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-<title>On the Design of Complementary Filters for Control - Computation with Matlab</title>
+<title>Robust and Optimal Sensor Fusion - Matlab Computation</title>
 <meta name="generator" content="Org mode" />
 <meta name="author" content="Thomas Dehaeze" />
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@@ -274,57 +274,91 @@ for the JavaScript code in this tag.
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 </div><div id="content">
-<h1 class="title">On the Design of Complementary Filters for Control - Computation with Matlab</h1>
+<h1 class="title">Robust and Optimal Sensor Fusion - Matlab Computation</h1>
 <div id="table-of-contents">
 <h2>Table of Contents</h2>
 <div id="text-table-of-contents">
 <ul>
-<li><a href="#orga1a1b9d">1. Optimal Sensor Fusion for noise characteristics</a>
+<li><a href="#org863f00e">1. Optimal Sensor Fusion for noise characteristics</a>
 <ul>
-<li><a href="#orga3597f6">1.1. Architecture</a></li>
-<li><a href="#orgcac1aa3">1.2. Noise of the sensors</a></li>
-<li><a href="#orgdb1ca76">1.3. H-Two Synthesis</a></li>
-<li><a href="#org375cb3a">1.4. Analysis</a></li>
+<li><a href="#org6a8ef27">1.1. Architecture</a></li>
+<li><a href="#org24dd492">1.2. Noise of the sensors</a></li>
+<li><a href="#orgd87c1f0">1.3. H-Two Synthesis</a></li>
+<li><a href="#orgea5c8b1">1.4. Analysis</a></li>
+<li><a href="#orga01f5f2">1.5. H-Infinity Synthesis</a>
+<ul>
+<li><a href="#org0af63e9">1.5.1. First idea</a></li>
+<li><a href="#org6952735">1.5.2. Second idea</a></li>
+<li><a href="#orgd67bea5">1.5.3. Conclusion</a></li>
 </ul>
 </li>
-<li><a href="#org2ebd870">2. Robustness to sensor dynamics uncertainty</a>
-<ul>
-<li><a href="#orgf63dc07">2.1. Unknown sensor dynamics dynamics</a></li>
-<li><a href="#org3244858">2.2. Design the complementary filters in order to limit the phase and gain uncertainty of the super sensor</a></li>
-<li><a href="#org20ae980">2.3. First Basic Example with gain mismatch</a></li>
-<li><a href="#org41f2b11">2.4. <span class="todo TODO">TODO</span> More Complete example with model uncertainty</a></li>
 </ul>
 </li>
-<li><a href="#orgf04a9a4">3. Complementary filters using analytical formula</a>
+<li><a href="#org52c42b7">2. Robustness to sensor dynamics uncertainty</a>
 <ul>
-<li><a href="#org7d339a9">3.1. Analytical 1st order complementary filters</a></li>
-<li><a href="#org8edbedb">3.2. Second Order Complementary Filters</a></li>
-<li><a href="#org5fd0c3c">3.3. Third Order Complementary Filters</a></li>
+<li><a href="#orga0447f7">2.1. Unknown sensor dynamics dynamics</a></li>
+<li><a href="#org4b0c13d">2.2. Design the complementary filters in order to limit the phase and gain uncertainty of the super sensor</a></li>
+<li><a href="#orge9f4bde">2.3. First Basic Example with gain mismatch</a></li>
+<li><a href="#org551c5ec">2.4. <span class="todo TODO">TODO</span> More Complete example with model uncertainty</a></li>
 </ul>
 </li>
-<li><a href="#org88004e2">4. H-Infinity synthesis of complementary filters</a>
+<li><a href="#org861c0eb">3. Equivalent Super Sensor</a>
 <ul>
-<li><a href="#orgc51b2bd">4.1. Synthesis Architecture</a></li>
-<li><a href="#orga135818">4.2. Weights</a></li>
-<li><a href="#orgbd3b8d8">4.3. H-Infinity Synthesis</a></li>
-<li><a href="#org0d62ef6">4.4. Obtained Complementary Filters</a></li>
+<li><a href="#org51db5bb">3.1. Sensor Fusion Architecture</a></li>
+<li><a href="#orgc301d63">3.2. Equivalent Configuration</a></li>
+<li><a href="#orge929495">3.3. Model the uncertainty of the super sensor</a></li>
+<li><a href="#org762c80b">3.4. Model the noise of the super sensor</a></li>
+<li><a href="#org01817fc">3.5. First guess</a></li>
 </ul>
 </li>
-<li><a href="#orga65dbb6">5. Feedback Control Architecture to generate Complementary Filters</a>
+<li><a href="#orgacde10f">4. Optimal And Robust Sensor Fusion in Practice</a>
 <ul>
-<li><a href="#orgec7fc24">5.1. Architecture</a></li>
-<li><a href="#org448cd20">5.2. Loop Gain Design</a></li>
-<li><a href="#org1ec0e07">5.3. Complementary Filters Obtained</a></li>
+<li><a href="#orgd36ee6e">4.1. Measurement of the noise characteristics of the sensors</a>
+<ul>
+<li><a href="#org36e1a94">4.1.1. Huddle Test</a></li>
+<li><a href="#orge63237d">4.1.2. Weights that represents the noises' PSD</a></li>
+<li><a href="#org3c415dc">4.1.3. Comparison of the noises' PSD</a></li>
+<li><a href="#orgb6d27a5">4.1.4. Computation of the coherence, power spectral density and cross spectral density of signals</a></li>
 </ul>
 </li>
-<li><a href="#org3b324a8">6. Analytical Formula found in the literature</a>
-<ul>
-<li><a href="#org8d499d4">6.1. Analytical Formula</a></li>
-<li><a href="#org60e2ce0">6.2. Matlab</a></li>
-<li><a href="#org1038fe9">6.3. Discussion</a></li>
+<li><a href="#org491f8d8">4.2. Estimate the dynamic uncertainty of the sensors</a></li>
+<li><a href="#orgaeae7ef">4.3. Optimal and Robust synthesis of the complementary filters</a></li>
+</ul>
+</li>
+<li><a href="#org06754da">5. Methods of complementary filter synthesis</a>
+<ul>
+<li><a href="#orgb77dafe">5.1. Complementary filters using analytical formula</a>
+<ul>
+<li><a href="#orga47e3d3">5.1.1. Analytical 1st order complementary filters</a></li>
+<li><a href="#orgd1a9522">5.1.2. Second Order Complementary Filters</a></li>
+<li><a href="#orgbc26e97">5.1.3. Third Order Complementary Filters</a></li>
+</ul>
+</li>
+<li><a href="#org630c88a">5.2. H-Infinity synthesis of complementary filters</a>
+<ul>
+<li><a href="#org9eecdbe">5.2.1. Synthesis Architecture</a></li>
+<li><a href="#orgad3b0d3">5.2.2. Weights</a></li>
+<li><a href="#org6fad77a">5.2.3. H-Infinity Synthesis</a></li>
+<li><a href="#orgae7d589">5.2.4. Obtained Complementary Filters</a></li>
+</ul>
+</li>
+<li><a href="#org7728066">5.3. Feedback Control Architecture to generate Complementary Filters</a>
+<ul>
+<li><a href="#org00ee41c">5.3.1. Architecture</a></li>
+<li><a href="#org69ae5c6">5.3.2. Loop Gain Design</a></li>
+<li><a href="#orgecfc029">5.3.3. Complementary Filters Obtained</a></li>
+</ul>
+</li>
+<li><a href="#org3bfd6fb">5.4. Analytical Formula found in the literature</a>
+<ul>
+<li><a href="#orge18079d">5.4.1. Analytical Formula</a></li>
+<li><a href="#org85b2b53">5.4.2. Matlab</a></li>
+<li><a href="#org3e6b82e">5.4.3. Discussion</a></li>
+</ul>
+</li>
+<li><a href="#orgd5b4c24">5.5. Comparison of the different methods of synthesis</a></li>
 </ul>
 </li>
-<li><a href="#org647d1a0">7. Comparison of the different methods of synthesis</a></li>
 </ul>
 </div>
 </div>
@@ -344,31 +378,31 @@ To achieve this, the sensors included in the filter should complement one anothe
 </blockquote>
 
 <ul class="org-ul">
-<li>in section <a href="#orge83d075">1</a>, the optimal design of the complementary filters in order to obtain the lowest resulting "super sensor" noise is studied</li>
+<li>in section <a href="#orgfdd6f69">1</a>, the optimal design of the complementary filters in order to obtain the lowest resulting "super sensor" noise is studied</li>
 </ul>
 
 <p>
 When blending two sensors using complementary filters with unknown dynamics, phase lag may be introduced that renders the close-loop system unstable.
 </p>
 <ul class="org-ul">
-<li>in section <a href="#orgb2ab616">2</a>, the blending robustness to sensor dynamic uncertainty is studied.</li>
+<li>in section <a href="#org6776beb">2</a>, the blending robustness to sensor dynamic uncertainty is studied.</li>
 </ul>
 
 <p>
 Then, three design methods for generating two complementary filters are proposed:
 </p>
 <ul class="org-ul">
-<li>in section <a href="#org654b360">3</a>, analytical formulas are proposed</li>
-<li>in section <a href="#org924f36f">4</a>, the \(\mathcal{H}_\infty\) synthesis is used</li>
-<li>in section <a href="#org3e654dc">5</a>, the classical feedback architecture is used</li>
-<li>in section <a href="#org22f6020">6</a>, analytical formulas found in the literature are listed</li>
+<li>in section <a href="#org535ff31">5.1</a>, analytical formulas are proposed</li>
+<li>in section <a href="#orgd45833d">5.2</a>, the \(\mathcal{H}_\infty\) synthesis is used</li>
+<li>in section <a href="#orgabbb632">5.3</a>, the classical feedback architecture is used</li>
+<li>in section <a href="#org53de8db">5.4</a>, analytical formulas found in the literature are listed</li>
 </ul>
 
-<div id="outline-container-orga1a1b9d" class="outline-2">
-<h2 id="orga1a1b9d"><span class="section-number-2">1</span> Optimal Sensor Fusion for noise characteristics</h2>
+<div id="outline-container-org863f00e" class="outline-2">
+<h2 id="org863f00e"><span class="section-number-2">1</span> Optimal Sensor Fusion for noise characteristics</h2>
 <div class="outline-text-2" id="text-1">
 <p>
-<a id="orge83d075"></a>
+<a id="orgfdd6f69"></a>
 </p>
 <p>
 The idea is to combine sensors that works in different frequency range using complementary filters.
@@ -389,11 +423,11 @@ All the files (data and Matlab scripts) are accessible <a href="data/optimal_com
 </div>
 </div>
 
-<div id="outline-container-orga3597f6" class="outline-3">
-<h3 id="orga3597f6"><span class="section-number-3">1.1</span> Architecture</h3>
+<div id="outline-container-org6a8ef27" class="outline-3">
+<h3 id="org6a8ef27"><span class="section-number-3">1.1</span> Architecture</h3>
 <div class="outline-text-3" id="text-1-1">
 <p>
-Let's consider the sensor fusion architecture shown on figure <a href="#org80c1b07">1</a> where two sensors 1 and 2 are measuring the same quantity \(x\) with different noise characteristics determined by \(W_1\) and \(W_2\).
+Let's consider the sensor fusion architecture shown on figure <a href="#orgb190e10">1</a> where two sensors 1 and 2 are measuring the same quantity \(x\) with different noise characteristics determined by \(W_1\) and \(W_2\).
 </p>
 
 <p>
@@ -401,19 +435,18 @@ Let's consider the sensor fusion architecture shown on figure <a href="#org80c1b
 </p>
 
 
-<div id="org80c1b07" class="figure">
-<p><object type="image/svg+xml" data="figs-tikz/fusion_two_noisy_sensors_with_dyn.svg" class="org-svg">
-Sorry, your browser does not support SVG.</object>
+<div id="orgb190e10" class="figure">
+<p><img src="figs-tikz/fusion_two_noisy_sensors_with_dyn.png" alt="fusion_two_noisy_sensors_with_dyn.png" />
 </p>
 <p><span class="figure-number">Figure 1: </span>Fusion of two sensors</p>
 </div>
 
 <p>
-We consider that the two sensor dynamics \(G_1\) and \(G_2\) are ideal (\(G_1 = G_2 = 1\)). We obtain the architecture of figure <a href="#org442f9da">2</a>.
+We consider that the two sensor dynamics \(G_1\) and \(G_2\) are ideal (\(G_1 = G_2 = 1\)). We obtain the architecture of figure <a href="#orge7d9e68">2</a>.
 </p>
 
 
-<div id="org442f9da" class="figure">
+<div id="orge7d9e68" class="figure">
 <p><img src="figs-tikz/fusion_two_noisy_sensors.png" alt="fusion_two_noisy_sensors.png" />
 </p>
 <p><span class="figure-number">Figure 2: </span>Fusion of two sensors with ideal dynamics</p>
@@ -448,8 +481,8 @@ For that, we will use the \(\mathcal{H}_2\) Synthesis.
 </div>
 </div>
 
-<div id="outline-container-orgcac1aa3" class="outline-3">
-<h3 id="orgcac1aa3"><span class="section-number-3">1.2</span> Noise of the sensors</h3>
+<div id="outline-container-org24dd492" class="outline-3">
+<h3 id="org24dd492"><span class="section-number-3">1.2</span> Noise of the sensors</h3>
 <div class="outline-text-3" id="text-1-2">
 <p>
 Let's define the noise characteristics of the two sensors by choosing \(W_1\) and \(W_2\):
@@ -469,7 +502,7 @@ W2 = <span class="org-rainbow-delimiters-depth-1">(</span><span class="org-rainb
 </div>
 
