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<?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">
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<!-- 2020-08-17 lun. 17:49 -->
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<title>Robust and Optimal Sensor Fusion - Tikz Figures</title>
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<meta name="author" content="Thomas Dehaeze" />
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<a accesskey="h" href="../index.html"> UP </a>
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<a accesskey="H" href="../index.html"> HOME </a>
</div><div id="content">
<h1 class="title">Robust and Optimal Sensor Fusion - Tikz Figures</h1>
<div id="table-of-contents">
<h2>Table of Contents</h2>
<div id="text-table-of-contents">
<ul>
<li><a href="#org4e2a932">1. Sensor Fusion Architecture</a></li>
<li><a href="#org1353381">2. Architecture used for \(\mathcal{H}_2\) synthesis of complementary filters</a></li>
<li><a href="#org1189340">3. Sensor fusion architecture with sensor dynamics uncertainty</a></li>
<li><a href="#org635364c">4. Uncertainty set of the super sensor dynamics</a></li>
<li><a href="#org8471952">5. Architecture used for \(\mathcal{H}_\infty\) synthesis of complementary filters</a></li>
<li><a href="#orgd2abe63">6. Sensor fusion architecture with sensor dynamics uncertainty and noise</a></li>
<li><a href="#org25fbe7c">7. Mixed H2/H-Infinity Synthesis</a></li>
</ul>
</div>
</div>
<p>
Configuration file is accessible <a href="config.html">here</a>.
</p>
<div id="outline-container-org4e2a932" class="outline-2">
<h2 id="org4e2a932"><span class="section-number-2">1</span> Sensor Fusion Architecture</h2>
<div class="outline-text-2" id="text-1">
<div class="org-src-container">
<pre class="src src-latex">\begin{tikzpicture}
\node[branch] (x) at (0, 0);
\node[block, above right=1.0 and 0.5 of x](G1){$G_1(s)$};
\node[block, below right=1.0 and 0.5 of x](G2){$G_2(s)$};
\node[addb, right=0.4 of G1](add1){};
\node[addb, right=0.4 of G2](add2){};
\node[block, right=1.1 of add1](H1){$H_1(s)$};
\node[block, right=1.1 of add2](H2){$H_2(s)$};
\node[block, above=0.5 of add1](N1){$N_1(s)$};
\node[block, above=0.5 of add2](N2){$N_2(s)$};
\node[addb, right=4.8 of x](add){};
\draw[] ($(x)+(-0.7, 0)$) node[above right]{$x$} -- (x.center);
\draw[-&gt;] (x.center) |- (G1.west);
\draw[-&gt;] (x.center) |- (G2.west);
\draw[-&gt;] (G1.east) -- (add1.west);
\draw[-&gt;] (G2.east) -- (add2.west);
\draw[-&gt;] (N1.south) -- (add1.north)node[above right]{$n_1$};
\draw[-&gt;] (N2.south) -- (add2.north)node[above right]{$n_2$};
\draw[&lt;-] (N1.north) -- ++(0, 0.6)node[below right](n1){$\tilde{n}_1$};
\draw[&lt;-] (N2.north) -- ++(0, 0.6)node[below right](n2){$\tilde{n}_2$};
\draw[-&gt;] (add1.east) -- (H1.west)node[above left]{$\hat{x}_1$};
\draw[-&gt;] (add2.east) -- (H2.west)node[above left]{$\hat{x}_2$};
