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<div id="content">
<h1 class="title">Tikz - Sensor Fusion Paper</h1>
<div id="table-of-contents">
<h2>Table of Contents</h2>
<div id="text-table-of-contents">
<ul>
<li><a href="#org667f301">1. Change some default</a></li>
<li><a href="#org1fa2a89">2. Sensor Fusion with complementary filters</a></li>
<li><a href="#org4155710">3. Equivalent configuration</a></li>
<li><a href="#orgd71025e">4. Equivalent configuration - classical Feedback</a></li>
<li><a href="#orgcc18c86">5. Equivalent configuration - classical Feedback with pre-filter</a></li>
<li><a href="#org0dcfd4e">6. Equivalent configuration - bis</a></li>
<li><a href="#org848380b">7. H-Infinity - Complementary filters - Generalized plant</a></li>
<li><a href="#org46ea807">8. H-Infinity - Complementary filters</a></li>
<li><a href="#orgc79b93a">9. Input Multiplicative Uncertainty</a></li>
<li><a href="#org594469f">10. One mass</a></li>
<li><a href="#orgd97f866">11. One mass - Control</a></li>
<li><a href="#org5ceffca">12. Piezoelectric Actuator</a></li>
<li><a href="#org8aa730c">13. Upper bounds</a></li>
<li><a href="#org4a37146">14. Bode plot of the mechanical system</a></li>
<li><a href="#orgfb7564e">15. Bode plot of the mechanical system - Uncertainty</a></li>
<li><a href="#org6e09d6f">16. Bode plot of the controller</a></li>
<li><a href="#org0fe4a21">17. Bode plot of the loop gain</a></li>
<li><a href="#orgb069f38">18. Upper bounds with weights</a></li>
<li><a href="#orgcb17586">19. Upper bounds with Complementary Filters</a></li>
<li><a href="#orgfa30d42">20. Upper bounds with weights and Complementary Filters</a></li>
<li><a href="#org0492f6b">21. Robustness - Nyquist</a></li>
<li><a href="#org0e6c625">22. Robustness - Loop Gain</a></li>
<li><a href="#orgd400711">23. Robust performance</a></li>
</ul>
</div>
</div>
<p>
Configuration file is accessible <a href="config.html">here</a>.
</p>
<div id="outline-container-org667f301" class="outline-2">
<h2 id="org667f301"><span class="section-number-2">1</span> Change some default</h2>
<div class="outline-text-2" id="text-1">
<div class="org-src-container">
<pre class="src src-latex" id="orgcbdd6dc"><span class="org-font-latex-sedate">\tikzset</span>{block/.default={0.8cm}{0.6cm}}
<span class="org-font-latex-sedate">\tikzset</span>{addb/.append style={scale=0.7}}
<span class="org-font-latex-sedate">\tikzset</span>{node distance=0.6}
<span class="org-font-latex-sedate">\def\cdist</span>{0.7}
<span class="org-font-latex-sedate">\definecolor</span>{T}{rgb}{0.230, 0.299, 0.754}<span class="org-comment">%</span>
<span class="org-font-latex-sedate">\definecolor</span>{S}{rgb}{0.706, 0.016, 0.150}<span class="org-comment">%</span>
</pre>
</div>
</div>
</div>
<div id="outline-container-org1fa2a89" class="outline-2">
<h2 id="org1fa2a89"><span class="section-number-2">2</span> Sensor Fusion with complementary filters</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">\tikzset</span>{block/.default={0.8cm}{0.6cm}}
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<span class="org-font-latex-sedate">\def\cdist</span>{0.7}
<span class="org-font-latex-sedate">\definecolor</span>{T}{rgb}{0.230, 0.299, 0.754}<span class="org-comment">%</span>
<span class="org-font-latex-sedate">\definecolor</span>{S}{rgb}{0.706, 0.016, 0.150}<span class="org-comment">%</span>
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</pre>
</div>
<div id="org821a188" class="figure">
<p><img src="figs/sf_arch.png" alt="sf_arch.png" />
</p>
<p><span class="figure-number">Figure 1: </span>Sensor Fusion with complementary filters (<a href="./figs/sf_arch.png">png</a>, <a href="./figs/sf_arch.pdf">pdf</a>, <a href="./figs/sf_arch.tex">tex</a>).</p>
</div>
</div>
</div>
<div id="outline-container-org4155710" class="outline-2">
<h2 id="org4155710"><span class="section-number-2">3</span> Equivalent configuration</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">\tikzset</span>{block/.default={0.8cm}{0.6cm}}
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<span class="org-font-latex-sedate">\draw</span>[&lt;-] (addn.east) -- ++(<span class="org-font-latex-sedate">\cdist</span>, 0) coordinate[](endpos) node[above left]{<span class="org-font-latex-math">$n$</span>};
<span class="org-font-latex-sedate">\draw</span>[-&gt;] (adddy.east) -- (G-|endpos) node[above left]{<span class="org-font-latex-math">$y$</span>};
<span class="org-font-latex-sedate">\draw</span>[-&gt;] (adddy-|addn) node[branch]{} -- (addn.north);
<span class="org-font-latex-sedate">\draw</span>[&lt;-] (addfb.west) -- ++(-<span class="org-font-latex-sedate">\cdist</span>, 0) node[above right]{<span class="org-font-latex-math">$r$</span>};
<span class="org-font-latex-sedate">\draw</span>[-&gt;] (addn.west) -- (Hl.east) node[above right]{<span class="org-font-latex-math">$y</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_m</span></span><span class="org-font-latex-math">$</span>};
<span class="org-font-latex-sedate">\draw</span>[&lt;-] (adddy.north) -- ++(0, <span class="org-font-latex-sedate">\cdist</span>) node[below right]{<span class="org-font-latex-math">$d</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_y</span></span><span class="org-font-latex-math">$</span>};
<span class="org-font-latex-sedate">\draw</span>[-&gt;] (Hl.west) -| (addfb.south) node[below right]{};
<span class="org-font-latex-sedate">\node</span>[fit={(<span class="org-font-latex-math">$(addK.west|-Hh.south)+(-0.1, 0)$</span>) (K.north-|sffb)}, inner sep=5pt, draw, dashed, color=gray, label={<span class="org-font-latex-math">$K$</span>}] (Kfb) {};
<span class="org-font-latex-sedate"><span class="org-keyword">\end</span></span>{<span class="org-function-name">tikzpicture</span>}
</pre>
</div>
<div id="org6e7bf4f" class="figure">
<p><img src="figs/sf_arch_eq.png" alt="sf_arch_eq.png" />
</p>
<p><span class="figure-number">Figure 2: </span>Equivalent configuration (<a href="./figs/sf_arch_eq.png">png</a>, <a href="./figs/sf_arch_eq.pdf">pdf</a>, <a href="./figs/sf_arch_eq.tex">tex</a>).</p>
</div>
</div>
</div>
<div id="outline-container-orgd71025e" class="outline-2">
<h2 id="orgd71025e"><span class="section-number-2">4</span> Equivalent configuration - classical Feedback</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">\tikzset</span>{block/.default={0.8cm}{0.6cm}}
<span class="org-font-latex-sedate">\tikzset</span>{addb/.append style={scale=0.7}}
<span class="org-font-latex-sedate">\tikzset</span>{node distance=0.6}
<span class="org-font-latex-sedate">\def\cdist</span>{0.7}
<span class="org-font-latex-sedate">\definecolor</span>{T}{rgb}{0.230, 0.299, 0.754}<span class="org-comment">%</span>
<span class="org-font-latex-sedate">\definecolor</span>{S}{rgb}{0.706, 0.016, 0.150}<span class="org-comment">%</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">\node</span>[addb={+}{}{}{}{-}] (addfb) at (0, 0){};
<span class="org-font-latex-sedate">\node</span>[block, right=of addfb] (K){<span class="org-font-latex-math">$K$</span>};
<span class="org-font-latex-sedate">\node</span>[block, right=of K] (G){<span class="org-font-latex-math">$G</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">^\prime</span></span><span class="org-font-latex-math">$</span>};
<span class="org-font-latex-sedate">\node</span>[addb={+}{}{}{}{}, right=of G] (adddy){};
<span class="org-font-latex-sedate">\node</span>[addb={+}{}{}{}{}, below right=and 0.5 of adddy] (addn) {};
