font-latex-sedate"><span class="org-keyword">\end</span></span>{<span class="org-function-name">tikzpicture</span>}
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
|
||||
<div id="org5b69e5a" class="figure">
|
||||
<p><img src="figs/sensor_fusion_dynamic_uncertainty.png" alt="sensor_fusion_dynamic_uncertainty.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 7: </span>Sensor fusion architecture with sensor dynamics uncertainty</p>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-orgb45d1ac" class="outline-2">
|
||||
<h2 id="orgb45d1ac"><span class="section-number-2">8.</span> Uncertainty set of the super sensor dynamics</h2>
|
||||
<div class="outline-text-2" id="text-8">
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-latex"><span class="org-font-latex-sedate">\definecolor</span>{myblue}{rgb}{0, 0.447, 0.741}
|
||||
<span class="org-font-latex-sedate">\definecolor</span>{myred}{rgb}{0.8500, 0.325, 0.098}
|
||||
|
||||
<span class="org-font-latex-sedate"><span class="org-keyword">\begin</span></span>{<span class="org-function-name">tikzpicture</span>}
|
||||
<span class="org-font-latex-sedate"><span class="org-keyword">\begin</span></span>{<span class="org-function-name">scope</span>}[shift={(4, 0)}]
|
||||
|
||||
<span class="org-comment">% Uncertainty Circle</span>
|
||||
<span class="org-font-latex-sedate">\node</span>[draw, circle, fill=black!20!white, minimum size=3.6cm] (c) at (0, 0) {};
|
||||
<span class="org-font-latex-sedate">\path</span>[draw, fill=myblue!20!white] (0, 0) circle [radius=1.0];
|
||||
<span class="org-font-latex-sedate">\path</span>[draw, fill=myred!20!white] (135:1.0) circle [radius=0.8];
|
||||
<span class="org-font-latex-sedate">\path</span>[draw, dashed] (0, 0) circle [radius=1.0];
|
||||
|
||||
<span class="org-comment">% Center of Circle</span>
|
||||
<span class="org-font-latex-sedate">\node</span>[below] at (0, 0){<span class="org-font-latex-math">$1$</span>};
|
||||
|
||||
<span class="org-font-latex-sedate">\draw</span>[<->] (0, 0) node[branch]{} -- coordinate[midway](r1) ++(45:1.0);
|
||||
<span class="org-font-latex-sedate">\draw</span>[<->] (135:1.0)node[branch]{} -- coordinate[midway](r2) ++(135:0.8);
|
||||
|
||||
<span class="org-font-latex-sedate">\node</span>[] (l1) at (2, 1.5) {<span class="org-font-latex-math">$|w</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">1 H</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">1|$</span>};
|
||||
<span class="org-font-latex-sedate">\draw</span>[->, out=-90, in=0] (l1.south) to (r1);
|
||||
|
||||
<span class="org-font-latex-sedate">\node</span>[] (l2) at (-3.2, 1.2) {<span class="org-font-latex-math">$|w</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">2 H</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">2|$</span>};
|
||||
<span class="org-font-latex-sedate">\draw</span>[->, out=0, in=-180] (l2.east) to (r2);
|
||||
|
||||
<span class="org-font-latex-sedate">\draw</span>[<->] (0, 0) -- coordinate[near end](r3) ++(200:1.8);
|
||||
<span class="org-font-latex-sedate">\node</span>[] (l3) at (-2.5, -1.5) {<span class="org-font-latex-math">$|w</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">1 H</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">1| + |w</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">2 H</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">2|$</span>};
|
||||
<span class="org-font-latex-sedate">\draw</span>[->, out=90, in=-90] (l3.north) to (r3);
|
||||
<span class="org-font-latex-sedate"><span class="org-keyword">\end</span></span>{<span class="org-function-name">scope</span>}
|
||||
|
||||
<span class="org-comment">% Real and Imaginary Axis</span>
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (-0.5, 0) -- (7.0, 0) node[below left]{Re};
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (0, -1.7) -- (0, 1.7) node[below left]{Im};
|
||||
|
||||
<span class="org-font-latex-sedate">\draw</span>[dashed] (0, 0) -- (tangent cs:node=c,point={(0, 0)},solution=2);
|
||||
<span class="org-font-latex-sedate">\draw</span>[dashed] (1, 0) arc (0:28:1) node[midway, right]{<span class="org-font-latex-math">$</span><span class="org-font-latex-sedate"><span class="org-font-latex-math">\Delta</span></span><span class="org-font-latex-math"> </span><span class="org-font-latex-sedate"><span class="org-font-latex-math">\phi</span></span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-sedate"><span class="org-font-latex-math">\text</span></span><span class="org-font-latex-math">{max}$</span>};
|
||||
<span class="org-font-latex-sedate"><span class="org-keyword">\end</span></span>{<span class="org-function-name">tikzpicture</span>}
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
|
||||
<div id="orgc80ec14" class="figure">
|
||||
<p><img src="figs/uncertainty_set_super_sensor.png" alt="uncertainty_set_super_sensor.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 8: </span>Uncertainty region of the super sensor dynamics in the complex plane (grey circle). The contribution of both sensors 1 and 2 to the total uncertainty are represented respectively by a blue circle and a red circle. The frequency dependency \(\omega\) is here omitted.</p>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-orgd194580" class="outline-2">
|
||||
<h2 id="orgd194580"><span class="section-number-2">9.</span> Generalized plant used for \(\mathcal{H}_\infty\) synthesis of complementary filters</h2>
|
||||
<div class="outline-text-2" id="text-9">
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-latex"><span class="org-font-latex-sedate"><span class="org-keyword">\begin</span></span>{<span class="org-function-name">tikzpicture</span>}
|
||||
<span class="org-font-latex-sedate">\node</span>[block={4.0cm}{3.0cm}, fill=black!10!white] (P) {};
|
||||
<span class="org-font-latex-sedate">\node</span>[above] at (P.north) {<span class="org-font-latex-math">$P(s)$</span>};
|
||||
|
||||
<span class="org-font-latex-sedate">\node</span>[block, below=0.2 of P, opacity=0] (H2) {<span class="org-font-latex-math">$H</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">2(s)$</span>};
|
||||
|
||||
<span class="org-font-latex-sedate">\coordinate</span>[] (inputw) at (<span class="org-font-latex-math">$(P.south west)!0.75!(P.north west) + (-0.7, 0)$</span>);
|
||||
<span class="org-font-latex-sedate">\coordinate</span>[] (inputu) at (<span class="org-font-latex-math">$(P.south west)!0.35!(P.north west) + (-0.7, 0)$</span>);
|
||||
|
||||
<span class="org-font-latex-sedate">\coordinate</span>[] (output1) at (<span class="org-font-latex-math">$(P.south east)!0.75!(P.north east) + ( 0.7, 0)$</span>);
|
||||
<span class="org-font-latex-sedate">\coordinate</span>[] (output2) at (<span class="org-font-latex-math">$(P.south east)!0.35!(P.north east) + ( 0.7, 0)$</span>);
|
||||
<span class="org-font-latex-sedate">\coordinate</span>[] (outputv) at (<span class="org-font-latex-math">$(P.south east)!0.1!(P.north east) + ( 0.7, 0)$</span>);
|
||||
|
||||
<span class="org-font-latex-sedate">\node</span>[block, left=1.4 of output1] (W1){<span class="org-font-latex-math">$W</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">1(s)$</span>};
|
||||
<span class="org-font-latex-sedate">\node</span>[block, left=1.4 of output2] (W2){<span class="org-font-latex-math">$W</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">2(s)$</span>};
|
||||
<span class="org-font-latex-sedate">\node</span>[addb={+}{}{}{}{-}, left=of W1] (sub) {};
|
||||
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (inputw) node[above right]{<span class="org-font-latex-math">$w$</span>} -- (sub.west);
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (inputu) node[above right]{<span class="org-font-latex-math">$u$</span>} -- (W2.west);
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (inputu-|sub) node[branch]{} -- (sub.south);
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (sub.east) -- (W1.west);
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (<span class="org-font-latex-math">$(sub.west)+(-0.6, 0)$</span>) node[branch]{} |- (outputv) node[above left]{<span class="org-font-latex-math">$v$</span>};
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (W1.east) -- (output1)node[above left]{<span class="org-font-latex-math">$z</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">1$</span>};
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (W2.east) -- (output2)node[above left]{<span class="org-font-latex-math">$z</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">2$</span>};
|
||||
<span class="org-font-latex-sedate"><span class="org-keyword">\end</span></span>{<span class="org-function-name">tikzpicture</span>}
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
|
||||
<div id="org725e946" class="figure">
|
||||
<p><img src="figs/h_infinity_robust_fusion_plant.png" alt="h_infinity_robust_fusion_plant.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 9: </span>Generalized plant used for \(\mathcal{H}_\infty\) synthesis of complementary filters</p>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-orgebe7ec5" class="outline-2">
|
||||
<h2 id="orgebe7ec5"><span class="section-number-2">10.</span> Architecture used for \(\mathcal{H}_\infty\) synthesis of complementary filters</h2>
|
||||
<div class="outline-text-2" id="text-10">
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-latex"><span class="org-font-latex-sedate"><span class="org-keyword">\begin</span></span>{<span class="org-function-name">tikzpicture</span>}
|
||||
<span class="org-font-latex-sedate">\node</span>[block={4.0cm}{3.0cm}, fill=black!10!white] (P) {};
|
||||
<span class="org-font-latex-sedate">\node</span>[above] at (P.north) {<span class="org-font-latex-math">$P(s)$</span>};
|
||||
|
||||
<span class="org-font-latex-sedate">\node</span>[block, below=0.2 of P] (H2) {<span class="org-font-latex-math">$H</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">2(s)$</span>};
|
||||
|
||||
<span class="org-font-latex-sedate">\coordinate</span>[] (inputw) at (<span class="org-font-latex-math">$(P.south west)!0.75!(P.north west) + (-0.7, 0)$</span>);
|
||||
<span class="org-font-latex-sedate">\coordinate</span>[] (inputu) at (<span class="org-font-latex-math">$(P.south west)!0.35!(P.north west) + (-0.4, 0)$</span>);
|
||||
|
||||
<span class="org-font-latex-sedate">\coordinate</span>[] (output1) at (<span class="org-font-latex-math">$(P.south east)!0.75!(P.north east) + ( 0.7, 0)$</span>);
|
||||
<span class="org-font-latex-sedate">\coordinate</span>[] (output2) at (<span class="org-font-latex-math">$(P.south east)!0.35!(P.north east) + ( 0.7, 0)$</span>);
|
||||
<span class="org-font-latex-sedate">\coordinate</span>[] (outputv) at (<span class="org-font-latex-math">$(P.south east)!0.1!(P.north east) + ( 0.4, 0)$</span>);
|
||||
|
||||
<span class="org-font-latex-sedate">\node</span>[block, left=1.4 of output1] (W1){<span class="org-font-latex-math">$W</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">1(s)$</span>};
|
||||
<span class="org-font-latex-sedate">\node</span>[block, left=1.4 of output2] (W2){<span class="org-font-latex-math">$W</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">2(s)$</span>};
|
||||
<span class="org-font-latex-sedate">\node</span>[addb={+}{}{}{}{-}, left=of W1] (sub) {};
|
||||
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (inputw) node[above right]{<span class="org-font-latex-math">$w$</span>} -- (sub.west);
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (inputu-|sub) node[branch]{} -- (sub.south);
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (sub.east) -- (W1.west);
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (<span class="org-font-latex-math">$(sub.west)+(-0.6, 0)$</span>) node[branch]{} |- (outputv) |- (H2.east);
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (H2.west) -| (inputu) -- (W2.west);
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (W1.east) -- (output1)node[above left]{<span class="org-font-latex-math">$z</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">1$</span>};
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (W2.east) -- (output2)node[above left]{<span class="org-font-latex-math">$z</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">2$</span>};
|
||||
<span class="org-font-latex-sedate"><span class="org-keyword">\end</span></span>{<span class="org-function-name">tikzpicture</span>}
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
|
||||
<div id="org47d276f" class="figure">
|
||||
<p><img src="figs/h_infinity_robust_fusion_fb.png" alt="h_infinity_robust_fusion_fb.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 10: </span>Generalized plant with the synthesized filter</p>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-orgf9a2f39" class="outline-2">
|
||||
<h2 id="orgf9a2f39"><span class="section-number-2">11.</span> LIGO Sensor Fusion Architecture</h2>
|
||||
<div class="outline-text-2" id="text-11">
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-latex"><span class="org-font-latex-sedate">\definecolor</span>{myblue}{rgb}{0, 0.447, 0.741}
|
||||
<span class="org-font-latex-sedate">\definecolor</span>{myred}{rgb}{0.8500, 0.325, 0.098}
|
||||
|
||||
<span class="org-font-latex-sedate"><span class="org-keyword">\begin</span></span>{<span class="org-function-name">tikzpicture</span>}
|
||||
<span class="org-font-latex-sedate">\node</span>[block, align=center] (position) at (0, 2.2) {Position<span class="org-font-latex-warning">\\</span>Sensor};
|
||||
<span class="org-font-latex-sedate">\node</span>[block, align=center] (seismometer) at (0, 1.0) {Seismometer};
|
||||
<span class="org-font-latex-sedate">\node</span>[block, align=center] (geophone) at (0,-0.6) {Geophone};
|
||||
|
||||
<span class="org-font-latex-sedate">\node</span>[branch, left=0.4 of seismometer] (x);
|
||||
|
||||
<span class="org-font-latex-sedate">\node</span>[block, right=1.1 of seismometer](H1){<span class="org-font-latex-math">$L</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">2(s)$</span>};
|
||||
<span class="org-font-latex-sedate">\node</span>[block](H2) at (H1|-geophone) {<span class="org-font-latex-math">$H</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">2(s)$</span>};
|
||||
|
||||
<span class="org-font-latex-sedate">\node</span>[addb] (add) at (4, 0){};
|
||||
<span class="org-font-latex-sedate">\node</span>[block, right=1.1 of add](H2p) {<span class="org-font-latex-math">$H</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">1(s)$</span>};
|
||||
<span class="org-font-latex-sedate">\node</span>[block] (H1p) at (H2p|-position) {<span class="org-font-latex-math">$L</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">1(s)$</span>};
|
||||
|
||||
<span class="org-font-latex-sedate">\node</span>[addb] (addp) at (7, 1.0){};
|
||||
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (<span class="org-font-latex-math">$(x)+(-1.0, 0)$</span>) -- (seismometer.west);
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (x.center) |- (position.west);
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (x.center) |- (geophone.west);
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (position.east) -- (H1p.west);
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (seismometer.east) -- (H1.west);
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (geophone.east) -- (H2.west);
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (H1) -| (add.north);
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (H2) -| (add.south);
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (add.east) -- (H2p.west);
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (H1p) -| (addp.north);
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (H2p) -| (addp.south);
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (addp.east) -- ++(1.0, 0);
|
||||
|
||||
<span class="org-font-latex-sedate"><span class="org-keyword">\begin</span></span>{<span class="org-function-name">scope</span>}[on background layer]
|
||||
<span class="org-font-latex-sedate">\node</span>[fit={(x.west|-geophone.south) (position.north-|addp.east)}, fill=black!10!white, draw, inner sep=6pt] (supersensor) {};
|
||||
<span class="org-font-latex-sedate">\node</span>[below] at (supersensor.north) {Super Sensor};
|
||||
|
||||
<span class="org-font-latex-sedate">\node</span>[fit={(x.west|-seismometer.north) (add.east|-geophone.south)}, fill=black!20!white, draw, inner sep=3pt] (superinertialsensor) {};
|
||||
<span class="org-font-latex-sedate">\node</span>[] at (superinertialsensor.center) {"Inertial" Super Sensor};
|
||||
<span class="org-font-latex-sedate"><span class="org-keyword">\end</span></span>{<span class="org-function-name">scope</span>}
|
||||
<span class="org-font-latex-sedate"><span class="org-keyword">\end</span></span>{<span class="org-function-name">tikzpicture</span>}
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
|
||||
<div id="org81fe1cf" class="figure">
|
||||
<p><img src="figs/ligo_super_sensor_architecture.png" alt="ligo_super_sensor_architecture.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 11: </span>Simplified block diagram of the sensor blending strategy for the first stage at the LIGO</p>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-orga2611bb" class="outline-2">
|
||||
<h2 id="orga2611bb"><span class="section-number-2">12.</span> Closed-Loop Complementary Filters</h2>
|
||||
<div class="outline-text-2" id="text-12">
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-latex"><span class="org-font-latex-sedate"><span class="org-keyword">\begin</span></span>{<span class="org-function-name">tikzpicture</span>}
|
||||
<span class="org-font-latex-sedate">\node</span>[addb={+}{}{}{}{-}] (addfb) at (0, 0){};
|
||||
<span class="org-font-latex-sedate">\node</span>[block, right=1 of addfb] (L){<span class="org-font-latex-math">$L(s)$</span>};
|
||||
<span class="org-font-latex-sedate">\node</span>[addb={+}{}{}{}{}, right=1 of L] (adddy){};
|
||||
|
||||
<span class="org-font-latex-sedate">\draw</span>[<-] (addfb.west) -- ++(-1, 0) node[above right]{<span class="org-font-latex-math">$</span><span class="org-font-latex-sedate"><span class="org-font-latex-math">\hat</span></span><span class="org-font-latex-math">{x}</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">2$</span>};
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (addfb.east) -- (L.west);
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (L.east) -- (adddy.west);
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (adddy.east) -- ++(1.4, 0) node[above left]{<span class="org-font-latex-math">$</span><span class="org-font-latex-sedate"><span class="org-font-latex-math">\hat</span></span><span class="org-font-latex-math">{x}$</span>};
