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<a accesskey="h" href="../index.html"> UP </a>
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<a accesskey="H" href="../index.html"> HOME </a>
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</div><div id="content">
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<h1 class="title">Robust and Optimal Sensor Fusion - Matlab Computation</h1>
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<div id="table-of-contents">
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<h2>Table of Contents</h2>
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<div id="text-table-of-contents">
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<ul>
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<li><a href="#org44d9894">1. Optimal Super Sensor Noise: \(\mathcal{H}_2\) Synthesis</a>
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<ul>
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<li><a href="#org57fc74a">1.1. Architecture</a></li>
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<li><a href="#org4e01148">1.2. Noise of the sensors</a></li>
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<li><a href="#org2009eeb">1.3. H-Two Synthesis</a></li>
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<li><a href="#org5bce7a5">1.4. Obtained Super Sensor’s noise uncertainty</a></li>
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<li><a href="#org3c6ed73">1.5. Conclusion</a></li>
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</ul>
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</li>
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<li><a href="#orgf8fcf68">2. Robust Sensor Fusion: \(\mathcal{H}_\infty\) Synthesis</a>
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<ul>
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<li><a href="#org95e625c">2.1. Super Sensor Dynamical Uncertainty</a></li>
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<li><a href="#orgf77bae9">2.2. Dynamical uncertainty of the individual sensors</a></li>
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<li><a href="#orgb8c1069">2.3. Synthesis objective</a></li>
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<li><a href="#org3b2fc2b">2.4. Requirements as an \(\mathcal{H}_\infty\) norm</a></li>
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<li><a href="#org1559f0c">2.5. Weighting Function used to bound the super sensor uncertainty</a></li>
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<li><a href="#org0942905">2.6. \(\mathcal{H}_\infty\) Synthesis</a></li>
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<li><a href="#orgd70c5ba">2.7. Super sensor uncertainty</a></li>
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<li><a href="#org822276a">2.8. Super sensor noise</a></li>
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<li><a href="#org78697e8">2.9. Conclusion</a></li>
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</ul>
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</li>
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<li><a href="#org885e445">3. Optimal and Robust Sensor Fusion: Mixed \(\mathcal{H}_2/\mathcal{H}_\infty\) Synthesis</a>
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<ul>
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<li><a href="#orgbe7b403">3.1. Mixed \(\mathcal{H}_2\) / \(\mathcal{H}_\infty\) Synthesis - Introduction</a></li>
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<li><a href="#org3013d41">3.2. Noise characteristics and Uncertainty of the individual sensors</a></li>
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<li><a href="#org39502ab">3.3. Weighting Functions on the uncertainty of the super sensor</a></li>
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<li><a href="#orgd08d347">3.4. Mixed Synthesis Architecture</a></li>
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<li><a href="#orgec6e5c4">3.5. Mixed \(\mathcal{H}_2\) / \(\mathcal{H}_\infty\) Synthesis</a></li>
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<li><a href="#orgcf41bc0">3.6. Obtained Super Sensor’s noise</a></li>
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<li><a href="#org39e64c2">3.7. Obtained Super Sensor’s Uncertainty</a></li>
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<li><a href="#orgdf7555f">3.8. Conclusion</a></li>
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||
</ul>
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</li>
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</ul>
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</div>
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</div>
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<p>
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In this document, the optimal and robust design of complementary filters is studied.
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</p>
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<p>
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Two sensors are considered with both different noise characteristics and dynamical uncertainties represented by multiplicative input uncertainty.
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</p>
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<ul class="org-ul">
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<li>Section <a href="#orgdd2b0ca">1</a>: the \(\mathcal{H}_2\) synthesis is used to design complementary filters such that the RMS value of the super sensor’s noise is minimized</li>
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<li>Section <a href="#org9f38ce2">2</a>: the \(\mathcal{H}_\infty\) synthesis is used to design complementary filters such that the super sensor’s uncertainty is bonded to acceptable values</li>
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<li>Section <a href="#org24cb1b0">3</a>: the mixed \(\mathcal{H}_2/\mathcal{H}_\infty\) synthesis is used to both limit the super sensor’s uncertainty and to lower the RMS value of the super sensor’s noise</li>
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||
</ul>
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||
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||
<div id="outline-container-org44d9894" class="outline-2">
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<h2 id="org44d9894"><span class="section-number-2">1</span> Optimal Super Sensor Noise: \(\mathcal{H}_2\) Synthesis</h2>
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<div class="outline-text-2" id="text-1">
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||
<p>
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||
<a id="orgdd2b0ca"></a>
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||
</p>
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<p>
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||
The idea is to combine sensors that works in different frequency range using complementary filters.
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||
</p>
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<p>
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||
Doing so, one “super sensor” is obtained that can have better noise characteristics than the individual sensors over a large frequency range.
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</p>
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<p>
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The complementary filters have to be designed in order to minimize the effect noise of each sensor on the super sensor noise.
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||
</p>
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<div class="note">
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||
<p>
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The Matlab scripts is accessible <a href="matlab/optimal_comp_filters.m">here</a>.
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||
</p>
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||
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</div>
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</div>
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<div id="outline-container-org57fc74a" class="outline-3">
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<h3 id="org57fc74a"><span class="section-number-3">1.1</span> Architecture</h3>
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<div class="outline-text-3" id="text-1-1">
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||
<p>
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||
Let’s consider the sensor fusion architecture shown on figure <a href="#orgd6622a2">1</a> where two sensors (sensor 1 and sensor 2) are measuring the same quantity \(x\) with different noise characteristics determined by \(N_1(s)\) and \(N_2(s)\).
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</p>
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<p>
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\(\tilde{n}_1\) and \(\tilde{n}_2\) are normalized white noise:
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||
</p>
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||
\begin{equation}
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||
\label{orga7ad7f8}
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||
\Phi_{\tilde{n}_1}(\omega) = \Phi_{\tilde{n}_2}(\omega) = 1
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||
\end{equation}
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<div id="orgd6622a2" class="figure">
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||
<p><img src="figs-tikz/fusion_two_noisy_sensors_weights.png" alt="fusion_two_noisy_sensors_weights.png" />
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||
</p>
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<p><span class="figure-number">Figure 1: </span>Fusion of two sensors</p>
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</div>
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<p>
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We consider that the two sensor dynamics \(G_1(s)\) and \(G_2(s)\) are ideal:
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||
</p>
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||
\begin{equation}
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||
\label{org024c8f5}
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||
G_1(s) = G_2(s) = 1
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||
\end{equation}
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<p>
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||
We obtain the architecture of figure <a href="#org0581836">2</a>.
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||
</p>
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||
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||
<div id="org0581836" class="figure">
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||
<p><img src="figs-tikz/sensor_fusion_noisy_perfect_dyn.png" alt="sensor_fusion_noisy_perfect_dyn.png" />
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||
</p>
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||
<p><span class="figure-number">Figure 2: </span>Fusion of two sensors with ideal dynamics</p>
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||
</div>
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||
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||
<p>
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||
\(H_1(s)\) and \(H_2(s)\) are complementary filters:
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||
</p>
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||
\begin{equation}
|
||
\label{orga71be68}
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||
H_1(s) + H_2(s) = 1
|
||
\end{equation}
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||
|
||
<p>
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||
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 estimation \(\hat{x}\).