 
-<div id="org8380e17" class="figure">
+<div id="org97b9ef1" class="figure">
 <p><img src="figs/nosie_characteristics_sensors.png" alt="nosie_characteristics_sensors.png" />
 </p>
 <p><span class="figure-number">Figure 3: </span>Noise Characteristics of the two sensors (<a href="./figs/nosie_characteristics_sensors.png">png</a>, <a href="./figs/nosie_characteristics_sensors.pdf">pdf</a>)</p>
@@ -477,15 +510,15 @@ W2 = <span class="org-rainbow-delimiters-depth-1">(</span><span class="org-rainb
 </div>
 </div>
 
-<div id="outline-container-orgdb1ca76" class="outline-3">
-<h3 id="orgdb1ca76"><span class="section-number-3">1.3</span> H-Two Synthesis</h3>
+<div id="outline-container-orgd87c1f0" class="outline-3">
+<h3 id="orgd87c1f0"><span class="section-number-3">1.3</span> H-Two Synthesis</h3>
 <div class="outline-text-3" id="text-1-3">
 <p>
-We use the generalized plant architecture shown on figure <a href="#orgfd9fdc8">4</a>.
+We use the generalized plant architecture shown on figure <a href="#org6d57cf2">4</a>.
 </p>
 
 
-<div id="orgfd9fdc8" class="figure">
+<div id="org6d57cf2" class="figure">
 <p><img src="figs-tikz/h_infinity_optimal_comp_filters.png" alt="h_infinity_optimal_comp_filters.png" />
 </p>
 <p><span class="figure-number">Figure 4: </span>\(\mathcal{H}_2\) Synthesis - Generalized plant used for the optimal generation of complementary filters</p>
@@ -503,7 +536,7 @@ Thus, if we minimize the \(\mathcal{H}_2\) norm of this transfer function, we mi
 </p>
 
 <p>
-We define the generalized plant \(P\) on matlab as shown on figure <a href="#orgfd9fdc8">4</a>.
+We define the generalized plant \(P\) on matlab as shown on figure <a href="#org6d57cf2">4</a>.
 </p>
 <div class="org-src-container">
 <pre class="src src-matlab">P = <span class="org-rainbow-delimiters-depth-1">[</span><span class="org-highlight-numbers-number">0</span>   W2  <span class="org-highlight-numbers-number">1</span>;
@@ -533,30 +566,30 @@ Finally, we define \(H_2 = 1 - H_1\).
 </div>
 </div>
 
-<div id="outline-container-org375cb3a" class="outline-3">
-<h3 id="org375cb3a"><span class="section-number-3">1.4</span> Analysis</h3>
+<div id="outline-container-orgea5c8b1" class="outline-3">
+<h3 id="orgea5c8b1"><span class="section-number-3">1.4</span> Analysis</h3>
 <div class="outline-text-3" id="text-1-4">
 <p>
-The complementary filters obtained are shown on figure <a href="#org4addc9f">5</a>. The PSD of the <a href="#org52b2e97">6</a>.
-Finally, the RMS value of \(\hat{x}\) is shown on table <a href="#orgccc886e">1</a>.
+The complementary filters obtained are shown on figure <a href="#org625fbd2">5</a>. The PSD of the <a href="#org01d6fa9">6</a>.
+Finally, the RMS value of \(\hat{x}\) is shown on table <a href="#org3eb5a44">1</a>.
 The optimal sensor fusion has permitted to reduced the RMS value of the estimation error by a factor 8 compare to when using only one sensor.
 </p>
 
 
-<div id="org4addc9f" class="figure">
+<div id="org625fbd2" class="figure">
 <p><img src="figs/htwo_comp_filters.png" alt="htwo_comp_filters.png" />
 </p>
 <p><span class="figure-number">Figure 5: </span>Obtained complementary filters using the \(\mathcal{H}_2\) Synthesis (<a href="./figs/htwo_comp_filters.png">png</a>, <a href="./figs/htwo_comp_filters.pdf">pdf</a>)</p>
 </div>
 
 
-<div id="org52b2e97" class="figure">
+<div id="org01d6fa9" class="figure">
 <p><img src="figs/psd_sensors_htwo_synthesis.png" alt="psd_sensors_htwo_synthesis.png" />
 </p>
 <p><span class="figure-number">Figure 6: </span>Power Spectral Density of the estimated \(\hat{x}\) using the two sensors alone and using the optimally fused signal (<a href="./figs/psd_sensors_htwo_synthesis.png">png</a>, <a href="./figs/psd_sensors_htwo_synthesis.pdf">pdf</a>)</p>
 </div>
 
-<table id="orgccc886e" border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
+<table id="org3eb5a44" border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
 <caption class="t-above"><span class="table-number">Table 1:</span> RMS value of the estimation error when using the sensor individually and when using the two sensor merged using the optimal complementary filters</caption>
 
 <colgroup>
@@ -589,20 +622,320 @@ The optimal sensor fusion has permitted to reduced the RMS value of the estimati
 </table>
 </div>
 </div>
+
+<div id="outline-container-orga01f5f2" class="outline-3">
+<h3 id="orga01f5f2"><span class="section-number-3">1.5</span> H-Infinity Synthesis</h3>
+<div class="outline-text-3" id="text-1-5">
+</div>
+<div id="outline-container-org0af63e9" class="outline-4">
+<h4 id="org0af63e9"><span class="section-number-4">1.5.1</span> First idea</h4>
+<div class="outline-text-4" id="text-1-5-1">
+<p>
+Another objective that we may have is that the noise of the super sensor \(n_{SS}\) is following the minimum of the noise of the two sensors \(n_1\) and \(n_2\):
+\[ \Gamma_{n_{ss}}(\omega) = \min(\Gamma_{n_1}(\omega),\ \Gamma_{n_2}(\omega)) \]
+</p>
+
+<p>
+In order to obtain that ideal case, we need that the complementary filters be designed such that:
+</p>
+\begin{align*}
+  & H_1(j\omega) = 1 \text{ and } H_2(j\omega) = 0 \text{ at frequencies where } \Gamma_{n_1}(\omega) < \Gamma_{n_2}(\omega) \\
+  & H_1(j\omega) = 0 \text{ and } H_2(j\omega) = 1 \text{ at frequencies where } \Gamma_{n_1}(\omega) > \Gamma_{n_2}(\omega)
+\end{align*}
+
+<p>
+Which is indeed <b>impossible in practice</b>.
+</p>
+
+<p>
+We could try to use filters with the highest possible order to approximate that.
+However, we may have robustness and implementation problems.
+</p>
+
+<div class="org-src-container">
+<pre class="src src-matlab">omegac = <span class="org-highlight-numbers-number">100</span><span class="org-type">*</span><span class="org-highlight-numbers-number">2</span><span class="org-type">*</span><span class="org-constant">pi</span>; G0 = <span class="org-highlight-numbers-number">1e</span><span class="org-type">-</span><span class="org-highlight-numbers-number">5</span>; Ginf = <span class="org-highlight-numbers-number">1e</span><span class="org-type">-</span><span class="org-highlight-numbers-number">4</span>;
+N1 = <span class="org-rainbow-delimiters-depth-1">(</span>Ginf<span class="org-type">*</span>s<span class="org-type">/</span>omegac <span class="org-type">+</span> G0<span class="org-rainbow-delimiters-depth-1">)</span><span class="org-type">/</span><span class="org-rainbow-delimiters-depth-1">(</span>s<span class="org-type">/</span>omegac <span class="org-type">+</span> <span class="org-highlight-numbers-number">1</span><span class="org-rainbow-delimiters-depth-1">)</span><span class="org-type">/</span><span class="org-rainbow-delimiters-depth-1">(</span><span class="org-highlight-numbers-number">1</span> <span class="org-type">+</span> s<span class="org-type">/</span><span class="org-highlight-numbers-number">2</span><span class="org-type">/</span><span class="org-constant">pi</span><span class="org-type">/</span><span class="org-highlight-numbers-number">4000</span><span class="org-rainbow-delimiters-depth-1">)</span>;
+
+omegac = <span class="org-highlight-numbers-number">1</span><span class="org-type">*</span><span class="org-highlight-numbers-number">2</span><span class="org-type">*</span><span class="org-constant">pi</span>; G0 = <span class="org-highlight-numbers-number">1e</span><span class="org-type">-</span><span class="org-highlight-numbers-number">3</span>; Ginf = <span class="org-highlight-numbers-number">1e</span><span class="org-type">-</span><span class="org-highlight-numbers-number">8</span>;
+N2 = <span class="org-rainbow-delimiters-depth-1">(</span><span class="org-rainbow-delimiters-depth-2">(</span>sqrt<span class="org-rainbow-delimiters-depth-3">(</span>Ginf<span class="org-rainbow-delimiters-depth-3">)</span><span class="org-type">*</span>s<span class="org-type">/</span>omegac <span class="org-type">+</span> sqrt<span class="org-rainbow-delimiters-depth-3">(</span>G0<span class="org-rainbow-delimiters-depth-3">)</span><span class="org-rainbow-delimiters-depth-2">)</span><span class="org-type">/</span><span class="org-rainbow-delimiters-depth-2">(</span>s<span class="org-type">/</span>omegac <span class="org-type">+</span> <span class="org-highlight-numbers-number">1</span><span class="org-rainbow-delimiters-depth-2">)</span><span class="org-rainbow-delimiters-depth-1">)</span><span class="org-type">^</span><span class="org-highlight-numbers-number">2</span><span class="org-type">/</span><span class="org-rainbow-delimiters-depth-1">(</span><span class="org-highlight-numbers-number">1</span> <span class="org-type">+</span> s<span class="org-type">/</span><span class="org-highlight-numbers-number">2</span><span class="org-type">/</span><span class="org-constant">pi</span><span class="org-type">/</span><span class="org-highlight-numbers-number">4000</span><span class="org-rainbow-delimiters-depth-1">)</span><span class="org-type">^</span><span class="org-highlight-numbers-number">2</span>;
+</pre>
 </div>
 
