From e7a25abb54134a94fb58625a723c445411f00fbe Mon Sep 17 00:00:00 2001
From: Thomas Dehaeze <dehaeze.thomas@gmail.com>
Date: Wed, 2 Dec 2020 20:04:05 +0100
Subject: [PATCH] Add meta viewport HTML header

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-<!-- 2020-12-02 mer. 19:53 -->
+<!-- 2020-12-02 mer. 20:03 -->
 <meta http-equiv="Content-Type" content="text/html;charset=utf-8" />
 <title>A brief and practical introduction to \(\mathcal{H}_\infty\) Control</title>
 <meta name="generator" content="Org mode" />
 <meta name="author" content="Dehaeze Thomas" />
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@@ -30,56 +31,56 @@
 <h2>Table of Contents</h2>
 <div id="text-table-of-contents">
 <ul>
-<li><a href="#org0c6d88b">1. Introduction to Model Based Control</a>
+<li><a href="#org4c8f6a6">1. Introduction to Model Based Control</a>
 <ul>
-<li><a href="#org5fb1a23">1.1. Model Based Control - Methodology</a></li>
-<li><a href="#orgff93d8b">1.2. From Classical Control to Robust Control</a></li>
-<li><a href="#orged790f0">1.3. Example System</a></li>
+<li><a href="#orgfcfdf29">1.1. Model Based Control - Methodology</a></li>
+<li><a href="#orgec35f84">1.2. From Classical Control to Robust Control</a></li>
+<li><a href="#orge75c14c">1.3. Example System</a></li>
 </ul>
 </li>
-<li><a href="#org042f0e7">2. Classical Open Loop Shaping</a>
+<li><a href="#org63a6e7c">2. Classical Open Loop Shaping</a>
 <ul>
-<li><a href="#orgdd836d3">2.1. Introduction to Loop Shaping</a></li>
-<li><a href="#org7ec1198">2.2. Example of Manual Open Loop Shaping</a></li>
-<li><a href="#orgb2a458f">2.3. \(\mathcal{H}_\infty\) Loop Shaping Synthesis</a></li>
-<li><a href="#orgf390eb2">2.4. Example of the \(\mathcal{H}_\infty\) Loop Shaping Synthesis</a></li>
+<li><a href="#org3f96bef">2.1. Introduction to Loop Shaping</a></li>
+<li><a href="#orga55eafa">2.2. Example of Manual Open Loop Shaping</a></li>
+<li><a href="#orgaa840e2">2.3. \(\mathcal{H}_\infty\) Loop Shaping Synthesis</a></li>
+<li><a href="#org6e8b6e8">2.4. Example of the \(\mathcal{H}_\infty\) Loop Shaping Synthesis</a></li>
 </ul>
 </li>
-<li><a href="#org214059b">3. A first Step into the \(\mathcal{H}_\infty\) world</a>
+<li><a href="#orgac445c2">3. A first Step into the \(\mathcal{H}_\infty\) world</a>
 <ul>
-<li><a href="#org4f9dc47">3.1. The \(\mathcal{H}_\infty\) Norm</a></li>
-<li><a href="#org260f673">3.2. \(\mathcal{H}_\infty\) Synthesis</a></li>
-<li><a href="#org71b47ed">3.3. The Generalized Plant</a></li>
-<li><a href="#orgc597f47">3.4. The \(\mathcal{H}_\infty\) Synthesis applied on the Generalized plant</a></li>
-<li><a href="#orgd735b55">3.5. From a Classical Feedback Architecture to a Generalized Plant</a></li>
+<li><a href="#org29b6052">3.1. The \(\mathcal{H}_\infty\) Norm</a></li>
+<li><a href="#orga7a5510">3.2. \(\mathcal{H}_\infty\) Synthesis</a></li>
+<li><a href="#org810d672">3.3. The Generalized Plant</a></li>
+<li><a href="#org5cbe4b7">3.4. The \(\mathcal{H}_\infty\) Synthesis applied on the Generalized plant</a></li>
+<li><a href="#orga713c1e">3.5. From a Classical Feedback Architecture to a Generalized Plant</a></li>
 </ul>
 </li>
-<li><a href="#orgea992d5">4. Modern Interpretation of Control Specifications</a>
+<li><a href="#org96ba1fd">4. Modern Interpretation of Control Specifications</a>
 <ul>
-<li><a href="#org9916c4e">4.1. Closed Loop Transfer Functions</a></li>
-<li><a href="#org9653453">4.2. Sensitivity Function</a></li>
-<li><a href="#org5ee7596">4.3. Robustness: Module Margin</a></li>
-<li><a href="#orgb77093f">4.4. Other Requirements</a></li>
+<li><a href="#org9d2f211">4.1. Closed Loop Transfer Functions</a></li>
+<li><a href="#org340b845">4.2. Sensitivity Function</a></li>
+<li><a href="#org3040671">4.3. Robustness: Module Margin</a></li>
+<li><a href="#orge8044fe">4.4. Other Requirements</a></li>
 </ul>
 </li>
-<li><a href="#org5613659">5. \(\mathcal{H}_\infty\) Shaping of closed-loop transfer functions</a>
+<li><a href="#orgbc1be70">5. \(\mathcal{H}_\infty\) Shaping of closed-loop transfer functions</a>
 <ul>
-<li><a href="#org0331165">5.1. How to Shape closed-loop transfer function? Using Weighting Functions!</a></li>
-<li><a href="#orgc705f0d">5.2. Design of Weighting Functions</a></li>
-<li><a href="#orgb481e60">5.3. Shaping the Sensitivity Function</a></li>
-<li><a href="#orgb96dfcb">5.4. Shaping multiple closed-loop transfer functions</a></li>
+<li><a href="#orga514a74">5.1. How to Shape closed-loop transfer function? Using Weighting Functions!</a></li>
+<li><a href="#org1057ca5">5.2. Design of Weighting Functions</a></li>
+<li><a href="#org7ac2800">5.3. Shaping the Sensitivity Function</a></li>
+<li><a href="#orgebe5d7a">5.4. Shaping multiple closed-loop transfer functions</a></li>
 </ul>
 </li>
-<li><a href="#orgb5da593">6. Mixed-Sensitivity \(\mathcal{H}_\infty\) Control - Example</a>
+<li><a href="#orgdfad810">6. Mixed-Sensitivity \(\mathcal{H}_\infty\) Control - Example</a>
 <ul>
-<li><a href="#org366bd83">6.1. Control Problem</a></li>
-<li><a href="#org7004a2e">6.2. Control Design Procedure</a></li>
-<li><a href="#org23a1031">6.3. Step 1 - Shaping of \(S\)</a></li>
-<li><a href="#org759fd41">6.4. Step 2 - Shaping of \(KS\)</a></li>
-<li><a href="#org5f57abd">6.5. Step 3 - Shaping of \(T\)</a></li>
+<li><a href="#org22bd096">6.1. Control Problem</a></li>
+<li><a href="#org02f4803">6.2. Control Design Procedure</a></li>
+<li><a href="#org741a631">6.3. Step 1 - Shaping of \(S\)</a></li>
+<li><a href="#orga7db01d">6.4. Step 2 - Shaping of \(KS\)</a></li>
+<li><a href="#orge4c24c4">6.5. Step 3 - Shaping of \(T\)</a></li>
 </ul>
 </li>
-<li><a href="#orgb1a415e">7. Conclusion</a></li>
+<li><a href="#org46694c3">7. Conclusion</a></li>
 </ul>
 </div>
 </div>
@@ -103,33 +104,33 @@ When possible, Matlab scripts used for the example/exercises are provided such t
 The general structure of this document is as follows:
 </p>
 <ul class="org-ul">
-<li>A short introduction to <i>model based control</i> is given in Section <a href="#orgb67e46e">1</a></li>
-<li>Classical <i>open</i> loop shaping method is presented in Section <a href="#orga360568">2</a>.
+<li>A short introduction to <i>model based control</i> is given in Section <a href="#org679d84c">1</a></li>
+<li>Classical <i>open</i> loop shaping method is presented in Section <a href="#orge2537e5">2</a>.
 It is also shown that \(\mathcal{H}_\infty\) synthesis can be used for <i>open</i> loop shaping</li>
-<li>Important concepts indispensable for \(\mathcal{H}_\infty\) control such as the \(\mathcal{H}_\infty\) norm and the generalized plant are introduced in Section <a href="#org4bf732f">3</a></li>
-<li>A very important step in \(\mathcal{H}_\infty\) control is to express the control specifications (performances, robustness, etc.) as an \(\mathcal{H}_\infty\) optimization problem. Such procedure is described in Section <a href="#orgb800b43">4</a></li>
+<li>Important concepts indispensable for \(\mathcal{H}_\infty\) control such as the \(\mathcal{H}_\infty\) norm and the generalized plant are introduced in Section <a href="#org252b5a9">3</a></li>
+<li>A very important step in \(\mathcal{H}_\infty\) control is to express the control specifications (performances, robustness, etc.) as an \(\mathcal{H}_\infty\) optimization problem. Such procedure is described in Section <a href="#orgb74964f">4</a></li>
 <li>One of the most useful use of the \(\mathcal{H}_\infty\) control is the shaping of closed-loop transfer functions.
-Such technique is presented in Section <a href="#org19e43fa">5</a></li>
-<li>Finally, complete examples of the use of \(\mathcal{H}_\infty\) Control for practical problems are provided in Section <a href="#org099ff7f">6</a>.</li>
+Such technique is presented in Section <a href="#orgf618d2f">5</a></li>
+<li>Finally, complete examples of the use of \(\mathcal{H}_\infty\) Control for practical problems are provided in Section <a href="#org8a12722">6</a>.</li>
 </ul>
 
-<div id="outline-container-org0c6d88b" class="outline-2">
-<h2 id="org0c6d88b"><span class="section-number-2">1</span> Introduction to Model Based Control</h2>
+<div id="outline-container-org4c8f6a6" class="outline-2">
+<h2 id="org4c8f6a6"><span class="section-number-2">1</span> Introduction to Model Based Control</h2>
 <div class="outline-text-2" id="text-1">
 <p>
-<a id="orgb67e46e"></a>
+<a id="org679d84c"></a>
 </p>
 </div>
 
-<div id="outline-container-org5fb1a23" class="outline-3">
-<h3 id="org5fb1a23"><span class="section-number-3">1.1</span> Model Based Control - Methodology</h3>
+<div id="outline-container-orgfcfdf29" class="outline-3">
+<h3 id="orgfcfdf29"><span class="section-number-3">1.1</span> Model Based Control - Methodology</h3>
 <div class="outline-text-3" id="text-1-1">
 <p>
-<a id="org55d5803"></a>
+<a id="orga9bbb9f"></a>
 </p>
 
 <p>
-The typical methodology for <b>Model Based Control</b> techniques is schematically shown in Figure <a href="#orga44920d">1</a>.
+The typical methodology for <b>Model Based Control</b> techniques is schematically shown in Figure <a href="#orgc689f68">1</a>.
 </p>
 