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||
</p>
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||
|
||
<p>
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||
We have that the Power Spectral Density (PSD) of \(\hat{x}\) is:
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||
\[ \Phi_{\hat{x}}(\omega) = |H_1(j\omega) N_1(j\omega)|^2 \Phi_{\tilde{n}_1}(\omega) + |H_2(j\omega) N_2(j\omega)|^2 \Phi_{\tilde{n}_2}(\omega), \quad \forall \omega \]
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||
</p>
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||
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||
<p>
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||
And the goal is the minimize the Root Mean Square (RMS) value of \(\hat{x}\):
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||
</p>
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||
\begin{equation}
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||
\label{orgc926b79}
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||
\sigma_{\hat{x}} = \sqrt{\int_0^\infty \Phi_{\hat{x}}(\omega) d\omega}
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||
\end{equation}
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||
</div>
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||
</div>
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||
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<div id="outline-container-org4e01148" class="outline-3">
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||
<h3 id="org4e01148"><span class="section-number-3">1.2</span> Noise of the sensors</h3>
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||
<div class="outline-text-3" id="text-1-2">
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||
<p>
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||
Let’s define the noise characteristics of the two sensors by choosing \(N_1\) and \(N_2\):
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||
</p>
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||
<ul class="org-ul">
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||
<li>Sensor 1 characterized by \(N_1(s)\) has low noise at low frequency (for instance a geophone)</li>
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||
<li>Sensor 2 characterized by \(N_2(s)\) has low noise at high frequency (for instance an accelerometer)</li>
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||
</ul>
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||
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||
<div class="org-src-container">
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||
<pre class="src src-matlab">omegac = 100*2*pi; G0 = 1e-5; Ginf = 1e-4;
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N1 = (Ginf*s/omegac + G0)/(s/omegac + 1)/(1 + s/2/pi/100);
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||
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||
omegac = 1*2*pi; G0 = 1e-3; Ginf = 1e-8;
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N2 = ((sqrt(Ginf)*s/omegac + sqrt(G0))/(s/omegac + 1))^2/(1 + s/2/pi/4000)^2;
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||
</pre>
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||
</div>
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||
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||
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||
<div id="org29561dd" class="figure">
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||
<p><img src="figs/noise_characteristics_sensors.png" alt="noise_characteristics_sensors.png" />
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||
</p>
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||
<p><span class="figure-number">Figure 3: </span>Noise Characteristics of the two sensors (<a href="./figs/noise_characteristics_sensors.png">png</a>, <a href="./figs/noise_characteristics_sensors.pdf">pdf</a>)</p>
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||
</div>
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||
</div>
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||
</div>
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||
|
||
<div id="outline-container-org2009eeb" class="outline-3">
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||
<h3 id="org2009eeb"><span class="section-number-3">1.3</span> H-Two Synthesis</h3>
|
||
<div class="outline-text-3" id="text-1-3">
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||
<p>
|
||
As \(\tilde{n}_1\) and \(\tilde{n}_2\) are normalized white noise: \(\Phi_{\tilde{n}_1}(\omega) = \Phi_{\tilde{n}_2}(\omega) = 1\) and we have:
|
||
\[ \sigma_{\hat{x}} = \sqrt{\int_0^\infty |H_1 N_1|^2(\omega) + |H_2 N_2|^2(\omega) d\omega} = \left\| \begin{matrix} H_1 N_1 \\ H_2 N_2 \end{matrix} \right\|_2 \]
|
||
Thus, the goal is to design \(H_1(s)\) and \(H_2(s)\) such that \(H_1(s) + H_2(s) = 1\) and such that \(\left\| \begin{matrix} H_1 N_1 \\ H_2 N_2 \end{matrix} \right\|_2\) is minimized.
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||
</p>
|
||
|
||
<p>
|
||
For that, we use the \(\mathcal{H}_2\) Synthesis.
|
||
</p>
|
||
|
||
<p>
|
||
We use the generalized plant architecture shown on figure <a href="#orgc952e5a">4</a>.
|
||
</p>
|
||
|
||
|
||
<div id="orgc952e5a" 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>
|
||
</div>
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||
|
||
\begin{equation*}
|
||
\begin{pmatrix}
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||
z \\ v
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||
\end{pmatrix} = \begin{pmatrix}
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||
0 & N_2 & 1 \\
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||
N_1 & -N_2 & 0
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||
\end{pmatrix} \begin{pmatrix}
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||
w_1 \\ w_2 \\ u
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||
\end{pmatrix}
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||
\end{equation*}
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||
|
||
<p>
|
||
The transfer function from \([n_1, n_2]\) to \(\hat{x}\) is:
|
||
\[ \begin{bmatrix} N_1 H_1 \\ N_2 (1 - H_1) \end{bmatrix} \]
|
||
If we define \(H_2 = 1 - H_1\), we obtain:
|
||
\[ \begin{bmatrix} N_1 H_1 \\ N_2 H_2 \end{bmatrix} \]
|
||
</p>
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||
|
||
<p>
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||
Thus, if we minimize the \(\mathcal{H}_2\) norm of this transfer function, we minimize the RMS value of \(\hat{x}\).
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||
</p>
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||
|
||
<p>
|
||
We define the generalized plant \(P\) on matlab as shown on figure <a href="#orgc952e5a">4</a>.
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||
</p>
|
||
<div class="org-src-container">
|
||
<pre class="src src-matlab">P = [0 N2 1;
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N1 -N2 0];
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||
</pre>
|
||
</div>
|
||
|
||
<p>
|
||
And we do the \(\mathcal{H}_2\) synthesis using the <code>h2syn</code> command.
|
||
</p>
|
||
<div class="org-src-container">
|
||
<pre class="src src-matlab">[H1, ~, gamma] = h2syn(P, 1, 1);
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||
</pre>
|
||
</div>
|
||
|
||
<p>
|
||
Finally, we define \(H_2(s) = 1 - H_1(s)\).
|
||
</p>
|
||
<div class="org-src-container">
|
||
<pre class="src src-matlab">H2 = 1 - H1;
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||
</pre>
|
||
</div>
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||
|
||
<p>
|
||
The complementary filters obtained are shown on figure <a href="#orgc176e1b">5</a>.
|
||
</p>
|
||
|
||
<p>
|
||
The PSD of the noise of the individual sensor and of the super sensor are shown in Fig. <a href="#org95e2127">6</a>.
|
||
</p>
|
||
|
||
<p>
|
||
The Cumulative Power Spectrum (CPS) is shown on Fig. <a href="#org3be7555">7</a>.
|
||
</p>
|
||
|
||
<p>
|
||
The obtained RMS value of the super sensor is lower than the RMS value of the individual sensors.
