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"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
<html xmlns="http://www.w3.org/1999/xhtml" lang="en" xml:lang="en">
<head>
<!-- 2020-10-05 lun. 11:45 -->
<!-- 2020-10-05 lun. 15:08 -->
<meta http-equiv="Content-Type" content="text/html;charset=utf-8" />
<title>Robust and Optimal Sensor Fusion - Matlab Computation</title>
<meta name="generator" content="Org mode" />
@ -35,50 +35,50 @@
<h2>Table of Contents</h2>
<div id="text-table-of-contents">
<ul>
<li><a href="#org27de0ea">1. Sensor Description</a>
<li><a href="#org0a492b7">1. Sensor Description</a>
<ul>
<li><a href="#org615e955">1.1. Sensor Dynamics</a></li>
<li><a href="#org693d3bf">1.2. Sensor Model Uncertainty</a></li>
<li><a href="#org60da040">1.3. Sensor Noise</a></li>
<li><a href="#org0b1325d">1.4. Save Model</a></li>
<li><a href="#org9575585">1.1. Sensor Dynamics</a></li>
<li><a href="#orgec8c81d">1.2. Sensor Model Uncertainty</a></li>
<li><a href="#org81d9a34">1.3. Sensor Noise</a></li>
<li><a href="#org4e23f6c">1.4. Save Model</a></li>
</ul>
</li>
<li><a href="#org0b389f3">2. Introduction to Sensor Fusion</a>
<li><a href="#org2cab1a2">2. Introduction to Sensor Fusion</a>
<ul>
<li><a href="#org1f9e1b3">2.1. Sensor Fusion Architecture</a></li>
<li><a href="#org6ce496e">2.2. Super Sensor Noise</a></li>
<li><a href="#org8a5c291">2.3. Super Sensor Dynamical Uncertainty</a></li>
<li><a href="#org0cbc92d">2.1. Sensor Fusion Architecture</a></li>
<li><a href="#orge9e0bd4">2.2. Super Sensor Noise</a></li>
<li><a href="#orgefb4347">2.3. Super Sensor Dynamical Uncertainty</a></li>
</ul>
</li>
<li><a href="#org7cb91ba">3. Optimal Super Sensor Noise: \(\mathcal{H}_2\) Synthesis</a>
<li><a href="#org5896b60">3. Optimal Super Sensor Noise: \(\mathcal{H}_2\) Synthesis</a>
<ul>
<li><a href="#orga0474b9">3.1. \(\mathcal{H}_2\) Synthesis</a></li>
<li><a href="#org1273dcd">3.2. Super Sensor Noise</a></li>
<li><a href="#orge9c2d41">3.3. Discrepancy between sensor dynamics and model</a></li>
<li><a href="#orgf5a8a84">3.1. \(\mathcal{H}_2\) Synthesis</a></li>
<li><a href="#org72159df">3.2. Super Sensor Noise</a></li>
<li><a href="#org8ffba19">3.3. Discrepancy between sensor dynamics and model</a></li>
</ul>
</li>
<li><a href="#orgef8f365">4. Robust Sensor Fusion: \(\mathcal{H}_\infty\) Synthesis</a>
<li><a href="#org26ea7b1">4. Robust Sensor Fusion: \(\mathcal{H}_\infty\) Synthesis</a>
<ul>
<li><a href="#org6f283c6">4.1. Weighting Function used to bound the super sensor uncertainty</a></li>
<li><a href="#org027886f">4.2. \(\mathcal{H}_\infty\) Synthesis</a></li>
<li><a href="#org0ad8fe8">4.3. Super sensor uncertainty</a></li>
<li><a href="#orgd5efe47">4.4. Super sensor noise</a></li>
<li><a href="#org0355c08">4.5. Conclusion</a></li>
<li><a href="#org8dfd9d2">4.1. Weighting Function used to bound the super sensor uncertainty</a></li>
<li><a href="#org7422ade">4.2. \(\mathcal{H}_\infty\) Synthesis</a></li>
<li><a href="#orga0267c6">4.3. Super sensor uncertainty</a></li>
<li><a href="#org979fede">4.4. Super sensor noise</a></li>
<li><a href="#orgda33992">4.5. Conclusion</a></li>
</ul>
</li>
<li><a href="#org3654cee">5. Optimal and Robust Sensor Fusion: Mixed \(\mathcal{H}_2/\mathcal{H}_\infty\) Synthesis</a>
<li><a href="#org15afe90">5. Optimal and Robust Sensor Fusion: Mixed \(\mathcal{H}_2/\mathcal{H}_\infty\) Synthesis</a>
<ul>
<li><a href="#org4d41c02">5.1. Mixed \(\mathcal{H}_2\) / \(\mathcal{H}_\infty\) Synthesis</a></li>
<li><a href="#orgc93b489">5.2. Obtained Super Sensor&rsquo;s noise</a></li>
<li><a href="#org5adb8ec">5.3. Obtained Super Sensor&rsquo;s Uncertainty</a></li>
<li><a href="#org76f00f7">5.4. Conclusion</a></li>
<li><a href="#org0f81a91">5.1. Mixed \(\mathcal{H}_2\) / \(\mathcal{H}_\infty\) Synthesis</a></li>
<li><a href="#org417aabd">5.2. Obtained Super Sensor&rsquo;s noise</a></li>
<li><a href="#org2dce888">5.3. Obtained Super Sensor&rsquo;s Uncertainty</a></li>
<li><a href="#org47da78c">5.4. Conclusion</a></li>
</ul>
</li>
<li><a href="#orgf68e579">6. Matlab Functions</a>
<li><a href="#org0afe5ef">6. Matlab Functions</a>
<ul>
<li><a href="#orgdec213a">6.1. <code>createWeight</code></a></li>
<li><a href="#orgad116a6">6.2. <code>plotMagUncertainty</code></a></li>
<li><a href="#orga641eed">6.3. <code>plotPhaseUncertainty</code></a></li>
<li><a href="#orge81e522">6.1. <code>createWeight</code></a></li>
<li><a href="#org37ec2b4">6.2. <code>plotMagUncertainty</code></a></li>
<li><a href="#org9f73572">6.3. <code>plotPhaseUncertainty</code></a></li>
</ul>
</li>
</ul>
@ -89,27 +89,27 @@
This document is arranged as follows:
</p>
<ul class="org-ul">
<li>Section <a href="#org740b45e">1</a>: the sensors are described (dynamics, uncertainty, noise)</li>
<li>Section <a href="#orge79447b">2</a>: the sensor fusion architecture is described and the super sensor noise and dynamical uncertainty are derived</li>
<li>Section <a href="#org4d1175a">3</a>: the \(\mathcal{H}_2\) synthesis is used to design complementary filters such that the RMS value of the super sensor&rsquo;s noise is minimized</li>
<li>Section <a href="#org92f543f">4</a>: the \(\mathcal{H}_\infty\) synthesis is used to design complementary filters such that the super sensor&rsquo;s uncertainty is bonded to acceptable values</li>
<li>Section <a href="#orga0c0443">5</a>: the mixed \(\mathcal{H}_2/\mathcal{H}_\infty\) synthesis is used to both limit the super sensor&rsquo;s uncertainty and to lower the RMS value of the super sensor&rsquo;s noise</li>
<li>Section <a href="#org4f93e35">6</a>: Matlab functions used for the analysis are described</li>
<li>Section <a href="#orgee25d07">1</a>: the sensors are described (dynamics, uncertainty, noise)</li>
<li>Section <a href="#orga64daad">2</a>: the sensor fusion architecture is described and the super sensor noise and dynamical uncertainty are derived</li>
<li>Section <a href="#orgdd6b9ce">3</a>: the \(\mathcal{H}_2\) synthesis is used to design complementary filters such that the RMS value of the super sensor&rsquo;s noise is minimized</li>
<li>Section <a href="#org5d93f37">4</a>: the \(\mathcal{H}_\infty\) synthesis is used to design complementary filters such that the super sensor&rsquo;s uncertainty is bonded to acceptable values</li>
<li>Section <a href="#org9f98c16">5</a>: the mixed \(\mathcal{H}_2/\mathcal{H}_\infty\) synthesis is used to both limit the super sensor&rsquo;s uncertainty and to lower the RMS value of the super sensor&rsquo;s noise</li>
<li>Section <a href="#orgf41dc8d">6</a>: Matlab functions used for the analysis are described</li>
</ul>
<div id="outline-container-org27de0ea" class="outline-2">
<h2 id="org27de0ea"><span class="section-number-2">1</span> Sensor Description</h2>
<div id="outline-container-org0a492b7" class="outline-2">
<h2 id="org0a492b7"><span class="section-number-2">1</span> Sensor Description</h2>
<div class="outline-text-2" id="text-1">
<p>
<a id="org740b45e"></a>
<a id="orgee25d07"></a>
</p>
<p>
In Figure <a href="#org6428f94">1</a> is shown a schematic of a sensor model that is used in the following study.
In Figure <a href="#org35e2340">1</a> is shown a schematic of a sensor model that is used in the following study.
In this example, the measured quantity \(x\) is the velocity of an object.
</p>
<table id="orge5eb43f" border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
<caption class="t-above"><span class="table-number">Table 1:</span> Description of signals in Figure <a href="#org6428f94">1</a></caption>
<table id="org2fe6194" border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
<caption class="t-above"><span class="table-number">Table 1:</span> Description of signals in Figure <a href="#org35e2340">1</a></caption>
<colgroup>
<col class="org-left" />
@ -176,8 +176,8 @@ In this example, the measured quantity \(x\) is the velocity of an object.
</tbody>
</table>
<table id="org9dd0355" border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
<caption class="t-above"><span class="table-number">Table 2:</span> Description of Systems in Figure <a href="#org6428f94">1</a></caption>
<table id="org281ecb3" border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
<caption class="t-above"><span class="table-number">Table 2:</span> Description of Systems in Figure <a href="#org35e2340">1</a></caption>
<colgroup>
<col class="org-left" />
@ -221,18 +221,18 @@ In this example, the measured quantity \(x\) is the velocity of an object.
</table>
<div id="org6428f94" class="figure">
<p><img src="figs-tikz/sensor_model_noise_uncertainty.png" alt="sensor_model_noise_uncertainty.png" />
<div id="org35e2340" class="figure">
<p><img src="figs-paper/sensor_model_noise_uncertainty.png" alt="sensor_model_noise_uncertainty.png" />
</p>
<p><span class="figure-number">Figure 1: </span>Sensor Model</p>
</div>
</div>
<div id="outline-container-org615e955" class="outline-3">
<h3 id="org615e955"><span class="section-number-3">1.1</span> Sensor Dynamics</h3>
<div id="outline-container-org9575585" class="outline-3">
<h3 id="org9575585"><span class="section-number-3">1.1</span> Sensor Dynamics</h3>
<div class="outline-text-3" id="text-1-1">
<p>
<a id="orgc593869"></a>
<a id="org52415aa"></a>
Let&rsquo;s consider two sensors measuring the velocity of an object.