 <p>
@@ -146,7 +147,7 @@ It consists of three steps:
 </ol>
 
 
-<div id="orga44920d" class="figure">
+<div id="orgc689f68" class="figure">
 <p><img src="figs/control-procedure.png" alt="control-procedure.png" />
 </p>
 <p><span class="figure-number">Figure 1: </span>Typical Methodoly for Model Based Control</p>
@@ -158,20 +159,20 @@ In this document, we will suppose a model of the plant is available (step 1 alre
 
 
 <p>
-In Section <a href="#orga360568">2</a>, steps 2 and 3 will be described for a control techniques called <b>classical (open-)loop shaping</b>.
+In Section <a href="#orge2537e5">2</a>, steps 2 and 3 will be described for a control techniques called <b>classical (open-)loop shaping</b>.
 </p>
 
 <p>
-Then, steps 2 and 3 for the <b>\(\mathcal{H}_\infty\) Loop Shaping</b> of closed-loop transfer functions will be discussed in Sections <a href="#orgb800b43">4</a>, <a href="#org19e43fa">5</a> and <a href="#org099ff7f">6</a>.
+Then, steps 2 and 3 for the <b>\(\mathcal{H}_\infty\) Loop Shaping</b> of closed-loop transfer functions will be discussed in Sections <a href="#orgb74964f">4</a>, <a href="#orgf618d2f">5</a> and <a href="#org8a12722">6</a>.
 </p>
 </div>
 </div>
 
-<div id="outline-container-orgff93d8b" class="outline-3">
-<h3 id="orgff93d8b"><span class="section-number-3">1.2</span> From Classical Control to Robust Control</h3>
+<div id="outline-container-orgec35f84" class="outline-3">
+<h3 id="orgec35f84"><span class="section-number-3">1.2</span> From Classical Control to Robust Control</h3>
 <div class="outline-text-3" id="text-1-2">
 <p>
-<a id="org036350e"></a>
+<a id="org92f98d7"></a>
 </p>
 
 <p>
@@ -203,14 +204,14 @@ This robust control theory is the subject of this document.
 
 
 <p>
-The three presented control methods are compared in Table <a href="#org4838a5e">1</a>.
+The three presented control methods are compared in Table <a href="#org14ca96f">1</a>.
 </p>
 
 <p>
 Note that in parallel, there have been numerous other developments, including non-linear control, adaptive control, machine-learning control just to name a few.
 </p>
 
-<table id="org4838a5e" border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
+<table id="org14ca96f" border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
 <caption class="t-above"><span class="table-number">Table 1:</span> Table summurazing the main differences between classical, modern and robust control</caption>
 
 <colgroup>
@@ -358,11 +359,11 @@ Note that in parallel, there have been numerous other developments, including no
 </div>
 </div>
 
-<div id="outline-container-orged790f0" class="outline-3">
-<h3 id="orged790f0"><span class="section-number-3">1.3</span> Example System</h3>
+<div id="outline-container-orge75c14c" class="outline-3">
+<h3 id="orge75c14c"><span class="section-number-3">1.3</span> Example System</h3>
 <div class="outline-text-3" id="text-1-3">
 <p>
-<a id="org1f983b6"></a>
+<a id="org7ae910e"></a>
 </p>
 
 <p>
@@ -371,24 +372,24 @@ Most of them will be applied on a physical system presented in this section.
 </p>
 
 <p>
-This system is shown in Figure <a href="#orgfc1874e">2</a>.
+This system is shown in Figure <a href="#orgdf183f5">2</a>.
 It could represent an active suspension stage supporting a payload.
 The <i>inertial</i> motion of the payload is measured using an inertial sensor and this is feedback to a force actuator.
 Such system could be used to actively isolate the payload (disturbance rejection problem) or to make it follow a trajectory (tracking problem).
 </p>
 
 <p>
-The notations used on Figure <a href="#orgfc1874e">2</a> are listed and described in Table <a href="#org9fdaa92">2</a>.
+The notations used on Figure <a href="#orgdf183f5">2</a> are listed and described in Table <a href="#org5c4d383">2</a>.
 </p>
 
 
-<div id="orgfc1874e" class="figure">
+<div id="orgdf183f5" class="figure">
 <p><img src="figs/mech_sys_1dof_inertial_contr.png" alt="mech_sys_1dof_inertial_contr.png" />
 </p>
 <p><span class="figure-number">Figure 2: </span>Test System consisting of a payload with a mass \(m\) on top of an active system with a stiffness \(k\), damping \(c\) and an actuator. A feedback controller \(K(s)\) is added to position / isolate the payload.</p>
 </div>
 
-<table id="org9fdaa92" border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
+<table id="org5c4d383" border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
 <caption class="t-above"><span class="table-number">Table 2:</span> Example system variables</caption>
 
 <colgroup>
@@ -474,7 +475,7 @@ The notations used on Figure <a href="#orgfc1874e">2</a> are listed and describe
 </tbody>
 </table>
 
-<div class="exercice" id="orgfa20d65">
+<div class="exercice" id="org9fd3701">
 <p>
 Derive the following open-loop transfer functions:
 </p>
@@ -507,14 +508,14 @@ You can follow this generic procedure:
 </div>
 
 <p>
-Having obtained \(G(s)\) and \(G_d(s)\), we can transform the system shown in Figure <a href="#orgfc1874e">2</a> into a classical feedback architecture as shown in Figure <a href="#orgf84f71d">6</a>.
+Having obtained \(G(s)\) and \(G_d(s)\), we can transform the system shown in Figure <a href="#orgdf183f5">2</a> into a classical feedback architecture as shown in Figure <a href="#orge4af8ce">6</a>.
 </p>
 
 
-<div id="org094d1b8" class="figure">
+<div id="orgddbd5f2" class="figure">
 <p><img src="figs/classical_feedback_test_system.png" alt="classical_feedback_test_system.png" />
 </p>
-<p><span class="figure-number">Figure 3: </span>Block diagram corresponding to the example system of Figure <a href="#orgfc1874e">2</a></p>
+<p><span class="figure-number">Figure 3: </span>Block diagram corresponding to the example system of Figure <a href="#orgdf183f5">2</a></p>
 </div>
 
 
@@ -538,18 +539,18 @@ And now the system dynamics \(G(s)\) and \(G_d(s)\).
 </div>
 
 <p>
-The Bode plots of \(G(s)\) and \(G_d(s)\) are shown in Figures <a href="#org9cb0641">4</a> and <a href="#org3684bd8">5</a>.
+The Bode plots of \(G(s)\) and \(G_d(s)\) are shown in Figures <a href="#org06f36e5">4</a> and <a href="#org229b7c8">5</a>.
 </p>
 
 
-<div id="org9cb0641" class="figure">
+<div id="org06f36e5" class="figure">
 <p><img src="figs/bode_plot_example_afm.png" alt="bode_plot_example_afm.png" />
 </p>
 <p><span class="figure-number">Figure 4: </span>Bode plot of the plant \(G(s)\)</p>
 </div>
 
 
-<div id="org3684bd8" class="figure">
+<div id="org229b7c8" class="figure">
 <p><img src="figs/bode_plot_example_Gd.png" alt="bode_plot_example_Gd.png" />
 </p>
 <p><span class="figure-number">Figure 5: </span>Magnitude of the disturbance transfer function \(G_d(s)\)</p>
@@ -558,40 +559,40 @@ The Bode plots of \(G(s)\) and \(G_d(s)\) are shown in Figures <a href="#org9cb0
 </div>
 </div>
 
-<div id="outline-container-org042f0e7" class="outline-2">
-<h2 id="org042f0e7"><span class="section-number-2">2</span> Classical Open Loop Shaping</h2>
+<div id="outline-container-org63a6e7c" class="outline-2">
+<h2 id="org63a6e7c"><span class="section-number-2">2</span> Classical Open Loop Shaping</h2>
 <div class="outline-text-2" id="text-2">
 <p>
-<a id="orga360568"></a>
+<a id="orge2537e5"></a>
 </p>
 <p>
-After an introduction to classical Loop Shaping in Section <a href="#org1108294">2.1</a>, a practical example is given in Section <a href="#orgbe90ac6">2.2</a>.
+After an introduction to classical Loop Shaping in Section <a href="#orgfa8b378">2.1</a>, a practical example is given in Section <a href="#orgbc5ea32">2.2</a>.
 Such Loop Shaping is usually performed manually with tools coming from the classical control theory.
 </p>
 
 <p>
 However, the \(\mathcal{H}_\infty\) synthesis can be used to automate the Loop Shaping process.
-This is presented in Section <a href="#org54bf645">2.3</a> and applied on the same example in Section <a href="#org8c3037b">2.4</a>.
+This is presented in Section <a href="#orgf1d238c">2.3</a> and applied on the same example in Section <a href="#org0c8f8d2">2.4</a>.
 </p>
 </div>
 
-<div id="outline-container-orgdd836d3" class="outline-3">
-<h3 id="orgdd836d3"><span class="section-number-3">2.1</span> Introduction to Loop Shaping</h3>
+<div id="outline-container-org3f96bef" class="outline-3">
+<h3 id="org3f96bef"><span class="section-number-3">2.1</span> Introduction to Loop Shaping</h3>
 <div class="outline-text-3" id="text-2-1">
 <p>
-<a id="org1108294"></a>
+<a id="orgfa8b378"></a>
 </p>
 
-<div class="definition" id="org25a343b">
+<div class="definition" id="org876ef44">
 <p>
 <b>Loop Shaping</b> refers to a control design procedure that involves explicitly shaping the magnitude of the <b>Loop Transfer Function</b> \(L(s)\).
 </p>
 
 </div>
 
-<div class="definition" id="org67649d4">
+<div class="definition" id="org708e423">
 <p>
-The <b>Loop Gain</b> (or Loop transfer function) \(L(s)\) usually refers to as the product of the controller and the plant (see Figure <a href="#orgf84f71d">6</a>):
+The <b>Loop Gain</b> (or Loop transfer function) \(L(s)\) usually refers to as the product of the controller and the plant (see Figure <a href="#orge4af8ce">6</a>):
 </p>
 \begin{equation}
   L(s) = G(s) \cdot K(s) \label{eq:loop_gain}
@@ -602,7 +603,7 @@ Its name comes from the fact that this is actually the &ldquo;gain around the lo
 </p>
 
 
-<div id="orgf84f71d" class="figure">
+<div id="orge4af8ce" class="figure">
 <p><img src="figs/open_loop_shaping.png" alt="open_loop_shaping.png" />
 </p>
 <p><span class="figure-number">Figure 6: </span>Classical Feedback Architecture</p>
@@ -626,12 +627,12 @@ It is widely used and generally successful as many characteristics of the closed
 <p>
 The shaping of the Loop Gain is done manually by combining several leads, lags, notches&#x2026;
 This process is very much simplified by the fact that the loop gain \(L(s)\) depends <b>linearly</b> on \(K(s)\) \eqref{eq:loop_gain}.
-A typical wanted Loop Shape is shown in Figure <a href="#org88a421a">7</a>.
+A typical wanted Loop Shape is shown in Figure <a href="#org641110f">7</a>.
 Another interesting Loop shape called &ldquo;Bode Step&rdquo; is described in <a href="#citeproc_bib_item_1">[1]</a>.
 </p>
 