|
||
</p>
|
||
|
||
|
||
<div id="orgc176e1b" 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 class="org-src-container">
|
||
<pre class="src src-matlab">PSD_S1 = abs(squeeze(freqresp(N1, freqs, 'Hz'))).^2;
|
||
PSD_S2 = abs(squeeze(freqresp(N2, freqs, 'Hz'))).^2;
|
||
PSD_H2 = abs(squeeze(freqresp(N1*H1, freqs, 'Hz'))).^2+abs(squeeze(freqresp(N2*H2, freqs, 'Hz'))).^2;
|
||
</pre>
|
||
</div>
|
||
|
||
|
||
<div id="org95e2127" 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>
|
||
|
||
<div class="org-src-container">
|
||
<pre class="src src-matlab">CPS_S1 = 1/pi*cumtrapz(2*pi*freqs, PSD_S1);
|
||
CPS_S2 = 1/pi*cumtrapz(2*pi*freqs, PSD_S2);
|
||
CPS_H2 = 1/pi*cumtrapz(2*pi*freqs, PSD_H2);
|
||
</pre>
|
||
</div>
|
||
|
||
|
||
<div id="org3be7555" class="figure">
|
||
<p><img src="figs/cps_h2_synthesis.png" alt="cps_h2_synthesis.png" />
|
||
</p>
|
||
<p><span class="figure-number">Figure 7: </span>Cumulative Power Spectrum of individual sensors and super sensor using the \(\mathcal{H}_2\) synthesis (<a href="./figs/cps_h2_synthesis.png">png</a>, <a href="./figs/cps_h2_synthesis.pdf">pdf</a>)</p>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
|
||
<div id="outline-container-org5bce7a5" class="outline-3">
|
||
<h3 id="org5bce7a5"><span class="section-number-3">1.4</span> Obtained Super Sensor’s noise uncertainty</h3>
|
||
<div class="outline-text-3" id="text-1-4">
|
||
<p>
|
||
We would like to verify if the obtained sensor fusion architecture is robust to change in the sensor dynamics.
|
||
</p>
|
||
|
||
<p>
|
||
To study the dynamical uncertainty on the super sensor, we defined some multiplicative uncertainty on both sensor dynamics.
|
||
Two weights \(w_1(s)\) and \(w_2(s)\) are used to described the amplitude of the dynamical uncertainty.
|
||
</p>
|
||
|
||
<div class="org-src-container">
|
||
<pre class="src src-matlab">omegac = 100*2*pi; G0 = 0.1; Ginf = 10;
|
||
w1 = (Ginf*s/omegac + G0)/(s/omegac + 1);
|
||
|
||
omegac = 0.2*2*pi; G0 = 5; Ginf = 0.1;
|
||
w2 = (Ginf*s/omegac + G0)/(s/omegac + 1);
|
||
omegac = 5000*2*pi; G0 = 1; Ginf = 50;
|
||
w2 = w2*(Ginf*s/omegac + G0)/(s/omegac + 1);
|
||
</pre>
|
||
</div>
|
||
|
||
<p>
|
||
The sensor uncertain models are defined below.
|
||
</p>
|
||
<div class="org-src-container">
|
||
<pre class="src src-matlab">G1 = 1 + w1*ultidyn('Delta',[1 1]);
|
||
G2 = 1 + w2*ultidyn('Delta',[1 1]);
|
||
</pre>
|
||
</div>
|
||
|
||
<p>
|
||
The super sensor uncertain model is defined below using the complementary filters obtained with the \(\mathcal{H}_2\) synthesis.
|
||
The dynamical uncertainty bounds of the super sensor is shown in Fig. <a href="#orgfde2308">8</a>.
|
||
Right Half Plane zero might be introduced in the super sensor dynamics which will render the feedback system unstable.
|
||
</p>
|
||
|
||
<div class="org-src-container">
|
||
<pre class="src src-matlab">Gss = G1*H1 + G2*H2;
|
||
</pre>
|
||
</div>
|
||
|
||
|
||
<div id="orgfde2308" class="figure">
|
||
<p><img src="figs/uncertainty_super_sensor_H2_syn.png" alt="uncertainty_super_sensor_H2_syn.png" />
|
||
</p>
|
||
<p><span class="figure-number">Figure 8: </span>Uncertianty regions of both individual sensors and of the super sensor when using the \(\mathcal{H}_2\) synthesis (<a href="./figs/uncertainty_super_sensor_H2_syn.png">png</a>, <a href="./figs/uncertainty_super_sensor_H2_syn.pdf">pdf</a>)</p>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
|
||
<div id="outline-container-org3c6ed73" class="outline-3">
|
||
<h3 id="org3c6ed73"><span class="section-number-3">1.5</span> Conclusion</h3>
|
||
<div class="outline-text-3" id="text-1-5">
|
||
<p>
|
||
From the above complementary filter design with the \(\mathcal{H}_2\) and \(\mathcal{H}_\infty\) synthesis, 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>
|
||
|
||
<p>
|
||
However, the synthesis does not take into account the robustness of the sensor fusion.
|
||
</p>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
|
||
<div id="outline-container-orgf8fcf68" class="outline-2">
|
||
<h2 id="orgf8fcf68"><span class="section-number-2">2</span> Robust Sensor Fusion: \(\mathcal{H}_\infty\) Synthesis</h2>
|
||
<div class="outline-text-2" id="text-2">
|
||
<p>
|
||
<a id="org9f38ce2"></a>
|
||
</p>
|
||
<p>
|
||
We initially considered perfectly known sensor dynamics so that it can be perfectly inverted.
|
||
</p>
|
||
|
||
<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 Fig. <a href="#org607f91f">9</a>.
|
||
</p>
|
||
|
||
|
||
<div id="org607f91f" class="figure">
|
||
<p><img src="figs-tikz/sensor_fusion_dynamic_uncertainty.png" alt="sensor_fusion_dynamic_uncertainty.png" />
|
||
</p>
|
||
<p><span class="figure-number">Figure 9: </span>Sensor fusion architecture with sensor dynamics uncertainty</p>
|
||
</div>
|
||
|
||
<p>
|
||
The objective here is to design complementary filters \(H_1(s)\) and \(H_2(s)\) in order to minimize the dynamical uncertainty of the super sensor.
|
||
</p>
|
||
<div class="note">
|
||
<p>
|
||
The Matlab scripts is accessible <a href="matlab/comp_filter_robustness.m">here</a>.
|
||
</p>
|
||
|
||
</div>
|
||
</div>
|
||
|
||
<div id="outline-container-org95e625c" class="outline-3">
|
||
<h3 id="org95e625c"><span class="section-number-3">2.1</span> Super Sensor Dynamical Uncertainty</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="#org607f91f">9</a>.
|
||
</p>
|
||
|
||
|
||
<div id="org8eaa465" class="figure">
|
||
<p><img src="figs-tikz/sensor_fusion_dynamic_uncertainty.png" alt="sensor_fusion_dynamic_uncertainty.png" />
|
||
</p>
|
||
<p><span class="figure-number">Figure 10: </span>Fusion of two sensors with input multiplicative uncertainty</p>
|
||
</div>
|
||
|
||
<p>
|
||
The dynamics of the super sensor is represented by
|
||
</p>
|
||
\begin{align*}
|
||
\frac{\hat{x}}{x} &= (1 + w_1 \Delta_1) H_1 + (1 + w_2 \Delta_2) H_2 \\
|
||
&= 1 + w_1 H_1 \Delta_1 + w_2 H_2 \Delta_2
|
||
\end{align*}
|
||
<p>
|
||
with \(\Delta_i\) is any transfer function satisfying \(\| \Delta_i \|_\infty < 1\).
|
||
</p>
|
||
|
||
<p>
|
||
We see that as soon as we have some uncertainty in the sensor dynamics, we have that the complementary filters have some effect on the transfer function from \(x\) to \(\hat{x}\).
|
||
</p>
|
||
|
||
<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 \(|w_1(j\omega) H_1(j\omega)| + |w_2(j\omega) H_2(j\omega)|\) (figure <a href="#org5f4c4b1">11</a>).