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (<span class="org-font-latex-math">$(adddy.east) + (0.5, 0)$</span>) node[branch]{} -- ++(0, -0.8) coordinate(botc) -| (addfb.south);
|
||||
<span class="org-font-latex-sedate">\draw</span>[<-] (adddy.north) -- ++(0, 1) node[below right]{<span class="org-font-latex-math">$</span><span class="org-font-latex-sedate"><span class="org-font-latex-math">\hat</span></span><span class="org-font-latex-math">{x}</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">1$</span>};
|
||||
|
||||
<span class="org-font-latex-sedate"><span class="org-keyword">\begin</span></span>{<span class="org-function-name">scope</span>}[on background layer]
|
||||
<span class="org-font-latex-sedate">\node</span>[fit={(L.north-|addfb.west) (botc)}, fill=black!10!white, draw, inner sep=6pt] (supersensor) {};
|
||||
<span class="org-font-latex-sedate"><span class="org-keyword">\end</span></span>{<span class="org-function-name">scope</span>}
|
||||
<span class="org-font-latex-sedate"><span class="org-keyword">\end</span></span>{<span class="org-function-name">tikzpicture</span>}
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
|
||||
<div id="org57ca8ce" class="figure">
|
||||
<p><img src="figs/feedback_sensor_fusion.png" alt="feedback_sensor_fusion.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 12: </span>“Closed-Loop” complementary filters</p>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org5af72e4" class="outline-2">
|
||||
<h2 id="org5af72e4"><span class="section-number-2">13.</span> Closed-Loop Fusion Architecture</h2>
|
||||
<div class="outline-text-2" id="text-13">
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-latex"><span class="org-font-latex-sedate"><span class="org-keyword">\begin</span></span>{<span class="org-function-name">tikzpicture</span>}
|
||||
<span class="org-font-latex-sedate">\node</span>[addb={+}{}{}{}{-}] (addfb) at (0, 0){};
|
||||
<span class="org-font-latex-sedate">\node</span>[block, right=1 of addfb] (L){<span class="org-font-latex-math">$L(s)$</span>};
|
||||
<span class="org-font-latex-sedate">\node</span>[addb={+}{}{}{}{}, right=1 of L] (adddy){};
|
||||
|
||||
<span class="org-font-latex-sedate">\node</span>[block, left=1.2 of addfb] (sensor2){Sensor 2};
|
||||
<span class="org-font-latex-sedate">\node</span>[block, above=0.4 of sensor2] (sensor1){Sensor 1};
|
||||
<span class="org-font-latex-sedate">\node</span>[branch, left=0.6 of sensor2] (x){};
|
||||
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (addfb.east) -- (L.west);
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (L.east) -- (adddy.west);
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (adddy.east) -- ++(1.4, 0) node[above left]{<span class="org-font-latex-math">$</span><span class="org-font-latex-sedate"><span class="org-font-latex-math">\hat</span></span><span class="org-font-latex-math">{x}$</span>};
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (<span class="org-font-latex-math">$(adddy.east) + (0.5, 0)$</span>) node[branch]{} -- ++(0, -0.8) coordinate(botc) -| (addfb.south);
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (x.center) |- (sensor1.west);
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (<span class="org-font-latex-math">$(x)-(0.8,0)$</span>) node[above right]{<span class="org-font-latex-math">$x$</span>} -- (sensor2.west);
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (sensor2.east)node[above right=0 and 0.25]{<span class="org-font-latex-math">$</span><span class="org-font-latex-sedate"><span class="org-font-latex-math">\hat</span></span><span class="org-font-latex-math">{x}</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">2$</span>} -- (addfb.west);
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (sensor1.east)node[above right=0 and 0.25]{<span class="org-font-latex-math">$</span><span class="org-font-latex-sedate"><span class="org-font-latex-math">\hat</span></span><span class="org-font-latex-math">{x}</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">1$</span>} -| (adddy.north);
|
||||
|
||||
<span class="org-font-latex-sedate"><span class="org-keyword">\begin</span></span>{<span class="org-function-name">scope</span>}[on background layer]
|
||||
<span class="org-font-latex-sedate">\node</span>[fit={(x|-sensor1.north) (botc)}, fill=black!10!white, draw, inner sep=9pt] (supersensor) {};
|
||||
<span class="org-font-latex-sedate">\node</span>[fit={(sensor1.north-|addfb.west) (botc)}, fill=black!20!white, draw, inner sep=6pt] (feedbackfilter) {};
|
||||
<span class="org-font-latex-sedate">\node</span>[fit={(sensor2.west|-botc) (sensor1.north east)}, fill=black!20!white, draw, inner sep=6pt] (sensors) {};
|
||||
<span class="org-font-latex-sedate">\node</span>[above, align=center] at (sensors.south) {{<span class="org-font-latex-sedate"><span class="org-keyword">\tiny</span></span><span class="org-type"> Normalized</span>}<span class="org-font-latex-warning">\\</span>[-0.5em]{<span class="org-font-latex-sedate"><span class="org-keyword">\tiny</span></span><span class="org-type"> sensors</span>}};
|
||||
<span class="org-font-latex-sedate">\node</span>[below, align=center] at (feedbackfilter.north) {{<span class="org-font-latex-sedate"><span class="org-keyword">\tiny</span></span><span class="org-type"> "Closed-Loop"</span>}<span class="org-font-latex-warning">\\</span>[-0.5em]{<span class="org-font-latex-sedate"><span class="org-keyword">\tiny</span></span><span class="org-type"> complementary filters</span>}};
|
||||
<span class="org-font-latex-sedate"><span class="org-keyword">\end</span></span>{<span class="org-function-name">scope</span>}
|
||||
<span class="org-font-latex-sedate"><span class="org-keyword">\end</span></span>{<span class="org-function-name">tikzpicture</span>}
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
|
||||
<div id="org8886809" class="figure">
|
||||
<p><img src="figs/feedback_sensor_fusion_arch.png" alt="feedback_sensor_fusion_arch.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 13: </span>Classical feedback architecture used for sensor fusion</p>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org4e79ee7" class="outline-2">
|
||||
<h2 id="org4e79ee7"><span class="section-number-2">14.</span> Feedback Loop Sensor Fusion Architecture</h2>
|
||||
<div class="outline-text-2" id="text-14">
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-latex"><span class="org-font-latex-sedate"><span class="org-keyword">\begin</span></span>{<span class="org-function-name">tikzpicture</span>}
|
||||
<span class="org-font-latex-sedate">\node</span>[block] (W2) at (0,0) {<span class="org-font-latex-math">$W</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">2(s)$</span>};
|
||||
<span class="org-font-latex-sedate">\node</span>[addb={+}{}{}{}{-}, right=0.8 of W2] (addfb){};
|
||||
<span class="org-font-latex-sedate">\node</span>[addb={+}{}{}{}{}, right=4.5 of W2] (adddy){};
|
||||
<span class="org-font-latex-sedate">\node</span>[block, above=0.8 of adddy] (W1){<span class="org-font-latex-math">$W</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">1(s)$</span>};
|
||||
|
||||
<span class="org-font-latex-sedate">\draw</span>[<-] (W2.west) -- ++(-0.8, 0) node[above right]{<span class="org-font-latex-math">$w</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">2$</span>};
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (W2.east) -- (addfb.west) node[above left]{<span class="org-font-latex-math">$</span><span class="org-font-latex-sedate"><span class="org-font-latex-math">\tilde</span></span><span class="org-font-latex-math">{w}</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">2$</span>};
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (addfb.east) -- ++(1, 0) node[above left]{<span class="org-font-latex-math">$v$</span>};
|
||||
<span class="org-font-latex-sedate">\draw</span>[<-] (adddy.west) -- ++(-1, 0) node[above right]{<span class="org-font-latex-math">$u$</span>};
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (adddy.east) -- ++(1.4, 0) node[above left]{<span class="org-font-latex-math">$z$</span>};
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (W1.south) -- (adddy.north) node[above right]{<span class="org-font-latex-math">$</span><span class="org-font-latex-sedate"><span class="org-font-latex-math">\tilde</span></span><span class="org-font-latex-math">{w}</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">1$</span>};
|
||||
<span class="org-font-latex-sedate">\draw</span>[<-] (W1.north) -- ++(0, 0.8) node[below right]{<span class="org-font-latex-math">$w</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">1$</span>};
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (<span class="org-font-latex-math">$(adddy.east) + (0.5, 0)$</span>) node[branch]{} -- ++(0, -0.8) -| (addfb.south);
|
||||
<span class="org-font-latex-sedate"><span class="org-keyword">\end</span></span>{<span class="org-function-name">tikzpicture</span>}
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
|
||||
<div id="org705140c" class="figure">
|
||||
<p><img src="figs/feedback_synthesis_architecture.png" alt="feedback_synthesis_architecture.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 14: </span>Feedback architecture with included weights</p>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-orgb59c88b" class="outline-2">
|
||||
<h2 id="orgb59c88b"><span class="section-number-2">15.</span> Feedback Sensor Fusion - Generalized Plant</h2>
|
||||
<div class="outline-text-2" id="text-15">
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-latex"><span class="org-font-latex-sedate"><span class="org-keyword">\begin</span></span>{<span class="org-function-name">tikzpicture</span>}
|
||||
<span class="org-font-latex-sedate">\node</span>[block={4.5cm}{3.0cm}, fill=black!10!white] (P) {};
|
||||
<span class="org-font-latex-sedate">\node</span>[above] at (P.north) {<span class="org-font-latex-math">$P</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">L(s)$</span>};
|
||||
|
||||
<span class="org-font-latex-sedate">\coordinate</span>[] (inputw1) at (<span class="org-font-latex-math">$(P.south west)!0.75!(P.north west) + (-0.7, 0)$</span>);
|
||||
<span class="org-font-latex-sedate">\coordinate</span>[] (inputw2) at (<span class="org-font-latex-math">$(P.south west)!0.40!(P.north west) + (-0.7, 0)$</span>);
|
||||
<span class="org-font-latex-sedate">\coordinate</span>[] (inputu) at (<span class="org-font-latex-math">$(P.south west)!0.15!(P.north west) + (-0.7, 0)$</span>);
|
||||
|
||||
<span class="org-font-latex-sedate">\coordinate</span>[] (outputz) at (<span class="org-font-latex-math">$(P.south east)!0.75!(P.north east) + ( 0.7, 0)$</span>);
|
||||
<span class="org-font-latex-sedate">\coordinate</span>[] (outputv) at (<span class="org-font-latex-math">$(P.south east)!0.40!(P.north east) + ( 0.7, 0)$</span>);
|
||||
|
||||
<span class="org-font-latex-sedate">\node</span>[block, right=1.2 of inputw2] (W2){<span class="org-font-latex-math">$W</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">2(s)$</span>};
|
||||
<span class="org-font-latex-sedate">\node</span>[block, right=1.2 of inputw1] (W1){<span class="org-font-latex-math">$W</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">1(s)$</span>};
|
||||
<span class="org-font-latex-sedate">\node</span>[addb={+}{}{}{}{}, right=0.8 of W1] (add) {};
|
||||
<span class="org-font-latex-sedate">\node</span>[addb={+}{}{-}{}{}, right=1.8 of W2] (sub) {};
|
||||
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (inputw2) node[above right]{<span class="org-font-latex-math">$w</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">2$</span>} -- (W2.west);
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (inputw1) node[above right]{<span class="org-font-latex-math">$w</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">1$</span>} -- (W1.west);
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (inputu) node[above right]{<span class="org-font-latex-math">$u$</span>} -| (add.south);
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (W2.east) -- (sub.west);
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (W1.east) -- (add.west);
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (add.east) -- (outputz)node[above left]{<span class="org-font-latex-math">$z$</span>};
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (sub.east) -- (outputv)node[above left]{<span class="org-font-latex-math">$v$</span>};
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (add-|sub) node[branch]{} -- (sub.north);
|
||||
<span class="org-font-latex-sedate"><span class="org-keyword">\end</span></span>{<span class="org-function-name">tikzpicture</span>}
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
|
||||
<div id="org32b295e" class="figure">
|
||||
<p><img src="figs/feedback_synthesis_architecture_generalized_plant.png" alt="feedback_synthesis_architecture_generalized_plant.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 15: </span>Generalized plant used for the \(\mathcal{H}_\infty\) synthesis of “closed-loop” complementary filters</p>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-orge593f6d" class="outline-2">
|
||||
<h2 id="orge593f6d"><span class="section-number-2">16.</span> Sensor Fusion - Sequential</h2>
|
||||
<div class="outline-text-2" id="text-16">
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-latex"><span class="org-font-latex-sedate"><span class="org-keyword">\begin</span></span>{<span class="org-function-name">tikzpicture</span>}
|
||||
<span class="org-font-latex-sedate">\node</span>[branch] (x) at (0, 0);
|
||||
|
||||
<span class="org-font-latex-sedate">\node</span>[block, right=0.4 of x] (sensor2) {Sensor 2};
|
||||
<span class="org-font-latex-sedate">\node</span>[block, above=0.4 of sensor2] (sensor1) {Sensor 1};
|
||||
<span class="org-font-latex-sedate">\node</span>[block, below=0.4 of sensor2] (sensor3) {Sensor 3};
|
||||
|
||||
<span class="org-font-latex-sedate">\node</span>[block, right=1.1 of sensor1](H1){<span class="org-font-latex-math">$H</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">1(s)$</span>};
|
||||
<span class="org-font-latex-sedate">\node</span>[block, right=1.1 of sensor2](H2){<span class="org-font-latex-math">$H</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">2(s)$</span>};
|
||||
<span class="org-font-latex-sedate">\node</span>[addb] (add) at (<span class="org-font-latex-math">$0.5*(H1.east)+0.5*(H2.east)+(0.6, 0)$</span>){};
|
||||
|
||||
<span class="org-font-latex-sedate">\node</span>[block, right=0.8 of add](H1p) {<span class="org-font-latex-math">$H</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">1</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">^</span></span><span class="org-font-latex-sedate"><span class="org-font-latex-math">\prime</span></span><span class="org-font-latex-math">(s)$</span>};
|
||||
<span class="org-font-latex-sedate">\node</span>[block] (H2p) at (H1p|-sensor3) {<span class="org-font-latex-math">$H</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">2</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">^</span></span><span class="org-font-latex-sedate"><span class="org-font-latex-math">\prime</span></span><span class="org-font-latex-math">(s)$</span>};
|
||||
|
||||
<span class="org-font-latex-sedate">\node</span>[addb] (addp) at (<span class="org-font-latex-math">$0.5*(H1p.east)+0.5*(H2p.east)+(0.6, 0)$</span>){};
|
||||
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (<span class="org-font-latex-math">$(x)+(-0.8, 0)$</span>) node[above right]{<span class="org-font-latex-math">$x$</span>} -- (sensor2.west);
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (x.center) |- (sensor1.west);
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (x.center) |- (sensor3.west);
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (sensor1.east) -- (H1.west) node[above left]{<span class="org-font-latex-math">$</span><span class="org-font-latex-sedate"><span class="org-font-latex-math">\hat</span></span><span class="org-font-latex-math">{x}</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">1$</span>};
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (sensor2.east) -- (H2.west) node[above left]{<span class="org-font-latex-math">$</span><span class="org-font-latex-sedate"><span class="org-font-latex-math">\hat</span></span><span class="org-font-latex-math">{x}</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">2$</span>};
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (sensor3.east) -- (H2p.west) node[above left]{<span class="org-font-latex-math">$</span><span class="org-font-latex-sedate"><span class="org-font-latex-math">\hat</span></span><span class="org-font-latex-math">{x}</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">3$</span>};
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (H1) -| (add.north);
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (H2) -| (add.south);
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (add.east) -- (H1p.west) node[above left]{<span class="org-font-latex-math">$</span><span class="org-font-latex-sedate"><span class="org-font-latex-math">\hat</span></span><span class="org-font-latex-math">{x}</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">{12}$</span>};
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (H1p) -| (addp.north);
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (H2p) -| (addp.south);
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (addp.east) -- ++(0.8, 0) node[above left]{<span class="org-font-latex-math">$</span><span class="org-font-latex-sedate"><span class="org-font-latex-math">\hat</span></span><span class="org-font-latex-math">{x}$</span>};
|
||||
|
||||
<span class="org-font-latex-sedate"><span class="org-keyword">\begin</span></span>{<span class="org-function-name">scope</span>}[on background layer]
|
||||
<span class="org-font-latex-sedate">\node</span>[fit={(x.west|-sensor3.south) (sensor1.north-|addp.east)}, fill=black!10!white, draw, inner sep=6pt] (supersensor) {};
|
||||
|
||||
<span class="org-font-latex-sedate">\node</span>[fit={(x.west|-sensor1.north) (add.east|-sensor2.south)}, fill=black!20!white, draw, inner sep=3pt] (superinertialsensor) {};
|
||||
<span class="org-font-latex-sedate"><span class="org-keyword">\end</span></span>{<span class="org-function-name">scope</span>}
|
||||
<span class="org-font-latex-sedate"><span class="org-keyword">\end</span></span>{<span class="org-function-name">tikzpicture</span>}
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
|
||||
