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+<title>Complementary Filters Shaping Using \(\mathcal{H}_\infty\) Synthesis - Matlab Computation</title>
+<meta name="generator" content="Org mode" />
+<meta name="author" content="Thomas Dehaeze" />
+<link rel="stylesheet" type="text/css" href="../css/htmlize.css"/>
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+<div id="org-div-home-and-up">
+ <a accesskey="h" href="../index.html"> UP </a>
+ |
+ <a accesskey="H" href="../index.html"> HOME </a>
+</div><div id="content">
+<h1 class="title">Complementary Filters Shaping Using \(\mathcal{H}_\infty\) Synthesis - Matlab Computation</h1>
+<div id="table-of-contents">
+<h2>Table of Contents</h2>
+<div id="text-table-of-contents">
+<ul>
+<li><a href="#orgcd571e1">1. H-Infinity synthesis of complementary filters</a>
+<ul>
+<li><a href="#orgea65f42">1.1. Synthesis Architecture</a></li>
+<li><a href="#org8a8a446">1.2. Design of Weighting Function</a></li>
+<li><a href="#orgfe22503">1.3. H-Infinity Synthesis</a></li>
+<li><a href="#orgf9471db">1.4. Obtained Complementary Filters</a></li>
+</ul>
+</li>
+<li><a href="#org0e59118">2. Generating 3 complementary filters</a>
+<ul>
+<li><a href="#orgfbd127e">2.1. Theory</a></li>
+<li><a href="#orgb689a85">2.2. Weights</a></li>
+<li><a href="#org24d1ff1">2.3. H-Infinity Synthesis</a></li>
+<li><a href="#orgef81b1e">2.4. Obtained Complementary Filters</a></li>
+</ul>
+</li>
+<li><a href="#org2c0b99c">3. Implement complementary filters for LIGO</a>
+<ul>
+<li><a href="#org6ea0071">3.1. Specifications</a></li>
+<li><a href="#org8f811fb">3.2. FIR Filter</a></li>
+<li><a href="#org9a34a27">3.3. Weights</a></li>
+<li><a href="#orgc9f5d68">3.4. H-Infinity Synthesis</a></li>
+<li><a href="#orgd83a581">3.5. Compare FIR and H-Infinity Filters</a></li>
+</ul>
+</li>
+</ul>
+</div>
+</div>
+
+<p>
+In this document, the design of complementary filters is studied.
+</p>
+
+<p>
+One use of complementary filter is described below:
+</p>
+<blockquote>
+<p>
+The basic idea of a complementary filter involves taking two or more sensors, filtering out unreliable frequencies for each sensor, and combining the filtered outputs to get a better estimate throughout the entire bandwidth of the system.
+To achieve this, the sensors included in the filter should complement one another by performing better over specific parts of the system bandwidth.
+</p>
+</blockquote>
+
+<p>
+This document is divided into several sections:
+</p>
+<ul class="org-ul">
+<li>in section <a href="#org954a80e">1</a>, the \(\mathcal{H}_\infty\) synthesis is used for generating two complementary filters</li>
+<li>in section <a href="#orga319f0c">2</a>, a method using the \(\mathcal{H}_\infty\) synthesis is proposed to shape three of more complementary filters</li>
+<li>in section <a href="#orgb69dc24">3</a>, the \(\mathcal{H}_\infty\) synthesis is used and compared with FIR complementary filters used for LIGO</li>
+</ul>
+
+<div class="note" id="org3c0f990">
+<p>
+Add the Matlab code use to obtain the results presented in the paper are accessible <a href="matlab.zip">here</a> and presented below.
+</p>
+
+</div>
+
+<div id="outline-container-orgcd571e1" class="outline-2">
+<h2 id="orgcd571e1"><span class="section-number-2">1</span> H-Infinity synthesis of complementary filters</h2>
+<div class="outline-text-2" id="text-1">
+<p>
+<a id="org954a80e"></a>
+</p>
+<div class="note" id="orgca3b1ba">
+<p>
+The Matlab file corresponding to this section is accessible <a href="matlab/h_inf_synthesis_complementary_filters.m">here</a>.
+</p>
+
+</div>
+</div>
+
+<div id="outline-container-orgea65f42" class="outline-3">
+<h3 id="orgea65f42"><span class="section-number-3">1.1</span> Synthesis Architecture</h3>
+<div class="outline-text-3" id="text-1-1">
+<p>
+We here synthesize two complementary filters using the \(\mathcal{H}_\infty\) synthesis.
+The goal is to specify upper bounds on the norms of the two complementary filters \(H_1(s)\) and \(H_2(s)\) while ensuring their complementary property (\(H_1(s) + H_2(s) = 1\)).
+</p>
+
+<p>
+In order to do so, we use the generalized plant shown on figure <a href="#org46317e2">1</a> where \(W_1(s)\) and \(W_2(s)\) are weighting transfer functions that will be used to shape \(H_1(s)\) and \(H_2(s)\) respectively.
+</p>
+
+
+<div id="org46317e2" 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 1: </span>\(\mathcal{H}_\infty\) synthesis of the complementary filters</p>
+</div>
+
+<p>
+The \(\mathcal{H}_\infty\) synthesis applied on this generalized plant will give a transfer function \(H_2\) (figure <a href="#org46317e2">1</a>) such that the \(\mathcal{H}_\infty\) norm of the transfer function from \(w\) to \([z_1,\ z_2]\) is less than one:
+\[ \left\| \begin{array}{c} (1 - H_2(s)) W_1(s) \\ H_2(s) W_2(s) \end{array} \right\|_\infty < 1 \]
+</p>
+
+<p>
+Thus, if the above condition is verified, we can define \(H_1(s) = 1 - H_2(s)\) and we have that:
+\[ \left\| \begin{array}{c} H_1(s) W_1(s) \\ H_2(s) W_2(s) \end{array} \right\|_\infty < 1 \]
+Which is almost (with an maximum error of \(\sqrt{2}\)) equivalent to:
+</p>
+\begin{align*}
+  |H_1(j\omega)| &< \frac{1}{|W_1(j\omega)|}, \quad \forall \omega \\
+  |H_2(j\omega)| &< \frac{1}{|W_2(j\omega)|}, \quad \forall \omega
+\end{align*}
+
+<p>
+We then see that \(W_1(s)\) and \(W_2(s)\) can be used to shape both \(H_1(s)\) and \(H_2(s)\) while ensuring their complementary property by the definition of \(H_1(s) = 1 - H_2(s)\).
+</p>
+</div>
+</div>
+
+<div id="outline-container-org8a8a446" class="outline-3">
+<h3 id="org8a8a446"><span class="section-number-3">1.2</span> Design of Weighting Function</h3>
+<div class="outline-text-3" id="text-1-2">
+<p>
+A formula is proposed to help the design of the weighting functions:
+</p>
+\begin{equation}
+  W(s) = \left( \frac{
+           \frac{1}{\omega_0} \sqrt{\frac{1 - \left(\frac{G_0}{G_c}\right)^{\frac{2}{n}}}{1 - \left(\frac{G_c}{G_\infty}\right)^{\frac{2}{n}}}} s + \left(\frac{G_0}{G_c}\right)^{\frac{1}{n}}
+         }{
+           \left(\frac{1}{G_\infty}\right)^{\frac{1}{n}} \frac{1}{\omega_0} \sqrt{\frac{1 - \left(\frac{G_0}{G_c}\right)^{\frac{2}{n}}}{1 - \left(\frac{G_c}{G_\infty}\right)^{\frac{2}{n}}}} s + \left(\frac{1}{G_c}\right)^{\frac{1}{n}}
+         }\right)^n
+\end{equation}
+
+<p>
+The parameters permits to specify:
+</p>
+<ul class="org-ul">
+<li>the low frequency gain: \(G_0 = lim_{\omega \to 0} |W(j\omega)|\)</li>
+<li>the high frequency gain: \(G_\infty = lim_{\omega \to \infty} |W(j\omega)|\)</li>
+<li>the absolute gain at \(\omega_0\): \(G_c = |W(j\omega_0)|\)</li>
+<li>the absolute slope between high and low frequency: \(n\)</li>
+</ul>
+
+<p>
+The general shape of a weighting function generated using the formula is shown in figure <a href="#org4ee6999">2</a>.
+</p>
+
+
+<div id="org4ee6999" class="figure">
+<p><img src="figs-tikz/weight_formula.png" alt="weight_formula.png" />
+</p>
+<p><span class="figure-number">Figure 2: </span>Amplitude of the proposed formula for the weighting functions</p>
+</div>
+
+<div class="org-src-container">
+<pre class="src src-matlab">n = 2; w0 = 2<span class="org-type">*</span><span class="org-constant">pi</span><span class="org-type">*</span>11; G0 = 1<span class="org-type">/</span>10; G1 = 1000; Gc = 1<span class="org-type">/</span>2;
+W1 = (((1<span class="org-type">/</span>w0)<span class="org-type">*</span>sqrt((1<span class="org-type">-</span>(G0<span class="org-type">/</span>Gc)<span class="org-type">^</span>(2<span class="org-type">/</span>n))<span class="org-type">/</span>(1<span class="org-type">-</span>(Gc<span class="org-type">/</span>G1)<span class="org-type">^</span>(2<span class="org-type">/</span>n)))<span class="org-type">*</span>s <span class="org-type">+</span> (G0<span class="org-type">/</span>Gc)<span class="org-type">^</span>(1<span class="org-type">/</span>n))<span class="org-type">/</span>((1<span class="org-type">/</span>G1)<span class="org-type">^</span>(1<span class="org-type">/</span>n)<span class="org-type">*</span>(1<span class="org-type">/</span>w0)<span class="org-type">*</span>sqrt((1<span class="org-type">-</span>(G0<span class="org-type">/</span>Gc)<span class="org-type">^</span>(2<span class="org-type">/</span>n))<span class="org-type">/</span>(1<span class="org-type">-</span>(Gc<span class="org-type">/</span>G1)<span class="org-type">^</span>(2<span class="org-type">/</span>n)))<span class="org-type">*</span>s <span class="org-type">+</span> (1<span class="org-type">/</span>Gc)<span class="org-type">^</span>(1<span class="org-type">/</span>n)))<span class="org-type">^</span>n;
+
+n = 3; w0 = 2<span class="org-type">*</span><span class="org-constant">pi</span><span class="org-type">*</span>10; G0 = 1000; G1 = 0.1; Gc = 1<span class="org-type">/</span>2;
+W2 = (((1<span class="org-type">/</span>w0)<span class="org-type">*</span>sqrt((1<span class="org-type">-</span>(G0<span class="org-type">/</span>Gc)<span class="org-type">^</span>(2<span class="org-type">/</span>n))<span class="org-type">/</span>(1<span class="org-type">-</span>(Gc<span class="org-type">/</span>G1)<span class="org-type">^</span>(2<span class="org-type">/</span>n)))<span class="org-type">*</span>s <span class="org-type">+</span> (G0<span class="org-type">/</span>Gc)<span class="org-type">^</span>(1<span class="org-type">/</span>n))<span class="org-type">/</span>((1<span class="org-type">/</span>G1)<span class="org-type">^</span>(1<span class="org-type">/</span>n)<span class="org-type">*</span>(1<span class="org-type">/</span>w0)<span class="org-type">*</span>sqrt((1<span class="org-type">-</span>(G0<span class="org-type">/</span>Gc)<span class="org-type">^</span>(2<span class="org-type">/</span>n))<span class="org-type">/</span>(1<span class="org-type">-</span>(Gc<span class="org-type">/</span>G1)<span class="org-type">^</span>(2<span class="org-type">/</span>n)))<span class="org-type">*</span>s <span class="org-type">+</span> (1<span class="org-type">/</span>Gc)<span class="org-type">^</span>(1<span class="org-type">/</span>n)))<span class="org-type">^</span>n;
+</pre>
+</div>
+
+
+<div id="org79474f8" class="figure">
+<p><img src="figs/weights_W1_W2.png" alt="weights_W1_W2.png" />
+</p>
+<p><span class="figure-number">Figure 3: </span>Weights on the complementary filters \(W_1\) and \(W_2\) and the associated performance weights</p>
+</div>
+</div>
+</div>
+
+<div id="outline-container-orgfe22503" class="outline-3">
+<h3 id="orgfe22503"><span class="section-number-3">1.3</span> H-Infinity Synthesis</h3>
+<div class="outline-text-3" id="text-1-3">
+<p>
+We define the generalized plant \(P\) on matlab.
+</p>
+<div class="org-src-container">
+<pre class="src src-matlab">P = [W1 <span class="org-type">-</span>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, <span class="org-type">~</span>, gamma, <span class="org-type">~</span>] = hinfsyn(P, 1, 1,<span class="org-string">'TOLGAM'</span>, 0.001, <span class="org-string">'METHOD'</span>, <span class="org-string">'ric'</span>, <span class="org-string">'DISPLAY'</span>, <span class="org-string">'on'</span>);
+</pre>
+</div>
+
+<pre class="example" id="org9fcd456">
+[H2, ~, gamma, ~] = hinfsyn(P, 1, 1,'TOLGAM', 0.001, 'METHOD', 'ric', 'DISPLAY', 'on');
+Resetting value of Gamma min based on D_11, D_12, D_21 terms
+
+Test bounds:      0.1000 &lt;  gamma  &lt;=   1050.0000
+
+  gamma    hamx_eig  xinf_eig  hamy_eig   yinf_eig   nrho_xy   p/f
+1.050e+03   2.8e+01   2.4e-07   4.1e+00    0.0e+00    0.0000    p
+  525.050   2.8e+01   2.4e-07   4.1e+00    0.0e+00    0.0000    p
+  262.575   2.8e+01   2.4e-07   4.1e+00    0.0e+00    0.0000    p
+  131.337   2.8e+01   2.4e-07   4.1e+00   -1.0e-13    0.0000    p
+   65.719   2.8e+01   2.4e-07   4.1e+00   -9.5e-14    0.0000    p
+   32.909   2.8e+01   2.4e-07   4.1e+00    0.0e+00    0.0000    p
+   16.505   2.8e+01   2.4e-07   4.1e+00   -1.0e-13    0.0000    p
+    8.302   2.8e+01   2.4e-07   4.1e+00   -7.2e-14    0.0000    p
+    4.201   2.8e+01   2.4e-07   4.1e+00   -2.5e-25    0.0000    p
+    2.151   2.7e+01   2.4e-07   4.1e+00   -3.8e-14    0.0000    p
+    1.125   2.6e+01   2.4e-07   4.1e+00   -5.4e-24    0.0000    p
+    0.613   2.3e+01 -3.7e+01#  4.1e+00    0.0e+00    0.0000    f
+    0.869   2.6e+01 -3.7e+02#  4.1e+00    0.0e+00    0.0000    f
+    0.997   2.6e+01 -1.1e+04#  4.1e+00    0.0e+00    0.0000    f
+    1.061   2.6e+01   2.4e-07   4.1e+00    0.0e+00    0.0000    p
+    1.029   2.6e+01   2.4e-07   4.1e+00    0.0e+00    0.0000    p
+    1.013   2.6e+01   2.4e-07   4.1e+00    0.0e+00    0.0000    p
+    1.005   2.6e+01   2.4e-07   4.1e+00    0.0e+00    0.0000    p
+    1.001   2.6e+01 -3.1e+04#  4.1e+00   -3.8e-14    0.0000    f
+    1.003   2.6e+01 -2.8e+05#  4.1e+00    0.0e+00    0.0000    f
+    1.004   2.6e+01   2.4e-07   4.1e+00   -5.8e-24    0.0000    p
+    1.004   2.6e+01   2.4e-07   4.1e+00    0.0e+00    0.0000    p
+
+ Gamma value achieved:     1.0036
+</pre>
+
+<p>
+We then define the high pass filter \(H_1 = 1 - H_2\). The bode plot of both \(H_1\) and \(H_2\) is shown on figure <a href="#orgbc23afe">4</a>.
+</p>
+
+<div class="org-src-container">
+<pre class="src src-matlab">H1 = 1 <span class="org-type">-</span> H2;
+</pre>
+</div>
+</div>
+</div>
+
+<div id="outline-container-orgf9471db" class="outline-3">
+<h3 id="orgf9471db"><span class="section-number-3">1.4</span> Obtained Complementary Filters</h3>
+<div class="outline-text-3" id="text-1-4">
+<p>
+The obtained complementary filters are shown on figure <a href="#orgbc23afe">4</a>.
+</p>
+
+
+<div id="orgbc23afe" class="figure">
+<p><img src="figs/hinf_filters_results.png" alt="hinf_filters_results.png" />
+</p>
+<p><span class="figure-number">Figure 4: </span>Obtained complementary filters using \(\mathcal{H}_\infty\) synthesis</p>
+</div>
+</div>
+</div>
+</div>
+
+<div id="outline-container-org0e59118" class="outline-2">
+<h2 id="org0e59118"><span class="section-number-2">2</span> Generating 3 complementary filters</h2>
+<div class="outline-text-2" id="text-2">
+<p>
+<a id="orga319f0c"></a>
+</p>
+<div class="note" id="org2ba2674">
+<p>
+The Matlab file corresponding to this section is accessible <a href="matlab/three_comp_filters.m">here</a>.
+</p>
+
+</div>
+</div>
+
+<div id="outline-container-orgfbd127e" class="outline-3">
+<h3 id="orgfbd127e"><span class="section-number-3">2.1</span> Theory</h3>
+<div class="outline-text-3" id="text-2-1">
+<p>
+We want:
+</p>
+\begin{align*}
+  & |H_1(j\omega)| < 1/|W_1(j\omega)|, \quad \forall\omega\\
+  & |H_2(j\omega)| < 1/|W_2(j\omega)|, \quad \forall\omega\\
+  & |H_3(j\omega)| < 1/|W_3(j\omega)|, \quad \forall\omega\\
+  & H_1(s) + H_2(s) + H_3(s) = 1
+\end{align*}
+
+<p>
+For that, we use the \(\mathcal{H}_\infty\) synthesis with the architecture shown on figure <a href="#org353bdb6">5</a>.
+</p>
+
+
+<div id="org353bdb6" class="figure">
+<p><img src="figs-tikz/comp_filter_three_hinf.png" alt="comp_filter_three_hinf.png" />
+</p>
+<p><span class="figure-number">Figure 5: </span>Generalized architecture for generating 3 complementary filters</p>
+</div>
+
+<p>
+The \(\mathcal{H}_\infty\) objective is:
+</p>
+\begin{align*}
+  & |(1 - H_2(j\omega) - H_3(j\omega)) W_1(j\omega)| < 1, \quad \forall\omega\\
+  & |H_2(j\omega) W_2(j\omega)| < 1, \quad \forall\omega\\
+  & |H_3(j\omega) W_3(j\omega)| < 1, \quad \forall\omega\\
+\end{align*}
+
+<p>
+And thus if we choose \(H_1 = 1 - H_2 - H_3\) we have solved the problem.