<span class="org-font-latex-sedate">\node</span>[block] (Hh) at (G|-addn) {<span class="org-font-latex-math">$H</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_L</span></span><span class="org-font-latex-math">$</span>};
<span class="org-font-latex-sedate">\draw</span>[-&gt;] (addfb.east) -- (K.west) node[above left]{};
<span class="org-font-latex-sedate">\draw</span>[-&gt;] (K.east) -- (G.west) node[above left]{<span class="org-font-latex-math">$u$</span>};
<span class="org-font-latex-sedate">\draw</span>[-&gt;] (G.east) -- (adddy.west);
<span class="org-font-latex-sedate">\draw</span>[&lt;-] (addn.east) -- ++(<span class="org-font-latex-sedate">\cdist</span>, 0) coordinate[](endpos) node[above left]{<span class="org-font-latex-math">$n$</span>};
<span class="org-font-latex-sedate">\draw</span>[-&gt;] (G-|addn)node[branch]{} -- (addn.north);
<span class="org-font-latex-sedate">\draw</span>[-&gt;] (adddy.east) -- (G-|endpos) node[above left]{<span class="org-font-latex-math">$y$</span>};
<span class="org-font-latex-sedate">\draw</span>[&lt;-] (addfb.west) -- ++(-<span class="org-font-latex-sedate">\cdist</span>, 0) node[above right]{<span class="org-font-latex-math">$r$</span>};
<span class="org-font-latex-sedate">\draw</span>[-&gt;] (addn.west) -- (Hh.east);
<span class="org-font-latex-sedate">\draw</span>[-&gt;] (Hh.west) -| (addfb.south);
<span class="org-font-latex-sedate">\draw</span>[&lt;-] (adddy.north) -- ++(0, <span class="org-font-latex-sedate">\cdist</span>) node[below right]{<span class="org-font-latex-math">$d</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_y</span></span><span class="org-font-latex-math">$</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="org610cf51" class="figure">
<p><img src="figs/sf_arch_class.png" alt="sf_arch_class.png" />
</p>
<p><span class="figure-number">Figure 3: </span>Equivalent configuration - classical Feedback (<a href="./figs/sf_arch_class.png">png</a>, <a href="./figs/sf_arch_class.pdf">pdf</a>, <a href="./figs/sf_arch_class.tex">tex</a>).</p>
</div>
</div>
</div>
<div id="outline-container-orgcc18c86" class="outline-2">
<h2 id="orgcc18c86"><span class="section-number-2">5</span> Equivalent configuration - classical Feedback with pre-filter</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">\tikzset</span>{block/.default={0.8cm}{0.6cm}}
<span class="org-font-latex-sedate">\tikzset</span>{addb/.append style={scale=0.7}}
<span class="org-font-latex-sedate">\tikzset</span>{node distance=0.6}
<span class="org-font-latex-sedate">\def\cdist</span>{0.7}
<span class="org-font-latex-sedate">\definecolor</span>{T}{rgb}{0.230, 0.299, 0.754}<span class="org-comment">%</span>
<span class="org-font-latex-sedate">\definecolor</span>{S}{rgb}{0.706, 0.016, 0.150}<span class="org-comment">%</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">\node</span>[addb={+}{}{}{}{-}] (addfb) at (0, 0){};
<span class="org-font-latex-sedate">\node</span>[block, left=of addfb] (Kr){<span class="org-font-latex-math">$K</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_r</span></span><span class="org-font-latex-math">$</span>};
<span class="org-font-latex-sedate">\node</span>[block, right=of addfb] (K){<span class="org-font-latex-math">$K$</span>};
<span class="org-font-latex-sedate">\node</span>[block, right=of K] (G){<span class="org-font-latex-math">$G</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">^\prime</span></span><span class="org-font-latex-math">$</span>};
<span class="org-font-latex-sedate">\node</span>[addb={+}{}{}{}{}, right=of G] (adddy){};
<span class="org-font-latex-sedate">\node</span>[addb={+}{}{}{}{}, below right=0.7 and 0.3 of adddy] (addn) {};
<span class="org-font-latex-sedate">\node</span>[block, left=of addn] (Hh) {<span class="org-font-latex-math">$H</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_H</span></span><span class="org-font-latex-math">$</span>};
<span class="org-font-latex-sedate">\draw</span>[-&gt;] (addfb.east) -- (K.west) node[above left]{};
<span class="org-font-latex-sedate">\draw</span>[-&gt;] (K.east) -- (G.west) node[above left]{<span class="org-font-latex-math">$u$</span>};
<span class="org-font-latex-sedate">\draw</span>[-&gt;] (G.east) -- (adddy.west);
<span class="org-font-latex-sedate">\draw</span>[&lt;-] (addn.east) -- ++(<span class="org-font-latex-sedate">\cdist</span>, 0) coordinate[](endpos) node[above left]{<span class="org-font-latex-math">$n$</span>};
<span class="org-font-latex-sedate">\draw</span>[-&gt;] (G-|addn)node[branch]{} -- (addn.north);
<span class="org-font-latex-sedate">\draw</span>[-&gt;] (adddy.east) -- (G-|endpos) node[above left]{<span class="org-font-latex-math">$y$</span>};
<span class="org-font-latex-sedate">\draw</span>[&lt;-] (Kr.west) -- ++(-<span class="org-font-latex-sedate">\cdist</span>, 0) node[above right]{<span class="org-font-latex-math">$r$</span>};
<span class="org-font-latex-sedate">\draw</span>[-&gt;] (Kr.east) -- (addfb.west);
<span class="org-font-latex-sedate">\draw</span>[-&gt;] (addn.west) -- (Hh.east);
<span class="org-font-latex-sedate">\draw</span>[-&gt;] (Hh.west) -| (addfb.south);
<span class="org-font-latex-sedate">\draw</span>[&lt;-] (adddy.north) -- ++(0, <span class="org-font-latex-sedate">\cdist</span>) node[below right]{<span class="org-font-latex-math">$d</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_y</span></span><span class="org-font-latex-math">$</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="org4217e6e" class="figure">
<p><img src="figs/sf_arch_class_prefilter.png" alt="sf_arch_class_prefilter.png" />
</p>
<p><span class="figure-number">Figure 4: </span>Equivalent configuration - classical Feedback with pre-filter (<a href="./figs/sf_arch_class_prefilter.png">png</a>, <a href="./figs/sf_arch_class_prefilter.pdf">pdf</a>, <a href="./figs/sf_arch_class_prefilter.tex">tex</a>).</p>
</div>
</div>
</div>
<div id="outline-container-org0dcfd4e" class="outline-2">
<h2 id="org0dcfd4e"><span class="section-number-2">6</span> Equivalent configuration - bis</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">\tikzset</span>{block/.default={0.8cm}{0.6cm}}
<span class="org-font-latex-sedate">\tikzset</span>{addb/.append style={scale=0.7}}
<span class="org-font-latex-sedate">\tikzset</span>{node distance=0.6}
<span class="org-font-latex-sedate">\def\cdist</span>{0.7}
<span class="org-font-latex-sedate">\definecolor</span>{T}{rgb}{0.230, 0.299, 0.754}<span class="org-comment">%</span>
<span class="org-font-latex-sedate">\definecolor</span>{S}{rgb}{0.706, 0.016, 0.150}<span class="org-comment">%</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">\node</span>[addb={+}{}{}{}{-}] (addfb) at (0, 0){};
<span class="org-font-latex-sedate">\node</span>[block, left=of addfb] (Kr){<span class="org-font-latex-math">${H</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_H</span></span><span class="org-font-latex-math">}</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">^{-1}</span></span><span class="org-font-latex-math">$</span>};
<span class="org-font-latex-sedate">\node</span>[block, right=of addfb] (K){<span class="org-font-latex-math">$G</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">^{-1}</span></span><span class="org-font-latex-math"> {H</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_L</span></span><span class="org-font-latex-math">}</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">^{-1}</span></span><span class="org-font-latex-math"> H</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_H</span></span><span class="org-font-latex-math">$</span>};