-<div id="outline-container-org2ebd870" class="outline-2">
-<h2 id="org2ebd870"><span class="section-number-2">2</span> Robustness to sensor dynamics uncertainty</h2>
+<div class="org-src-container">
+<pre class="src src-matlab">n = <span class="org-highlight-numbers-number">5</span>; w0 = <span class="org-highlight-numbers-number">2</span><span class="org-type">*</span><span class="org-constant">pi</span><span class="org-type">*</span><span class="org-highlight-numbers-number">10</span>; G0 = <span class="org-highlight-numbers-number">1</span><span class="org-type">/</span><span class="org-highlight-numbers-number">10</span>; G1 = <span class="org-highlight-numbers-number">10000</span>; Gc = <span class="org-highlight-numbers-number">1</span><span class="org-type">/</span><span class="org-highlight-numbers-number">2</span>;
+W1 = <span class="org-rainbow-delimiters-depth-1">(</span><span class="org-rainbow-delimiters-depth-2">(</span><span class="org-rainbow-delimiters-depth-3">(</span><span class="org-highlight-numbers-number">1</span><span class="org-type">/</span>w0<span class="org-rainbow-delimiters-depth-3">)</span><span class="org-type">*</span>sqrt<span class="org-rainbow-delimiters-depth-3">(</span><span class="org-rainbow-delimiters-depth-4">(</span><span class="org-highlight-numbers-number">1</span><span class="org-type">-</span><span class="org-rainbow-delimiters-depth-5">(</span>G0<span class="org-type">/</span>Gc<span class="org-rainbow-delimiters-depth-5">)</span><span class="org-type">^</span><span class="org-rainbow-delimiters-depth-5">(</span><span class="org-highlight-numbers-number">2</span><span class="org-type">/</span>n<span class="org-rainbow-delimiters-depth-5">)</span><span class="org-rainbow-delimiters-depth-4">)</span><span class="org-type">/</span><span class="org-rainbow-delimiters-depth-4">(</span><span class="org-highlight-numbers-number">1</span><span class="org-type">-</span><span class="org-rainbow-delimiters-depth-5">(</span>Gc<span class="org-type">/</span>G1<span class="org-rainbow-delimiters-depth-5">)</span><span class="org-type">^</span><span class="org-rainbow-delimiters-depth-5">(</span><span class="org-highlight-numbers-number">2</span><span class="org-type">/</span>n<span class="org-rainbow-delimiters-depth-5">)</span><span class="org-rainbow-delimiters-depth-4">)</span><span class="org-rainbow-delimiters-depth-3">)</span><span class="org-type">*</span>s <span class="org-type">+</span> <span class="org-rainbow-delimiters-depth-3">(</span>G0<span class="org-type">/</span>Gc<span class="org-rainbow-delimiters-depth-3">)</span><span class="org-type">^</span><span class="org-rainbow-delimiters-depth-3">(</span><span class="org-highlight-numbers-number">1</span><span class="org-type">/</span>n<span class="org-rainbow-delimiters-depth-3">)</span><span class="org-rainbow-delimiters-depth-2">)</span><span class="org-type">/</span><span class="org-rainbow-delimiters-depth-2">(</span><span class="org-rainbow-delimiters-depth-3">(</span><span class="org-highlight-numbers-number">1</span><span class="org-type">/</span>G1<span class="org-rainbow-delimiters-depth-3">)</span><span class="org-type">^</span><span class="org-rainbow-delimiters-depth-3">(</span><span class="org-highlight-numbers-number">1</span><span class="org-type">/</span>n<span class="org-rainbow-delimiters-depth-3">)</span><span class="org-type">*</span><span class="org-rainbow-delimiters-depth-3">(</span><span class="org-highlight-numbers-number">1</span><span class="org-type">/</span>w0<span class="org-rainbow-delimiters-depth-3">)</span><span class="org-type">*</span>sqrt<span class="org-rainbow-delimiters-depth-3">(</span><span class="org-rainbow-delimiters-depth-4">(</span><span class="org-highlight-numbers-number">1</span><span class="org-type">-</span><span class="org-rainbow-delimiters-depth-5">(</span>G0<span class="org-type">/</span>Gc<span class="org-rainbow-delimiters-depth-5">)</span><span class="org-type">^</span><span class="org-rainbow-delimiters-depth-5">(</span><span class="org-highlight-numbers-number">2</span><span class="org-type">/</span>n<span class="org-rainbow-delimiters-depth-5">)</span><span class="org-rainbow-delimiters-depth-4">)</span><span class="org-type">/</span><span class="org-rainbow-delimiters-depth-4">(</span><span class="org-highlight-numbers-number">1</span><span class="org-type">-</span><span class="org-rainbow-delimiters-depth-5">(</span>Gc<span class="org-type">/</span>G1<span class="org-rainbow-delimiters-depth-5">)</span><span class="org-type">^</span><span class="org-rainbow-delimiters-depth-5">(</span><span class="org-highlight-numbers-number">2</span><span class="org-type">/</span>n<span class="org-rainbow-delimiters-depth-5">)</span><span class="org-rainbow-delimiters-depth-4">)</span><span class="org-rainbow-delimiters-depth-3">)</span><span class="org-type">*</span>s <span class="org-type">+</span> <span class="org-rainbow-delimiters-depth-3">(</span><span class="org-highlight-numbers-number">1</span><span class="org-type">/</span>Gc<span class="org-rainbow-delimiters-depth-3">)</span><span class="org-type">^</span><span class="org-rainbow-delimiters-depth-3">(</span><span class="org-highlight-numbers-number">1</span><span class="org-type">/</span>n<span class="org-rainbow-delimiters-depth-3">)</span><span class="org-rainbow-delimiters-depth-2">)</span><span class="org-rainbow-delimiters-depth-1">)</span><span class="org-type">^</span>n;
+
+n = <span class="org-highlight-numbers-number">5</span>; w0 = <span class="org-highlight-numbers-number">2</span><span class="org-type">*</span><span class="org-constant">pi</span><span class="org-type">*</span><span class="org-highlight-numbers-number">8</span>; G0 = <span class="org-highlight-numbers-number">1000</span>; G1 = <span class="org-highlight-numbers-number">0</span>.<span class="org-highlight-numbers-number">1</span>; Gc = <span class="org-highlight-numbers-number">1</span><span class="org-type">/</span><span class="org-highlight-numbers-number">2</span>;
+W2 = <span class="org-rainbow-delimiters-depth-1">(</span><span class="org-rainbow-delimiters-depth-2">(</span><span class="org-rainbow-delimiters-depth-3">(</span><span class="org-highlight-numbers-number">1</span><span class="org-type">/</span>w0<span class="org-rainbow-delimiters-depth-3">)</span><span class="org-type">*</span>sqrt<span class="org-rainbow-delimiters-depth-3">(</span><span class="org-rainbow-delimiters-depth-4">(</span><span class="org-highlight-numbers-number">1</span><span class="org-type">-</span><span class="org-rainbow-delimiters-depth-5">(</span>G0<span class="org-type">/</span>Gc<span class="org-rainbow-delimiters-depth-5">)</span><span class="org-type">^</span><span class="org-rainbow-delimiters-depth-5">(</span><span class="org-highlight-numbers-number">2</span><span class="org-type">/</span>n<span class="org-rainbow-delimiters-depth-5">)</span><span class="org-rainbow-delimiters-depth-4">)</span><span class="org-type">/</span><span class="org-rainbow-delimiters-depth-4">(</span><span class="org-highlight-numbers-number">1</span><span class="org-type">-</span><span class="org-rainbow-delimiters-depth-5">(</span>Gc<span class="org-type">/</span>G1<span class="org-rainbow-delimiters-depth-5">)</span><span class="org-type">^</span><span class="org-rainbow-delimiters-depth-5">(</span><span class="org-highlight-numbers-number">2</span><span class="org-type">/</span>n<span class="org-rainbow-delimiters-depth-5">)</span><span class="org-rainbow-delimiters-depth-4">)</span><span class="org-rainbow-delimiters-depth-3">)</span><span class="org-type">*</span>s <span class="org-type">+</span> <span class="org-rainbow-delimiters-depth-3">(</span>G0<span class="org-type">/</span>Gc<span class="org-rainbow-delimiters-depth-3">)</span><span class="org-type">^</span><span class="org-rainbow-delimiters-depth-3">(</span><span class="org-highlight-numbers-number">1</span><span class="org-type">/</span>n<span class="org-rainbow-delimiters-depth-3">)</span><span class="org-rainbow-delimiters-depth-2">)</span><span class="org-type">/</span><span class="org-rainbow-delimiters-depth-2">(</span><span class="org-rainbow-delimiters-depth-3">(</span><span class="org-highlight-numbers-number">1</span><span class="org-type">/</span>G1<span class="org-rainbow-delimiters-depth-3">)</span><span class="org-type">^</span><span class="org-rainbow-delimiters-depth-3">(</span><span class="org-highlight-numbers-number">1</span><span class="org-type">/</span>n<span class="org-rainbow-delimiters-depth-3">)</span><span class="org-type">*</span><span class="org-rainbow-delimiters-depth-3">(</span><span class="org-highlight-numbers-number">1</span><span class="org-type">/</span>w0<span class="org-rainbow-delimiters-depth-3">)</span><span class="org-type">*</span>sqrt<span class="org-rainbow-delimiters-depth-3">(</span><span class="org-rainbow-delimiters-depth-4">(</span><span class="org-highlight-numbers-number">1</span><span class="org-type">-</span><span class="org-rainbow-delimiters-depth-5">(</span>G0<span class="org-type">/</span>Gc<span class="org-rainbow-delimiters-depth-5">)</span><span class="org-type">^</span><span class="org-rainbow-delimiters-depth-5">(</span><span class="org-highlight-numbers-number">2</span><span class="org-type">/</span>n<span class="org-rainbow-delimiters-depth-5">)</span><span class="org-rainbow-delimiters-depth-4">)</span><span class="org-type">/</span><span class="org-rainbow-delimiters-depth-4">(</span><span class="org-highlight-numbers-number">1</span><span class="org-type">-</span><span class="org-rainbow-delimiters-depth-5">(</span>Gc<span class="org-type">/</span>G1<span class="org-rainbow-delimiters-depth-5">)</span><span class="org-type">^</span><span class="org-rainbow-delimiters-depth-5">(</span><span class="org-highlight-numbers-number">2</span><span class="org-type">/</span>n<span class="org-rainbow-delimiters-depth-5">)</span><span class="org-rainbow-delimiters-depth-4">)</span><span class="org-rainbow-delimiters-depth-3">)</span><span class="org-type">*</span>s <span class="org-type">+</span> <span class="org-rainbow-delimiters-depth-3">(</span><span class="org-highlight-numbers-number">1</span><span class="org-type">/</span>Gc<span class="org-rainbow-delimiters-depth-3">)</span><span class="org-type">^</span><span class="org-rainbow-delimiters-depth-3">(</span><span class="org-highlight-numbers-number">1</span><span class="org-type">/</span>n<span class="org-rainbow-delimiters-depth-3">)</span><span class="org-rainbow-delimiters-depth-2">)</span><span class="org-rainbow-delimiters-depth-1">)</span><span class="org-type">^</span>n;
+</pre>
+</div>
+
+<div class="org-src-container">
+<pre class="src src-matlab">P = <span class="org-rainbow-delimiters-depth-1">[</span>W1 <span class="org-type">-</span>W1;
+     <span class="org-highlight-numbers-number">0</span>   W2;
+     <span class="org-highlight-numbers-number">1</span>   <span class="org-highlight-numbers-number">0</span><span class="org-rainbow-delimiters-depth-1">]</span>;
+</pre>
+</div>
+
+<p>
+And we do the \(\mathcal{H}_\infty\) synthesis using the <code>hinfsyn</code> command.
+</p>
+<div class="org-src-container">
+<pre class="src src-matlab"><span class="org-rainbow-delimiters-depth-1">[</span>H2, <span class="org-type">~</span>, gamma, <span class="org-type">~</span><span class="org-rainbow-delimiters-depth-1">]</span> = hinfsyn<span class="org-rainbow-delimiters-depth-1">(</span>P, <span class="org-highlight-numbers-number">1</span>, <span class="org-highlight-numbers-number">1</span>,'TOLGAM', <span class="org-highlight-numbers-number">0</span>.<span class="org-highlight-numbers-number">001</span>, 'METHOD', 'ric', 'DISPLAY', 'on'<span class="org-rainbow-delimiters-depth-1">)</span>;
+</pre>
+</div>
+
+<pre class="example">
+[H2, ~, gamma, ~] = hinfsyn(P, 1, 1,'TOLGAM', 0.001, 'METHOD', 'ric', 'DISPLAY', 'on');
+Resetting value of Gamma min based on D_11, D_12, D_21 terms
+
+Test bounds:      0.1000 &lt;  gamma  &lt;=  10500.0000
+
+  gamma    hamx_eig  xinf_eig  hamy_eig   yinf_eig   nrho_xy   p/f
+1.050e+04   2.1e+01 -3.0e-07   7.8e+00   -1.3e-15    0.0000    p
+5.250e+03   2.1e+01 -1.5e-08   7.8e+00   -5.8e-14    0.0000    p
+2.625e+03   2.1e+01   2.5e-10   7.8e+00   -3.7e-12    0.0000    p
+1.313e+03   2.1e+01 -3.2e-11   7.8e+00   -7.3e-14    0.0000    p
+  656.344   2.1e+01 -2.2e-10   7.8e+00   -1.1e-15    0.0000    p
+  328.222   2.1e+01 -1.1e-10   7.8e+00   -1.2e-15    0.0000    p
+  164.161   2.1e+01 -2.4e-08   7.8e+00   -8.9e-16    0.0000    p