<div id="org6fe58a4" class="figure">
|
||||
<p><img src="figs/sensor_fusion_three_sequential.png" alt="sensor_fusion_three_sequential.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 16: </span>Sequential fusion</p>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org3b323e8" class="outline-2">
|
||||
<h2 id="org3b323e8"><span class="section-number-2">17.</span> Sensor Fusion - Parallel</h2>
|
||||
<div class="outline-text-2" id="text-17">
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-latex"><span class="org-font-latex-sedate"><span class="org-keyword">\begin</span></span>{<span class="org-function-name">tikzpicture</span>}
|
||||
<span class="org-font-latex-sedate">\node</span>[branch] (x) at (0, 0);
|
||||
<span class="org-font-latex-sedate">\node</span>[block, right=0.4 of x] (sensor2) {Sensor 2};
|
||||
<span class="org-font-latex-sedate">\node</span>[block, above=0.3 of sensor2] (sensor1) {Sensor 1};
|
||||
<span class="org-font-latex-sedate">\node</span>[block, below=0.3 of sensor2] (sensor3) {Sensor 3};
|
||||
|
||||
<span class="org-font-latex-sedate">\node</span>[block, right=1.1 of sensor1](H1){<span class="org-font-latex-math">$H</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">1(s)$</span>};
|
||||
<span class="org-font-latex-sedate">\node</span>[block, right=1.1 of sensor2](H2){<span class="org-font-latex-math">$H</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">2(s)$</span>};
|
||||
<span class="org-font-latex-sedate">\node</span>[block, right=1.1 of sensor3](H3){<span class="org-font-latex-math">$H</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">3(s)$</span>};
|
||||
|
||||
<span class="org-font-latex-sedate">\node</span>[addb, right=0.6 of H2](add){};
|
||||
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (x.center) |- (sensor1.west);
|
||||
<span class="org-font-latex-sedate">\draw</span>[] (<span class="org-font-latex-math">$(x)+(-0.8, 0)$</span>) node[above right]{<span class="org-font-latex-math">$x$</span>} -- (sensor2.west);
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (x.center) |- (sensor3.west);
|
||||
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (sensor1.east) -- (H1.west) node[above left]{<span class="org-font-latex-math">$</span><span class="org-font-latex-sedate"><span class="org-font-latex-math">\hat</span></span><span class="org-font-latex-math">{x}</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">1$</span>};
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (sensor2.east) -- (H2.west) node[above left]{<span class="org-font-latex-math">$</span><span class="org-font-latex-sedate"><span class="org-font-latex-math">\hat</span></span><span class="org-font-latex-math">{x}</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">2$</span>};
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (sensor3.east) -- (H3.west) node[above left]{<span class="org-font-latex-math">$</span><span class="org-font-latex-sedate"><span class="org-font-latex-math">\hat</span></span><span class="org-font-latex-math">{x}</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">3$</span>};
|
||||
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (H1) -| (add.north);
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (H2) -- (add.west);
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (H3) -| (add.south);
|
||||
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (add.east) -- ++(0.8, 0) node[above left]{<span class="org-font-latex-math">$</span><span class="org-font-latex-sedate"><span class="org-font-latex-math">\hat</span></span><span class="org-font-latex-math">{x}$</span>};
|
||||
|
||||
<span class="org-font-latex-sedate"><span class="org-keyword">\begin</span></span>{<span class="org-function-name">scope</span>}[on background layer]
|
||||
<span class="org-font-latex-sedate">\node</span>[fit={(H3.south-|x) (H1.north-|add.east)}, fill=black!10!white, draw, inner sep=6pt] (supersensor) {};
|
||||
<span class="org-font-latex-sedate"><span class="org-keyword">\end</span></span>{<span class="org-function-name">scope</span>}
|
||||
<span class="org-font-latex-sedate"><span class="org-keyword">\end</span></span>{<span class="org-function-name">tikzpicture</span>}
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
|
||||
<div id="org6ec849a" class="figure">
|
||||
<p><img src="figs/sensor_fusion_three_parallel.png" alt="sensor_fusion_three_parallel.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 17: </span>Parallel fusion</p>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org9baa5b5" class="outline-2">
|
||||
<h2 id="org9baa5b5"><span class="section-number-2">18.</span> Architecture for \(\mathcal{H}_\infty\) synthesis of three complementary filters</h2>
|
||||
<div class="outline-text-2" id="text-18">
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-latex"> <span class="org-font-latex-sedate"><span class="org-keyword">\begin</span></span>{<span class="org-function-name">tikzpicture</span>}
|
||||
<span class="org-font-latex-sedate">\node</span>[block={5.0cm}{4.5cm}, fill=black!10!white] (P) {};
|
||||
<span class="org-font-latex-sedate">\node</span>[above] at (P.north) {<span class="org-font-latex-math">$P</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">3(s)$</span>};
|
||||
|
||||
<span class="org-font-latex-sedate">\coordinate</span>[] (inputw) at (<span class="org-font-latex-math">$(P.south west)!0.8!(P.north west) + (-0.7, 0)$</span>);
|
||||
<span class="org-font-latex-sedate">\coordinate</span>[] (inputu) at (<span class="org-font-latex-math">$(P.south west)!0.4!(P.north west) + (-0.7, 0)$</span>);
|
||||
|
||||
<span class="org-font-latex-sedate">\coordinate</span>[] (output1) at (<span class="org-font-latex-math">$(P.south east)!0.8!(P.north east) + (0.7, 0)$</span>);
|
||||
<span class="org-font-latex-sedate">\coordinate</span>[] (output2) at (<span class="org-font-latex-math">$(P.south east)!0.55!(P.north east) + (0.7, 0)$</span>);
|
||||
<span class="org-font-latex-sedate">\coordinate</span>[] (output3) at (<span class="org-font-latex-math">$(P.south east)!0.3!(P.north east) + (0.7, 0)$</span>);
|
||||
<span class="org-font-latex-sedate">\coordinate</span>[] (outputv) at (<span class="org-font-latex-math">$(P.south east)!0.1!(P.north east) + (0.7, 0)$</span>);
|
||||
|
||||
<span class="org-font-latex-sedate">\node</span>[block, left=1.4 of output1] (W1){<span class="org-font-latex-math">$W</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">1(s)$</span>};
|
||||
<span class="org-font-latex-sedate">\node</span>[block, left=1.4 of output2] (W2){<span class="org-font-latex-math">$W</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">2(s)$</span>};
|
||||
<span class="org-font-latex-sedate">\node</span>[block, left=1.4 of output3] (W3){<span class="org-font-latex-math">$W</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">3(s)$</span>};
|
||||
<span class="org-font-latex-sedate">\node</span>[addb={+}{}{}{}{-}, left=of W1] (sub1) {};
|
||||
<span class="org-font-latex-sedate">\node</span>[addb={+}{}{}{}{-}, left=of sub1] (sub2) {};
|
||||
|
||||
<span class="org-font-latex-sedate">\node</span>[block, below=0.3 of P, opacity=0] (H) {<span class="org-font-latex-math">$</span><span class="org-font-latex-sedate"><span class="org-keyword"><span class="org-font-latex-math">\begin</span></span></span><span class="org-font-latex-math">{</span><span class="org-function-name"><span class="org-font-latex-math">bmatrix</span></span><span class="org-font-latex-math">}H</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">2(s) \\ H</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">3(s)</span><span class="org-font-latex-sedate"><span class="org-keyword"><span class="org-font-latex-math">\end</span></span></span><span class="org-font-latex-math">{</span><span class="org-function-name"><span class="org-font-latex-math">bmatrix</span></span><span class="org-font-latex-math">}$</span>};
|
||||
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (inputw) node[above right](w){<span class="org-font-latex-math">$w$</span>} -- (sub2.west);
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (W3-|sub1)node[branch]{} -- (sub1.south);
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (W2-|sub2)node[branch]{} -- (sub2.south);
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (<span class="org-font-latex-math">$(sub2.west)+(-0.5, 0)$</span>) node[branch]{} |- (outputv);
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (inputu|-W2) -- (W2.west);
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (inputu|-W3) -- (W3.west);
|
||||
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (sub2.east) -- (sub1.west);
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (sub1.east) -- (W1.west);
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (W1.east) -- (output1)node[above left](z){<span class="org-font-latex-math">$z</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">1$</span>};
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (W2.east) -- (output2)node[above left]{<span class="org-font-latex-math">$z</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">2$</span>};
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (W3.east) -- (output3)node[above left]{<span class="org-font-latex-math">$z</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">3$</span>};
|
||||
<span class="org-font-latex-sedate">\node</span>[above] at (W2-|w){<span class="org-font-latex-math">$u</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">1$</span>};
|
||||
<span class="org-font-latex-sedate">\node</span>[above] at (W3-|w){<span class="org-font-latex-math">$u</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">2$</span>};
|
||||
<span class="org-font-latex-sedate">\node</span>[above] at (outputv-|z){<span class="org-font-latex-math">$v$</span>};
|
||||
<span class="org-font-latex-sedate"><span class="org-keyword">\end</span></span>{<span class="org-function-name">tikzpicture</span>}
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
|
||||
<div id="org7b3470a" class="figure">
|
||||
<p><img src="figs/comp_filter_three_hinf_gen_plant.png" alt="comp_filter_three_hinf_gen_plant.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 18: </span>Generalized plant for the \(\mathcal{H}_\infty\) synthesis of three complementary filters</p>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
<div id="outline-container-org10ffb09" class="outline-2">
|
||||
<h2 id="org10ffb09"><span class="section-number-2">19.</span> Architecture for \(\mathcal{H}_\infty\) synthesis of three complementary filters</h2>
|
||||
<div class="outline-text-2" id="text-19">
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-latex"> <span class="org-font-latex-sedate"><span class="org-keyword">\begin</span></span>{<span class="org-function-name">tikzpicture</span>}
|
||||
<span class="org-font-latex-sedate">\node</span>[block={5.0cm}{4.5cm}, fill=black!10!white] (P) {};
|
||||
<span class="org-font-latex-sedate">\node</span>[above] at (P.north) {<span class="org-font-latex-math">$P</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">3(s)$</span>};
|
||||
|
||||
<span class="org-font-latex-sedate">\coordinate</span>[] (inputw) at (<span class="org-font-latex-math">$(P.south west)!0.8!(P.north west) + (-0.7, 0)$</span>);
|
||||
<span class="org-font-latex-sedate">\coordinate</span>[] (inputu) at (<span class="org-font-latex-math">$(P.south west)!0.4!(P.north west) + (-0.7, 0)$</span>);
|
||||
|
||||
<span class="org-font-latex-sedate">\coordinate</span>[] (output1) at (<span class="org-font-latex-math">$(P.south east)!0.8!(P.north east) + (0.7, 0)$</span>);
|
||||
<span class="org-font-latex-sedate">\coordinate</span>[] (output2) at (<span class="org-font-latex-math">$(P.south east)!0.55!(P.north east) + (0.7, 0)$</span>);
|
||||
<span class="org-font-latex-sedate">\coordinate</span>[] (output3) at (<span class="org-font-latex-math">$(P.south east)!0.3!(P.north east) + (0.7, 0)$</span>);
|
||||
<span class="org-font-latex-sedate">\coordinate</span>[] (outputv) at (<span class="org-font-latex-math">$(P.south east)!0.1!(P.north east) + (0.7, 0)$</span>);
|
||||
|
||||
<span class="org-font-latex-sedate">\node</span>[block, left=1.4 of output1] (W1){<span class="org-font-latex-math">$W</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">1(s)$</span>};
|
||||
<span class="org-font-latex-sedate">\node</span>[block, left=1.4 of output2] (W2){<span class="org-font-latex-math">$W</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">2(s)$</span>};
|
||||
<span class="org-font-latex-sedate">\node</span>[block, left=1.4 of output3] (W3){<span class="org-font-latex-math">$W</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">3(s)$</span>};
|
||||
<span class="org-font-latex-sedate">\node</span>[addb={+}{}{}{}{-}, left=of W1] (sub1) {};
|
||||
<span class="org-font-latex-sedate">\node</span>[addb={+}{}{}{}{-}, left=of sub1] (sub2) {};
|
||||
|
||||
<span class="org-font-latex-sedate">\node</span>[block, below=0.3 of P] (H) {<span class="org-font-latex-math">$</span><span class="org-font-latex-sedate"><span class="org-keyword"><span class="org-font-latex-math">\begin</span></span></span><span class="org-font-latex-math">{</span><span class="org-function-name"><span class="org-font-latex-math">bmatrix</span></span><span class="org-font-latex-math">}H</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">2(s) \\ H</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">3(s)</span><span class="org-font-latex-sedate"><span class="org-keyword"><span class="org-font-latex-math">\end</span></span></span><span class="org-font-latex-math">{</span><span class="org-function-name"><span class="org-font-latex-math">bmatrix</span></span><span class="org-font-latex-math">}$</span>};
|
||||
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (inputw) node[above right](w){<span class="org-font-latex-math">$w$</span>} -- (sub2.west);
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (W3-|sub1)node[branch]{} -- (sub1.south);
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (W2-|sub2)node[branch]{} -- (sub2.south);
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (<span class="org-font-latex-math">$(sub2.west)+(-0.5, 0)$</span>) node[branch]{} |- (outputv) |- (H.east);
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (<span class="org-font-latex-math">$(H.south west)!0.7!(H.north west)$</span>) -| (inputu|-W2) -- (W2.west);
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (<span class="org-font-latex-math">$(H.south west)!0.3!(H.north west)$</span>) -| (<span class="org-font-latex-math">$(inputu|-W3)+(0.4, 0)$</span>) -- (W3.west);
|
||||
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (sub2.east) -- (sub1.west);
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (sub1.east) -- (W1.west);
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (W1.east) -- (output1)node[above left](z){<span class="org-font-latex-math">$z</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">1$</span>};
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (W2.east) -- (output2)node[above left]{<span class="org-font-latex-math">$z</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">2$</span>};
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (W3.east) -- (output3)node[above left]{<span class="org-font-latex-math">$z</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">3$</span>};
|
||||
<span class="org-font-latex-sedate">\node</span>[above] at (W2-|w){<span class="org-font-latex-math">$u</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">1$</span>};
|
||||
<span class="org-font-latex-sedate">\node</span>[above] at (W3-|w){<span class="org-font-latex-math">$u</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">2$</span>};
|
||||
<span class="org-font-latex-sedate">\node</span>[above] at (outputv-|z){<span class="org-font-latex-math">$v$</span>};
|
||||
<span class="org-font-latex-sedate"><span class="org-keyword">\end</span></span>{<span class="org-function-name">tikzpicture</span>}
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
|
||||
<div id="org71495ac" class="figure">
|
||||
<p><img src="figs/comp_filter_three_hinf_fb.png" alt="comp_filter_three_hinf_fb.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 19: </span>Generalized plant with the synthesized filter for the \(\mathcal{H}_\infty\) synthesis of three complementary filters</p>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
<div id="postamble" class="status">
|
||||
<p class="author">Author: Thomas Dehaeze</p>
|
||||
<p class="date">Created: 2021-09-01 mer. 10:15</p>
|
||||
</div>
|
||||
</body>
|
||||
</html>
|
||||
@ -1,4 +1,4 @@
|
||||
#+TITLE: Complementary Filters Shaping Using $\mathcal{H}_\infty$ Synthesis - Tikz Figures
|
||||
#+TITLE: A new method of designing complementary filters for sensor fusion using the $\mathcal{H}_\infty$ synthesis - Tikz Figures
|
||||
:DRAWER:
|
||||
#+HTML_LINK_HOME: ../index.html
|
||||
#+HTML_LINK_UP: ../index.html
|
||||
@ -59,7 +59,7 @@ Configuration file is accessible [[file:config.org][here]].
|
||||
#+end_src
|
||||
|
||||
#+name: fig:sensor_fusion_overview
|
||||
#+caption: Sensor Fusion Architecture - Overview
|
||||
#+caption: Schematic of a sensor fusion architecture using complementary filters
|
||||
#+RESULTS:
|
||||
[[file:figs/sensor_fusion_overview.png]]
|
||||
|
||||
@ -82,7 +82,7 @@ Configuration file is accessible [[file:config.org][here]].
|
||||
#+end_src
|
||||
|
||||
#+name: fig:sensor_model
|
||||
#+caption: Basic Sensor Model
|
||||
#+caption: Basic sensor model consisting of a noise input $n_i$ and a linear time invariant transfer function $G_i(s)$
|
||||
#+RESULTS:
|
||||
[[file:figs/sensor_model.png]]
|
||||
|
||||
@ -110,7 +110,7 @@ Configuration file is accessible [[file:config.org][here]].
|
||||
#+end_src
|
||||
|
||||
#+name: fig:sensor_model_calibrated
|
||||
#+caption: Calibrated Sensor
|
||||
#+caption: Normalized sensors using the inverse of an estimate $\hat{G}_i(s)$ of the sensor dynamics
|
||||
#+RESULTS:
|
||||
[[file:figs/sensor_model_calibrated.png]]
|
||||
|
||||
@ -164,7 +164,7 @@ Configuration file is accessible [[file:config.org][here]].
|
||||
#+end_src
|
||||
|
||||
#+name: fig:fusion_super_sensor
|
||||
#+caption: Sensor Fusion Architecture ([[./figs/fusion_super_sensor.png][png]], [[./figs/fusion_super_sensor.pdf][pdf]], [[./figs/fusion_super_sensor.tex][tex]]).
|
||||
#+caption: Sensor fusion architecture with two normalized sensors
|
||||
#+RESULTS:
|
||||
[[file:figs/fusion_super_sensor.png]]
|
||||
|
||||
@ -200,7 +200,7 @@ Configuration file is accessible [[file:config.org][here]].