 
-<div id="org88a421a" class="figure">
+<div id="org641110f" class="figure">
 <p><img src="figs/open_loop_shaping_shape.png" alt="open_loop_shaping_shape.png" />
 </p>
 <p><span class="figure-number">Figure 7: </span>Typical Wanted Shape for the Loop Gain \(L(s)\)</p>
@@ -640,21 +641,21 @@ Another interesting Loop shape called &ldquo;Bode Step&rdquo; is described in <a
 <p>
 The shaping of <b>closed-loop</b> transfer functions is obviously not as simple as they don&rsquo;t depend linearly on \(K(s)\).
 But this is were the \(\mathcal{H}_\infty\) Synthesis will be useful!
-More details on that in Sections <a href="#orgb800b43">4</a> and <a href="#org19e43fa">5</a>.
+More details on that in Sections <a href="#orgb74964f">4</a> and <a href="#orgf618d2f">5</a>.
 </p>
 </div>
 </div>
 
-<div id="outline-container-org7ec1198" class="outline-3">
-<h3 id="org7ec1198"><span class="section-number-3">2.2</span> Example of Manual Open Loop Shaping</h3>
+<div id="outline-container-orga55eafa" class="outline-3">
+<h3 id="orga55eafa"><span class="section-number-3">2.2</span> Example of Manual Open Loop Shaping</h3>
 <div class="outline-text-3" id="text-2-2">
 <p>
-<a id="orgbe90ac6"></a>
+<a id="orgbc5ea32"></a>
 </p>
 
-<div class="exampl" id="orgf1a99c8">
+<div class="exampl" id="org1894ecc">
 <p>
-Let&rsquo;s take our example system described in Section <a href="#org1f983b6">1.3</a> and design a controller using the Open-Loop shaping synthesis approach.
+Let&rsquo;s take our example system described in Section <a href="#org7ae910e">1.3</a> and design a controller using the Open-Loop shaping synthesis approach.
 The specifications are:
 </p>
 <ol class="org-ol">
@@ -666,7 +667,7 @@ The specifications are:
 
 </div>
 
-<div class="exercice" id="org6af928e">
+<div class="exercice" id="org882ae5c">
 <p>
 Using <code>SISOTOOL</code>, design a controller that fulfills the specifications.
 </p>
@@ -705,7 +706,7 @@ Let&rsquo;s say we came up with the following controller.
 </div>
 
 <p>
-The bode plot of the Loop Gain is shown in Figure <a href="#org60a0147">8</a> and we can verify that we have the wanted stability margins using the <code>margin</code> command:
+The bode plot of the Loop Gain is shown in Figure <a href="#orged16b46">8</a> and we can verify that we have the wanted stability margins using the <code>margin</code> command:
 </p>
 <div class="org-src-container">
 <pre class="src src-matlab">[Gm, Pm, <span class="org-type">~</span>, Wc] = margin(G<span class="org-type">*</span>K)
@@ -745,7 +746,7 @@ The bode plot of the Loop Gain is shown in Figure <a href="#org60a0147">8</a> an
 </table>
 
 
-<div id="org60a0147" class="figure">
+<div id="orged16b46" class="figure">
 <p><img src="figs/loop_gain_manual_afm.png" alt="loop_gain_manual_afm.png" />
 </p>
 <p><span class="figure-number">Figure 8: </span>Bode Plot of the obtained Loop Gain \(L(s) = G(s) K(s)\)</p>
@@ -753,11 +754,11 @@ The bode plot of the Loop Gain is shown in Figure <a href="#org60a0147">8</a> an
 </div>
 </div>
 
-<div id="outline-container-orgb2a458f" class="outline-3">
-<h3 id="orgb2a458f"><span class="section-number-3">2.3</span> \(\mathcal{H}_\infty\) Loop Shaping Synthesis</h3>
+<div id="outline-container-orgaa840e2" class="outline-3">
+<h3 id="orgaa840e2"><span class="section-number-3">2.3</span> \(\mathcal{H}_\infty\) Loop Shaping Synthesis</h3>
 <div class="outline-text-3" id="text-2-3">
 <p>
-<a id="org54bf645"></a>
+<a id="orgf1d238c"></a>
 </p>
 
 <p>
@@ -780,7 +781,7 @@ where:
 <li><code>K</code> is the synthesize controller</li>
 </ul>
 
-<div class="seealso" id="org24e35b1">
+<div class="seealso" id="org3c5dca5">
 <p>
 Matlab documentation of <code>loopsyn</code> (<a href="https://www.mathworks.com/help/robust/ref/loopsyn.html">link</a>).
 </p>
@@ -797,23 +798,23 @@ Even though we will not go into details and explain how such synthesis is workin
 </div>
 </div>
 
-<div id="outline-container-orgf390eb2" class="outline-3">
-<h3 id="orgf390eb2"><span class="section-number-3">2.4</span> Example of the \(\mathcal{H}_\infty\) Loop Shaping Synthesis</h3>
+<div id="outline-container-org6e8b6e8" class="outline-3">
+<h3 id="org6e8b6e8"><span class="section-number-3">2.4</span> Example of the \(\mathcal{H}_\infty\) Loop Shaping Synthesis</h3>
 <div class="outline-text-3" id="text-2-4">
 <p>
-<a id="org8c3037b"></a>
+<a id="org0c8f8d2"></a>
 </p>
 
 <p>
 To apply the \(\mathcal{H}_\infty\) Loop Shaping Synthesis, the wanted shape of the loop gain should be determined from the specifications.
-This is summarized in Table <a href="#orgba6d313">3</a>.
+This is summarized in Table <a href="#orge0fa26a">3</a>.
 </p>
 
 <p>
-Such shape corresponds to the typical wanted Loop gain Shape shown in Figure <a href="#org88a421a">7</a>.
+Such shape corresponds to the typical wanted Loop gain Shape shown in Figure <a href="#org641110f">7</a>.
 </p>
 
-<table id="orgba6d313" border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
+<table id="orge0fa26a" border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
 <caption class="t-above"><span class="table-number">Table 3:</span> Wanted Loop Shape corresponding to each specification</caption>
 
 <colgroup>
@@ -882,18 +883,18 @@ The \(\mathcal{H}_\infty\) open loop shaping synthesis is then performed using t
 </div>
 
 <p>
-The obtained Loop Gain is shown in Figure <a href="#orgab9fab2">9</a> and matches the specified one by a factor \(\gamma \approx 2\).
+The obtained Loop Gain is shown in Figure <a href="#org072658b">9</a> and matches the specified one by a factor \(\gamma \approx 2\).
 </p>
 
 
-<div id="orgab9fab2" class="figure">
+<div id="org072658b" class="figure">
 <p><img src="figs/open_loop_shaping_hinf_L.png" alt="open_loop_shaping_hinf_L.png" />
 </p>
 <p><span class="figure-number">Figure 9: </span>Obtained Open Loop Gain \(L(s) = G(s) K(s)\) and comparison with the wanted Loop gain \(L_w\)</p>
 </div>
 
 
-<div class="important" id="org14d61b6">
+<div class="important" id="org9a13893">
 <p>
 When using the \(\mathcal{H}_\infty\) Synthesis, it is usually recommended to analyze the obtained controller.
 </p>
@@ -905,7 +906,7 @@ This is usually done by breaking down the controller into simple elements such a
 </div>
 
 <p>
-Let&rsquo;s briefly analyze the obtained controller which bode plot is shown in Figure <a href="#org41aaa1a">10</a>:
+Let&rsquo;s briefly analyze the obtained controller which bode plot is shown in Figure <a href="#orgc38d60f">10</a>:
 </p>
 <ul class="org-ul">
 <li>two integrators are used at low frequency to have the wanted low frequency high gain</li>
@@ -914,7 +915,7 @@ Let&rsquo;s briefly analyze the obtained controller which bode plot is shown in
 </ul>
 
 
-<div id="org41aaa1a" class="figure">
+<div id="orgc38d60f" class="figure">
 <p><img src="figs/open_loop_shaping_hinf_K.png" alt="open_loop_shaping_hinf_K.png" />
 </p>
 <p><span class="figure-number">Figure 10: </span>Obtained controller \(K\) using the open-loop \(\mathcal{H}_\infty\) shaping</p>
@@ -922,10 +923,10 @@ Let&rsquo;s briefly analyze the obtained controller which bode plot is shown in
 
 
 <p>
-Let&rsquo;s finally compare the obtained stability margins of the \(\mathcal{H}_\infty\) controller and of the manually developed controller in Table <a href="#org851641e">4</a>.
+Let&rsquo;s finally compare the obtained stability margins of the \(\mathcal{H}_\infty\) controller and of the manually developed controller in Table <a href="#orgec6a8bd">4</a>.
 </p>
 
-<table id="org851641e" border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
+<table id="orgec6a8bd" border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
 <caption class="t-above"><span class="table-number">Table 4:</span> Comparison of the characteristics obtained with the two methods</caption>
 
 <colgroup>
@@ -966,38 +967,38 @@ Let&rsquo;s finally compare the obtained stability margins of the \(\mathcal{H}_
 </div>
 </div>
 
-<div id="outline-container-org214059b" class="outline-2">
-<h2 id="org214059b"><span class="section-number-2">3</span> A first Step into the \(\mathcal{H}_\infty\) world</h2>
+<div id="outline-container-orgac445c2" class="outline-2">
+<h2 id="orgac445c2"><span class="section-number-2">3</span> A first Step into the \(\mathcal{H}_\infty\) world</h2>
 <div class="outline-text-2" id="text-3">
 <p>
-<a id="org4bf732f"></a>
+<a id="org252b5a9"></a>
 </p>
 <p>
 In this section, the \(\mathcal{H}_\infty\) Synthesis method, which is based on the optimization of the \(\mathcal{H}_\infty\) norm of transfer functions, is introduced.
 </p>
 
 <p>
-After the \(\mathcal{H}_\infty\) norm is defined in Section <a href="#orgc25050f">3.1</a>, the \(\mathcal{H}_\infty\) synthesis procedure is described in Section <a href="#org3172a68">3.2</a> .
+After the \(\mathcal{H}_\infty\) norm is defined in Section <a href="#org6448a1d">3.1</a>, the \(\mathcal{H}_\infty\) synthesis procedure is described in Section <a href="#orgb18bcd4">3.2</a> .
 </p>
 
 <p>
-The generalized plant, a very useful tool to describe a control problem, is presented in Section <a href="#orgf28bf1e">3.3</a>.
-The \(\mathcal{H}_\infty\) is then applied to this generalized plant in Section <a href="#orga117abd">3.4</a>.
+The generalized plant, a very useful tool to describe a control problem, is presented in Section <a href="#orgfabc275">3.3</a>.
+The \(\mathcal{H}_\infty\) is then applied to this generalized plant in Section <a href="#org5bd05eb">3.4</a>.
 </p>
 
 <p>
-Finally, an example showing how to convert a typical feedback control architecture into a generalized plant is given in Section <a href="#org2a6e144">3.5</a>.
+Finally, an example showing how to convert a typical feedback control architecture into a generalized plant is given in Section <a href="#orgce041bb">3.5</a>.
 </p>
 </div>
 