</p>
@ -263,14 +263,14 @@ G2 = g_pos<span class="org-type">/</span>s<span class="org-type">/</span>(1 <spa
<p>
These nominal dynamics are also taken as the model of the sensor dynamics.
The true sensor dynamics has some uncertainty associated to it and described in section <a href="#orgba186f9">1.2</a>.
The true sensor dynamics has some uncertainty associated to it and described in section <a href="#orgf04439d">1.2</a>.
</p>
<p>
Both sensor dynamics in \([\frac{V}{m/s}]\) are shown in Figure <a href="#org8ac4b85">2</a>.
Both sensor dynamics in \([\frac{V}{m/s}]\) are shown in Figure <a href="#org2d1ef0b">2</a>.
</p>
<div id="org8ac4b85" class="figure">
<div id="org2d1ef0b" class="figure">
<p><img src="figs/sensors_nominal_dynamics.png" alt="sensors_nominal_dynamics.png" />
</p>
<p><span class="figure-number">Figure 2: </span>Sensor nominal dynamics from the velocity of the object to the output voltage</p>
@ -278,12 +278,12 @@ Both sensor dynamics in \([\frac{V}{m/s}]\) are shown in Figure <a href="#org8ac
</div>
</div>
<div id="outline-container-org693d3bf" class="outline-3">
<h3 id="org693d3bf"><span class="section-number-3">1.2</span> Sensor Model Uncertainty</h3>
<div id="outline-container-orgec8c81d" class="outline-3">
<h3 id="orgec8c81d"><span class="section-number-3">1.2</span> Sensor Model Uncertainty</h3>
<div class="outline-text-3" id="text-1-2">
<p>
<a id="orgba186f9"></a>
The uncertainty on the sensor dynamics is described by multiplicative uncertainty (Figure <a href="#org6428f94">1</a>).
<a id="orgf04439d"></a>
The uncertainty on the sensor dynamics is described by multiplicative uncertainty (Figure <a href="#org35e2340">1</a>).
</p>
<p>
@ -295,7 +295,7 @@ The true sensor dynamics \(G_i(s)\) is then described by \eqref{eq:sensor_dynami
\end{equation}
<p>
The weights \(W_i(s)\) representing the dynamical uncertainty are defined below and their magnitude is shown in Figure <a href="#orgd098571">3</a>.
The weights \(W_i(s)\) representing the dynamical uncertainty are defined below and their magnitude is shown in Figure <a href="#org557e062">3</a>.
</p>
<div class="org-src-container">
<pre class="src src-matlab">W1 = createWeight(<span class="org-string">'n'</span>, 2, <span class="org-string">'w0'</span>, 2<span class="org-type">*</span><span class="org-constant">pi</span><span class="org-type">*</span>3, <span class="org-string">'G0'</span>, 2, <span class="org-string">'G1'</span>, 0.1, <span class="org-string">'Gc'</span>, 1) <span class="org-type">*</span> ...
@ -306,18 +306,18 @@ W2 = createWeight(<span class="org-string">'n'</span>, 2, <span class="org-strin
</div>
<p>
The bode plot of the sensors nominal dynamics as well as their defined dynamical spread are shown in Figure <a href="#org4527c71">4</a>.
The bode plot of the sensors nominal dynamics as well as their defined dynamical spread are shown in Figure <a href="#orgc218675">4</a>.
</p>
<div id="orgd098571" class="figure">
<div id="org557e062" class="figure">
<p><img src="figs/sensors_uncertainty_weights.png" alt="sensors_uncertainty_weights.png" />
</p>
<p><span class="figure-number">Figure 3: </span>Magnitude of the multiplicative uncertainty weights \(|W_i(j\omega)|\)</p>
</div>
<div id="org4527c71" class="figure">
<div id="orgc218675" class="figure">
<p><img src="figs/sensors_nominal_dynamics_and_uncertainty.png" alt="sensors_nominal_dynamics_and_uncertainty.png" />
</p>
<p><span class="figure-number">Figure 4: </span>Nominal Sensor Dynamics \(\hat{G}_i\) (solid lines) as well as the spread of the dynamical uncertainty (background color)</p>
@ -325,12 +325,12 @@ The bode plot of the sensors nominal dynamics as well as their defined dynamical
</div>
</div>
<div id="outline-container-org60da040" class="outline-3">
<h3 id="org60da040"><span class="section-number-3">1.3</span> Sensor Noise</h3>
<div id="outline-container-org81d9a34" class="outline-3">
<h3 id="org81d9a34"><span class="section-number-3">1.3</span> Sensor Noise</h3>
<div class="outline-text-3" id="text-1-3">
<p>
<a id="orgb5f9d77"></a>
The noise of the sensors \(n_i\) are modelled by shaping a white noise with unitary PSD \(\tilde{n}_i\) \eqref{eq:unitary_noise_psd} with a LTI transfer function \(N_i(s)\) (Figure <a href="#org6428f94">1</a>).
<a id="org71b587e"></a>
The noise of the sensors \(n_i\) are modelled by shaping a white noise with unitary PSD \(\tilde{n}_i\) \eqref{eq:unitary_noise_psd} with a LTI transfer function \(N_i(s)\) (Figure <a href="#org35e2340">1</a>).
</p>
\begin{equation}
\Phi_{\tilde{n}_i}(\omega) = 1 \label{eq:unitary_noise_psd}
@ -344,7 +344,7 @@ The Power Spectral Density of the sensor noise \(\Phi_{n_i}(\omega)\) is then co
\end{equation}
<p>
The weights \(N_1\) and \(N_2\) representing the amplitude spectral density of the sensor noises are defined below and shown in Figure <a href="#orgf397a85">5</a>.
The weights \(N_1\) and \(N_2\) representing the amplitude spectral density of the sensor noises are defined below and shown in Figure <a href="#org51f9788">5</a>.
</p>
<div class="org-src-container">
<pre class="src src-matlab">omegac = 0.15<span class="org-type">*</span>2<span class="org-type">*</span><span class="org-constant">pi</span>; G0 = 1e<span class="org-type">-</span>1; Ginf = 1e<span class="org-type">-</span>6;
@ -356,7 +356,7 @@ N2 = (Ginf<span class="org-type">*</span>s<span class="org-type">/</span>omegac
</div>
<div id="orgf397a85" class="figure">
<div id="org51f9788" class="figure">
<p><img src="figs/sensors_noise.png" alt="sensors_noise.png" />
</p>
<p><span class="figure-number">Figure 5: </span>Amplitude spectral density of the sensors \(\sqrt{\Phi_{n_i}(\omega)} = |N_i(j\omega)|\)</p>
@ -364,8 +364,8 @@ N2 = (Ginf<span class="org-type">*</span>s<span class="org-type">/</span>omegac
</div>
</div>
<div id="outline-container-org0b1325d" class="outline-3">
<h3 id="org0b1325d"><span class="section-number-3">1.4</span> Save Model</h3>
<div id="outline-container-org4e23f6c" class="outline-3">
<h3 id="org4e23f6c"><span class="section-number-3">1.4</span> Save Model</h3>
<div class="outline-text-3" id="text-1-4">
<p>
All the dynamical systems representing the sensors are saved for further use.
@ -379,28 +379,28 @@ All the dynamical systems representing the sensors are saved for further use.
</div>
</div>
<div id="outline-container-org0b389f3" class="outline-2">
<h2 id="org0b389f3"><span class="section-number-2">2</span> Introduction to Sensor Fusion</h2>
<div id="outline-container-org2cab1a2" class="outline-2">
<h2 id="org2cab1a2"><span class="section-number-2">2</span> Introduction to Sensor Fusion</h2>
<div class="outline-text-2" id="text-2">
<p>
<a id="orge79447b"></a>
<a id="orga64daad"></a>
</p>
</div>
<div id="outline-container-org1f9e1b3" class="outline-3">
<h3 id="org1f9e1b3"><span class="section-number-3">2.1</span> Sensor Fusion Architecture</h3>
<div id="outline-container-org0cbc92d" class="outline-3">
<h3 id="org0cbc92d"><span class="section-number-3">2.1</span> Sensor Fusion Architecture</h3>
<div class="outline-text-3" id="text-2-1">
<p>
<a id="org0b744b4"></a>
<a id="org31e00a0"></a>
</p>
<p>
The two sensors presented in Section <a href="#org740b45e">1</a> are now merged together using complementary filters \(H_1(s)\) and \(H_2(s)\) to form a super sensor (Figure <a href="#org5caaf16">6</a>).
The two sensors presented in Section <a href="#orgee25d07">1</a> are now merged together using complementary filters \(H_1(s)\) and \(H_2(s)\) to form a super sensor (Figure <a href="#org48a16fd">6</a>).
</p>
<div id="org5caaf16" class="figure">
<p><img src="figs-tikz/sensor_fusion_noise_arch.png" alt="sensor_fusion_noise_arch.png" />
<div id="org48a16fd" class="figure">
<p><img src="figs-paper/sensor_fusion_noise_arch.png" alt="sensor_fusion_noise_arch.png" />
</p>
<p><span class="figure-number">Figure 6: </span>Sensor Fusion Architecture</p>
</div>
@ -423,11 +423,11 @@ The super sensor estimate \(\hat{x}\) is given by \eqref{eq:super_sensor_estimat
</div>
</div>
<div id="outline-container-org6ce496e" class="outline-3">
<h3 id="org6ce496e"><span class="section-number-3">2.2</span> Super Sensor Noise</h3>
<div id="outline-container-orge9e0bd4" class="outline-3">
<h3 id="orge9e0bd4"><span class="section-number-3">2.2</span> Super Sensor Noise</h3>
<div class="outline-text-3" id="text-2-2">
<p>
<a id="org3cb79e5"></a>
<a id="orgff055a3"></a>
</p>
<p>
@ -458,15 +458,15 @@ And the Root Mean Square (RMS) value of the super sensor noise \(\sigma_n\) is g
</div>
</div>
<div id="outline-container-org8a5c291" class="outline-3">
<h3 id="org8a5c291"><span class="section-number-3">2.3</span> Super Sensor Dynamical Uncertainty</h3>
<div id="outline-container-orgefb4347" class="outline-3">
<h3 id="orgefb4347"><span class="section-number-3">2.3</span> Super Sensor Dynamical Uncertainty</h3>
<div class="outline-text-3" id="text-2-3">
<p>
<a id="org2ff763e"></a>
<a id="org859b213"></a>
</p>
<p>
If we consider some dynamical uncertainty (the true system dynamics \(G_i\) not being perfectly equal to our model \(\hat{G}_i\)) that we model by the use of multiplicative uncertainty (Figure <a href="#org3a5e8c1">7</a>), the super sensor dynamics is then equals to:
If we consider some dynamical uncertainty (the true system dynamics \(G_i\) not being perfectly equal to our model \(\hat{G}_i\)) that we model by the use of multiplicative uncertainty (Figure <a href="#org54538d7">7</a>), the super sensor dynamics is then equals to:
</p>
\begin{equation}
@ -478,19 +478,19 @@ If we consider some dynamical uncertainty (the true system dynamics \(G_i\) not
\end{equation}
<div id="org3a5e8c1" class="figure">
<p><img src="figs-tikz/sensor_model_uncertainty.png" alt="sensor_model_uncertainty.png" />
<div id="org54538d7" class="figure">
<p><img src="figs-paper/sensor_model_uncertainty.png" alt="sensor_model_uncertainty.png" />
</p>
<p><span class="figure-number">Figure 7: </span>Sensor Model including Dynamical Uncertainty</p>
</div>
<p>
The uncertainty set of the transfer function from \(\hat{x}\) to \(x\) at frequency \(\omega\) is bounded in the complex plane by a circle centered on 1 and with a radius equal to \(|W_1(j\omega) H_1(j\omega)| + |W_2(j\omega) H_2(j\omega)|\) as shown in Figure <a href="#orge995373">8</a>.