|
||
</p>
|
||
|
||
<p>
|
||
We then have that the angle introduced by the super sensor is bounded by \(\arcsin(\epsilon)\):
|
||
\[ \angle \frac{\hat{x}}{x}(j\omega) \le \arcsin \Big(|w_1(j\omega) H_1(j\omega)| + |w_2(j\omega) H_2(j\omega)|\Big) \]
|
||
</p>
|
||
|
||
|
||
<div id="org5f4c4b1" 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 11: </span>Maximum phase variation</p>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
|
||
<div id="outline-container-orgf77bae9" class="outline-3">
|
||
<h3 id="orgf77bae9"><span class="section-number-3">2.2</span> Dynamical uncertainty of the individual sensors</h3>
|
||
<div class="outline-text-3" id="text-2-2">
|
||
<p>
|
||
Let say we want to merge two sensors:
|
||
</p>
|
||
<ul class="org-ul">
|
||
<li>sensor 1 that has unknown dynamics above 10Hz: \(|w_1(j\omega)| > 1\) for \(\omega > 10\text{ Hz}\)</li>
|
||
<li>sensor 2 that has unknown dynamics below 1Hz and above 1kHz \(|w_2(j\omega)| > 1\) for \(\omega < 1\text{ Hz}\) and \(\omega > 1\text{ kHz}\)</li>
|
||
</ul>
|
||
|
||
<p>
|
||
We define the weights that are used to characterize the dynamic uncertainty of the sensors.
|
||
</p>
|
||
|
||
<div class="org-src-container">
|
||
<pre class="src src-matlab">omegac = 100*2*pi; G0 = 0.1; Ginf = 10;
|
||
w1 = (Ginf*s/omegac + G0)/(s/omegac + 1);
|
||
|
||
omegac = 0.2*2*pi; G0 = 5; Ginf = 0.1;
|
||
w2 = (Ginf*s/omegac + G0)/(s/omegac + 1);
|
||
omegac = 5000*2*pi; G0 = 1; Ginf = 50;
|
||
w2 = w2*(Ginf*s/omegac + G0)/(s/omegac + 1);
|
||
</pre>
|
||
</div>
|
||
|
||
<p>
|
||
From the weights, we define the uncertain transfer functions of the sensors. Some of the uncertain dynamics of both sensors are shown on Fig. <a href="#orgb3ffcc1">12</a> with the upper and lower bounds on the magnitude and on the phase.
|
||
</p>
|
||
<div class="org-src-container">
|
||
<pre class="src src-matlab">G1 = 1 + w1*ultidyn('Delta',[1 1]);
|
||
G2 = 1 + w2*ultidyn('Delta',[1 1]);
|
||
</pre>
|
||
</div>
|
||
|
||
|
||
<div id="orgb3ffcc1" class="figure">
|
||
<p><img src="figs/uncertainty_dynamics_sensors.png" alt="uncertainty_dynamics_sensors.png" />
|
||
</p>
|
||
<p><span class="figure-number">Figure 12: </span>Dynamic uncertainty of the two sensors (<a href="./figs/uncertainty_dynamics_sensors.png">png</a>, <a href="./figs/uncertainty_dynamics_sensors.pdf">pdf</a>)</p>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
|
||
<div id="outline-container-orgb8c1069" class="outline-3">
|
||
<h3 id="orgb8c1069"><span class="section-number-3">2.3</span> Synthesis objective</h3>
|
||
<div class="outline-text-3" id="text-2-3">
|
||
<p>
|
||
The uncertainty region of the super sensor dynamics is represented by a circle in the complex plane as shown in Fig. <a href="#org5f4c4b1">11</a>.
|
||
</p>
|
||
|
||
<p>
|
||
At each frequency \(\omega\), the radius of the circle is \(|w_1(j\omega) H_1(j\omega)| + |w_2(j\omega) H_2(j\omega)|\).
|
||
</p>
|
||
|
||
<p>
|
||
Thus, the phase shift \(\Delta\phi(\omega)\) due to the super sensor uncertainty is bounded by:
|
||
\[ |\Delta\phi(\omega)| \leq \arcsin\big( |w_1(j\omega) H_1(j\omega)| + |w_2(j\omega) H_2(j\omega)| \big) \]
|
||
</p>
|
||
|
||
<p>
|
||
Let’s define some allowed frequency depend phase shift \(\Delta\phi_\text{max}(\omega) > 0\) such that:
|
||
\[ |\Delta\phi(\omega)| < \Delta\phi_\text{max}(\omega), \quad \forall\omega \]
|
||
</p>
|
||
|
||
|
||
<p>
|
||
If \(H_1(s)\) and \(H_2(s)\) are designed such that
|
||
\[ |w_1(j\omega) H_1(j\omega)| + |w_2(j\omega) H_2(j\omega)| < \sin\big( \Delta\phi_\text{max}(\omega) \big) \]
|
||
</p>
|
||
|
||
<p>
|
||
The maximum phase shift due to dynamic uncertainty at frequency \(\omega\) will be \(\Delta\phi_\text{max}(\omega)\).
|
||
</p>
|
||
</div>
|
||
</div>
|
||
|
||
<div id="outline-container-org3b2fc2b" class="outline-3">
|
||
<h3 id="org3b2fc2b"><span class="section-number-3">2.4</span> Requirements as an \(\mathcal{H}_\infty\) norm</h3>
|
||
<div class="outline-text-3" id="text-2-4">
|
||
<p>
|
||
We now try to express this requirement in terms of an \(\mathcal{H}_\infty\) norm.
|
||
</p>
|
||
|
||
<p>
|
||
Let’s define one weight \(w_\phi(s)\) that represents the maximum wanted phase uncertainty:
|
||
\[ |w_{\phi}(j\omega)|^{-1} \approx \sin(\Delta\phi_{\text{max}}(\omega)), \quad \forall\omega \]
|
||
</p>
|
||
|
||
<p>
|
||
Then:
|
||
</p>
|
||
\begin{align*}
|
||
& |w_1(j\omega) H_1(j\omega)| + |w_2(j\omega) H_2(j\omega)| < \sin\big( \Delta\phi_\text{max}(\omega) \big), \quad \forall\omega \\
|
||
\Longleftrightarrow & |w_1(j\omega) H_1(j\omega)| + |w_2(j\omega) H_2(j\omega)| < |w_\phi(j\omega)|^{-1}, \quad \forall\omega \\
|
||
\Longleftrightarrow & \left| w_1(j\omega) H_1(j\omega) w_\phi(j\omega) \right| + \left| w_2(j\omega) H_2(j\omega) w_\phi(j\omega) \right| < 1, \quad \forall\omega
|
||
\end{align*}
|
||
|
||
<p>
|
||
Which is approximately equivalent to (with an error of maximum \(\sqrt{2}\)):
|
||
</p>
|
||
\begin{equation}
|
||
\label{org559e8db}
|
||
\left\| \begin{matrix} w_1(s) w_\phi(s) H_1(s) \\ w_2(s) w_\phi(s) H_2(s) \end{matrix} \right\|_\infty < 1
|
||
\end{equation}
|
||
|
||
<p>
|
||
One should not forget that at frequency where both sensors has unknown dynamics (\(|w_1(j\omega)| > 1\) and \(|w_2(j\omega)| > 1\)), the super sensor dynamics will also be unknown and the phase uncertainty cannot be bounded.