+</p>
+</div>
+</div>
+
+<div id="outline-container-orgb689a85" class="outline-3">
+<h3 id="orgb689a85"><span class="section-number-3">2.2</span> Weights</h3>
+<div class="outline-text-3" id="text-2-2">
+<p>
+First we define the weights.
+</p>
+<div class="org-src-container">
+<pre class="src src-matlab">n = 2; w0 = 2<span class="org-type">*</span><span class="org-constant">pi</span><span class="org-type">*</span>1; G0 = 1<span class="org-type">/</span>10; G1 = 1000; Gc = 1<span class="org-type">/</span>2;
+W1 = (((1<span class="org-type">/</span>w0)<span class="org-type">*</span>sqrt((1<span class="org-type">-</span>(G0<span class="org-type">/</span>Gc)<span class="org-type">^</span>(2<span class="org-type">/</span>n))<span class="org-type">/</span>(1<span class="org-type">-</span>(Gc<span class="org-type">/</span>G1)<span class="org-type">^</span>(2<span class="org-type">/</span>n)))<span class="org-type">*</span>s <span class="org-type">+</span> (G0<span class="org-type">/</span>Gc)<span class="org-type">^</span>(1<span class="org-type">/</span>n))<span class="org-type">/</span>((1<span class="org-type">/</span>G1)<span class="org-type">^</span>(1<span class="org-type">/</span>n)<span class="org-type">*</span>(1<span class="org-type">/</span>w0)<span class="org-type">*</span>sqrt((1<span class="org-type">-</span>(G0<span class="org-type">/</span>Gc)<span class="org-type">^</span>(2<span class="org-type">/</span>n))<span class="org-type">/</span>(1<span class="org-type">-</span>(Gc<span class="org-type">/</span>G1)<span class="org-type">^</span>(2<span class="org-type">/</span>n)))<span class="org-type">*</span>s <span class="org-type">+</span> (1<span class="org-type">/</span>Gc)<span class="org-type">^</span>(1<span class="org-type">/</span>n)))<span class="org-type">^</span>n;
+
+W2 = 0.22<span class="org-type">*</span>(1 <span class="org-type">+</span> s<span class="org-type">/</span>2<span class="org-type">/</span><span class="org-constant">pi</span><span class="org-type">/</span>1)<span class="org-type">^</span>2<span class="org-type">/</span>(sqrt(1e<span class="org-type">-</span>4) <span class="org-type">+</span> s<span class="org-type">/</span>2<span class="org-type">/</span><span class="org-constant">pi</span><span class="org-type">/</span>1)<span class="org-type">^</span>2<span class="org-type">*</span>(1 <span class="org-type">+</span> s<span class="org-type">/</span>2<span class="org-type">/</span><span class="org-constant">pi</span><span class="org-type">/</span>10)<span class="org-type">^</span>2<span class="org-type">/</span>(1 <span class="org-type">+</span> s<span class="org-type">/</span>2<span class="org-type">/</span><span class="org-constant">pi</span><span class="org-type">/</span>1000)<span class="org-type">^</span>2;
+
+n = 3; w0 = 2<span class="org-type">*</span><span class="org-constant">pi</span><span class="org-type">*</span>10; G0 = 1000; G1 = 0.1; Gc = 1<span class="org-type">/</span>2;
+W3 = (((1<span class="org-type">/</span>w0)<span class="org-type">*</span>sqrt((1<span class="org-type">-</span>(G0<span class="org-type">/</span>Gc)<span class="org-type">^</span>(2<span class="org-type">/</span>n))<span class="org-type">/</span>(1<span class="org-type">-</span>(Gc<span class="org-type">/</span>G1)<span class="org-type">^</span>(2<span class="org-type">/</span>n)))<span class="org-type">*</span>s <span class="org-type">+</span> (G0<span class="org-type">/</span>Gc)<span class="org-type">^</span>(1<span class="org-type">/</span>n))<span class="org-type">/</span>((1<span class="org-type">/</span>G1)<span class="org-type">^</span>(1<span class="org-type">/</span>n)<span class="org-type">*</span>(1<span class="org-type">/</span>w0)<span class="org-type">*</span>sqrt((1<span class="org-type">-</span>(G0<span class="org-type">/</span>Gc)<span class="org-type">^</span>(2<span class="org-type">/</span>n))<span class="org-type">/</span>(1<span class="org-type">-</span>(Gc<span class="org-type">/</span>G1)<span class="org-type">^</span>(2<span class="org-type">/</span>n)))<span class="org-type">*</span>s <span class="org-type">+</span> (1<span class="org-type">/</span>Gc)<span class="org-type">^</span>(1<span class="org-type">/</span>n)))<span class="org-type">^</span>n;
+</pre>
+</div>
+
+
+<div id="org743463d" class="figure">
+<p><img src="figs/three_weighting_functions.png" alt="three_weighting_functions.png" />
+</p>
+<p><span class="figure-number">Figure 6: </span>Three weighting functions used for the \(\mathcal{H}_\infty\) synthesis of the complementary filters</p>
+</div>
+</div>
+</div>
+
+<div id="outline-container-org24d1ff1" class="outline-3">
+<h3 id="org24d1ff1"><span class="section-number-3">2.3</span> H-Infinity Synthesis</h3>
+<div class="outline-text-3" id="text-2-3">
+<p>
+Then we create the generalized plant <code>P</code>.
+</p>
+<div class="org-src-container">
+<pre class="src src-matlab">P = [W1 <span class="org-type">-</span>W1 <span class="org-type">-</span>W1;
+     0   W2  0 ;
+     0   0   W3;
+     1   0   0];
+</pre>
+</div>
+
+<p>
+And we do the \(\mathcal{H}_\infty\) synthesis.
+</p>
+<div class="org-src-container">
+<pre class="src src-matlab">[H, <span class="org-type">~</span>, gamma, <span class="org-type">~</span>] = hinfsyn(P, 1, 2,<span class="org-string">'TOLGAM'</span>, 0.001, <span class="org-string">'METHOD'</span>, <span class="org-string">'ric'</span>, <span class="org-string">'DISPLAY'</span>, <span class="org-string">'on'</span>);
+</pre>
+</div>
+
+<pre class="example" id="org4dc4918">
+[H, ~, gamma, ~] = hinfsyn(P, 1, 2,'TOLGAM', 0.001, 'METHOD', 'ric', 'DISPLAY', 'on');
+Resetting value of Gamma min based on D_11, D_12, D_21 terms
+
+Test bounds:      0.1000 &lt;  gamma  &lt;=   1050.0000
+
+  gamma    hamx_eig  xinf_eig  hamy_eig   yinf_eig   nrho_xy   p/f
+1.050e+03   3.2e+00   4.5e-13   6.3e-02   -1.2e-11    0.0000    p
+  525.050   3.2e+00   1.3e-13   6.3e-02    0.0e+00    0.0000    p
+  262.575   3.2e+00   2.1e-12   6.3e-02   -1.5e-13    0.0000    p
+  131.337   3.2e+00   1.1e-12   6.3e-02   -7.2e-29    0.0000    p
+   65.719   3.2e+00   2.0e-12   6.3e-02    0.0e+00    0.0000    p
+   32.909   3.2e+00   7.4e-13   6.3e-02   -5.9e-13    0.0000    p
+   16.505   3.2e+00   1.4e-12   6.3e-02    0.0e+00    0.0000    p
+    8.302   3.2e+00   1.6e-12   6.3e-02    0.0e+00    0.0000    p
+    4.201   3.2e+00   1.6e-12   6.3e-02    0.0e+00    0.0000    p
+    2.151   3.2e+00   1.6e-12   6.3e-02    0.0e+00    0.0000    p
+    1.125   3.2e+00   2.8e-12   6.3e-02    0.0e+00    0.0000    p
+    0.613   3.0e+00 -2.5e+03#  6.3e-02    0.0e+00    0.0000    f
+    0.869   3.1e+00 -2.9e+01#  6.3e-02    0.0e+00    0.0000    f
+    0.997   3.2e+00   1.9e-12   6.3e-02    0.0e+00    0.0000    p
+    0.933   3.1e+00 -6.9e+02#  6.3e-02    0.0e+00    0.0000    f
+    0.965   3.1e+00 -3.0e+03#  6.3e-02    0.0e+00    0.0000    f
+    0.981   3.1e+00 -8.6e+03#  6.3e-02    0.0e+00    0.0000    f
+    0.989   3.2e+00 -2.7e+04#  6.3e-02    0.0e+00    0.0000    f
+    0.993   3.2e+00 -5.7e+05#  6.3e-02    0.0e+00    0.0000    f
+    0.995   3.2e+00   2.2e-12   6.3e-02    0.0e+00    0.0000    p
+    0.994   3.2e+00   1.6e-12   6.3e-02    0.0e+00    0.0000    p
+    0.994   3.2e+00   1.0e-12   6.3e-02    0.0e+00    0.0000    p
+
+ Gamma value achieved:     0.9936
+</pre>
+</div>
+</div>
+
+<div id="outline-container-orgef81b1e" class="outline-3">
+<h3 id="orgef81b1e"><span class="section-number-3">2.4</span> Obtained Complementary Filters</h3>
+<div class="outline-text-3" id="text-2-4">
+<p>
+The obtained filters are:
+</p>
+<div class="org-src-container">
+<pre class="src src-matlab">H2 = tf(H(1));
+H3 = tf(H(2));
+H1 = 1 <span class="org-type">-</span> H2 <span class="org-type">-</span> H3;
+</pre>
+</div>
+
+
+<div id="org7b51da2" class="figure">
+<p><img src="figs/three_complementary_filters_results.png" alt="three_complementary_filters_results.png" />
+</p>
+<p><span class="figure-number">Figure 7: </span>The three complementary filters obtained after \(\mathcal{H}_\infty\) synthesis</p>
+</div>
+</div>
+</div>
+</div>
+
+<div id="outline-container-org2c0b99c" class="outline-2">
+<h2 id="org2c0b99c"><span class="section-number-2">3</span> Implement complementary filters for LIGO</h2>
+<div class="outline-text-2" id="text-3">
+<p>
+<a id="orgb69dc24"></a>
+</p>
+<div class="note" id="orgce8d730">
+<p>
+The Matlab file corresponding to this section is accessible <a href="matlab/comp_filters_ligo.m">here</a>.
+</p>
+
+</div>
+
+<p>
+Let&rsquo;s try to design complementary filters that are corresponding to the complementary filters design for the LIGO and described in (<a href="#citeproc_bib_item_1">Hua 2005</a>).
+</p>
+
+<p>
+The FIR complementary filters designed in (<a href="#citeproc_bib_item_1">Hua 2005</a>) are of order 512.
+</p>
+</div>
+
+<div id="outline-container-org6ea0071" class="outline-3">
+<h3 id="org6ea0071"><span class="section-number-3">3.1</span> Specifications</h3>
+<div class="outline-text-3" id="text-3-1">
+<p>
+The specifications for the filters are:
+</p>
+<ol class="org-ol">
+<li>From \(0\) to \(0.008\text{ Hz}\),the magnitude of the filter’s transfer function should be less than or equal to \(8 \times 10^{-3}\)</li>
+<li>From \(0.008\text{ Hz}\) to \(0.04\text{ Hz}\), it attenuates the input signal proportional to frequency cubed</li>
+<li>Between \(0.04\text{ Hz}\) and \(0.1\text{ Hz}\), the magnitude of the transfer function should be less than 3</li>
+<li>Above \(0.1\text{ Hz}\), the maximum of the magnitude of the complement filter should be as close to zero as possible. In our system, we would like to have the magnitude of the complementary filter to be less than \(0.1\). As the filters obtained in (<a href="#citeproc_bib_item_1">Hua 2005</a>) have a magnitude of \(0.045\), we will set that as our requirement</li>
+</ol>
+
+<p>
+The specifications are translated in upper bounds of the complementary filters are shown on figure <a href="#orgcfe60ed">8</a>.
+</p>
+
+
+<div id="orgcfe60ed" class="figure">
+<p><img src="figs/ligo_specifications.png" alt="ligo_specifications.png" />
+</p>
+<p><span class="figure-number">Figure 8: </span>Specification for the LIGO complementary filters</p>
+</div>
+</div>
+</div>
+
+<div id="outline-container-org8f811fb" class="outline-3">
+<h3 id="org8f811fb"><span class="section-number-3">3.2</span> FIR Filter</h3>
+<div class="outline-text-3" id="text-3-2">
+<p>
+We here try to implement the FIR complementary filter synthesis as explained in (<a href="#citeproc_bib_item_1">Hua 2005</a>).
+For that, we use the <a href="http://cvxr.com/cvx/">CVX matlab Toolbox</a>.
+</p>
+
+<p>
+We setup the CVX toolbox and use the <code>SeDuMi</code> solver.
+</p>
+<div class="org-src-container">
+<pre class="src src-matlab">cvx_startup;
+cvx_solver sedumi;
+</pre>
+</div>
+
+<p>
+We define the frequency vectors on which we will constrain the norm of the FIR filter.
+</p>
+<div class="org-src-container">
+<pre class="src src-matlab">w1 = 0<span class="org-type">:</span>4.06e<span class="org-type">-</span>4<span class="org-type">:</span>0.008;
+w2 = 0.008<span class="org-type">:</span>4.06e<span class="org-type">-</span>4<span class="org-type">:</span>0.04;
+w3 = 0.04<span class="org-type">:</span>8.12e<span class="org-type">-</span>4<span class="org-type">:</span>0.1;
+w4 = 0.1<span class="org-type">:</span>8.12e<span class="org-type">-</span>4<span class="org-type">:</span>0.83;
+</pre>
+</div>
+
+<p>
+We then define the order of the FIR filter.
+</p>
+<div class="org-src-container">
+<pre class="src src-matlab">n = 512;
+</pre>
+</div>
+
+<div class="org-src-container">
+<pre class="src src-matlab">A1 = [ones(length(w1),1),  cos(kron(w1<span class="org-type">'.*</span>(2<span class="org-type">*</span><span class="org-constant">pi</span>),[1<span class="org-type">:</span>n<span class="org-type">-</span>1]))];
+A2 = [ones(length(w2),1),  cos(kron(w2<span class="org-type">'.*</span>(2<span class="org-type">*</span><span class="org-constant">pi</span>),[1<span class="org-type">:</span>n<span class="org-type">-</span>1]))];
+A3 = [ones(length(w3),1),  cos(kron(w3<span class="org-type">'.*</span>(2<span class="org-type">*</span><span class="org-constant">pi</span>),[1<span class="org-type">:</span>n<span class="org-type">-</span>1]))];
+A4 = [ones(length(w4),1),  cos(kron(w4<span class="org-type">'.*</span>(2<span class="org-type">*</span><span class="org-constant">pi</span>),[1<span class="org-type">:</span>n<span class="org-type">-</span>1]))];
+
+B1 = [zeros(length(w1),1), sin(kron(w1<span class="org-type">'.*</span>(2<span class="org-type">*</span><span class="org-constant">pi</span>),[1<span class="org-type">:</span>n<span class="org-type">-</span>1]))];
+B2 = [zeros(length(w2),1), sin(kron(w2<span class="org-type">'.*</span>(2<span class="org-type">*</span><span class="org-constant">pi</span>),[1<span class="org-type">:</span>n<span class="org-type">-</span>1]))];
+B3 = [zeros(length(w3),1), sin(kron(w3<span class="org-type">'.*</span>(2<span class="org-type">*</span><span class="org-constant">pi</span>),[1<span class="org-type">:</span>n<span class="org-type">-</span>1]))];
+B4 = [zeros(length(w4),1), sin(kron(w4<span class="org-type">'.*</span>(2<span class="org-type">*</span><span class="org-constant">pi</span>),[1<span class="org-type">:</span>n<span class="org-type">-</span>1]))];
+</pre>
+</div>
+
+<p>
+We run the convex optimization.
+</p>
+<div class="org-src-container">
+<pre class="src src-matlab">cvx_begin
+
+variable y(n<span class="org-type">+</span>1,1)
+
+<span class="org-comment">% t</span>
+maximize(<span class="org-type">-</span>y(1))
+
+<span class="org-keyword">for</span> <span class="org-variable-name"><span class="org-constant">i</span></span> = <span class="org-constant">1:length(w1)</span>
+    norm([0 A1(<span class="org-constant">i</span>,<span class="org-type">:</span>); 0 B1(<span class="org-constant">i</span>,<span class="org-type">:</span>)]<span class="org-type">*</span>y) <span class="org-type">&lt;=</span> 8e<span class="org-type">-</span>3;
+<span class="org-keyword">end</span>
+
+<span class="org-keyword">for</span>  <span class="org-variable-name"><span class="org-constant">i</span></span> = <span class="org-constant">1:length(w2)</span>
+    norm([0 A2(<span class="org-constant">i</span>,<span class="org-type">:</span>); 0 B2(<span class="org-constant">i</span>,<span class="org-type">:</span>)]<span class="org-type">*</span>y) <span class="org-type">&lt;=</span> 8e<span class="org-type">-</span>3<span class="org-type">*</span>(2<span class="org-type">*</span><span class="org-constant">pi</span><span class="org-type">*</span>w2(<span class="org-constant">i</span>)<span class="org-type">/</span>(0.008<span class="org-type">*</span>2<span class="org-type">*</span><span class="org-constant">pi</span>))<span class="org-type">^</span>3;
+<span class="org-keyword">end</span>
+
+<span class="org-keyword">for</span> <span class="org-variable-name"><span class="org-constant">i</span></span> = <span class="org-constant">1:length(w3)</span>
+    norm([0 A3(<span class="org-constant">i</span>,<span class="org-type">:</span>); 0 B3(<span class="org-constant">i</span>,<span class="org-type">:</span>)]<span class="org-type">*</span>y) <span class="org-type">&lt;=</span> 3;
+<span class="org-keyword">end</span>
+
+<span class="org-keyword">for</span> <span class="org-variable-name"><span class="org-constant">i</span></span> = <span class="org-constant">1:length(w4)</span>
+    norm([[1 0]<span class="org-type">'-</span> [0 A4(<span class="org-constant">i</span>,<span class="org-type">:</span>); 0 B4(<span class="org-constant">i</span>,<span class="org-type">:</span>)]<span class="org-type">*</span>y]) <span class="org-type">&lt;=</span> y(1);
+<span class="org-keyword">end</span>
+
+cvx_end
+
+h = y(2<span class="org-type">:</span>end);
+</pre>
+</div>
+
+<pre class="example" id="orgd562951">
+cvx_begin
+variable y(n+1,1)
+% t
+maximize(-y(1))
+for i = 1:length(w1)
+    norm([0 A1(i,:); 0 B1(i,:)]*y) &lt;= 8e-3;
+end
+for  i = 1:length(w2)
+    norm([0 A2(i,:); 0 B2(i,:)]*y) &lt;= 8e-3*(2*pi*w2(i)/(0.008*2*pi))^3;
+end
+for i = 1:length(w3)
+    norm([0 A3(i,:); 0 B3(i,:)]*y) &lt;= 3;
+end
+for i = 1:length(w4)
+    norm([[1 0]'- [0 A4(i,:); 0 B4(i,:)]*y]) &lt;= y(1);
+end
+cvx_end
+
+Calling SeDuMi 1.34: 4291 variables, 1586 equality constraints
+   For improved efficiency, SeDuMi is solving the dual problem.