\draw[-&gt;] (H1) -| (add.north);
\draw[-&gt;] (H2) -| (add.south);
\draw[-&gt;] (add.east) -- ++(0.7, 0) node[above left]{$\hat{x}$};
\begin{scope}[on background layer]
\node[fit={($(G2.south-|x)+(-0.2, -0.3)$) ($(n1.north east-|add.east)+(0.2, 0.3)$)}, fill=black!10!white, draw, dashed, inner sep=0pt] (supersensor) {};
\node[below left] at (supersensor.north east) {Super Sensor};
\node[fit={($(G1.south west)+(-0.3, -0.1)$) ($(n1.north east)+(0.1, 0.1)$)}, fill=black!20!white, draw, dashed, inner sep=0pt] (sensor1) {};
\node[below right] at (sensor1.north west) {Sensor 1};
\node[fit={($(G2.south west)+(-0.3, -0.1)$) ($(n2.north east)+(0.1, 0.1)$)}, fill=black!20!white, draw, dashed, inner sep=0pt] (sensor2) {};
\node[below right] at (sensor2.north west) {Sensor 2};
\end{scope}
\end{tikzpicture}
</pre>
</div>
<div id="orge2c7af4" class="figure">
<p><img src="figs/sensor_fusion_noise_arch.png" alt="sensor_fusion_noise_arch.png" />
</p>
<p><span class="figure-number">Figure 1: </span>Sensor Fusion Architecture (<a href="./figs/sensor_fusion_noise_arch.png">png</a>, <a href="./figs/sensor_fusion_noise_arch.pdf">pdf</a>).</p>
</div>
</div>
</div>
<div id="outline-container-org1353381" class="outline-2">
<h2 id="org1353381"><span class="section-number-2">2</span> Architecture used for \(\mathcal{H}_2\) synthesis of complementary filters</h2>
<div class="outline-text-2" id="text-2">
<div class="org-src-container">
<pre class="src src-latex">\begin{tikzpicture}
\node[block={4.0cm}{2.5cm}, fill=black!20!white, dashed] (P) {};
\node[above] at (P.north) {$P(s)$};
\coordinate[] (inputw) at ($(P.south west)!0.75!(P.north west) + (-0.7, 0)$);
\coordinate[] (inputu) at ($(P.south west)!0.35!(P.north west) + (-0.7, 0)$);
\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 ($(P.south east)!0.1!(P.north east) + ( 0.7, 0)$);
\node[block, left=1.4 of output1] (N1){$N_1(s)$};
\node[block, left=1.4 of output2] (N2){$N_2(s)$};
\node[addb={+}{}{}{}{-}, left=of N1] (sub) {};
\node[block, below=0.3 of P] (H2) {$H_2(s)$};
\draw[-&gt;] (inputw) node[above right]{$w$} -- (sub.west);
\draw[-&gt;] (H2.west) -| ($(inputu)+(0.35, 0)$) node[above]{$u$} -- (N2.west);
\draw[-&gt;] (inputu-|sub) node[branch]{} -- (sub.south);
\draw[-&gt;] (sub.east) -- (N1.west);
\draw[-&gt;] ($(sub.west)+(-0.6, 0)$) node[branch]{} |- ($(outputv)+(-0.35, 0)$) node[above]{$v$} |- (H2.east);
\draw[-&gt;] (N1.east) -- (output1)node[above left]{$z_1$};
\draw[-&gt;] (N2.east) -- (output2)node[above left]{$z_2$};
\end{tikzpicture}
</pre>
</div>
<div id="org8ddefad" class="figure">
<p><img src="figs/h_two_optimal_fusion.png" alt="h_two_optimal_fusion.png" />
</p>
<p><span class="figure-number">Figure 2: </span>Architecture used for \(\mathcal{H}_\infty\) synthesis of complementary filters (<a href="./figs/h_two_optimal_fusion.png">png</a>, <a href="./figs/h_two_optimal_fusion.pdf">pdf</a>).</p>