|
||||
#+end_src
|
||||
|
||||
#+name: fig:sensor_model_uncertainty
|
||||
#+caption: Input Uncertainty
|
||||
#+caption: Sensor with multiplicative input uncertainty
|
||||
#+RESULTS:
|
||||
[[file:figs/sensor_model_uncertainty.png]]
|
||||
|
||||
@ -229,7 +229,7 @@ Configuration file is accessible [[file:config.org][here]].
|
||||
#+end_src
|
||||
|
||||
#+name: fig:sensor_model_uncertainty_simplified
|
||||
#+caption: Input Uncertainty
|
||||
#+caption: Simplified sensor model
|
||||
#+RESULTS:
|
||||
[[file:figs/sensor_model_uncertainty_simplified.png]]
|
||||
|
||||
@ -291,7 +291,7 @@ Configuration file is accessible [[file:config.org][here]].
|
||||
#+end_src
|
||||
|
||||
#+name: fig:sensor_fusion_dynamic_uncertainty
|
||||
#+caption: Sensor fusion architecture with sensor dynamics uncertainty ([[./figs/sensor_fusion_dynamic_uncertainty.png][png]], [[./figs/sensor_fusion_dynamic_uncertainty.pdf][pdf]], [[./figs/sensor_fusion_dynamic_uncertainty.tex][tex]]).
|
||||
#+caption: Sensor fusion architecture with sensor dynamics uncertainty
|
||||
#+RESULTS:
|
||||
[[file:figs/sensor_fusion_dynamic_uncertainty.png]]
|
||||
|
||||
@ -331,21 +331,23 @@ Configuration file is accessible [[file:config.org][here]].
|
||||
\draw[->] (0, -1.7) -- (0, 1.7) node[below left]{Im};
|
||||
|
||||
\draw[dashed] (0, 0) -- (tangent cs:node=c,point={(0, 0)},solution=2);
|
||||
\draw[dashed] (1, 0) arc (0:28:1) node[midway, right]{$\Delta \phi$};
|
||||
\draw[dashed] (1, 0) arc (0:28:1) node[midway, right]{$\Delta \phi_\text{max}$};
|
||||
\end{tikzpicture}
|
||||
#+end_src
|
||||
|
||||
#+name: fig:uncertainty_set_super_sensor
|
||||
#+caption: Uncertainty region of the super sensor dynamics in the complex plane (solid circle), of the sensor 1 (dotted circle) and of the sensor 2 (dashed circle) ([[./figs/uncertainty_set_super_sensor.png][png]], [[./figs/uncertainty_set_super_sensor.pdf][pdf]], [[./figs/uncertainty_set_super_sensor.tex][tex]]).
|
||||
#+caption: Uncertainty region of the super sensor dynamics in the complex plane (grey circle). The contribution of both sensors 1 and 2 to the total uncertainty are represented respectively by a blue circle and a red circle. The frequency dependency $\omega$ is here omitted.
|
||||
#+RESULTS:
|
||||
[[file:figs/uncertainty_set_super_sensor.png]]
|
||||
|
||||
* Architecture used for $\mathcal{H}_\infty$ synthesis of complementary filters
|
||||
#+begin_src latex :file h_infinity_robust_fusion.pdf :tangle figs/h_infinity_robust_fusion.tex :exports both
|
||||
* Generalized plant used for $\mathcal{H}_\infty$ synthesis of complementary filters
|
||||
#+begin_src latex :file h_infinity_robust_fusion_plant.pdf :tangle figs/h_infinity_robust_fusion_plant.tex :exports both
|
||||
\begin{tikzpicture}
|
||||
\node[block={4.0cm}{3.0cm}, fill=black!10!white] (P) {};
|
||||
\node[above] at (P.north) {$P(s)$};
|
||||
|
||||
\node[block, below=0.2 of P, opacity=0] (H2) {$H_2(s)$};
|
||||
|
||||
\coordinate[] (inputw) at ($(P.south west)!0.75!(P.north west) + (-0.7, 0)$);
|
||||
\coordinate[] (inputu) at ($(P.south west)!0.35!(P.north west) + (-0.7, 0)$);
|
||||
|
||||
@ -357,8 +359,6 @@ Configuration file is accessible [[file:config.org][here]].
|
||||
\node[block, left=1.4 of output2] (W2){$W_2(s)$};
|
||||
\node[addb={+}{}{}{}{-}, left=of W1] (sub) {};
|
||||
|
||||
% \node[block, below=0.3 of P] (H2) {$H_2(s)$};
|
||||
|
||||
\draw[->] (inputw) node[above right]{$w$} -- (sub.west);
|
||||
\draw[->] (inputu) node[above right]{$u$} -- (W2.west);
|
||||
\draw[->] (inputu-|sub) node[branch]{} -- (sub.south);
|
||||
@ -369,10 +369,44 @@ Configuration file is accessible [[file:config.org][here]].
|
||||
\end{tikzpicture}
|
||||
#+end_src
|
||||
|
||||
#+name: fig:h_infinity_robust_fusion
|
||||
#+caption: Architecture used for $\mathcal{H}_\infty$ synthesis of complementary filters ([[./figs/h_infinity_robust_fusion.png][png]], [[./figs/h_infinity_robust_fusion.pdf][pdf]], [[./figs/h_infinity_robust_fusion.tex][tex]]).
|
||||
#+name: fig:h_infinity_robust_fusion_plant
|
||||
#+caption: Generalized plant used for $\mathcal{H}_\infty$ synthesis of complementary filters
|
||||
#+RESULTS:
|
||||
[[file:figs/h_infinity_robust_fusion.png]]
|
||||
[[file:figs/h_infinity_robust_fusion_plant.png]]
|
||||
|
||||
* Architecture used for $\mathcal{H}_\infty$ synthesis of complementary filters
|
||||
#+begin_src latex :file h_infinity_robust_fusion_fb.pdf :tangle figs/h_infinity_robust_fusion_fb.tex :exports both
|
||||
\begin{tikzpicture}
|
||||
\node[block={4.0cm}{3.0cm}, fill=black!10!white] (P) {};
|
||||
\node[above] at (P.north) {$P(s)$};
|
||||
|
||||
\node[block, below=0.2 of P] (H2) {$H_2(s)$};
|
||||
|
||||
\coordinate[] (inputw) at ($(P.south west)!0.75!(P.north west) + (-0.7, 0)$);
|
||||
\coordinate[] (inputu) at ($(P.south west)!0.35!(P.north west) + (-0.4, 0)$);
|
||||
|
||||
\coordinate[] (output1) at ($(P.south east)!0.75!(P.north east) + ( 0.7, 0)$);
|
||||
\coordinate[] (output2) at ($(P.south east)!0.35!(P.north east) + ( 0.7, 0)$);
|
||||
\coordinate[] (outputv) at ($(P.south east)!0.1!(P.north east) + ( 0.4, 0)$);
|
||||
|
||||
\node[block, left=1.4 of output1] (W1){$W_1(s)$};
|
||||
\node[block, left=1.4 of output2] (W2){$W_2(s)$};
|
||||
\node[addb={+}{}{}{}{-}, left=of W1] (sub) {};
|
||||
|
||||
\draw[->] (inputw) node[above right]{$w$} -- (sub.west);
|
||||
\draw[->] (inputu-|sub) node[branch]{} -- (sub.south);
|
||||
\draw[->] (sub.east) -- (W1.west);
|
||||
\draw[->] ($(sub.west)+(-0.6, 0)$) node[branch]{} |- (outputv) |- (H2.east);
|
||||
\draw[->] (H2.west) -| (inputu) -- (W2.west);
|
||||
\draw[->] (W1.east) -- (output1)node[above left]{$z_1$};
|
||||
\draw[->] (W2.east) -- (output2)node[above left]{$z_2$};
|
||||
\end{tikzpicture}
|
||||
#+end_src
|
||||
|
||||
#+name: fig:h_infinity_robust_fusion_fb
|
||||
#+caption: Generalized plant with the synthesized filter
|
||||
#+RESULTS:
|
||||
[[file:figs/h_infinity_robust_fusion_fb.png]]
|
||||
|
||||
* LIGO Sensor Fusion Architecture
|
||||
#+begin_src latex :file ligo_super_sensor_architecture.pdf :tangle figs/ligo_super_sensor_architecture.tex :exports both
|
||||
@ -419,7 +453,7 @@ Configuration file is accessible [[file:config.org][here]].
|
||||
#+end_src
|
||||
|
||||
#+name: fig:ligo_super_sensor_architecture
|
||||
#+caption: ([[./figs/ligo_super_sensor_architecture.png][png]], [[./figs/ligo_super_sensor_architecture.pdf][pdf]], [[./figs/ligo_super_sensor_architecture.tex][tex]]).
|
||||
#+caption: Simplified block diagram of the sensor blending strategy for the first stage at the LIGO
|
||||
#+RESULTS:
|
||||
[[file:figs/ligo_super_sensor_architecture.png]]
|
||||
|
||||
@ -439,13 +473,12 @@ Configuration file is accessible [[file:config.org][here]].
|
||||
|
||||
\begin{scope}[on background layer]
|
||||
\node[fit={(L.north-|addfb.west) (botc)}, fill=black!10!white, draw, inner sep=6pt] (supersensor) {};
|
||||
% \node[below] at (supersensor.north) {Super Sensor};
|
||||
\end{scope}
|
||||
\end{tikzpicture}
|
||||
#+end_src
|
||||
|
||||
#+name: fig:feedback_sensor_fusion
|
||||
#+caption: "Closed-Loop" complementary filters ([[./figs/feedback_sensor_fusion.png][png]], [[./figs/feedback_sensor_fusion.pdf][pdf]], [[./figs/feedback_sensor_fusion.tex][tex]]).
|
||||
#+caption: "Closed-Loop" complementary filters
|
||||
#+RESULTS:
|
||||
[[file:figs/feedback_sensor_fusion.png]]
|
||||
|
||||
@ -475,37 +508,36 @@ Configuration file is accessible [[file:config.org][here]].
|
||||
\node[fit={(sensor2.west|-botc) (sensor1.north east)}, fill=black!20!white, draw, inner sep=6pt] (sensors) {};
|
||||
\node[above, align=center] at (sensors.south) {{\tiny Normalized}\\[-0.5em]{\tiny sensors}};
|
||||
\node[below, align=center] at (feedbackfilter.north) {{\tiny "Closed-Loop"}\\[-0.5em]{\tiny complementary filters}};
|
||||
% \node[above, align=center] at (supersensor.north) {Super Sensor};
|
||||
\end{scope}
|
||||
\end{tikzpicture}
|
||||
#+end_src
|
||||
|
||||
#+name: fig:feedback_sensor_fusion_arch
|
||||
#+caption: ([[./figs/feedback_sensor_fusion_arch.png][png]], [[./figs/feedback_sensor_fusion_arch.pdf][pdf]], [[./figs/feedback_sensor_fusion_arch.tex][tex]]).
|
||||
#+caption: Classical feedback architecture used for sensor fusion
|
||||
#+RESULTS:
|
||||
[[file:figs/feedback_sensor_fusion_arch.png]]
|
||||
|
||||
* TODO Feedback Loop Sensor Fusion Architecture
|
||||
* Feedback Loop Sensor Fusion Architecture
|
||||
#+begin_src latex :file feedback_synthesis_architecture.pdf :tangle figs/feedback_synthesis_architecture.tex
|
||||
\begin{tikzpicture}
|
||||
\node[block] (W2) at (0,0) {$W_2$};
|
||||
\node[addb={+}{}{}{}{-}, right=1 of W2] (addfb){};
|
||||
\node[block] (W2) at (0,0) {$W_2(s)$};
|
||||
\node[addb={+}{}{}{}{-}, right=0.8 of W2] (addfb){};
|
||||
\node[addb={+}{}{}{}{}, right=4.5 of W2] (adddy){};
|
||||
\node[block, above=0.8 of adddy] (W1){$W_1$};
|
||||
\node[block, above=0.8 of adddy] (W1){$W_1(s)$};
|
||||
|
||||
\draw[<-] (W2.west) -- ++(-1, 0) node[above right]{$w_2$};
|
||||
\draw[<-] (W2.west) -- ++(-0.8, 0) node[above right]{$w_2$};
|
||||
\draw[->] (W2.east) -- (addfb.west) node[above left]{$\tilde{w}_2$};
|
||||
\draw[->] (addfb.east) -- ++(1, 0) node[above left]{$v$};
|
||||
\draw[<-] (adddy.west) -- ++(-1, 0) node[above right]{$u$};
|
||||
\draw[->] (adddy.east) -- ++(1.4, 0) node[above left]{$z$};
|
||||
\draw[->] (W1.south) -- (adddy.north) node[above right]{$\tilde{w}_1$};
|
||||
\draw[<-] (W1.north) -- ++(0, 1) node[below right]{$w_1$};
|
||||
\draw[<-] (W1.north) -- ++(0, 0.8) node[below right]{$w_1$};
|
||||
\draw[->] ($(adddy.east) + (0.5, 0)$) node[branch]{} -- ++(0, -0.8) -| (addfb.south);
|
||||
\end{tikzpicture}
|
||||
#+end_src
|
||||
|
||||
#+name: fig:feedback_synthesis_architecture
|
||||
#+caption: ([[./figs/feedback_synthesis_architecture.png][png]], [[./figs/feedback_synthesis_architecture.pdf][pdf]], [[./figs/feedback_synthesis_architecture.tex][tex]]).
|
||||
#+caption: Feedback architecture with included weights
|
||||
#+RESULTS:
|
||||
[[file:figs/feedback_synthesis_architecture.png]]
|
||||
|
||||
@ -539,10 +571,54 @@ Configuration file is accessible [[file:config.org][here]].
|
||||
#+end_src
|
||||
|
||||
#+name: fig:feedback_synthesis_architecture_generalized_plant
|
||||
#+caption: ([[./figs/feedback_synthesis_architecture_generalized_plant.png][png]], [[./figs/feedback_synthesis_architecture_generalized_plant.pdf][pdf]], [[./figs/feedback_synthesis_architecture_generalized_plant.tex][tex]]).
|
||||
#+caption: Generalized plant used for the $\mathcal{H}_\infty$ synthesis of "closed-loop" complementary filters
|
||||
#+RESULTS:
|
||||
[[file:figs/feedback_synthesis_architecture_generalized_plant.png]]
|
||||
|
||||
* Sensor Fusion - Sequential
|
||||
#+begin_src latex :file sensor_fusion_three_sequential.pdf
|
||||
\begin{tikzpicture}
|
||||
\node[branch] (x) at (0, 0);
|
||||
|
||||
\node[block, right=0.4 of x] (sensor2) {Sensor 2};
|
||||
\node[block, above=0.4 of sensor2] (sensor1) {Sensor 1};
|
||||
\node[block, below=0.4 of sensor2] (sensor3) {Sensor 3};
|
||||
|
||||
\node[block, right=1.1 of sensor1](H1){$H_1(s)$};
|
||||
\node[block, right=1.1 of sensor2](H2){$H_2(s)$};
|
||||
\node[addb] (add) at ($0.5*(H1.east)+0.5*(H2.east)+(0.6, 0)$){};
|
||||
|
||||
\node[block, right=0.8 of add](H1p) {$H_1^\prime(s)$};
|
||||
\node[block] (H2p) at (H1p|-sensor3) {$H_2^\prime(s)$};
|
||||
|
||||
\node[addb] (addp) at ($0.5*(H1p.east)+0.5*(H2p.east)+(0.6, 0)$){};
|
||||
|
||||
\draw[->] ($(x)+(-0.8, 0)$) node[above right]{$x$} -- (sensor2.west);
|
||||
\draw[->] (x.center) |- (sensor1.west);
|
||||
\draw[->] (x.center) |- (sensor3.west);
|
||||
\draw[->] (sensor1.east) -- (H1.west) node[above left]{$\hat{x}_1$};
|
||||
\draw[->] (sensor2.east) -- (H2.west) node[above left]{$\hat{x}_2$};
|
||||
\draw[->] (sensor3.east) -- (H2p.west) node[above left]{$\hat{x}_3$};
|
||||
\draw[->] (H1) -| (add.north);
|
||||
\draw[->] (H2) -| (add.south);
|
||||
\draw[->] (add.east) -- (H1p.west) node[above left]{$\hat{x}_{12}$};
|
||||
\draw[->] (H1p) -| (addp.north);
|
||||
\draw[->] (H2p) -| (addp.south);
|
||||
\draw[->] (addp.east) -- ++(0.8, 0) node[above left]{$\hat{x}$};
|
||||
|
||||
\begin{scope}[on background layer]
|
||||
\node[fit={(x.west|-sensor3.south) (sensor1.north-|addp.east)}, fill=black!10!white, draw, inner sep=6pt] (supersensor) {};
|
||||
|
||||
\node[fit={(x.west|-sensor1.north) (add.east|-sensor2.south)}, fill=black!20!white, draw, inner sep=3pt] (superinertialsensor) {};
|
||||
\end{scope}
|
||||
\end{tikzpicture}
|
||||
#+end_src
|
||||
|
||||
#+name: fig:sensor_fusion_three_sequential
|
||||
#+caption: Sequential fusion
|
||||
#+RESULTS:
|
||||
[[file:figs/sensor_fusion_three_sequential.png]]
|
||||
|
||||
* Sensor Fusion - Parallel
|
||||
#+begin_src latex :file sensor_fusion_three_parallel.pdf
|
||||
\begin{tikzpicture}
|
||||
@ -578,57 +654,56 @@ Configuration file is accessible [[file:config.org][here]].