-<div id="outline-container-org4f9dc47" class="outline-3">
-<h3 id="org4f9dc47"><span class="section-number-3">3.1</span> The \(\mathcal{H}_\infty\) Norm</h3>
+<div id="outline-container-org29b6052" class="outline-3">
+<h3 id="org29b6052"><span class="section-number-3">3.1</span> The \(\mathcal{H}_\infty\) Norm</h3>
 <div class="outline-text-3" id="text-3-1">
 <p>
-<a id="orgc25050f"></a>
+<a id="org6448a1d"></a>
 </p>
 
-<div class="definition" id="org2b3a0f3">
+<div class="definition" id="org5625fc9">
 <p>
 The \(\mathcal{H}_\infty\) norm of a multi-input multi-output system \(G(s)\) is defined as the peak of the maximum singular value of its frequency response
 </p>
@@ -1014,7 +1015,7 @@ For a single-input single-output system \(G(s)\), it is simply the peak value of
 
 </div>
 
-<div class="exampl" id="org4b679b4">
+<div class="exampl" id="orgfa10a58">
 <p>
 Let&rsquo;s compute the \(\mathcal{H}_\infty\) norm of our test plant \(G(s)\) using the <code>hinfnorm</code> function:
 </p>
@@ -1029,11 +1030,11 @@ Let&rsquo;s compute the \(\mathcal{H}_\infty\) norm of our test plant \(G(s)\) u
 
 
 <p>
-We can see in Figure <a href="#org2d61fed">11</a> that indeed, the \(\mathcal{H}_\infty\) norm of \(G(s)\) does corresponds to the peak value of \(|G(j\omega)|\).
+We can see in Figure <a href="#org91f6418">11</a> that indeed, the \(\mathcal{H}_\infty\) norm of \(G(s)\) does corresponds to the peak value of \(|G(j\omega)|\).
 </p>
 
 
-<div id="org2d61fed" class="figure">
+<div id="org91f6418" class="figure">
 <p><img src="figs/hinfinity_norm_siso_bode.png" alt="hinfinity_norm_siso_bode.png" />
 </p>
 <p><span class="figure-number">Figure 11: </span>Example of the \(\mathcal{H}_\infty\) norm of a SISO system</p>
@@ -1043,14 +1044,14 @@ We can see in Figure <a href="#org2d61fed">11</a> that indeed, the \(\mathcal{H}
 </div>
 </div>
 
-<div id="outline-container-org260f673" class="outline-3">
-<h3 id="org260f673"><span class="section-number-3">3.2</span> \(\mathcal{H}_\infty\) Synthesis</h3>
+<div id="outline-container-orga7a5510" class="outline-3">
+<h3 id="orga7a5510"><span class="section-number-3">3.2</span> \(\mathcal{H}_\infty\) Synthesis</h3>
 <div class="outline-text-3" id="text-3-2">
 <p>
-<a id="org3172a68"></a>
+<a id="orgb18bcd4"></a>
 </p>
 
-<div class="definition" id="orgb4b4cdc">
+<div class="definition" id="orgb221a1f">
 <p>
 The \(\mathcal{H}_\infty\) synthesis is a method that uses an <b>algorithm</b> (LMI optimization, Riccati equation) to find a controller that stabilizes the system and that <b>minimizes</b> the \(\mathcal{H}_\infty\) norms of defined transfer functions.
 </p>
@@ -1058,16 +1059,16 @@ The \(\mathcal{H}_\infty\) synthesis is a method that uses an <b>algorithm</b> (
 </div>
 
 <p>
-Why optimizing the \(\mathcal{H}_\infty\) norm of transfer functions is a pertinent choice will become clear when we will translate the typical control specifications into the \(\mathcal{H}_\infty\) norm of transfer functions in Section <a href="#orgb800b43">4</a>.
+Why optimizing the \(\mathcal{H}_\infty\) norm of transfer functions is a pertinent choice will become clear when we will translate the typical control specifications into the \(\mathcal{H}_\infty\) norm of transfer functions in Section <a href="#orgb74964f">4</a>.
 </p>
 
-<div class="important" id="orgc07fd7a">
+<div class="important" id="org1c210f4">
 <p>
 Then applying the \(\mathcal{H}_\infty\) synthesis to a plant, the engineer work usually consists of the following steps:
 </p>
 <ol class="org-ol">
 <li>Write the problem as standard \(\mathcal{H}_\infty\) problem using the generalized plant (described in the next section)</li>
-<li>Translate the specifications as \(\mathcal{H}_\infty\) norms of transfer functions (Section <a href="#orgb800b43">4</a>)</li>
+<li>Translate the specifications as \(\mathcal{H}_\infty\) norms of transfer functions (Section <a href="#orgb74964f">4</a>)</li>
 <li>Make the synthesis and analyze the obtained controller</li>
 </ol>
 
@@ -1090,11 +1091,11 @@ Note that there are many ways to use the \(\mathcal{H}_\infty\) Synthesis:
 </div>
 </div>
 
-<div id="outline-container-org71b47ed" class="outline-3">
-<h3 id="org71b47ed"><span class="section-number-3">3.3</span> The Generalized Plant</h3>
+<div id="outline-container-org810d672" class="outline-3">
+<h3 id="org810d672"><span class="section-number-3">3.3</span> The Generalized Plant</h3>
 <div class="outline-text-3" id="text-3-3">
 <p>
-<a id="orgf28bf1e"></a>
+<a id="orgfabc275"></a>
 </p>
 
 <p>
@@ -1103,7 +1104,7 @@ This consist of deriving the <b>Generalized Plant</b> for the current problem.
 </p>
 
 <p>
-The generalized plant, usually noted \(P(s)\), is shown in Figure <a href="#org899a647">12</a>.
+The generalized plant, usually noted \(P(s)\), is shown in Figure <a href="#org7bb4b83">12</a>.
 It has two <i>sets</i> of inputs \([w,\,u]\) and two <i>sets</i> of outputs \([z\,v]\) such that:
 </p>
 \begin{equation}
@@ -1111,22 +1112,22 @@ It has two <i>sets</i> of inputs \([w,\,u]\) and two <i>sets</i> of outputs \([z
 \end{equation}
 
 <p>
-The meaning of these inputs and outputs are summarized in Table <a href="#org4b4500e">5</a>.
+The meaning of these inputs and outputs are summarized in Table <a href="#orgfa6c458">5</a>.
 </p>
 
 <p>
-A practical example about how to derive the generalized plant for a classical control problem is given in Section <a href="#org2a6e144">3.5</a>.
+A practical example about how to derive the generalized plant for a classical control problem is given in Section <a href="#orgce041bb">3.5</a>.
 </p>
 
-<div class="important" id="orgd6ce6f0">
+<div class="important" id="orge299357">
 
-<div id="org899a647" class="figure">
+<div id="org7bb4b83" class="figure">
 <p><img src="figs/general_plant.png" alt="general_plant.png" />
 </p>
 <p><span class="figure-number">Figure 12: </span>Inputs and Outputs of the generalized Plant</p>
 </div>
 
-<table id="org4b4500e" border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
+<table id="orgfa6c458" border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
 <caption class="t-above"><span class="table-number">Table 5:</span> Notations for the general configuration</caption>
 
 <colgroup>
@@ -1172,18 +1173,18 @@ A practical example about how to derive the generalized plant for a classical co
 </div>
 </div>
 
-<div id="outline-container-orgc597f47" class="outline-3">
-<h3 id="orgc597f47"><span class="section-number-3">3.4</span> The \(\mathcal{H}_\infty\) Synthesis applied on the Generalized plant</h3>
+<div id="outline-container-org5cbe4b7" class="outline-3">
+<h3 id="org5cbe4b7"><span class="section-number-3">3.4</span> The \(\mathcal{H}_\infty\) Synthesis applied on the Generalized plant</h3>
 <div class="outline-text-3" id="text-3-4">
 <p>
-<a id="orga117abd"></a>
+<a id="org5bd05eb"></a>
 </p>
 
 <p>
 Once the generalized plant is obtained, the \(\mathcal{H}_\infty\) synthesis problem can be stated as follows:
 </p>
 
-<div class="important" id="orge33b7f4">
+<div class="important" id="org299bcfb">
 <dl class="org-dl">
 <dt>\(\mathcal{H}_\infty\) Synthesis applied on the generalized plant</dt><dd></dd>
 </dl>
@@ -1196,11 +1197,11 @@ After \(K\) is found, the system is <i>robustified</i> by adjusting the response
 </p>
 
 <p>
-The obtained controller \(K\) and the generalized plant are connected as shown in Figure <a href="#orgcd7f15a">13</a>.
+The obtained controller \(K\) and the generalized plant are connected as shown in Figure <a href="#org58a89af">13</a>.
 </p>
 
 
-<div id="orgcd7f15a" class="figure">
+<div id="org58a89af" class="figure">
 <p><img src="figs/general_control_names.png" alt="general_control_names.png" />
 </p>
 <p><span class="figure-number">Figure 13: </span>General Control Configuration</p>
@@ -1226,29 +1227,29 @@ where:
 </ul>
 
 <p>
-Note that the general control configure of Figure <a href="#orgcd7f15a">13</a>, as its name implies, is quite <i>general</i> and can represent feedback control as well as feedforward control architectures.
+Note that the general control configure of Figure <a href="#org58a89af">13</a>, as its name implies, is quite <i>general</i> and can represent feedback control as well as feedforward control architectures.
 </p>
 </div>
 </div>
 
-<div id="outline-container-orgd735b55" class="outline-3">
-<h3 id="orgd735b55"><span class="section-number-3">3.5</span> From a Classical Feedback Architecture to a Generalized Plant</h3>
+<div id="outline-container-orga713c1e" class="outline-3">
+<h3 id="orga713c1e"><span class="section-number-3">3.5</span> From a Classical Feedback Architecture to a Generalized Plant</h3>
 <div class="outline-text-3" id="text-3-5">
 <p>
-<a id="org2a6e144"></a>
+<a id="orgce041bb"></a>
 </p>
 
 <p>
-The procedure to convert a typical control architecture as the one shown in Figure <a href="#org121846f">14</a> to a generalized Plant is as follows:
+The procedure to convert a typical control architecture as the one shown in Figure <a href="#orge5a6300">14</a> to a generalized Plant is as follows:
 </p>
 <ol class="org-ol">
 <li>Define signals of the generalized plant: \(w\), \(z\), \(u\) and \(v\)</li>
-<li>Remove \(K\) and rearrange the inputs and outputs to match the generalized configuration shown in Figure <a href="#org899a647">12</a></li>
+<li>Remove \(K\) and rearrange the inputs and outputs to match the generalized configuration shown in Figure <a href="#org7bb4b83">12</a></li>
 </ol>
 
-<div class="exercice" id="orgbc619a7">
+<div class="exercice" id="orgdd29bd1">
 <p>
-Consider the feedback control architecture shown in Figure <a href="#org121846f">14</a>.
+Consider the feedback control architecture shown in Figure <a href="#orge5a6300">14</a>.
 Suppose we want to design \(K\) using the general \(\mathcal{H}_\infty\) synthesis, and suppose the signals to be minimized are the control input \(u\) and the tracking error \(\epsilon\).
 </p>
 <ol class="org-ol">
@@ -1257,7 +1258,7 @@ Suppose we want to design \(K\) using the general \(\mathcal{H}_\infty\) synthes
 </ol>
 