+------------------------------------------------------------
+SeDuMi 1.34 (beta) by AdvOL, 2005-2008 and Jos F. Sturm, 1998-2003.
+Alg = 2: xz-corrector, Adaptive Step-Differentiation, theta = 0.250, beta = 0.500
+eqs m = 1586, order n = 3220, dim = 4292, blocks = 1073
+nnz(A) = 1100727 + 0, nnz(ADA) = 1364794, nnz(L) = 683190
+ it :     b*y       gap    delta  rate   t/tP*  t/tD*   feas cg cg  prec
+  0 :            4.11E+02 0.000
+  1 :  -2.58E+00 1.25E+02 0.000 0.3049 0.9000 0.9000   4.87  1  1  3.0E+02
+  2 :  -2.36E+00 3.90E+01 0.000 0.3118 0.9000 0.9000   1.83  1  1  6.6E+01
+  3 :  -1.69E+00 1.31E+01 0.000 0.3354 0.9000 0.9000   1.76  1  1  1.5E+01
+  4 :  -8.60E-01 7.10E+00 0.000 0.5424 0.9000 0.9000   2.48  1  1  4.8E+00
+  5 :  -4.91E-01 5.44E+00 0.000 0.7661 0.9000 0.9000   3.12  1  1  2.5E+00
+  6 :  -2.96E-01 3.88E+00 0.000 0.7140 0.9000 0.9000   2.62  1  1  1.4E+00
+  7 :  -1.98E-01 2.82E+00 0.000 0.7271 0.9000 0.9000   2.14  1  1  8.5E-01
+  8 :  -1.39E-01 2.00E+00 0.000 0.7092 0.9000 0.9000   1.78  1  1  5.4E-01
+  9 :  -9.99E-02 1.30E+00 0.000 0.6494 0.9000 0.9000   1.51  1  1  3.3E-01
+ 10 :  -7.57E-02 8.03E-01 0.000 0.6175 0.9000 0.9000   1.31  1  1  2.0E-01
+ 11 :  -5.99E-02 4.22E-01 0.000 0.5257 0.9000 0.9000   1.17  1  1  1.0E-01
+ 12 :  -5.28E-02 2.45E-01 0.000 0.5808 0.9000 0.9000   1.08  1  1  5.9E-02
+ 13 :  -4.82E-02 1.28E-01 0.000 0.5218 0.9000 0.9000   1.05  1  1  3.1E-02
+ 14 :  -4.56E-02 5.65E-02 0.000 0.4417 0.9045 0.9000   1.02  1  1  1.4E-02
+ 15 :  -4.43E-02 2.41E-02 0.000 0.4265 0.9004 0.9000   1.01  1  1  6.0E-03
+ 16 :  -4.37E-02 8.90E-03 0.000 0.3690 0.9070 0.9000   1.00  1  1  2.3E-03
+ 17 :  -4.35E-02 3.24E-03 0.000 0.3641 0.9164 0.9000   1.00  1  1  9.5E-04
+ 18 :  -4.34E-02 1.55E-03 0.000 0.4788 0.9086 0.9000   1.00  1  1  4.7E-04
+ 19 :  -4.34E-02 8.77E-04 0.000 0.5653 0.9169 0.9000   1.00  1  1  2.8E-04
+ 20 :  -4.34E-02 5.05E-04 0.000 0.5754 0.9034 0.9000   1.00  1  1  1.6E-04
+ 21 :  -4.34E-02 2.94E-04 0.000 0.5829 0.9136 0.9000   1.00  1  1  9.9E-05
+ 22 :  -4.34E-02 1.63E-04 0.015 0.5548 0.9000 0.0000   1.00  1  1  6.6E-05
+ 23 :  -4.33E-02 9.42E-05 0.000 0.5774 0.9053 0.9000   1.00  1  1  3.9E-05
+ 24 :  -4.33E-02 6.27E-05 0.000 0.6658 0.9148 0.9000   1.00  1  1  2.6E-05
+ 25 :  -4.33E-02 3.75E-05 0.000 0.5972 0.9187 0.9000   1.00  1  1  1.6E-05
+ 26 :  -4.33E-02 1.89E-05 0.000 0.5041 0.9117 0.9000   1.00  1  1  8.6E-06
+ 27 :  -4.33E-02 9.72E-06 0.000 0.5149 0.9050 0.9000   1.00  1  1  4.5E-06
+ 28 :  -4.33E-02 2.94E-06 0.000 0.3021 0.9194 0.9000   1.00  1  1  1.5E-06
+ 29 :  -4.33E-02 9.73E-07 0.000 0.3312 0.9189 0.9000   1.00  2  2  5.3E-07
+ 30 :  -4.33E-02 2.82E-07 0.000 0.2895 0.9063 0.9000   1.00  2  2  1.6E-07
+ 31 :  -4.33E-02 8.05E-08 0.000 0.2859 0.9049 0.9000   1.00  2  2  4.7E-08
+ 32 :  -4.33E-02 1.43E-08 0.000 0.1772 0.9059 0.9000   1.00  2  2  8.8E-09
+
+iter seconds digits       c*x               b*y
+ 32     49.4   6.8 -4.3334083581e-02 -4.3334090214e-02
+|Ax-b| =   3.7e-09, [Ay-c]_+ =   1.1E-10, |x|=  1.0e+00, |y|=  2.6e+00
+
+Detailed timing (sec)
+   Pre          IPM          Post
+3.902E+00    4.576E+01    1.035E-02
+Max-norms: ||b||=1, ||c|| = 3,
+Cholesky |add|=0, |skip| = 0, ||L.L|| = 4.26267.
+------------------------------------------------------------
+Status: Solved
+Optimal value (cvx_optval): -0.0433341
+h = y(2:end);
+</pre>
+
+<p>
+Finally, we compute the filter response over the frequency vector defined and the result is shown on figure <a href="#org0d25e01">9</a> which is very close to the filters obtain in (<a href="#citeproc_bib_item_1">Hua 2005</a>).
+</p>
+
+<div class="org-src-container">
+<pre class="src src-matlab">w = [w1 w2 w3 w4];
+H = [exp(<span class="org-type">-</span><span class="org-constant">j</span><span class="org-type">*</span>kron(w<span class="org-type">'.*</span>2<span class="org-type">*</span><span class="org-constant">pi</span>,[0<span class="org-type">:</span>n<span class="org-type">-</span>1]))]<span class="org-type">*</span>h;
+</pre>
+</div>
+
+
+<div id="org0d25e01" class="figure">
+<p><img src="figs/fir_filter_ligo.png" alt="fir_filter_ligo.png" />
+</p>
+<p><span class="figure-number">Figure 9: </span>FIR Complementary filters obtain after convex optimization (<a href="./figs/fir_filter_ligo.png">png</a>, <a href="./figs/fir_filter_ligo.pdf">pdf</a>)</p>
+</div>
+</div>
+</div>
+
+<div id="outline-container-org9a34a27" class="outline-3">
+<h3 id="org9a34a27"><span class="section-number-3">3.3</span> Weights</h3>
+<div class="outline-text-3" id="text-3-3">
+<p>
+We design weights that will be used for the \(\mathcal{H}_\infty\) synthesis of the complementary filters.
+These weights will determine the order of the obtained filters.
+Here are the requirements on the filters:
+</p>
+<ul class="org-ul">
+<li>reasonable order</li>
+<li>to be as close as possible to the specified upper bounds</li>
+<li>stable minimum phase</li>
+</ul>
+
+<p>
+The bode plot of the weights is shown on figure <a href="#orge27ef5e">10</a>.
+</p>
+
+
+<div id="orge27ef5e" class="figure">
+<p><img src="figs/ligo_weights.png" alt="ligo_weights.png" />
+</p>
+<p><span class="figure-number">Figure 10: </span>Weights for the \(\mathcal{H}_\infty\) synthesis</p>
+</div>
+</div>
+</div>
+
+<div id="outline-container-orgc9f5d68" class="outline-3">
+<h3 id="orgc9f5d68"><span class="section-number-3">3.4</span> H-Infinity Synthesis</h3>
+<div class="outline-text-3" id="text-3-4">
+<p>
+We define the generalized plant as shown on figure <a href="#org46317e2">1</a>.
+</p>
+<div class="org-src-container">
+<pre class="src src-matlab">P = [0   wL;
+     wH <span class="org-type">-</span>wH;
+     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">[Hl, <span class="org-type">~</span>, gamma, <span class="org-type">~</span>] = hinfsyn(P, 1, 1,<span class="org-string">'TOLGAM'</span>, 0.001, <span class="org-string">'METHOD'</span>, <span class="org-string">'ric'</span>, <span class="org-string">'DISPLAY'</span>, <span class="org-string">'on'</span>);
+</pre>
+</div>
+
+<pre class="example" id="org51b2320">
+[Hl, ~, 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:      0.3276 &lt;  gamma  &lt;=      1.8063
+
+  gamma    hamx_eig  xinf_eig  hamy_eig   yinf_eig   nrho_xy   p/f
+    1.806   1.4e-02 -1.7e-16   3.6e-03   -4.8e-12    0.0000    p
+    1.067   1.3e-02 -4.2e-14   3.6e-03   -1.9e-12    0.0000    p
+    0.697   1.3e-02 -3.0e-01#  3.6e-03   -3.5e-11    0.0000    f
+    0.882   1.3e-02 -9.5e-01#  3.6e-03   -1.2e-34    0.0000    f
+    0.975   1.3e-02 -2.7e+00#  3.6e-03   -1.6e-12    0.0000    f
+    1.021   1.3e-02 -8.7e+00#  3.6e-03   -4.5e-16    0.0000    f
+    1.044   1.3e-02 -6.5e-14   3.6e-03   -3.0e-15    0.0000    p
+    1.032   1.3e-02 -1.8e+01#  3.6e-03    0.0e+00    0.0000    f
+    1.038   1.3e-02 -3.8e+01#  3.6e-03    0.0e+00    0.0000    f
+    1.041   1.3e-02 -8.3e+01#  3.6e-03   -2.9e-33    0.0000    f
+    1.042   1.3e-02 -1.9e+02#  3.6e-03   -3.4e-11    0.0000    f
+    1.043   1.3e-02 -5.3e+02#  3.6e-03   -7.5e-13    0.0000    f
+
+ Gamma value achieved:     1.0439
+</pre>
+
+<p>
+The high pass filter is defined as \(H_H = 1 - H_L\).
+</p>
+<div class="org-src-container">
+<pre class="src src-matlab">Hh = 1 <span class="org-type">-</span> Hl;
+</pre>
+</div>
+
+<p>
+The size of the filters is shown below.
+</p>
+
+<pre class="example" id="orgede384d">
+size(Hh), size(Hl)
+State-space model with 1 outputs, 1 inputs, and 27 states.
+State-space model with 1 outputs, 1 inputs, and 27 states.
+</pre>
+
+<p>
+The bode plot of the obtained filters as shown on figure <a href="#org941eede">11</a>.
+</p>
+
+
+<div id="org941eede" class="figure">
+<p><img src="figs/hinf_synthesis_ligo_results.png" alt="hinf_synthesis_ligo_results.png" />
+</p>
+<p><span class="figure-number">Figure 11: </span>Obtained complementary filters using the \(\mathcal{H}_\infty\) synthesis</p>
+</div>
+</div>
+</div>
+
+<div id="outline-container-orgd83a581" class="outline-3">
+<h3 id="orgd83a581"><span class="section-number-3">3.5</span> Compare FIR and H-Infinity Filters</h3>
+<div class="outline-text-3" id="text-3-5">
+<p>
+Let&rsquo;s now compare the FIR filters designed in (<a href="#citeproc_bib_item_1">Hua 2005</a>) and the one obtained with the \(\mathcal{H}_\infty\) synthesis on figure <a href="#org4f70e8a">12</a>.
+</p>
+
+
+<div id="org4f70e8a" class="figure">
+<p><img src="figs/comp_fir_ligo_hinf.png" alt="comp_fir_ligo_hinf.png" />
+</p>
+<p><span class="figure-number">Figure 12: </span>Comparison between the FIR filters developped for LIGO and the \(\mathcal{H}_\infty\) complementary filters</p>
+</div>
+</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>Hua, Wensheng. 2005. “Low Frequency Vibration Isolation and Alignment System for Advanced LIGO.” stanford university.</div>
+</div>
+</div>
+<div id="postamble" class="status">
+<p class="author">Author: Thomas Dehaeze</p>
+<p class="date">Created: 2020-10-26 lun. 18:59</p>
+</div>
+</body>
+</html>
diff --git a/matlab/index.org b/matlab/index.org
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+#+TITLE: Complementary Filters Shaping Using $\mathcal{H}_\infty$ Synthesis - Matlab Computation
+:DRAWER:
+#+HTML_LINK_HOME: ../index.html
+#+HTML_LINK_UP: ../index.html
+
+#+LATEX_CLASS: cleanreport
+#+LATEX_CLASS_OPTIONS: [tocnp, secbreak, minted]
+
+#+HTML_HEAD: <link rel="stylesheet" type="text/css" href="../css/htmlize.css"/>
+#+HTML_HEAD: <link rel="stylesheet" type="text/css" href="../css/readtheorg.css"/>
+#+HTML_HEAD: <script src="../js/jquery.min.js"></script>
+#+HTML_HEAD: <script src="../js/bootstrap.min.js"></script>
+#+HTML_HEAD: <script src="../js/jquery.stickytableheaders.min.js"></script>
+#+HTML_HEAD: <script src="../js/readtheorg.js"></script>
+
+#+PROPERTY: header-args:matlab  :session *MATLAB*
+#+PROPERTY: header-args:matlab+ :tangle matlab/comp_filters_design.m
+#+PROPERTY: header-args:matlab+ :comments org
+#+PROPERTY: header-args:matlab+ :exports both
+#+PROPERTY: header-args:matlab+ :results none
+#+PROPERTY: header-args:matlab+ :eval no-export
+#+PROPERTY: header-args:matlab+ :noweb yes
+#+PROPERTY: header-args:matlab+ :mkdirp yes
+#+PROPERTY: header-args:matlab+ :output-dir figs
+:END:
+
+* Introduction                                                       :ignore:
+In this document, the design of complementary filters is studied.
+
+One use of complementary filter is described below:
+#+begin_quote
+  The basic idea of a complementary filter involves taking two or more sensors, filtering out unreliable frequencies for each sensor, and combining the filtered outputs to get a better estimate throughout the entire bandwidth of the system.
+  To achieve this, the sensors included in the filter should complement one another by performing better over specific parts of the system bandwidth.
+#+end_quote
+
+This document is divided into several sections:
+- in section [[sec:h_inf_synthesis_complementary_filters]], the $\mathcal{H}_\infty$ synthesis is used for generating two complementary filters
+- in section [[sec:three_comp_filters]], a method using the $\mathcal{H}_\infty$ synthesis is proposed to shape three of more complementary filters
+- in section [[sec:comp_filters_ligo]], the $\mathcal{H}_\infty$ synthesis is used and compared with FIR complementary filters used for LIGO
+
+#+begin_note
+  Add the Matlab code use to obtain the results presented in the paper are accessible [[file:matlab.zip][here]] and presented below.
+#+end_note
+
+* H-Infinity synthesis of complementary filters
+:PROPERTIES:
+:header-args:matlab+: :tangle matlab/h_inf_synthesis_complementary_filters.m
+:header-args:matlab+: :comments org :mkdirp yes
+:END:
+<<sec:h_inf_synthesis_complementary_filters>>
+
+** Introduction                                                      :ignore:
+#+begin_note
+  The Matlab file corresponding to this section is accessible [[file:matlab/h_inf_synthesis_complementary_filters.m][here]].
+#+end_note
+
+** Matlab Init                                              :noexport:ignore:
+#+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name)
+  <<matlab-dir>>
+#+end_src
+
+#+begin_src matlab :exports none :results silent :noweb yes
+  <<matlab-init>>
+#+end_src
+
+#+begin_src matlab
+  freqs = logspace(-1, 3, 1000);
+#+end_src
+
+** Synthesis Architecture
+We here synthesize two complementary filters using the $\mathcal{H}_\infty$ synthesis.
+The goal is to specify upper bounds on the norms of the two complementary filters $H_1(s)$ and $H_2(s)$ while ensuring their complementary property ($H_1(s) + H_2(s) = 1$).
+
+In order to do so, we use the generalized plant shown on figure [[fig:h_infinity_robst_fusion]] where $W_1(s)$ and $W_2(s)$ are weighting transfer functions that will be used to shape $H_1(s)$ and $H_2(s)$ respectively.
+
+#+name: fig:h_infinity_robst_fusion
+#+caption: $\mathcal{H}_\infty$ synthesis of the complementary filters
+[[file:figs-tikz/h_infinity_robust_fusion.png]]
+
+The $\mathcal{H}_\infty$ synthesis applied on this generalized plant will give a transfer function $H_2$ (figure [[fig:h_infinity_robst_fusion]]) such that the $\mathcal{H}_\infty$ norm of the transfer function from $w$ to $[z_1,\ z_2]$ is less than one:
+\[ \left\| \begin{array}{c} (1 - H_2(s)) W_1(s) \\ H_2(s) W_2(s) \end{array} \right\|_\infty < 1 \]
+
+Thus, if the above condition is verified, we can define $H_1(s) = 1 - H_2(s)$ and we have that:
+\[ \left\| \begin{array}{c} H_1(s) W_1(s) \\ H_2(s) W_2(s) \end{array} \right\|_\infty < 1 \]
+Which is almost (with an maximum error of $\sqrt{2}$) equivalent to:
+\begin{align*}
+  |H_1(j\omega)| &< \frac{1}{|W_1(j\omega)|}, \quad \forall \omega \\
+  |H_2(j\omega)| &< \frac{1}{|W_2(j\omega)|}, \quad \forall \omega
+\end{align*}
+
+We then see that $W_1(s)$ and $W_2(s)$ can be used to shape both $H_1(s)$ and $H_2(s)$ while ensuring their complementary property by the definition of $H_1(s) = 1 - H_2(s)$.