|
||||
#+end_src
|
||||
|
||||
#+name: fig:sensor_fusion_three_parallel
|
||||
#+caption: Sensor Fusion Architecture - Overview
|
||||
#+caption: Parallel fusion
|
||||
#+RESULTS:
|
||||
[[file:figs/sensor_fusion_three_parallel.png]]
|
||||
|
||||
* Sensor Fusion - Sequential
|
||||
#+begin_src latex :file sensor_fusion_three_sequential.pdf
|
||||
\begin{tikzpicture}
|
||||
\node[branch] (x) at (0, 0);
|
||||
* Architecture for $\mathcal{H}_\infty$ synthesis of three complementary filters
|
||||
#+begin_src latex :file comp_filter_three_hinf_gen_plant.pdf :tangle figs/comp_filter_three_hinf_gen_plant.tex
|
||||
\begin{tikzpicture}
|
||||
\node[block={5.0cm}{4.5cm}, fill=black!10!white] (P) {};
|
||||
\node[above] at (P.north) {$P_3(s)$};
|
||||
|
||||
\node[block, right=0.4 of x] (sensor2) {Sensor 2};
|
||||
\node[block, above=0.4 of sensor2] (sensor1) {Sensor 1};
|
||||
\node[block, below=0.4 of sensor2] (sensor3) {Sensor 3};
|
||||
\coordinate[] (inputw) at ($(P.south west)!0.8!(P.north west) + (-0.7, 0)$);
|
||||
\coordinate[] (inputu) at ($(P.south west)!0.4!(P.north west) + (-0.7, 0)$);
|
||||
|
||||
\node[block, right=1.1 of sensor1](H1){$H_1(s)$};
|
||||
\node[block, right=1.1 of sensor2](H2){$H_2(s)$};
|
||||
\node[addb] (add) at ($0.5*(H1.east)+0.5*(H2.east)+(0.6, 0)$){};
|
||||
\coordinate[] (output1) at ($(P.south east)!0.8!(P.north east) + (0.7, 0)$);
|
||||
\coordinate[] (output2) at ($(P.south east)!0.55!(P.north east) + (0.7, 0)$);
|
||||
\coordinate[] (output3) at ($(P.south east)!0.3!(P.north east) + (0.7, 0)$);
|
||||
\coordinate[] (outputv) at ($(P.south east)!0.1!(P.north east) + (0.7, 0)$);
|
||||
|
||||
\node[block, right=0.8 of add](H1p) {$H_1^\prime(s)$};
|
||||
\node[block] (H2p) at (H1p|-sensor3) {$H_2^\prime(s)$};
|
||||
\node[block, left=1.4 of output1] (W1){$W_1(s)$};
|
||||
\node[block, left=1.4 of output2] (W2){$W_2(s)$};
|
||||
\node[block, left=1.4 of output3] (W3){$W_3(s)$};
|
||||
\node[addb={+}{}{}{}{-}, left=of W1] (sub1) {};
|
||||
\node[addb={+}{}{}{}{-}, left=of sub1] (sub2) {};
|
||||
|
||||
\node[addb] (addp) at ($0.5*(H1p.east)+0.5*(H2p.east)+(0.6, 0)$){};
|
||||
\node[block, below=0.3 of P, opacity=0] (H) {$\begin{bmatrix}H_2(s) \\ H_3(s)\end{bmatrix}$};
|
||||
|
||||
\draw[->] ($(x)+(-0.8, 0)$) node[above right]{$x$} -- (sensor2.west);
|
||||
\draw[->] (x.center) |- (sensor1.west);
|
||||
\draw[->] (x.center) |- (sensor3.west);
|
||||
\draw[->] (sensor1.east) -- (H1.west) node[above left]{$\hat{x}_1$};
|
||||
\draw[->] (sensor2.east) -- (H2.west) node[above left]{$\hat{x}_2$};
|
||||
\draw[->] (sensor3.east) -- (H2p.west) node[above left]{$\hat{x}_3$};
|
||||
\draw[->] (H1) -| (add.north);
|
||||
\draw[->] (H2) -| (add.south);
|
||||
\draw[->] (add.east) -- (H1p.west) node[above left]{$\hat{x}_{12}$};
|
||||
\draw[->] (H1p) -| (addp.north);
|
||||
\draw[->] (H2p) -| (addp.south);
|
||||
\draw[->] (addp.east) -- ++(0.8, 0) node[above left]{$\hat{x}$};
|
||||
\draw[->] (inputw) node[above right](w){$w$} -- (sub2.west);
|
||||
\draw[->] (W3-|sub1)node[branch]{} -- (sub1.south);
|
||||
\draw[->] (W2-|sub2)node[branch]{} -- (sub2.south);
|
||||
\draw[->] ($(sub2.west)+(-0.5, 0)$) node[branch]{} |- (outputv);
|
||||
\draw[->] (inputu|-W2) -- (W2.west);
|
||||
\draw[->] (inputu|-W3) -- (W3.west);
|
||||
|
||||
\begin{scope}[on background layer]
|
||||
\node[fit={(x.west|-sensor3.south) (sensor1.north-|addp.east)}, fill=black!10!white, draw, inner sep=6pt] (supersensor) {};
|
||||
% \node[below] at (supersensor.north) {Super Sensor};
|
||||
|
||||
\node[fit={(x.west|-sensor1.north) (add.east|-sensor2.south)}, fill=black!20!white, draw, inner sep=3pt] (superinertialsensor) {};
|
||||
% \node[] at (superinertialsensor.center) {"Inertial" Super Sensor};
|
||||
\end{scope}
|
||||
\end{tikzpicture}
|
||||
\draw[->] (sub2.east) -- (sub1.west);
|
||||
\draw[->] (sub1.east) -- (W1.west);
|
||||
\draw[->] (W1.east) -- (output1)node[above left](z){$z_1$};
|
||||
\draw[->] (W2.east) -- (output2)node[above left]{$z_2$};
|
||||
\draw[->] (W3.east) -- (output3)node[above left]{$z_3$};
|
||||
\node[above] at (W2-|w){$u_1$};
|
||||
\node[above] at (W3-|w){$u_2$};
|
||||
\node[above] at (outputv-|z){$v$};
|
||||
\end{tikzpicture}
|
||||
#+end_src
|
||||
|
||||
#+name: fig:sensor_fusion_three_sequential
|
||||
#+caption: Sensor Fusion Architecture - Overview
|
||||
#+name: fig:comp_filter_three_hinf_gen_plant
|
||||
#+caption: Generalized plant for the $\mathcal{H}_\infty$ synthesis of three complementary filters
|
||||
#+RESULTS:
|
||||
[[file:figs/sensor_fusion_three_sequential.png]]
|
||||
[[file:figs/comp_filter_three_hinf_gen_plant.png]]
|
||||
* Architecture for $\mathcal{H}_\infty$ synthesis of three complementary filters
|
||||
#+begin_src latex :file comp_filter_three_hinf.pdf :tangle figs/comp_filter_three_hinf.tex
|
||||
#+begin_src latex :file comp_filter_three_hinf_fb.pdf :tangle figs/comp_filter_three_hinf_fb.tex
|
||||
\begin{tikzpicture}
|
||||
\node[block={5.0cm}{4.5cm}, fill=black!10!white] (P) {};
|
||||
\node[above] at (P.north) {$P_3(s)$};
|
||||
@ -667,7 +742,7 @@ Configuration file is accessible [[file:config.org][here]].
|
||||
\end{tikzpicture}
|
||||
#+end_src
|
||||
|
||||
#+name: fig:comp_filter_three_hinf
|
||||
#+caption: Architecture for $\mathcal{H}_\infty$ synthesis of three complementary filters ([[./figs/comp_filter_three_hinf.png][png]], [[./figs/comp_filter_three_hinf.pdf][pdf]], [[./figs/comp_filter_three_hinf.tex][tex]]).
|
||||
#+name: fig:comp_filter_three_hinf_fb
|
||||
#+caption: Generalized plant with the synthesized filter for the $\mathcal{H}_\infty$ synthesis of three complementary filters
|
||||
#+RESULTS:
|
||||
[[file:figs/comp_filter_three_hinf.png]]
|
||||
[[file:figs/comp_filter_three_hinf_fb.png]]
|
||||
716
tikz/index.html
716
tikz/index.html
@ -1,716 +0,0 @@
|
||||
<?xml version="1.0" encoding="utf-8"?>
|
||||
<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN"
|
||||
"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
|
||||
<html xmlns="http://www.w3.org/1999/xhtml" lang="en" xml:lang="en">
|
||||
<head>
|
||||
<!-- 2020-11-12 jeu. 10:44 -->
|
||||
<meta http-equiv="Content-Type" content="text/html;charset=utf-8" />
|
||||
<title>Complementary Filters Shaping Using \(\mathcal{H}_\infty\) Synthesis - Tikz Figures</title>
|
||||
<meta name="generator" content="Org mode" />
|
||||
<meta name="author" content="Thomas Dehaeze" />
|
||||
<link rel="stylesheet" type="text/css" href="https://research.tdehaeze.xyz/css/style.css"/>
|
||||
<script type="text/javascript" src="https://research.tdehaeze.xyz/js/script.js"></script>
|
||||
<script>MathJax = {
|
||||
tex: {
|
||||
tags: 'ams',
|
||||
macros: {bm: ["\\boldsymbol{#1}",1],}
|
||||
}
|
||||
};
|
||||
</script>
|
||||
<script type="text/javascript" src="https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-mml-chtml.js"></script>
|
||||
</head>
|
||||
<body>
|
||||
<div id="org-div-home-and-up">
|
||||
<a accesskey="h" href="../index.html"> UP </a>
|
||||
|
|
||||
<a accesskey="H" href="../index.html"> HOME </a>
|
||||
</div><div id="content">
|
||||
<h1 class="title">Complementary Filters Shaping Using \(\mathcal{H}_\infty\) Synthesis - Tikz Figures</h1>
|
||||
<div id="table-of-contents">
|
||||
<h2>Table of Contents</h2>
|
||||
<div id="text-table-of-contents">
|
||||
<ul>
|
||||
<li><a href="#org9059d4e">1. Fig 1: Sensor Fusion Architecture</a></li>
|
||||
<li><a href="#orgf8987c9">2. Fig 2: Sensor fusion architecture with sensor dynamics uncertainty</a></li>
|
||||
<li><a href="#org1801a90">3. Fig 3: Uncertainty set of the super sensor dynamics</a></li>
|
||||
<li><a href="#org355bde0">4. Fig 4: Architecture used for \(\mathcal{H}_\infty\) synthesis of complementary filters</a></li>
|
||||
<li><a href="#org879ecb3">5. Fig 5: Magnitude of a weighting function generated using the proposed formula</a></li>
|
||||
<li><a href="#org2b1f7ae">6. Fig 6: Frequency response of the weighting functions and complementary filters obtained using \(\mathcal{H}_\infty\) synthesis</a></li>
|
||||
<li><a href="#org84a7e33">7. Fig 7: Architecture for \(\mathcal{H}_\infty\) synthesis of three complementary filters</a></li>
|
||||
<li><a href="#org20bfce3">8. Fig 8: Frequency response of the weighting functions and three complementary filters obtained using \(\mathcal{H}_\infty\) synthesis</a></li>
|
||||
<li><a href="#orgb08c277">9. Fig 9: Specifications and weighting functions magnitude used for \(\mathcal{H}_\infty\) synthesis</a></li>
|
||||
<li><a href="#org74b4762">10. Fig 10: Comparison of the FIR filters (solid) with the filters obtained with \(\mathcal{H}_\infty\) synthesis (dashed)</a></li>
|
||||
</ul>
|
||||
</div>
|
||||
</div>
|
||||
<p>
|
||||
Configuration file is accessible <a href="config.html">here</a>.
|
||||
</p>
|
||||
|
||||
<div id="outline-container-org9059d4e" class="outline-2">
|
||||
<h2 id="org9059d4e"><span class="section-number-2">1</span> Fig 1: Sensor Fusion Architecture</h2>
|
||||
<div class="outline-text-2" id="text-1">
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-latex"><span class="org-font-latex-sedate"><span class="org-keyword">\begin</span></span>{<span class="org-function-name">tikzpicture</span>}
|
||||
<span class="org-font-latex-sedate">\node</span>[branch] (x) at (0, 0);
|
||||
<span class="org-font-latex-sedate">\node</span>[block, above right=0.5 and 0.5 of x](G1){<span class="org-font-latex-math">$G</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">1(s)$</span>};
|
||||
<span class="org-font-latex-sedate">\node</span>[block, below right=0.5 and 0.5 of x](G2){<span class="org-font-latex-math">$G</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">2(s)$</span>};
|
||||
<span class="org-font-latex-sedate">\node</span>[addb, right=0.8 of G1](add1){};
|
||||
<span class="org-font-latex-sedate">\node</span>[addb, right=0.8 of G2](add2){};
|
||||
<span class="org-font-latex-sedate">\node</span>[block, right=0.8 of add1](H1){<span class="org-font-latex-math">$H</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">1(s)$</span>};
|
||||
<span class="org-font-latex-sedate">\node</span>[block, right=0.8 of add2](H2){<span class="org-font-latex-math">$H</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">2(s)$</span>};
|
||||
<span class="org-font-latex-sedate">\node</span>[addb, right=5 of x](add){};
|
||||
|
||||
<span class="org-font-latex-sedate">\draw</span>[] (<span class="org-font-latex-math">$(x)+(-0.7, 0)$</span>) node[above right]{<span class="org-font-latex-math">$x$</span>} -- (x.center);
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (x.center) |- (G1.west);
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (x.center) |- (G2.west);
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (G1.east) -- (add1.west);
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (G2.east) -- (add2.west);
|
||||
<span class="org-font-latex-sedate">\draw</span>[<-] (add1.north) -- ++(0, 0.8)node[below right](n1){<span class="org-font-latex-math">$n</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">1$</span>};
|
||||
<span class="org-font-latex-sedate">\draw</span>[<-] (add2.north) -- ++(0, 0.8)node[below right](n2){<span class="org-font-latex-math">$n</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">2$</span>};
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (add1.east) -- (H1.west);
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (add2.east) -- (H2.west);
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (H1) -| (add.north);
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (H2) -| (add.south);
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (add.east) -- ++(0.7, 0) node[above left]{<span class="org-font-latex-math">$</span><span class="org-font-latex-sedate"><span class="org-font-latex-math">\hat</span></span><span class="org-font-latex-math">{x}$</span>};
|
||||
|
||||
<span class="org-font-latex-sedate"><span class="org-keyword">\begin</span></span>{<span class="org-function-name">scope</span>}[on background layer]
|
||||
<span class="org-font-latex-sedate">\node</span>[fit={(<span class="org-font-latex-math">$(G2.south-|x)+(-0.2, -0.3)$</span>) (<span class="org-font-latex-math">$(n1.north east-|add.east)+(0.2, 0.3)$</span>)}, fill=black!10!white, draw, dashed, inner sep=0pt] (supersensor) {};
|
||||
<span class="org-font-latex-sedate">\node</span>[below left] at (supersensor.north east) {Super Sensor};
|
||||
|
||||
<span class="org-font-latex-sedate">\node</span>[fit={(<span class="org-font-latex-math">$(G1.south west)+(-0.3, -0.1)$</span>) (<span class="org-font-latex-math">$(n1.north east)+(0.0, 0.1)$</span>)}, fill=black!20!white, draw, dashed, inner sep=0pt] (sensor1) {};
|
||||
<span class="org-font-latex-sedate">\node</span>[below right] at (sensor1.north west) {Sensor 1};
|
||||
<span class="org-font-latex-sedate">\node</span>[fit={(<span class="org-font-latex-math">$(G2.south west)+(-0.3, -0.1)$</span>) (<span class="org-font-latex-math">$(n2.north east)+(0.0, 0.1)$</span>)}, fill=black!20!white, draw, dashed, inner sep=0pt] (sensor2) {};
|
||||
<span class="org-font-latex-sedate">\node</span>[below right] at (sensor2.north west) {Sensor 2};
|
||||
<span class="org-font-latex-sedate"><span class="org-keyword">\end</span></span>{<span class="org-function-name">scope</span>}
|
||||
<span class="org-font-latex-sedate"><span class="org-keyword">\end</span></span>{<span class="org-function-name">tikzpicture</span>}
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
|
||||
<div id="org9c10944" class="figure">
|
||||
<p><img src="figs/fusion_super_sensor.png" alt="fusion_super_sensor.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 1: </span>Sensor Fusion Architecture (<a href="./figs/fusion_super_sensor.png">png</a>, <a href="./figs/fusion_super_sensor.pdf">pdf</a>, <a href="./figs/fusion_super_sensor.tex">tex</a>).</p>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-orgf8987c9" class="outline-2">
|
||||
<h2 id="orgf8987c9"><span class="section-number-2">2</span> Fig 2: Sensor fusion architecture with sensor dynamics uncertainty</h2>
|
||||
<div class="outline-text-2" id="text-2">
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-latex"><span class="org-font-latex-sedate"><span class="org-keyword">\begin</span></span>{<span class="org-function-name">tikzpicture</span>}
|
||||
<span class="org-font-latex-sedate">\node</span>[branch] (x) at (0, 0);
|
||||
<span class="org-font-latex-sedate">\node</span>[addb, above right=0.8 and 4 of x](add1){};
|
||||
<span class="org-font-latex-sedate">\node</span>[addb, below right=0.8 and 4 of x](add2){};
|
||||
<span class="org-font-latex-sedate">\node</span>[block, above left=0.2 and 0.1 of add1](delta1){<span class="org-font-latex-math">$</span><span class="org-font-latex-sedate"><span class="org-font-latex-math">\Delta</span></span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">1(s)$</span>};
|
||||
<span class="org-font-latex-sedate">\node</span>[block, above left=0.2 and 0.1 of add2](delta2){<span class="org-font-latex-math">$</span><span class="org-font-latex-sedate"><span class="org-font-latex-math">\Delta</span></span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">2(s)$</span>};
|
||||
<span class="org-font-latex-sedate">\node</span>[block, left=0.5 of delta1](W1){<span class="org-font-latex-math">$w</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">1(s)$</span>};
|
||||
<span class="org-font-latex-sedate">\node</span>[block, left=0.5 of delta2](W2){<span class="org-font-latex-math">$w</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">2(s)$</span>};
|
||||
<span class="org-font-latex-sedate">\node</span>[block, right=0.5 of add1](H1){<span class="org-font-latex-math">$H</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">1(s)$</span>};
|
||||