+   82.130   2.1e+01   2.0e-10   7.8e+00   -9.1e-31    0.0000    p
+   41.115   2.1e+01 -6.8e-09   7.8e+00   -4.1e-13    0.0000    p
+   20.608   2.1e+01   3.3e-10   7.8e+00   -1.4e-12    0.0000    p
+   10.354   2.1e+01 -9.8e-09   7.8e+00   -1.8e-15    0.0000    p
+    5.227   2.1e+01 -4.1e-09   7.8e+00   -2.5e-12    0.0000    p
+    2.663   2.1e+01   2.7e-10   7.8e+00   -4.0e-14    0.0000    p
+    1.382   2.1e+01 -3.2e+05#  7.8e+00   -3.5e-14    0.0000    f
+    2.023   2.1e+01 -5.0e-10   7.8e+00    0.0e+00    0.0000    p
+    1.702   2.1e+01 -2.4e+07#  7.8e+00   -1.6e-13    0.0000    f
+    1.862   2.1e+01 -6.0e+08#  7.8e+00   -1.0e-12    0.0000    f
+    1.942   2.1e+01 -2.8e-09   7.8e+00   -8.1e-14    0.0000    p
+    1.902   2.1e+01 -2.5e-09   7.8e+00   -1.1e-13    0.0000    p
+    1.882   2.1e+01 -9.3e-09   7.8e+00   -2.0e-15    0.0001    p
+    1.872   2.1e+01 -1.3e+09#  7.8e+00   -3.6e-22    0.0000    f
+    1.877   2.1e+01 -2.6e+09#  7.8e+00   -1.2e-13    0.0000    f
+    1.880   2.1e+01 -5.6e+09#  7.8e+00   -1.4e-13    0.0000    f
+    1.881   2.1e+01 -1.2e+10#  7.8e+00   -3.3e-12    0.0000    f
+    1.882   2.1e+01 -3.2e+10#  7.8e+00   -8.5e-14    0.0001    f
+
+ Gamma value achieved:     1.8824
+</pre>
+
+<div class="org-src-container">
+<pre class="src src-matlab">H1 = <span class="org-highlight-numbers-number">1</span> <span class="org-type">-</span> H2;
+</pre>
+</div>
+
+<table border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
+
+
+<colgroup>
+<col  class="org-left" />
+
+<col  class="org-right" />
+</colgroup>
+<thead>
+<tr>
+<th scope="col" class="org-left">&#xa0;</th>
+<th scope="col" class="org-right">rms value</th>
+</tr>
+</thead>
+<tbody>
+<tr>
+<td class="org-left">Sensor 1</td>
+<td class="org-right">1.1e-02</td>
+</tr>
+
+<tr>
+<td class="org-left">Sensor 2</td>
+<td class="org-right">1.3e-03</td>
+</tr>
+
+<tr>
+<td class="org-left">Optimal Sensor Fusion</td>
+<td class="org-right">3.2e-04</td>
+</tr>
+</tbody>
+</table>
+</div>
+</div>
+
+<div id="outline-container-org6952735" class="outline-4">
+<h4 id="org6952735"><span class="section-number-4">1.5.2</span> Second idea</h4>
+<div class="outline-text-4" id="text-1-5-2">
+<p>
+We have that:
+\[ \Phi_{n_{ss}} = \left|H_1\right|^2 \Phi_{n_1} + \left|H_2\right|^2 \Phi_{n_2} \]
+</p>
+
+<p>
+We model the Power Spectral Densities of the two sensors by weighting functions \(N_1(s)\) and \(N_2(s)\).
+\[ \Phi_{n_{ss}} = \left|H_1 N_1\right|^2 + \left|H_2\right N_2|^2 \]
+</p>
+
+<p>
+Then, at frequencies where \(|H_1| < |H_2|\) we would like that \(|N_1| = 1\) and \(|N_2| = 0\) as we discussed before.
+Then \(|H_1 N_1|^2 + |H_2 N_2|^2 = |H_1 N_1|^2\).
+</p>
+
+<p>
+However, if we design \(H_2\) such that when \(|N_1| < |N_2|\), we have that:
+\[ |H_2 N_2|^2 = \epsilon |H_1 N_1|^2 \]
+And we have:
+</p>
+\begin{align*}
+  \Phi_{n_{ss}} &= \left|H_1 N_1\right|^2 + |H_2 N_2|^2 \\
+                &= (1 + \epsilon)\left|H_1 N_1\right|^2 \\
+                &\approx \left|H_1 N_1\right|^2
+\end{align*}
+
+<p>
+We have then that:
+\[ |H_2 N_2| = \sqrt{\epsilon} |H_1 N_1| \]
+We suppose that \(|H_1| = 1\):
+\[ |H_2| = \sqrt{\epsilon} \frac{|N_1|}{|N_2|} \]
+We can use that for the maximum magnitude of \(|H_2|\) at frequencies where \(|N_1| < |N_2|\)
+</p>
+
+<p>
+Similarly, we have that the maximum magnitude of \(|H_1|\) at frequencies where \(|N_1| > |N_2|\) is:
+\[ |H_1| = \sqrt{\epsilon} \frac{|N_2|}{|N_1|} \]
+</p>
+
+<p>
+We could take \(\epsilon = 1\), then we increase the noise of the super sensor just by a factor \(\sqrt{2}\) over the all bandwidth from the idea case.
+</p>
+
+<p>
+This could be used as weighting functions for the \(\mathcal{H}_\infty\) synthesis of the complementary filters.
+</p>
+
+<div class="org-src-container">
+<pre class="src src-matlab">omegac = <span class="org-highlight-numbers-number">100</span><span class="org-type">*</span><span class="org-highlight-numbers-number">2</span><span class="org-type">*</span><span class="org-constant">pi</span>; G0 = <span class="org-highlight-numbers-number">1e</span><span class="org-type">-</span><span class="org-highlight-numbers-number">5</span>; Ginf = <span class="org-highlight-numbers-number">1e</span><span class="org-type">-</span><span class="org-highlight-numbers-number">4</span>;
+N1 = <span class="org-rainbow-delimiters-depth-1">(</span>Ginf<span class="org-type">*</span>s<span class="org-type">/</span>omegac <span class="org-type">+</span> G0<span class="org-rainbow-delimiters-depth-1">)</span><span class="org-type">/</span><span class="org-rainbow-delimiters-depth-1">(</span>s<span class="org-type">/</span>omegac <span class="org-type">+</span> <span class="org-highlight-numbers-number">1</span><span class="org-rainbow-delimiters-depth-1">)</span><span class="org-type">/</span><span class="org-rainbow-delimiters-depth-1">(</span><span class="org-highlight-numbers-number">1</span> <span class="org-type">+</span> s<span class="org-type">/</span><span class="org-highlight-numbers-number">2</span><span class="org-type">/</span><span class="org-constant">pi</span><span class="org-type">/</span><span class="org-highlight-numbers-number">4000</span><span class="org-rainbow-delimiters-depth-1">)</span>;
+
+omegac = <span class="org-highlight-numbers-number">1</span><span class="org-type">*</span><span class="org-highlight-numbers-number">2</span><span class="org-type">*</span><span class="org-constant">pi</span>; G0 = <span class="org-highlight-numbers-number">1e</span><span class="org-type">-</span><span class="org-highlight-numbers-number">3</span>; Ginf = <span class="org-highlight-numbers-number">1e</span><span class="org-type">-</span><span class="org-highlight-numbers-number">8</span>;
+N2 = <span class="org-rainbow-delimiters-depth-1">(</span><span class="org-rainbow-delimiters-depth-2">(</span>sqrt<span class="org-rainbow-delimiters-depth-3">(</span>Ginf<span class="org-rainbow-delimiters-depth-3">)</span><span class="org-type">*</span>s<span class="org-type">/</span>omegac <span class="org-type">+</span> sqrt<span class="org-rainbow-delimiters-depth-3">(</span>G0<span class="org-rainbow-delimiters-depth-3">)</span><span class="org-rainbow-delimiters-depth-2">)</span><span class="org-type">/</span><span class="org-rainbow-delimiters-depth-2">(</span>s<span class="org-type">/</span>omegac <span class="org-type">+</span> <span class="org-highlight-numbers-number">1</span><span class="org-rainbow-delimiters-depth-2">)</span><span class="org-rainbow-delimiters-depth-1">)</span><span class="org-type">^</span><span class="org-highlight-numbers-number">2</span><span class="org-type">/</span><span class="org-rainbow-delimiters-depth-1">(</span><span class="org-highlight-numbers-number">1</span> <span class="org-type">+</span> s<span class="org-type">/</span><span class="org-highlight-numbers-number">2</span><span class="org-type">/</span><span class="org-constant">pi</span><span class="org-type">/</span><span class="org-highlight-numbers-number">4000</span><span class="org-rainbow-delimiters-depth-1">)</span><span class="org-type">^</span><span class="org-highlight-numbers-number">2</span>;
+</pre>
+</div>
+
+<div class="org-src-container">
+<pre class="src src-matlab">W1 = N1<span class="org-type">/</span>N2;
+W2 = N2<span class="org-type">/</span>N1;
+
+W1 = W1<span class="org-type">/</span><span class="org-rainbow-delimiters-depth-1">(</span><span class="org-highlight-numbers-number">1</span> <span class="org-type">+</span> s<span class="org-type">/</span><span class="org-highlight-numbers-number">2</span><span class="org-type">/</span><span class="org-constant">pi</span><span class="org-type">/</span><span class="org-highlight-numbers-number">1000</span><span class="org-rainbow-delimiters-depth-1">)</span>; <span class="org-comment">% this is added so that it is proper</span>
+</pre>
+</div>
+
+<div class="org-src-container">
+<pre class="src src-matlab">P = <span class="org-rainbow-delimiters-depth-1">[</span>W1 <span class="org-type">-</span>W1;
+     <span class="org-highlight-numbers-number">0</span>   W2;
+     <span class="org-highlight-numbers-number">1</span>   <span class="org-highlight-numbers-number">0</span><span class="org-rainbow-delimiters-depth-1">]</span>;
+</pre>
+</div>
+
+<p>
+And we do the \(\mathcal{H}_\infty\) synthesis using the <code>hinfsyn</code> command.
+</p>
+<div class="org-src-container">
+<pre class="src src-matlab"><span class="org-rainbow-delimiters-depth-1">[</span>H2, <span class="org-type">~</span>, gamma, <span class="org-type">~</span><span class="org-rainbow-delimiters-depth-1">]</span> = hinfsyn<span class="org-rainbow-delimiters-depth-1">(</span>P, <span class="org-highlight-numbers-number">1</span>, <span class="org-highlight-numbers-number">1</span>,'TOLGAM', <span class="org-highlight-numbers-number">0</span>.<span class="org-highlight-numbers-number">001</span>, 'METHOD', 'ric', 'DISPLAY', 'on'<span class="org-rainbow-delimiters-depth-1">)</span>;
+</pre>
+</div>
+
+<pre class="example">
+[H2, ~, gamma, ~] = hinfsyn(P, 1, 1,'TOLGAM', 0.001, 'METHOD', 'ric', 'DISPLAY', 'on');
+Test bounds:      0.0000 &lt;  gamma  &lt;=   2625.0000
+
+  gamma    hamx_eig  xinf_eig  hamy_eig   yinf_eig   nrho_xy   p/f
+2.625e+03   1.8e+01 -2.3e-12   6.3e+00   -1.7e-14    0.0000    p
+1.313e+03   1.8e+01 -3.3e-12   6.3e+00   -5.9e-13    0.0000    p
+  656.250   1.8e+01 -1.8e-12   6.3e+00   -1.2e-13    0.0000    p
+  328.125   1.8e+01 -8.3e-13   6.3e+00   -7.5e-14    0.0000    p
+  164.063   1.8e+01 -3.8e-13   6.3e+00   -6.1e-14    0.0000    p
+   82.031   1.8e+01 -3.3e-22   6.3e+00   -5.4e-14    0.0000    p
+   41.016   1.8e+01 -3.4e-12   6.3e+00   -1.4e-16    0.0000    p
+   20.508   1.8e+01 -6.4e-14   6.3e+00   -4.4e-13    0.0000    p
+   10.254   1.8e+01 -1.4e-13   6.3e+00   -4.9e-14    0.0000    p
+    5.127   1.7e+01 -1.1e-12   6.3e+00   -1.6e-13    0.0000    p
+    2.563   1.7e+01 -6.5e-13   6.3e+00   -7.3e-14    0.0000    p
+    1.282   1.4e+01 -8.9e+04#  6.3e+00   -1.6e-14    0.0000    f
+    1.923   1.6e+01 -4.6e+05#  6.3e+00   -2.1e-13    0.0000    f
+    2.243   1.7e+01 -1.9e+06#  6.3e+00   -9.9e-14    0.0000    f
+    2.403   1.7e+01 -3.7e-11   6.3e+00   -2.1e-12    0.0000    p
+    2.323   1.7e+01 -4.6e+06#  6.3e+00   -5.8e-14    0.0000    f
+    2.363   1.7e+01 -1.2e+07#  6.3e+00   -3.3e-14    0.0000    f
+    2.383   1.7e+01 -6.8e+07#  6.3e+00   -1.2e-13    0.0000    f
+    2.393   1.7e+01 -5.5e-11   6.3e+00   -1.8e-12    0.0000    p
+    2.388   1.7e+01   1.3e-22   6.3e+00   -2.5e-15    0.0000    p
+    2.386   1.7e+01 -1.5e+08#  6.3e+00   -1.4e-13    0.0000    f
+    2.387   1.7e+01 -4.0e+08#  6.3e+00   -1.0e-13    0.0000    f
+    2.388   1.7e+01 -2.1e+09#  6.3e+00   -4.2e-13    0.0000    f
+
+ Gamma value achieved:     2.3882
+</pre>
+
+<div class="org-src-container">
+<pre class="src src-matlab">H1 = <span class="org-highlight-numbers-number">1</span> <span class="org-type">-</span> H2;
+</pre>
+</div>
+
+<table border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
+
+
+<colgroup>
+<col  class="org-left" />
+
+<col  class="org-right" />
+</colgroup>
+<thead>
+<tr>
+<th scope="col" class="org-left">&#xa0;</th>
+<th scope="col" class="org-right">rms value</th>
+</tr>
+</thead>
+<tbody>
+<tr>
+<td class="org-left">Sensor 1</td>
+<td class="org-right">1.1e-02</td>
+</tr>
+
+<tr>
+<td class="org-left">Sensor 2</td>
+<td class="org-right">1.3e-03</td>
+</tr>
+
+<tr>
+<td class="org-left">Optimal Sensor Fusion</td>
+<td class="org-right">1.9e-04</td>
+</tr>
+</tbody>
+</table>
+</div>
+</div>
+
+<div id="outline-container-orgd67bea5" class="outline-4">
+<h4 id="orgd67bea5"><span class="section-number-4">1.5.3</span> Conclusion</h4>
+<div class="outline-text-4" id="text-1-5-3">
+<p>
+From the above two experiments with \(\mathcal{H}_\infty\) it still seems that the \(\mathcal{H}_2\) synthesis gives the complementary filters that permits to obtain the minimal super sensor noise (when measuring with the \(\mathcal{H}_2\) norm)
+</p>
+</div>
+</div>
+</div>
+</div>
+
+<div id="outline-container-org52c42b7" class="outline-2">
+<h2 id="org52c42b7"><span class="section-number-2">2</span> Robustness to sensor dynamics uncertainty</h2>
 <div class="outline-text-2" id="text-2">
 <p>
-<a id="orgb2ab616"></a>
+<a id="org6776beb"></a>
 </p>
 <p>
-Let's first consider ideal sensors where \(G_1 = 1\) and \(G_2 = 1\) (figure <a href="#orgd0780c6">7</a>).
+Let's first consider ideal sensors where \(G_1 = 1\) and \(G_2 = 1\) (figure <a href="#orgac3aa1e">7</a>).
 </p>
 