 
-<div id="org121846f" class="figure">
+<div id="orge5a6300" class="figure">
 <p><img src="figs/classical_feedback_tracking.png" alt="classical_feedback_tracking.png" />
 </p>
 <p><span class="figure-number">Figure 14: </span>Classical Feedback Control Architecture (Tracking)</p>
@@ -1276,17 +1277,17 @@ Usually, we want to minimize the tracking errors \(\epsilon\) and the control si
 </ul>
 
 <p>
-Then, Remove \(K\) and rearrange the inputs and outputs as in Figure <a href="#org899a647">12</a>.
+Then, Remove \(K\) and rearrange the inputs and outputs as in Figure <a href="#org7bb4b83">12</a>.
 </p>
 </details>
 
 <details><summary>Answer</summary>
 <p>
-The obtained generalized plant shown in Figure <a href="#orgcdeabb7">15</a>.
+The obtained generalized plant shown in Figure <a href="#orgd51dc13">15</a>.
 </p>
 
 
-<div id="orgcdeabb7" class="figure">
+<div id="orgd51dc13" class="figure">
 <p><img src="figs/mixed_sensitivity_ref_tracking.png" alt="mixed_sensitivity_ref_tracking.png" />
 </p>
 <p><span class="figure-number">Figure 15: </span>Generalized plant of the Classical Feedback Control Architecture (Tracking)</p>
@@ -1310,21 +1311,21 @@ P.OutputName = {<span class="org-string">'e'</span>, <span class="org-string">'u
 </div>
 </div>
 
-<div id="outline-container-orgea992d5" class="outline-2">
-<h2 id="orgea992d5"><span class="section-number-2">4</span> Modern Interpretation of Control Specifications</h2>
+<div id="outline-container-org96ba1fd" class="outline-2">
+<h2 id="org96ba1fd"><span class="section-number-2">4</span> Modern Interpretation of Control Specifications</h2>
 <div class="outline-text-2" id="text-4">
 <p>
-<a id="orgb800b43"></a>
+<a id="orgb74964f"></a>
 </p>
 <ul class="org-ul">
-<li>Section <a href="#org6733abe">4.1</a></li>
-<li>Section <a href="#orgb382bc9">4.2</a></li>
-<li>Section <a href="#org7528a51">4.3</a></li>
-<li>Section <a href="#org264ea4e">4.4</a></li>
+<li>Section <a href="#orgf702768">4.1</a></li>
+<li>Section <a href="#orgf334105">4.2</a></li>
+<li>Section <a href="#org021b969">4.3</a></li>
+<li>Section <a href="#orgae4d505">4.4</a></li>
 </ul>
 
 <p>
-As shown in Section <a href="#orga360568">2</a>, the loop gain \(L(s) = G(s) K(s)\) is a useful and easy tool when manually designing controllers.
+As shown in Section <a href="#orge2537e5">2</a>, the loop gain \(L(s) = G(s) K(s)\) is a useful and easy tool when manually designing controllers.
 This is mainly due to the fact that \(L(s)\) is very easy to shape as it depends <i>linearly</i> on \(K(s)\).
 Moreover, important quantities such as the stability margins and the control bandwidth can be estimated from the shape/phase of \(L(s)\).
 </p>
@@ -1334,11 +1335,11 @@ However, the loop gain \(L(s)\) does <b>not</b> directly give the performances o
 </p>
 
 <p>
-If we consider the feedback system shown in Figure <a href="#org979796b">16</a>, we can link to the following specifications to closed-loop transfer functions.
-This is summarized in Table <a href="#org958c1ae">6</a>.
+If we consider the feedback system shown in Figure <a href="#orgb6d4090">16</a>, we can link to the following specifications to closed-loop transfer functions.
+This is summarized in Table <a href="#orgd678c6d">6</a>.
 </p>
 
-<table id="org958c1ae" border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
+<table id="orgd678c6d" border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
 <caption class="t-above"><span class="table-number">Table 6:</span> Typical Specification and associated closed-loop transfer function</caption>
 
 <colgroup>
@@ -1380,33 +1381,33 @@ This is summarized in Table <a href="#org958c1ae">6</a>.
 
 <tr>
 <td class="org-left">Robustness (stability margins)</td>
-<td class="org-left">Module margin (see Section <a href="#org7528a51">4.3</a>)</td>
+<td class="org-left">Module margin (see Section <a href="#org021b969">4.3</a>)</td>
 </tr>
 </tbody>
 </table>
 
 
-<div id="org979796b" class="figure">
+<div id="orgb6d4090" class="figure">
 <p><img src="figs/gang_of_four_feedback.png" alt="gang_of_four_feedback.png" />
 </p>
 <p><span class="figure-number">Figure 16: </span>Simple Feedback Architecture</p>
 </div>
 </div>
 
-<div id="outline-container-org9916c4e" class="outline-3">
-<h3 id="org9916c4e"><span class="section-number-3">4.1</span> Closed Loop Transfer Functions</h3>
+<div id="outline-container-org9d2f211" class="outline-3">
+<h3 id="org9d2f211"><span class="section-number-3">4.1</span> Closed Loop Transfer Functions</h3>
 <div class="outline-text-3" id="text-4-1">
 <p>
-<a id="org6733abe"></a>
+<a id="orgf702768"></a>
 </p>
 
 <p>
 As the performances of a controlled system depend on the <b>closed</b> loop transfer functions, it is very important to derive these closed-loop transfer functions as a function of the plant \(G(s)\) and controller \(K(s)\).
 </p>
 
-<div class="exercice" id="orgf463be6">
+<div class="exercice" id="orga698290">
 <p>
-Write the output signals \([\epsilon, u, y]\) as a function of the systems \(K(s), G(s)\) and of the input signals \([r, d, n]\) as shown in Figure <a href="#org979796b">16</a>.
+Write the output signals \([\epsilon, u, y]\) as a function of the systems \(K(s), G(s)\) and of the input signals \([r, d, n]\) as shown in Figure <a href="#orgb6d4090">16</a>.
 </p>
 
 <details><summary>Hint</summary>
@@ -1443,9 +1444,9 @@ The following equations should be obtained:
 
 </div>
 
-<div class="important" id="org4c56973">
+<div class="important" id="org179dc2a">
 <p>
-We can see that they are 4 different transfer functions describing the behavior of the system in Figure <a href="#org979796b">16</a>.
+We can see that they are 4 different transfer functions describing the behavior of the system in Figure <a href="#orgb6d4090">16</a>.
 These called the <b>Gang of Four</b>:
 </p>
 \begin{align}
@@ -1457,7 +1458,7 @@ These called the <b>Gang of Four</b>:
 
 </div>
 
-<div class="seealso" id="org42a539e">
+<div class="seealso" id="orge6b7c7d">
 <p>
 If a feedforward controller is included, a <b>Gang of Six</b> transfer functions can be defined.
 More on that in this <a href="https://www.youtube.com/watch?v=b_8v8scghh8">short video</a>.
@@ -1481,11 +1482,11 @@ Similarly, to reduce the effect of measurement noise \(n\) on the output \(y\),
 </div>
 </div>
 
-<div id="outline-container-org9653453" class="outline-3">
-<h3 id="org9653453"><span class="section-number-3">4.2</span> Sensitivity Function</h3>
+<div id="outline-container-org340b845" class="outline-3">
+<h3 id="org340b845"><span class="section-number-3">4.2</span> Sensitivity Function</h3>
 <div class="outline-text-3" id="text-4-2">
 <p>
-<a id="orgb382bc9"></a>
+<a id="orgf334105"></a>
 </p>
 
 <p>
@@ -1496,14 +1497,14 @@ In this section, we will see how the shape of the sensitivity function will impa
 
 <p>
 Suppose we have developed a &ldquo;<i>reference</i>&rdquo; controller \(K_r(s)\) and made three small changes to obtained three controllers \(K_1(s)\), \(K_2(s)\) and \(K_3(s)\).
-The obtained sensitivity functions are shown in Figure <a href="#orgd3fa40e">17</a> and the corresponding step responses are shown in Figure <a href="#org25e45b1">18</a>.
+The obtained sensitivity functions are shown in Figure <a href="#org88e12a6">17</a> and the corresponding step responses are shown in Figure <a href="#org54bde8f">18</a>.
 </p>
 
 <p>
-The comparison of the sensitivity functions shapes and their effect on the step response is summarized in Table <a href="#org88583bb">7</a>.
+The comparison of the sensitivity functions shapes and their effect on the step response is summarized in Table <a href="#orgbab6bd8">7</a>.
 </p>
 
-<table id="org88583bb" border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
+<table id="orgbab6bd8" border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
 <caption class="t-above"><span class="table-number">Table 7:</span> Comparison of the sensitivity function shape and the corresponding step response for the three controller variations</caption>
 
 <colgroup>
@@ -1542,20 +1543,20 @@ The comparison of the sensitivity functions shapes and their effect on the step
 </table>
 
 
-<div id="orgd3fa40e" class="figure">
+<div id="org88e12a6" class="figure">
 <p><img src="figs/sensitivity_shape_effect.png" alt="sensitivity_shape_effect.png" />
 </p>
 <p><span class="figure-number">Figure 17: </span>Sensitivity function magnitude \(|S(j\omega)|\) corresponding to the reference controller \(K_r(s)\) and the three modified controllers \(K_i(s)\)</p>
 </div>
 
 
-<div id="org25e45b1" class="figure">
+<div id="org54bde8f" class="figure">
 <p><img src="figs/sensitivity_shape_effect_step.png" alt="sensitivity_shape_effect_step.png" />
 </p>
 <p><span class="figure-number">Figure 18: </span>Step response (response from \(r\) to \(y\)) for the different controllers</p>
 </div>
 
-<div class="definition" id="org61c2dcc">
+<div class="definition" id="orgf2a5a4c">
 <dl class="org-dl">
 <dt>Closed-Loop Bandwidth</dt><dd><p>
 The closed-loop bandwidth \(\omega_b\) is the frequency where \(|S(j\omega)|\) first crosses \(1/\sqrt{2} = -3dB\) from below.
@@ -1568,9 +1569,9 @@ In general, a large bandwidth corresponds to a faster rise time.
 