<span class="org-font-latex-sedate">\node</span>[block, right=of K] (G){<span class="org-font-latex-math">$G</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">^\prime</span></span><span class="org-font-latex-math">$</span>};
<span class="org-font-latex-sedate">\node</span>[addb={+}{}{}{}{}, right=of G] (adddy){};
<span class="org-font-latex-sedate">\node</span>[addb={+}{}{}{}{}, below right=0.7 and 0.3 of adddy] (addn) {};
<span class="org-font-latex-sedate">\draw</span>[-&gt;] (addfb.east) -- (K.west) node[above left]{};
<span class="org-font-latex-sedate">\draw</span>[-&gt;] (K.east) -- (G.west) node[above left]{<span class="org-font-latex-math">$u$</span>};
<span class="org-font-latex-sedate">\draw</span>[-&gt;] (G.east) -- (adddy.west);
<span class="org-font-latex-sedate">\draw</span>[&lt;-] (addn.east) -- ++(<span class="org-font-latex-sedate">\cdist</span>, 0) coordinate[](endpos) node[above left]{<span class="org-font-latex-math">$n$</span>};
<span class="org-font-latex-sedate">\draw</span>[-&gt;] (G-|addn)node[branch]{} -- (addn.north);
<span class="org-font-latex-sedate">\draw</span>[-&gt;] (adddy.east) -- (G-|endpos) node[above left]{<span class="org-font-latex-math">$y$</span>};
<span class="org-font-latex-sedate">\draw</span>[&lt;-] (Kr.west) -- ++(-<span class="org-font-latex-sedate">\cdist</span>, 0) node[above right]{<span class="org-font-latex-math">$r$</span>};
<span class="org-font-latex-sedate">\draw</span>[-&gt;] (Kr.east) -- (addfb.west);
<span class="org-font-latex-sedate">\draw</span>[-&gt;] (addn.west) -| (addfb.south);
<span class="org-font-latex-sedate">\draw</span>[&lt;-] (adddy.north) -- ++(0, <span class="org-font-latex-sedate">\cdist</span>) node[below right]{<span class="org-font-latex-math">$d</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_y</span></span><span class="org-font-latex-math">$</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="orge78aa8f" class="figure">
<p><img src="figs/sf_arch_class_bis.png" alt="sf_arch_class_bis.png" />
</p>
<p><span class="figure-number">Figure 5: </span>Equivalent configuration - bis (<a href="./figs/sf_arch_class_bis.png">png</a>, <a href="./figs/sf_arch_class_bis.pdf">pdf</a>, <a href="./figs/sf_arch_class_bis.tex">tex</a>).</p>
</div>
</div>
</div>
<div id="outline-container-org848380b" class="outline-2">
<h2 id="org848380b"><span class="section-number-2">7</span> H-Infinity - Complementary filters - Generalized plant</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">\tikzset</span>{block/.default={0.8cm}{0.6cm}}
<span class="org-font-latex-sedate">\tikzset</span>{addb/.append style={scale=0.7}}
<span class="org-font-latex-sedate">\tikzset</span>{node distance=0.6}
<span class="org-font-latex-sedate">\def\cdist</span>{0.7}
<span class="org-font-latex-sedate">\definecolor</span>{T}{rgb}{0.230, 0.299, 0.754}<span class="org-comment">%</span>
<span class="org-font-latex-sedate">\definecolor</span>{S}{rgb}{0.706, 0.016, 0.150}<span class="org-comment">%</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">\node</span>[block={4.0cm}{3.0cm}, dashed] (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-sedate">\coordinate</span>[] (inputw) at (<span class="org-font-latex-math">$(P.south west)!0.8!(P.north west) + (-</span><span class="org-font-latex-sedate"><span class="org-font-latex-math">\cdist</span></span><span class="org-font-latex-math">, 0)$</span>);
<span class="org-font-latex-sedate">\coordinate</span>[] (inputu) at (<span class="org-font-latex-math">$(P.south west)!0.4!(P.north west) + (-</span><span class="org-font-latex-sedate"><span class="org-font-latex-math">\cdist</span></span><span class="org-font-latex-math">, 0)$</span>);
<span class="org-font-latex-sedate">\coordinate</span>[] (outputh) at (<span class="org-font-latex-math">$(P.south east)!0.8!(P.north east) + ( </span><span class="org-font-latex-sedate"><span class="org-font-latex-math">\cdist</span></span><span class="org-font-latex-math">, 0)$</span>);
<span class="org-font-latex-sedate">\coordinate</span>[] (outputl) at (<span class="org-font-latex-math">$(P.south east)!0.4!(P.north east) + ( </span><span class="org-font-latex-sedate"><span class="org-font-latex-math">\cdist</span></span><span class="org-font-latex-math">, 0)$</span>);
<span class="org-font-latex-sedate">\coordinate</span>[] (outputv) at (<span class="org-font-latex-math">$(P.south east)!0.1!(P.north east) + ( </span><span class="org-font-latex-sedate"><span class="org-font-latex-math">\cdist</span></span><span class="org-font-latex-math">, 0)$</span>);
<span class="org-font-latex-sedate">\node</span>[block, left=2*<span class="org-font-latex-sedate">\cdist</span> of outputl] (WL){<span class="org-font-latex-math">$w</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_L</span></span><span class="org-font-latex-math">$</span>};
<span class="org-font-latex-sedate">\node</span>[block, left=2*<span class="org-font-latex-sedate">\cdist</span> of outputh] (WH){<span class="org-font-latex-math">$w</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_H</span></span><span class="org-font-latex-math">$</span>};
<span class="org-font-latex-sedate">\node</span>[addb={+}{}{}{}{-}, left=of WH] (sub) {};
<span class="org-font-latex-sedate">\draw</span>[-&gt;] (inputw) node[above right]{<span class="org-font-latex-math">$w$</span>} -- (sub.west);
<span class="org-font-latex-sedate">\draw</span>[-&gt;] (inputu) node[above right]{<span class="org-font-latex-math">$u$</span>} -- (WL.west);
<span class="org-font-latex-sedate">\draw</span>[-&gt;] (inputu-|sub) node[branch]{} -- (sub.south);
<span class="org-font-latex-sedate">\draw</span>[-&gt;] (sub.east) -- (WH.west);
<span class="org-font-latex-sedate">\draw</span>[-&gt;] (<span class="org-font-latex-math">$(inputw)+(2*</span><span class="org-font-latex-sedate"><span class="org-font-latex-math">\cdist</span></span><span class="org-font-latex-math">, 0)$</span>) node[branch]{} |- (outputv) node[above left]{<span class="org-font-latex-math">$v$</span>};
<span class="org-font-latex-sedate">\draw</span>[-&gt;] (WH.east) -- (outputh)node[above left]{<span class="org-font-latex-math">$z</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_H</span></span><span class="org-font-latex-math">$</span>};
<span class="org-font-latex-sedate">\draw</span>[-&gt;] (WL.east) -- (outputl)node[above left]{<span class="org-font-latex-math">$z</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_L</span></span><span class="org-font-latex-math">$</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="org461659b" class="figure">
<p><img src="figs/sf_hinf_filters_plant_b.png" alt="sf_hinf_filters_plant_b.png" />
</p>
<p><span class="figure-number">Figure 6: </span>H-Infinity - Complementary filters - Generalized plant (<a href="./figs/sf_hinf_filters_plant_b.png">png</a>, <a href="./figs/sf_hinf_filters_plant_b.pdf">pdf</a>, <a href="./figs/sf_hinf_filters_plant_b.tex">tex</a>).</p>
</div>
</div>
</div>
<div id="outline-container-org46ea807" class="outline-2">
<h2 id="org46ea807"><span class="section-number-2">8</span> H-Infinity - Complementary filters</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">\tikzset</span>{block/.default={0.8cm}{0.6cm}}
<span class="org-font-latex-sedate">\tikzset</span>{addb/.append style={scale=0.7}}
<span class="org-font-latex-sedate">\tikzset</span>{node distance=0.6}
<span class="org-font-latex-sedate">\def\cdist</span>{0.7}
<span class="org-font-latex-sedate">\definecolor</span>{T}{rgb}{0.230, 0.299, 0.754}<span class="org-comment">%</span>