+
+** Design of Weighting Function
+A formula is proposed to help the design of the weighting functions:
+\begin{equation}
+  W(s) = \left( \frac{
+           \frac{1}{\omega_0} \sqrt{\frac{1 - \left(\frac{G_0}{G_c}\right)^{\frac{2}{n}}}{1 - \left(\frac{G_c}{G_\infty}\right)^{\frac{2}{n}}}} s + \left(\frac{G_0}{G_c}\right)^{\frac{1}{n}}
+         }{
+           \left(\frac{1}{G_\infty}\right)^{\frac{1}{n}} \frac{1}{\omega_0} \sqrt{\frac{1 - \left(\frac{G_0}{G_c}\right)^{\frac{2}{n}}}{1 - \left(\frac{G_c}{G_\infty}\right)^{\frac{2}{n}}}} s + \left(\frac{1}{G_c}\right)^{\frac{1}{n}}
+         }\right)^n
+\end{equation}
+
+The parameters permits to specify:
+- the low frequency gain: $G_0 = lim_{\omega \to 0} |W(j\omega)|$
+- the high frequency gain: $G_\infty = lim_{\omega \to \infty} |W(j\omega)|$
+- the absolute gain at $\omega_0$: $G_c = |W(j\omega_0)|$
+- the absolute slope between high and low frequency: $n$
+
+The general shape of a weighting function generated using the formula is shown in figure [[fig:weight_formula]].
+
+#+name: fig:weight_formula
+#+caption: Amplitude of the proposed formula for the weighting functions
+[[file:figs-tikz/weight_formula.png]]
+
+#+begin_src matlab
+  n = 2; w0 = 2*pi*11; G0 = 1/10; G1 = 1000; Gc = 1/2;
+  W1 = (((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;
+
+  n = 3; w0 = 2*pi*10; G0 = 1000; G1 = 0.1; Gc = 1/2;
+  W2 = (((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;
+#+end_src
+
+#+begin_src matlab :exports none
+  figure;
+  hold on;
+  set(gca,'ColorOrderIndex',1)
+  plot(freqs, 1./abs(squeeze(freqresp(W1, freqs, 'Hz'))), '--', 'DisplayName', '$|W_1|^{-1}$');
+  set(gca,'ColorOrderIndex',2)
+  plot(freqs, 1./abs(squeeze(freqresp(W2, freqs, 'Hz'))), '--', 'DisplayName', '$|W_2|^{-1}$');
+  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
+  xlabel('Frequency [Hz]'); ylabel('Magnitude');
+  hold off;
+  xlim([freqs(1), freqs(end)]);
+  ylim([5e-4, 20]);
+  xticks([0.1, 1, 10, 100, 1000]);
+  legend('location', 'northeast', 'FontSize', 8);
+#+end_src
+
+#+begin_src matlab :tangle no :exports results :results file replace
+  exportFig('figs/weights_W1_W2.pdf', 'width', 'wide', 'height', 'normal');
+#+end_src
+
+#+name: fig:weights_W1_W2
+#+caption: Weights on the complementary filters $W_1$ and $W_2$ and the associated performance weights
+#+RESULTS:
+[[file:figs/weights_W1_W2.png]]
+
+** H-Infinity Synthesis
+We define the generalized plant $P$ on matlab.
+#+begin_src matlab
+  P = [W1 -W1;
+       0   W2;
+       1   0];
+#+end_src
+
+And we do the $\mathcal{H}_\infty$ synthesis using the =hinfsyn= command.
+#+begin_src matlab :results output replace :exports both
+  [H2, ~, gamma, ~] = hinfsyn(P, 1, 1,'TOLGAM', 0.001, 'METHOD', 'ric', 'DISPLAY', 'on');
+#+end_src
+
+#+RESULTS:
+#+begin_example
+[H2, ~, gamma, ~] = hinfsyn(P, 1, 1,'TOLGAM', 0.001, 'METHOD', 'ric', 'DISPLAY', 'on');
+Resetting value of Gamma min based on D_11, D_12, D_21 terms
+
+Test bounds:      0.1000 <  gamma  <=   1050.0000
+
+  gamma    hamx_eig  xinf_eig  hamy_eig   yinf_eig   nrho_xy   p/f
+1.050e+03   2.8e+01   2.4e-07   4.1e+00    0.0e+00    0.0000    p
+  525.050   2.8e+01   2.4e-07   4.1e+00    0.0e+00    0.0000    p
+  262.575   2.8e+01   2.4e-07   4.1e+00    0.0e+00    0.0000    p
+  131.337   2.8e+01   2.4e-07   4.1e+00   -1.0e-13    0.0000    p
+   65.719   2.8e+01   2.4e-07   4.1e+00   -9.5e-14    0.0000    p
+   32.909   2.8e+01   2.4e-07   4.1e+00    0.0e+00    0.0000    p
+   16.505   2.8e+01   2.4e-07   4.1e+00   -1.0e-13    0.0000    p
+    8.302   2.8e+01   2.4e-07   4.1e+00   -7.2e-14    0.0000    p
+    4.201   2.8e+01   2.4e-07   4.1e+00   -2.5e-25    0.0000    p
+    2.151   2.7e+01   2.4e-07   4.1e+00   -3.8e-14    0.0000    p
+    1.125   2.6e+01   2.4e-07   4.1e+00   -5.4e-24    0.0000    p
+    0.613   2.3e+01 -3.7e+01#  4.1e+00    0.0e+00    0.0000    f
+    0.869   2.6e+01 -3.7e+02#  4.1e+00    0.0e+00    0.0000    f
+    0.997   2.6e+01 -1.1e+04#  4.1e+00    0.0e+00    0.0000    f
+    1.061   2.6e+01   2.4e-07   4.1e+00    0.0e+00    0.0000    p
+    1.029   2.6e+01   2.4e-07   4.1e+00    0.0e+00    0.0000    p
+    1.013   2.6e+01   2.4e-07   4.1e+00    0.0e+00    0.0000    p
+    1.005   2.6e+01   2.4e-07   4.1e+00    0.0e+00    0.0000    p
+    1.001   2.6e+01 -3.1e+04#  4.1e+00   -3.8e-14    0.0000    f
+    1.003   2.6e+01 -2.8e+05#  4.1e+00    0.0e+00    0.0000    f
+    1.004   2.6e+01   2.4e-07   4.1e+00   -5.8e-24    0.0000    p
+    1.004   2.6e+01   2.4e-07   4.1e+00    0.0e+00    0.0000    p
+
+ Gamma value achieved:     1.0036
+#+end_example
+
+We then define the high pass filter $H_1 = 1 - H_2$. The bode plot of both $H_1$ and $H_2$ is shown on figure [[fig:hinf_filters_results]].
+
+#+begin_src matlab
+  H1 = 1 - H2;
+#+end_src
+
+** Obtained Complementary Filters
+The obtained complementary filters are shown on figure [[fig:hinf_filters_results]].
+
+#+begin_src matlab :exports none
+  figure;
+  tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None');
+
+  % Magnitude
+  ax1 = nexttile([2, 1]);
+  hold on;
+  set(gca,'ColorOrderIndex',1)
+  plot(freqs, 1./abs(squeeze(freqresp(W1, freqs, 'Hz'))), '--', 'DisplayName', '$w_1$');
+  set(gca,'ColorOrderIndex',2)
+  plot(freqs, 1./abs(squeeze(freqresp(W2, freqs, 'Hz'))), '--', 'DisplayName', '$w_2$');
+
+  set(gca,'ColorOrderIndex',1)
+  plot(freqs, abs(squeeze(freqresp(H1, freqs, 'Hz'))), '-', 'DisplayName', '$H_1$');
+  set(gca,'ColorOrderIndex',2)
+  plot(freqs, abs(squeeze(freqresp(H2, freqs, 'Hz'))), '-', 'DisplayName', '$H_2$');
+  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
+  hold off;
+  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
+  ylabel('Magnitude');
+  set(gca, 'XTickLabel',[]);
+  ylim([1e-4, 20]);
+  yticks([1e-4, 1e-3, 1e-2, 1e-1, 1, 1e1]);
+  legend('location', 'southeast', 'FontSize', 8, 'NumColumns', 2);
+
+  % Phase
+  ax2 = nexttile;
+  hold on;
+  set(gca,'ColorOrderIndex',1)
+  plot(freqs, 180/pi*phase(squeeze(freqresp(H1, freqs, 'Hz'))), '-');
+  set(gca,'ColorOrderIndex',2)
+  plot(freqs, 180/pi*phase(squeeze(freqresp(H2, freqs, 'Hz'))), '-');
+  hold off;
+  xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
+  set(gca, 'XScale', 'log');
+  yticks([-180:90:180]);
+
+  linkaxes([ax1,ax2],'x');
+  xlim([freqs(1), freqs(end)]);
+#+end_src
+
+#+begin_src matlab :tangle no :exports results :results file replace
+  exportFig('figs/hinf_filters_results.pdf', 'width', 'wide', 'height', 'tall');
+#+end_src
+
+#+name: fig:hinf_filters_results
+#+caption: Obtained complementary filters using $\mathcal{H}_\infty$ synthesis
+#+RESULTS:
+[[file:figs/hinf_filters_results.png]]
+
+* Generating 3 complementary filters
+:PROPERTIES:
+:header-args:matlab+: :tangle matlab/three_comp_filters.m
+:header-args:matlab+: :comments org :mkdirp yes
+:END:
+<<sec:three_comp_filters>>
+
+** Introduction                                                      :ignore:
+#+begin_note
+  The Matlab file corresponding to this section is accessible [[file:matlab/three_comp_filters.m][here]].
+#+end_note
+
+** Matlab Init                                              :noexport:ignore:
+#+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name)
+  <<matlab-dir>>
+#+end_src
+
+#+begin_src matlab :exports none :results silent :noweb yes
+  <<matlab-init>>
+#+end_src
+
+#+begin_src matlab
+  freqs = logspace(-2, 4, 1000);
+#+end_src
+
+** Theory
+We want:
+\begin{align*}
+  & |H_1(j\omega)| < 1/|W_1(j\omega)|, \quad \forall\omega\\
+  & |H_2(j\omega)| < 1/|W_2(j\omega)|, \quad \forall\omega\\
+  & |H_3(j\omega)| < 1/|W_3(j\omega)|, \quad \forall\omega\\
+  & H_1(s) + H_2(s) + H_3(s) = 1
+\end{align*}
+
+For that, we use the $\mathcal{H}_\infty$ synthesis with the architecture shown on figure [[fig:comp_filter_three_hinf]].
+
+#+name: fig:comp_filter_three_hinf
+#+caption: Generalized architecture for generating 3 complementary filters
+[[file:figs-tikz/comp_filter_three_hinf.png]]
+
+The $\mathcal{H}_\infty$ objective is:
+\begin{align*}
+  & |(1 - H_2(j\omega) - H_3(j\omega)) W_1(j\omega)| < 1, \quad \forall\omega\\
+  & |H_2(j\omega) W_2(j\omega)| < 1, \quad \forall\omega\\
+  & |H_3(j\omega) W_3(j\omega)| < 1, \quad \forall\omega\\
+\end{align*}
+
+And thus if we choose $H_1 = 1 - H_2 - H_3$ we have solved the problem.
+
+** Weights
+First we define the weights.
+#+begin_src matlab
+  n = 2; w0 = 2*pi*1; G0 = 1/10; G1 = 1000; Gc = 1/2;
+  W1 = (((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;
+
+  W2 = 0.22*(1 + s/2/pi/1)^2/(sqrt(1e-4) + s/2/pi/1)^2*(1 + s/2/pi/10)^2/(1 + s/2/pi/1000)^2;
+
+  n = 3; w0 = 2*pi*10; G0 = 1000; G1 = 0.1; Gc = 1/2;
+  W3 = (((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;
+#+end_src
+
+#+begin_src matlab :exports none
+  figure;
+  hold on;
+  set(gca,'ColorOrderIndex',1)
+  plot(freqs, 1./abs(squeeze(freqresp(W1, freqs, 'Hz'))), '--', 'DisplayName', '$|W_1|^{-1}$');
+  set(gca,'ColorOrderIndex',2)
+  plot(freqs, 1./abs(squeeze(freqresp(W2, freqs, 'Hz'))), '--', 'DisplayName', '$|W_2|^{-1}$');
+  set(gca,'ColorOrderIndex',3)
+  plot(freqs, 1./abs(squeeze(freqresp(W3, freqs, 'Hz'))), '--', 'DisplayName', '$|W_3|^{-1}$');
+  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
+  xlabel('Frequency [Hz]'); ylabel('Magnitude');
+  hold off;
+  xlim([freqs(1), freqs(end)]);
+  legend('location', 'northeast', 'FontSize', 8);
+#+end_src
+
+#+begin_src matlab :tangle no :exports results :results file replace
+  exportFig('figs/three_weighting_functions.pdf', 'width', 'wide', 'height', 'normal');
+#+end_src
+
+#+name: fig:three_weighting_functions
+#+caption: Three weighting functions used for the $\mathcal{H}_\infty$ synthesis of the complementary filters
+#+RESULTS:
+[[file:figs/three_weighting_functions.png]]
+
+** H-Infinity Synthesis
+Then we create the generalized plant =P=.
+#+begin_src matlab
+  P = [W1 -W1 -W1;
+       0   W2  0 ;
+       0   0   W3;
+       1   0   0];
+#+end_src
+
+And we do the $\mathcal{H}_\infty$ synthesis.
+#+begin_src matlab :results output replace :exports both
+  [H, ~, gamma, ~] = hinfsyn(P, 1, 2,'TOLGAM', 0.001, 'METHOD', 'ric', 'DISPLAY', 'on');
+#+end_src
+
+#+RESULTS:
+#+begin_example
+[H, ~, gamma, ~] = hinfsyn(P, 1, 2,'TOLGAM', 0.001, 'METHOD', 'ric', 'DISPLAY', 'on');
+Resetting value of Gamma min based on D_11, D_12, D_21 terms
+
+Test bounds:      0.1000 <  gamma  <=   1050.0000
+
+  gamma    hamx_eig  xinf_eig  hamy_eig   yinf_eig   nrho_xy   p/f
+1.050e+03   3.2e+00   4.5e-13   6.3e-02   -1.2e-11    0.0000    p
+  525.050   3.2e+00   1.3e-13   6.3e-02    0.0e+00    0.0000    p
+  262.575   3.2e+00   2.1e-12   6.3e-02   -1.5e-13    0.0000    p
+  131.337   3.2e+00   1.1e-12   6.3e-02   -7.2e-29    0.0000    p
+   65.719   3.2e+00   2.0e-12   6.3e-02    0.0e+00    0.0000    p
+   32.909   3.2e+00   7.4e-13   6.3e-02   -5.9e-13    0.0000    p
+   16.505   3.2e+00   1.4e-12   6.3e-02    0.0e+00    0.0000    p
+    8.302   3.2e+00   1.6e-12   6.3e-02    0.0e+00    0.0000    p
+    4.201   3.2e+00   1.6e-12   6.3e-02    0.0e+00    0.0000    p
+    2.151   3.2e+00   1.6e-12   6.3e-02    0.0e+00    0.0000    p
+    1.125   3.2e+00   2.8e-12   6.3e-02    0.0e+00    0.0000    p
+    0.613   3.0e+00 -2.5e+03#  6.3e-02    0.0e+00    0.0000    f
+    0.869   3.1e+00 -2.9e+01#  6.3e-02    0.0e+00    0.0000    f
+    0.997   3.2e+00   1.9e-12   6.3e-02    0.0e+00    0.0000    p
+    0.933   3.1e+00 -6.9e+02#  6.3e-02    0.0e+00    0.0000    f
+    0.965   3.1e+00 -3.0e+03#  6.3e-02    0.0e+00    0.0000    f
+    0.981   3.1e+00 -8.6e+03#  6.3e-02    0.0e+00    0.0000    f
+    0.989   3.2e+00 -2.7e+04#  6.3e-02    0.0e+00    0.0000    f
+    0.993   3.2e+00 -5.7e+05#  6.3e-02    0.0e+00    0.0000    f
+    0.995   3.2e+00   2.2e-12   6.3e-02    0.0e+00    0.0000    p
+    0.994   3.2e+00   1.6e-12   6.3e-02    0.0e+00    0.0000    p
+    0.994   3.2e+00   1.0e-12   6.3e-02    0.0e+00    0.0000    p
+
+ Gamma value achieved:     0.9936
+#+end_example
+
+** Obtained Complementary Filters
+The obtained filters are:
+#+begin_src matlab
+  H2 = tf(H(1));
+  H3 = tf(H(2));
+  H1 = 1 - H2 - H3;
+#+end_src
+
+#+begin_src matlab :exports none
+  figure;
+  tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None');
+
+  % Magnitude
+  ax1 = nexttile([2, 1]);
+  hold on;
+  set(gca,'ColorOrderIndex',1)
+  plot(freqs, 1./abs(squeeze(freqresp(W1, freqs, 'Hz'))), '--', 'DisplayName', '$|W_1|^{-1}$');
+  set(gca,'ColorOrderIndex',2)
+  plot(freqs, 1./abs(squeeze(freqresp(W2, freqs, 'Hz'))), '--', 'DisplayName', '$|W_2|^{-1}$');
+  set(gca,'ColorOrderIndex',3)
+  plot(freqs, 1./abs(squeeze(freqresp(W3, freqs, 'Hz'))), '--', 'DisplayName', '$|W_3|^{-1}$');
+  set(gca,'ColorOrderIndex',1)
+  plot(freqs, abs(squeeze(freqresp(H1, freqs, 'Hz'))), '-', 'DisplayName', '$H_1$');
+  set(gca,'ColorOrderIndex',2)
+  plot(freqs, abs(squeeze(freqresp(H2, freqs, 'Hz'))), '-', 'DisplayName', '$H_2$');
+  set(gca,'ColorOrderIndex',3)
+  plot(freqs, abs(squeeze(freqresp(H3, freqs, 'Hz'))), '-', 'DisplayName', '$H_3$');
+  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
+  hold off;
+  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
+  ylabel('Magnitude');
+  set(gca, 'XTickLabel',[]);
+  ylim([1e-4, 20]);
+  legend('location', 'northeast', 'FontSize', 8, 'NumColumns', 2);
+
+  % Phase
+  ax2 = nexttile;
+  hold on;
+  set(gca,'ColorOrderIndex',1)
+  plot(freqs, 180/pi*phase(squeeze(freqresp(H1, freqs, 'Hz'))));
+  set(gca,'ColorOrderIndex',2)
+  plot(freqs, 180/pi*phase(squeeze(freqresp(H2, freqs, 'Hz'))));
+  set(gca,'ColorOrderIndex',3)
+  plot(freqs, 180/pi*phase(squeeze(freqresp(H3, freqs, 'Hz'))));
+  hold off;
+  xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
+  set(gca, 'XScale', 'log');
+  yticks([-360:90:360]); ylim([-270, 270]);
+
+  linkaxes([ax1,ax2],'x');
+  xlim([freqs(1), freqs(end)]);
+#+end_src
+
+#+begin_src matlab :tangle no :exports results :results file replace
+  exportFig('figs/three_complementary_filters_results.pdf', 'width', 'wide', 'height', 'tall');
+#+end_src
+
+#+name: fig:three_complementary_filters_results
+#+caption: The three complementary filters obtained after $\mathcal{H}_\infty$ synthesis
+#+RESULTS:
+[[file:figs/three_complementary_filters_results.png]]
+
+* Implement complementary filters for LIGO
+:PROPERTIES:
+:header-args:matlab+: :tangle matlab/comp_filters_ligo.m
+:header-args:matlab+: :comments org :mkdirp yes
+:END:
+<<sec:comp_filters_ligo>>
+
+** Introduction                                                      :ignore:
+#+begin_note
+  The Matlab file corresponding to this section is accessible [[file:matlab/comp_filters_ligo.m][here]].