 
-<div id="orgd0780c6" class="figure">
+<div id="orgac3aa1e" class="figure">
 <p><img src="figs/fusion_two_noisy_sensors_with_dyn_bis.png" alt="fusion_two_noisy_sensors_with_dyn_bis.png" />
 </p>
 <p><span class="figure-number">Figure 7: </span>Fusion of two sensors</p>
@@ -632,16 +965,16 @@ All the files (data and Matlab scripts) are accessible <a href="data/comp_filter
 </div>
 </div>
 
-<div id="outline-container-orgf63dc07" class="outline-3">
-<h3 id="orgf63dc07"><span class="section-number-3">2.1</span> Unknown sensor dynamics dynamics</h3>
+<div id="outline-container-orga0447f7" class="outline-3">
+<h3 id="orga0447f7"><span class="section-number-3">2.1</span> Unknown sensor dynamics dynamics</h3>
 <div class="outline-text-3" id="text-2-1">
 <p>
 In practical systems, the sensor dynamics has always some level of uncertainty.
-Let's represent that with multiplicative input uncertainty as shown on figure <a href="#orga6c7451">8</a>.
+Let's represent that with multiplicative input uncertainty as shown on figure <a href="#org6cad0f2">8</a>.
 </p>
 
 
-<div id="orga6c7451" class="figure">
+<div id="org6cad0f2" class="figure">
 <p><img src="figs-tikz/fusion_gain_mismatch.png" alt="fusion_gain_mismatch.png" />
 </p>
 <p><span class="figure-number">Figure 8: </span>Fusion of two sensors with input multiplicative uncertainty</p>
@@ -686,7 +1019,7 @@ Which is approximately the same as requiring
 </p>
 
 <p>
-The uncertainty set of the transfer function from \(\hat{x}\) to \(x\) is bounded in the complex plane by a circle centered on 1 and with a radius equal to \(\epsilon\) (figure <a href="#org5dfc170">9</a>).
+The uncertainty set of the transfer function from \(\hat{x}\) to \(x\) is bounded in the complex plane by a circle centered on 1 and with a radius equal to \(\epsilon\) (figure <a href="#orgc4f401a">9</a>).
 </p>
 
 <p>
@@ -695,7 +1028,7 @@ We then have that the angle introduced by the super sensor is bounded by \(\arcs
 </p>
 
 
-<div id="org5dfc170" class="figure">
+<div id="orgc4f401a" class="figure">
 <p><img src="figs-tikz/uncertainty_gain_phase_variation.png" alt="uncertainty_gain_phase_variation.png" />
 </p>
 <p><span class="figure-number">Figure 9: </span>Maximum phase variation</p>
@@ -707,8 +1040,8 @@ Thus, we choose should choose \(\epsilon\) so that the maximum phase uncertainty
 </div>
 </div>
 
-<div id="outline-container-org3244858" class="outline-3">
-<h3 id="org3244858"><span class="section-number-3">2.2</span> Design the complementary filters in order to limit the phase and gain uncertainty of the super sensor</h3>
+<div id="outline-container-org4b0c13d" class="outline-3">
+<h3 id="org4b0c13d"><span class="section-number-3">2.2</span> Design the complementary filters in order to limit the phase and gain uncertainty of the super sensor</h3>
 <div class="outline-text-3" id="text-2-2">
 <p>
 Let's say the two sensors dynamics \(H_1\) and \(H_2\) have been identified with the associated uncertainty weights \(W_1\) and \(W_2\).
@@ -741,8 +1074,8 @@ This is of primary importance in order to ensure the stability of the feedback l
 </div>
 </div>
 
-<div id="outline-container-org20ae980" class="outline-3">
-<h3 id="org20ae980"><span class="section-number-3">2.3</span> First Basic Example with gain mismatch</h3>
+<div id="outline-container-orge9f4bde" class="outline-3">
+<h3 id="orge9f4bde"><span class="section-number-3">2.3</span> First Basic Example with gain mismatch</h3>
 <div class="outline-text-3" id="text-2-3">
 <p>
 Let's consider two ideal sensors except one sensor has not an expected gain of one but a gain of \(0.6\).
@@ -754,19 +1087,19 @@ G2 = <span class="org-highlight-numbers-number">0</span>.<span class="org-highli
 </div>
 
 <p>
-Let's design two complementary filters as shown on figure <a href="#orgc3e7049">10</a>.
+Let's design two complementary filters as shown on figure <a href="#orgfa47686">10</a>.
 The complementary filters shown in blue does not present a bump as the red ones but provides less sensor separation at high and low frequencies.
 </p>
 
 
-<div id="orgc3e7049" class="figure">
+<div id="orgfa47686" class="figure">
 <p><img src="figs/comp_filters_robustness_test.png" alt="comp_filters_robustness_test.png" />
 </p>
 <p><span class="figure-number">Figure 10: </span>The two complementary filters designed for the robustness test (<a href="./figs/comp_filters_robustness_test.png">png</a>, <a href="./figs/comp_filters_robustness_test.pdf">pdf</a>)</p>
 </div>
 
 <p>
-We then compute the bode plot of the super sensor transfer function \(H_1*G_1 + H_2*G_2\) for both complementary filters pair (figure <a href="#org124e3b5">11</a>).
+We then compute the bode plot of the super sensor transfer function \(H_1*G_1 + H_2*G_2\) for both complementary filters pair (figure <a href="#org56b6e89">11</a>).
 </p>
 
 <p>
@@ -774,7 +1107,7 @@ We see that the blue complementary filters with a lower maximum norm permits to
 </p>
 
 
-<div id="org124e3b5" class="figure">
+<div id="org56b6e89" class="figure">
 <p><img src="figs/tf_super_sensor_comp.png" alt="tf_super_sensor_comp.png" />
 </p>
 <p><span class="figure-number">Figure 11: </span>Comparison of the obtained super sensor transfer functions (<a href="./figs/tf_super_sensor_comp.png">png</a>, <a href="./figs/tf_super_sensor_comp.pdf">pdf</a>)</p>
@@ -782,15 +1115,368 @@ We see that the blue complementary filters with a lower maximum norm permits to
 </div>
 </div>
 