<span class="org-font-latex-sedate">\definecolor</span>{S}{rgb}{0.706, 0.016, 0.150}<span class="org-comment">%</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">\node</span>[block={4.0cm}{3.0cm}, dashed] (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-sedate">\coordinate</span>[] (inputw) at (<span class="org-font-latex-math">$(P.south west)!0.8!(P.north west) + (-</span><span class="org-font-latex-sedate"><span class="org-font-latex-math">\cdist</span></span><span class="org-font-latex-math">, 0)$</span>);
<span class="org-font-latex-sedate">\coordinate</span>[] (inputu) at (<span class="org-font-latex-math">$(P.south west)!0.4!(P.north west) + (-</span><span class="org-font-latex-sedate"><span class="org-font-latex-math">\cdist</span></span><span class="org-font-latex-math">, 0)$</span>);
<span class="org-font-latex-sedate">\coordinate</span>[] (outputh) at (<span class="org-font-latex-math">$(P.south east)!0.8!(P.north east) + ( </span><span class="org-font-latex-sedate"><span class="org-font-latex-math">\cdist</span></span><span class="org-font-latex-math">, 0)$</span>);
<span class="org-font-latex-sedate">\coordinate</span>[] (outputl) at (<span class="org-font-latex-math">$(P.south east)!0.4!(P.north east) + ( </span><span class="org-font-latex-sedate"><span class="org-font-latex-math">\cdist</span></span><span class="org-font-latex-math">, 0)$</span>);
<span class="org-font-latex-sedate">\coordinate</span>[] (outputv) at (<span class="org-font-latex-math">$(P.south east)!0.1!(P.north east) + ( </span><span class="org-font-latex-sedate"><span class="org-font-latex-math">\cdist</span></span><span class="org-font-latex-math">, 0)$</span>);
<span class="org-font-latex-sedate">\node</span>[block, left=2*<span class="org-font-latex-sedate">\cdist</span> of outputl] (WL){<span class="org-font-latex-math">$w</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_L</span></span><span class="org-font-latex-math">$</span>};
<span class="org-font-latex-sedate">\node</span>[block, left=2*<span class="org-font-latex-sedate">\cdist</span> of outputh] (WH){<span class="org-font-latex-math">$w</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_H</span></span><span class="org-font-latex-math">$</span>};
<span class="org-font-latex-sedate">\node</span>[addb={+}{}{}{}{-}, left=of WH] (sub) {};
<span class="org-font-latex-sedate">\node</span>[block, below=<span class="org-font-latex-sedate">\cdist</span> of P] (HL) {<span class="org-font-latex-math">$H</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_L</span></span><span class="org-font-latex-math">$</span>};
<span class="org-font-latex-sedate">\draw</span>[-&gt;] (inputw) node[above right]{<span class="org-font-latex-math">$w$</span>} -- (sub.west);
<span class="org-font-latex-sedate">\draw</span>[-&gt;] (HL.west) -| (<span class="org-font-latex-math">$(inputu)+(0.5*</span><span class="org-font-latex-sedate"><span class="org-font-latex-math">\cdist</span></span><span class="org-font-latex-math">, 0)$</span>) -- (WL.west);
<span class="org-font-latex-sedate">\draw</span>[-&gt;] (inputu-|sub) node[branch]{} -- (sub.south);
<span class="org-font-latex-sedate">\draw</span>[-&gt;] (sub.east) -- (WH.west);
<span class="org-font-latex-sedate">\draw</span>[-&gt;] (<span class="org-font-latex-math">$(inputw)+(2*</span><span class="org-font-latex-sedate"><span class="org-font-latex-math">\cdist</span></span><span class="org-font-latex-math">, 0)$</span>) node[branch]{} |- (<span class="org-font-latex-math">$(outputv)+(-0.5*</span><span class="org-font-latex-sedate"><span class="org-font-latex-math">\cdist</span></span><span class="org-font-latex-math">, 0)$</span>) |- (HL.east);
<span class="org-font-latex-sedate">\draw</span>[-&gt;] (WH.east) -- (outputh)node[above left]{<span class="org-font-latex-math">$z</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_H</span></span><span class="org-font-latex-math">$</span>};
<span class="org-font-latex-sedate">\draw</span>[-&gt;] (WL.east) -- (outputl)node[above left]{<span class="org-font-latex-math">$z</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_L</span></span><span class="org-font-latex-math">$</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="orgb6ab595" class="figure">
<p><img src="figs/sf_hinf_filters_b.png" alt="sf_hinf_filters_b.png" />
</p>
<p><span class="figure-number">Figure 7: </span>H-Infinity - Complementary filters (<a href="./figs/sf_hinf_filters_b.png">png</a>, <a href="./figs/sf_hinf_filters_b.pdf">pdf</a>, <a href="./figs/sf_hinf_filters_b.tex">tex</a>).</p>
</div>
</div>
</div>
<div id="outline-container-orgc79b93a" class="outline-2">
<h2 id="orgc79b93a"><span class="section-number-2">9</span> Input Multiplicative Uncertainty</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">\tikzset</span>{block/.default={0.8cm}{0.6cm}}
<span class="org-font-latex-sedate">\tikzset</span>{addb/.append style={scale=0.7}}
<span class="org-font-latex-sedate">\tikzset</span>{node distance=0.6}
<span class="org-font-latex-sedate">\def\cdist</span>{0.7}
<span class="org-font-latex-sedate">\definecolor</span>{T}{rgb}{0.230, 0.299, 0.754}<span class="org-comment">%</span>
<span class="org-font-latex-sedate">\definecolor</span>{S}{rgb}{0.706, 0.016, 0.150}<span class="org-comment">%</span>
<span class="org-font-latex-sedate"><span class="org-keyword">\begin</span></span>{<span class="org-function-name">tikzpicture</span>}
<span class="org-comment">% Blocs</span>
<span class="org-font-latex-sedate">\node</span>[block] (G) {<span class="org-font-latex-math">$G$</span>};
<span class="org-font-latex-sedate">\node</span>[addb, left=of G] (addi) {};
<span class="org-font-latex-sedate">\node</span>[block, above left=0.5 and 0.2 of addi] (deltai) {<span class="org-font-latex-math">$</span><span class="org-font-latex-sedate"><span class="org-font-latex-math">\Delta</span></span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_I</span></span><span class="org-font-latex-math">$</span>};
<span class="org-font-latex-sedate">\node</span>[block, left=of deltai] (wi) {<span class="org-font-latex-math">$w</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_I</span></span><span class="org-font-latex-math">$</span>};
<span class="org-comment">% Connections and labels</span>
<span class="org-font-latex-sedate">\draw</span>[-&gt;] (<span class="org-font-latex-math">$(G.west)+(-4, 0)$</span>)coordinate[](start) node[branch]{} |- (wi.west);
<span class="org-font-latex-sedate">\draw</span>[-&gt;] (<span class="org-font-latex-math">$(start)+(-</span><span class="org-font-latex-sedate"><span class="org-font-latex-math">\cdist</span></span><span class="org-font-latex-math">, 0)$</span>) -- (addi.west);
<span class="org-font-latex-sedate">\draw</span>[-&gt;] (wi.east) -- (deltai.west);
<span class="org-font-latex-sedate">\draw</span>[-&gt;] (deltai.east) -| (addi.north);
<span class="org-font-latex-sedate">\draw</span>[-&gt;] (addi.east) -- (G.west);
<span class="org-font-latex-sedate">\draw</span>[-&gt;] (G.east) -- ++(0.8, 0);
<span class="org-font-latex-sedate">\node</span>[fit={(start|-wi.north) (G.south east)}, inner sep=10pt, draw, dashed, color=gray, label={<span class="org-font-latex-math">$G</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_p</span></span><span class="org-font-latex-math">$</span>}] (Gp) {};
<span class="org-font-latex-sedate"><span class="org-keyword">\end</span></span>{<span class="org-function-name">tikzpicture</span>}
</pre>
</div>
<div id="org8e1de7c" class="figure">
<p><img src="figs/input_uncertainty.png" alt="input_uncertainty.png" />