<span class="org-font-latex-sedate">\node</span>[block, right=0.5 of add2](H2){<span class="org-font-latex-math">$H</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">2(s)$</span>};
|
||||
<span class="org-font-latex-sedate">\node</span>[addb, right=6 of x](add){};
|
||||
|
||||
<span class="org-font-latex-sedate">\draw</span>[] (<span class="org-font-latex-math">$(x)+(-0.7, 0)$</span>) node[above right]{<span class="org-font-latex-math">$x$</span>} -- (x.center);
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (x.center) |- (add1.west);
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (x.center) |- (add2.west);
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (<span class="org-font-latex-math">$(add1-|W1.west)+(-0.5, 0)$</span>)node[branch](S1){} |- (W1.west);
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (<span class="org-font-latex-math">$(add2-|W2.west)+(-0.5, 0)$</span>)node[branch](S1){} |- (W2.west);
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (W1.east) -- (delta1.west);
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (W2.east) -- (delta2.west);
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (delta1.east) -| (add1.north);
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (delta2.east) -| (add2.north);
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (add1.east) -- (H1.west);
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (add2.east) -- (H2.west);
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (H1.east) -| (add.north);
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (H2.east) -| (add.south);
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (add.east) -- ++(0.7, 0) node[above left]{<span class="org-font-latex-math">$</span><span class="org-font-latex-sedate"><span class="org-font-latex-math">\hat</span></span><span class="org-font-latex-math">{x}$</span>};
|
||||
|
||||
<span class="org-font-latex-sedate"><span class="org-keyword">\begin</span></span>{<span class="org-function-name">scope</span>}[on background layer]
|
||||
<span class="org-font-latex-sedate">\node</span>[block, fit={(<span class="org-font-latex-math">$(W1.north-|S1)+(-0.2, 0.2)$</span>) (<span class="org-font-latex-math">$(add1.south east)+(0.2, -0.3)$</span>)}, fill=black!20!white, dashed, inner sep=0pt] (sensor1) {};
|
||||
<span class="org-font-latex-sedate">\node</span>[above right] at (sensor1.south west) {Sensor 1};
|
||||
<span class="org-font-latex-sedate">\node</span>[block, fit={(<span class="org-font-latex-math">$(W2.north-|S1)+(-0.2, 0.2)$</span>) (<span class="org-font-latex-math">$(add2.south east)+(0.2, -0.3)$</span>)}, fill=black!20!white, dashed, inner sep=0pt] (sensor2) {};
|
||||
<span class="org-font-latex-sedate">\node</span>[above right] at (sensor2.south west) {Sensor 2};
|
||||
<span class="org-font-latex-sedate"><span class="org-keyword">\end</span></span>{<span class="org-function-name">scope</span>}
|
||||
<span class="org-font-latex-sedate"><span class="org-keyword">\end</span></span>{<span class="org-function-name">tikzpicture</span>}
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
|
||||
<div id="orgfc65b94" class="figure">
|
||||
<p><img src="figs/sensor_fusion_dynamic_uncertainty.png" alt="sensor_fusion_dynamic_uncertainty.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 2: </span>Sensor fusion architecture with sensor dynamics uncertainty (<a href="./figs/sensor_fusion_dynamic_uncertainty.png">png</a>, <a href="./figs/sensor_fusion_dynamic_uncertainty.pdf">pdf</a>, <a href="./figs/sensor_fusion_dynamic_uncertainty.tex">tex</a>).</p>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org1801a90" class="outline-2">
|
||||
<h2 id="org1801a90"><span class="section-number-2">3</span> Fig 3: Uncertainty set of the super sensor dynamics</h2>
|
||||
<div class="outline-text-2" id="text-3">
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-latex"><span class="org-font-latex-sedate"><span class="org-keyword">\begin</span></span>{<span class="org-function-name">tikzpicture</span>}
|
||||
<span class="org-font-latex-sedate"><span class="org-keyword">\begin</span></span>{<span class="org-function-name">scope</span>}[shift={(4, 0)}]
|
||||
|
||||
<span class="org-comment">% Uncertainty Circle</span>
|
||||
<span class="org-font-latex-sedate">\node</span>[draw, circle, fill=black!20!white, minimum size=3.6cm] (c) at (0, 0) {};
|
||||
<span class="org-font-latex-sedate">\path</span>[draw, dotted] (0, 0) circle [radius=1.0];
|
||||
<span class="org-font-latex-sedate">\path</span>[draw, dashed] (135:1.0) circle [radius=0.8];
|
||||
|
||||
<span class="org-comment">% Center of Circle</span>
|
||||
<span class="org-font-latex-sedate">\node</span>[below] at (0, 0){<span class="org-font-latex-math">$1$</span>};
|
||||
|
||||
<span class="org-font-latex-sedate">\draw</span>[<->, dashed] (0, 0) node[branch]{} -- coordinate[midway](r1) ++(45:1.0);
|
||||
<span class="org-font-latex-sedate">\draw</span>[<->, dashed] (135:1.0)node[branch]{} -- coordinate[midway](r2) ++(90:0.8);
|
||||
|
||||
<span class="org-font-latex-sedate">\node</span>[] (l1) at (2, 1.5) {<span class="org-font-latex-math">$|w</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">1 H</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">1|$</span>};
|
||||
<span class="org-font-latex-sedate">\draw</span>[->, dashed, out=-90, in=0] (l1.south) to (r1);
|
||||
|
||||
<span class="org-font-latex-sedate">\node</span>[] (l2) at (-2.5, 1.5) {<span class="org-font-latex-math">$|w</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">2 H</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">2|$</span>};
|
||||
<span class="org-font-latex-sedate">\draw</span>[->, dashed, out=0, in=-180] (l2.east) to (r2);
|
||||
|
||||
<span class="org-font-latex-sedate">\draw</span>[<->, dashed] (0, 0) -- coordinate[near end](r3) ++(200:1.8);
|
||||
<span class="org-font-latex-sedate">\node</span>[] (l3) at (-2.5, -1.5) {<span class="org-font-latex-math">$|w</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">1 H</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">1| + |w</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">2 H</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">2|$</span>};
|
||||
<span class="org-font-latex-sedate">\draw</span>[->, dashed, out=90, in=-90] (l3.north) to (r3);
|
||||
<span class="org-font-latex-sedate"><span class="org-keyword">\end</span></span>{<span class="org-function-name">scope</span>}
|
||||
|
||||
<span class="org-comment">% Real and Imaginary Axis</span>
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (-0.5, 0) -- (7.0, 0) node[below left]{Re};
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (0, -1.7) -- (0, 1.7) node[below left]{Im};
|
||||
|
||||
<span class="org-font-latex-sedate">\draw</span>[dashed] (0, 0) -- (tangent cs:node=c,point={(0, 0)},solution=2);
|
||||
<span class="org-font-latex-sedate">\draw</span>[dashed] (1, 0) arc (0:28:1) node[midway, right]{<span class="org-font-latex-math">$</span><span class="org-font-latex-sedate"><span class="org-font-latex-math">\Delta</span></span><span class="org-font-latex-math"> </span><span class="org-font-latex-sedate"><span class="org-font-latex-math">\phi</span></span><span class="org-font-latex-math">$</span>};
|
||||
<span class="org-font-latex-sedate"><span class="org-keyword">\end</span></span>{<span class="org-function-name">tikzpicture</span>}
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
|
||||
<div id="orga1a6d2e" class="figure">
|
||||
<p><img src="figs/uncertainty_set_super_sensor.png" alt="uncertainty_set_super_sensor.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 3: </span>Uncertainty region of the super sensor dynamics in the complex plane (solid circle), of the sensor 1 (dotted circle) and of the sensor 2 (dashed circle) (<a href="./figs/uncertainty_set_super_sensor.png">png</a>, <a href="./figs/uncertainty_set_super_sensor.pdf">pdf</a>, <a href="./figs/uncertainty_set_super_sensor.tex">tex</a>).</p>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org355bde0" class="outline-2">
|
||||
<h2 id="org355bde0"><span class="section-number-2">4</span> Fig 4: Architecture used for \(\mathcal{H}_\infty\) synthesis of complementary filters</h2>
|
||||
<div class="outline-text-2" id="text-4">
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-latex"><span class="org-font-latex-sedate"><span class="org-keyword">\begin</span></span>{<span class="org-function-name">tikzpicture</span>}
|
||||
<span class="org-font-latex-sedate">\node</span>[block={4.0cm}{2.5cm}, fill=black!20!white, dashed] (P) {};
|
||||
<span class="org-font-latex-sedate">\node</span>[above] at (P.north) {<span class="org-font-latex-math">$P(s)$</span>};
|
||||
|
||||
<span class="org-font-latex-sedate">\coordinate</span>[] (inputw) at (<span class="org-font-latex-math">$(P.south west)!0.75!(P.north west) + (-0.7, 0)$</span>);
|
||||
<span class="org-font-latex-sedate">\coordinate</span>[] (inputu) at (<span class="org-font-latex-math">$(P.south west)!0.35!(P.north west) + (-0.7, 0)$</span>);
|
||||
|
||||
<span class="org-font-latex-sedate">\coordinate</span>[] (output1) at (<span class="org-font-latex-math">$(P.south east)!0.75!(P.north east) + ( 0.7, 0)$</span>);
|
||||
<span class="org-font-latex-sedate">\coordinate</span>[] (output2) at (<span class="org-font-latex-math">$(P.south east)!0.35!(P.north east) + ( 0.7, 0)$</span>);
|
||||
<span class="org-font-latex-sedate">\coordinate</span>[] (outputv) at (<span class="org-font-latex-math">$(P.south east)!0.1!(P.north east) + ( 0.7, 0)$</span>);
|
||||
|
||||
<span class="org-font-latex-sedate">\node</span>[block, left=1.4 of output1] (W1){<span class="org-font-latex-math">$W</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">1(s)$</span>};
|
||||
<span class="org-font-latex-sedate">\node</span>[block, left=1.4 of output2] (W2){<span class="org-font-latex-math">$W</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">2(s)$</span>};
|
||||
<span class="org-font-latex-sedate">\node</span>[addb={+}{}{}{}{-}, left=of W1] (sub) {};
|
||||
|
||||
<span class="org-font-latex-sedate">\node</span>[block, below=0.3 of P] (H2) {<span class="org-font-latex-math">$H</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">2(s)$</span>};
|
||||
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (inputw) node[above right]{<span class="org-font-latex-math">$w$</span>} -- (sub.west);
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (H2.west) -| (<span class="org-font-latex-math">$(inputu)+(0.35, 0)$</span>) node[above]{<span class="org-font-latex-math">$u$</span>} -- (W2.west);
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (inputu-|sub) node[branch]{} -- (sub.south);
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (sub.east) -- (W1.west);
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (<span class="org-font-latex-math">$(sub.west)+(-0.6, 0)$</span>) node[branch]{} |- (<span class="org-font-latex-math">$(outputv)+(-0.35, 0)$</span>) node[above]{<span class="org-font-latex-math">$v$</span>} |- (H2.east);
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (W1.east) -- (output1)node[above left]{<span class="org-font-latex-math">$z</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">1$</span>};
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (W2.east) -- (output2)node[above left]{<span class="org-font-latex-math">$z</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">2$</span>};
|
||||
<span class="org-font-latex-sedate"><span class="org-keyword">\end</span></span>{<span class="org-function-name">tikzpicture</span>}
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
|
||||
<div id="org903a05c" class="figure">
|
||||
<p><img src="figs/h_infinity_robust_fusion.png" alt="h_infinity_robust_fusion.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 4: </span>Architecture used for \(\mathcal{H}_\infty\) synthesis of complementary filters (<a href="./figs/h_infinity_robust_fusion.png">png</a>, <a href="./figs/h_infinity_robust_fusion.pdf">pdf</a>, <a href="./figs/h_infinity_robust_fusion.tex">tex</a>).</p>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org879ecb3" class="outline-2">
|
||||
<h2 id="org879ecb3"><span class="section-number-2">5</span> Fig 5: Magnitude of a weighting function generated using the proposed formula</h2>
|
||||
<div class="outline-text-2" id="text-5">
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-latex"><span class="org-font-latex-sedate"><span class="org-keyword">\setlength</span></span><span class="org-font-latex-sedate"><span class="org-variable-name">\fwidth</span></span>{<span class="org-variable-name">6.5cm</span>}
|
||||
<span class="org-font-latex-sedate"><span class="org-keyword">\setlength</span></span><span class="org-font-latex-sedate"><span class="org-variable-name">\fheight</span></span>{<span class="org-variable-name">3.5cm</span>}
|
||||
|
||||
<span class="org-font-latex-sedate"><span class="org-keyword">\begin</span></span>{<span class="org-function-name">tikzpicture</span>}
|
||||
<span class="org-font-latex-sedate"><span class="org-keyword">\begin</span></span>{<span class="org-function-name">axis</span>}[<span class="org-comment">%</span>
|
||||
width=1.0<span class="org-font-latex-sedate">\fwidth</span>,
|
||||
height=1.0<span class="org-font-latex-sedate">\fheight</span>,
|
||||
at={(0.0<span class="org-font-latex-sedate">\fwidth</span>, 0.0<span class="org-font-latex-sedate">\fheight</span>)},
|
||||
scale only axis,
|
||||
xmode=log,
|
||||
xmin=0.1,
|
||||
xmax=100,
|
||||
xtick={0.1,1,10, 100},
|
||||
xminorticks=true,
|
||||
ymode=log,
|
||||
ymin=0.0005,
|
||||
ymax=20,
|
||||
ytick={0.001, 0.01, 0.1, 1, 10},
|
||||
yminorticks=true,
|
||||
ylabel={Magnitude},
|
||||
xlabel={Frequency [Hz]},
|
||||
xminorgrids,
|
||||
yminorgrids,
|
||||
]
|
||||
|
||||
<span class="org-font-latex-sedate">\addplot</span> [color=black, line width=1.5pt, forget plot]
|
||||
table [x=freqs, y=ampl, col sep=comma] {/home/thomas/Cloud/thesis/papers/dehaeze19_desig_compl_filte/matlab/matweight_formula.csv};
|
||||
|
||||
<span class="org-font-latex-sedate">\addplot</span> [color=black, dashed, line width=1.5pt]
|
||||
table[row sep=crcr]{<span class="org-comment">%</span>
|
||||
1 10<span class="org-font-latex-warning">\\</span>
|
||||
100 10<span class="org-font-latex-warning">\\</span>
|
||||
};
|
||||
<span class="org-font-latex-sedate">\addplot</span> [color=black, dashed, line width=1.5pt]
|
||||
table[row sep=crcr]{<span class="org-comment">%</span>
|
||||
0.1 0.001<span class="org-font-latex-warning">\\</span>
|
||||
3 0.001<span class="org-font-latex-warning">\\</span>
|
||||
};
|
||||
|
||||
<span class="org-font-latex-sedate">\addplot</span> [color=black, line width=1.5pt]
|
||||
table[row sep=crcr]{<span class="org-comment">%</span>
|
||||
0.1 1<span class="org-font-latex-warning">\\</span>
|
||||
100 1<span class="org-font-latex-warning">\\</span>
|
||||
};
|
||||
|
||||
<span class="org-font-latex-sedate">\addplot</span> [color=black, dashed, line width=1.5pt]
|
||||
table[row sep=crcr]{<span class="org-comment">%</span>
|
||||
10 2<span class="org-font-latex-warning">\\</span>
|
||||
10 1<span class="org-font-latex-warning">\\</span>
|
||||
};
|
||||
|
||||
<span class="org-font-latex-sedate">\node</span>[below] at (2, 10) {<span class="org-font-latex-math">$G_</span><span class="org-font-latex-sedate"><span class="org-font-latex-math">\infty</span></span><span class="org-font-latex-math">$</span>};
|
||||
<span class="org-font-latex-sedate">\node</span>[above] at (2, 0.001) {<span class="org-font-latex-math">$G_0$</span>};
|
||||
|
||||
<span class="org-font-latex-sedate">\node</span>[branch] at (10, 2){};
|
||||
<span class="org-font-latex-sedate">\draw</span>[dashed, line cap=round] (7, 2) -- (20, 2) node[right]{<span class="org-font-latex-math">$G_c$</span>};