-<div id="outline-container-org41f2b11" class="outline-3">
-<h3 id="org41f2b11"><span class="section-number-3">2.4</span> <span class="todo TODO">TODO</span> More Complete example with model uncertainty</h3>
+<div id="outline-container-org551c5ec" class="outline-3">
+<h3 id="org551c5ec"><span class="section-number-3">2.4</span> <span class="todo TODO">TODO</span> More Complete example with model uncertainty</h3>
 </div>
 </div>
-<div id="outline-container-orgf04a9a4" class="outline-2">
-<h2 id="orgf04a9a4"><span class="section-number-2">3</span> Complementary filters using analytical formula</h2>
+<div id="outline-container-org861c0eb" class="outline-2">
+<h2 id="org861c0eb"><span class="section-number-2">3</span> Equivalent Super Sensor</h2>
 <div class="outline-text-2" id="text-3">
 <p>
-<a id="org654b360"></a>
+The goal here is to find the parameters of a single sensor that would best represent a super sensor.
+</p>
+</div>
+
+<div id="outline-container-org51db5bb" class="outline-3">
+<h3 id="org51db5bb"><span class="section-number-3">3.1</span> Sensor Fusion Architecture</h3>
+<div class="outline-text-3" id="text-3-1">
+<p>
+Let consider figure <a href="#orgeb35c78">12</a> where two sensors are merged.
+The dynamic uncertainty of each sensor is represented by a weight \(w_i(s)\), the frequency characteristics each of the sensor noise is represented by the weights \(N_i(s)\).
+The noise sources \(\tilde{n}_i\) are considered to be white noise: \(\Phi_{\tilde{n}_i}(\omega) = 1, \ \forall\omega\).
+</p>
+
+
+<div id="orgeb35c78" class="figure">
+<p><img src="figs-tikz/sensor_fusion_full.png" alt="sensor_fusion_full.png" />
+</p>
+<p><span class="figure-number">Figure 12: </span>Sensor fusion architecture (<a href="./figs/sensor_fusion_full.png">png</a>, <a href="./figs/sensor_fusion_full.pdf">pdf</a>).</p>
+</div>
+
+
+\begin{align*}
+  \hat{x} &= H_1(s) N_1(s) \tilde{n}_1 + H_2(s) N_2(s) \tilde{n}_2 \\
+          &\quad \quad + \Big(H_1(s) \big(1 + w_1(s) \Delta_1(s)\big) + H_2(s) \big(1 + w_2(s) \Delta_2(s)\big)\Big) x \\
+          &= H_1(s) N_1(s) \tilde{n}_1 + H_2(s) N_2(s) \tilde{n}_2 \\
+          &\quad \quad + \big(1 + H_1(s) w_1(s) \Delta_1(s) + H_2(s) w_2(s) \Delta_2(s)\big) x
+\end{align*}
+
+<p>
+To the dynamics of the super sensor is:
+</p>
+\begin{equation}
+  \frac{\hat{x}}{x} = 1 + H_1(s) w_1(s) \Delta_1(s) + H_2(s) w_2(s) \Delta_2(s)
+\end{equation}
+
+<p>
+And the noise of the super sensor is:
+</p>
+\begin{equation}
+  n_{ss} = H_1(s) N_1(s) \tilde{n}_1 + H_2(s) N_2(s) \tilde{n}_2
+\end{equation}
+</div>
+</div>
+
+<div id="outline-container-orgc301d63" class="outline-3">
+<h3 id="orgc301d63"><span class="section-number-3">3.2</span> Equivalent Configuration</h3>
+<div class="outline-text-3" id="text-3-2">
+<p>
+We try to determine \(w_{ss}(s)\) and \(N_{ss}(s)\) such that the sensor on figure <a href="#org663de1e">13</a> is equivalent to the super sensor of figure <a href="#orgeb35c78">12</a>.
+</p>
+
+
+<div id="org663de1e" class="figure">
+<p><img src="figs-tikz/sensor_fusion_equivalent.png" alt="sensor_fusion_equivalent.png" />
+</p>
+<p><span class="figure-number">Figure 13: </span>Equivalent Super Sensor (<a href="./figs/sensor_fusion_equivalent.png">png</a>, <a href="./figs/sensor_fusion_equivalent.pdf">pdf</a>).</p>
+</div>
+</div>
+</div>
+
+<div id="outline-container-orge929495" class="outline-3">
+<h3 id="orge929495"><span class="section-number-3">3.3</span> Model the uncertainty of the super sensor</h3>
+<div class="outline-text-3" id="text-3-3">
+<p>
+At each frequency \(\omega\), the uncertainty set of the super sensor shown on figure <a href="#orgeb35c78">12</a> is a circle centered on \(1\) with a radius equal to \(|H_1(j\omega) w_1(j\omega)| + |H_2(j\omega) w_2(j\omega)|\) on the complex plane.
+The uncertainty set of the sensor shown on figure <a href="#org663de1e">13</a> is a circle centered on \(1\) with a radius equal to \(|w_{ss}(j\omega)|\) on the complex plane.
+</p>
+
+<p>
+Ideally, we want to find a weight \(w_{ss}(s)\) so that:
+</p>
+<div class="important">
+<p>
+\[ |w_{ss}(j\omega)| = |H_1(j\omega) w_1(j\omega)| + |H_2(j\omega) w_2(j\omega)|, \quad \forall\omega \]
+</p>
+
+</div>
+</div>
+</div>
+
+<div id="outline-container-org762c80b" class="outline-3">
+<h3 id="org762c80b"><span class="section-number-3">3.4</span> Model the noise of the super sensor</h3>
+<div class="outline-text-3" id="text-3-4">
+<p>
+The PSD of the estimation \(\hat{x}\) when \(x = 0\) of the configuration shown on figure <a href="#orgeb35c78">12</a> is:
+</p>
+\begin{align*}
+  \Phi_{\hat{x}}(\omega) &= | H_1(j\omega) N_1(j\omega) |^2 \Phi_{\tilde{n}_1} + | H_2(j\omega) N_2(j\omega) |^2 \Phi_{\tilde{n}_2} \\
+                         &= | H_1(j\omega) N_1(j\omega) |^2 + | H_2(j\omega) N_2(j\omega) |^2
+\end{align*}
+
+<p>
+The PSD of the estimation \(\hat{x}\) when \(x = 0\) of the configuration shown on figure <a href="#org663de1e">13</a> is:
+</p>
+\begin{align*}
+  \Phi_{\hat{x}}(\omega) &= | N_{ss}(j\omega) |^2 \Phi_{\tilde{n}} \\
+                         &= | N_{ss}(j\omega) |^2
+\end{align*}
+
+<p>
+Ideally, we want to find a weight \(N_{ss}(s)\) such that:
+</p>
+<div class="important">
+<p>
+\[ |N_{ss}(j\omega)|^2 = | H_1(j\omega) N_1(j\omega) |^2 + | H_2(j\omega) N_2(j\omega) |^2 \quad \forall\omega \]
+</p>
+
+</div>
+</div>
+</div>
+
+<div id="outline-container-org01817fc" class="outline-3">
+<h3 id="org01817fc"><span class="section-number-3">3.5</span> First guess</h3>
+<div class="outline-text-3" id="text-3-5">
+<p>
+We could choose
+</p>
+\begin{align*}
+  w_{ss}(s) &= H_1(s) w_1(s) + H_2(s) w_2(s) \\
+  N_{ss}(s) &= H_1(s) N_1(s) + H_2(s) N_2(s)
+\end{align*}
+
+<p>
+But we would have:
+</p>
+\begin{align*}
+  |w_{ss}(j\omega)| &= |H_1(j\omega) w_1(j\omega) + H_2(j\omega) w_2(j\omega)|, \quad \forall\omega \\
+                    &\neq |H_1(j\omega) w_1(j\omega)| + |H_2(j\omega) w_2(j\omega)|, \quad \forall\omega
+\end{align*}
+<p>
+and
+</p>
+\begin{align*}
+  |N_{ss}(j\omega)|^2 &= | H_1(j\omega) N_1(j\omega) + H_2(j\omega) N_2(j\omega) |^2 \quad \forall\omega \\
+                      &\neq | H_1(j\omega) N_1(j\omega)|^2 + |H_2(j\omega) N_2(j\omega) |^2 \quad \forall\omega \\
+\end{align*}
+</div>
+</div>
+</div>
+
+<div id="outline-container-orgacde10f" class="outline-2">
+<h2 id="orgacde10f"><span class="section-number-2">4</span> Optimal And Robust Sensor Fusion in Practice</h2>
+<div class="outline-text-2" id="text-4">
+<p>
+Here are the steps in order to apply optimal and robust sensor fusion:
+</p>
+
+<ul class="org-ul">
+<li>Measure the noise characteristics of the sensors to be merged (necessary for "optimal" part of the fusion)</li>
+<li>Measure/Estimate the dynamic uncertainty of the sensors (necessary for "robust" part of the fusion)</li>
+<li>Apply H2/H-infinity synthesis of the complementary filters</li>
+</ul>
+</div>
+
+<div id="outline-container-orgd36ee6e" class="outline-3">
+<h3 id="orgd36ee6e"><span class="section-number-3">4.1</span> Measurement of the noise characteristics of the sensors</h3>
+<div class="outline-text-3" id="text-4-1">
+</div>
+<div id="outline-container-org36e1a94" class="outline-4">
+<h4 id="org36e1a94"><span class="section-number-4">4.1.1</span> Huddle Test</h4>
+<div class="outline-text-4" id="text-4-1-1">
+<p>
+The technique to estimate the sensor noise is taken from <a class='org-ref-reference' href="#barzilai98_techn_measur_noise_sensor_presen">barzilai98_techn_measur_noise_sensor_presen</a>.
+</p>
+
+<p>
+Let's consider two sensors (sensor 1 and sensor 2) that are measuring the same quantity \(x\) as shown in figure <a href="#orgd3469c0">14</a>.
+</p>
+
+
+<div id="orgd3469c0" class="figure">
+<p><img src="figs-tikz/huddle_test.png" alt="huddle_test.png" />
+</p>
+<p><span class="figure-number">Figure 14: </span>Huddle test block diagram</p>
+</div>
+
+<p>
+Each sensor has uncorrelated noise \(n_1\) and \(n_2\) and internal dynamics \(G_1(s)\) and \(G_2(s)\) respectively.
+</p>
+
+<p>
+We here suppose that each sensor has the same magnitude of instrumental noise: \(n_1 = n_2 = n\).
+We also assume that their dynamics is ideal: \(G_1(s) = G_2(s) = 1\).
+</p>
+
+<p>
+We then have:
+</p>
+\begin{equation}
+\label{orgdaace50}
+  \gamma_{\hat{x}_1\hat{x}_2}^2(\omega) = \frac{1}{1 + 2 \left( \frac{|\Phi_n(\omega)|}{|\Phi_{\hat{x}}(\omega)|} \right) + \left( \frac{|\Phi_n(\omega)|}{|\Phi_{\hat{x}}(\omega)|} \right)^2}
+\end{equation}
+
+<p>
+Since the input signal \(x\) and the instrumental noise \(n\) are incoherent:
+</p>
+\begin{equation}
+\label{org4188c50}
+  |\Phi_{\hat{x}}(\omega)| = |\Phi_n(\omega)| + |\Phi_x(\omega)|
+\end{equation}
+
+<p>
+From equations \eqref{orgdaace50} and \eqref{org4188c50}, we finally obtain
+</p>
+<div class="important">
+\begin{equation}
+\label{orgf2631e0}
+  |\Phi_n(\omega)| = |\Phi_{\hat{x}}(\omega)| \left( 1 - \sqrt{\gamma_{\hat{x}_1\hat{x}_2}^2(\omega)} \right)
+\end{equation}
+
+</div>
+</div>
+</div>
+
+<div id="outline-container-orge63237d" class="outline-4">
+<h4 id="orge63237d"><span class="section-number-4">4.1.2</span> Weights that represents the noises' PSD</h4>
+<div class="outline-text-4" id="text-4-1-2">
+<p>
+For further complementary filter synthesis, it is preferred to consider a normalized noise source \(\tilde{n}\) that has a PSD equal to one (\(\Phi_{\tilde{n}}(\omega) = 1\)) and to use a weighting filter \(N(s)\) in order to represent the frequency dependence of the noise.
+</p>
+
+<p>
+The weighting filter \(N(s)\) should be designed such that:
+</p>
+\begin{align*}
+                      & \Phi_n(\omega) \approx |N(j\omega)|^2 \Phi_{\tilde{n}}(\omega) \quad \forall \omega \\
+  \Longleftrightarrow & |N(j\omega)| \approx \sqrt{\Phi_n(\omega)} \quad \forall \omega
+\end{align*}
+
+<p>
+These weighting filters can then be used to compare the noise level of sensors for the synthesis of complementary filters.
+</p>
+
+<p>
+The sensor with a normalized noise input is shown in figure <a href="#org51cf0cf">15</a>.
+</p>
+
+
+<div id="org51cf0cf" class="figure">
+<p><img src="figs-tikz/one_sensor_normalized_noise.png" alt="one_sensor_normalized_noise.png" />
+</p>
+<p><span class="figure-number">Figure 15: </span>One sensor with normalized noise</p>
+</div>
+</div>
+</div>
+
+<div id="outline-container-org3c415dc" class="outline-4">
+<h4 id="org3c415dc"><span class="section-number-4">4.1.3</span> Comparison of the noises' PSD</h4>
+<div class="outline-text-4" id="text-4-1-3">
+<p>
+Once the noise of the sensors to be merged have been characterized, the power spectral density of both sensors have to be compared.
+</p>
+
+<p>
+Ideally, the PSD of the noise are such that:
+</p>
+\begin{align*}
+  \Phi_{n_1}(\omega) &< \Phi_{n_2}(\omega) \text{ for } \omega < \omega_m \\
+  \Phi_{n_1}(\omega) &> \Phi_{n_2}(\omega) \text{ for } \omega > \omega_m
+\end{align*}
+</div>
+</div>
+
+<div id="outline-container-orgb6d27a5" class="outline-4">
+<h4 id="orgb6d27a5"><span class="section-number-4">4.1.4</span> Computation of the coherence, power spectral density and cross spectral density of signals</h4>
+<div class="outline-text-4" id="text-4-1-4">
+<p>
+The coherence between signals \(x\) and \(y\) is defined as follow
+\[ \gamma^2_{xy}(\omega) = \frac{|\Phi_{xy}(\omega)|^2}{|\Phi_{x}(\omega)| |\Phi_{y}(\omega)|} \]
+where \(|\Phi_x(\omega)|\) is the output Power Spectral Density (PSD) of signal \(x\) and \(|\Phi_{xy}(\omega)|\) is the Cross Spectral Density (CSD) of signal \(x\) and \(y\).
+</p>
+
+<p>
+The PSD and CSD are defined as follow:
+</p>
+\begin{align}
+  |\Phi_x(\omega)|    &= \frac{2}{n_d T} \sum^{n_d}_{n=1} \left| X_k(\omega, T) \right|^2 \\
+  |\Phi_{xy}(\omega)| &= \frac{2}{n_d T} \sum^{n_d}_{n=1} [ X_k^*(\omega, T) ] [ Y_k(\omega, T) ]
+\end{align}
+<p>
+where:
+</p>
+<ul class="org-ul">
+<li>\(n_d\) is the number for records averaged</li>
+<li>\(T\) is the length of each record</li>
+<li>\(X_k(\omega, T)\) is the finite Fourier transform of the \(k^{\text{th}}\) record</li>
+<li>\(X_k^*(\omega, T)\) is its complex conjugate</li>
+</ul>
+</div>
+</div>
+</div>
+
+<div id="outline-container-org491f8d8" class="outline-3">
+<h3 id="org491f8d8"><span class="section-number-3">4.2</span> Estimate the dynamic uncertainty of the sensors</h3>
+<div class="outline-text-3" id="text-4-2">
+<p>
+Let's consider one sensor represented on figure <a href="#org60b957e">16</a>.
+</p>
+
+<p>
+The dynamic uncertainty is represented by an input multiplicative uncertainty where \(w(s)\) is a weight that represents the level of the uncertainty.
+</p>
+
+<p>
+The goal is to accurately determine \(w(s)\) for the sensors that have to be merged.
+</p>
+
+
+<div id="org60b957e" class="figure">
+<p><img src="figs-tikz/one_sensor_dyn_uncertainty.png" alt="one_sensor_dyn_uncertainty.png" />
+</p>
+<p><span class="figure-number">Figure 16: </span>Sensor with dynamic uncertainty</p>
+</div>
+</div>
+</div>
+
+<div id="outline-container-orgaeae7ef" class="outline-3">
+<h3 id="orgaeae7ef"><span class="section-number-3">4.3</span> Optimal and Robust synthesis of the complementary filters</h3>
+<div class="outline-text-3" id="text-4-3">
+<p>
+Once the noise characteristics and dynamic uncertainty of both sensors have been determined and we have determined the following weighting functions:
+</p>
+<ul class="org-ul">
+<li>\(w_1(s)\) and \(w_2(s)\) representing the dynamic uncertainty of both sensors</li>
+<li>\(N_1(s)\) and \(N_2(s)\) representing the noise characteristics of both sensors</li>
+</ul>
+
+<p>
+The goal is to design complementary filters \(H_1(s)\) and \(H_2(s)\) shown in figure <a href="#orgeb35c78">12</a> such that:
+</p>
+<ul class="org-ul">
+<li>the uncertainty on the super sensor dynamics is minimized</li>
+<li>the noise sources \(\tilde{n}_1\) and \(\tilde{n}_2\) has the lowest possible effect on the estimation \(\hat{x}\)</li>
+</ul>
+
+
+<div id="org1089f62" class="figure">
+<p><img src="figs-tikz/sensor_fusion_full.png" alt="sensor_fusion_full.png" />
+</p>
+<p><span class="figure-number">Figure 17: </span>Sensor fusion architecture with sensor dynamics uncertainty</p>
+</div>
+</div>
+</div>
+</div>
+
+<div id="outline-container-org06754da" class="outline-2">
+<h2 id="org06754da"><span class="section-number-2">5</span> Methods of complementary filter synthesis</h2>
+<div class="outline-text-2" id="text-5">
+</div>
+<div id="outline-container-orgb77dafe" class="outline-3">
+<h3 id="orgb77dafe"><span class="section-number-3">5.1</span> Complementary filters using analytical formula</h3>
+<div class="outline-text-3" id="text-5-1">
+<p>
+<a id="org535ff31"></a>
 </p>
 <div class="note">
 <p>
@@ -800,9 +1486,9 @@ All the files (data and Matlab scripts) are accessible <a href="data/comp_filter
 </div>
 </div>
 
-<div id="outline-container-org7d339a9" class="outline-3">
-<h3 id="org7d339a9"><span class="section-number-3">3.1</span> Analytical 1st order complementary filters</h3>
-<div class="outline-text-3" id="text-3-1">
+<div id="outline-container-orga47e3d3" class="outline-4">
+<h4 id="orga47e3d3"><span class="section-number-4">5.1.1</span> Analytical 1st order complementary filters</h4>
+<div class="outline-text-4" id="text-5-1-1">
 <p>
 First order complementary filters are defined with following equations:
 </p>
@@ -812,7 +1498,7 @@ First order complementary filters are defined with following equations:
 \end{align}
 
 <p>
-Their bode plot is shown figure <a href="#orgd35e336">12</a>.
+Their bode plot is shown figure <a href="#org2e9e17d">18</a>.
 </p>
 
 <div class="org-src-container">
@@ -824,17 +1510,17 @@ Hl1 = <span class="org-highlight-numbers-number">1</span><span class="org-type">
 </div>
 
 
-<div id="orgd35e336" class="figure">
+<div id="org2e9e17d" class="figure">
 <p><img src="figs/comp_filter_1st_order.png" alt="comp_filter_1st_order.png" />
 </p>
-<p><span class="figure-number">Figure 12: </span>Bode plot of first order complementary filter (<a href="./figs/comp_filter_1st_order.png">png</a>, <a href="./figs/comp_filter_1st_order.pdf">pdf</a>)</p>
+<p><span class="figure-number">Figure 18: </span>Bode plot of first order complementary filter (<a href="./figs/comp_filter_1st_order.png">png</a>, <a href="./figs/comp_filter_1st_order.pdf">pdf</a>)</p>
 </div>
 </div>
 </div>
 
-<div id="outline-container-org8edbedb" class="outline-3">
-<h3 id="org8edbedb"><span class="section-number-3">3.2</span> Second Order Complementary Filters</h3>
-<div class="outline-text-3" id="text-3-2">
+<div id="outline-container-orgd1a9522" class="outline-4">
+<h4 id="orgd1a9522"><span class="section-number-4">5.1.2</span> Second Order Complementary Filters</h4>
+<div class="outline-text-4" id="text-5-1-2">
 <p>
 We here use analytical formula for the complementary filters \(H_L\) and \(H_H\).
 </p>
@@ -859,15 +1545,15 @@ where:
 </ul>
 
 <p>
-This is illustrated on figure <a href="#org551e9a2">13</a>.
+This is illustrated on figure <a href="#org5c74869">19</a>.
 The slope of those filters at high and low frequencies is \(-2\) and \(2\) respectively for \(H_L\) and \(H_H\).
 </p>
 
 
-<div id="org551e9a2" class="figure">
+<div id="org5c74869" class="figure">
 <p><img src="figs/comp_filters_param_alpha.png" alt="comp_filters_param_alpha.png" />
 </p>
-<p><span class="figure-number">Figure 13: </span>Effect of the parameter \(\alpha\) on the shape of the generated second order complementary filters (<a href="./figs/comp_filters_param_alpha.png">png</a>, <a href="./figs/comp_filters_param_alpha.pdf">pdf</a>)</p>
+<p><span class="figure-number">Figure 19: </span>Effect of the parameter \(\alpha\) on the shape of the generated second order complementary filters (<a href="./figs/comp_filters_param_alpha.png">png</a>, <a href="./figs/comp_filters_param_alpha.pdf">pdf</a>)</p>
 </div>
 