</p>
<p><span class="figure-number">Figure 8: </span>Input Multiplicative Uncertainty (<a href="./figs/input_uncertainty.png">png</a>, <a href="./figs/input_uncertainty.pdf">pdf</a>, <a href="./figs/input_uncertainty.tex">tex</a>).</p>
</div>
</div>
</div>
<div id="outline-container-org594469f" class="outline-2">
<h2 id="org594469f"><span class="section-number-2">10</span> One mass</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">\tikzset</span>{block/.default={0.8cm}{0.6cm}}
<span class="org-font-latex-sedate">\tikzset</span>{addb/.append style={scale=0.7}}
<span class="org-font-latex-sedate">\tikzset</span>{node distance=0.6}
<span class="org-font-latex-sedate">\def\cdist</span>{0.7}
<span class="org-font-latex-sedate">\definecolor</span>{T}{rgb}{0.230, 0.299, 0.754}<span class="org-comment">%</span>
<span class="org-font-latex-sedate">\definecolor</span>{S}{rgb}{0.706, 0.016, 0.150}<span class="org-comment">%</span>
<span class="org-font-latex-sedate"><span class="org-keyword">\begin</span></span>{<span class="org-function-name">tikzpicture</span>}
<span class="org-comment">% ====================</span>
<span class="org-comment">% Parameters</span>
<span class="org-comment">% ====================</span>
<span class="org-font-latex-sedate">\def\massw</span>{2.2} <span class="org-comment">% Width of the masses</span>
<span class="org-font-latex-sedate">\def\massh</span>{0.8} <span class="org-comment">% Height of the masses</span>
<span class="org-font-latex-sedate">\def\spaceh</span>{1.2} <span class="org-comment">% Height of the springs/dampers</span>
<span class="org-font-latex-sedate">\def\dispw</span>{0.3} <span class="org-comment">% Width of the dashed line for the displacement</span>
<span class="org-font-latex-sedate">\def\disph</span>{0.5} <span class="org-comment">% Height of the arrow for the displacements</span>
<span class="org-font-latex-sedate">\def\bracs</span>{0.05} <span class="org-comment">% Brace spacing vertically</span>
<span class="org-font-latex-sedate">\def\brach</span>{-10pt} <span class="org-comment">% Brace shift horizontaly</span>
<span class="org-comment">% ====================</span>
<span class="org-comment">% ====================</span>
<span class="org-comment">% Ground</span>
<span class="org-comment">% ====================</span>
<span class="org-font-latex-sedate">\draw</span> (-0.5*<span class="org-font-latex-sedate">\massw</span>, 0) -- (0.5*<span class="org-font-latex-sedate">\massw</span>, 0);
<span class="org-font-latex-sedate">\draw</span>[dashed] (0.5*<span class="org-font-latex-sedate">\massw</span>, 0) -- ++(<span class="org-font-latex-sedate">\dispw</span>, 0);
<span class="org-font-latex-sedate">\draw</span>[-&gt;] (0.5*<span class="org-font-latex-sedate">\massw+0</span>.5*<span class="org-font-latex-sedate">\dispw</span>, 0) -- ++(0, <span class="org-font-latex-sedate">\disph</span>) node[right]{<span class="org-font-latex-math">$w$</span>};
<span class="org-comment">% ====================</span>
<span class="org-font-latex-sedate"><span class="org-keyword">\begin</span></span>{<span class="org-function-name">scope</span>}[shift={(0, 0)}]
<span class="org-comment">% Mass</span>
<span class="org-font-latex-sedate">\draw</span>[fill=white] (-0.5*<span class="org-font-latex-sedate">\massw</span>, <span class="org-font-latex-sedate">\spaceh</span>) rectangle (0.5*<span class="org-font-latex-sedate">\massw</span>, <span class="org-font-latex-sedate">\spaceh+\massh</span>) node[pos=0.5]{<span class="org-font-latex-math">$m$</span>};
<span class="org-comment">% Spring, Damper, and Actuator</span>
<span class="org-font-latex-sedate">\draw</span>[spring] (-0.4*<span class="org-font-latex-sedate">\massw</span>, 0) -- (-0.4*<span class="org-font-latex-sedate">\massw</span>, <span class="org-font-latex-sedate">\spaceh</span>) node[midway, left=0.1]{<span class="org-font-latex-math">$k$</span>};
<span class="org-font-latex-sedate">\draw</span>[damper] (0, 0) -- ( 0, <span class="org-font-latex-sedate">\spaceh</span>) node[midway, left=0.2]{<span class="org-font-latex-math">$c$</span>};
<span class="org-font-latex-sedate">\draw</span>[actuator] ( 0.4*<span class="org-font-latex-sedate">\massw</span>, 0) -- ( 0.4*<span class="org-font-latex-sedate">\massw</span>, <span class="org-font-latex-sedate">\spaceh</span>) node[midway, left=0.1](F){<span class="org-font-latex-math">$F$</span>};
<span class="org-comment">% Displacements</span>
<span class="org-font-latex-sedate">\draw</span>[dashed] (0.5*<span class="org-font-latex-sedate">\massw</span>, <span class="org-font-latex-sedate">\spaceh</span>) -- ++(<span class="org-font-latex-sedate">\dispw</span>, 0);
<span class="org-font-latex-sedate">\draw</span>[-&gt;] (0.5*<span class="org-font-latex-sedate">\massw+0</span>.5*<span class="org-font-latex-sedate">\dispw</span>, <span class="org-font-latex-sedate">\spaceh</span>) -- ++(0, <span class="org-font-latex-sedate">\disph</span>) node[right]{<span class="org-font-latex-math">$x$</span>};
<span class="org-comment">% Legend</span>
<span class="org-comment">% </span><span class="org-font-latex-sedate"><span class="org-comment">\draw</span></span><span class="org-comment">[decorate, decoration={brace, amplitude=8pt}, xshift=</span><span class="org-font-latex-sedate"><span class="org-comment">\brach</span></span><span class="org-comment">] %</span>
<span class="org-comment">% (-0.5*</span><span class="org-font-latex-sedate"><span class="org-comment">\massw</span></span><span class="org-comment">, </span><span class="org-font-latex-sedate"><span class="org-comment">\bracs</span></span><span class="org-comment">) -- (-0.5*</span><span class="org-font-latex-sedate"><span class="org-comment">\massw</span></span><span class="org-comment">, </span><span class="org-font-latex-sedate"><span class="org-comment">\spaceh+\massh-\bracs</span></span><span class="org-comment">) %</span>
<span class="org-comment">% node[midway,rotate=90,anchor=south,yshift=10pt]{};</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="org870d8a8" class="figure">
<p><img src="figs/mech_sys_alone.png" alt="mech_sys_alone.png" />
</p>
<p><span class="figure-number">Figure 9: </span>One mass (<a href="./figs/mech_sys_alone.png">png</a>, <a href="./figs/mech_sys_alone.pdf">pdf</a>, <a href="./figs/mech_sys_alone.tex">tex</a>).</p>
</div>
</div>
</div>
<div id="outline-container-orgd97f866" class="outline-2">
<h2 id="orgd97f866"><span class="section-number-2">11</span> One mass - Control</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">\tikzset</span>{block/.default={0.8cm}{0.6cm}}
<span class="org-font-latex-sedate">\tikzset</span>{addb/.append style={scale=0.7}}
<span class="org-font-latex-sedate">\tikzset</span>{node distance=0.6}
<span class="org-font-latex-sedate">\def\cdist</span>{0.7}
<span class="org-font-latex-sedate">\definecolor</span>{T}{rgb}{0.230, 0.299, 0.754}<span class="org-comment">%</span>
<span class="org-font-latex-sedate">\definecolor</span>{S}{rgb}{0.706, 0.016, 0.150}<span class="org-comment">%</span>
<span class="org-font-latex-sedate"><span class="org-keyword">\begin</span></span>{<span class="org-function-name">tikzpicture</span>}
<span class="org-comment">% ====================</span>
<span class="org-comment">% Parameters</span>
<span class="org-comment">% ====================</span>
<span class="org-font-latex-sedate">\def\massw</span>{2.2} <span class="org-comment">% Width of the masses</span>
<span class="org-font-latex-sedate">\def\massh</span>{0.8} <span class="org-comment">% Height of the masses</span>