The uncertainty set of the transfer function from \(\hat{x}\) to \(x\) at frequency \(\omega\) is bounded in the complex plane by a circle centered on 1 and with a radius equal to \(|W_1(j\omega) H_1(j\omega)| + |W_2(j\omega) H_2(j\omega)|\) as shown in Figure <a href="#orgb11ef23">8</a>.
</p>
<div id="orge995373" class="figure">
<p><img src="figs-tikz/uncertainty_set_super_sensor.png" alt="uncertainty_set_super_sensor.png" />
<div id="orgb11ef23" class="figure">
<p><img src="figs-paper/uncertainty_set_super_sensor.png" alt="uncertainty_set_super_sensor.png" />
</p>
<p><span class="figure-number">Figure 8: </span>Super Sensor model uncertainty displayed in the complex plane</p>
</div>
@ -498,11 +498,11 @@ The uncertainty set of the transfer function from \(\hat{x}\) to \(x\) at freque
</div>
</div>
<div id="outline-container-org7cb91ba" class="outline-2">
<h2 id="org7cb91ba"><span class="section-number-2">3</span> Optimal Super Sensor Noise: \(\mathcal{H}_2\) Synthesis</h2>
<div id="outline-container-org5896b60" class="outline-2">
<h2 id="org5896b60"><span class="section-number-2">3</span> Optimal Super Sensor Noise: \(\mathcal{H}_2\) Synthesis</h2>
<div class="outline-text-2" id="text-3">
<p>
<a id="org4d1175a"></a>
<a id="orgdd6b9ce"></a>
</p>
<p>
In this section, the complementary filters \(H_1(s)\) and \(H_2(s)\) are designed in order to minimize the RMS value of super sensor noise \(\sigma_n\).
@ -520,24 +520,24 @@ The RMS value of the super sensor noise is (neglecting the model uncertainty):
<p>
The goal is to design \(H_1(s)\) and \(H_2(s)\) such that \(H_1(s) + H_2(s) = 1\) (complementary property) and such that \(\left\| \begin{matrix} H_1 N_1 \\ H_2 N_2 \end{matrix} \right\|_2\) is minimized (minimized RMS value of the super sensor noise).
This is done using the \(\mathcal{H}_2\) synthesis in Section <a href="#org15426e5">3.1</a>.
This is done using the \(\mathcal{H}_2\) synthesis in Section <a href="#org5bc9386">3.1</a>.
</p>
</div>
<div id="outline-container-orga0474b9" class="outline-3">
<h3 id="orga0474b9"><span class="section-number-3">3.1</span> \(\mathcal{H}_2\) Synthesis</h3>
<div id="outline-container-orgf5a8a84" class="outline-3">
<h3 id="orgf5a8a84"><span class="section-number-3">3.1</span> \(\mathcal{H}_2\) Synthesis</h3>
<div class="outline-text-3" id="text-3-1">
<p>
<a id="org15426e5"></a>
<a id="org5bc9386"></a>
</p>
<p>
Consider the generalized plant \(P_{\mathcal{H}_2}\) shown in Figure <a href="#orgb96eccb">9</a> and described by Equation \eqref{eq:H2_generalized_plant}.
Consider the generalized plant \(P_{\mathcal{H}_2}\) shown in Figure <a href="#orgd218886">9</a> and described by Equation \eqref{eq:H2_generalized_plant}.
</p>
<div id="orgb96eccb" class="figure">
<p><img src="figs-tikz/h_two_optimal_fusion.png" alt="h_two_optimal_fusion.png" />
<div id="orgd218886" class="figure">
<p><img src="figs-paper/h_two_optimal_fusion.png" alt="h_two_optimal_fusion.png" />
</p>
<p><span class="figure-number">Figure 9: </span>Architecture used for \(\mathcal{H}_\infty\) synthesis of complementary filters</p>
</div>
@ -592,10 +592,10 @@ Finally, \(H_1(s)\) is defined as follows
</div>
<p>
The obtained complementary filters are shown in Figure <a href="#orga2bc39b">10</a>.
The obtained complementary filters are shown in Figure <a href="#org8c7ba6b">10</a>.
</p>
<div id="orga2bc39b" class="figure">
<div id="org8c7ba6b" class="figure">
<p><img src="figs/htwo_comp_filters.png" alt="htwo_comp_filters.png" />
</p>
<p><span class="figure-number">Figure 10: </span>Obtained complementary filters using the \(\mathcal{H}_2\) Synthesis</p>
@ -603,11 +603,11 @@ The obtained complementary filters are shown in Figure <a href="#orga2bc39b">10<
</div>
</div>
<div id="outline-container-org1273dcd" class="outline-3">
<h3 id="org1273dcd"><span class="section-number-3">3.2</span> Super Sensor Noise</h3>
<div id="outline-container-org72159df" class="outline-3">
<h3 id="org72159df"><span class="section-number-3">3.2</span> Super Sensor Noise</h3>
<div class="outline-text-3" id="text-3-2">
<p>
<a id="org0a41807"></a>
<a id="orgc7cc0a8"></a>
</p>
<p>
@ -622,13 +622,13 @@ PSD_H2 = abs(squeeze(freqresp(N1<span class="org-type">*</span>H1, freqs, <span
</div>
<p>
The obtained ASD are shown in Figure <a href="#org9628dea">11</a>.
The obtained ASD are shown in Figure <a href="#orge481bcd">11</a>.
</p>
<p>
The RMS value of the individual sensors and of the super sensor are listed in Table <a href="#org124b0f8">3</a>.
The RMS value of the individual sensors and of the super sensor are listed in Table <a href="#org3918d27">3</a>.
</p>
<table id="org124b0f8" border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
<table id="org3918d27" border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
<caption class="t-above"><span class="table-number">Table 3:</span> RMS value of the individual sensor noise and of the super sensor using the \(\mathcal{H}_2\) Synthesis</caption>
<colgroup>
@ -661,7 +661,7 @@ The RMS value of the individual sensors and of the super sensor are listed in Ta
</table>
<div id="org9628dea" class="figure">
<div id="orge481bcd" class="figure">
<p><img src="figs/psd_sensors_htwo_synthesis.png" alt="psd_sensors_htwo_synthesis.png" />
</p>
<p><span class="figure-number">Figure 11: </span>Power Spectral Density of the estimated \(\hat{x}\) using the two sensors alone and using the optimally fused signal</p>
@ -670,19 +670,19 @@ The RMS value of the individual sensors and of the super sensor are listed in Ta
<p>
A time domain simulation is now performed.
The measured velocity \(x\) is set to be a sweep sine with an amplitude of \(0.1\ [m/s]\).
The velocity estimates from the two sensors and from the super sensors are shown in Figure <a href="#orga63fd84">12</a>.
The resulting noises are displayed in Figure <a href="#orgf8fd218">13</a>.
The velocity estimates from the two sensors and from the super sensors are shown in Figure <a href="#org9c2a7e4">12</a>.
The resulting noises are displayed in Figure <a href="#org4cc42bf">13</a>.
</p>
<div id="orga63fd84" class="figure">
<div id="org9c2a7e4" class="figure">
<p><img src="figs/super_sensor_time_domain_h2.png" alt="super_sensor_time_domain_h2.png" />
</p>
<p><span class="figure-number">Figure 12: </span>Noise of individual sensors and noise of the super sensor</p>
</div>
<div id="orgf8fd218" class="figure">
<div id="org4cc42bf" class="figure">
<p><img src="figs/sensor_noise_H2_time_domain.png" alt="sensor_noise_H2_time_domain.png" />
</p>
<p><span class="figure-number">Figure 13: </span>Noise of the two sensors \(n_1, n_2\) and noise of the super sensor \(n\)</p>
@ -690,15 +690,15 @@ The resulting noises are displayed in Figure <a href="#orgf8fd218">13</a>.
</div>
</div>
<div id="outline-container-orge9c2d41" class="outline-3">
<h3 id="orge9c2d41"><span class="section-number-3">3.3</span> Discrepancy between sensor dynamics and model</h3>
<div id="outline-container-org8ffba19" class="outline-3">
<h3 id="org8ffba19"><span class="section-number-3">3.3</span> Discrepancy between sensor dynamics and model</h3>
<div class="outline-text-3" id="text-3-3">
<p>
If we consider sensor dynamical uncertainty as explained in Section <a href="#orgba186f9">1.2</a>, we can compute what would be the super sensor dynamical uncertainty when using the complementary filters obtained using the \(\mathcal{H}_2\) Synthesis.
If we consider sensor dynamical uncertainty as explained in Section <a href="#orgf04439d">1.2</a>, we can compute what would be the super sensor dynamical uncertainty when using the complementary filters obtained using the \(\mathcal{H}_2\) Synthesis.
</p>
<p>
The super sensor dynamical uncertainty is shown in Figure <a href="#orgfce6557">14</a>.
The super sensor dynamical uncertainty is shown in Figure <a href="#org865879b">14</a>.
</p>
<p>
@ -706,7 +706,7 @@ It is shown that the phase uncertainty is not bounded between 100Hz and 200Hz.
As a result the super sensor signal can not be used for feedback applications about 100Hz.