</div>
</div>
</div>
<div id="outline-container-org1189340" class="outline-2">
<h2 id="org1189340"><span class="section-number-2">3</span> Sensor fusion architecture with sensor dynamics uncertainty</h2>
<div class="outline-text-2" id="text-3">
<div class="org-src-container">
<pre class="src src-latex">\begin{tikzpicture}
\node[branch] (x) at (0, 0);
\node[addb, above right=0.8 and 4 of x](add1){};
\node[addb, below right=0.8 and 4 of x](add2){};
\node[block, above left=0.2 and 0.1 of add1](delta1){$\Delta_1(s)$};
\node[block, above left=0.2 and 0.1 of add2](delta2){$\Delta_2(s)$};
\node[block, left=0.5 of delta1](W1){$w_1(s)$};
\node[block, left=0.5 of delta2](W2){$w_2(s)$};
\node[block, right=0.5 of add1](H1){$H_1(s)$};
\node[block, right=0.5 of add2](H2){$H_2(s)$};
\node[addb, right=6 of x](add){};
\draw[] ($(x)+(-0.7, 0)$) node[above right]{$x$} -- (x.center);
\draw[-&gt;] (x.center) |- (add1.west);
\draw[-&gt;] (x.center) |- (add2.west);
\draw[-&gt;] ($(add1-|W1.west)+(-0.5, 0)$)node[branch](S1){} |- (W1.west);
\draw[-&gt;] ($(add2-|W2.west)+(-0.5, 0)$)node[branch](S1){} |- (W2.west);
\draw[-&gt;] (W1.east) -- (delta1.west);
\draw[-&gt;] (W2.east) -- (delta2.west);
\draw[-&gt;] (delta1.east) -| (add1.north);
\draw[-&gt;] (delta2.east) -| (add2.north);
\draw[-&gt;] (add1.east) -- (H1.west);
\draw[-&gt;] (add2.east) -- (H2.west);
\draw[-&gt;] (H1.east) -| (add.north);
\draw[-&gt;] (H2.east) -| (add.south);
\draw[-&gt;] (add.east) -- ++(0.7, 0) node[above left]{$\hat{x}$};
\begin{scope}[on background layer]
\node[fit={($(H2.south-|x)+(-0.2, -0.3)$) ($(delta1.north east-|add.east)+(0.2, 0.4)$)}, fill=black!10!white, draw, dashed, inner sep=0pt] (supersensor) {};
\node[below left] at (supersensor.north east) {Super Sensor};
\node[block, fit={($(W1.north-|S1)+(-0.2, 0.2)$) ($(add1.south east)+(0.2, -0.3)$)}, fill=black!20!white, dashed, inner sep=0pt] (sensor1) {};
\node[above right] at (sensor1.south west) {Sensor 1};
\node[block, fit={($(W2.north-|S1)+(-0.2, 0.2)$) ($(add2.south east)+(0.2, -0.3)$)}, fill=black!20!white, dashed, inner sep=0pt] (sensor2) {};
\node[above right] at (sensor2.south west) {Sensor 2};
\end{scope}
\end{tikzpicture}
</pre>
</div>
<div id="org6316ce9" class="figure">
<p><img src="figs/sensor_fusion_arch_uncertainty.png" alt="sensor_fusion_arch_uncertainty.png" />
</p>
<p><span class="figure-number">Figure 3: </span>Sensor fusion architecture with sensor dynamics uncertainty (<a href="./figs/sensor_fusion_arch_uncertainty.png">png</a>, <a href="./figs/sensor_fusion_arch_uncertainty.pdf">pdf</a>).</p>
</div>
</div>
</div>
<div id="outline-container-org635364c" class="outline-2">
<h2 id="org635364c"><span class="section-number-2">4</span> Uncertainty set of the super sensor dynamics</h2>
<div class="outline-text-2" id="text-4">
<div class="org-src-container">
<pre class="src src-latex">\begin{tikzpicture}