<span class="org-font-latex-sedate">\def\spaceh</span>{1.2} <span class="org-comment">% Height of the springs/dampers</span>
<span class="org-font-latex-sedate">\def\dispw</span>{0.3} <span class="org-comment">% Width of the dashed line for the displacement</span>
<span class="org-font-latex-sedate">\def\disph</span>{0.5} <span class="org-comment">% Height of the arrow for the displacements</span>
<span class="org-font-latex-sedate">\def\bracs</span>{0.05} <span class="org-comment">% Brace spacing vertically</span>
<span class="org-font-latex-sedate">\def\brach</span>{-10pt} <span class="org-comment">% Brace shift horizontaly</span>
<span class="org-comment">% ====================</span>
<span class="org-comment">% ====================</span>
<span class="org-comment">% Ground</span>
<span class="org-comment">% ====================</span>
<span class="org-font-latex-sedate">\draw</span> (-0.5*<span class="org-font-latex-sedate">\massw</span>, 0) -- (0.5*<span class="org-font-latex-sedate">\massw</span>, 0);
<span class="org-font-latex-sedate">\draw</span>[dashed] (0.5*<span class="org-font-latex-sedate">\massw</span>, 0) -- ++(<span class="org-font-latex-sedate">\dispw</span>, 0);
<span class="org-font-latex-sedate">\draw</span>[-&gt;] (0.5*<span class="org-font-latex-sedate">\massw+0</span>.5*<span class="org-font-latex-sedate">\dispw</span>, 0) -- ++(0, <span class="org-font-latex-sedate">\disph</span>) node[below right]{<span class="org-font-latex-math">$w$</span>};
<span class="org-comment">% ====================</span>
<span class="org-font-latex-sedate"><span class="org-keyword">\begin</span></span>{<span class="org-function-name">scope</span>}[shift={(0, 0)}]
<span class="org-comment">% Mass</span>
<span class="org-font-latex-sedate">\draw</span>[fill=white] (-0.5*<span class="org-font-latex-sedate">\massw</span>, <span class="org-font-latex-sedate">\spaceh</span>) rectangle (0.5*<span class="org-font-latex-sedate">\massw</span>, <span class="org-font-latex-sedate">\spaceh+\massh</span>) node[pos=0.5]{<span class="org-font-latex-math">$m$</span>};
<span class="org-comment">% Spring, Damper, and Actuator</span>
<span class="org-font-latex-sedate">\draw</span>[spring] (-0.4*<span class="org-font-latex-sedate">\massw</span>, 0) -- (-0.4*<span class="org-font-latex-sedate">\massw</span>, <span class="org-font-latex-sedate">\spaceh</span>) node[midway, left=0.1]{<span class="org-font-latex-math">$k$</span>};
<span class="org-font-latex-sedate">\draw</span>[damper] (0, 0) -- ( 0, <span class="org-font-latex-sedate">\spaceh</span>) node[midway, left=0.2]{<span class="org-font-latex-math">$c$</span>};
<span class="org-font-latex-sedate">\draw</span>[actuator] ( 0.4*<span class="org-font-latex-sedate">\massw</span>, 0) -- ( 0.4*<span class="org-font-latex-sedate">\massw</span>, <span class="org-font-latex-sedate">\spaceh</span>) coordinate[midway, right=0.15](F);
<span class="org-comment">% Displacements</span>
<span class="org-font-latex-sedate">\draw</span>[dashed] (0.5*<span class="org-font-latex-sedate">\massw</span>, <span class="org-font-latex-sedate">\spaceh</span>) -- ++(<span class="org-font-latex-sedate">\dispw</span>, 0);
<span class="org-font-latex-sedate">\draw</span>[-&gt;] (0.5*<span class="org-font-latex-sedate">\massw+0</span>.5*<span class="org-font-latex-sedate">\dispw</span>, <span class="org-font-latex-sedate">\spaceh</span>) -- ++(0, <span class="org-font-latex-sedate">\disph</span>) node[right](x){<span class="org-font-latex-math">$x$</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">\node</span>[block, right=1 of F] (Kfb) {<span class="org-font-latex-math">$K$</span>};
<span class="org-font-latex-sedate">\node</span>[addb={+}{}{-}{}{}, right=2*<span class="org-font-latex-sedate">\cdist</span> of Kfb] (add) {};
<span class="org-font-latex-sedate">\node</span>[addb] (addn) at (x-|Kfb) {};
<span class="org-font-latex-sedate">\node</span>[block, right=of addn] (Hl) {<span class="org-font-latex-math">$H</span><span class="org-font-latex-math"><span class="org-font-latex-script-char">_L</span></span><span class="org-font-latex-math">$</span>};
<span class="org-font-latex-sedate">\draw</span>[-&gt;] (x) -- (addn.west);
<span class="org-font-latex-sedate">\draw</span>[-&gt;] (addn.east) -- (Hl.west);
<span class="org-font-latex-sedate">\draw</span>[-&gt;] (Hl.east) -| (add.north);
<span class="org-font-latex-sedate">\draw</span>[-&gt;] (add.west) -- (Kfb.east);
<span class="org-font-latex-sedate">\draw</span>[-&gt;] (Kfb.west) -- (F) node[above right]{<span class="org-font-latex-math">$F$</span>};
<span class="org-font-latex-sedate">\draw</span>[&lt;-] (addn.north) -- ++(0,<span class="org-font-latex-sedate">\cdist</span>) node[below right]{<span class="org-font-latex-math">$n$</span>};
<span class="org-font-latex-sedate">\draw</span>[&lt;-] (add.east) -- ++(<span class="org-font-latex-sedate">\cdist</span>,0) node[above left]{<span class="org-font-latex-math">$r$</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="orga4cd5fa" class="figure">
<p><img src="figs/mech_sys_alone_ctrl.png" alt="mech_sys_alone_ctrl.png" />
</p>
<p><span class="figure-number">Figure 10: </span>One mass - Control (<a href="./figs/mech_sys_alone_ctrl.png">png</a>, <a href="./figs/mech_sys_alone_ctrl.pdf">pdf</a>, <a href="./figs/mech_sys_alone_ctrl.tex">tex</a>).</p>
</div>
</div>
</div>
<div id="outline-container-org5ceffca" class="outline-2">
<h2 id="org5ceffca"><span class="section-number-2">12</span> Piezoelectric Actuator</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">\tikzset</span>{block/.default={0.8cm}{0.6cm}}
<span class="org-font-latex-sedate">\tikzset</span>{addb/.append style={scale=0.7}}
<span class="org-font-latex-sedate">\tikzset</span>{node distance=0.6}
<span class="org-font-latex-sedate">\def\cdist</span>{0.7}
<span class="org-font-latex-sedate">\definecolor</span>{T}{rgb}{0.230, 0.299, 0.754}<span class="org-comment">%</span>
<span class="org-font-latex-sedate">\definecolor</span>{S}{rgb}{0.706, 0.016, 0.150}<span class="org-comment">%</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">\node</span>[piezo={2}{3}{10}] (piezo) at (0, 0){};
<span class="org-font-latex-sedate">\node</span>[draw, fill=white, anchor=south, minimum width=3cm, minimum height=1.5cm] (mass) at (<span class="org-font-latex-math">$(piezo.north)+(0, 0.5)$</span>) {Mass};
<span class="org-font-latex-sedate">\draw</span>[] (<span class="org-font-latex-math">$(piezo.south)+(-1.5, -0.5)$</span>) -- ++(3, 0);
<span class="org-font-latex-sedate">\draw</span> (<span class="org-font-latex-math">$0.8*(piezo.north west)+0.2*(piezo.north east)$</span>) -- ++(0, 0.5);
<span class="org-font-latex-sedate">\draw</span> (<span class="org-font-latex-math">$0.2*(piezo.north west)+0.8*(piezo.north east)$</span>) -- ++(0, 0.5);
<span class="org-font-latex-sedate">\draw</span> (<span class="org-font-latex-math">$0.8*(piezo.south west)+0.2*(piezo.south east)$</span>) -- ++(0, -0.5);
<span class="org-font-latex-sedate">\draw</span> (<span class="org-font-latex-math">$0.2*(piezo.south west)+0.8*(piezo.south east)$</span>) -- ++(0, -0.5);
<span class="org-font-latex-sedate"><span class="org-keyword">\end</span></span>{<span class="org-function-name">tikzpicture</span>}
</pre>
</div>
<div id="orga0430e9" class="figure">
<p><img src="figs/1dof_isolation_piezo.png" alt="1dof_isolation_piezo.png" />
</p>
<p><span class="figure-number">Figure 11: </span>Piezoelectric Actuator (<a href="./figs/1dof_isolation_piezo.png">png</a>, <a href="./figs/1dof_isolation_piezo.pdf">pdf</a>, <a href="./figs/1dof_isolation_piezo.tex">tex</a>).</p>