+#+end_note
+
+Let's try to design complementary filters that are corresponding to the complementary filters design for the LIGO and described in cite:hua05_low_ligo.
+
+The FIR complementary filters designed in cite:hua05_low_ligo are of order 512.
+
+** Matlab Init                                              :noexport:ignore:
+#+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name)
+  <<matlab-dir>>
+#+end_src
+
+#+begin_src matlab :exports none :results silent :noweb yes
+  <<matlab-init>>
+#+end_src
+
+#+begin_src matlab
+  freqs = logspace(-3, 0, 1000);
+#+end_src
+
+** Specifications
+The specifications for the filters are:
+1. From $0$ to $0.008\text{ Hz}$,the magnitude of the filter’s transfer function should be less than or equal to $8 \times 10^{-3}$
+2. From $0.008\text{ Hz}$ to $0.04\text{ Hz}$, it attenuates the input signal proportional to frequency cubed
+3. Between $0.04\text{ Hz}$ and $0.1\text{ Hz}$, the magnitude of the transfer function should be less than 3
+4. Above $0.1\text{ Hz}$, the maximum of the magnitude of the complement filter should be as close to zero as possible. In our system, we would like to have the magnitude of the complementary filter to be less than $0.1$. As the filters obtained in cite:hua05_low_ligo have a magnitude of $0.045$, we will set that as our requirement
+
+The specifications are translated in upper bounds of the complementary filters are shown on figure [[fig:ligo_specifications]].
+
+#+begin_src matlab :exports none
+  figure;
+  hold on;
+  set(gca,'ColorOrderIndex',1)
+  plot([0.0001, 0.008], [8e-3, 8e-3], ':', 'DisplayName', 'Spec. on $H_H$');
+  set(gca,'ColorOrderIndex',1)
+  plot([0.008 0.04], [8e-3, 1], ':', 'HandleVisibility', 'off');
+  set(gca,'ColorOrderIndex',1)
+  plot([0.04 0.1], [3, 3], ':', 'HandleVisibility', 'off');
+  set(gca,'ColorOrderIndex',2)
+  plot([0.1, 10], [0.045, 0.045], ':', 'DisplayName', 'Spec. on $H_L$');
+  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
+  xlabel('Frequency [Hz]'); ylabel('Magnitude');
+  hold off;
+  xlim([freqs(1), freqs(end)]);
+  ylim([1e-4, 10]);
+  legend('location', 'southeast', 'FontSize', 8);
+#+end_src
+
+#+begin_src matlab :tangle no :exports results :results file replace
+  exportFig('figs/ligo_specifications.pdf', 'width', 'wide', 'height', 'normal');
+#+end_src
+
+#+name: fig:ligo_specifications
+#+caption: Specification for the LIGO complementary filters
+#+RESULTS:
+[[file:figs/ligo_specifications.png]]
+
+** TODO FIR Filter
+We here try to implement the FIR complementary filter synthesis as explained in cite:hua05_low_ligo.
+For that, we use the [[http://cvxr.com/cvx/][CVX matlab Toolbox]].
+
+We setup the CVX toolbox and use the =SeDuMi= solver.
+#+begin_src matlab
+  cvx_startup;
+  cvx_solver sedumi;
+#+end_src
+
+We define the frequency vectors on which we will constrain the norm of the FIR filter.
+#+begin_src matlab
+  w1 = 0:4.06e-4:0.008;
+  w2 = 0.008:4.06e-4:0.04;
+  w3 = 0.04:8.12e-4:0.1;
+  w4 = 0.1:8.12e-4:0.83;
+#+end_src
+
+We then define the order of the FIR filter.
+#+begin_src matlab
+  n = 512;
+#+end_src
+
+#+begin_src matlab
+  A1 = [ones(length(w1),1),  cos(kron(w1'.*(2*pi),[1:n-1]))];
+  A2 = [ones(length(w2),1),  cos(kron(w2'.*(2*pi),[1:n-1]))];
+  A3 = [ones(length(w3),1),  cos(kron(w3'.*(2*pi),[1:n-1]))];
+  A4 = [ones(length(w4),1),  cos(kron(w4'.*(2*pi),[1:n-1]))];
+
+  B1 = [zeros(length(w1),1), sin(kron(w1'.*(2*pi),[1:n-1]))];
+  B2 = [zeros(length(w2),1), sin(kron(w2'.*(2*pi),[1:n-1]))];
+  B3 = [zeros(length(w3),1), sin(kron(w3'.*(2*pi),[1:n-1]))];
+  B4 = [zeros(length(w4),1), sin(kron(w4'.*(2*pi),[1:n-1]))];
+#+end_src
+
+We run the convex optimization.
+#+begin_src matlab :results output replace :wrap example
+  cvx_begin
+
+  variable y(n+1,1)
+
+  % t
+  maximize(-y(1))
+
+  for i = 1:length(w1)
+      norm([0 A1(i,:); 0 B1(i,:)]*y) <= 8e-3;
+  end
+
+  for  i = 1:length(w2)
+      norm([0 A2(i,:); 0 B2(i,:)]*y) <= 8e-3*(2*pi*w2(i)/(0.008*2*pi))^3;
+  end
+
+  for i = 1:length(w3)
+      norm([0 A3(i,:); 0 B3(i,:)]*y) <= 3;
+  end
+
+  for i = 1:length(w4)
+      norm([[1 0]'- [0 A4(i,:); 0 B4(i,:)]*y]) <= y(1);
+  end
+
+  cvx_end
+
+  h = y(2:end);
+#+end_src
+
+#+RESULTS:
+#+begin_example
+cvx_begin
+variable y(n+1,1)
+% t
+maximize(-y(1))
+for i = 1:length(w1)
+    norm([0 A1(i,:); 0 B1(i,:)]*y) <= 8e-3;
+end
+for  i = 1:length(w2)
+    norm([0 A2(i,:); 0 B2(i,:)]*y) <= 8e-3*(2*pi*w2(i)/(0.008*2*pi))^3;
+end
+for i = 1:length(w3)
+    norm([0 A3(i,:); 0 B3(i,:)]*y) <= 3;
+end
+for i = 1:length(w4)
+    norm([[1 0]'- [0 A4(i,:); 0 B4(i,:)]*y]) <= y(1);
+end
+cvx_end
+
+Calling SeDuMi 1.34: 4291 variables, 1586 equality constraints
+   For improved efficiency, SeDuMi is solving the dual problem.
+------------------------------------------------------------
+SeDuMi 1.34 (beta) by AdvOL, 2005-2008 and Jos F. Sturm, 1998-2003.
+Alg = 2: xz-corrector, Adaptive Step-Differentiation, theta = 0.250, beta = 0.500
+eqs m = 1586, order n = 3220, dim = 4292, blocks = 1073
+nnz(A) = 1100727 + 0, nnz(ADA) = 1364794, nnz(L) = 683190
+ it :     b*y       gap    delta  rate   t/tP*  t/tD*   feas cg cg  prec
+  0 :            4.11E+02 0.000
+  1 :  -2.58E+00 1.25E+02 0.000 0.3049 0.9000 0.9000   4.87  1  1  3.0E+02
+  2 :  -2.36E+00 3.90E+01 0.000 0.3118 0.9000 0.9000   1.83  1  1  6.6E+01
+  3 :  -1.69E+00 1.31E+01 0.000 0.3354 0.9000 0.9000   1.76  1  1  1.5E+01
+  4 :  -8.60E-01 7.10E+00 0.000 0.5424 0.9000 0.9000   2.48  1  1  4.8E+00
+  5 :  -4.91E-01 5.44E+00 0.000 0.7661 0.9000 0.9000   3.12  1  1  2.5E+00
+  6 :  -2.96E-01 3.88E+00 0.000 0.7140 0.9000 0.9000   2.62  1  1  1.4E+00
+  7 :  -1.98E-01 2.82E+00 0.000 0.7271 0.9000 0.9000   2.14  1  1  8.5E-01
+  8 :  -1.39E-01 2.00E+00 0.000 0.7092 0.9000 0.9000   1.78  1  1  5.4E-01
+  9 :  -9.99E-02 1.30E+00 0.000 0.6494 0.9000 0.9000   1.51  1  1  3.3E-01
+ 10 :  -7.57E-02 8.03E-01 0.000 0.6175 0.9000 0.9000   1.31  1  1  2.0E-01
+ 11 :  -5.99E-02 4.22E-01 0.000 0.5257 0.9000 0.9000   1.17  1  1  1.0E-01
+ 12 :  -5.28E-02 2.45E-01 0.000 0.5808 0.9000 0.9000   1.08  1  1  5.9E-02
+ 13 :  -4.82E-02 1.28E-01 0.000 0.5218 0.9000 0.9000   1.05  1  1  3.1E-02
+ 14 :  -4.56E-02 5.65E-02 0.000 0.4417 0.9045 0.9000   1.02  1  1  1.4E-02
+ 15 :  -4.43E-02 2.41E-02 0.000 0.4265 0.9004 0.9000   1.01  1  1  6.0E-03
+ 16 :  -4.37E-02 8.90E-03 0.000 0.3690 0.9070 0.9000   1.00  1  1  2.3E-03
+ 17 :  -4.35E-02 3.24E-03 0.000 0.3641 0.9164 0.9000   1.00  1  1  9.5E-04
+ 18 :  -4.34E-02 1.55E-03 0.000 0.4788 0.9086 0.9000   1.00  1  1  4.7E-04
+ 19 :  -4.34E-02 8.77E-04 0.000 0.5653 0.9169 0.9000   1.00  1  1  2.8E-04
+ 20 :  -4.34E-02 5.05E-04 0.000 0.5754 0.9034 0.9000   1.00  1  1  1.6E-04
+ 21 :  -4.34E-02 2.94E-04 0.000 0.5829 0.9136 0.9000   1.00  1  1  9.9E-05
+ 22 :  -4.34E-02 1.63E-04 0.015 0.5548 0.9000 0.0000   1.00  1  1  6.6E-05
+ 23 :  -4.33E-02 9.42E-05 0.000 0.5774 0.9053 0.9000   1.00  1  1  3.9E-05
+ 24 :  -4.33E-02 6.27E-05 0.000 0.6658 0.9148 0.9000   1.00  1  1  2.6E-05
+ 25 :  -4.33E-02 3.75E-05 0.000 0.5972 0.9187 0.9000   1.00  1  1  1.6E-05
+ 26 :  -4.33E-02 1.89E-05 0.000 0.5041 0.9117 0.9000   1.00  1  1  8.6E-06
+ 27 :  -4.33E-02 9.72E-06 0.000 0.5149 0.9050 0.9000   1.00  1  1  4.5E-06
+ 28 :  -4.33E-02 2.94E-06 0.000 0.3021 0.9194 0.9000   1.00  1  1  1.5E-06
+ 29 :  -4.33E-02 9.73E-07 0.000 0.3312 0.9189 0.9000   1.00  2  2  5.3E-07
+ 30 :  -4.33E-02 2.82E-07 0.000 0.2895 0.9063 0.9000   1.00  2  2  1.6E-07
+ 31 :  -4.33E-02 8.05E-08 0.000 0.2859 0.9049 0.9000   1.00  2  2  4.7E-08
+ 32 :  -4.33E-02 1.43E-08 0.000 0.1772 0.9059 0.9000   1.00  2  2  8.8E-09
+
+iter seconds digits       c*x               b*y
+ 32     49.4   6.8 -4.3334083581e-02 -4.3334090214e-02
+|Ax-b| =   3.7e-09, [Ay-c]_+ =   1.1E-10, |x|=  1.0e+00, |y|=  2.6e+00
+
+Detailed timing (sec)
+   Pre          IPM          Post
+3.902E+00    4.576E+01    1.035E-02
+Max-norms: ||b||=1, ||c|| = 3,
+Cholesky |add|=0, |skip| = 0, ||L.L|| = 4.26267.
+------------------------------------------------------------
+Status: Solved
+Optimal value (cvx_optval): -0.0433341
+h = y(2:end);
+#+end_example
+
+Finally, we compute the filter response over the frequency vector defined and the result is shown on figure [[fig:fir_filter_ligo]] which is very close to the filters obtain in cite:hua05_low_ligo.
+
+#+begin_src matlab
+  w = [w1 w2 w3 w4];
+  H = [exp(-j*kron(w'.*2*pi,[0:n-1]))]*h;
+#+end_src
+
+#+begin_src matlab :exports none
+  figure;
+  tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None');
+
+  % Magnitude
+  ax1 = nexttile([2, 1]);
+  hold on;
+  plot(w, abs(H), 'k-');
+  plot(w, abs(1-H), 'k--');
+  hold off;
+  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
+  ylabel('Magnitude');
+  set(gca, 'XTickLabel',[]);
+  ylim([5e-3, 5]);
+
+  % Phase
+  ax2 = nexttile;
+  hold on;
+  plot(w, 180/pi*angle(H), 'k-');
+  plot(w, 180/pi*angle(1-H), 'k--');
+  hold off;
+  xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
+  set(gca, 'XScale', 'log');
+  yticks([-180:90:180]); ylim([-180, 180]);
+
+  linkaxes([ax1,ax2],'x');
+  xlim([1e-3, 1]);
+#+end_src
+
+#+begin_src matlab :tangle no :exports results :results file replace
+  exportFig('figs/fir_filter_ligo.pdf', 'width', 'wide', 'height', 'normal');
+#+end_src
+
+#+name: fig:fir_filter_ligo
+#+caption: FIR Complementary filters obtain after convex optimization
+#+RESULTS:
+[[file:figs/fir_filter_ligo.png]]
+
+** Weights
+We design weights that will be used for the $\mathcal{H}_\infty$ synthesis of the complementary filters.
+These weights will determine the order of the obtained filters.
+Here are the requirements on the filters:
+- reasonable order
+- to be as close as possible to the specified upper bounds
+- stable minimum phase
+
+The bode plot of the weights is shown on figure [[fig:ligo_weights]].
+
+#+begin_src matlab :exports none
+  w1 = 2*pi*0.008; x1 = 0.35;
+  w2 = 2*pi*0.04;  x2 = 0.5;
+  w3 = 2*pi*0.05;  x3 = 0.5;
+
+  % Slope of +3 from w1
+  wH = 0.008*(s^2/w1^2 + 2*x1/w1*s + 1)*(s/w1 + 1);
+  % Little bump from w2 to w3
+  wH = wH*(s^2/w2^2 + 2*x2/w2*s + 1)/(s^2/w3^2 + 2*x3/w3*s + 1);
+  % No Slope at high frequencies
+  wH = wH/(s^2/w3^2 + 2*x3/w3*s + 1)/(s/w3 + 1);
+  % Little bump between w2 and w3
+  w0 = 2*pi*0.045; xi = 0.1; A = 2; n = 1;
+  wH = wH*((s^2 + 2*w0*xi*A^(1/n)*s + w0^2)/(s^2 + 2*w0*xi*s + w0^2))^n;
+
+  wH = 1/wH;
+  wH = minreal(ss(wH));
+#+end_src
+
+#+begin_src matlab :exports none
+  n = 20; Rp = 1; Wp = 2*pi*0.102;
+  [b,a] = cheby1(n, Rp, Wp, 'high', 's');
+  wL = 0.04*tf(a, b);
+
+  wL = 1/wL;
+  wL = minreal(ss(wL));
+#+end_src
+
+#+begin_src matlab :exports none
+  figure;
+  hold on;
+  set(gca,'ColorOrderIndex',1);
+  plot(freqs, abs(squeeze(freqresp(inv(wH), freqs, 'Hz'))), '-', 'DisplayName', '$|w_H|^{-1}$');
+  set(gca,'ColorOrderIndex',2);
+  plot(freqs, abs(squeeze(freqresp(inv(wL), freqs, 'Hz'))), '-', 'DisplayName', '$|w_L|^{-1}$');
+
+  plot([0.0001, 0.008], [8e-3, 8e-3], 'k--', 'DisplayName', 'Spec.');
+  plot([0.008 0.04], [8e-3, 1], 'k--', 'HandleVisibility', 'off');
+  plot([0.04 0.1], [3, 3], 'k--', 'HandleVisibility', 'off');
+  plot([0.1, 10], [0.045, 0.045], 'k--', 'HandleVisibility', 'off');
+
+  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
+  xlabel('Frequency [Hz]'); ylabel('Magnitude');
+  hold off;
+  xlim([freqs(1), freqs(end)]);
+  ylim([1e-3, 10]);
+  legend('location', 'southeast', 'FontSize', 8);
+#+end_src
+
+#+begin_src matlab :tangle no :exports results :results file replace
+  exportFig('figs/ligo_weights.pdf', 'width', 'wide', 'height', 'normal');
+#+end_src
+
+#+name: fig:ligo_weights
+#+caption: Weights for the $\mathcal{H}_\infty$ synthesis
+#+RESULTS:
+[[file:figs/ligo_weights.png]]
+
+** H-Infinity Synthesis
+We define the generalized plant as shown on figure [[fig:h_infinity_robst_fusion]].