</p>
<div id="orgfce6557" class="figure">
<div id="org865879b" class="figure">
<p><img src="figs/super_sensor_dynamical_uncertainty_H2.png" alt="super_sensor_dynamical_uncertainty_H2.png" />
</p>
<p><span class="figure-number">Figure 14: </span>Super sensor dynamical uncertainty when using the \(\mathcal{H}_2\) Synthesis</p>
@ -715,11 +715,11 @@ As a result the super sensor signal can not be used for feedback applications ab
</div>
</div>
<div id="outline-container-orgef8f365" class="outline-2">
<h2 id="orgef8f365"><span class="section-number-2">4</span> Robust Sensor Fusion: \(\mathcal{H}_\infty\) Synthesis</h2>
<div id="outline-container-org26ea7b1" class="outline-2">
<h2 id="org26ea7b1"><span class="section-number-2">4</span> Robust Sensor Fusion: \(\mathcal{H}_\infty\) Synthesis</h2>
<div class="outline-text-2" id="text-4">
<p>
<a id="org92f543f"></a>
<a id="org5d93f37"></a>
</p>
<p>
We initially considered perfectly known sensor dynamics so that it can be perfectly inverted.
@ -727,18 +727,18 @@ We initially considered perfectly known sensor dynamics so that it can be perfec
<p>
We now take into account the fact that the sensor dynamics is only partially known.
To do so, we model the uncertainty that we have on the sensor dynamics by multiplicative input uncertainty as shown in Figure <a href="#org98d72a2">15</a>.
To do so, we model the uncertainty that we have on the sensor dynamics by multiplicative input uncertainty as shown in Figure <a href="#org2b71ca6">15</a>.
</p>
<div id="org98d72a2" class="figure">
<p><img src="figs-tikz/sensor_fusion_arch_uncertainty.png" alt="sensor_fusion_arch_uncertainty.png" />
<div id="org2b71ca6" class="figure">
<p><img src="figs-paper/sensor_fusion_arch_uncertainty.png" alt="sensor_fusion_arch_uncertainty.png" />
</p>
<p><span class="figure-number">Figure 15: </span>Sensor fusion architecture with sensor dynamics uncertainty</p>
</div>
<p>
As explained in Section <a href="#orgba186f9">1.2</a>, at each frequency \(\omega\), the dynamical uncertainty of the super sensor can be represented in the complex plane by a circle with a radius equals to \(|H_1(j\omega) W_1(j\omega)| + |H_2(j\omega) W_2(j\omega)|\) and centered on 1.
As explained in Section <a href="#orgf04439d">1.2</a>, at each frequency \(\omega\), the dynamical uncertainty of the super sensor can be represented in the complex plane by a circle with a radius equals to \(|H_1(j\omega) W_1(j\omega)| + |H_2(j\omega) W_2(j\omega)|\) and centered on 1.
</p>
<p>
@ -760,7 +760,7 @@ In order to specify a wanted upper bound on the dynamical uncertainty, a weight
\end{align}
<p>
The choice of \(W_u\) is presented in Section <a href="#org510f718">4.1</a>.
The choice of \(W_u\) is presented in Section <a href="#orgefed264">4.1</a>.
</p>
@ -778,15 +778,15 @@ The objective is to design \(H_1(s)\) and \(H_2(s)\) such that \(H_1(s) + H_2(s)
</p>
<p>
This is done using the \(\mathcal{H}_\infty\) synthesis in Section <a href="#org48c47d7">4.2</a>.
This is done using the \(\mathcal{H}_\infty\) synthesis in Section <a href="#org2c990ce">4.2</a>.
</p>
</div>
<div id="outline-container-org6f283c6" class="outline-3">
<h3 id="org6f283c6"><span class="section-number-3">4.1</span> Weighting Function used to bound the super sensor uncertainty</h3>
<div id="outline-container-org8dfd9d2" class="outline-3">
<h3 id="org8dfd9d2"><span class="section-number-3">4.1</span> Weighting Function used to bound the super sensor uncertainty</h3>
<div class="outline-text-3" id="text-4-1">
<p>
<a id="org510f718"></a>
<a id="orgefed264"></a>
</p>
<p>
@ -799,7 +799,7 @@ This is done using the \(\mathcal{H}_\infty\) synthesis in Section <a href="#org
\end{align}
<p>
The uncertainty bounds of the two individual sensor as well as the wanted maximum uncertainty bounds of the super sensor are shown in Figure <a href="#org3a8c93b">16</a>.
The uncertainty bounds of the two individual sensor as well as the wanted maximum uncertainty bounds of the super sensor are shown in Figure <a href="#orgdffee80">16</a>.
</p>
<div class="org-src-container">
@ -810,7 +810,7 @@ Wu = createWeight(<span class="org-string">'n'</span>, 2, <span class="org-strin
</div>
<div id="org3a8c93b" class="figure">
<div id="orgdffee80" class="figure">
<p><img src="figs/weight_uncertainty_bounds_Wu.png" alt="weight_uncertainty_bounds_Wu.png" />
</p>
<p><span class="figure-number">Figure 16: </span>Uncertainty region of the two sensors as well as the wanted maximum uncertainty of the super sensor (dashed lines)</p>
@ -818,20 +818,20 @@ Wu = createWeight(<span class="org-string">'n'</span>, 2, <span class="org-strin
</div>
</div>
<div id="outline-container-org027886f" class="outline-3">
<h3 id="org027886f"><span class="section-number-3">4.2</span> \(\mathcal{H}_\infty\) Synthesis</h3>
<div id="outline-container-org7422ade" class="outline-3">
<h3 id="org7422ade"><span class="section-number-3">4.2</span> \(\mathcal{H}_\infty\) Synthesis</h3>
<div class="outline-text-3" id="text-4-2">
<p>
<a id="org48c47d7"></a>
<a id="org2c990ce"></a>
</p>
<p>
The generalized plant \(P_{\mathcal{H}_\infty}\) used for the \(\mathcal{H}_\infty\) Synthesis of the complementary filters is shown in Figure <a href="#orgaac3e7e">17</a> and is described by Equation \eqref{eq:Hinf_generalized_plant}.
The generalized plant \(P_{\mathcal{H}_\infty}\) used for the \(\mathcal{H}_\infty\) Synthesis of the complementary filters is shown in Figure <a href="#org59777f5">17</a> and is described by Equation \eqref{eq:Hinf_generalized_plant}.
</p>
<div id="orgaac3e7e" class="figure">
<p><img src="figs-tikz/h_infinity_robust_fusion.png" alt="h_infinity_robust_fusion.png" />
<div id="org59777f5" class="figure">
<p><img src="figs-paper/h_infinity_robust_fusion.png" alt="h_infinity_robust_fusion.png" />
</p>
<p><span class="figure-number">Figure 17: </span>Architecture used for \(\mathcal{H}_\infty\) synthesis of complementary filters</p>
</div>
@ -897,11 +897,11 @@ The \(\mathcal{H}_\infty\) is successful as the \(\mathcal{H}_\infty\) norm of t
</div>
<p>
The obtained complementary filters as well as the wanted upper bounds are shown in Figure <a href="#org4673b81">18</a>.
The obtained complementary filters as well as the wanted upper bounds are shown in Figure <a href="#orga1806e3">18</a>.
</p>
<div id="org4673b81" class="figure">
<div id="orga1806e3" class="figure">
<p><img src="figs/hinf_comp_filters.png" alt="hinf_comp_filters.png" />
</p>
<p><span class="figure-number">Figure 18: </span>Obtained complementary filters using the \(\mathcal{H}_\infty\) Synthesis</p>
@ -909,11 +909,11 @@ The obtained complementary filters as well as the wanted upper bounds are shown
</div>
</div>
<div id="outline-container-org0ad8fe8" class="outline-3">
<h3 id="org0ad8fe8"><span class="section-number-3">4.3</span> Super sensor uncertainty</h3>
<div id="outline-container-orga0267c6" class="outline-3">
<h3 id="orga0267c6"><span class="section-number-3">4.3</span> Super sensor uncertainty</h3>
<div class="outline-text-3" id="text-4-3">
<p>
The super sensor dynamical uncertainty is displayed in Figure <a href="#org9f35650">19</a>.
The super sensor dynamical uncertainty is displayed in Figure <a href="#orge75f5ef">19</a>.
It is confirmed that the super sensor dynamical uncertainty is less than the maximum allowed uncertainty defined by the norm of \(W_u(s)\).
</p>
@ -922,7 +922,7 @@ The \(\mathcal{H}_\infty\) synthesis thus allows to design filters such that the
</p>
<div id="org9f35650" class="figure">
<div id="orge75f5ef" class="figure">
<p><img src="figs/super_sensor_dynamical_uncertainty_Hinf.png" alt="super_sensor_dynamical_uncertainty_Hinf.png" />
</p>
<p><span class="figure-number">Figure 19: </span>Super sensor dynamical uncertainty (solid curve) when using the \(\mathcal{H}_\infty\) Synthesis</p>
@ -930,12 +930,12 @@ The \(\mathcal{H}_\infty\) synthesis thus allows to design filters such that the
</div>
</div>
<div id="outline-container-orgd5efe47" class="outline-3">
<h3 id="orgd5efe47"><span class="section-number-3">4.4</span> Super sensor noise</h3>
<div id="outline-container-org979fede" class="outline-3">
<h3 id="org979fede"><span class="section-number-3">4.4</span> Super sensor noise</h3>
<div class="outline-text-3" id="text-4-4">
<p>
We now compute the obtain Power Spectral Density of the super sensor&rsquo;s noise.
The Amplitude Spectral Densities are shown in Figure <a href="#orgf375b8c">20</a>.
The Amplitude Spectral Densities are shown in Figure <a href="#org5aac73f">20</a>.
</p>
<div class="org-src-container">
@ -947,18 +947,18 @@ PSD_Hinf = abs(squeeze(freqresp(N1<span class="org-type">*</span>H1, freqs, <spa
</div>
<p>
The obtained RMS of the super sensor noise in the \(\mathcal{H}_2\) and \(\mathcal{H}_\infty\) case are shown in Table <a href="#org2c81207">4</a>.
The obtained RMS of the super sensor noise in the \(\mathcal{H}_2\) and \(\mathcal{H}_\infty\) case are shown in Table <a href="#org3e08452">4</a>.
As expected, the super sensor obtained from the \(\mathcal{H}_\infty\) synthesis is much noisier than the super sensor obtained from the \(\mathcal{H}_2\) synthesis.