\begin{scope}[shift={(4, 0)}]
% Uncertainty Circle
\node[draw, circle, fill=black!20!white, minimum size=3.6cm] (c) at (0, 0) {};
\path[draw, dotted] (0, 0) circle [radius=1.0];
\path[draw, dashed] (135:1.0) circle [radius=0.8];
% Center of Circle
\node[below] at (0, 0){$1$};
\draw[&lt;-&gt;, dashed] (0, 0) node[branch]{} -- coordinate[midway](r1) ++(45:1.0);
\draw[&lt;-&gt;, dashed] (135:1.0)node[branch]{} -- coordinate[midway](r2) ++(90:0.8);
\node[] (l1) at (2, 1.5) {$|w_1 H_1|$};
\draw[-&gt;, dashed, out=-90, in=0] (l1.south) to (r1);
\node[] (l2) at (-2.5, 1.5) {$|w_2 H_2|$};
\draw[-&gt;, dashed, out=0, in=-180] (l2.east) to (r2);
\draw[&lt;-&gt;, dashed] (0, 0) -- coordinate[near end](r3) ++(200:1.8);
\node[] (l3) at (-2.5, -1.5) {$|w_1 H_1| + |w_2 H_2|$};
\draw[-&gt;, dashed, out=90, in=-90] (l3.north) to (r3);
\end{scope}
% Real and Imaginary Axis
\draw[-&gt;] (-0.5, 0) -- (7.0, 0) node[below left]{Re};
\draw[-&gt;] (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]{$\Delta \phi$};
\end{tikzpicture}
</pre>
</div>
<div id="orgdf90733" class="figure">
<p><img src="figs/uncertainty_set_super_sensor.png" alt="uncertainty_set_super_sensor.png" />
</p>
<p><span class="figure-number">Figure 4: </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>).</p>
</div>
</div>
</div>
<div id="outline-container-org8471952" class="outline-2">
<h2 id="org8471952"><span class="section-number-2">5</span> Architecture used for \(\mathcal{H}_\infty\) synthesis of complementary filters</h2>
<div class="outline-text-2" id="text-5">
<div class="org-src-container">
<pre class="src src-latex">\begin{tikzpicture}
\node[block={4.0cm}{2.5cm}, fill=black!20!white, dashed] (P) {};
\node[above] at (P.north) {$P(s)$};
\coordinate[] (inputw) at ($(P.south west)!0.75!(P.north west) + (-0.7, 0)$);
\coordinate[] (inputu) at ($(P.south west)!0.35!(P.north west) + (-0.7, 0)$);
\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 ($(P.south east)!0.1!(P.north east) + ( 0.7, 0)$);
\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) {};
\node[block, below=0.3 of P] (H2) {$H_2(s)$};
\draw[-&gt;] (inputw) node[above right]{$w$} -- (sub.west);
\draw[-&gt;] (H2.west) -| ($(inputu)+(0.35, 0)$) node[above]{$u$} -- (W2.west);
\draw[-&gt;] (inputu-|sub) node[branch]{} -- (sub.south);
\draw[-&gt;] (sub.east) -- (W1.west);
\draw[-&gt;] ($(sub.west)+(-0.6, 0)$) node[branch]{} |- ($(outputv)+(-0.35, 0)$) node[above]{$v$} |- (H2.east);
\draw[-&gt;] (W1.east) -- (output1)node[above left]{$z_1$};
\draw[-&gt;] (W2.east) -- (output2)node[above left]{$z_2$};
\end{tikzpicture}
</pre>
</div>
<div id="orgc7070d9" class="figure">
<p><img src="figs/h_infinity_robust_fusion.png" alt="h_infinity_robust_fusion.png" />
</p>
<p><span class="figure-number">Figure 5: </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>).</p>
</div>
</div>
</div>