</div>
</div>
</div>
<div id="outline-container-org8aa730c" class="outline-2">
<h2 id="org8aa730c"><span class="section-number-2">13</span> Upper bounds</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">\tikzset</span>{block/.default={0.8cm}{0.6cm}}
<span class="org-font-latex-sedate">\tikzset</span>{addb/.append style={scale=0.7}}
<span class="org-font-latex-sedate">\tikzset</span>{node distance=0.6}
<span class="org-font-latex-sedate">\def\cdist</span>{0.7}
<span class="org-font-latex-sedate">\definecolor</span>{T}{rgb}{0.230, 0.299, 0.754}<span class="org-comment">%</span>
<span class="org-font-latex-sedate">\definecolor</span>{S}{rgb}{0.706, 0.016, 0.150}<span class="org-comment">%</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">\fwidth</span></span>{<span class="org-variable-name">7cm</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">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>
name=axis,
width=<span class="org-font-latex-sedate">\fwidth</span>,
height=<span class="org-font-latex-sedate">\fheight</span>,
at={(0, 0)},
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=0.1,
xmax=1000,
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=0.001,
ymax=10,
yminorticks=true,
ylabel={Magnitude},
axis background/.style={fill=white},
xmajorgrids,
xminorgrids,
ymajorgrids,
yminorgrids
]
<span class="org-font-latex-sedate">\addplot</span> [color=T, line width=1.5pt, forget plot, upperbound]
table[row sep=crcr]{<span class="org-comment">%</span>
0.1 4.7609074288834<span class="org-font-latex-warning">\\</span>
0.457784053837662 4.74112847250108<span class="org-font-latex-warning">\\</span>
0.781435060784454 4.70208061093208<span class="org-font-latex-warning">\\</span>
1.0991097009295 4.64559165613624<span class="org-font-latex-warning">\\</span>
1.42283045721435 4.57144906349989<span class="org-font-latex-warning">\\</span>
1.75891659032773 4.47927616460061<span class="org-font-latex-warning">\\</span>
2.09566239948043 4.37423445322045<span class="org-font-latex-warning">\\</span>
2.45126006203334 4.25274519592358<span class="org-font-latex-warning">\\</span>
2.8408836901833 4.11095719445138<span class="org-font-latex-warning">\\</span>
3.26222200971167 3.95170788088451<span class="org-font-latex-warning">\\</span>
3.71167181947577 3.77948905141281<span class="org-font-latex-warning">\\</span>
4.18428850790158 3.59988757314696<span class="org-font-latex-warning">\\</span>
4.71708469091702 3.40317136463527<span class="org-font-latex-warning">\\</span>
5.31772317785097 3.19223684289429<span class="org-font-latex-warning">\\</span>
5.99484250318941 2.97087790006113<span class="org-font-latex-warning">\\</span>
6.82077673286568 2.72585419958062<span class="org-font-latex-warning">\\</span>
7.76050333513357 2.47949503869403<span class="org-font-latex-warning">\\</span>
8.9114823228402 2.2201435349723<span class="org-font-latex-warning">\\</span>
10.3279473191895 1.95491825546796<span class="org-font-latex-warning">\\</span>
12.0804213467733 1.69160934903556<span class="org-font-latex-warning">\\</span>
14.3932264471941 1.42447068049222<span class="org-font-latex-warning">\\</span>
17.3076553419573 1.17667603248786<span class="org-font-latex-warning">\\</span>
21.1995345753607 0.943491337665097<span class="org-font-latex-warning">\\</span>
26.2070669648385 0.740920673515988<span class="org-font-latex-warning">\\</span>
32.6974974451177 0.569453372601831<span class="org-font-latex-warning">\\</span>
41.1731993116168 0.427919272009563<span class="org-font-latex-warning">\\</span>
51.8459354389291 0.317863515154134<span class="org-font-latex-warning">\\</span>
65.8898955079995 0.230558428152838<span class="org-font-latex-warning">\\</span>
85.2964449974102 0.1611173349527<span class="org-font-latex-warning">\\</span>
114.566872863487 0.105607825522203<span class="org-font-latex-warning">\\</span>
228.74908173557 0.0390036255196556<span class="org-font-latex-warning">\\</span>
272.543253128103 0.0307810979565345<span class="org-font-latex-warning">\\</span>
315.863540826782 0.025500739341156<span class="org-font-latex-warning">\\</span>
359.381366380463 0.0218605610825003<span class="org-font-latex-warning">\\</span>
405.142317111465 0.0191453206052172<span class="org-font-latex-warning">\\</span>
452.538627817017 0.0171066280251345<span class="org-font-latex-warning">\\</span>
505.479682119124 0.0154393411147139<span class="org-font-latex-warning">\\</span>
559.432570616938 0.0141825204910006<span class="org-font-latex-warning">\\</span>
619.144175597784 0.0131421696026314<span class="org-font-latex-warning">\\</span>
685.229159528406 0.012282656024432<span class="org-font-latex-warning">\\</span>
758.367791499719 0.0115737646620205<span class="org-font-latex-warning">\\</span>
847.08682665574 0.0109423524789141<span class="org-font-latex-warning">\\</span>
946.1848194722 0.0104322816237634<span class="org-font-latex-warning">\\</span>
1000 0.0102154679782451<span class="org-font-latex-warning">\\</span>
};
<span class="org-font-latex-sedate">\addplot</span> [color=T, line width=1.5pt, forget plot, upperbound]
table[row sep=crcr]{<span class="org-comment">%</span>
20 0.7<span class="org-font-latex-warning">\\</span>
50 0.7<span class="org-font-latex-warning">\\</span>
};
<span class="org-font-latex-sedate">\addplot</span> [color=T, line width=1.5pt, forget plot, upperbound]
table[row sep=crcr]{<span class="org-comment">%</span>
50 0.1<span class="org-font-latex-warning">\\</span>
500 0.1<span class="org-font-latex-warning">\\</span>
};
<span class="org-font-latex-sedate">\addplot</span> [color=T, line width=1.5pt, forget plot, upperbound]
table[row sep=crcr]{<span class="org-comment">%</span>
500 0.01<span class="org-font-latex-warning">\\</span>
1000 0.01<span class="org-font-latex-warning">\\</span>
};
<span class="org-font-latex-sedate">\addplot</span> [color=S, line width=1.5pt, forget plot, upperbound]
table[row sep=crcr]{<span class="org-comment">%</span>
0.02 1e-05<span class="org-font-latex-warning">\\</span>
2 0.1<span class="org-font-latex-warning">\\</span>
};
<span class="org-font-latex-sedate">\node</span>[draw, fill=white, align=center, anchor=west] (noise) at (20, 0.01){Noise<span class="org-font-latex-warning">\\</span>Attenuation};
<span class="org-font-latex-sedate">\draw</span>[-&gt;] (noise.north) -- (100, 0.1-|noise.north);
<span class="org-font-latex-sedate">\draw</span>[-&gt;] (noise.east) -- (500, 0.01);
<span class="org-font-latex-sedate">\node</span>[draw, fill=white, align=center, anchor=north east] (bandwidth) at (900, 8) {Bandwidth<span class="org-font-latex-warning">\\</span>Limitation};
<span class="org-font-latex-sedate">\draw</span>[-&gt;] (bandwidth.west) -- (20, 0.7);
<span class="org-font-latex-sedate">\node</span>[draw, fill=white, align=center, anchor=south west] (robust) at (0.2, 1) {Robust<span class="org-font-latex-warning">\\</span>Stability};
<span class="org-font-latex-sedate">\draw</span>[-&gt;] (robust.east) -- (8,2|-robust.east);
<span class="org-font-latex-sedate">\node</span>[draw, fill=white, align=center, anchor=south west] (dist) at (0.11, 0.1) {Disturbance<span class="org-font-latex-warning">\\</span>Rejection};
<span class="org-font-latex-sedate">\draw</span>[-&gt;] (dist.south) -- (1, 0.025);
<span class="org-font-latex-sedate"><span class="org-keyword">\end</span></span>{<span class="org-function-name">axis</span>}