</p>
<div id="orgf375b8c" class="figure">
<div id="org5aac73f" class="figure">
<p><img src="figs/psd_sensors_hinf_synthesis.png" alt="psd_sensors_hinf_synthesis.png" />
</p>
<p><span class="figure-number">Figure 20: </span>Power Spectral Density of the estimated \(\hat{x}\) using the two sensors alone and using the</p>
</div>
<table id="org2c81207" border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
<table id="org3e08452" border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
<caption class="t-above"><span class="table-number">Table 4:</span> Comparison of the obtained RMS noise of the super sensor</caption>
<colgroup>
@ -987,8 +987,8 @@ As expected, the super sensor obtained from the \(\mathcal{H}_\infty\) synthesis
</div>
</div>
<div id="outline-container-org0355c08" class="outline-3">
<h3 id="org0355c08"><span class="section-number-3">4.5</span> Conclusion</h3>
<div id="outline-container-orgda33992" class="outline-3">
<h3 id="orgda33992"><span class="section-number-3">4.5</span> Conclusion</h3>
<div class="outline-text-3" id="text-4-5">
<p>
Using the \(\mathcal{H}_\infty\) synthesis, the dynamical uncertainty of the super sensor can be bounded to acceptable values.
@ -1001,23 +1001,23 @@ However, the RMS of the super sensor noise is not optimized as it was the case w
</div>
</div>
<div id="outline-container-org3654cee" class="outline-2">
<h2 id="org3654cee"><span class="section-number-2">5</span> Optimal and Robust Sensor Fusion: Mixed \(\mathcal{H}_2/\mathcal{H}_\infty\) Synthesis</h2>
<div id="outline-container-org15afe90" class="outline-2">
<h2 id="org15afe90"><span class="section-number-2">5</span> Optimal and Robust Sensor Fusion: Mixed \(\mathcal{H}_2/\mathcal{H}_\infty\) Synthesis</h2>
<div class="outline-text-2" id="text-5">
<p>
<a id="orga0c0443"></a>
<a id="org9f98c16"></a>
</p>
<p>
The (optima) \(\mathcal{H}_2\) synthesis and the (robust) \(\mathcal{H}_\infty\) synthesis are now combined to form an Optimal and Robust synthesis of complementary filters for sensor fusion.
</p>
<p>
The sensor fusion architecture is shown in Figure <a href="#org6b7e130">21</a> (\(\hat{G}_i\) are omitted for space reasons).
The sensor fusion architecture is shown in Figure <a href="#org3cc874e">21</a> (\(\hat{G}_i\) are omitted for space reasons).
</p>
<div id="org6b7e130" class="figure">
<p><img src="figs-tikz/sensor_fusion_arch_full.png" alt="sensor_fusion_arch_full.png" />
<div id="org3cc874e" class="figure">
<p><img src="figs-paper/sensor_fusion_arch_full.png" alt="sensor_fusion_arch_full.png" />
</p>
<p><span class="figure-number">Figure 21: </span>Sensor fusion architecture with sensor dynamics uncertainty</p>
</div>
@ -1031,18 +1031,18 @@ The goal is to design complementary filters such that:
</ul>
<p>
To do so, we can use the Mixed \(\mathcal{H}_2/\mathcal{H}_\infty\) Synthesis presented in Section <a href="#org7eb3cad">5.1</a>.
To do so, we can use the Mixed \(\mathcal{H}_2/\mathcal{H}_\infty\) Synthesis presented in Section <a href="#orgbbc8594">5.1</a>.
</p>
</div>
<div id="outline-container-org4d41c02" class="outline-3">
<h3 id="org4d41c02"><span class="section-number-3">5.1</span> Mixed \(\mathcal{H}_2\) / \(\mathcal{H}_\infty\) Synthesis</h3>
<div id="outline-container-org0f81a91" class="outline-3">
<h3 id="org0f81a91"><span class="section-number-3">5.1</span> Mixed \(\mathcal{H}_2\) / \(\mathcal{H}_\infty\) Synthesis</h3>
<div class="outline-text-3" id="text-5-1">
<p>
<a id="org7eb3cad"></a>
<a id="orgbbc8594"></a>
</p>
<p>
The synthesis architecture that is used here is shown in Figure <a href="#orgb136ff8">22</a>.
The synthesis architecture that is used here is shown in Figure <a href="#orga971cdb">22</a>.
</p>
<p>
@ -1054,8 +1054,8 @@ The filter \(H_2(s)\) is synthesized such that it:
</ul>
<div id="orgb136ff8" class="figure">
<p><img src="figs-tikz/mixed_h2_hinf_synthesis.png" alt="mixed_h2_hinf_synthesis.png" />
<div id="orga971cdb" class="figure">
<p><img src="figs-paper/mixed_h2_hinf_synthesis.png" alt="mixed_h2_hinf_synthesis.png" />
</p>
<p><span class="figure-number">Figure 22: </span>Mixed \(\mathcal{H}_2/\mathcal{H}_\infty\) Synthesis</p>
</div>
@ -1098,11 +1098,11 @@ And the mixed \(\mathcal{H}_2/\mathcal{H}_\infty\) synthesis is performed.
</div>
<p>
The obtained complementary filters are shown in Figure <a href="#org1bf2ce4">23</a>.
The obtained complementary filters are shown in Figure <a href="#org30e8a3f">23</a>.
</p>
<div id="org1bf2ce4" class="figure">
<div id="org30e8a3f" class="figure">
<p><img src="figs/htwo_hinf_comp_filters.png" alt="htwo_hinf_comp_filters.png" />
</p>
<p><span class="figure-number">Figure 23: </span>Obtained complementary filters after mixed \(\mathcal{H}_2/\mathcal{H}_\infty\) synthesis</p>
@ -1110,19 +1110,19 @@ The obtained complementary filters are shown in Figure <a href="#org1bf2ce4">23<
</div>
</div>
<div id="outline-container-orgc93b489" class="outline-3">
<h3 id="orgc93b489"><span class="section-number-3">5.2</span> Obtained Super Sensor&rsquo;s noise</h3>
<div id="outline-container-org417aabd" class="outline-3">
<h3 id="org417aabd"><span class="section-number-3">5.2</span> Obtained Super Sensor&rsquo;s noise</h3>
<div class="outline-text-3" id="text-5-2">
<p>
The Amplitude Spectral Density of the super sensor&rsquo;s noise is shown in Figure <a href="#org6c93dd1">24</a>.
The Amplitude Spectral Density of the super sensor&rsquo;s noise is shown in Figure <a href="#orgfdfbf76">24</a>.
</p>
<p>
A time domain simulation is shown in Figure <a href="#orgb1e4b20">25</a>.
A time domain simulation is shown in Figure <a href="#orgdf5905f">25</a>.
</p>
<p>
The RMS values of the super sensor noise for the presented three synthesis are listed in Table <a href="#orga287858">5</a>.
The RMS values of the super sensor noise for the presented three synthesis are listed in Table <a href="#org3b7df55">5</a>.
</p>
<div class="org-src-container">
@ -1134,20 +1134,20 @@ PSD_H2Hinf = abs(squeeze(freqresp(N1<span class="org-type">*</span>H1, freqs, <s
</div>
<div id="org6c93dd1" class="figure">
<div id="orgfdfbf76" class="figure">
<p><img src="figs/psd_sensors_htwo_hinf_synthesis.png" alt="psd_sensors_htwo_hinf_synthesis.png" />
</p>
<p><span class="figure-number">Figure 24: </span>Power Spectral Density of the Super Sensor obtained with the mixed \(\mathcal{H}_2/\mathcal{H}_\infty\) synthesis</p>
</div>
<div id="orgb1e4b20" class="figure">
<div id="orgdf5905f" class="figure">
<p><img src="figs/super_sensor_time_domain_h2_hinf.png" alt="super_sensor_time_domain_h2_hinf.png" />
</p>
<p><span class="figure-number">Figure 25: </span>Noise of individual sensors and noise of the super sensor</p>
</div>
<table id="orga287858" border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
<table id="org3b7df55" border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
<caption class="t-above"><span class="table-number">Table 5:</span> Comparison of the obtained RMS noise of the super sensor</caption>
<colgroup>
@ -1181,15 +1181,15 @@ PSD_H2Hinf = abs(squeeze(freqresp(N1<span class="org-type">*</span>H1, freqs, <s
</div>
</div>
<div id="outline-container-org5adb8ec" class="outline-3">
<h3 id="org5adb8ec"><span class="section-number-3">5.3</span> Obtained Super Sensor&rsquo;s Uncertainty</h3>
<div id="outline-container-org2dce888" class="outline-3">
<h3 id="org2dce888"><span class="section-number-3">5.3</span> Obtained Super Sensor&rsquo;s Uncertainty</h3>
<div class="outline-text-3" id="text-5-3">
<p>
The uncertainty on the super sensor&rsquo;s dynamics is shown in Figure <a href="#org82b5806">26</a>.
The uncertainty on the super sensor&rsquo;s dynamics is shown in Figure <a href="#orgb2d28c5">26</a>.
</p>
<div id="org82b5806" class="figure">
<div id="orgb2d28c5" class="figure">
<p><img src="figs/super_sensor_dynamical_uncertainty_Htwo_Hinf.png" alt="super_sensor_dynamical_uncertainty_Htwo_Hinf.png" />
</p>
<p><span class="figure-number">Figure 26: </span>Super sensor dynamical uncertainty (solid curve) when using the mixed \(\mathcal{H}_2/\mathcal{H}_\infty\) Synthesis</p>
@ -1197,8 +1197,8 @@ The uncertainty on the super sensor&rsquo;s dynamics is shown in Figure <a href=
</div>
</div>
<div id="outline-container-org76f00f7" class="outline-3">
<h3 id="org76f00f7"><span class="section-number-3">5.4</span> Conclusion</h3>
<div id="outline-container-org47da78c" class="outline-3">
<h3 id="org47da78c"><span class="section-number-3">5.4</span> Conclusion</h3>
<div class="outline-text-3" id="text-5-4">
<p>
The mixed \(\mathcal{H}_2/\mathcal{H}_\infty\) synthesis of the complementary filters allows to:
@ -1211,18 +1211,18 @@ The mixed \(\mathcal{H}_2/\mathcal{H}_\infty\) synthesis of the complementary fi
</div>
</div>
<div id="outline-container-orgf68e579" class="outline-2">
<h2 id="orgf68e579"><span class="section-number-2">6</span> Matlab Functions</h2>
<div id="outline-container-org0afe5ef" class="outline-2">
<h2 id="org0afe5ef"><span class="section-number-2">6</span> Matlab Functions</h2>
<div class="outline-text-2" id="text-6">
<p>
<a id="org4f93e35"></a>
<a id="orgf41dc8d"></a>
</p>
</div>
<div id="outline-container-orgdec213a" class="outline-3">
<h3 id="orgdec213a"><span class="section-number-3">6.1</span> <code>createWeight</code></h3>
<div id="outline-container-orge81e522" class="outline-3">
<h3 id="orge81e522"><span class="section-number-3">6.1</span> <code>createWeight</code></h3>
<div class="outline-text-3" id="text-6-1">
<p>
<a id="orgb6d8184"></a>
<a id="org89bad6d"></a>
</p>
<p>
@ -1274,11 +1274,11 @@ This Matlab function is accessible <a href="src/createWeight.m">here</a>.