+#+begin_src matlab
+  P = [0   wL;
+       wH -wH;
+       1   0];
+#+end_src
+
+And we do the $\mathcal{H}_\infty$ synthesis using the =hinfsyn= command.
+#+begin_src matlab :results output replace :exports both :wrap example
+  [Hl, ~, gamma, ~] = hinfsyn(P, 1, 1,'TOLGAM', 0.001, 'METHOD', 'ric', 'DISPLAY', 'on');
+#+end_src
+
+#+RESULTS:
+#+begin_example
+[Hl, ~, 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:      0.3276 <  gamma  <=      1.8063
+
+  gamma    hamx_eig  xinf_eig  hamy_eig   yinf_eig   nrho_xy   p/f
+    1.806   1.4e-02 -1.7e-16   3.6e-03   -4.8e-12    0.0000    p
+    1.067   1.3e-02 -4.2e-14   3.6e-03   -1.9e-12    0.0000    p
+    0.697   1.3e-02 -3.0e-01#  3.6e-03   -3.5e-11    0.0000    f
+    0.882   1.3e-02 -9.5e-01#  3.6e-03   -1.2e-34    0.0000    f
+    0.975   1.3e-02 -2.7e+00#  3.6e-03   -1.6e-12    0.0000    f
+    1.021   1.3e-02 -8.7e+00#  3.6e-03   -4.5e-16    0.0000    f
+    1.044   1.3e-02 -6.5e-14   3.6e-03   -3.0e-15    0.0000    p
+    1.032   1.3e-02 -1.8e+01#  3.6e-03    0.0e+00    0.0000    f
+    1.038   1.3e-02 -3.8e+01#  3.6e-03    0.0e+00    0.0000    f
+    1.041   1.3e-02 -8.3e+01#  3.6e-03   -2.9e-33    0.0000    f
+    1.042   1.3e-02 -1.9e+02#  3.6e-03   -3.4e-11    0.0000    f
+    1.043   1.3e-02 -5.3e+02#  3.6e-03   -7.5e-13    0.0000    f
+
+ Gamma value achieved:     1.0439
+#+end_example
+
+The high pass filter is defined as $H_H = 1 - H_L$.
+#+begin_src matlab
+  Hh = 1 - Hl;
+#+end_src
+
+#+begin_src matlab :exports none
+  Hh = minreal(Hh);
+  Hl = minreal(Hl);
+#+end_src
+
+The size of the filters is shown below.
+
+#+begin_src matlab :exports results :results output replace :wrap example
+  size(Hh), size(Hl)
+#+end_src
+
+#+RESULTS:
+#+begin_example
+size(Hh), size(Hl)
+State-space model with 1 outputs, 1 inputs, and 27 states.
+State-space model with 1 outputs, 1 inputs, and 27 states.
+#+end_example
+
+The bode plot of the obtained filters as shown on figure [[fig:hinf_synthesis_ligo_results]].
+
+#+begin_src matlab :exports none
+  figure;
+  hold on;
+  set(gca,'ColorOrderIndex',1);
+  plot([0.0001, 0.008], [8e-3, 8e-3], ':', 'DisplayName', 'Spec. on $H_H$');
+  set(gca,'ColorOrderIndex',1);
+  plot([0.008 0.04], [8e-3, 1], ':', 'HandleVisibility', 'off');
+  set(gca,'ColorOrderIndex',1);
+  plot([0.04 0.1], [3, 3], ':', 'HandleVisibility', 'off');
+
+  set(gca,'ColorOrderIndex',2);
+  plot([0.1, 10], [0.045, 0.045], ':', 'DisplayName', 'Spec. on $H_L$');
+
+  set(gca,'ColorOrderIndex',1);
+  plot(freqs, abs(squeeze(freqresp(Hh, freqs, 'Hz'))), '-', 'DisplayName', '$H_H$');
+  set(gca,'ColorOrderIndex',2);
+  plot(freqs, abs(squeeze(freqresp(Hl, freqs, 'Hz'))), '-', 'DisplayName', '$H_L$');
+
+  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
+  xlabel('Frequency [Hz]'); ylabel('Magnitude');
+  hold off;
+  xlim([freqs(1), freqs(end)]);
+  ylim([1e-3, 10]);
+  legend('location', 'southeast', 'FontSize', 8);
+#+end_src
+
+#+begin_src matlab :tangle no :exports results :results file replace
+  exportFig('figs/hinf_synthesis_ligo_results.pdf', 'width', 'wide', 'height', 'normal');
+#+end_src
+
+#+name: fig:hinf_synthesis_ligo_results
+#+caption: Obtained complementary filters using the $\mathcal{H}_\infty$ synthesis
+#+RESULTS:
+[[file:figs/hinf_synthesis_ligo_results.png]]
+
+** TODO Compare FIR and H-Infinity Filters
+Let's now compare the FIR filters designed in cite:hua05_low_ligo and the one obtained with the $\mathcal{H}_\infty$ synthesis on figure [[fig:comp_fir_ligo_hinf]].
+
+#+begin_src matlab :exports none
+  figure;
+  tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None');
+
+  % Magnitude
+  ax1 = nexttile([2, 1]);
+  hold on;
+  set(gca,'ColorOrderIndex',1);
+  plot(freqs, abs(squeeze(freqresp(Hh, freqs, 'Hz'))), '-', ...
+       'DisplayName', '$H_H(s)$ - $\mathcal{H}_\infty$');
+  set(gca,'ColorOrderIndex',2);
+  plot(freqs, abs(squeeze(freqresp(Hl, freqs, 'Hz'))), '-', ...
+       'DisplayName', '$H_L(s)$ - $\mathcal{H}_\infty$');
+
+  set(gca,'ColorOrderIndex',1);
+  plot(w, abs(H), '--', ...
+       'DisplayName', '$H_H(s)$ - FIR');
+  set(gca,'ColorOrderIndex',2);
+  plot(w, abs(1-H), '--', ...
+       'DisplayName', '$H_L(s)$ - FIR');
+  hold off;
+  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
+  ylabel('Magnitude');
+  set(gca, 'XTickLabel',[]);
+  ylim([5e-3, 10]);
+  legend('location', 'southeast', 'FontSize', 8, 'NumColumns', 2);
+
+
+  % Phase
+  ax2 = nexttile;
+  hold on;
+  set(gca,'ColorOrderIndex',1);
+  plot(freqs, 180/pi*angle(squeeze(freqresp(Hh, freqs, 'Hz'))), '-');
+  set(gca,'ColorOrderIndex',2);
+  plot(freqs, 180/pi*angle(squeeze(freqresp(Hl, freqs, 'Hz'))), '-');
+
+  set(gca,'ColorOrderIndex',1);
+  plot(w, 180/pi*angle(H), '--');
+  set(gca,'ColorOrderIndex',2);
+  plot(w, 180/pi*angle(1-H), '--');
+  set(gca, 'XScale', 'log');
+  xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
+  hold off;
+  yticks([-180:90:180]); ylim([-180, 180]);
+
+  linkaxes([ax1,ax2],'x');
+  xlim([freqs(1), freqs(end)]);
+#+end_src
+
+#+begin_src matlab :tangle no :exports results :results file replace
+  exportFig('figs/comp_fir_ligo_hinf.pdf', 'width', 'wide', 'height', 'tall');
+#+end_src
+
+#+name: fig:comp_fir_ligo_hinf
+#+caption: Comparison between the FIR filters developped for LIGO and the $\mathcal{H}_\infty$ complementary filters
+#+RESULTS:
+[[file:figs/comp_fir_ligo_hinf.png]]
+
+* Bibliography                                                        :ignore:
+bibliographystyle:unsrt
+bibliography:ref.bib
diff --git a/matlab/mat/comp_ligo_fir.csv b/matlab/mat/comp_ligo_fir.csv
new file mode 100644
index 0000000..10d13d2
--- /dev/null
+++ b/matlab/mat/comp_ligo_fir.csv
@@ -0,0 +1,1074 @@
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diff --git a/matlab/mat/comp_ligo_hinf.csv b/matlab/mat/comp_ligo_hinf.csv
new file mode 100644
index 0000000..2c7bc71
--- /dev/null
+++ b/matlab/mat/comp_ligo_hinf.csv
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diff --git a/matlab/mat/hinf_filters_results.csv b/matlab/mat/hinf_filters_results.csv
new file mode 100644
index 0000000..4cf51d8
--- /dev/null
+++ b/matlab/mat/hinf_filters_results.csv
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diff --git a/matlab/mat/hinf_three_results.csv b/matlab/mat/hinf_three_results.csv
new file mode 100644
index 0000000..72d805a
--- /dev/null
+++ b/matlab/mat/hinf_three_results.csv
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diff --git a/matlab/mat/hinf_three_weights.csv b/matlab/mat/hinf_three_weights.csv
new file mode 100644
index 0000000..4a99c57
--- /dev/null
+++ b/matlab/mat/hinf_three_weights.csv
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diff --git a/matlab/mat/hinf_weights.csv b/matlab/mat/hinf_weights.csv
new file mode 100644
index 0000000..417cbe1
--- /dev/null
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diff --git a/matlab/mat/ligo_weights.csv b/matlab/mat/ligo_weights.csv
new file mode 100644
index 0000000..8762de9
--- /dev/null
+++ b/matlab/mat/ligo_weights.csv
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diff --git a/matlab/matlab.zip b/matlab/matlab.zip
new file mode 100644
index 0000000..8d190b1
Binary files /dev/null and b/matlab/matlab.zip differ
diff --git a/matlab/matlab/comp_filters_ligo.m b/matlab/matlab/comp_filters_ligo.m
new file mode 100644
index 0000000..fc1c8f9
--- /dev/null
+++ b/matlab/matlab/comp_filters_ligo.m
@@ -0,0 +1,395 @@
+%% Clear Workspace and Close figures
+clear; close all; clc;
+
+%% Intialize Laplace variable
+s = zpk('s');
+
+freqs = logspace(-3, 0, 1000);
+
+% Specifications
+% The specifications for the filters are:
+% 1. From $0$ to $0.008\text{ Hz}$,the magnitude of the filter’s transfer function should be less than or equal to $8 \times 10^{-3}$
+% 2. From $0.008\text{ Hz}$ to $0.04\text{ Hz}$, it attenuates the input signal proportional to frequency cubed
+% 3. Between $0.04\text{ Hz}$ and $0.1\text{ Hz}$, the magnitude of the transfer function should be less than 3
+% 4. Above $0.1\text{ Hz}$, the maximum of the magnitude of the complement filter should be as close to zero as possible. In our system, we would like to have the magnitude of the complementary filter to be less than $0.1$. As the filters obtained in cite:hua05_low_ligo have a magnitude of $0.045$, we will set that as our requirement
+
+% The specifications are translated in upper bounds of the complementary filters are shown on figure [[fig:ligo_specifications]].
+
+
+figure;
+hold on;
+set(gca,'ColorOrderIndex',1)
+plot([0.0001, 0.008], [8e-3, 8e-3], ':', 'DisplayName', 'Spec. on $H_H$');
+set(gca,'ColorOrderIndex',1)
+plot([0.008 0.04], [8e-3, 1], ':', 'HandleVisibility', 'off');
+set(gca,'ColorOrderIndex',1)
+plot([0.04 0.1], [3, 3], ':', 'HandleVisibility', 'off');
+set(gca,'ColorOrderIndex',2)
+plot([0.1, 10], [0.045, 0.045], ':', 'DisplayName', 'Spec. on $H_L$');
+set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
+xlabel('Frequency [Hz]'); ylabel('Magnitude');
+hold off;
+xlim([freqs(1), freqs(end)]);
+ylim([1e-4, 10]);
+legend('location', 'northeast');
+
+% FIR Filter
+% We here try to implement the FIR complementary filter synthesis as explained in cite:hua05_low_ligo.
+% For that, we use the [[http://cvxr.com/cvx/][CVX matlab Toolbox]].
+
+% We setup the CVX toolbox and use the =SeDuMi= solver.
+
+cvx_startup;
+cvx_solver sedumi;
+
+
+
+% We define the frequency vectors on which we will constrain the norm of the FIR filter.
+
+w1 = 0:4.06e-4:0.008;
+w2 = 0.008:4.06e-4:0.04;
+w3 = 0.04:8.12e-4:0.1;
+w4 = 0.1:8.12e-4:0.83;
+
+
+
+% We then define the order of the FIR filter.
+
+n = 512;
+
+A1 = [ones(length(w1),1),  cos(kron(w1'.*(2*pi),[1:n-1]))];
+A2 = [ones(length(w2),1),  cos(kron(w2'.*(2*pi),[1:n-1]))];
+A3 = [ones(length(w3),1),  cos(kron(w3'.*(2*pi),[1:n-1]))];
+A4 = [ones(length(w4),1),  cos(kron(w4'.*(2*pi),[1:n-1]))];
+
+B1 = [zeros(length(w1),1), sin(kron(w1'.*(2*pi),[1:n-1]))];
+B2 = [zeros(length(w2),1), sin(kron(w2'.*(2*pi),[1:n-1]))];
+B3 = [zeros(length(w3),1), sin(kron(w3'.*(2*pi),[1:n-1]))];
+B4 = [zeros(length(w4),1), sin(kron(w4'.*(2*pi),[1:n-1]))];
+
+
+
+% We run the convex optimization.
+
+cvx_begin
+
+variable y(n+1,1)
+
+% t
+maximize(-y(1))
+
+for i = 1:length(w1)
+    norm([0 A1(i,:); 0 B1(i,:)]*y) <= 8e-3;
+end
+
+for  i = 1:length(w2)
+    norm([0 A2(i,:); 0 B2(i,:)]*y) <= 8e-3*(2*pi*w2(i)/(0.008*2*pi))^3;
+end
+
+for i = 1:length(w3)
+    norm([0 A3(i,:); 0 B3(i,:)]*y) <= 3;
+end
+
+for i = 1:length(w4)
+    norm([[1 0]'- [0 A4(i,:); 0 B4(i,:)]*y]) <= y(1);
+end
+
+cvx_end
+
+h = y(2:end);
+
+
+
+% #+RESULTS:
+% #+begin_example
+% cvx_begin
+% variable y(n+1,1)
+% % t
+% maximize(-y(1))
+% for i = 1:length(w1)
+%     norm([0 A1(i,:); 0 B1(i,:)]*y) <= 8e-3;
+% end
+% for  i = 1:length(w2)
+%     norm([0 A2(i,:); 0 B2(i,:)]*y) <= 8e-3*(2*pi*w2(i)/(0.008*2*pi))^3;
+% end
+% for i = 1:length(w3)
+%     norm([0 A3(i,:); 0 B3(i,:)]*y) <= 3;
+% end
+% for i = 1:length(w4)
+%     norm([[1 0]'- [0 A4(i,:); 0 B4(i,:)]*y]) <= y(1);
+% end
+% cvx_end
+
+% Calling SeDuMi 1.34: 4291 variables, 1586 equality constraints
+%    For improved efficiency, SeDuMi is solving the dual problem.
+% ------------------------------------------------------------
+% SeDuMi 1.34 (beta) by AdvOL, 2005-2008 and Jos F. Sturm, 1998-2003.
+% Alg = 2: xz-corrector, Adaptive Step-Differentiation, theta = 0.250, beta = 0.500
+% eqs m = 1586, order n = 3220, dim = 4292, blocks = 1073
+% nnz(A) = 1100727 + 0, nnz(ADA) = 1364794, nnz(L) = 683190
+%  it :     b*y       gap    delta  rate   t/tP*  t/tD*   feas cg cg  prec
+%   0 :            4.11E+02 0.000
+%   1 :  -2.58E+00 1.25E+02 0.000 0.3049 0.9000 0.9000   4.87  1  1  3.0E+02
+%   2 :  -2.36E+00 3.90E+01 0.000 0.3118 0.9000 0.9000   1.83  1  1  6.6E+01
+%   3 :  -1.69E+00 1.31E+01 0.000 0.3354 0.9000 0.9000   1.76  1  1  1.5E+01
+%   4 :  -8.60E-01 7.10E+00 0.000 0.5424 0.9000 0.9000   2.48  1  1  4.8E+00
+%   5 :  -4.91E-01 5.44E+00 0.000 0.7661 0.9000 0.9000   3.12  1  1  2.5E+00
+%   6 :  -2.96E-01 3.88E+00 0.000 0.7140 0.9000 0.9000   2.62  1  1  1.4E+00
+%   7 :  -1.98E-01 2.82E+00 0.000 0.7271 0.9000 0.9000   2.14  1  1  8.5E-01
+%   8 :  -1.39E-01 2.00E+00 0.000 0.7092 0.9000 0.9000   1.78  1  1  5.4E-01
+%   9 :  -9.99E-02 1.30E+00 0.000 0.6494 0.9000 0.9000   1.51  1  1  3.3E-01
+%  10 :  -7.57E-02 8.03E-01 0.000 0.6175 0.9000 0.9000   1.31  1  1  2.0E-01
+%  11 :  -5.99E-02 4.22E-01 0.000 0.5257 0.9000 0.9000   1.17  1  1  1.0E-01
+%  12 :  -5.28E-02 2.45E-01 0.000 0.5808 0.9000 0.9000   1.08  1  1  5.9E-02
+%  13 :  -4.82E-02 1.28E-01 0.000 0.5218 0.9000 0.9000   1.05  1  1  3.1E-02
+%  14 :  -4.56E-02 5.65E-02 0.000 0.4417 0.9045 0.9000   1.02  1  1  1.4E-02
+%  15 :  -4.43E-02 2.41E-02 0.000 0.4265 0.9004 0.9000   1.01  1  1  6.0E-03
+%  16 :  -4.37E-02 8.90E-03 0.000 0.3690 0.9070 0.9000   1.00  1  1  2.3E-03
+%  17 :  -4.35E-02 3.24E-03 0.000 0.3641 0.9164 0.9000   1.00  1  1  9.5E-04
+%  18 :  -4.34E-02 1.55E-03 0.000 0.4788 0.9086 0.9000   1.00  1  1  4.7E-04
+%  19 :  -4.34E-02 8.77E-04 0.000 0.5653 0.9169 0.9000   1.00  1  1  2.8E-04
+%  20 :  -4.34E-02 5.05E-04 0.000 0.5754 0.9034 0.9000   1.00  1  1  1.6E-04
+%  21 :  -4.34E-02 2.94E-04 0.000 0.5829 0.9136 0.9000   1.00  1  1  9.9E-05
+%  22 :  -4.34E-02 1.63E-04 0.015 0.5548 0.9000 0.0000   1.00  1  1  6.6E-05
+%  23 :  -4.33E-02 9.42E-05 0.000 0.5774 0.9053 0.9000   1.00  1  1  3.9E-05
+%  24 :  -4.33E-02 6.27E-05 0.000 0.6658 0.9148 0.9000   1.00  1  1  2.6E-05
+%  25 :  -4.33E-02 3.75E-05 0.000 0.5972 0.9187 0.9000   1.00  1  1  1.6E-05
+%  26 :  -4.33E-02 1.89E-05 0.000 0.5041 0.9117 0.9000   1.00  1  1  8.6E-06
+%  27 :  -4.33E-02 9.72E-06 0.000 0.5149 0.9050 0.9000   1.00  1  1  4.5E-06
+%  28 :  -4.33E-02 2.94E-06 0.000 0.3021 0.9194 0.9000   1.00  1  1  1.5E-06
+%  29 :  -4.33E-02 9.73E-07 0.000 0.3312 0.9189 0.9000   1.00  2  2  5.3E-07
+%  30 :  -4.33E-02 2.82E-07 0.000 0.2895 0.9063 0.9000   1.00  2  2  1.6E-07
+%  31 :  -4.33E-02 8.05E-08 0.000 0.2859 0.9049 0.9000   1.00  2  2  4.7E-08
+%  32 :  -4.33E-02 1.43E-08 0.000 0.1772 0.9059 0.9000   1.00  2  2  8.8E-09
+
+% iter seconds digits       c*x               b*y
+%  32     49.4   6.8 -4.3334083581e-02 -4.3334090214e-02
+% |Ax-b| =   3.7e-09, [Ay-c]_+ =   1.1E-10, |x|=  1.0e+00, |y|=  2.6e+00
+
+% Detailed timing (sec)
+%    Pre          IPM          Post
+% 3.902E+00    4.576E+01    1.035E-02
+% Max-norms: ||b||=1, ||c|| = 3,
+% Cholesky |add|=0, |skip| = 0, ||L.L|| = 4.26267.