|
||
Thus, at these frequencies, \(|w_\phi|\) should be smaller than \(1\).
|
||
</p>
|
||
</div>
|
||
</div>
|
||
|
||
<div id="outline-container-org1559f0c" class="outline-3">
|
||
<h3 id="org1559f0c"><span class="section-number-3">2.5</span> Weighting Function used to bound the super sensor uncertainty</h3>
|
||
<div class="outline-text-3" id="text-2-5">
|
||
<p>
|
||
Let’s define \(w_\phi(s)\) in order to bound the maximum allowed phase uncertainty \(\Delta\phi_\text{max}\) of the super sensor dynamics.
|
||
The magnitude \(|w_\phi(j\omega)|\) is shown in Fig. <a href="#org90016c8">13</a> and the corresponding maximum allowed phase uncertainty of the super sensor dynamics of shown in Fig. <a href="#orgf2ea8be">14</a>.
|
||
</p>
|
||
|
||
<div class="org-src-container">
|
||
<pre class="src src-matlab">Dphi = 20; % [deg]
|
||
|
||
n = 4; w0 = 2*pi*900; G0 = 1/sin(Dphi*pi/180); Ginf = 1/100; Gc = 1;
|
||
wphi = (((1/w0)*sqrt((1-(G0/Gc)^(2/n))/(1-(Gc/Ginf)^(2/n)))*s + (G0/Gc)^(1/n))/((1/Ginf)^(1/n)*(1/w0)*sqrt((1-(G0/Gc)^(2/n))/(1-(Gc/Ginf)^(2/n)))*s + (1/Gc)^(1/n)))^n;
|
||
|
||
W1 = w1*wphi;
|
||
W2 = w2*wphi;
|
||
</pre>
|
||
</div>
|
||
|
||
|
||
<div id="org90016c8" class="figure">
|
||
<p><img src="figs/magnitude_wphi.png" alt="magnitude_wphi.png" />
|
||
</p>
|
||
<p><span class="figure-number">Figure 13: </span>Magnitude of the weght \(w_\phi(s)\) that is used to bound the uncertainty of the super sensor (<a href="./figs/magnitude_wphi.png">png</a>, <a href="./figs/magnitude_wphi.pdf">pdf</a>)</p>
|
||
</div>
|
||
|
||
|
||
<div id="orgf2ea8be" class="figure">
|
||
<p><img src="figs/maximum_wanted_phase_uncertainty.png" alt="maximum_wanted_phase_uncertainty.png" />
|
||
</p>
|
||
<p><span class="figure-number">Figure 14: </span>Maximum wanted phase uncertainty using this weight (<a href="./figs/maximum_wanted_phase_uncertainty.png">png</a>, <a href="./figs/maximum_wanted_phase_uncertainty.pdf">pdf</a>)</p>
|
||
</div>
|
||
|
||
<p>
|
||
The obtained upper bounds on the complementary filters in order to limit the phase uncertainty of the super sensor are represented in Fig. <a href="#org6bf03e5">15</a>.
|
||
</p>
|
||
|
||
|
||
<div id="org6bf03e5" class="figure">
|
||
<p><img src="figs/upper_bounds_comp_filter_max_phase_uncertainty.png" alt="upper_bounds_comp_filter_max_phase_uncertainty.png" />
|
||
</p>
|
||
<p><span class="figure-number">Figure 15: </span>Upper bounds on the complementary filters set in order to limit the maximum phase uncertainty of the super sensor to 30 degrees until 500Hz (<a href="./figs/upper_bounds_comp_filter_max_phase_uncertainty.png">png</a>, <a href="./figs/upper_bounds_comp_filter_max_phase_uncertainty.pdf">pdf</a>)</p>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
|
||
<div id="outline-container-org0942905" class="outline-3">
|
||
<h3 id="org0942905"><span class="section-number-3">2.6</span> \(\mathcal{H}_\infty\) Synthesis</h3>
|
||
<div class="outline-text-3" id="text-2-6">
|
||
<p>
|
||
The \(\mathcal{H}_\infty\) synthesis architecture used for the complementary filters is shown in Fig. <a href="#orgb6417fc">16</a>.
|
||
</p>
|
||
|
||
|
||
<div id="orgb6417fc" class="figure">
|
||
<p><img src="figs-tikz/h_infinity_robust_fusion.png" alt="h_infinity_robust_fusion.png" />
|
||
</p>
|
||
<p><span class="figure-number">Figure 16: </span>Architecture used for \(\mathcal{H}_\infty\) synthesis of complementary filters</p>
|
||
</div>
|
||
|
||
<p>
|
||
The generalized plant is defined below.
|
||
</p>
|
||
<div class="org-src-container">
|
||
<pre class="src src-matlab">P = [W1 -W1;
|
||
0 W2;
|
||
1 0];
|
||
</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">[H2, ~, gamma, ~] = hinfsyn(P, 1, 1,'TOLGAM', 0.001, 'METHOD', 'ric', 'DISPLAY', 'on');
|
||
</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: 0.0447 < gamma <= 1.3318
|
||
|
||
gamma hamx_eig xinf_eig hamy_eig yinf_eig nrho_xy p/f
|
||
1.332 1.3e+01 -1.0e-14 1.3e+00 -2.6e-18 0.0000 p
|
||
0.688 1.3e-11# ******** 1.3e+00 -6.7e-15 ******** f
|
||
1.010 1.1e+01 -1.5e-14 1.3e+00 -2.5e-14 0.0000 p
|
||
0.849 6.9e-11# ******** 1.3e+00 -2.3e-14 ******** f
|
||
0.930 5.2e-12# ******** 1.3e+00 -6.1e-18 ******** f
|
||
0.970 5.6e-11# ******** 1.3e+00 -2.3e-14 ******** f
|
||
0.990 5.0e-11# ******** 1.3e+00 -1.7e-17 ******** f
|
||
1.000 2.1e-10# ******** 1.3e+00 0.0e+00 ******** f
|
||
1.005 1.9e-10# ******** 1.3e+00 -3.7e-14 ******** f
|
||
1.008 1.1e+01 -9.1e-15 1.3e+00 0.0e+00 0.0000 p
|
||
1.006 1.2e-09# ******** 1.3e+00 -6.9e-16 ******** f
|
||
1.007 1.1e+01 -4.6e-15 1.3e+00 -1.8e-16 0.0000 p
|
||
|
||
Gamma value achieved: 1.0069
|
||
</pre>
|
||
|
||
<p>
|
||
And \(H_1(s)\) is defined as the complementary of \(H_2(s)\).
|
||
</p>
|
||
<div class="org-src-container">
|
||
<pre class="src src-matlab">H1 = 1 - H2;
|
||
</pre>
|
||
</div>
|
||
|
||
<p>
|
||
The obtained complementary filters are shown in Fig. <a href="#orgec29b6b">17</a>.