</div>
</div>
<div id="outline-container-orgad116a6" class="outline-3">
<h3 id="orgad116a6"><span class="section-number-3">6.2</span> <code>plotMagUncertainty</code></h3>
<div id="outline-container-org37ec2b4" class="outline-3">
<h3 id="org37ec2b4"><span class="section-number-3">6.2</span> <code>plotMagUncertainty</code></h3>
<div class="outline-text-3" id="text-6-2">
<p>
<a id="orgd8e37bd"></a>
<a id="org8739875"></a>
</p>
<p>
@ -1329,11 +1329,11 @@ p.FaceAlpha = args.opacity;
</div>
</div>
<div id="outline-container-orga641eed" class="outline-3">
<h3 id="orga641eed"><span class="section-number-3">6.3</span> <code>plotPhaseUncertainty</code></h3>
<div id="outline-container-org9f73572" class="outline-3">
<h3 id="org9f73572"><span class="section-number-3">6.3</span> <code>plotPhaseUncertainty</code></h3>
<div class="outline-text-3" id="text-6-3">
<p>
<a id="org5f016a4"></a>
<a id="org49e21eb"></a>
</p>
<p>
@ -1397,7 +1397,7 @@ p.FaceAlpha = args.opacity;
</div>
<div id="postamble" class="status">
<p class="author">Author: Thomas Dehaeze</p>
<p class="date">Created: 2020-10-05 lun. 11:45</p>
<p class="date">Created: 2020-10-05 lun. 15:08</p>
</div>
</body>
</html>

22
matlab/index.org
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@ -100,7 +100,7 @@ In this example, the measured quantity $x$ is the velocity of an object.
#+name: fig:sensor_model_noise_uncertainty
#+caption: Sensor Model
#+RESULTS:
[[file:figs-tikz/sensor_model_noise_uncertainty.png]]
[[file:figs-paper/sensor_model_noise_uncertainty.png]]
** Matlab Init :noexport:ignore:
#+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name)
@ -325,7 +325,7 @@ The two sensors presented in Section [[sec:sensor_description]] are now merged t
#+name: fig:sensor_fusion_noise_arch
#+caption: Sensor Fusion Architecture
[[file:figs-tikz/sensor_fusion_noise_arch.png]]
[[file:figs-paper/sensor_fusion_noise_arch.png]]
The complementary property of $H_1(s)$ and $H_2(s)$ means that the sum of their transfer function is equal to $1$ eqref:eq:complementary_property.
@ -377,13 +377,13 @@ If we consider some dynamical uncertainty (the true system dynamics $G_i$ not be
#+name: fig:sensor_model_uncertainty
#+caption: Sensor Model including Dynamical Uncertainty
[[file:figs-tikz/sensor_model_uncertainty.png]]
[[file:figs-paper/sensor_model_uncertainty.png]]
The uncertainty set of the transfer function from $\hat{x}$ to $x$ at frequency $\omega$ is bounded in the complex plane by a circle centered on 1 and with a radius equal to $|W_1(j\omega) H_1(j\omega)| + |W_2(j\omega) H_2(j\omega)|$ as shown in Figure [[fig:uncertainty_set_super_sensor]].
#+name: fig:uncertainty_set_super_sensor
#+caption: Super Sensor model uncertainty displayed in the complex plane
[[file:figs-tikz/uncertainty_set_super_sensor.png]]
[[file:figs-paper/uncertainty_set_super_sensor.png]]
* Optimal Super Sensor Noise: $\mathcal{H}_2$ Synthesis
:PROPERTIES:
@ -427,7 +427,7 @@ Consider the generalized plant $P_{\mathcal{H}_2}$ shown in Figure [[fig:h_two_o
#+name: fig:h_two_optimal_fusion
#+caption: Architecture used for $\mathcal{H}_\infty$ synthesis of complementary filters
[[file:figs-tikz/h_two_optimal_fusion.png]]
[[file:figs-paper/h_two_optimal_fusion.png]]
\begin{equation} \label{eq:H2_generalized_plant}
\begin{pmatrix}
@ -696,7 +696,7 @@ To do so, we model the uncertainty that we have on the sensor dynamics by multip
#+name: fig:sensor_fusion_arch_uncertainty
#+caption: Sensor fusion architecture with sensor dynamics uncertainty
[[file:figs-tikz/sensor_fusion_arch_uncertainty.png]]
[[file:figs-paper/sensor_fusion_arch_uncertainty.png]]
As explained in Section [[sec:sensor_uncertainty]], at each frequency $\omega$, the dynamical uncertainty of the super sensor can be represented in the complex plane by a circle with a radius equals to $|H_1(j\omega) W_1(j\omega)| + |H_2(j\omega) W_2(j\omega)|$ and centered on 1.
@ -818,7 +818,7 @@ The generalized plant $P_{\mathcal{H}_\infty}$ used for the $\mathcal{H}_\infty$
#+name: fig:h_infinity_robust_fusion
#+caption: Architecture used for $\mathcal{H}_\infty$ synthesis of complementary filters
[[file:figs-tikz/h_infinity_robust_fusion.png]]
[[file:figs-paper/h_infinity_robust_fusion.png]]
\begin{equation} \label{eq:Hinf_generalized_plant}
\begin{pmatrix}
@ -1017,7 +1017,7 @@ As expected, the super sensor obtained from the $\mathcal{H}_\infty$ synthesis i
#+end_src
#+name: fig:psd_sensors_hinf_synthesis
#+caption: Power Spectral Density of the estimated $\hat{x}$ using the two sensors alone and using the
#+caption: Power Spectral Density of the estimated $\hat{x}$ using the two sensors alone and using the $\mathcal{H}_\infty$ synthesis
#+RESULTS:
[[file:figs/psd_sensors_hinf_synthesis.png]]
@ -1054,7 +1054,7 @@ The sensor fusion architecture is shown in Figure [[fig:sensor_fusion_arch_full]
#+name: fig:sensor_fusion_arch_full
#+caption: Sensor fusion architecture with sensor dynamics uncertainty
[[file:figs-tikz/sensor_fusion_arch_full.png]]
[[file:figs-paper/sensor_fusion_arch_full.png]]
The goal is to design complementary filters such that:
- the maximum uncertainty of the super sensor is bounded to acceptable values (defined by $W_u(s)$)
@ -1087,7 +1087,7 @@ The filter $H_2(s)$ is synthesized such that it:
#+name: fig:mixed_h2_hinf_synthesis
#+caption: Mixed $\mathcal{H}_2/\mathcal{H}_\infty$ Synthesis
[[file:figs-tikz/mixed_h2_hinf_synthesis.png]]
[[file:figs-paper/mixed_h2_hinf_synthesis.png]]
Let's see that
with $H_1(s)= 1 - H_2(s)$
@ -1213,7 +1213,7 @@ The RMS values of the super sensor noise for the presented three synthesis are l
#+end_src
#+name: fig:psd_sensors_htwo_hinf_synthesis
#+CAPTION: Power Spectral Density of the Super Sensor obtained with the mixed $\mathcal{H}_2/\mathcal{H}_\infty$ synthesis
#+caption: Power Spectral Density of the Super Sensor obtained with the mixed $\mathcal{H}_2/\mathcal{H}_\infty$ synthesis
#+RESULTS:
[[file:figs/psd_sensors_htwo_hinf_synthesis.png]]

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@ -221,13 +221,46 @@ We then have that the $\mathcal{H}_2$ synthesis applied on $P_{\mathcal{H}_2}$ g
** Example
#+name: fig:figure_name
#+caption: Figure caption
#+name: fig:sensors_nominal_dynamics
#+caption: Sensor nominal dynamics from the velocity of the object to the output voltage
#+attr_latex: :scale 1
[[file:figs/sensors_nominal_dynamics.pdf]]
#+name: fig:sensors_noise
#+caption: Amplitude spectral density of the sensors $\sqrt{\Phi_{n_i}(\omega)} = |N_i(j\omega)|$
#+attr_latex: :scale 1
[[file:figs/sensors_noise.pdf]]
#+name: fig:htwo_comp_filters
#+caption: Obtained complementary filters using the $\mathcal{H}_2$ Synthesis
#+attr_latex: :scale 1
[[file:figs/htwo_comp_filters.pdf]]
#+name: fig:psd_sensors_htwo_synthesis
#+caption: Power Spectral Density of the estimated $\hat{x}$ using the two sensors alone and using the optimally fused signal
#+attr_latex: :scale 1
[[file:figs/psd_sensors_htwo_synthesis.pdf]]
#+name: fig:super_sensor_time_domain_h2
#+caption: Noise of individual sensors and noise of the super sensor
#+attr_latex: :scale 1
[[file:figs/super_sensor_time_domain_h2.pdf]]
** Robustness Problem
#+name: fig:sensors_nominal_dynamics_and_uncertainty
#+caption: Nominal Sensor Dynamics $\hat{G}_i$ (solid lines) as well as the spread of the dynamical uncertainty (background color)
#+attr_latex: :scale 1
[[file:figs/sensors_nominal_dynamics_and_uncertainty.pdf]]
#+name: fig:super_sensor_dynamical_uncertainty_H2
#+caption: Super sensor dynamical uncertainty when using the $\mathcal{H}_2$ Synthesis
#+attr_latex: :scale 1
[[file:figs/super_sensor_dynamical_uncertainty_H2.pdf]]
* Robust Sensor Fusion: $\mathcal{H}_\infty$ Synthesis
<<sec:robust_fusion>>
@ -334,6 +367,33 @@ The $\mathcal{H}_\infty$ norm of Eq. eqref:eq:Hinf_norm is equals to $\sigma_n$
** Example
#+name: fig:sensors_uncertainty_weights
#+caption: Magnitude of the multiplicative uncertainty weights $|W_i(j\omega)|$
#+attr_latex: :scale 1
[[file:figs/sensors_uncertainty_weights.pdf]]
#+name: fig:weight_uncertainty_bounds_Wu
#+caption: Uncertainty region of the two sensors as well as the wanted maximum uncertainty of the super sensor (dashed lines)
#+attr_latex: :scale 1
[[file:figs/weight_uncertainty_bounds_Wu.pdf]]
#+name: fig:hinf_comp_filters
#+caption: Obtained complementary filters using the $\mathcal{H}_\infty$ Synthesis
#+attr_latex: :scale 1
[[file:figs/hinf_comp_filters.pdf]]
#+name: fig:super_sensor_dynamical_uncertainty_Hinf