|
||||
<span class="org-font-latex-sedate">\draw</span>[dashed, line cap=round] (10, 2) -- (10, 1) node[below]{<span class="org-font-latex-math">$</span><span class="org-font-latex-sedate"><span class="org-font-latex-math">\omega</span></span><span class="org-font-latex-math">_c$</span>};
|
||||
|
||||
<span class="org-font-latex-sedate">\node</span>[right] at (3, 0.1) {<span class="org-font-latex-math">$+n$</span>};
|
||||
|
||||
<span class="org-font-latex-sedate"><span class="org-keyword">\end</span></span>{<span class="org-function-name">axis</span>}
|
||||
<span class="org-font-latex-sedate"><span class="org-keyword">\end</span></span>{<span class="org-function-name">tikzpicture</span>}
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
|
||||
<div id="org1aeddba" class="figure">
|
||||
<p><img src="figs/weight_formula.png" alt="weight_formula.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 5: </span>Magnitude of a weighting function generated using the proposed formula (<a href="./figs/weight_formula.png">png</a>, <a href="./figs/weight_formula.pdf">pdf</a>, <a href="./figs/weight_formula.tex">tex</a>).</p>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org2b1f7ae" class="outline-2">
|
||||
<h2 id="org2b1f7ae"><span class="section-number-2">6</span> Fig 6: Frequency response of the weighting functions and complementary filters obtained using \(\mathcal{H}_\infty\) synthesis</h2>
|
||||
<div class="outline-text-2" id="text-6">
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-latex"><span class="org-font-latex-sedate"><span class="org-keyword">\setlength</span></span><span class="org-font-latex-sedate"><span class="org-variable-name">\fwidth</span></span>{<span class="org-variable-name">6.5cm</span>}
|
||||
<span class="org-font-latex-sedate"><span class="org-keyword">\setlength</span></span><span class="org-font-latex-sedate"><span class="org-variable-name">\fheight</span></span>{<span class="org-variable-name">6cm</span>}
|
||||
|
||||
<span class="org-font-latex-sedate"><span class="org-keyword">\begin</span></span>{<span class="org-function-name">tikzpicture</span>}
|
||||
<span class="org-font-latex-sedate"><span class="org-keyword">\begin</span></span>{<span class="org-function-name">axis</span>}[<span class="org-comment">%</span>
|
||||
width=1.0<span class="org-font-latex-sedate">\fwidth</span>,
|
||||
height=0.5<span class="org-font-latex-sedate">\fheight</span>,
|
||||
at={(0.0<span class="org-font-latex-sedate">\fwidth</span>, 0.47<span class="org-font-latex-sedate">\fheight</span>)},
|
||||
scale only axis,
|
||||
xmode=log,
|
||||
xmin=0.1,
|
||||
xmax=1000,
|
||||
xtick={0.1, 1, 10, 100, 1000},
|
||||
xticklabels={{}},
|
||||
xminorticks=true,
|
||||
ymode=log,
|
||||
ymin=0.0005,
|
||||
ymax=20,
|
||||
ytick={0.001, 0.01, 0.1, 1, 10},
|
||||
yminorticks=true,
|
||||
ylabel={Magnitude},
|
||||
xminorgrids,
|
||||
yminorgrids,
|
||||
]
|
||||
<span class="org-font-latex-sedate">\addplot</span> [color=mycolor1, line width=1.5pt, forget plot]
|
||||
table [x=freqs, y=H1, col sep=comma] {/home/thomas/Cloud/thesis/papers/dehaeze19_desig_compl_filte/matlab/mathinf_filters_results.csv};
|
||||
|
||||
<span class="org-font-latex-sedate">\addplot</span> [color=mycolor2, line width=1.5pt, forget plot]
|
||||
table [x=freqs, y=H2, col sep=comma] {/home/thomas/Cloud/thesis/papers/dehaeze19_desig_compl_filte/matlab/mathinf_filters_results.csv};
|
||||
|
||||
<span class="org-font-latex-sedate">\addplot</span> [color=mycolor1, dashed, line width=1.5pt, forget plot]
|
||||
table [x=freqs, y=W1, col sep=comma] {/home/thomas/Cloud/thesis/papers/dehaeze19_desig_compl_filte/matlab/mathinf_weights.csv};
|
||||
|
||||
<span class="org-font-latex-sedate">\addplot</span> [color=mycolor2, dashed, line width=1.5pt, forget plot]
|
||||
table [x=freqs, y=W2, col sep=comma] {/home/thomas/Cloud/thesis/papers/dehaeze19_desig_compl_filte/matlab/mathinf_weights.csv};
|
||||
<span class="org-font-latex-sedate"><span class="org-keyword">\end</span></span>{<span class="org-function-name">axis</span>}
|
||||
|
||||
<span class="org-font-latex-sedate"><span class="org-keyword">\begin</span></span>{<span class="org-function-name">axis</span>}[<span class="org-comment">%</span>
|
||||
width=1.0<span class="org-font-latex-sedate">\fwidth</span>,
|
||||
height=0.45<span class="org-font-latex-sedate">\fheight</span>,
|
||||
at={(0.0<span class="org-font-latex-sedate">\fwidth</span>, 0.0<span class="org-font-latex-sedate">\fheight</span>)},
|
||||
scale only axis,
|
||||
xmode=log,
|
||||
xmin=0.1,
|
||||
xmax=1000,
|
||||
xtick={0.1, 1, 10, 100, 1000},
|
||||
xminorticks=true,
|
||||
xlabel={Frequency [Hz]},
|
||||
ymin=-200,
|
||||
ymax=200,
|
||||
ytick={-180, -90, 0, 90, 180},
|
||||
ylabel={Phase [deg]},
|
||||
xminorgrids,
|
||||
legend style={at={(1,1.1)}, outer sep=2pt , anchor=north east, legend cell align=left, align=left, draw=black, nodes={scale=0.7, transform shape}},
|
||||
]
|
||||
<span class="org-font-latex-sedate">\addlegendimage</span>{color=mycolor1, dashed, line width=1.5pt}
|
||||
<span class="org-font-latex-sedate">\addlegendentry</span>{<span class="org-font-latex-math">$W_1^{-1}$</span>};
|
||||
<span class="org-font-latex-sedate">\addlegendimage</span>{color=mycolor2, dashed, line width=1.5pt}
|
||||
<span class="org-font-latex-sedate">\addlegendentry</span>{<span class="org-font-latex-math">$W_2^{-1}$</span>};
|
||||
<span class="org-font-latex-sedate">\addplot</span> [color=mycolor1, line width=1.5pt]
|
||||
table [x=freqs, y=H1p, col sep=comma] {/home/thomas/Cloud/thesis/papers/dehaeze19_desig_compl_filte/matlab/mathinf_filters_results.csv};
|
||||
<span class="org-font-latex-sedate">\addlegendentry</span>{<span class="org-font-latex-math">$H_1$</span>};
|
||||
<span class="org-font-latex-sedate">\addplot</span> [color=mycolor2, line width=1.5pt]
|
||||
table [x=freqs, y=H2p, col sep=comma] {/home/thomas/Cloud/thesis/papers/dehaeze19_desig_compl_filte/matlab/mathinf_filters_results.csv};
|
||||
<span class="org-font-latex-sedate">\addlegendentry</span>{<span class="org-font-latex-math">$H_2$</span>};
|
||||
<span class="org-font-latex-sedate"><span class="org-keyword">\end</span></span>{<span class="org-function-name">axis</span>}
|
||||
<span class="org-font-latex-sedate"><span class="org-keyword">\end</span></span>{<span class="org-function-name">tikzpicture</span>}
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
|
||||
<div id="orgd2fb3fa" class="figure">
|
||||
<p><img src="figs/hinf_synthesis_results.png" alt="hinf_synthesis_results.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 6: </span>Frequency response of the weighting functions and complementary filters obtained using \(\mathcal{H}_\infty\) synthesis (<a href="./figs/hinf_synthesis_results.png">png</a>, <a href="./figs/hinf_synthesis_results.pdf">pdf</a>, <a href="./figs/hinf_synthesis_results.tex">tex</a>).</p>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org84a7e33" class="outline-2">
|
||||
<h2 id="org84a7e33"><span class="section-number-2">7</span> Fig 7: Architecture for \(\mathcal{H}_\infty\) synthesis of three complementary filters</h2>
|
||||
<div class="outline-text-2" id="text-7">
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-latex"><span class="org-font-latex-sedate"><span class="org-keyword">\begin</span></span>{<span class="org-function-name">tikzpicture</span>}
|
||||
<span class="org-font-latex-sedate">\node</span>[block={5.0cm}{3.5cm}, fill=black!20!white, dashed] (P) {};
|
||||
<span class="org-font-latex-sedate">\node</span>[above] at (P.north) {<span class="org-font-latex-math">$P(s)$</span>};
|
||||
|
||||
<span class="org-font-latex-sedate">\coordinate</span>[] (inputw) at (<span class="org-font-latex-math">$(P.south west)!0.8!(P.north west) + (-0.7, 0)$</span>);
|
||||
<span class="org-font-latex-sedate">\coordinate</span>[] (inputu) at (<span class="org-font-latex-math">$(P.south west)!0.4!(P.north west) + (-0.7, 0)$</span>);
|
||||
|
||||
<span class="org-font-latex-sedate">\coordinate</span>[] (output1) at (<span class="org-font-latex-math">$(P.south east)!0.8!(P.north east) + (0.7, 0)$</span>);
|
||||
<span class="org-font-latex-sedate">\coordinate</span>[] (output2) at (<span class="org-font-latex-math">$(P.south east)!0.55!(P.north east) + (0.7, 0)$</span>);
|
||||
<span class="org-font-latex-sedate">\coordinate</span>[] (output3) at (<span class="org-font-latex-math">$(P.south east)!0.3!(P.north east) + (0.7, 0)$</span>);
|
||||
<span class="org-font-latex-sedate">\coordinate</span>[] (outputv) at (<span class="org-font-latex-math">$(P.south east)!0.1!(P.north east) + (0.7, 0)$</span>);
|
||||
|
||||
<span class="org-font-latex-sedate">\node</span>[block, left=1.4 of output1] (W1){<span class="org-font-latex-math">$W</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">1(s)$</span>};
|
||||
<span class="org-font-latex-sedate">\node</span>[block, left=1.4 of output2] (W2){<span class="org-font-latex-math">$W</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">2(s)$</span>};
|
||||
<span class="org-font-latex-sedate">\node</span>[block, left=1.4 of output3] (W3){<span class="org-font-latex-math">$W</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">3(s)$</span>};
|
||||
<span class="org-font-latex-sedate">\node</span>[addb={+}{}{}{}{-}, left=of W1] (sub1) {};
|
||||
<span class="org-font-latex-sedate">\node</span>[addb={+}{}{}{}{-}, left=of sub1] (sub2) {};
|
||||
|
||||
<span class="org-font-latex-sedate">\node</span>[block, below=0.3 of P] (H) {<span class="org-font-latex-math">$</span><span class="org-font-latex-sedate"><span class="org-keyword"><span class="org-font-latex-math">\begin</span></span></span><span class="org-font-latex-math">{</span><span class="org-function-name"><span class="org-font-latex-math">bmatrix</span></span><span class="org-font-latex-math">}H</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">2(s) \\ H</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">3(s)</span><span class="org-font-latex-sedate"><span class="org-keyword"><span class="org-font-latex-math">\end</span></span></span><span class="org-font-latex-math">{</span><span class="org-function-name"><span class="org-font-latex-math">bmatrix</span></span><span class="org-font-latex-math">}$</span>};
|
||||
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (inputw) node[above right](w){<span class="org-font-latex-math">$w$</span>} -- (sub2.west);
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (W3-|sub1)node[branch]{} -- (sub1.south);
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (W2-|sub2)node[branch]{} -- (sub2.south);
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (<span class="org-font-latex-math">$(sub2.west)+(-0.5, 0)$</span>) node[branch]{} |- (outputv) |- (H.east);
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (<span class="org-font-latex-math">$(H.south west)!0.7!(H.north west)$</span>) -| (inputu|-W2) -- (W2.west);
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (<span class="org-font-latex-math">$(H.south west)!0.3!(H.north west)$</span>) -| (<span class="org-font-latex-math">$(inputu|-W3)+(0.4, 0)$</span>) -- (W3.west);
|
||||
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (sub2.east) -- (sub1.west);
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (sub1.east) -- (W1.west);
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (W1.east) -- (output1)node[above left](z){<span class="org-font-latex-math">$z</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">1$</span>};
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (W2.east) -- (output2)node[above left]{<span class="org-font-latex-math">$z</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">2$</span>};
|
||||
<span class="org-font-latex-sedate">\draw</span>[->] (W3.east) -- (output3)node[above left]{<span class="org-font-latex-math">$z</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">3$</span>};
|
||||
<span class="org-font-latex-sedate">\node</span>[above] at (W2-|w){<span class="org-font-latex-math">$u</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">1$</span>};
|
||||
<span class="org-font-latex-sedate">\node</span>[above] at (W3-|w){<span class="org-font-latex-math">$u</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_</span></span><span class="org-font-latex-math">2$</span>};
|
||||
<span class="org-font-latex-sedate">\node</span>[above] at (outputv-|z){<span class="org-font-latex-math">$v$</span>};
|
||||
<span class="org-font-latex-sedate"><span class="org-keyword">\end</span></span>{<span class="org-function-name">tikzpicture</span>}
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
|
||||
<div id="orgb697644" class="figure">
|
||||
<p><img src="figs/comp_filter_three_hinf.png" alt="comp_filter_three_hinf.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 7: </span>Architecture for \(\mathcal{H}_\infty\) synthesis of three complementary filters (<a href="./figs/comp_filter_three_hinf.png">png</a>, <a href="./figs/comp_filter_three_hinf.pdf">pdf</a>, <a href="./figs/comp_filter_three_hinf.tex">tex</a>).</p>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org20bfce3" class="outline-2">
|
||||
<h2 id="org20bfce3"><span class="section-number-2">8</span> Fig 8: Frequency response of the weighting functions and three complementary filters obtained using \(\mathcal{H}_\infty\) synthesis</h2>
|
||||
<div class="outline-text-2" id="text-8">
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-latex"><span class="org-font-latex-sedate"><span class="org-keyword">\setlength</span></span><span class="org-font-latex-sedate"><span class="org-variable-name">\fwidth</span></span>{<span class="org-variable-name">6.5cm</span>}
|
||||
<span class="org-font-latex-sedate"><span class="org-keyword">\setlength</span></span><span class="org-font-latex-sedate"><span class="org-variable-name">\fheight</span></span>{<span class="org-variable-name">6cm</span>}
|
||||
|
||||
<span class="org-font-latex-sedate"><span class="org-keyword">\begin</span></span>{<span class="org-function-name">tikzpicture</span>}
|
||||
<span class="org-font-latex-sedate"><span class="org-keyword">\begin</span></span>{<span class="org-function-name">axis</span>}[<span class="org-comment">%</span>
|
||||
width=1.0<span class="org-font-latex-sedate">\fwidth</span>,
|
||||
height=0.55<span class="org-font-latex-sedate">\fheight</span>,
|
||||
at={(0.0<span class="org-font-latex-sedate">\fwidth</span>, 0.42<span class="org-font-latex-sedate">\fheight</span>)},
|
||||
scale only axis,
|
||||
xmode=log,
|
||||
xmin=0.1,
|
||||
xmax=100,
|
||||
xticklabels={{}},
|
||||
xminorticks=true,
|
||||
ymode=log,
|
||||
ymin=0.0005,
|
||||
ymax=20,
|
||||
ytick={0.001, 0.01, 0.1, 1, 10},
|
||||
yminorticks=true,
|
||||
ylabel={Magnitude},
|
||||
xminorgrids,
|
||||
yminorgrids,
|
||||
legend columns=2,
|
||||
legend style={
|
||||
/tikz/column 2/.style={
|
||||
column sep=5pt,
|
||||
},
|
||||
at={(1,0)}, outer sep=2pt , anchor=south east, legend cell align=left, align=left, draw=black, nodes={scale=0.7, transform shape}
|
||||
},
|
||||
]
|
||||
<span class="org-font-latex-sedate">\addplot</span> [color=mycolor1, dashed, line width=1.5pt]
|
||||
table [x=freqs, y=W1, col sep=comma] {/home/thomas/Cloud/thesis/papers/dehaeze19_desig_compl_filte/matlab/mathinf_three_weights.csv};
|
||||
<span class="org-font-latex-sedate">\addlegendentry</span>{<span class="org-font-latex-math">${W_1}^{-1}$</span>};
|
||||
<span class="org-font-latex-sedate">\addplot</span> [color=mycolor1, line width=1.5pt]
|
||||
table [x=freqs, y=H1, col sep=comma] {/home/thomas/Cloud/thesis/papers/dehaeze19_desig_compl_filte/matlab/mathinf_three_results.csv};
|
||||
<span class="org-font-latex-sedate">\addlegendentry</span>{<span class="org-font-latex-math">$H_1$</span>};