|
||
</p>
|
||
|
||
<div id="orgec29b6b" class="figure">
|
||
<p><img src="figs/comp_filter_hinf_uncertainty.png" alt="comp_filter_hinf_uncertainty.png" />
|
||
</p>
|
||
<p><span class="figure-number">Figure 17: </span>Obtained complementary filters (<a href="./figs/comp_filter_hinf_uncertainty.png">png</a>, <a href="./figs/comp_filter_hinf_uncertainty.pdf">pdf</a>)</p>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
|
||
<div id="outline-container-orgd70c5ba" class="outline-3">
|
||
<h3 id="orgd70c5ba"><span class="section-number-3">2.7</span> Super sensor uncertainty</h3>
|
||
<div class="outline-text-3" id="text-2-7">
|
||
<p>
|
||
We can now compute the uncertainty of the super sensor. The result is shown in Fig. <a href="#org4dd04dd">18</a>.
|
||
</p>
|
||
|
||
<div class="org-src-container">
|
||
<pre class="src src-matlab">Gss = G1*H1 + G2*H2;
|
||
</pre>
|
||
</div>
|
||
|
||
|
||
<div id="org4dd04dd" class="figure">
|
||
<p><img src="figs/super_sensor_uncertainty_bode_plot.png" alt="super_sensor_uncertainty_bode_plot.png" />
|
||
</p>
|
||
<p><span class="figure-number">Figure 18: </span>Uncertainty on the dynamics of the super sensor (<a href="./figs/super_sensor_uncertainty_bode_plot.png">png</a>, <a href="./figs/super_sensor_uncertainty_bode_plot.pdf">pdf</a>)</p>
|
||
</div>
|
||
|
||
<p>
|
||
The uncertainty of the super sensor cannot be made smaller than both the individual sensor. Ideally, it would follow the minimum uncertainty of both sensors.
|
||
</p>
|
||
|
||
<p>
|
||
We here just used very wimple weights.
|
||
For instance, we could improve the dynamical uncertainty of the super sensor by making \(|w_\phi(j\omega)|\) smaller bellow 2Hz where the dynamical uncertainty of the sensor 1 is small.
|
||
</p>
|
||
</div>
|
||
</div>
|
||
|
||
<div id="outline-container-org822276a" class="outline-3">
|
||
<h3 id="org822276a"><span class="section-number-3">2.8</span> Super sensor noise</h3>
|
||
<div class="outline-text-3" id="text-2-8">
|
||
<p>
|
||
We now compute the obtain Power Spectral Density of the super sensor’s noise.
|
||
The noise characteristics of both individual sensor are defined below.
|
||
</p>
|
||
|
||
<div class="org-src-container">
|
||
<pre class="src src-matlab">omegac = 100*2*pi; G0 = 1e-5; Ginf = 1e-4;
|
||
N1 = (Ginf*s/omegac + G0)/(s/omegac + 1)/(1 + s/2/pi/100);
|
||
|
||
omegac = 1*2*pi; G0 = 1e-3; Ginf = 1e-8;
|
||
N2 = ((sqrt(Ginf)*s/omegac + sqrt(G0))/(s/omegac + 1))^2/(1 + s/2/pi/4000)^2;
|
||
</pre>
|
||
</div>
|
||
|
||
<p>
|
||
The PSD of both sensor and of the super sensor is shown in Fig. <a href="#orgbaedbae">19</a>.
|
||
The CPS of both sensor and of the super sensor is shown in Fig. <a href="#orgb734af5">20</a>.
|
||
</p>
|
||
|
||
|
||
<div id="orgbaedbae" class="figure">
|
||
<p><img src="figs/psd_sensors_hinf_synthesis.png" alt="psd_sensors_hinf_synthesis.png" />
|
||
</p>
|
||
<p><span class="figure-number">Figure 19: </span>Power Spectral Density of the obtained super sensor using the \(\mathcal{H}_\infty\) synthesis (<a href="./figs/psd_sensors_hinf_synthesis.png">png</a>, <a href="./figs/psd_sensors_hinf_synthesis.pdf">pdf</a>)</p>
|
||
</div>
|
||
|
||
|
||
<div id="orgb734af5" class="figure">
|
||
<p><img src="figs/cps_sensors_hinf_synthesis.png" alt="cps_sensors_hinf_synthesis.png" />
|
||
</p>
|
||
<p><span class="figure-number">Figure 20: </span>Cumulative Power Spectrum of the obtained super sensor using the \(\mathcal{H}_\infty\) synthesis (<a href="./figs/cps_sensors_hinf_synthesis.png">png</a>, <a href="./figs/cps_sensors_hinf_synthesis.cps">cps</a>)</p>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
|
||
<div id="outline-container-org78697e8" class="outline-3">
|
||
<h3 id="org78697e8"><span class="section-number-3">2.9</span> Conclusion</h3>
|
||
<div class="outline-text-3" id="text-2-9">
|
||
<p>
|
||
Using the \(\mathcal{H}_\infty\) synthesis, the dynamical uncertainty of the super sensor can be bounded to acceptable values.
|
||
</p>
|
||
|
||
<p>
|
||
However, the RMS of the super sensor noise is not optimized as it was the case with the \(\mathcal{H}_2\) synthesis
|
||
</p>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
|
||
<div id="outline-container-org885e445" class="outline-2">
|
||
<h2 id="org885e445"><span class="section-number-2">3</span> Optimal and Robust Sensor Fusion: Mixed \(\mathcal{H}_2/\mathcal{H}_\infty\) Synthesis</h2>
|
||
<div class="outline-text-2" id="text-3">
|
||
<p>
|
||
<a id="org24cb1b0"></a>
|
||
</p>
|
||
<div class="note">
|
||
<p>
|
||
The Matlab scripts is accessible <a href="matlab/mixed_synthesis_sensor_fusion.m">here</a>.
|
||
</p>
|
||
|
||
</div>
|
||
</div>
|
||
<div id="outline-container-orgbe7b403" class="outline-3">
|
||
<h3 id="orgbe7b403"><span class="section-number-3">3.1</span> Mixed \(\mathcal{H}_2\) / \(\mathcal{H}_\infty\) Synthesis - Introduction</h3>
|
||
<div class="outline-text-3" id="text-3-1">
|
||
<p>
|
||
The goal is to design complementary filters such that:
|
||
</p>
|
||
<ul class="org-ul">
|
||
<li>the maximum uncertainty of the super sensor is bounded</li>
|
||
<li>the RMS value of the super sensor noise is minimized</li>
|
||
</ul>
|
||
|
||
<p>
|
||
To do so, we can use the Mixed \(\mathcal{H}_2\) / \(\mathcal{H}_\infty\) Synthesis.
|
||
</p>
|
||
|
||
<p>
|
||
The Matlab function for that is <code>h2hinfsyn</code> (<a href="https://fr.mathworks.com/help/robust/ref/h2hinfsyn.html">doc</a>).
|
||
</p>
|
||
</div>
|
||
</div>
|
||
|
||
<div id="outline-container-org3013d41" class="outline-3">
|
||
<h3 id="org3013d41"><span class="section-number-3">3.2</span> Noise characteristics and Uncertainty of the individual sensors</h3>
|
||
<div class="outline-text-3" id="text-3-2">
|
||
<p>
|
||
We define the weights that are used to characterize the dynamic uncertainty of the sensors. This will be used for the \(\mathcal{H}_\infty\) part of the synthesis.