 </div>
 
-<div class="important" id="org4d9835e">
+<div class="important" id="org4a21201">
 <p>
-From the simple analysis above, we can draw a first estimation of the wanted shape for the sensitivity function in Figure <a href="#orgd10d428">19</a>.
+From the simple analysis above, we can draw a first estimation of the wanted shape for the sensitivity function in Figure <a href="#org6cfe20b">19</a>.
 </p>
 
 <p>
@@ -1585,7 +1586,7 @@ This will become clear in the next section about the <b>module margin</b>.</li>
 </ul>
 
 
-<div id="orgd10d428" class="figure">
+<div id="org6cfe20b" class="figure">
 <p><img src="figs/h-infinity-spec-S.png" alt="h-infinity-spec-S.png" />
 </p>
 <p><span class="figure-number">Figure 19: </span>Typical wanted shape of the Sensitivity transfer function</p>
@@ -1595,18 +1596,18 @@ This will become clear in the next section about the <b>module margin</b>.</li>
 </div>
 </div>
 
-<div id="outline-container-org5ee7596" class="outline-3">
-<h3 id="org5ee7596"><span class="section-number-3">4.3</span> Robustness: Module Margin</h3>
+<div id="outline-container-org3040671" class="outline-3">
+<h3 id="org3040671"><span class="section-number-3">4.3</span> Robustness: Module Margin</h3>
 <div class="outline-text-3" id="text-4-3">
 <p>
-<a id="org7528a51"></a>
+<a id="org021b969"></a>
 </p>
 
 <p>
 Let&rsquo;s start by an example demonstrating why the phase and gain margins might not be good indicators of robustness.
 </p>
 
-<div class="exampl" id="org12f4616">
+<div class="exampl" id="org2744e10">
 <p>
 Let&rsquo;s consider the following plant \(G_t(s)\):
 </p>
@@ -1620,7 +1621,7 @@ Gt = 1<span class="org-type">/</span>k<span class="org-type">*</span>(s<span cla
 </div>
 
 <p>
-Let&rsquo;s say we have designed a controller \(K_t(s)\) that gives the loop gain shown in Figure <a href="#orgcb10b9c">20</a>.
+Let&rsquo;s say we have designed a controller \(K_t(s)\) that gives the loop gain shown in Figure <a href="#org65c5d85">20</a>.
 </p>
 
 <div class="org-src-container">
@@ -1629,7 +1630,7 @@ Let&rsquo;s say we have designed a controller \(K_t(s)\) that gives the loop gai
 </div>
 
 <p>
-The following characteristics can be determined from Figure <a href="#orgcb10b9c">20</a>:
+The following characteristics can be determined from Figure <a href="#org65c5d85">20</a>:
 </p>
 <ul class="org-ul">
 <li>bandwidth of \(\approx 10\, \text{Hz}\)</li>
@@ -1642,7 +1643,7 @@ This might indicate very good robustness properties of the closed-loop system.
 </p>
 
 
-<div id="orgcb10b9c" class="figure">
+<div id="org65c5d85" class="figure">
 <p><img src="figs/phase_gain_margin_model_plant.png" alt="phase_gain_margin_model_plant.png" />
 </p>
 <p><span class="figure-number">Figure 20: </span>Bode plot of the obtained Loop Gain \(L(s)\)</p>
@@ -1658,7 +1659,7 @@ Now let&rsquo;s suppose the &ldquo;real&rdquo; plant \(G_r(s)\) as a slightly lo
 </div>
 
 <p>
-The obtained &ldquo;real&rdquo; loop gain is shown in Figure <a href="#org59883d5">21</a>.
+The obtained &ldquo;real&rdquo; loop gain is shown in Figure <a href="#org0764509">21</a>.
 At a frequency little bit above 100Hz, the phase of the loop gain reaches -180 degrees while its magnitude is more than one which indicated instability.
 </p>
 
@@ -1676,7 +1677,7 @@ It is confirmed by checking the stability of the closed loop system:
 
 
 
-<div id="org59883d5" class="figure">
+<div id="org0764509" class="figure">
 <p><img src="figs/phase_gain_margin_real_plant.png" alt="phase_gain_margin_real_plant.png" />
 </p>
 <p><span class="figure-number">Figure 21: </span>Bode plots of \(L(s)\) (loop gain corresponding the nominal plant) and \(L_r(s)\) (loop gain corresponding to the real plant)</p>
@@ -1696,7 +1697,7 @@ This is due to the fact that the gain and phase margin are robustness indicators
 Let&rsquo;s now determine a new robustness indicator based on the Nyquist Stability Criteria.
 </p>
 
-<div class="definition" id="orgcc68e06">
+<div class="definition" id="org2000f74">
 <dl class="org-dl">
 <dt>Nyquist Stability Criteria (for stable systems)</dt><dd>If the open-loop transfer function \(L(s)\) is stable, then the closed-loop system is unstable for any encirclement of the point \(−1\) on the Nyquist plot.</dd>
 
@@ -1705,7 +1706,7 @@ Let&rsquo;s now determine a new robustness indicator based on the Nyquist Stabil
 
 </div>
 
-<div class="seealso" id="orgb4677ae">
+<div class="seealso" id="org57f5161">
 <p>
 For more information about the <i>general</i> Nyquist Stability Criteria, you may want to look at <a href="https://www.youtube.com/watch?v=sof3meN96MA">this</a> video.
 </p>
@@ -1717,21 +1718,21 @@ From the Nyquist stability criteria, it is clear that we want \(L(j\omega)\) to
 This minimum distance is called the <b>module margin</b>.
 </p>
 
-<div class="definition" id="orgb850be1">
+<div class="definition" id="org0b46fbb">
 <dl class="org-dl">
 <dt>Module Margin</dt><dd>The Module Margin \(\Delta M\) is defined as the <b>minimum distance</b> between the point \(-1\) and the loop gain \(L(j\omega)\) in the complex plane.</dd>
 </dl>
 
 </div>
 
-<div class="exampl" id="org3f5f2ac">
+<div class="exampl" id="orgad55ef4">
 <p>
-A typical Nyquist plot is shown in Figure <a href="#org18e423b">22</a>.
+A typical Nyquist plot is shown in Figure <a href="#org2e9b339">22</a>.
 The gain, phase and module margins are graphically shown to have an idea of what they represent.
 </p>
 
 
-<div id="org18e423b" class="figure">
+<div id="org2e9b339" class="figure">
 <p><img src="figs/module_margin_example.png" alt="module_margin_example.png" />
 </p>
 <p><span class="figure-number">Figure 22: </span>Nyquist plot with visual indication of the Gain margin \(\Delta G\), Phase margin \(\Delta \phi\) and Module margin \(\Delta M\)</p>
@@ -1740,7 +1741,7 @@ The gain, phase and module margins are graphically shown to have an idea of what
 </div>
 
 <p>
-As expected from Figure <a href="#org18e423b">22</a>, there is a close relationship between the module margin and the gain and phase margins.
+As expected from Figure <a href="#org2e9b339">22</a>, there is a close relationship between the module margin and the gain and phase margins.
 We can indeed show that for a given value of the module margin \(\Delta M\), we have:
 </p>
 \begin{equation}
@@ -1759,7 +1760,7 @@ Let&rsquo;s now try to express the Module margin \(\Delta M\) as an \(\mathcal{H
          &= \frac{1}{\|S\|_\infty}
 \end{align*}
 
-<div class="important" id="org29883cb">
+<div class="important" id="orge696804">
 <p>
 The \(\mathcal{H}_\infty\) norm of the sensitivity function \(\|S\|_\infty\) is a measure of the Module margin \(\Delta M\) and therefore an indicator of the system robustness.
 </p>
@@ -1779,14 +1780,14 @@ Note that this is why large peak value of \(|S(j\omega)|\) usually indicate robu
 And we know understand why setting an upper bound on the magnitude of \(S\) is generally a good idea.
 </p>
 
-<div class="exampl" id="org07b0353">
+<div class="exampl" id="org9b652e8">
 <p>
 Typical, we require \(\|S\|_\infty < 2 (6dB)\) which implies \(\Delta G \ge 2\) and \(\Delta \phi \ge 29^o\)
 </p>
 
 </div>
 
-<div class="seealso" id="orgd014332">
+<div class="seealso" id="org119dd41">
 <p>
 To learn more about module/disk margin, you can check out <a href="https://www.youtube.com/watch?v=XazdN6eZF80">this</a> video.
 </p>
@@ -1795,11 +1796,11 @@ To learn more about module/disk margin, you can check out <a href="https://www.y
 </div>
 </div>
 
-<div id="outline-container-orgb77093f" class="outline-3">
-<h3 id="orgb77093f"><span class="section-number-3">4.4</span> Other Requirements</h3>
+<div id="outline-container-orge8044fe" class="outline-3">
+<h3 id="orge8044fe"><span class="section-number-3">4.4</span> Other Requirements</h3>
 <div class="outline-text-3" id="text-4-4">
 <p>
-<a id="org264ea4e"></a>
+<a id="orgae4d505"></a>
 </p>
 
 <p>
@@ -1828,7 +1829,7 @@ with:
 Noise Attenuation: typical wanted shape for \(T\)
 </p>
 
-<table id="orgfa3589c" border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
+<table id="orgb94a744" border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
 <caption class="t-above"><span class="table-number">Table 8:</span> Typical Specifications and corresponding wanted norms of open and closed loop tansfer functions</caption>
 
 <colgroup>
@@ -1881,11 +1882,11 @@ Noise Attenuation: typical wanted shape for \(T\)
 </div>
 </div>
 
-<div id="outline-container-org5613659" class="outline-2">
-<h2 id="org5613659"><span class="section-number-2">5</span> \(\mathcal{H}_\infty\) Shaping of closed-loop transfer functions</h2>
+<div id="outline-container-orgbc1be70" class="outline-2">
+<h2 id="orgbc1be70"><span class="section-number-2">5</span> \(\mathcal{H}_\infty\) Shaping of closed-loop transfer functions</h2>
 <div class="outline-text-2" id="text-5">
 <p>
-<a id="org19e43fa"></a>
+<a id="orgf618d2f"></a>
 </p>
 <p>
 In the previous sections, we have seen that the performances of the system depends on the <b>shape</b> of the closed-loop transfer function.
@@ -1902,22 +1903,22 @@ But don&rsquo;t worry, the \(\mathcal{H}_\infty\) synthesis will do this job for
 
 <p>
 This
-Section <a href="#org128c370">5.1</a>
-Section <a href="#orgfd9d7c1">5.2</a>
-Section <a href="#org5776302">5.3</a>
-Section <a href="#org3426451">5.4</a>
+Section <a href="#orgde38648">5.1</a>
+Section <a href="#org066dedd">5.2</a>
+Section <a href="#org4adc4a9">5.3</a>
+Section <a href="#org377446a">5.4</a>
 </p>
 </div>
 
-<div id="outline-container-org0331165" class="outline-3">
-<h3 id="org0331165"><span class="section-number-3">5.1</span> How to Shape closed-loop transfer function? Using Weighting Functions!</h3>
+<div id="outline-container-orga514a74" class="outline-3">
+<h3 id="orga514a74"><span class="section-number-3">5.1</span> How to Shape closed-loop transfer function? Using Weighting Functions!</h3>
 <div class="outline-text-3" id="text-5-1">
 <p>
-<a id="org128c370"></a>
+<a id="orgde38648"></a>
 </p>
 
 <p>
-If the \(\mathcal{H}_\infty\) synthesis is applied on the generalized plant \(P(s)\) shown in Figure <a href="#org0dc082d">23</a>, it will generate a controller \(K(s)\) such that the \(\mathcal{H}_\infty\) norm of closed-loop transfer function from \(r\) to \(\epsilon\) is minimized.
+If the \(\mathcal{H}_\infty\) synthesis is applied on the generalized plant \(P(s)\) shown in Figure <a href="#org2fa4651">23</a>, it will generate a controller \(K(s)\) such that the \(\mathcal{H}_\infty\) norm of closed-loop transfer function from \(r\) to \(\epsilon\) is minimized.
 This closed-loop transfer function actually correspond to the sensitivity function.
 Therefore, it will minimize the the \(\mathcal{H}_\infty\) norm of the sensitivity function: \(\|S\|_\infty\).
 </p>
@@ -1927,16 +1928,16 @@ However, as the \(\mathcal{H}_\infty\) norm is the maximum peak value of the tra
 </p>
 