#+caption: Super sensor dynamical uncertainty (solid curve) when using the $\mathcal{H}_\infty$ Synthesis
#+attr_latex: :scale 1
[[file:figs/super_sensor_dynamical_uncertainty_Hinf.pdf]]
#+name: fig:psd_sensors_hinf_synthesis
#+caption: Power Spectral Density of the estimated $\hat{x}$ using the two sensors alone and using the $\mathcal{H}_\infty$ synthesis
#+attr_latex: :scale 1
[[file:figs/psd_sensors_hinf_synthesis.pdf]]
* Optimal and Robust Sensor Fusion: Mixed $\mathcal{H}_2/\mathcal{H}_\infty$ Synthesis
<<sec:optimal_robust_fusion>>
@ -414,6 +474,26 @@ The synthesis objective is to:
** Example
#+name: fig:htwo_hinf_comp_filters
#+caption: Obtained complementary filters after mixed $\mathcal{H}_2/\mathcal{H}_\infty$ synthesis
#+attr_latex: :scale 1
[[file:figs/htwo_hinf_comp_filters.pdf]]
#+name: fig:psd_sensors_htwo_hinf_synthesis
#+caption: Power Spectral Density of the Super Sensor obtained with the mixed $\mathcal{H}_2/\mathcal{H}_\infty$ synthesis
#+attr_latex: :scale 1
[[file:figs/psd_sensors_htwo_hinf_synthesis.pdf]]
#+name: fig:super_sensor_time_domain_h2_hinf
#+caption: Noise of individual sensors and noise of the super sensor
#+attr_latex: :scale 1
[[file:figs/super_sensor_time_domain_h2_hinf.pdf]]
#+name: fig:super_sensor_dynamical_uncertainty_Htwo_Hinf
#+caption: Super sensor dynamical uncertainty (solid curve) when using the mixed $\mathcal{H}_2/\mathcal{H}_\infty$ Synthesis
#+attr_latex: :scale 1
[[file:figs/super_sensor_dynamical_uncertainty_Htwo_Hinf.pdf]]
* Experimental Validation
<<sec:experimental_validation>>

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% Created 2020-09-23 mer. 14:15
% Created 2020-10-05 lun. 15:33
% Intended LaTeX compiler: pdflatex
\documentclass[conference]{IEEEtran}
\usepackage[utf8]{inputenc}
@ -35,7 +35,7 @@
\def\BibTeX{{\rm B\kern-.05em{\sc i\kern-.025em b}\kern-.08em T\kern-.1667em\lower.7ex\hbox{E}\kern-.125emX}}
\usepackage{showframe}
\author{\IEEEauthorblockN{Dehaeze Thomas} \IEEEauthorblockA{\textit{European Synchrotron Radiation Facility} \\ Grenoble, France\\ \textit{Precision Mechatronics Laboratory} \\ \textit{University of Liege}, Belgium \\ thomas.dehaeze@esrf.fr }\and \IEEEauthorblockN{Collette Christophe} \IEEEauthorblockA{\textit{BEAMS Department}\\ \textit{Free University of Brussels}, Belgium\\ \textit{Precision Mechatronics Laboratory} \\ \textit{University of Liege}, Belgium \\ ccollett@ulb.ac.be }}
\date{2020-09-23}
\date{2020-10-05}
\title{Optimal and Robust Sensor Fusion}
\begin{document}
@ -50,7 +50,7 @@ Complementary Filters, Sensor Fusion, H-Infinity Synthesis
\end{IEEEkeywords}
\section{Introduction}
\label{sec:org88afd51}
\label{sec:org26a7400}
\label{sec:introduction}
\begin{itemize}
@ -61,12 +61,11 @@ Complementary Filters, Sensor Fusion, H-Infinity Synthesis
\end{itemize}
\section{Optimal Super Sensor Noise: \(\mathcal{H}_2\) Synthesis}
\label{sec:org5853545}
\label{sec:org49e80fd}
\label{sec:optimal_fusion}
\subsection{Sensor Model}
\label{sec:org565ea86}
\label{sec:org9555932}
Let's consider a sensor measuring a physical quantity \(x\) (Figure \ref{fig:sensor_model_noise}).
The sensor has an internal dynamics which is here modelled with a Linear Time Invariant (LTI) system transfer function \(G_i(s)\).
@ -102,8 +101,7 @@ In order to obtain an estimate \(\hat{x}_i\) of \(x\), a model \(\hat{G}_i\) of
\end{figure}
\subsection{Sensor Fusion Architecture}
\label{sec:org1ae73e8}
\label{sec:orga12ae12}
Let's now consider two sensors measuring the same physical quantity \(x\) but with different dynamics \((G_1, G_2)\) and noise characteristics \((N_1, N_2)\) (Figure \ref{fig:sensor_fusion_noise_arch}).
The noise sources \(\tilde{n}_1\) and \(\tilde{n}_2\) are considered to be uncorrelated.
@ -140,7 +138,7 @@ In such case, the super sensor estimate \(\hat{x}\) is equal to \(x\) plus the n
\end{equation}
\subsection{Super Sensor Noise}
\label{sec:orgb2e8dd6}
\label{sec:org924b750}
Let's note \(n\) the super sensor noise.
\begin{equation}
n = \left( H_1 N_1 \right) \tilde{n}_1 + \left( H_2 N_2 \right) \tilde{n}_2
@ -154,7 +152,7 @@ As the noise of both sensors are considered to be uncorrelated, the PSD of the s
It is clear that the PSD of the super sensor depends on the norm of the complementary filters.
\subsection{\(\mathcal{H}_2\) Synthesis of Complementary Filters}
\label{sec:orga4cf5f1}
\label{sec:org042a601}
The goal is to design \(H_1(s)\) and \(H_2(s)\) such that the effect of the noise sources \(\tilde{n}_1\) and \(\tilde{n}_2\) has the smallest possible effect on the noise \(n\) of the estimation \(\hat{x}\).
And the goal is the minimize the Root Mean Square (RMS) value of \(n\):
@ -170,8 +168,8 @@ This can be cast into an \(\mathcal{H}_2\) synthesis problem by considering the
\begin{pmatrix}
z_1 \\ z_2 \\ v
\end{pmatrix} = \underbrace{\begin{bmatrix}
N_1 & N_1 \\
0 & N_2 \\
N_1 & -N_1 \\
0 & N_2 \\
1 & 0
\end{bmatrix}}_{P_{\mathcal{H}_2}} \begin{pmatrix}
w \\ u
@ -198,17 +196,62 @@ We then have that the \(\mathcal{H}_2\) synthesis applied on \(P_{\mathcal{H}_2}
\end{figure}
\subsection{Example}
\label{sec:org74634c9}
\label{sec:org98c54c2}
\begin{figure}[htbp]
\centering
\includegraphics[scale=1]{figs/sensors_nominal_dynamics.pdf}
\caption{\label{fig:sensors_nominal_dynamics}Sensor nominal dynamics from the velocity of the object to the output voltage}
\end{figure}
\begin{figure}[htbp]
\centering
\includegraphics[scale=1]{figs/sensors_noise.pdf}
\caption{\label{fig:sensors_noise}Amplitude spectral density of the sensors \(\sqrt{\Phi_{n_i}(\omega)} = |N_i(j\omega)|\)}
\end{figure}
\begin{figure}[htbp]
\centering
\includegraphics[scale=1]{figs/htwo_comp_filters.pdf}
\caption{\label{fig:htwo_comp_filters}Obtained complementary filters using the \(\mathcal{H}_2\) Synthesis}
\end{figure}
\begin{figure}[htbp]
\centering
\includegraphics[scale=1]{figs/psd_sensors_htwo_synthesis.pdf}
\caption{\label{fig:psd_sensors_htwo_synthesis}Power Spectral Density of the estimated \(\hat{x}\) using the two sensors alone and using the optimally fused signal}
\end{figure}
\begin{figure}[htbp]
\centering
\includegraphics[scale=1]{figs/super_sensor_time_domain_h2.pdf}
\caption{\label{fig:super_sensor_time_domain_h2}Noise of individual sensors and noise of the super sensor}
\end{figure}
\subsection{Robustness Problem}
\label{sec:org5fda5c1}
\label{sec:org81a0772}
\begin{figure}[htbp]
\centering
\includegraphics[scale=1]{figs/sensors_nominal_dynamics_and_uncertainty.pdf}
\caption{\label{fig:sensors_nominal_dynamics_and_uncertainty}Nominal Sensor Dynamics \(\hat{G}_i\) (solid lines) as well as the spread of the dynamical uncertainty (background color)}
\end{figure}
\begin{figure}[htbp]
\centering
\includegraphics[scale=1]{figs/super_sensor_dynamical_uncertainty_H2.pdf}
\caption{\label{fig:super_sensor_dynamical_uncertainty_H2}Super sensor dynamical uncertainty when using the \(\mathcal{H}_2\) Synthesis}
\end{figure}
\section{Robust Sensor Fusion: \(\mathcal{H}_\infty\) Synthesis}
\label{sec:orgc88050f}
\label{sec:org78ced60}
\label{sec:robust_fusion}
\subsection{Representation of Sensor Dynamical Uncertainty}
\label{sec:orgb09aa5a}
\label{sec:org9df3b01}
In Section \ref{sec:optimal_fusion}, the model \(\hat{G}_i(s)\) of the sensor was considered to be perfect.
In reality, there are always uncertainty (neglected dynamics) associated with the estimation of the sensor dynamics.
@ -228,7 +271,7 @@ The sensor can then be represented as shown in Figure \ref{fig:sensor_model_unce
\end{figure}
\subsection{Sensor Fusion Architecture}
\label{sec:org1d92a74}
\label{sec:orgf4531ff}
Let's consider the sensor fusion architecture shown in Figure \ref{fig:sensor_fusion_arch_uncertainty} where the dynamical uncertainties of both sensors are included.
The super sensor estimate is then:
@ -253,8 +296,7 @@ As \(H_1\) and \(H_2\) are complementary filters, we finally have:
\end{figure}
\subsection{Super Sensor Dynamical Uncertainty}
\label{sec:org81db1d8}
\label{sec:orgf5bb33e}
The uncertainty set of the transfer function from \(\hat{x}\) to \(x\) at frequency \(\omega\) is bounded in the complex plane by a circle centered on 1 and with a radius equal to \(|W_1(j\omega) H_1(j\omega)| + |W_2(j\omega) H_2(j\omega)|\) as shown in Figure \ref{fig:uncertainty_set_super_sensor}.
@ -269,7 +311,7 @@ And we can see that the dynamical uncertainty of the super sensor is equal to th
At frequencies where \(\left|W_i(j\omega)\right| > 1\) the uncertainty exceeds \(100\%\) and sensor fusion is impossible.