|
||
</p>
|
||
<div class="org-src-container">
|
||
<pre class="src src-matlab">omegac = 100*2*pi; G0 = 0.1; Ginf = 10;
|
||
w1 = (Ginf*s/omegac + G0)/(s/omegac + 1);
|
||
|
||
omegac = 0.2*2*pi; G0 = 5; Ginf = 0.1;
|
||
w2 = (Ginf*s/omegac + G0)/(s/omegac + 1);
|
||
omegac = 5000*2*pi; G0 = 1; Ginf = 50;
|
||
w2 = w2*(Ginf*s/omegac + G0)/(s/omegac + 1);
|
||
</pre>
|
||
</div>
|
||
|
||
<p>
|
||
We define the noise characteristics of the two sensors by choosing \(N_1\) and \(N_2\). This will be used for the \(\mathcal{H}_2\) part of the synthesis.
|
||
</p>
|
||
<div class="org-src-container">
|
||
<pre class="src src-matlab">omegac = 100*2*pi; G0 = 1e-5; Ginf = 1e-4;
|
||
N1 = (Ginf*s/omegac + G0)/(s/omegac + 1)/(1 + s/2/pi/100);
|
||
|
||
omegac = 1*2*pi; G0 = 1e-3; Ginf = 1e-8;
|
||
N2 = ((sqrt(Ginf)*s/omegac + sqrt(G0))/(s/omegac + 1))^2/(1 + s/2/pi/4000)^2;
|
||
</pre>
|
||
</div>
|
||
|
||
<p>
|
||
Both dynamical uncertainty and noise characteristics of the individual sensors are shown in Fig. <a href="#org1270265">21</a>.
|
||
</p>
|
||
|
||
|
||
<div id="org1270265" class="figure">
|
||
<p><img src="figs/mixed_synthesis_noise_uncertainty_sensors.png" alt="mixed_synthesis_noise_uncertainty_sensors.png" />
|
||
</p>
|
||
<p><span class="figure-number">Figure 21: </span>Noise characteristsics and Dynamical uncertainty of the individual sensors (<a href="./figs/mixed_synthesis_noise_uncertainty_sensors.png">png</a>, <a href="./figs/mixed_synthesis_noise_uncertainty_sensors.pdf">pdf</a>)</p>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
|
||
<div id="outline-container-org39502ab" class="outline-3">
|
||
<h3 id="org39502ab"><span class="section-number-3">3.3</span> Weighting Functions on the uncertainty of the super sensor</h3>
|
||
<div class="outline-text-3" id="text-3-3">
|
||
<p>
|
||
We design weights for the \(\mathcal{H}_\infty\) part of the synthesis in order to limit the dynamical uncertainty of the super sensor.
|
||
The maximum wanted multiplicative uncertainty is shown in Fig. <a href="#org56b5728">22</a>. The idea here is that we don’t really need low uncertainty at low frequency but only near the crossover frequency that is suppose to be around 300Hz here.
|
||
</p>
|
||
|
||
<div class="org-src-container">
|
||
<pre class="src src-matlab">n = 4; w0 = 2*pi*900; G0 = 9; G1 = 1; Gc = 1.1;
|
||
H = (((1/w0)*sqrt((1-(G0/Gc)^(2/n))/(1-(Gc/G1)^(2/n)))*s + (G0/Gc)^(1/n))/((1/G1)^(1/n)*(1/w0)*sqrt((1-(G0/Gc)^(2/n))/(1-(Gc/G1)^(2/n)))*s + (1/Gc)^(1/n)))^n;
|
||
wphi = 0.2*(s+3.142e04)/(s+628.3)/H;
|
||
</pre>
|
||
</div>
|
||
|
||
|
||
<div id="org56b5728" class="figure">
|
||
<p><img src="figs/mixed_syn_hinf_weight.png" alt="mixed_syn_hinf_weight.png" />
|
||
</p>
|
||
<p><span class="figure-number">Figure 22: </span>Wanted maximum module uncertainty of the super sensor (<a href="./figs/mixed_syn_hinf_weight.png">png</a>, <a href="./figs/mixed_syn_hinf_weight.pdf">pdf</a>)</p>
|
||
</div>
|
||
|
||
<p>
|
||
The equivalent Magnitude and Phase uncertainties are shown in Fig. <a href="#orgaf5d6a0">23</a>.
|
||
</p>
|
||
|
||
|
||
<div id="orgaf5d6a0" class="figure">
|
||
<p><img src="figs/mixed_syn_objective_hinf.png" alt="mixed_syn_objective_hinf.png" />
|
||
</p>
|
||
<p><span class="figure-number">Figure 23: </span>\(\mathcal{H}_\infty\) synthesis objective part of the mixed-synthesis (<a href="./figs/mixed_syn_objective_hinf.png">png</a>, <a href="./figs/mixed_syn_objective_hinf.pdf">pdf</a>)</p>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
|
||
<div id="outline-container-orgd08d347" class="outline-3">
|
||
<h3 id="orgd08d347"><span class="section-number-3">3.4</span> Mixed Synthesis Architecture</h3>
|
||
<div class="outline-text-3" id="text-3-4">
|
||
<p>
|
||
The synthesis architecture that is used here is shown in Fig. <a href="#orgc007636">24</a>.
|
||
</p>
|
||
|
||
<p>
|
||
The controller \(K\) is synthesized such that it:
|
||
</p>
|
||
<ul class="org-ul">
|
||
<li>Keeps the \(\mathcal{H}_\infty\) norm \(G\) of the transfer function from \(w\) to \(z_\infty\) bellow some specified value</li>
|
||
<li>Keeps the \(\mathcal{H}_2\) norm \(H\) of the transfer function from \(w\) to \(z_2\) bellow some specified value</li>
|
||
<li>Minimizes a trade-off criterion of the form \(W_1 G^2 + W_2 H^2\) where \(W_1\) and \(W_2\) are specified values</li>
|
||
</ul>
|
||
|
||
|
||
<div id="orgc007636" class="figure">
|
||
<p><img src="figs-tikz/mixed_h2_hinf_synthesis.png" alt="mixed_h2_hinf_synthesis.png" />
|
||
</p>
|
||
<p><span class="figure-number">Figure 24: </span>Mixed H2/H-Infinity Synthesis</p>
|
||
</div>
|
||
|
||
<p>
|
||
Here, we define \(P\) such that:
|
||
</p>
|
||
\begin{align*}
|
||
\left\| \frac{z_\infty}{w} \right\|_\infty &= \left\| \begin{matrix}W_1(s) H_1(s) \\ W_2(s) H_2(s)\end{matrix} \right\|_\infty \\
|
||
\left\| \frac{z_2}{w} \right\|_2 &= \left\| \begin{matrix}N_1(s) H_1(s) \\ N_2(s) H_2(s)\end{matrix} \right\|_2
|
||
\end{align*}
|
||
|
||
<p>
|
||
Then:
|
||
</p>
|
||
<ul class="org-ul">
|
||
<li>we specify the maximum value for the \(\mathcal{H}_\infty\) norm between \(w\) and \(z_\infty\) to be \(1\)</li>
|
||
<li>we don’t specify any maximum value for the \(\mathcal{H}_2\) norm between \(w\) and \(z_2\)</li>
|
||
<li>we choose \(W_1 = 0\) and \(W_2 = 1\) such that the objective is to minimize the \(\mathcal{H}_2\) norm between \(w\) and \(z_2\)</li>
|
||
</ul>
|
||
|
||
<p>
|
||
The synthesis objective is to have:
|
||
\[ \left\| \frac{z_\infty}{w} \right\|_\infty = \left\| \begin{matrix}W_1(s) H_1(s) \\ W_2(s) H_2(s)\end{matrix} \right\|_\infty < 1 \]
|
||
and to minimize:
|
||
\[ \left\| \frac{z_2}{w} \right\|_2 = \left\| \begin{matrix}N_1(s) H_1(s) \\ N_2(s) H_2(s)\end{matrix} \right\|_2 \]
|
||
which is what we wanted.