<div id="outline-container-orgd2abe63" class="outline-2">
<h2 id="orgd2abe63"><span class="section-number-2">6</span> Sensor fusion architecture with sensor dynamics uncertainty and noise</h2>
<div class="outline-text-2" id="text-6">
<div class="org-src-container">
<pre class="src src-latex">\begin{tikzpicture}
\node[branch] (x) at (0, 0);
\node[addb, above right=1.2 and 3.7 of x](add1){};
\node[addb, below right=1.2 and 3.7 of x](add2){};
\node[block, above left=0.2 and 0.1 of add1](delta1){$\Delta_1(s)$};
\node[block, above left=0.2 and 0.1 of add2](delta2){$\Delta_2(s)$};
\node[block, left=0.5 of delta1](W1){$w_1(s)$};
\node[block, left=0.5 of delta2](W2){$w_2(s)$};
\node[addb, right=0.5 of add1](addn1){};
\node[addb, right=0.5 of add2](addn2){};
\node[block, above=0.5 of addn1](N1) {$N_1(s)$};
\node[block, above=0.5 of addn2](N2) {$N_2(s)$};
\node[block, right=1.2 of addn1](H1){$H_1(s)$};
\node[block, right=1.2 of addn2](H2){$H_2(s)$};
\node[addb, right=7.5 of x](add){};
\draw[] ($(x)+(-0.7, 0)$) node[above right]{$x$} -- (x.center);
\draw[-&gt;] (x.center) |- (add1.west);
\draw[-&gt;] (x.center) |- (add2.west);
\draw[-&gt;] ($(add1-|W1.west)+(-0.5, 0)$)node[branch](S1){} |- (W1.west);
\draw[-&gt;] ($(add2-|W2.west)+(-0.5, 0)$)node[branch](S2){} |- (W2.west);
\draw[-&gt;] (W1.east) -- (delta1.west);
\draw[-&gt;] (W2.east) -- (delta2.west);
\draw[-&gt;] (delta1.east) -| (add1.north);
\draw[-&gt;] (delta2.east) -| (add2.north);
\draw[-&gt;] (add1.east) -- (addn1.west);
\draw[-&gt;] (add2.east) -- (addn2.west);
\draw[-&gt;] (addn1.east) -- (H1.west)node[above left]{$\hat{x}_1$};
\draw[-&gt;] (addn2.east) -- (H2.west)node[above left]{$\hat{x}_2$};
\draw[-&gt;] ($(N1.north)+(0,0.7)$) node[below right](n1){$\tilde{n}_1$} -- (N1.north);
\draw[-&gt;] ($(N2.north)+(0,0.7)$) node[below right](n2){$\tilde{n}_2$} -- (N2.north);
\draw[-&gt;] (N1.south) -- (addn1.north)node[above right]{$n_1$};
\draw[-&gt;] (N2.south) -- (addn2.north)node[above right]{$n_2$};
\draw[-&gt;] (H1.east) -| (add.north);
\draw[-&gt;] (H2.east) -| (add.south);
\draw[-&gt;] (add.east) -- ++(0.7, 0) node[above left]{$\hat{x}$};
\begin{scope}[on background layer]
\node[fit={($(add2.south-|x)+(-0.2, -0.4)$) ($(n1.north-|add.east)+(0.2, 0.3)$)}, fill=black!10!white, draw, dashed, inner sep=0pt] (supersensor) {};
\node[below left] at (supersensor.north east) {Super Sensor};
\node[block, fit={($(S1|-add1.south)+(-0.2, -0.2)$) ($(n1.north-|N1.east)+(0.2, 0.1)$)}, fill=black!20!white, dashed, inner sep=0pt] (sensor1) {};
\node[below right] at (sensor1.north west) {Sensor 1};
\node[block, fit={($(S2|-add2.south)+(-0.2, -0.2)$) ($(n2.north-|N2.east)+(0.2, 0.1)$)}, fill=black!20!white, dashed, inner sep=0pt] (sensor2) {};
\node[below right] at (sensor2.north west) {Sensor 2};
\end{scope}
\end{tikzpicture}
</pre>
</div>
<div id="org2a7478b" class="figure">