\subsection{\(\mathcal{H_\infty}\) Synthesis of Complementary Filters}
\label{sec:org0e2a7a8}
\label{sec:orgf07efa7}
In order for the fusion to be ``robust'', meaning no phase drop will be induced in the super sensor dynamics,
The goal is to design two complementary filters \(H_1(s)\) and \(H_2(s)\) such that the super sensor noise uncertainty is kept reasonably small.
@ -279,7 +321,7 @@ To define what by ``small'' we mean, we use a weighting filter \(W_u(s)\) such t
\left| W_1(j\omega)H_1(j\omega) \right| + \left| W_2(j\omega)H_2(j\omega) \right| < \frac{1}{\left| W_u(j\omega) \right|}, \quad \forall \omega
\end{equation}
This is actually almost equivalent (to within a factor \(\sqrt{2}\)) equivalent as to have:
This is actually almost equivalent as to have (within a factor \(\sqrt{2}\)):
\begin{equation}
\left\| \begin{matrix} W_u W_1 H_1 \\ W_u W_2 H_2 \end{matrix} \right\|_\infty < 1
\end{equation}
@ -289,9 +331,9 @@ This problem can thus be dealt with an \(\mathcal{H}_\infty\) synthesis problem
\begin{pmatrix}
z_1 \\ z_2 \\ v
\end{pmatrix} = \underbrace{\begin{bmatrix}
W_u W_1 & W_u W_1 \\
0 & W_u W_2 \\
1 & 0
W_u W_1 & -W_u W_1 \\
0 & W_u W_2 \\
1 & 0
\end{bmatrix}}_{P_{\mathcal{H}_\infty}} \begin{pmatrix}
w \\ u
\end{pmatrix}
@ -315,15 +357,58 @@ The \(\mathcal{H}_\infty\) norm of Eq. \eqref{eq:Hinf_norm} is equals to \(\sigm
\end{figure}
\subsection{Example}
\label{sec:org0122000}
\label{sec:org0ca6ef9}
\begin{figure}[htbp]
\centering
\includegraphics[scale=1]{figs/sensors_uncertainty_weights.pdf}
\caption{\label{fig:sensors_uncertainty_weights}Magnitude of the multiplicative uncertainty weights \(|W_i(j\omega)|\)}
\end{figure}
\begin{figure}[htbp]
\centering
\includegraphics[scale=1]{figs/weight_uncertainty_bounds_Wu.pdf}
\caption{\label{fig:weight_uncertainty_bounds_Wu}Uncertainty region of the two sensors as well as the wanted maximum uncertainty of the super sensor (dashed lines)}
\end{figure}
\begin{figure}[htbp]
\centering
\includegraphics[scale=1]{figs/hinf_comp_filters.pdf}
\caption{\label{fig:hinf_comp_filters}Obtained complementary filters using the \(\mathcal{H}_\infty\) Synthesis}
\end{figure}
\begin{figure}[htbp]
\centering
\includegraphics[scale=1]{figs/super_sensor_dynamical_uncertainty_Hinf.pdf}
\caption{\label{fig:super_sensor_dynamical_uncertainty_Hinf}Super sensor dynamical uncertainty (solid curve) when using the \(\mathcal{H}_\infty\) Synthesis}
\end{figure}
\begin{figure}[htbp]
\centering
\includegraphics[scale=1]{figs/psd_sensors_hinf_synthesis.pdf}
\caption{\label{fig:psd_sensors_hinf_synthesis}Power Spectral Density of the estimated \(\hat{x}\) using the two sensors alone and using the \(\mathcal{H}_\infty\) synthesis}
\end{figure}
\section{Optimal and Robust Sensor Fusion: Mixed \(\mathcal{H}_2/\mathcal{H}_\infty\) Synthesis}
\label{sec:orgdf5a196}
\label{sec:orgf642e73}
\label{sec:optimal_robust_fusion}
\subsection{Sensor Fusion Architecture}
\label{sec:orge16b510}
\subsection{Sensor with noise and model uncertainty}
\label{sec:org8949812}
We wish now to combine the two previous synthesis, that is to say
The sensors are now modelled by a white noise with unitary PSD \(\tilde{n}_i\) shaped by a LTI transfer function \(N_i(s)\).
The dynamical uncertainty of the sensor is modelled using multiplicative uncertainty
\begin{equation}
v_i = \hat{G}_i (1 + W_i \Delta_i) x + \hat{G_i} (1 + W_i \Delta_i) N_i \tilde{n}_i
\end{equation}
Multiplying by the inverse of the nominal model of the sensor dynamics gives an estimate \(\hat{x}_i\) of \(x\):
\begin{equation}
\hat{x} = (1 + W_i \Delta_i) x + (1 + W_i \Delta_i) N_i \tilde{n}_i
\end{equation}
\begin{figure}[htbp]
\centering
@ -331,6 +416,26 @@ The \(\mathcal{H}_\infty\) norm of Eq. \eqref{eq:Hinf_norm} is equals to \(\sigm
\caption{\label{fig:sensor_model_noise_uncertainty}Sensor Model including Noise and Dynamical Uncertainty}
\end{figure}
\subsection{Sensor Fusion Architecture}
\label{sec:orgcbc3d54}
For reason of space, the blocks \(\hat{G}_i\) and \(\hat{G}_i^{-1}\) are omitted.
\begin{equation}
\begin{aligned}
\hat{x} = &\Big( H_1 (1 + W_1 \Delta_1) + H_2 (1 + W_2 \Delta_2) \Big) x \\
&+ \Big( H_1 (1 + W_1 \Delta_1) N_1 \Big) \tilde{n}_1 + \Big( H_2 (1 + W_2 \Delta_2) N_2 \Big) \tilde{n}_2
\end{aligned}
\end{equation}
\begin{equation}
\begin{aligned}
\hat{x} = &\Big( 1 + H_1 W_1 \Delta_1 + H_2 W_2 \Delta_2 \Big) x \\
&+ \Big( H_1 (1 + W_1 \Delta_1) N_1 \Big) \tilde{n}_1 + \Big( H_2 (1 + W_2 \Delta_2) N_2 \Big) \tilde{n}_2
\end{aligned}
\end{equation}
The estimate \(\hat{x}\) of \(x\)
\begin{figure}[htbp]
\centering
@ -338,11 +443,34 @@ The \(\mathcal{H}_\infty\) norm of Eq. \eqref{eq:Hinf_norm} is equals to \(\sigm
\caption{\label{fig:sensor_fusion_arch_full}Super Sensor Fusion with both sensor noise and sensor model uncertainty}
\end{figure}
\subsection{Synthesis Objective}
\label{sec:orgb4b43b3}
\subsection{Mixed \(\mathcal{H}_2/\mathcal{H}_\infty\) Synthesis}
\label{sec:orgb9b52ad}
\label{sec:org9d3f160}
The synthesis objective is to generate two complementary filters \(H_1(s)\) and \(H_2(s)\) such that the uncertainty associated with the super sensor is kept reasonably small and such that the RMS value of super sensors noise is minimized.
To specify how small we want the super sensor dynamic spread, we use a weighting filter \(W_u(s)\) as was done in Section \ref{sec:robust_fusion}.
This synthesis problem can be solved using the mixed \(\mathcal{H}_2/\mathcal{H}_\infty\) synthesis on the following generalized plant:
\begin{equation}
\begin{pmatrix}
z_{\infty, 1} \\ z_{\infty, 2} \\ z_{2, 1} \\ z_{2, 2} \\ v
\end{pmatrix} = \underbrace{\begin{bmatrix}
W_u W_1 & W_u W_1 \\
0 & W_u W_2 \\
N_1 & N_1 \\
0 & N_2 \\
1 & 0
\end{bmatrix}}_{P_{\mathcal{H}_2/\mathcal{H}_\infty}} \begin{pmatrix}
w \\ u
\end{pmatrix}
\end{equation}
The synthesis objective is to:
\begin{itemize}
\item Keep the \(\mathcal{H}_\infty\) norm from \(w\) to \((z_{\infty,1}, z_{\infty,2})\) below \(1\)
\item Minimize the \(\mathcal{H}_2\) norm from \(w\) to \((z_{2,1}, z_{2,2})\)
\end{itemize}
\begin{figure}[htbp]
\centering
@ -351,30 +479,54 @@ The \(\mathcal{H}_\infty\) norm of Eq. \eqref{eq:Hinf_norm} is equals to \(\sigm
\end{figure}
\subsection{Example}
\label{sec:orgc881f20}
\label{sec:org85f304b}
\begin{figure}[htbp]
\centering
\includegraphics[scale=1]{figs/htwo_hinf_comp_filters.pdf}
\caption{\label{fig:htwo_hinf_comp_filters}Obtained complementary filters after mixed \(\mathcal{H}_2/\mathcal{H}_\infty\) synthesis}
\end{figure}
\begin{figure}[htbp]
\centering
\includegraphics[scale=1]{figs/psd_sensors_htwo_hinf_synthesis.pdf}
\caption{\label{fig:psd_sensors_htwo_hinf_synthesis}Power Spectral Density of the Super Sensor obtained with the mixed \(\mathcal{H}_2/\mathcal{H}_\infty\) synthesis}
\end{figure}
\begin{figure}[htbp]
\centering
\includegraphics[scale=1]{figs/super_sensor_time_domain_h2_hinf.pdf}
\caption{\label{fig:super_sensor_time_domain_h2_hinf}Noise of individual sensors and noise of the super sensor}
\end{figure}
\begin{figure}[htbp]
\centering
\includegraphics[scale=1]{figs/super_sensor_dynamical_uncertainty_Htwo_Hinf.pdf}
\caption{\label{fig:super_sensor_dynamical_uncertainty_Htwo_Hinf}Super sensor dynamical uncertainty (solid curve) when using the mixed \(\mathcal{H}_2/\mathcal{H}_\infty\) Synthesis}
\end{figure}
\section{Experimental Validation}
\label{sec:org05b79a0}
\label{sec:org49bf34a}
\label{sec:experimental_validation}
\subsection{Experimental Setup}
\label{sec:orgc3daf35}
\label{sec:orgdd8fce6}
\subsection{Sensor Noise and Dynamical Uncertainty}
\label{sec:org26fedf6}
\label{sec:org21add72}
\subsection{Mixed \(\mathcal{H}_2/\mathcal{H}_\infty\) Synthesis}
\label{sec:org72f2969}
\label{sec:org30521a3}
\subsection{Super Sensor Noise and Dynamical Uncertainty}
\label{sec:orgf66f78b}
\label{sec:org86cde79}
\section{Conclusion}
\label{sec:orge0f0a43}
\label{sec:org16245b7}
\label{sec:conclusion}
\section{Acknowledgment}
\label{sec:orgb16559e}
\label{sec:orgd992049}
\bibliography{ref}
\end{document}

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