+% ------------------------------------------------------------
+% Status: Solved
+% Optimal value (cvx_optval): -0.0433341
+% h = y(2:end);
+% #+end_example
+
+% Finally, we compute the filter response over the frequency vector defined and the result is shown on figure [[fig:fir_filter_ligo]] which is very close to the filters obtain in cite:hua05_low_ligo.
+
+
+w = [w1 w2 w3 w4];
+H = [exp(-j*kron(w'.*2*pi,[0:n-1]))]*h;
+
+figure;
+
+ax1 = subplot(2,1,1);
+hold on;
+plot(w, abs(H), 'k-');
+plot(w, abs(1-H), 'k--');
+hold off;
+set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
+ylabel('Magnitude');
+set(gca, 'XTickLabel',[]);
+ylim([5e-3, 5]);
+
+ax2 = subplot(2,1,2);
+hold on;
+plot(w, 180/pi*angle(H), 'k-');
+plot(w, 180/pi*angle(1-H), 'k--');
+hold off;
+xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
+set(gca, 'XScale', 'log');
+yticks([-540:90:360]);
+
+linkaxes([ax1,ax2],'x');
+xlim([1e-3, 1]);
+xticks([0.01, 0.1, 1, 10, 100, 1000]);
+
+% Weights
+% We design weights that will be used for the $\mathcal{H}_\infty$ synthesis of the complementary filters.
+% These weights will determine the order of the obtained filters.
+% Here are the requirements on the filters:
+% - reasonable order
+% - to be as close as possible to the specified upper bounds
+% - stable minimum phase
+
+% The bode plot of the weights is shown on figure [[fig:ligo_weights]].
+
+
+w1 = 2*pi*0.008; x1 = 0.35;
+w2 = 2*pi*0.04;  x2 = 0.5;
+w3 = 2*pi*0.05;  x3 = 0.5;
+
+% Slope of +3 from w1
+wH = 0.008*(s^2/w1^2 + 2*x1/w1*s + 1)*(s/w1 + 1);
+% Little bump from w2 to w3
+wH = wH*(s^2/w2^2 + 2*x2/w2*s + 1)/(s^2/w3^2 + 2*x3/w3*s + 1);
+% No Slope at high frequencies
+wH = wH/(s^2/w3^2 + 2*x3/w3*s + 1)/(s/w3 + 1);
+% Little bump between w2 and w3
+w0 = 2*pi*0.045; xi = 0.1; A = 2; n = 1;
+wH = wH*((s^2 + 2*w0*xi*A^(1/n)*s + w0^2)/(s^2 + 2*w0*xi*s + w0^2))^n;
+
+wH = 1/wH;
+wH = minreal(ss(wH));
+
+n = 20; Rp = 1; Wp = 2*pi*0.102;
+[b,a] = cheby1(n, Rp, Wp, 'high', 's');
+wL = 0.04*tf(a, b);
+
+wL = 1/wL;
+wL = minreal(ss(wL));
+
+figure;
+hold on;
+set(gca,'ColorOrderIndex',1);
+plot(freqs, abs(squeeze(freqresp(inv(wH), freqs, 'Hz'))), '-', 'DisplayName', '$|w_H|^{-1}$');
+set(gca,'ColorOrderIndex',2);
+plot(freqs, abs(squeeze(freqresp(inv(wL), freqs, 'Hz'))), '-', 'DisplayName', '$|w_L|^{-1}$');
+
+plot([0.0001, 0.008], [8e-3, 8e-3], 'k--', 'DisplayName', 'Spec.');
+plot([0.008 0.04], [8e-3, 1], 'k--', 'HandleVisibility', 'off');
+plot([0.04 0.1], [3, 3], 'k--', 'HandleVisibility', 'off');
+plot([0.1, 10], [0.045, 0.045], 'k--', 'HandleVisibility', 'off');
+
+set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
+xlabel('Frequency [Hz]'); ylabel('Magnitude');
+hold off;
+xlim([freqs(1), freqs(end)]);
+ylim([1e-3, 10]);
+legend('location', 'southeast');
+
+% H-Infinity Synthesis
+% We define the generalized plant as shown on figure [[fig:h_infinity_robst_fusion]].
+
+P = [0   wL;
+     wH -wH;
+     1   0];
+
+
+
+% And we do the $\mathcal{H}_\infty$ synthesis using the =hinfsyn= command.
+
+[Hl, ~, gamma, ~] = hinfsyn(P, 1, 1,'TOLGAM', 0.001, 'METHOD', 'ric', 'DISPLAY', 'on');
+
+
+
+% #+RESULTS:
+% #+begin_example
+% [Hl, ~, 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:      0.3276 <  gamma  <=      1.8063
+
+%   gamma    hamx_eig  xinf_eig  hamy_eig   yinf_eig   nrho_xy   p/f
+%     1.806   1.4e-02 -1.7e-16   3.6e-03   -4.8e-12    0.0000    p
+%     1.067   1.3e-02 -4.2e-14   3.6e-03   -1.9e-12    0.0000    p
+%     0.697   1.3e-02 -3.0e-01#  3.6e-03   -3.5e-11    0.0000    f
+%     0.882   1.3e-02 -9.5e-01#  3.6e-03   -1.2e-34    0.0000    f
+%     0.975   1.3e-02 -2.7e+00#  3.6e-03   -1.6e-12    0.0000    f
+%     1.021   1.3e-02 -8.7e+00#  3.6e-03   -4.5e-16    0.0000    f
+%     1.044   1.3e-02 -6.5e-14   3.6e-03   -3.0e-15    0.0000    p
+%     1.032   1.3e-02 -1.8e+01#  3.6e-03    0.0e+00    0.0000    f
+%     1.038   1.3e-02 -3.8e+01#  3.6e-03    0.0e+00    0.0000    f
+%     1.041   1.3e-02 -8.3e+01#  3.6e-03   -2.9e-33    0.0000    f
+%     1.042   1.3e-02 -1.9e+02#  3.6e-03   -3.4e-11    0.0000    f
+%     1.043   1.3e-02 -5.3e+02#  3.6e-03   -7.5e-13    0.0000    f
+
+%  Gamma value achieved:     1.0439
+% #+end_example
+
+% The high pass filter is defined as $H_H = 1 - H_L$.
+
+Hh = 1 - Hl;
+
+Hh = minreal(Hh);
+Hl = minreal(Hl);
+
+
+
+% The size of the filters is shown below.
+
+
+size(Hh), size(Hl)
+
+
+
+% #+RESULTS:
+% #+begin_example
+% size(Hh), size(Hl)
+% State-space model with 1 outputs, 1 inputs, and 27 states.
+% State-space model with 1 outputs, 1 inputs, and 27 states.
+% #+end_example
+
+% The bode plot of the obtained filters as shown on figure [[fig:hinf_synthesis_ligo_results]].
+
+
+figure;
+hold on;
+set(gca,'ColorOrderIndex',1);
+plot([0.0001, 0.008], [8e-3, 8e-3], ':', 'DisplayName', 'Spec. on $H_H$');
+set(gca,'ColorOrderIndex',1);
+plot([0.008 0.04], [8e-3, 1], ':', 'HandleVisibility', 'off');
+set(gca,'ColorOrderIndex',1);
+plot([0.04 0.1], [3, 3], ':', 'HandleVisibility', 'off');
+
+set(gca,'ColorOrderIndex',2);
+plot([0.1, 10], [0.045, 0.045], ':', 'DisplayName', 'Spec. on $H_L$');
+
+set(gca,'ColorOrderIndex',1);
+plot(freqs, abs(squeeze(freqresp(Hh, freqs, 'Hz'))), '-', 'DisplayName', '$H_H$');
+set(gca,'ColorOrderIndex',2);
+plot(freqs, abs(squeeze(freqresp(Hl, freqs, 'Hz'))), '-', 'DisplayName', '$H_L$');
+
+set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
+xlabel('Frequency [Hz]'); ylabel('Magnitude');
+hold off;
+xlim([freqs(1), freqs(end)]);
+ylim([1e-3, 10]);
+legend('location', 'southeast');
+
+% Compare FIR and H-Infinity Filters
+% Let's now compare the FIR filters designed in cite:hua05_low_ligo and the one obtained with the $\mathcal{H}_\infty$ synthesis on figure [[fig:comp_fir_ligo_hinf]].
+
+
+figure;
+ax1 = subplot(2,1,1);
+hold on;
+set(gca,'ColorOrderIndex',1);
+plot(freqs, abs(squeeze(freqresp(Hh, freqs, 'Hz'))), '-');
+set(gca,'ColorOrderIndex',2);
+plot(freqs, abs(squeeze(freqresp(Hl, freqs, 'Hz'))), '-');
+
+set(gca,'ColorOrderIndex',1);
+plot(w, abs(H), '--');
+set(gca,'ColorOrderIndex',2);
+plot(w, abs(1-H), '--');
+hold off;
+set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
+ylabel('Magnitude');
+set(gca, 'XTickLabel',[]);
+ylim([1e-3, 10]);
+
+ax2 = subplot(2,1,2);
+hold on;
+set(gca,'ColorOrderIndex',1);
+plot(freqs, 180/pi*angle(squeeze(freqresp(Hh, freqs, 'Hz'))), '-', 'DisplayName', '$\mathcal{H}_\infty$ filters');
+set(gca,'ColorOrderIndex',2);
+plot(freqs, 180/pi*angle(squeeze(freqresp(Hl, freqs, 'Hz'))), '-', 'HandleVisibility', 'off');
+
+set(gca,'ColorOrderIndex',1);
+plot(w, 180/pi*angle(H), '--', 'DisplayName', 'FIR filters');
+set(gca,'ColorOrderIndex',2);
+plot(w, 180/pi*angle(1-H), '--', 'HandleVisibility', 'off');
+set(gca, 'XScale', 'log');
+xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
+hold off;
+yticks([-540:90:360]);
+legend('location', 'northeast');
+
+linkaxes([ax1,ax2],'x');
+xlim([freqs(1), freqs(end)]);
+xticks([0.001, 0.01, 0.1, 1]);
diff --git a/matlab/matlab/h_inf_synthesis_complementary_filters.m b/matlab/matlab/h_inf_synthesis_complementary_filters.m
new file mode 100644
index 0000000..2eff502
--- /dev/null
+++ b/matlab/matlab/h_inf_synthesis_complementary_filters.m
@@ -0,0 +1,144 @@
+%% Clear Workspace and Close figures
+clear; close all; clc;
+
+%% Intialize Laplace variable
+s = zpk('s');
+
+freqs = logspace(-1, 3, 1000);
+
+% Design of Weighting Function
+% A formula is proposed to help the design of the weighting functions:
+% \begin{equation}
+%   W(s) = \left( \frac{
+%            \frac{1}{\omega_0} \sqrt{\frac{1 - \left(\frac{G_0}{G_c}\right)^{\frac{2}{n}}}{1 - \left(\frac{G_c}{G_\infty}\right)^{\frac{2}{n}}}} s + \left(\frac{G_0}{G_c}\right)^{\frac{1}{n}}
+%          }{
+%            \left(\frac{1}{G_\infty}\right)^{\frac{1}{n}} \frac{1}{\omega_0} \sqrt{\frac{1 - \left(\frac{G_0}{G_c}\right)^{\frac{2}{n}}}{1 - \left(\frac{G_c}{G_\infty}\right)^{\frac{2}{n}}}} s + \left(\frac{1}{G_c}\right)^{\frac{1}{n}}
+%          }\right)^n
+% \end{equation}
+
+% The parameters permits to specify:
+% - the low frequency gain: $G_0 = lim_{\omega \to 0} |W(j\omega)|$
+% - the high frequency gain: $G_\infty = lim_{\omega \to \infty} |W(j\omega)|$
+% - the absolute gain at $\omega_0$: $G_c = |W(j\omega_0)|$
+% - the absolute slope between high and low frequency: $n$
+
+% The general shape of a weighting function generated using the formula is shown in figure [[fig:weight_formula]].
+
+% #+name: fig:weight_formula
+% #+caption: Amplitude of the proposed formula for the weighting functions
+% [[file:figs-tikz/weight_formula.png]]
+
+
+n = 2; w0 = 2*pi*11; G0 = 1/10; G1 = 1000; Gc = 1/2;
+W1 = (((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;
+
+n = 3; w0 = 2*pi*10; G0 = 1000; G1 = 0.1; Gc = 1/2;
+W2 = (((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;
+
+figure;
+hold on;
+set(gca,'ColorOrderIndex',1)
+plot(freqs, 1./abs(squeeze(freqresp(W1, freqs, 'Hz'))), '--', 'DisplayName', '$|W_1|^{-1}$');
+set(gca,'ColorOrderIndex',2)
+plot(freqs, 1./abs(squeeze(freqresp(W2, freqs, 'Hz'))), '--', 'DisplayName', '$|W_2|^{-1}$');
+set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
+xlabel('Frequency [Hz]'); ylabel('Magnitude');
+hold off;
+xlim([freqs(1), freqs(end)]);
+ylim([5e-4, 20]);
+xticks([0.1, 1, 10, 100, 1000]);
+legend('location', 'northeast');
+
+% H-Infinity Synthesis
+% We define the generalized plant $P$ on matlab.
+
+P = [W1 -W1;
+     0   W2;
+     1   0];
+
+
+
+% And we do the $\mathcal{H}_\infty$ synthesis using the =hinfsyn= command.
+
+[H2, ~, gamma, ~] = hinfsyn(P, 1, 1,'TOLGAM', 0.001, 'METHOD', 'ric', 'DISPLAY', 'on');
+
+
+
+% #+RESULTS:
+% #+begin_example
+% [H2, ~, gamma, ~] = hinfsyn(P, 1, 1,'TOLGAM', 0.001, 'METHOD', 'ric', 'DISPLAY', 'on');
+% Resetting value of Gamma min based on D_11, D_12, D_21 terms
+
+% Test bounds:      0.1000 <  gamma  <=   1050.0000
+
+%   gamma    hamx_eig  xinf_eig  hamy_eig   yinf_eig   nrho_xy   p/f
+% 1.050e+03   2.8e+01   2.4e-07   4.1e+00    0.0e+00    0.0000    p
+%   525.050   2.8e+01   2.4e-07   4.1e+00    0.0e+00    0.0000    p
+%   262.575   2.8e+01   2.4e-07   4.1e+00    0.0e+00    0.0000    p
+%   131.337   2.8e+01   2.4e-07   4.1e+00   -1.0e-13    0.0000    p
+%    65.719   2.8e+01   2.4e-07   4.1e+00   -9.5e-14    0.0000    p
+%    32.909   2.8e+01   2.4e-07   4.1e+00    0.0e+00    0.0000    p
+%    16.505   2.8e+01   2.4e-07   4.1e+00   -1.0e-13    0.0000    p
+%     8.302   2.8e+01   2.4e-07   4.1e+00   -7.2e-14    0.0000    p
+%     4.201   2.8e+01   2.4e-07   4.1e+00   -2.5e-25    0.0000    p
+%     2.151   2.7e+01   2.4e-07   4.1e+00   -3.8e-14    0.0000    p
+%     1.125   2.6e+01   2.4e-07   4.1e+00   -5.4e-24    0.0000    p
+%     0.613   2.3e+01 -3.7e+01#  4.1e+00    0.0e+00    0.0000    f
+%     0.869   2.6e+01 -3.7e+02#  4.1e+00    0.0e+00    0.0000    f
+%     0.997   2.6e+01 -1.1e+04#  4.1e+00    0.0e+00    0.0000    f
+%     1.061   2.6e+01   2.4e-07   4.1e+00    0.0e+00    0.0000    p
+%     1.029   2.6e+01   2.4e-07   4.1e+00    0.0e+00    0.0000    p
+%     1.013   2.6e+01   2.4e-07   4.1e+00    0.0e+00    0.0000    p
+%     1.005   2.6e+01   2.4e-07   4.1e+00    0.0e+00    0.0000    p
+%     1.001   2.6e+01 -3.1e+04#  4.1e+00   -3.8e-14    0.0000    f
+%     1.003   2.6e+01 -2.8e+05#  4.1e+00    0.0e+00    0.0000    f
+%     1.004   2.6e+01   2.4e-07   4.1e+00   -5.8e-24    0.0000    p
+%     1.004   2.6e+01   2.4e-07   4.1e+00    0.0e+00    0.0000    p
+
+%  Gamma value achieved:     1.0036
+% #+end_example
+
+% We then define the high pass filter $H_1 = 1 - H_2$. The bode plot of both $H_1$ and $H_2$ is shown on figure [[fig:hinf_filters_results]].