<p><img src="figs/sensor_fusion_arch_full.png" alt="sensor_fusion_arch_full.png" />
</p>
<p><span class="figure-number">Figure 6: </span>Sensor fusion architecture with sensor dynamics uncertainty (<a href="./figs/sensor_fusion_arch_full.png">png</a>, <a href="./figs/sensor_fusion_arch_full.pdf">pdf</a>).</p>
</div>
</div>
</div>
<div id="outline-container-org25fbe7c" class="outline-2">
<h2 id="org25fbe7c"><span class="section-number-2">7</span> Mixed H2/H-Infinity Synthesis</h2>
<div class="outline-text-2" id="text-7">
<div class="org-src-container">
<pre class="src src-latex">\begin{tikzpicture}
\node[block={5.0cm}{5.0cm}, fill=black!20!white, dashed] (P) {};
\node[above] at (P.north) {$P(s)$};
\coordinate[] (inputw) at ($(P.south west)!0.85!(P.north west) + (-0.7, 0)$);
\coordinate[] (inputu) at ($(P.south west)!0.35!(P.north west) + (-0.7, 0)$);
\coordinate[] (output1) at ($(P.south east)!0.85!(P.north east) + ( 0.7, 0)$);
\coordinate[] (output2) at ($(P.south east)!0.7!(P.north east) + ( 0.7, 0)$);
\coordinate[] (output3) at ($(P.south east)!0.5!(P.north east) + ( 0.7, 0)$);
\coordinate[] (output4) at ($(P.south east)!0.35!(P.north east) + ( 0.7, 0)$);
\coordinate[] (outputv) at ($(P.south east)!0.15!(P.north east) + ( 0.7, 0)$);
\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=1.6 of W1] (sub1) {};
\node[block, left=1.4 of output3] (N1){$N_1(s)$};
\node[block, left=1.4 of output4] (N2){$N_2(s)$};
\node[addb={+}{}{}{}{-}, left=1 of N1] (sub2) {};
\node[block, below=0.3 of P] (H2) {$H_2(s)$};
\draw[-&gt;] (inputw) node[above right]{$w$} -- (sub1.west);
\draw[-&gt;] (H2.west) -| ($(inputu)+(0.35, 0)$) node[above]{$u$} -- (N2.west);
\draw[-&gt;] (inputu-|sub1) node[branch]{} -- (sub1.south);
\draw[-&gt;] (inputu-|sub2) node[branch]{} -- (sub2.south);
\draw[-&gt;] (sub1|-W2) node[branch]{} -- (W2.west);
\draw[-&gt;] (sub1.east) -- (W1.west);
\draw[-&gt;] (sub2.east) -- (N1.west);
\draw[-&gt;] ($(sub1.west)+(-0.6, 0)$) node[branch](w_branch){} |- ($(outputv)+(-0.35, 0)$) node[above]{$v$} |- (H2.east);
\draw[-&gt;] (w_branch|-sub2) node[branch]{} -- (sub2.west);
\draw[-&gt;] (W1.east) -- (output1)node[above left]{$z_1$};
\draw[-&gt;] (W2.east) -- (output2)node[above left]{$z_2$};
\draw[-&gt;] (N1.east) -- (output3)node[above left]{$z_3$};
\draw[-&gt;] (N2.east) -- (output4)node[above left]{$z_4$};
\end{tikzpicture}
</pre>
</div>
<div id="org855c0bc" class="figure">
<p><img src="figs/mixed_h2_hinf_synthesis.png" alt="mixed_h2_hinf_synthesis.png" />
</p>
<p><span class="figure-number">Figure 7: </span>Mixed H2/H-Infinity Synthesis (<a href="./figs/mixed_h2_hinf_synthesis.png">png</a>, <a href="./figs/mixed_h2_hinf_synthesis.pdf">pdf</a>).</p>
</div>
</div>
</div>
</div>
<div id="postamble" class="status">
<p class="author">Author: Thomas Dehaeze</p>
<p class="date">Created: 2020-08-17 lun. 17:49</p>
</div>
</body>
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