 <p>
@@ -882,17 +1568,17 @@ xlabel<span class="org-rainbow-delimiters-depth-1">(</span>'$<span class="org-ty
 </div>
 
 
-<div id="org9c950bd" class="figure">
+<div id="orgad8b0e4" class="figure">
 <p><img src="figs/param_alpha_hinf_norm.png" alt="param_alpha_hinf_norm.png" />
 </p>
-<p><span class="figure-number">Figure 14: </span>Evolution of the H-Infinity norm of the complementary filters with the parameter \(\alpha\) (<a href="./figs/param_alpha_hinf_norm.png">png</a>, <a href="./figs/param_alpha_hinf_norm.pdf">pdf</a>)</p>
+<p><span class="figure-number">Figure 20: </span>Evolution of the H-Infinity norm of the complementary filters with the parameter \(\alpha\) (<a href="./figs/param_alpha_hinf_norm.png">png</a>, <a href="./figs/param_alpha_hinf_norm.pdf">pdf</a>)</p>
 </div>
 </div>
 </div>
 
-<div id="outline-container-org5fd0c3c" class="outline-3">
-<h3 id="org5fd0c3c"><span class="section-number-3">3.3</span> Third Order Complementary Filters</h3>
-<div class="outline-text-3" id="text-3-3">
+<div id="outline-container-orgbc26e97" class="outline-4">
+<h4 id="orgbc26e97"><span class="section-number-4">5.1.3</span> Third Order Complementary Filters</h4>
+<div class="outline-text-4" id="text-5-1-3">
 <p>
 The following formula gives complementary filters with slopes of \(-3\) and \(3\):
 </p>
@@ -910,7 +1596,7 @@ The parameters are:
 </ul>
 
 <p>
-The filters are defined below and the result is shown on figure <a href="#org8791569">15</a>.
+The filters are defined below and the result is shown on figure <a href="#org7d35d91">21</a>.
 </p>
 
 <div class="org-src-container">
@@ -924,20 +1610,20 @@ Hl3_ana = <span class="org-rainbow-delimiters-depth-1">(</span><span class="org-
 </div>
 
 
-<div id="org8791569" class="figure">
+<div id="org7d35d91" class="figure">
 <p><img src="figs/complementary_filters_third_order.png" alt="complementary_filters_third_order.png" />
 </p>
-<p><span class="figure-number">Figure 15: </span>Third order complementary filters using the analytical formula (<a href="./figs/complementary_filters_third_order.png">png</a>, <a href="./figs/complementary_filters_third_order.pdf">pdf</a>)</p>
+<p><span class="figure-number">Figure 21: </span>Third order complementary filters using the analytical formula (<a href="./figs/complementary_filters_third_order.png">png</a>, <a href="./figs/complementary_filters_third_order.pdf">pdf</a>)</p>
 </div>
 </div>
 </div>
 </div>
 
-<div id="outline-container-org88004e2" class="outline-2">
-<h2 id="org88004e2"><span class="section-number-2">4</span> H-Infinity synthesis of complementary filters</h2>
-<div class="outline-text-2" id="text-4">
+<div id="outline-container-org630c88a" class="outline-3">
+<h3 id="org630c88a"><span class="section-number-3">5.2</span> H-Infinity synthesis of complementary filters</h3>
+<div class="outline-text-3" id="text-5-2">
 <p>
-<a id="org924f36f"></a>
+<a id="orgd45833d"></a>
 </p>
 <div class="note">
 <p>
@@ -947,27 +1633,27 @@ All the files (data and Matlab scripts) are accessible <a href="data/h_inf_synth
 </div>
 </div>
 
-<div id="outline-container-orgc51b2bd" class="outline-3">
-<h3 id="orgc51b2bd"><span class="section-number-3">4.1</span> Synthesis Architecture</h3>
-<div class="outline-text-3" id="text-4-1">
+<div id="outline-container-org9eecdbe" class="outline-4">
+<h4 id="org9eecdbe"><span class="section-number-4">5.2.1</span> Synthesis Architecture</h4>
+<div class="outline-text-4" id="text-5-2-1">
 <p>
 We here synthesize the complementary filters using the \(\mathcal{H}_\infty\) synthesis.
 The goal is to specify upper bounds on the norms of \(H_L\) and \(H_H\) while ensuring their complementary property (\(H_L + H_H = 1\)).
 </p>
 
 <p>
-In order to do so, we use the generalized plant shown on figure <a href="#orge2545fd">16</a> where \(w_L\) and \(w_H\) weighting transfer functions that will be used to shape \(H_L\) and \(H_H\) respectively.
+In order to do so, we use the generalized plant shown on figure <a href="#orgcf43d1a">22</a> where \(w_L\) and \(w_H\) weighting transfer functions that will be used to shape \(H_L\) and \(H_H\) respectively.
 </p>
 
 
-<div id="orge2545fd" class="figure">
+<div id="orgcf43d1a" class="figure">
 <p><img src="figs-tikz/sf_hinf_filters_plant_b.png" alt="sf_hinf_filters_plant_b.png" />
 </p>
-<p><span class="figure-number">Figure 16: </span>Generalized plant used for the \(\mathcal{H}_\infty\) synthesis of the complementary filters</p>
+<p><span class="figure-number">Figure 22: </span>Generalized plant used for the \(\mathcal{H}_\infty\) synthesis of the complementary filters</p>
 </div>
 
 <p>
-The \(\mathcal{H}_\infty\) synthesis applied on this generalized plant will give a transfer function \(H_L\) (figure <a href="#org130d2c0">17</a>) such that the \(\mathcal{H}_\infty\) norm of the transfer function from \(w\) to \([z_H,\ z_L]\) is less than one:
+The \(\mathcal{H}_\infty\) synthesis applied on this generalized plant will give a transfer function \(H_L\) (figure <a href="#org0c8df47">23</a>) such that the \(\mathcal{H}_\infty\) norm of the transfer function from \(w\) to \([z_H,\ z_L]\) is less than one:
 \[ \left\| \begin{array}{c} H_L w_L \\ (1 - H_L) w_H \end{array} \right\|_\infty < 1 \]
 </p>
 
@@ -986,17 +1672,17 @@ We then see that \(w_L\) and \(w_H\) can be used to shape both \(H_L\) and \(H_H
 </p>
 
 
-<div id="org130d2c0" class="figure">
+<div id="org0c8df47" class="figure">
 <p><img src="figs-tikz/sf_hinf_filters_b.png" alt="sf_hinf_filters_b.png" />
 </p>
-<p><span class="figure-number">Figure 17: </span>\(\mathcal{H}_\infty\) synthesis of the complementary filters</p>
+<p><span class="figure-number">Figure 23: </span>\(\mathcal{H}_\infty\) synthesis of the complementary filters</p>
 </div>
 </div>
 </div>
 
-<div id="outline-container-orga135818" class="outline-3">
-<h3 id="orga135818"><span class="section-number-3">4.2</span> Weights</h3>
-<div class="outline-text-3" id="text-4-2">
+<div id="outline-container-orgad3b0d3" class="outline-4">
+<h4 id="orgad3b0d3"><span class="section-number-4">5.2.2</span> Weights</h4>
+<div class="outline-text-4" id="text-5-2-2">
 <div class="org-src-container">
 <pre class="src src-matlab">omegab = <span class="org-highlight-numbers-number">2</span><span class="org-type">*</span><span class="org-constant">pi</span><span class="org-type">*</span><span class="org-highlight-numbers-number">9</span>;
 wH = <span class="org-rainbow-delimiters-depth-1">(</span>omegab<span class="org-rainbow-delimiters-depth-1">)</span><span class="org-type">^</span><span class="org-highlight-numbers-number">2</span><span class="org-type">/</span><span class="org-rainbow-delimiters-depth-1">(</span>s <span class="org-type">+</span> omegab<span class="org-type">*</span>sqrt<span class="org-rainbow-delimiters-depth-2">(</span><span class="org-highlight-numbers-number">1e</span><span class="org-type">-</span><span class="org-highlight-numbers-number">5</span><span class="org-rainbow-delimiters-depth-2">)</span><span class="org-rainbow-delimiters-depth-1">)</span><span class="org-type">^</span><span class="org-highlight-numbers-number">2</span>;
@@ -1006,17 +1692,17 @@ wL = <span class="org-rainbow-delimiters-depth-1">(</span>s <span class="org-typ
 </div>
 
 
-<div id="org3cb0b70" class="figure">
+<div id="orgbad7951" class="figure">
 <p><img src="figs/weights_wl_wh.png" alt="weights_wl_wh.png" />
 </p>
-<p><span class="figure-number">Figure 18: </span>Weights on the complementary filters \(w_L\) and \(w_H\) and the associated performance weights (<a href="./figs/weights_wl_wh.png">png</a>, <a href="./figs/weights_wl_wh.pdf">pdf</a>)</p>
+<p><span class="figure-number">Figure 24: </span>Weights on the complementary filters \(w_L\) and \(w_H\) and the associated performance weights (<a href="./figs/weights_wl_wh.png">png</a>, <a href="./figs/weights_wl_wh.pdf">pdf</a>)</p>
 </div>
 </div>
 </div>
 
-<div id="outline-container-orgbd3b8d8" class="outline-3">
-<h3 id="orgbd3b8d8"><span class="section-number-3">4.3</span> H-Infinity Synthesis</h3>
-<div class="outline-text-3" id="text-4-3">
+<div id="outline-container-org6fad77a" class="outline-4">
+<h4 id="org6fad77a"><span class="section-number-4">5.2.3</span> H-Infinity Synthesis</h4>
+<div class="outline-text-4" id="text-5-2-3">
 <p>
 We define the generalized plant \(P\) on matlab.
 </p>
@@ -1057,7 +1743,7 @@ Test bounds:      0.0000 &lt;  gamma  &lt;=      1.7285
 </pre>
 
 <p>
-We then define the high pass filter \(H_H = 1 - H_L\). The bode plot of both \(H_L\) and \(H_H\) is shown on figure <a href="#orgc1ec9ab">19</a>.
+We then define the high pass filter \(H_H = 1 - H_L\). The bode plot of both \(H_L\) and \(H_H\) is shown on figure <a href="#orgf485001">25</a>.
 </p>
 <div class="org-src-container">
 <pre class="src src-matlab">Hh_hinf = <span class="org-highlight-numbers-number">1</span> <span class="org-type">-</span> Hl_hinf;
@@ -1066,28 +1752,28 @@ We then define the high pass filter \(H_H = 1 - H_L\). The bode plot of both \(H
 </div>
 </div>
 
-<div id="outline-container-org0d62ef6" class="outline-3">
-<h3 id="org0d62ef6"><span class="section-number-3">4.4</span> Obtained Complementary Filters</h3>
-<div class="outline-text-3" id="text-4-4">
+<div id="outline-container-orgae7d589" class="outline-4">
+<h4 id="orgae7d589"><span class="section-number-4">5.2.4</span> Obtained Complementary Filters</h4>
+<div class="outline-text-4" id="text-5-2-4">
 <p>
-The obtained complementary filters are shown on figure <a href="#orgc1ec9ab">19</a>.
+The obtained complementary filters are shown on figure <a href="#orgf485001">25</a>.
 </p>
 
 
-<div id="orgc1ec9ab" class="figure">
+<div id="orgf485001" class="figure">
 <p><img src="figs/hinf_filters_results.png" alt="hinf_filters_results.png" />
 </p>
-<p><span class="figure-number">Figure 19: </span>Obtained complementary filters using \(\mathcal{H}_\infty\) synthesis (<a href="./figs/hinf_filters_results.png">png</a>, <a href="./figs/hinf_filters_results.pdf">pdf</a>)</p>
+<p><span class="figure-number">Figure 25: </span>Obtained complementary filters using \(\mathcal{H}_\infty\) synthesis (<a href="./figs/hinf_filters_results.png">png</a>, <a href="./figs/hinf_filters_results.pdf">pdf</a>)</p>
 </div>
 </div>
 </div>
 </div>
 
-<div id="outline-container-orga65dbb6" class="outline-2">
-<h2 id="orga65dbb6"><span class="section-number-2">5</span> Feedback Control Architecture to generate Complementary Filters</h2>
-<div class="outline-text-2" id="text-5">
+<div id="outline-container-org7728066" class="outline-3">
+<h3 id="org7728066"><span class="section-number-3">5.3</span> Feedback Control Architecture to generate Complementary Filters</h3>
+<div class="outline-text-3" id="text-5-3">
 <p>
-<a id="org3e654dc"></a>
+<a id="orgabbb632"></a>
 </p>
 <p>
 The idea is here to use the fact that in a classical feedback architecture, \(S + T = 1\), in order to design complementary filters.
@@ -1104,14 +1790,14 @@ All the files (data and Matlab scripts) are accessible <a href="data/feedback_ge
 </div>
 </div>
 
-<div id="outline-container-orgec7fc24" class="outline-3">
-<h3 id="orgec7fc24"><span class="section-number-3">5.1</span> Architecture</h3>
-<div class="outline-text-3" id="text-5-1">
+<div id="outline-container-org00ee41c" class="outline-4">
+<h4 id="org00ee41c"><span class="section-number-4">5.3.1</span> Architecture</h4>
+<div class="outline-text-4" id="text-5-3-1">
 