|
||
</p>
|
||
|
||
<p>
|
||
We define the generalized plant that will be used for the mixed synthesis.
|
||
</p>
|
||
<div class="org-src-container">
|
||
<pre class="src src-matlab">W1u = ss(w1*wphi); W2u = ss(w2*wphi); % Weight on the uncertainty
|
||
W1n = ss(N1); W2n = ss(N2); % Weight on the noise
|
||
|
||
P = [W1u -W1u;
|
||
0 W2u;
|
||
W1n -W1n;
|
||
0 W2n;
|
||
1 0];
|
||
</pre>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
|
||
<div id="outline-container-orgec6e5c4" class="outline-3">
|
||
<h3 id="orgec6e5c4"><span class="section-number-3">3.5</span> Mixed \(\mathcal{H}_2\) / \(\mathcal{H}_\infty\) Synthesis</h3>
|
||
<div class="outline-text-3" id="text-3-5">
|
||
<p>
|
||
The mixed \(\mathcal{H}_2/\mathcal{H}_\infty\) synthesis is performed below.
|
||
</p>
|
||
<div class="org-src-container">
|
||
<pre class="src src-matlab">Nmeas = 1; Ncon = 1; Nz2 = 2;
|
||
|
||
[H2,~,normz,~] = h2hinfsyn(P, Nmeas, Ncon, Nz2, [0, 1], 'HINFMAX', 1, 'H2MAX', Inf, 'DKMAX', 100, 'TOL', 0.01, 'DISPLAY', 'on');
|
||
|
||
H1 = 1 - H2;
|
||
</pre>
|
||
</div>
|
||
|
||
<p>
|
||
The obtained complementary filters are shown in Fig. <a href="#org13db4e3">25</a>.
|
||
</p>
|
||
|
||
|
||
<div id="org13db4e3" class="figure">
|
||
<p><img src="figs/comp_filters_mixed_synthesis.png" alt="comp_filters_mixed_synthesis.png" />
|
||
</p>
|
||
<p><span class="figure-number">Figure 25: </span>Obtained complementary filters after mixed \(\mathcal{H}_2/\mathcal{H}_\infty\) synthesis (<a href="./figs/comp_filters_mixed_synthesis.png">png</a>, <a href="./figs/comp_filters_mixed_synthesis.pdf">pdf</a>)</p>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
|
||
<div id="outline-container-orgcf41bc0" class="outline-3">
|
||
<h3 id="orgcf41bc0"><span class="section-number-3">3.6</span> Obtained Super Sensor’s noise</h3>
|
||
<div class="outline-text-3" id="text-3-6">
|
||
<p>
|
||
The PSD and CPS of the super sensor’s noise are shown in Fig. <a href="#org217e5b6">26</a> and Fig. <a href="#orgfc270f8">27</a> respectively.
|
||
</p>
|
||
|
||
|
||
<div id="org217e5b6" class="figure">
|
||
<p><img src="figs/psd_super_sensor_mixed_syn.png" alt="psd_super_sensor_mixed_syn.png" />
|
||
</p>
|
||
<p><span class="figure-number">Figure 26: </span>Power Spectral Density of the Super Sensor obtained with the mixed \(\mathcal{H}_2/\mathcal{H}_\infty\) synthesis (<a href="./figs/psd_super_sensor_mixed_syn.png">png</a>, <a href="./figs/psd_super_sensor_mixed_syn.pdf">pdf</a>)</p>
|
||
</div>
|
||
|
||
|
||
|
||
<div id="orgfc270f8" class="figure">
|
||
<p><img src="figs/cps_super_sensor_mixed_syn.png" alt="cps_super_sensor_mixed_syn.png" />
|
||
</p>
|
||
<p><span class="figure-number">Figure 27: </span>Cumulative Power Spectrum of the Super Sensor obtained with the mixed \(\mathcal{H}_2/\mathcal{H}_\infty\) synthesis (<a href="./figs/cps_super_sensor_mixed_syn.png">png</a>, <a href="./figs/cps_super_sensor_mixed_syn.pdf">pdf</a>)</p>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
|
||
<div id="outline-container-org39e64c2" class="outline-3">
|
||
<h3 id="org39e64c2"><span class="section-number-3">3.7</span> Obtained Super Sensor’s Uncertainty</h3>
|
||
<div class="outline-text-3" id="text-3-7">
|
||
<p>
|
||
The uncertainty on the super sensor’s dynamics is shown in Fig. <a href="#org5613508">28</a>.
|
||
</p>
|
||
|
||
|
||
<div id="org5613508" class="figure">
|
||
<p><img src="figs/super_sensor_dyn_uncertainty_mixed_syn.png" alt="super_sensor_dyn_uncertainty_mixed_syn.png" />
|
||
</p>
|
||
<p><span class="figure-number">Figure 28: </span>Super Sensor Dynamical Uncertainty obtained with the mixed synthesis (<a href="./figs/super_sensor_dyn_uncertainty_mixed_syn.png">png</a>, <a href="./figs/super_sensor_dyn_uncertainty_mixed_syn.pdf">pdf</a>)</p>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
|
||
<div id="outline-container-orgdf7555f" class="outline-3">
|
||
<h3 id="orgdf7555f"><span class="section-number-3">3.8</span> Conclusion</h3>
|
||
<div class="outline-text-3" id="text-3-8">
|
||
<p>
|
||
This synthesis methods allows both to:
|
||
</p>
|
||
<ul class="org-ul">
|
||
<li>limit the dynamical uncertainty of the super sensor</li>
|
||
<li>minimize the RMS value of the estimation</li>
|
||
</ul>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
|
||
<p>
|
||
|
||
</p>
|
||
|
||
<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><h2 class='citeproc-org-bib-h2'>Bibliography</h2>
|
||
<div class="csl-bib-body">
|
||
<div class="csl-entry"><a name="citeproc_bib_item_1"></a>Barzilai, Aaron, Tom VanZandt, and Tom Kenny. 1998. “Technique for Measurement of the Noise of a Sensor in the Presence of Large Background Signals.” <i>Review of Scientific Instruments</i> 69 (7):2767–72. <a href="https://doi.org/10.1063/1.1149013">https://doi.org/10.1063/1.1149013</a>.</div>
|
||
<div class="csl-entry"><a name="citeproc_bib_item_2"></a>Moore, Steven Ian, Andrew J. Fleming, and Yuen Kuan Yong. 2019. “Capacitive Instrumentation and Sensor Fusion for High-Bandwidth Nanopositioning.” <i>IEEE Sensors Letters</i> 3 (8):1–3. <a href="https://doi.org/10.1109/lsens.2019.2933065">https://doi.org/10.1109/lsens.2019.2933065</a>.</div>
|
||
</div>
|
||
</div>
|
||
<div id="postamble" class="status">
|
||
<p class="author">Author: Thomas Dehaeze</p>
|
||
<p class="date">Created: 2020-09-23 mer. 15:37</p>
|
||
</div>
|
||
</body>
|
||
</html>
|