 
-<div id="org0dc082d" class="figure">
+<div id="org2fa4651" class="figure">
 <p><img src="figs/loop_shaping_S_without_W.png" alt="loop_shaping_S_without_W.png" />
 </p>
 <p><span class="figure-number">Figure 23: </span>Generalized Plant</p>
 </div>
 
 
-<div class="important" id="org8baba74">
+<div class="important" id="org8fbda82">
 <p>
-The <i>trick</i> is to include a <b>weighting function</b> \(W_S(s)\) in the generalized plant as shown in Figure <a href="#orgc89e4e8">24</a>.
+The <i>trick</i> is to include a <b>weighting function</b> \(W_S(s)\) in the generalized plant as shown in Figure <a href="#org9dc36a5">24</a>.
 </p>
 
 <p>
@@ -1954,7 +1955,7 @@ Let&rsquo;s now show how this is equivalent as <b>shaping</b> the sensitivity fu
   \Leftrightarrow & \left| S(j\omega) \right| < \frac{1}{\left| W_s(j\omega) \right|} \quad \forall \omega \label{eq:sensitivity_shaping}
 \end{align}
 
-<div class="important" id="org150ff41">
+<div class="important" id="org6e4c5aa">
 <p>
 As shown in Equation \eqref{eq:sensitivity_shaping}, the \(\mathcal{H}_\infty\) synthesis applying on the <i>weighted</i> generalized plant allows to <b>shape</b> the magnitude of the sensitivity transfer function.
 </p>
@@ -1966,15 +1967,15 @@ Therefore, the choice of the weighting function \(W_s(s)\) is very important: it
 </div>
 
 
-<div id="orgc89e4e8" class="figure">
+<div id="org9dc36a5" class="figure">
 <p><img src="figs/loop_shaping_S_with_W.png" alt="loop_shaping_S_with_W.png" />
 </p>
 <p><span class="figure-number">Figure 24: </span>Weighted Generalized Plant</p>
 </div>
 
-<div class="exercice" id="org88f2695">
+<div class="exercice" id="org399f607">
 <p>
-Using matlab, compute the weighted generalized plant shown in Figure <a href="#org93fcf3e">25</a> as a function of \(G(s)\) and \(W_S(s)\).
+Using matlab, compute the weighted generalized plant shown in Figure <a href="#orga4342c7">25</a> as a function of \(G(s)\) and \(W_S(s)\).
 </p>
 
 <details><summary>Hint</summary>
@@ -2012,18 +2013,18 @@ The second solution is however more general, and can also be used when weights a
 </div>
 </div>
 
-<div id="outline-container-orgc705f0d" class="outline-3">
-<h3 id="orgc705f0d"><span class="section-number-3">5.2</span> Design of Weighting Functions</h3>
+<div id="outline-container-org1057ca5" class="outline-3">
+<h3 id="org1057ca5"><span class="section-number-3">5.2</span> Design of Weighting Functions</h3>
 <div class="outline-text-3" id="text-5-2">
 <p>
-<a id="orgfd9d7c1"></a>
+<a id="org066dedd"></a>
 </p>
 
 <p>
 Weighting function included in the generalized plant must be <b>proper</b>, <b>stable</b> and <b>minimum phase</b> transfer functions.
 </p>
 
-<div class="definition" id="org26b4006">
+<div class="definition" id="org231c8a7">
 <dl class="org-dl">
 <dt>proper</dt><dd>more poles than zeros, this implies \(\lim_{\omega \to \infty} |W(j\omega)| < \infty\)</dd>
 <dt>stable</dt><dd>no poles in the right half plane</dd>
@@ -2048,9 +2049,9 @@ with:
 <li><code>hfgain</code>: high frequency gain</li>
 </ul>
 
-<div class="exampl" id="orgc859ba7">
+<div class="exampl" id="orgfe21b48">
 <p>
-The Matlab code below produces a weighting function with the following characteristics (Figure <a href="#org93fcf3e">25</a>):
+The Matlab code below produces a weighting function with the following characteristics (Figure <a href="#orga4342c7">25</a>):
 </p>
 <ul class="org-ul">
 <li>Low frequency gain of 100</li>
@@ -2064,7 +2065,7 @@ The Matlab code below produces a weighting function with the following character
 </div>
 
 
-<div id="org93fcf3e" class="figure">
+<div id="orga4342c7" class="figure">
 <p><img src="figs/first_order_weight.png" alt="first_order_weight.png" />
 </p>
 <p><span class="figure-number">Figure 25: </span>Obtained Magnitude of the Weighting Function</p>
@@ -2072,7 +2073,7 @@ The Matlab code below produces a weighting function with the following character
 
 </div>
 
-<div class="seealso" id="org99adbda">
+<div class="seealso" id="org7a12ac9">
 <p>
 Quite often, higher orders weights are required.
 </p>
@@ -2104,7 +2105,7 @@ A Matlab function implementing Equation \eqref{eq:weight_formula_advanced} is sh
 </p>
 
 <div class="org-src-container">
-<label class="org-src-name"><span class="listing-number">Listing 1: </span>Matlab Function that can be used to generate Weighting functions</label><pre class="src src-matlab" id="org692762a"><span class="org-keyword">function</span> <span class="org-variable-name">[W]</span> = <span class="org-function-name">generateWeight</span>(<span class="org-variable-name">args</span>)
+<label class="org-src-name"><span class="listing-number">Listing 1: </span>Matlab Function that can be used to generate Weighting functions</label><pre class="src src-matlab" id="org28e7171"><span class="org-keyword">function</span> <span class="org-variable-name">[W]</span> = <span class="org-function-name">generateWeight</span>(<span class="org-variable-name">args</span>)
     arguments
         args.G0 (1,1) double {mustBeNumeric, mustBePositive} = 0.1
         args.G1 (1,1) double {mustBeNumeric, mustBePositive} = 10
@@ -2139,11 +2140,11 @@ W3 = generateWeight(<span class="org-string">'G0'</span>, 1e2, <span class="org-
 </div>
 
 <p>
-The obtained shapes are shown in Figure <a href="#orgb27906c">26</a>.
+The obtained shapes are shown in Figure <a href="#orgeb82834">26</a>.
 </p>
 
 
-<div id="orgb27906c" class="figure">
+<div id="orgeb82834" class="figure">
 <p><img src="figs/high_order_weight.png" alt="high_order_weight.png" />
 </p>
 <p><span class="figure-number">Figure 26: </span>Higher order weights using Equation \eqref{eq:weight_formula_advanced}</p>
@@ -2153,11 +2154,11 @@ The obtained shapes are shown in Figure <a href="#orgb27906c">26</a>.
 </div>
 </div>
 
-<div id="outline-container-orgb481e60" class="outline-3">
-<h3 id="orgb481e60"><span class="section-number-3">5.3</span> Shaping the Sensitivity Function</h3>
+<div id="outline-container-org7ac2800" class="outline-3">
+<h3 id="org7ac2800"><span class="section-number-3">5.3</span> Shaping the Sensitivity Function</h3>
 <div class="outline-text-3" id="text-5-3">
 <p>
-<a id="org5776302"></a>
+<a id="org4adc4a9"></a>
 </p>
 
 <p>
@@ -2170,17 +2171,17 @@ Let&rsquo;s design a controller using the \(\mathcal{H}_\infty\) synthesis that
 </ol>
 
 <p>
-As usual, the plant used is the one presented in Section <a href="#org1f983b6">1.3</a>.
+As usual, the plant used is the one presented in Section <a href="#org7ae910e">1.3</a>.
 </p>
 
-<div class="exercice" id="orge475a4d">
+<div class="exercice" id="org030d4ce">
 <p>
 Translate the requirements as upper bounds on the Sensitivity function and design the corresponding Weight using Matlab.
 </p>
 
 <details><summary>Hint</summary>
 <p>
-The typical wanted upper bound of the sensitivity function is shown in Figure <a href="#org995b01b">27</a>.
+The typical wanted upper bound of the sensitivity function is shown in Figure <a href="#org0c358be">27</a>.
 </p>
 
 <p>
@@ -2198,7 +2199,7 @@ Remember that the wanted upper bound of the sensitivity function is defined by t
 </p>
 
 
-<div id="org995b01b" class="figure">
+<div id="org0c358be" class="figure">
 <p><img src="figs/h-infinity-spec-S.png" alt="h-infinity-spec-S.png" />
 </p>
 <p><span class="figure-number">Figure 27: </span>Typical wanted shape of the Sensitivity transfer function</p>
@@ -2263,7 +2264,7 @@ And the \(\mathcal{H}_\infty\) synthesis is performed on the <i>weighted</i> gen
 </pre>
 </div>
 
-<pre class="example" id="org0b3a709">
+<pre class="example" id="orgefdde4f">
 Test bounds:  0.5 &lt;=  gamma  &lt;=  0.51
 
  gamma        X&gt;=0        Y&gt;=0       rho(XY)&lt;1    p/f
@@ -2284,10 +2285,10 @@ Best performance (actual): 0.503
 \end{aligned}
 
 <p>
-This is indeed what we can see by comparing \(|S|\) and \(|W_S|\) in Figure <a href="#orgfc5fe5c">28</a>.
+This is indeed what we can see by comparing \(|S|\) and \(|W_S|\) in Figure <a href="#orgb5931a1">28</a>.
 </p>
 
-<div class="important" id="org050fdf7">
+<div class="important" id="org3f015df">
 <p>
 Having \(\gamma < 1\) means that the \(\mathcal{H}_\infty\) synthesis found a controller such that the specified closed-loop transfer functions are bellow the specified upper bounds.
 </p>
@@ -2300,7 +2301,7 @@ It just means that at some frequency, one of the closed-loop transfer functions
 </div>
 
 
-<div id="orgfc5fe5c" class="figure">
+<div id="orgb5931a1" class="figure">
 <p><img src="figs/results_sensitivity_hinf.png" alt="results_sensitivity_hinf.png" />
 </p>
 <p><span class="figure-number">Figure 28: </span>Weighting function and obtained closed-loop sensitivity</p>
@@ -2308,16 +2309,16 @@ It just means that at some frequency, one of the closed-loop transfer functions
 </div>
 </div>
 
-<div id="outline-container-orgb96dfcb" class="outline-3">
-<h3 id="orgb96dfcb"><span class="section-number-3">5.4</span> Shaping multiple closed-loop transfer functions</h3>
+<div id="outline-container-orgebe5d7a" class="outline-3">
+<h3 id="orgebe5d7a"><span class="section-number-3">5.4</span> Shaping multiple closed-loop transfer functions</h3>
 <div class="outline-text-3" id="text-5-4">
 <p>
-<a id="org3426451"></a>
+<a id="org377446a"></a>
 </p>
 