-<div id="orgf3202e6" class="figure">
+<div id="org8b341f8" class="figure">
 <p><img src="figs-tikz/complementary_filters_feedback_architecture.png" alt="complementary_filters_feedback_architecture.png" />
 </p>
-<p><span class="figure-number">Figure 20: </span>Architecture used to generate the complementary filters</p>
+<p><span class="figure-number">Figure 26: </span>Architecture used to generate the complementary filters</p>
 </div>
 
 <p>
@@ -1135,9 +1821,9 @@ Which contains two integrator and a lead. \(\omega_c\) is used to tune the cross
 </div>
 </div>
 
-<div id="outline-container-org448cd20" class="outline-3">
-<h3 id="org448cd20"><span class="section-number-3">5.2</span> Loop Gain Design</h3>
-<div class="outline-text-3" id="text-5-2">
+<div id="outline-container-org69ae5c6" class="outline-4">
+<h4 id="org69ae5c6"><span class="section-number-4">5.3.2</span> Loop Gain Design</h4>
+<div class="outline-text-4" id="text-5-3-2">
 <p>
 Let's first define the loop gain \(L\).
 </p>
@@ -1150,17 +1836,17 @@ L = <span class="org-rainbow-delimiters-depth-1">(</span>wc<span class="org-type
 </div>
 
 
-<div id="orgc090be3" class="figure">
+<div id="org62cdf6a" class="figure">
 <p><img src="figs/loop_gain_bode_plot.png" alt="loop_gain_bode_plot.png" />
 </p>
-<p><span class="figure-number">Figure 21: </span>Bode plot of the loop gain \(L\) (<a href="./figs/loop_gain_bode_plot.png">png</a>, <a href="./figs/loop_gain_bode_plot.pdf">pdf</a>)</p>
+<p><span class="figure-number">Figure 27: </span>Bode plot of the loop gain \(L\) (<a href="./figs/loop_gain_bode_plot.png">png</a>, <a href="./figs/loop_gain_bode_plot.pdf">pdf</a>)</p>
 </div>
 </div>
 </div>
 
-<div id="outline-container-org1ec0e07" class="outline-3">
-<h3 id="org1ec0e07"><span class="section-number-3">5.3</span> Complementary Filters Obtained</h3>
-<div class="outline-text-3" id="text-5-3">
+<div id="outline-container-orgecfc029" class="outline-4">
+<h4 id="orgecfc029"><span class="section-number-4">5.3.3</span> Complementary Filters Obtained</h4>
+<div class="outline-text-4" id="text-5-3-3">
 <p>
 We then compute the resulting low pass and high pass filters.
 </p>
@@ -1171,26 +1857,26 @@ Hh = <span class="org-highlight-numbers-number">1</span><span class="org-type">/
 </div>
 
 
-<div id="org84c20e0" class="figure">
+<div id="orgce868cd" class="figure">
 <p><img src="figs/low_pass_high_pass_filters.png" alt="low_pass_high_pass_filters.png" />
 </p>
-<p><span class="figure-number">Figure 22: </span>Low pass and High pass filters \(H_L\) and \(H_H\) for different values of \(\alpha\) (<a href="./figs/low_pass_high_pass_filters.png">png</a>, <a href="./figs/low_pass_high_pass_filters.pdf">pdf</a>)</p>
+<p><span class="figure-number">Figure 28: </span>Low pass and High pass filters \(H_L\) and \(H_H\) for different values of \(\alpha\) (<a href="./figs/low_pass_high_pass_filters.png">png</a>, <a href="./figs/low_pass_high_pass_filters.pdf">pdf</a>)</p>
 </div>
 </div>
 </div>
 </div>
 
-<div id="outline-container-org3b324a8" class="outline-2">
-<h2 id="org3b324a8"><span class="section-number-2">6</span> Analytical Formula found in the literature</h2>
-<div class="outline-text-2" id="text-6">
+<div id="outline-container-org3bfd6fb" class="outline-3">
+<h3 id="org3bfd6fb"><span class="section-number-3">5.4</span> Analytical Formula found in the literature</h3>
+<div class="outline-text-3" id="text-5-4">
 <p>
-<a id="org22f6020"></a>
+<a id="org53de8db"></a>
 </p>
 </div>
 
-<div id="outline-container-org8d499d4" class="outline-3">
-<h3 id="org8d499d4"><span class="section-number-3">6.1</span> Analytical Formula</h3>
-<div class="outline-text-3" id="text-6-1">
+<div id="outline-container-orge18079d" class="outline-4">
+<h4 id="orge18079d"><span class="section-number-4">5.4.1</span> Analytical Formula</h4>
+<div class="outline-text-4" id="text-5-4-1">
 <p>
 <a class='org-ref-reference' href="#min15_compl_filter_desig_angle_estim">min15_compl_filter_desig_angle_estim</a>
 </p>
@@ -1238,9 +1924,9 @@ Hh = <span class="org-highlight-numbers-number">1</span><span class="org-type">/
 </div>
 </div>
 
-<div id="outline-container-org60e2ce0" class="outline-3">
-<h3 id="org60e2ce0"><span class="section-number-3">6.2</span> Matlab</h3>
-<div class="outline-text-3" id="text-6-2">
+<div id="outline-container-org85b2b53" class="outline-4">
+<h4 id="org85b2b53"><span class="section-number-4">5.4.2</span> Matlab</h4>
+<div class="outline-text-4" id="text-5-4-2">
 <div class="org-src-container">
 <pre class="src src-matlab">omega0 = <span class="org-highlight-numbers-number">1</span><span class="org-type">*</span><span class="org-highlight-numbers-number">2</span><span class="org-type">*</span><span class="org-constant">pi</span>; <span class="org-comment">% [rad/s]</span>
 tau = <span class="org-highlight-numbers-number">1</span><span class="org-type">/</span>omega0; <span class="org-comment">% [s]</span>
@@ -1257,17 +1943,17 @@ HL3 = <span class="org-rainbow-delimiters-depth-1">(</span><span class="org-high
 </div>
 
 
-<div id="org5dd1179" class="figure">
+<div id="orgf29fb6e" class="figure">
 <p><img src="figs/comp_filters_literature.png" alt="comp_filters_literature.png" />
 </p>
-<p><span class="figure-number">Figure 23: </span>Comparison of some complementary filters found in the literature (<a href="./figs/comp_filters_literature.png">png</a>, <a href="./figs/comp_filters_literature.pdf">pdf</a>)</p>
+<p><span class="figure-number">Figure 29: </span>Comparison of some complementary filters found in the literature (<a href="./figs/comp_filters_literature.png">png</a>, <a href="./figs/comp_filters_literature.pdf">pdf</a>)</p>
 </div>
 </div>
 </div>
 
-<div id="outline-container-org1038fe9" class="outline-3">
-<h3 id="org1038fe9"><span class="section-number-3">6.3</span> Discussion</h3>
-<div class="outline-text-3" id="text-6-3">
+<div id="outline-container-org3e6b82e" class="outline-4">
+<h4 id="org3e6b82e"><span class="section-number-4">5.4.3</span> Discussion</h4>
+<div class="outline-text-4" id="text-5-4-3">
 <p>
 Analytical Formula found in the literature provides either no parameter for tuning the robustness / performance trade-off.
 </p>
@@ -1275,11 +1961,11 @@ Analytical Formula found in the literature provides either no parameter for tuni
 </div>
 </div>
 
-<div id="outline-container-org647d1a0" class="outline-2">
-<h2 id="org647d1a0"><span class="section-number-2">7</span> Comparison of the different methods of synthesis</h2>
-<div class="outline-text-2" id="text-7">
+<div id="outline-container-orgd5b4c24" class="outline-3">
+<h3 id="orgd5b4c24"><span class="section-number-3">5.5</span> Comparison of the different methods of synthesis</h3>
+<div class="outline-text-3" id="text-5-5">
 <p>
-  <a id="orgcbb346f"></a>
+  <a id="orgab28999"></a>
 The generated complementary filters using \(\mathcal{H}_\infty\) and the analytical formulas are very close to each other. However there is some difference to note here:
 </p>
 <ul class="org-ul">
@@ -1288,11 +1974,13 @@ The generated complementary filters using \(\mathcal{H}_\infty\) and the analyti
 </ul>
 </div>
 </div>
+</div>
 
 <p>
 
 <h1 class='org-ref-bib-h1'>Bibliography</h1>
-<ul class='org-ref-bib'><li><a id="min15_compl_filter_desig_angle_estim">[min15_compl_filter_desig_angle_estim]</a> <a name="min15_compl_filter_desig_angle_estim"></a>Min & Jeung, Complementary Filter Design for Angle Estimation Using Mems Accelerometer and Gyroscope, <i>Department of Control and Instrumentation, Changwon National University, Changwon, Korea</i>,  641-773 (2015).</li>
+<ul class='org-ref-bib'><li><a id="barzilai98_techn_measur_noise_sensor_presen">[barzilai98_techn_measur_noise_sensor_presen]</a> <a name="barzilai98_techn_measur_noise_sensor_presen"></a>Aaron Barzilai, Tom VanZandt & Tom Kenny, Technique for Measurement of the Noise of a Sensor in the  Presence of Large Background Signals, <i>Review of Scientific Instruments</i>, <b>69(7)</b>, 2767-2772 (1998). <a href="https://doi.org/10.1063/1.1149013">link</a>. <a href="http://dx.doi.org/10.1063/1.1149013">doi</a>.</li>
+<li><a id="min15_compl_filter_desig_angle_estim">[min15_compl_filter_desig_angle_estim]</a> <a name="min15_compl_filter_desig_angle_estim"></a>Min & Jeung, Complementary Filter Design for Angle Estimation Using Mems Accelerometer and Gyroscope, <i>Department of Control and Instrumentation, Changwon National University, Changwon, Korea</i>,  641-773 (2015).</li>
 <li><a id="corke04_inert_visual_sensin_system_small_auton_helic">[corke04_inert_visual_sensin_system_small_auton_helic]</a> <a name="corke04_inert_visual_sensin_system_small_auton_helic"></a>Peter Corke, An Inertial and Visual Sensing System for a Small Autonomous Helicopter, <i>Journal of Robotic Systems</i>, <b>21(2)</b>, 43-51 (2004). <a href="https://doi.org/10.1002/rob.10127">link</a>. <a href="http://dx.doi.org/10.1002/rob.10127">doi</a>.</li>
 <li><a id="jensen13_basic_uas">[jensen13_basic_uas]</a> <a name="jensen13_basic_uas"></a>Austin Jensen, Cal Coopmans & YangQuan Chen, Basics and guidelines of complementary filters for small UAS  navigation, nil, in in: 2013 International Conference on Unmanned Aircraft Systems
                   (ICUAS), edited by (2013)</li>
@@ -1303,7 +1991,7 @@ The generated complementary filters using \(\mathcal{H}_\infty\) and the analyti
 </div>
 <div id="postamble" class="status">
 <p class="author">Author: Thomas Dehaeze</p>
-<p class="date">Created: 2019-08-21 mer. 13:17</p>
+<p class="date">Created: 2019-08-29 jeu. 11:26</p>
 <p class="validation"><a href="http://validator.w3.org/check?uri=referer">Validate</a></p>
 </div>
 </body>
diff --git a/matlab/index.org b/matlab/index.org
index 606be34..130f37f 100644
--- a/matlab/index.org
+++ b/matlab/index.org
@@ -835,8 +835,10 @@ The noise sources $\tilde{n}_i$ are considered to be white noise: $\Phi_{\tilde{
 
 
 \begin{align*}
-  \hat{x} &= H_1(s) N_1(s) \tilde{n}_1 + H_2(s) N_2(s) \tilde{n}_2 + \Big(H_1(s) \big(1 + w_1(s) \Delta_1(s)\big) + H_2(s) \big(1 + w_2(s) \Delta_2(s)\big)\Big) x \\
-          &= H_1(s) N_1(s) \tilde{n}_1 + H_2(s) N_2(s) \tilde{n}_2 + \big(1 + H_1(s) w_1(s) \Delta_1(s) + H_2(s) w_2(s) \Delta_2(s)\big) x
+  \hat{x} &= H_1(s) N_1(s) \tilde{n}_1 + H_2(s) N_2(s) \tilde{n}_2 \\
+          &\quad \quad + \Big(H_1(s) \big(1 + w_1(s) \Delta_1(s)\big) + H_2(s) \big(1 + w_2(s) \Delta_2(s)\big)\Big) x \\
+          &= H_1(s) N_1(s) \tilde{n}_1 + H_2(s) N_2(s) \tilde{n}_2 \\
+          &\quad \quad + \big(1 + H_1(s) w_1(s) \Delta_1(s) + H_2(s) w_2(s) \Delta_2(s)\big) x
 \end{align*}
 
 To the dynamics of the super sensor is:
diff --git a/matlab/ref.bib b/matlab/ref.bib
index bf21574..681b617 100644
--- a/matlab/ref.bib
+++ b/matlab/ref.bib
@@ -168,4 +168,17 @@
   year =         2008,
   doi =          {10.1109/tac.2008.923738},
   url =          {https://doi.org/10.1109/tac.2008.923738},
+}
+
+@article{barzilai98_techn_measur_noise_sensor_presen,
+  author =       {Aaron Barzilai and Tom VanZandt and Tom Kenny},
+  title =        {Technique for Measurement of the Noise of a Sensor in the
+                  Presence of Large Background Signals},
+  journal =      {Review of Scientific Instruments},
+  volume =       69,
+  number =       7,
+  pages =        {2767-2772},
+  year =         1998,
+  doi =          {10.1063/1.1149013},
+  url =          {https://doi.org/10.1063/1.1149013},
 }
\ No newline at end of file