<span class="org-comment">% </span><span class="org-font-latex-sedate"><span class="org-comment">\begin</span></span><span class="org-comment">{customlegend}[legend cell align=left, %&lt;= to align cells</span>
<span class="org-comment">% legend entries={ % &lt;= in the following there are the entries</span>
<span class="org-comment">% $S$,</span>
<span class="org-comment">% $T$</span>
<span class="org-comment">% },</span>
<span class="org-comment">% legend style={at={(axis.north east)}, outer sep=5pt, font=</span><span class="org-font-latex-sedate"><span class="org-comment">\footnotesize</span></span><span class="org-comment">}]</span>
<span class="org-comment">% </span><span class="org-font-latex-sedate"><span class="org-comment">\addlegendimage</span></span><span class="org-comment">{upperbound, S}</span>
<span class="org-comment">% </span><span class="org-font-latex-sedate"><span class="org-comment">\addlegendimage</span></span><span class="org-comment">{upperbound, T}</span>
<span class="org-comment">% </span><span class="org-font-latex-sedate"><span class="org-comment">\end</span></span><span class="org-comment">{customlegend}</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="orge0540f4" class="figure">
<p><img src="figs/spec_S_T.png" alt="spec_S_T.png" />
</p>
<p><span class="figure-number">Figure 12: </span>Upper bounds (<a href="./figs/spec_S_T.png">png</a>, <a href="./figs/spec_S_T.pdf">pdf</a>, <a href="./figs/spec_S_T.tex">tex</a>).</p>
</div>
</div>
</div>
<div id="outline-container-org4a37146" class="outline-2">
<h2 id="org4a37146"><span class="section-number-2">14</span> Bode plot of the mechanical system</h2>
<div class="outline-text-2" id="text-14">
<div id="org01b09d8" class="figure">
<p><img src="figs/bode_plot_mech_sys.png" alt="bode_plot_mech_sys.png" />
</p>
<p><span class="figure-number">Figure 13: </span>Bode plot of the mechanical system (<a href="./figs/bode_plot_mech_sys.png">png</a>, <a href="./figs/bode_plot_mech_sys.pdf">pdf</a>, <a href="./figs/bode_plot_mech_sys.tex">tex</a>).</p>
</div>
</div>
</div>
<div id="outline-container-orgfb7564e" class="outline-2">
<h2 id="orgfb7564e"><span class="section-number-2">15</span> Bode plot of the mechanical system - Uncertainty</h2>
<div class="outline-text-2" id="text-15">
<div id="orge95bf07" class="figure">
<p><img src="figs/bode_plot_mech_sys_uncertainty.png" alt="bode_plot_mech_sys_uncertainty.png" />
</p>
<p><span class="figure-number">Figure 14: </span>Bode plot of the mechanical system - Uncertainty (<a href="./figs/bode_plot_mech_sys_uncertainty.png">png</a>, <a href="./figs/bode_plot_mech_sys_uncertainty.pdf">pdf</a>, <a href="./figs/bode_plot_mech_sys_uncertainty.tex">tex</a>).</p>
</div>
</div>
</div>
<div id="outline-container-org6e09d6f" class="outline-2">
<h2 id="org6e09d6f"><span class="section-number-2">16</span> Bode plot of the controller</h2>
<div class="outline-text-2" id="text-16">
<div id="org9257f5a" class="figure">
<p><img src="figs/bode_Kfb.png" alt="bode_Kfb.png" />
</p>
<p><span class="figure-number">Figure 15: </span>Bode plot of the controller (<a href="./figs/bode_Kfb.png">png</a>, <a href="./figs/bode_Kfb.pdf">pdf</a>, <a href="./figs/bode_Kfb.tex">tex</a>).</p>
</div>
</div>
</div>
<div id="outline-container-org0fe4a21" class="outline-2">
<h2 id="org0fe4a21"><span class="section-number-2">17</span> Bode plot of the loop gain</h2>
<div class="outline-text-2" id="text-17">
<div id="orgb9041b0" class="figure">
<p><img src="figs/bode_plot_loop_gain.png" alt="bode_plot_loop_gain.png" />
</p>
<p><span class="figure-number">Figure 16: </span>Bode plot of the loop gain (<a href="./figs/bode_plot_loop_gain.png">png</a>, <a href="./figs/bode_plot_loop_gain.pdf">pdf</a>, <a href="./figs/bode_plot_loop_gain.tex">tex</a>).</p>
</div>
</div>
</div>
<div id="outline-container-orgb069f38" class="outline-2">
<h2 id="orgb069f38"><span class="section-number-2">18</span> Upper bounds with weights</h2>
<div class="outline-text-2" id="text-18">
<div id="orgbcacda1" class="figure">
<p><img src="figs/spec_S_T_weights.png" alt="spec_S_T_weights.png" />
</p>
<p><span class="figure-number">Figure 17: </span>Upper bounds with weights (<a href="./figs/spec_S_T_weights.png">png</a>, <a href="./figs/spec_S_T_weights.pdf">pdf</a>, <a href="./figs/spec_S_T_weights.tex">tex</a>).</p>
</div>
</div>
</div>
<div id="outline-container-orgcb17586" class="outline-2">
<h2 id="orgcb17586"><span class="section-number-2">19</span> Upper bounds with Complementary Filters</h2>
<div class="outline-text-2" id="text-19">
<div id="orgb717352" class="figure">
<p><img src="figs/hinf_filters_result_tikz.png" alt="hinf_filters_result_tikz.png" />
</p>
<p><span class="figure-number">Figure 18: </span>Upper bounds with Complementary Filters (<a href="./figs/hinf_filters_result_tikz.png">png</a>, <a href="./figs/hinf_filters_result_tikz.pdf">pdf</a>, <a href="./figs/hinf_filters_result_tikz.tex">tex</a>).</p>
</div>
</div>
</div>
<div id="outline-container-orgfa30d42" class="outline-2">
<h2 id="orgfa30d42"><span class="section-number-2">20</span> Upper bounds with weights and Complementary Filters</h2>
<div class="outline-text-2" id="text-20">
<div id="orgb42239e" class="figure">
<p><img src="figs/hinf_filters_result_weights.png" alt="hinf_filters_result_weights.png" />
</p>
<p><span class="figure-number">Figure 19: </span>Upper bounds with weights and Complementary Filters (<a href="./figs/hinf_filters_result_weights.png">png</a>, <a href="./figs/hinf_filters_result_weights.pdf">pdf</a>, <a href="./figs/hinf_filters_result_weights.tex">tex</a>).</p>
</div>
</div>
</div>
<div id="outline-container-org0492f6b" class="outline-2">
<h2 id="org0492f6b"><span class="section-number-2">21</span> Robustness - Nyquist</h2>
<div class="outline-text-2" id="text-21">
<div id="org590bd5d" class="figure">
<p><img src="figs/nyquist_robustness.png" alt="nyquist_robustness.png" />
</p>
<p><span class="figure-number">Figure 20: </span>Robustness - Nyquist (<a href="./figs/nyquist_robustness.png">png</a>, <a href="./figs/nyquist_robustness.pdf">pdf</a>, <a href="./figs/nyquist_robustness.tex">tex</a>).</p>
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<div id="outline-container-org0e6c625" class="outline-2">
<h2 id="org0e6c625"><span class="section-number-2">22</span> Robustness - Loop Gain</h2>
<div class="outline-text-2" id="text-22">
<div id="org771089c" class="figure">
<p><img src="figs/bode_plot_loop_gain_robustness.png" alt="bode_plot_loop_gain_robustness.png" />
</p>
<p><span class="figure-number">Figure 21: </span>Robustness - Loop Gain (<a href="./figs/sf_arch.png">png</a>, <a href="./figs/sf_arch.pdf">pdf</a>, <a href="./figs/sf_arch.tex">tex</a>).</p>
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<div id="outline-container-orgd400711" class="outline-2">
<h2 id="orgd400711"><span class="section-number-2">23</span> Robust performance</h2>
<div class="outline-text-2" id="text-23">
<div id="org910dc73" class="figure">
<p><img src="figs/robust_perf.png" alt="robust_perf.png" />
</p>
<p><span class="figure-number">Figure 22: </span>Robust performance (<a href="./figs/robust_perf.png">png</a>, <a href="./figs/robust_perf.pdf">pdf</a>, <a href="./figs/robust_perf.tex">tex</a>).</p>
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<div id="postamble" class="status">
<p class="author">Author: Thomas Dehaeze</p>
<p class="date">Created: 2019-07-22 lun. 11:55</p>
<p class="validation"><a href="http://validator.w3.org/check?uri=referer">Validate</a></p>
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