|
||||
|
||||
|
||||
<span class="org-font-latex-sedate">\addplot</span> [color=mycolor2, dashed, line width=1.5pt]
|
||||
table [x=freqs, y=W2, col sep=comma] {/home/thomas/Cloud/thesis/papers/dehaeze19_desig_compl_filte/matlab/mathinf_three_weights.csv};
|
||||
<span class="org-font-latex-sedate">\addlegendentry</span>{<span class="org-font-latex-math">${W_2}^{-1}$</span>};
|
||||
<span class="org-font-latex-sedate">\addplot</span> [color=mycolor2, line width=1.5pt]
|
||||
table [x=freqs, y=H2, col sep=comma] {/home/thomas/Cloud/thesis/papers/dehaeze19_desig_compl_filte/matlab/mathinf_three_results.csv};
|
||||
<span class="org-font-latex-sedate">\addlegendentry</span>{<span class="org-font-latex-math">$H_2$</span>};
|
||||
|
||||
<span class="org-font-latex-sedate">\addplot</span> [color=mycolor3, dashed, line width=1.5pt]
|
||||
table [x=freqs, y=W3, col sep=comma] {/home/thomas/Cloud/thesis/papers/dehaeze19_desig_compl_filte/matlab/mathinf_three_weights.csv};
|
||||
<span class="org-font-latex-sedate">\addlegendentry</span>{<span class="org-font-latex-math">${W_3}^{-1}$</span>};
|
||||
<span class="org-font-latex-sedate">\addplot</span> [color=mycolor3, line width=1.5pt]
|
||||
table [x=freqs, y=H3, col sep=comma] {/home/thomas/Cloud/thesis/papers/dehaeze19_desig_compl_filte/matlab/mathinf_three_results.csv};
|
||||
<span class="org-font-latex-sedate">\addlegendentry</span>{<span class="org-font-latex-math">$H_3$</span>};
|
||||
<span class="org-font-latex-sedate"><span class="org-keyword">\end</span></span>{<span class="org-function-name">axis</span>}
|
||||
|
||||
<span class="org-font-latex-sedate"><span class="org-keyword">\begin</span></span>{<span class="org-function-name">axis</span>}[<span class="org-comment">%</span>
|
||||
width=1.0<span class="org-font-latex-sedate">\fwidth</span>,
|
||||
height=0.4<span class="org-font-latex-sedate">\fheight</span>,
|
||||
at={(0.0<span class="org-font-latex-sedate">\fwidth</span>, 0.0<span class="org-font-latex-sedate">\fheight</span>)},
|
||||
scale only axis,
|
||||
xmode=log,
|
||||
xmin=0.1,
|
||||
xmax=100,
|
||||
xminorticks=true,
|
||||
xlabel={Frequency [Hz]},
|
||||
ymin=-240,
|
||||
ymax=240,
|
||||
ytick={-180, -90, 0, 90, 180},
|
||||
ylabel={Phase [deg]},
|
||||
xminorgrids,
|
||||
]
|
||||
|
||||
<span class="org-font-latex-sedate">\addplot</span> [color=mycolor1, line width=1.5pt]
|
||||
table [x=freqs, y=H1p, col sep=comma] {/home/thomas/Cloud/thesis/papers/dehaeze19_desig_compl_filte/matlab/mathinf_three_results.csv};
|
||||
|
||||
<span class="org-font-latex-sedate">\addplot</span> [color=mycolor2, line width=1.5pt]
|
||||
table [x=freqs, y=H2p, col sep=comma] {/home/thomas/Cloud/thesis/papers/dehaeze19_desig_compl_filte/matlab/mathinf_three_results.csv};
|
||||
|
||||
<span class="org-font-latex-sedate">\addplot</span> [color=mycolor3, line width=1.5pt]
|
||||
table [x=freqs, y=H3p, col sep=comma] {/home/thomas/Cloud/thesis/papers/dehaeze19_desig_compl_filte/matlab/mathinf_three_results.csv};
|
||||
<span class="org-font-latex-sedate"><span class="org-keyword">\end</span></span>{<span class="org-function-name">axis</span>}
|
||||
<span class="org-font-latex-sedate"><span class="org-keyword">\end</span></span>{<span class="org-function-name">tikzpicture</span>}
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
|
||||
<div id="orgdaa24c8" class="figure">
|
||||
<p><img src="figs/hinf_three_synthesis_results.png" alt="hinf_three_synthesis_results.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 8: </span>Frequency response of the weighting functions and three complementary filters obtained using \(\mathcal{H}_\infty\) synthesis (<a href="./figs/hinf_three_synthesis_results.png">png</a>, <a href="./figs/hinf_three_synthesis_results.pdf">pdf</a>, <a href="./figs/hinf_three_synthesis_results.tex">tex</a>).</p>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-orgb08c277" class="outline-2">
|
||||
<h2 id="orgb08c277"><span class="section-number-2">9</span> Fig 9: Specifications and weighting functions magnitude used for \(\mathcal{H}_\infty\) synthesis</h2>
|
||||
<div class="outline-text-2" id="text-9">
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-latex"><span class="org-font-latex-sedate"><span class="org-keyword">\setlength</span></span><span class="org-font-latex-sedate"><span class="org-variable-name">\fwidth</span></span>{<span class="org-variable-name">6.5cm</span>}
|
||||
<span class="org-font-latex-sedate"><span class="org-keyword">\setlength</span></span><span class="org-font-latex-sedate"><span class="org-variable-name">\fheight</span></span>{<span class="org-variable-name">3.2cm</span>}
|
||||
|
||||
<span class="org-font-latex-sedate"><span class="org-keyword">\begin</span></span>{<span class="org-function-name">tikzpicture</span>}
|
||||
<span class="org-font-latex-sedate"><span class="org-keyword">\begin</span></span>{<span class="org-function-name">axis</span>}[<span class="org-comment">%</span>
|
||||
width=1.0<span class="org-font-latex-sedate">\fwidth</span>,
|
||||
height=1.0<span class="org-font-latex-sedate">\fheight</span>,
|
||||
at={(0.0<span class="org-font-latex-sedate">\fwidth</span>, 0.0<span class="org-font-latex-sedate">\fheight</span>)},
|
||||
scale only axis,
|
||||
separate axis lines,
|
||||
every outer x axis line/.append style={black},
|
||||
every x tick label/.append style={font=<span class="org-font-latex-sedate">\color</span>{black}},
|
||||
every x tick/.append style={black},
|
||||
xmode=log,
|
||||
xmin=0.001,
|
||||
xmax=1,
|
||||
xminorticks=true,
|
||||
xlabel={Frequency [Hz]},
|
||||
every outer y axis line/.append style={black},
|
||||
every y tick label/.append style={font=<span class="org-font-latex-sedate">\color</span>{black}},
|
||||
every y tick/.append style={black},
|
||||
ymode=log,
|
||||
ymin=0.005,
|
||||
ymax=20,
|
||||
yminorticks=true,
|
||||
ylabel={Magnitude},
|
||||
axis background/.style={fill=white},
|
||||
xmajorgrids,
|
||||
xminorgrids,
|
||||
ymajorgrids,
|
||||
yminorgrids,
|
||||
legend style={at={(0,1)}, outer sep=2pt, anchor=north west, legend cell align=left, align=left, draw=black, nodes={scale=0.7, transform shape}}
|
||||
]
|
||||
|
||||
<span class="org-font-latex-sedate">\addplot</span> [color=mycolor1, line width=1.5pt]
|
||||
table [x=freqs, y=wHm, col sep=comma] {/home/thomas/Cloud/thesis/papers/dehaeze19_desig_compl_filte/matlab/matligo_weights.csv};
|
||||
<span class="org-font-latex-sedate">\addlegendentry</span>{<span class="org-font-latex-math">$|w_H|^{-1}$</span>}
|
||||
|
||||
<span class="org-font-latex-sedate">\addplot</span> [color=mycolor2, line width=1.5pt]
|
||||
table [x=freqs, y=wLm, col sep=comma] {/home/thomas/Cloud/thesis/papers/dehaeze19_desig_compl_filte/matlab/matligo_weights.csv};
|
||||
<span class="org-font-latex-sedate">\addlegendentry</span>{<span class="org-font-latex-math">$|w_L|^{-1}$</span>}
|
||||
|
||||
<span class="org-font-latex-sedate">\addplot</span> [color=black, dotted, line width=1.5pt]
|
||||
table[row sep=crcr]{<span class="org-comment">%</span>
|
||||
0.0005 0.008<span class="org-font-latex-warning">\\</span>
|
||||
0.008 0.008<span class="org-font-latex-warning">\\</span>
|
||||
};
|
||||
<span class="org-font-latex-sedate">\addlegendentry</span>{Specifications}
|
||||
|
||||
<span class="org-font-latex-sedate">\addplot</span> [color=black, dotted, line width=1.5pt, forget plot]
|
||||
table[row sep=crcr]{<span class="org-comment">%</span>
|
||||
0.008 0.008<span class="org-font-latex-warning">\\</span>
|
||||
0.04 1<span class="org-font-latex-warning">\\</span>
|
||||
};
|
||||
<span class="org-font-latex-sedate">\addplot</span> [color=black, dotted, line width=1.5pt, forget plot]
|
||||
table[row sep=crcr]{<span class="org-comment">%</span>
|
||||
0.04 3<span class="org-font-latex-warning">\\</span>
|
||||
0.1 3<span class="org-font-latex-warning">\\</span>
|
||||
};
|
||||
<span class="org-font-latex-sedate">\addplot</span> [color=black, dotted, line width=1.5pt]
|
||||
table[row sep=crcr]{<span class="org-comment">%</span>
|
||||
0.1 0.045<span class="org-font-latex-warning">\\</span>
|
||||
2 0.045<span class="org-font-latex-warning">\\</span>
|
||||
};
|
||||
<span class="org-font-latex-sedate"><span class="org-keyword">\end</span></span>{<span class="org-function-name">axis</span>}
|
||||
<span class="org-font-latex-sedate"><span class="org-keyword">\end</span></span>{<span class="org-function-name">tikzpicture</span>}
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
|
||||
<div id="org2ba2505" class="figure">
|
||||
<p><img src="figs/ligo_weights.png" alt="ligo_weights.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 9: </span>Specifications and weighting functions magnitude used for \(\mathcal{H}_\infty\) synthesis (<a href="./figs/ligo_weights.png">png</a>, <a href="./figs/ligo_weights.pdf">pdf</a>, <a href="./figs/ligo_weights.tex">tex</a>).</p>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org74b4762" class="outline-2">
|
||||
<h2 id="org74b4762"><span class="section-number-2">10</span> Fig 10: Comparison of the FIR filters (solid) with the filters obtained with \(\mathcal{H}_\infty\) synthesis (dashed)</h2>
|
||||
<div class="outline-text-2" id="text-10">
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-latex"><span class="org-font-latex-sedate"><span class="org-keyword">\setlength</span></span><span class="org-font-latex-sedate"><span class="org-variable-name">\fwidth</span></span>{<span class="org-variable-name">6.5cm</span>}
|
||||
<span class="org-font-latex-sedate"><span class="org-keyword">\setlength</span></span><span class="org-font-latex-sedate"><span class="org-variable-name">\fheight</span></span>{<span class="org-variable-name">6.8cm</span>}
|
||||
|
||||
<span class="org-font-latex-sedate"><span class="org-keyword">\begin</span></span>{<span class="org-function-name">tikzpicture</span>}
|
||||
<span class="org-font-latex-sedate"><span class="org-keyword">\begin</span></span>{<span class="org-function-name">axis</span>}[<span class="org-comment">%</span>
|
||||
width=1.0<span class="org-font-latex-sedate">\fwidth</span>,
|
||||
height=0.60<span class="org-font-latex-sedate">\fheight</span>,
|
||||
at={(0.0<span class="org-font-latex-sedate">\fwidth</span>, 0.32<span class="org-font-latex-sedate">\fheight</span>)},
|
||||
scale only axis,
|
||||
xmode=log,
|
||||
xmin=0.001,
|
||||
xmax=1,
|
||||
xtick={0.001,0.01,0.1,1},
|
||||
xticklabels={{}},
|
||||
xminorticks=true,
|
||||
ymode=log,
|
||||
ymin=0.002,
|
||||
ymax=5,
|
||||
ytick={0.001, 0.01, 0.1, 1, 10},
|
||||
yminorticks=true,
|
||||
ylabel={Magnitude},
|
||||
xminorgrids,
|
||||
yminorgrids,
|
||||
legend style={at={(1,0)}, outer sep=2pt, anchor=south east, legend cell align=left, align=left, draw=black, nodes={scale=0.7, transform shape}}
|
||||
]
|
||||
<span class="org-font-latex-sedate">\addplot</span> [color=mycolor1, line width=1.5pt]
|
||||
table [x=freqs, y=Hhm, col sep=comma] {/home/thomas/Cloud/thesis/papers/dehaeze19_desig_compl_filte/matlab/matcomp_ligo_hinf.csv};
|
||||
<span class="org-font-latex-sedate">\addlegendentry</span>{<span class="org-font-latex-math">$H_H(s)$</span> - <span class="org-font-latex-math">$</span><span class="org-font-latex-sedate"><span class="org-font-latex-math">\mathcal</span></span><span class="org-font-latex-math">{H}_</span><span class="org-font-latex-sedate"><span class="org-font-latex-math">\infty</span></span><span class="org-font-latex-math">$</span>}
|
||||
<span class="org-font-latex-sedate">\addplot</span> [color=mycolor1, dashed, line width=1.5pt]
|
||||
table [x=freqs, y=Hhm, col sep=comma] {/home/thomas/Cloud/thesis/papers/dehaeze19_desig_compl_filte/matlab/matcomp_ligo_fir.csv};
|
||||
<span class="org-font-latex-sedate">\addlegendentry</span>{<span class="org-font-latex-math">$H_H(s)$</span> - FIR}
|
||||
<span class="org-font-latex-sedate">\addplot</span> [color=mycolor2, line width=1.5pt]
|
||||
table [x=freqs, y=Hlm, col sep=comma] {/home/thomas/Cloud/thesis/papers/dehaeze19_desig_compl_filte/matlab/matcomp_ligo_hinf.csv};
|
||||
<span class="org-font-latex-sedate">\addlegendentry</span>{<span class="org-font-latex-math">$H_L(s)$</span> - <span class="org-font-latex-math">$</span><span class="org-font-latex-sedate"><span class="org-font-latex-math">\mathcal</span></span><span class="org-font-latex-math">{H}_</span><span class="org-font-latex-sedate"><span class="org-font-latex-math">\infty</span></span><span class="org-font-latex-math">$</span>}
|
||||
<span class="org-font-latex-sedate">\addplot</span> [color=mycolor2, dashed, line width=1.5pt]
|
||||
table [x=freqs, y=Hlm, col sep=comma] {/home/thomas/Cloud/thesis/papers/dehaeze19_desig_compl_filte/matlab/matcomp_ligo_fir.csv};
|
||||
<span class="org-font-latex-sedate">\addlegendentry</span>{<span class="org-font-latex-math">$H_L(s)$</span> - FIR}
|
||||
<span class="org-font-latex-sedate"><span class="org-keyword">\end</span></span>{<span class="org-function-name">axis</span>}
|
||||
|
||||
<span class="org-font-latex-sedate"><span class="org-keyword">\begin</span></span>{<span class="org-function-name">axis</span>}[<span class="org-comment">%</span>
|
||||
width=1.0<span class="org-font-latex-sedate">\fwidth</span>,
|
||||
height=0.3<span class="org-font-latex-sedate">\fheight</span>,
|
||||
at={(0.0<span class="org-font-latex-sedate">\fwidth</span>, 0.0<span class="org-font-latex-sedate">\fheight</span>)},
|
||||
scale only axis,
|
||||
xmode=log,
|
||||
xmin=0.001,
|
||||
xmax=1,
|
||||
xtick={0.001, 0.01, 0.1, 1},
|
||||
xminorticks=true,
|
||||
xlabel={Frequency [Hz]},
|
||||
ymin=-180,
|
||||
ymax=180,
|
||||
ytick={-180, -90, 0, 90, 180},
|
||||
ylabel={Phase [deg]},
|
||||
xminorgrids,
|
||||
]
|
||||
<span class="org-font-latex-sedate">\addplot</span> [color=mycolor1, line width=1.5pt, forget plot]
|
||||
table [x=freqs, y=Hhp, col sep=comma] {/home/thomas/Cloud/thesis/papers/dehaeze19_desig_compl_filte/matlab/matcomp_ligo_hinf.csv};
|
||||
<span class="org-font-latex-sedate">\addplot</span> [color=mycolor1, dashed, line width=1.5pt, forget plot]
|
||||
table [x=freqs, y=Hhp, col sep=comma] {/home/thomas/Cloud/thesis/papers/dehaeze19_desig_compl_filte/matlab/matcomp_ligo_fir.csv};
|
||||
<span class="org-font-latex-sedate">\addplot</span> [color=mycolor2, line width=1.5pt, forget plot]
|
||||
table [x=freqs, y=Hlp, col sep=comma] {/home/thomas/Cloud/thesis/papers/dehaeze19_desig_compl_filte/matlab/matcomp_ligo_hinf.csv};
|
||||
<span class="org-font-latex-sedate">\addplot</span> [color=mycolor2, dashed, line width=1.5pt, forget plot]
|
||||
table [x=freqs, y=Hlp, col sep=comma] {/home/thomas/Cloud/thesis/papers/dehaeze19_desig_compl_filte/matlab/matcomp_ligo_fir.csv};
|
||||
<span class="org-font-latex-sedate"><span class="org-keyword">\end</span></span>{<span class="org-function-name">axis</span>}
|
||||
<span class="org-font-latex-sedate"><span class="org-keyword">\end</span></span>{<span class="org-function-name">tikzpicture</span>}
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
|
||||
<div id="org6bce6a7" class="figure">
|
||||
<p><img src="figs/comp_fir_ligo_hinf.png" alt="comp_fir_ligo_hinf.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 10: </span>Comparison of the FIR filters (solid) with the filters obtained with \(\mathcal{H}_\infty\) synthesis (dashed) (<a href="./figs/comp_fir_ligo_hinf.png">png</a>, <a href="./figs/comp_fir_ligo_hinf.pdf">pdf</a>, <a href="./figs/comp_fir_ligo_hinf.tex">tex</a>).</p>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
<div id="postamble" class="status">
|
||||
<p class="author">Author: Thomas Dehaeze</p>
|
||||
<p class="date">Created: 2020-11-12 jeu. 10:44</p>
|
||||
</div>
|
||||
</body>
|
||||
</html>
|
||||
Loading…
Reference in New Issue
Block a user