+
+
+H1 = 1 - H2;
+
+% Obtained Complementary Filters
+% The obtained complementary filters are shown on figure [[fig:hinf_filters_results]].
+
+
+figure;
+
+ax1 = subplot(2,1,1);
+hold on;
+set(gca,'ColorOrderIndex',1)
+plot(freqs, 1./abs(squeeze(freqresp(W1, freqs, 'Hz'))), '--', 'DisplayName', '$w_1$');
+set(gca,'ColorOrderIndex',2)
+plot(freqs, 1./abs(squeeze(freqresp(W2, freqs, 'Hz'))), '--', 'DisplayName', '$w_2$');
+
+set(gca,'ColorOrderIndex',1)
+plot(freqs, abs(squeeze(freqresp(H1, freqs, 'Hz'))), '-', 'DisplayName', '$H_1$');
+set(gca,'ColorOrderIndex',2)
+plot(freqs, abs(squeeze(freqresp(H2, freqs, 'Hz'))), '-', 'DisplayName', '$H_2$');
+set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
+hold off;
+set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
+ylabel('Magnitude');
+set(gca, 'XTickLabel',[]);
+ylim([5e-4, 20]);
+legend('location', 'northeast');
+
+ax2 = subplot(2,1,2);
+hold on;
+set(gca,'ColorOrderIndex',1)
+plot(freqs, 180/pi*phase(squeeze(freqresp(H1, freqs, 'Hz'))), '-');
+set(gca,'ColorOrderIndex',2)
+plot(freqs, 180/pi*phase(squeeze(freqresp(H2, freqs, 'Hz'))), '-');
+hold off;
+xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
+set(gca, 'XScale', 'log');
+yticks([-360:90:360]);
+
+linkaxes([ax1,ax2],'x');
+xlim([freqs(1), freqs(end)]);
+xticks([0.1, 1, 10, 100, 1000]);
diff --git a/matlab/matlab/three_comp_filters.m b/matlab/matlab/three_comp_filters.m
new file mode 100644
index 0000000..6161383
--- /dev/null
+++ b/matlab/matlab/three_comp_filters.m
@@ -0,0 +1,95 @@
+%% Clear Workspace and Close figures
+clear; close all; clc;
+
+%% Intialize Laplace variable
+s = zpk('s');
+
+freqs = logspace(-2, 4, 1000);
+
+% Weights
+% First we define the weights.
+
+n = 2; w0 = 2*pi*1; G0 = 1/10; G1 = 1000; Gc = 1/2;
+W1 = (((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;
+
+W2 = 0.22*(1 + s/2/pi/1)^2/(sqrt(1e-4) + s/2/pi/1)^2*(1 + s/2/pi/10)^2/(1 + s/2/pi/1000)^2;
+
+n = 3; w0 = 2*pi*10; G0 = 1000; G1 = 0.1; Gc = 1/2;
+W3 = (((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;
+
+figure;
+hold on;
+set(gca,'ColorOrderIndex',1)
+plot(freqs, 1./abs(squeeze(freqresp(W1, freqs, 'Hz'))), '--', 'DisplayName', '$|W_1|^{-1}$');
+set(gca,'ColorOrderIndex',2)
+plot(freqs, 1./abs(squeeze(freqresp(W2, freqs, 'Hz'))), '--', 'DisplayName', '$|W_2|^{-1}$');
+set(gca,'ColorOrderIndex',3)
+plot(freqs, 1./abs(squeeze(freqresp(W3, freqs, 'Hz'))), '--', 'DisplayName', '$|W_3|^{-1}$');
+set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
+xlabel('Frequency [Hz]'); ylabel('Magnitude');
+hold off;
+xlim([freqs(1), freqs(end)]);
+xticks([0.01, 0.1, 1, 10, 100, 1000]);
+legend('location', 'northeast');
+
+% H-Infinity Synthesis
+% Then we create the generalized plant =P=.
+
+P = [W1 -W1 -W1;
+     0   W2  0 ;
+     0   0   W3;
+     1   0   0];
+
+
+
+% And we do the $\mathcal{H}_\infty$ synthesis.
+
+[H, ~, gamma, ~] = hinfsyn(P, 1, 2,'TOLGAM', 0.001, 'METHOD', 'ric', 'DISPLAY', 'on');
+
+% Obtained Complementary Filters
+% The obtained filters are:
+
+H2 = tf(H(1));
+H3 = tf(H(2));
+H1 = 1 - H2 - H3;
+
+figure;
+
+ax1 = subplot(2,1,1);
+hold on;
+set(gca,'ColorOrderIndex',1)
+plot(freqs, 1./abs(squeeze(freqresp(W1, freqs, 'Hz'))), '--', 'DisplayName', '$|W_1|^{-1}$');
+set(gca,'ColorOrderIndex',2)
+plot(freqs, 1./abs(squeeze(freqresp(W2, freqs, 'Hz'))), '--', 'DisplayName', '$|W_2|^{-1}$');
+set(gca,'ColorOrderIndex',3)
+plot(freqs, 1./abs(squeeze(freqresp(W3, freqs, 'Hz'))), '--', 'DisplayName', '$|W_3|^{-1}$');
+set(gca,'ColorOrderIndex',1)
+plot(freqs, abs(squeeze(freqresp(H1, freqs, 'Hz'))), '-', 'DisplayName', '$H_1$');
+set(gca,'ColorOrderIndex',2)
+plot(freqs, abs(squeeze(freqresp(H2, freqs, 'Hz'))), '-', 'DisplayName', '$H_2$');
+set(gca,'ColorOrderIndex',3)
+plot(freqs, abs(squeeze(freqresp(H3, freqs, 'Hz'))), '-', 'DisplayName', '$H_3$');
+set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
+hold off;
+set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
+ylabel('Magnitude');
+set(gca, 'XTickLabel',[]);
+ylim([5e-4, 20]);
+legend('location', 'northeast');
+
+ax2 = subplot(2,1,2);
+hold on;
+set(gca,'ColorOrderIndex',1)
+plot(freqs, 180/pi*phase(squeeze(freqresp(H1, freqs, 'Hz'))));
+set(gca,'ColorOrderIndex',2)
+plot(freqs, 180/pi*phase(squeeze(freqresp(H2, freqs, 'Hz'))));
+set(gca,'ColorOrderIndex',3)
+plot(freqs, 180/pi*phase(squeeze(freqresp(H3, freqs, 'Hz'))));
+hold off;
+xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
+set(gca, 'XScale', 'log');
+yticks([-360:90:360]);
+
+linkaxes([ax1,ax2],'x');
+xlim([freqs(1), freqs(end)]);
+xticks([0.1, 1, 10, 100, 1000]);
diff --git a/matlab/ref.bib b/matlab/ref.bib
new file mode 100644
index 0000000..bf21574
--- /dev/null
+++ b/matlab/ref.bib
@@ -0,0 +1,171 @@
+@article{collette15_sensor_fusion_method_high_perfor,
+  author = {C. Collette and F. Matichard},
+  title = {Sensor Fusion Methods for High Performance Active Vibration Isolation Systems},
+  journal = {Journal of Sound and Vibration},
+  volume = {342},
+  number = {nil},
+  pages = {1-21},
+  year = {2015},
+  doi = {10.1016/j.jsv.2015.01.006},
+  url = {https://doi.org/10.1016/j.jsv.2015.01.006},
+  keywords = {},
+}
+
+
+@phdthesis{hua05_low_ligo,
+  author =       {Hua, Wensheng},
+  school =       {stanford university},
+  title =        {Low frequency vibration isolation and alignment system for
+                  advanced LIGO},
+  year =         2005,
+}
+
+
+@inproceedings{hua04_polyp_fir_compl_filter_contr_system,
+  author = {Wensheng Hua and Dan B. Debra and Corwin T. Hardham and Brian T. Lantz and Joseph A. Giaime},
+  title = {Polyphase FIR Complementary Filters for Control Systems},
+  booktitle = {Proceedings of ASPE Spring Topical Meeting on Control of Precision Systems},
+  year = {2004},
+  pages = {109--114},
+}
+
+
+@article{matichard15_seism_isolat_advan_ligo,
+  author = {Matichard, F and Lantz, B and Mittleman, R and Mason, K and Kissel, J and Abbott, B and Biscans, S and McIver, J and Abbott, R and Abbott, S and others},
+  title = {Seismic Isolation of Advanced Ligo: Review of Strategy, Instrumentation and Performance},
+  journal = {Classical and Quantum Gravity},
+  volume = {32},
+  number = {18},
+  pages = {185003},
+  year = {2015},
+  publisher = {IOP Publishing},
+}
+
+
+@article{min15_compl_filter_desig_angle_estim,
+  author = {Min, Hyung Gi and Jeung, Eun Tae},
+  title = {Complementary Filter Design for Angle Estimation Using Mems Accelerometer and Gyroscope},
+  journal = {Department of Control and Instrumentation, Changwon National University, Changwon, Korea},
+  pages = {641--773},
+  year = {2015},
+}
+
+
+@article{corke04_inert_visual_sensin_system_small_auton_helic,
+  author = {Peter Corke},
+  title = {An Inertial and Visual Sensing System for a Small Autonomous Helicopter},
+  journal = {Journal of Robotic Systems},
+  volume = {21},
+  number = {2},
+  pages = {43-51},
+  year = {2004},
+  doi = {10.1002/rob.10127},
+  url = {https://doi.org/10.1002/rob.10127},
+}
+
+
+@inproceedings{jensen13_basic_uas,
+  author =       {Austin Jensen and Cal Coopmans and YangQuan Chen},
+  title =        {Basics and guidelines of complementary filters for small UAS
+                  navigation},
+  booktitle =    {2013 International Conference on Unmanned Aircraft Systems
+                  (ICUAS)},
+  year =         2013,
+  pages =        {nil},
+  doi =          {10.1109/icuas.2013.6564726},
+  url =          {https://doi.org/10.1109/icuas.2013.6564726},
+  month =        5,
+}
+
+
+@inproceedings{pascoal99_navig_system_desig_using_time,
+  author = {A. Pascoal and I. Kaminer and P. Oliveira},
+  title = {Navigation System Design Using Time-Varying Complementary Filters},
+  booktitle = {Guidance, Navigation, and Control Conference and Exhibit},
+  year = {1999},
+  pages = {nil},
+  doi = {10.2514/6.1999-4290},
+  url = {https://doi.org/10.2514/6.1999-4290},
+}
+
+
+@article{zimmermann92_high_bandw_orien_measur_contr,
+  author = {M. Zimmermann and W. Sulzer},
+  title = {High Bandwidth Orientation Measurement and Control Based on Complementary Filtering},
+  journal = {Robot Control 1991},
+  pages = {525-530},
+  year = {1992},
+  doi = {10.1016/b978-0-08-041276-4.50093-5},
+  url = {https://doi.org/10.1016/b978-0-08-041276-4.50093-5},
+  publisher = {Elsevier},
+  series = {Robot Control 1991},
+}
+
+
+@inproceedings{baerveldt97_low_cost_low_weigh_attit,
+  author = {A.-J. Baerveldt and R. Klang},
+  title = {A Low-Cost and Low-Weight Attitude Estimation System for an Autonomous Helicopter},
+  booktitle = {Proceedings of IEEE International Conference on Intelligent Engineering Systems},
+  year = {1997},
+  pages = {nil},
+  doi = {10.1109/ines.1997.632450},
+  url = {https://doi.org/10.1109/ines.1997.632450},
+  month = {-},
+}
+
+
+@article{shaw90_bandw_enhan_posit_measur_using_measur_accel,
+  author =       {F.R. Shaw and K. Srinivasan},
+  title =        {Bandwidth Enhancement of Position Measurements Using Measured
+                  Acceleration},
+  journal =      {Mechanical Systems and Signal Processing},
+  volume =       4,
+  number =       1,
+  pages =        {23-38},
+  year =         1990,
+  doi =          {10.1016/0888-3270(90)90038-m},
+  url =          {https://doi.org/10.1016/0888-3270(90)90038-m},
+}
+
+
+@article{brown72_integ_navig_system_kalman_filter,
+  author =       {R. G. Brown},
+  title =        {Integrated Navigation Systems and Kalman Filtering: a
+                  Perspective},
+  journal =      {Navigation},
+  volume =       19,
+  number =       4,
+  pages =        {355-362},
+  year =         1972,
+  doi =          {10.1002/j.2161-4296.1972.tb01706.x},
+  url =          {https://doi.org/10.1002/j.2161-4296.1972.tb01706.x},
+}
+
+
+@article{matichard15_seism_isolat_advan_ligo,
+  author =       {Matichard, F and Lantz, B and Mittleman, R and Mason, K and
+                  Kissel, J and Abbott, B and Biscans, S and McIver, J and
+                  Abbott, R and Abbott, S and others},
+  title =        {Seismic Isolation of Advanced Ligo: Review of Strategy,
+                  Instrumentation and Performance},
+  journal =      {Classical and Quantum Gravity},
+  volume =       32,
+  number =       18,
+  pages =        185003,
+  year =         2015,
+  publisher =    {IOP Publishing},
+}
+
+
+@article{mahony08_nonlin_compl_filter_special_orthog_group,
+  author =       {Robert Mahony and Tarek Hamel and Jean-Michel Pflimlin},
+  title =        {Nonlinear Complementary Filters on the Special Orthogonal
+                  Group},
+  journal =      {IEEE Transactions on Automatic Control},
+  volume =       53,
+  number =       5,
+  pages =        {1203-1218},
+  year =         2008,
+  doi =          {10.1109/tac.2008.923738},
+  url =          {https://doi.org/10.1109/tac.2008.923738},
+}
\ No newline at end of file
diff --git a/paper/paper.org b/paper/paper.org
index 4f32cd0..94b9993 100644
--- a/paper/paper.org
+++ b/paper/paper.org
@@ -320,7 +320,7 @@ Let's validate the proposed design method of complementary filters with a simple
 - the gain of both filters is equal to $10^{-3}$ away from the merging frequency
 
 The weighting functions $W_1(s)$ and $W_2(s)$ are designed using eqref:eq:weight_formula.
-The parameters used are summarized in table ref:tab:weights_params and the magnitude of the weighting functions is shown in Fig. ref:fig:hinf_synthesis_results.
+The parameters used are summarized in table ref:tab:weights_params and the magnitude of the weighting functions is shown in Fig. ref:fig:hinf_filters_results.
 
 #+name: tab:weights_params
 #+caption: Parameters used for $W_1(s)$ and $W_2(s)$
@@ -334,16 +334,16 @@ The parameters used are summarized in table ref:tab:weights_params and the magni
 | $G_c$                  | $0.5$    | $0.5$    |
 | $n$                    | $2$      | $3$      |
 
-The bode plots of the obtained complementary filters are shown in Fig. ref:fig:hinf_synthesis_results and their transfer functions in the Laplace domain are given below.
+The bode plots of the obtained complementary filters are shown in Fig. ref:fig:hinf_filters_results and their transfer functions in the Laplace domain are given below.
 \begin{align*}
   H_1(s) &= \frac{10^{-8} (s+6.6e^9) (s+3450)^2 (s^2 + 49s + 895)}{(s+6.6e^4) (s^2 + 106 s + 3e^3) (s^2 + 72s + 3580)}\\
   H_2(s) &= \frac{(s+6.6e^4) (s+160) (s+4)^3}{(s+6.6e^4) (s^2 + 106 s + 3e^3) (s^2 + 72s + 3580)}
 \end{align*}
 
-#+name: fig:hinf_synthesis_results
+#+name: fig:hinf_filters_results
 #+caption: Frequency response of the weighting functions and complementary filters obtained using $\mathcal{H}_\infty$ synthesis
 #+attr_latex: :scale 1
-[[file:figs/hinf_synthesis_results.pdf]]
+[[file:figs/hinf_filters_results.pdf]]
 
 ** Synthesis of Three Complementary Filters
 <<sec:hinf_three_comp_filters>>
@@ -380,13 +380,13 @@ By choosing $H_1(s) \triangleq 1 - H_2(s) - H_3(s)$, the proposed $\mathcal{H}_\
 
 *** Example of generated complementary filters                     :ignore:
 An example is given to validate the method where three sensors are used in different frequency bands (up to $\SI{1}{Hz}$, from $1$ to $\SI{10}{Hz}$ and above $\SI{10}{Hz}$ respectively).
-Three weighting functions are designed using eqref:eq:weight_formula and shown by dashed curves in Fig. ref:fig:hinf_three_synthesis_results.
-The bode plots of the obtained complementary filters are shown in Fig. ref:fig:hinf_three_synthesis_results.
+Three weighting functions are designed using eqref:eq:weight_formula and shown by dashed curves in Fig. ref:fig:three_complementary_filters_results.
+The bode plots of the obtained complementary filters are shown in Fig. ref:fig:three_complementary_filters_results.
 
-#+name: fig:hinf_three_synthesis_results
+#+name: fig:three_complementary_filters_results
 #+caption: Frequency response of the weighting functions and three complementary filters obtained using $\mathcal{H}_\infty$ synthesis
 #+attr_latex: :scale 1
-[[file:figs/hinf_three_synthesis_results.pdf]]
+[[file:figs/three_complementary_filters_results.pdf]]
 
 * Application: Design of Complementary Filters used in the Active Vibration Isolation System at the LIGO
 <<sec:application_ligo>>
diff --git a/paper/paper.pdf b/paper/paper.pdf
index 0183232..db9d458 100644
Binary files a/paper/paper.pdf and b/paper/paper.pdf differ
diff --git a/tikz/figs/comp_fir_ligo_hinf.pdf b/tikz/figs/comp_fir_ligo_hinf.pdf
index 1d2f27f..df38463 100644
Binary files a/tikz/figs/comp_fir_ligo_hinf.pdf and b/tikz/figs/comp_fir_ligo_hinf.pdf differ
diff --git a/tikz/figs/comp_fir_ligo_hinf.png b/tikz/figs/comp_fir_ligo_hinf.png
deleted file mode 100644
index 252f098..0000000
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diff --git a/tikz/figs/comp_fir_ligo_hinf.svg b/tikz/figs/comp_fir_ligo_hinf.svg
deleted file mode 100644
index b5a7413..0000000
--- a/tikz/figs/comp_fir_ligo_hinf.svg
+++ /dev/null
@@ -1,487 +0,0 @@
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deleted file mode 100644
index f1e90c6..0000000
--- a/tikz/figs/comp_fir_ligo_hinf.tex
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deleted file mode 100644
index 114854a..0000000
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