 <p>
-As was shown in Section <a href="#orgb800b43">4</a>, depending on the specifications, up to four closed-loop transfer function may be shaped (the Gang of four).
-This was summarized in Table <a href="#orgfa3589c">8</a>.
+As was shown in Section <a href="#orgb74964f">4</a>, depending on the specifications, up to four closed-loop transfer function may be shaped (the Gang of four).
+This was summarized in Table <a href="#orgb94a744">8</a>.
 </p>
 
 <p>
@@ -2325,7 +2326,7 @@ For instance to limit the control input \(u\), \(KS\) should be shaped while to
 </p>
 
 <p>
-When multiple closed-loop transfer function are shaped at the same time, it is refereed to as &ldquo;Mixed-Sensitivity \(\mathcal{H}_\infty\) Control&rdquo; and is the subject of Section <a href="#org099ff7f">6</a>.
+When multiple closed-loop transfer function are shaped at the same time, it is refereed to as &ldquo;Mixed-Sensitivity \(\mathcal{H}_\infty\) Control&rdquo; and is the subject of Section <a href="#org8a12722">6</a>.
 </p>
 
 <p>
@@ -2333,14 +2334,14 @@ Depending on the closed-loop transfer function being shaped, different general c
 </p>
 <details><summary>Shaping of S and KS</summary>
 
-<div id="orgd446819" class="figure">
+<div id="orgf7d9252" class="figure">
 <p><img src="figs/general_conf_shaping_S_KS.png" alt="general_conf_shaping_S_KS.png" />
 </p>
 <p><span class="figure-number">Figure 29: </span>Generalized Plant to shape \(S\) and \(KS\)</p>
 </div>
 
 <div class="org-src-container">
-<label class="org-src-name"><span class="listing-number">Listing 2: </span>General Plant definition corresponding to Figure <a href="#orgd446819">29</a></label><pre class="src src-matlab" id="org0b97731">P = [W1 <span class="org-type">-</span>G<span class="org-type">*</span>W1
+<label class="org-src-name"><span class="listing-number">Listing 2: </span>General Plant definition corresponding to Figure <a href="#orgf7d9252">29</a></label><pre class="src src-matlab" id="orge30f47b">P = [W1 <span class="org-type">-</span>G<span class="org-type">*</span>W1
      0   W2
      1  <span class="org-type">-</span>G];
 </pre>
@@ -2353,14 +2354,14 @@ Depending on the closed-loop transfer function being shaped, different general c
 </details>
 <details><summary>Shaping of S and T</summary>
 
-<div id="org874b337" class="figure">
+<div id="org2c3caa4" class="figure">
 <p><img src="figs/general_conf_shaping_S_T.png" alt="general_conf_shaping_S_T.png" />
 </p>
 <p><span class="figure-number">Figure 30: </span>Generalized Plant to shape \(S\) and \(T\)</p>
 </div>
 
 <div class="org-src-container">
-<label class="org-src-name"><span class="listing-number">Listing 3: </span>General Plant definition corresponding to Figure <a href="#org874b337">30</a></label><pre class="src src-matlab" id="org17e76c0">P = [W1 <span class="org-type">-</span>G<span class="org-type">*</span>W1
+<label class="org-src-name"><span class="listing-number">Listing 3: </span>General Plant definition corresponding to Figure <a href="#org2c3caa4">30</a></label><pre class="src src-matlab" id="org9e2738c">P = [W1 <span class="org-type">-</span>G<span class="org-type">*</span>W1
      0   G<span class="org-type">*</span>W2
      1   <span class="org-type">-</span>G];
 </pre>
@@ -2373,14 +2374,14 @@ Depending on the closed-loop transfer function being shaped, different general c
 </details>
 <details><summary>Shaping of S and GS</summary>
 
-<div id="org6a1853c" class="figure">
+<div id="org55a03e5" class="figure">
 <p><img src="figs/general_conf_shaping_S_GS.png" alt="general_conf_shaping_S_GS.png" />
 </p>
 <p><span class="figure-number">Figure 31: </span>Generalized Plant to shape \(S\) and \(GS\)</p>
 </div>
 
 <div class="org-src-container">
-<label class="org-src-name"><span class="listing-number">Listing 4: </span>General Plant definition corresponding to Figure <a href="#org6a1853c">31</a></label><pre class="src src-matlab" id="org9aef8ab">P = [W1   <span class="org-type">-</span>W1
+<label class="org-src-name"><span class="listing-number">Listing 4: </span>General Plant definition corresponding to Figure <a href="#org55a03e5">31</a></label><pre class="src src-matlab" id="org46e1b50">P = [W1   <span class="org-type">-</span>W1
      G<span class="org-type">*</span>W2 <span class="org-type">-</span>G<span class="org-type">*</span>W2
      G    <span class="org-type">-</span>G];
 </pre>
@@ -2392,14 +2393,14 @@ Depending on the closed-loop transfer function being shaped, different general c
 </details>
 <details><summary>Shaping of S, T and KS</summary>
 
-<div id="org5c0ca1b" class="figure">
+<div id="org7a79fdf" class="figure">
 <p><img src="figs/general_conf_shaping_S_T_KS.png" alt="general_conf_shaping_S_T_KS.png" />
 </p>
 <p><span class="figure-number">Figure 32: </span>Generalized Plant to shape \(S\), \(T\) and \(KS\)</p>
 </div>
 
 <div class="org-src-container">
-<label class="org-src-name"><span class="listing-number">Listing 5: </span>General Plant definition corresponding to Figure <a href="#org5c0ca1b">32</a></label><pre class="src src-matlab" id="orga2cf96a">P = [W1 <span class="org-type">-</span>G<span class="org-type">*</span>W1
+<label class="org-src-name"><span class="listing-number">Listing 5: </span>General Plant definition corresponding to Figure <a href="#org7a79fdf">32</a></label><pre class="src src-matlab" id="org12204ef">P = [W1 <span class="org-type">-</span>G<span class="org-type">*</span>W1
      0   W2
      0   G<span class="org-type">*</span>W3
      1   <span class="org-type">-</span>G];
@@ -2414,14 +2415,14 @@ Depending on the closed-loop transfer function being shaped, different general c
 </details>
 <details><summary>Shaping of S, T and GS</summary>
 
-<div id="orgbccafc1" class="figure">
+<div id="org4256751" class="figure">
 <p><img src="figs/general_conf_shaping_S_T_GS.png" alt="general_conf_shaping_S_T_GS.png" />
 </p>
 <p><span class="figure-number">Figure 33: </span>Generalized Plant to shape \(S\), \(T\) and \(GS\)</p>
 </div>
 
 <div class="org-src-container">
-<label class="org-src-name"><span class="listing-number">Listing 6: </span>General Plant definition corresponding to Figure <a href="#orgbccafc1">33</a></label><pre class="src src-matlab" id="orgd8cf3af">P = [W1   <span class="org-type">-</span>W1
+<label class="org-src-name"><span class="listing-number">Listing 6: </span>General Plant definition corresponding to Figure <a href="#org4256751">33</a></label><pre class="src src-matlab" id="org8819559">P = [W1   <span class="org-type">-</span>W1
      G<span class="org-type">*</span>W2 <span class="org-type">-</span>G<span class="org-type">*</span>W2
      0     W3
      G    <span class="org-type">-</span>G];
@@ -2436,14 +2437,14 @@ Depending on the closed-loop transfer function being shaped, different general c
 </details>
 <details><summary>Shaping of S, T, KS and GS</summary>
 
-<div id="org63e1222" class="figure">
+<div id="org59888c5" class="figure">
 <p><img src="figs/general_conf_shaping_S_T_KS_GS.png" alt="general_conf_shaping_S_T_KS_GS.png" />
 </p>
 <p><span class="figure-number">Figure 34: </span>Generalized Plant to shape \(S\), \(T\), \(KS\) and \(GS\)</p>
 </div>
 
 <div class="org-src-container">
-<label class="org-src-name"><span class="listing-number">Listing 7: </span>General Plant definition corresponding to Figure <a href="#org63e1222">34</a></label><pre class="src src-matlab" id="org425738e">P = [ W1  <span class="org-type">-</span>W1<span class="org-type">*</span>G<span class="org-type">*</span>W3 <span class="org-type">-</span>G<span class="org-type">*</span>W1
+<label class="org-src-name"><span class="listing-number">Listing 7: </span>General Plant definition corresponding to Figure <a href="#org59888c5">34</a></label><pre class="src src-matlab" id="orgc87786a">P = [ W1  <span class="org-type">-</span>W1<span class="org-type">*</span>G<span class="org-type">*</span>W3 <span class="org-type">-</span>G<span class="org-type">*</span>W1
       0    0        W2
       1   <span class="org-type">-</span>G<span class="org-type">*</span>W3    <span class="org-type">-</span>G];
 </pre>
@@ -2467,7 +2468,7 @@ When shaping multiple closed-loop transfer functions, one should be verify caref
 
 
 
-<div class="warning" id="org17bbf33">
+<div class="warning" id="orgbfc5482">
 <p>
 Mathematical relations are linking the closed-loop transfer functions.
 For instance, the sensitivity function \(S(s)\) and the complementary sensitivity function \(T(s)\) as link by the following well known relation:
@@ -2500,7 +2501,7 @@ The control bandwidth is clearly limited by physical constrains such as sampling
 Similar relationship can be found for \(T\), \(KS\) and \(GS\).
 </p>
 
-<div class="exercice" id="org363ac82">
+<div class="exercice" id="org08cec2a">
 <p>
 Determine the approximate norms of \(T\), \(KS\) and \(GS\) for large loop gains (\(|G(j\omega) K(j\omega)| \gg 1\)) and small loop gains (\(|G(j\omega) K(j\omega)| \ll 1\)).
 </p>
@@ -2518,7 +2519,7 @@ You can follows this procedure for \(T\), \(KS\) and \(GS\):
 
 <details><summary>Answer</summary>
 <p>
-The obtained constrains are shown in Figure <a href="#orgd172dae">35</a>.
+The obtained constrains are shown in Figure <a href="#org63c6f6a">35</a>.
 </p>
 </details>
 
@@ -2531,17 +2532,17 @@ However, in some frequency bands, the norms do not depend on the controller and
 
 <p>
 Therefore the weighting functions should only focus on certainty frequency range depending on the transfer function being shaped.
-These regions are summarized in Figure <a href="#orgd172dae">35</a>.
+These regions are summarized in Figure <a href="#org63c6f6a">35</a>.
 </p>
 
 
-<div id="orgd172dae" class="figure">
+<div id="org63c6f6a" class="figure">
 <p><img src="figs/h-infinity-4-blocs-constrains.png" alt="h-infinity-4-blocs-constrains.png" />
 </p>
 <p><span class="figure-number">Figure 35: </span>Shaping the Gang of Four: Limitations</p>
 </div>
 
-<div class="warning" id="org2274506">
+<div class="warning" id="org8d89294">
 <p>
 The order (resp. number of state) of the controller given by the \(\mathcal{H}_\infty\) synthesis is equal to the order (resp. number of state) of the weighted generalized plant.
 It is thus equal to the <b>sum</b> of the number of state of the non-weighted generalized plant and the number of state of all the weighting functions.
@@ -2560,16 +2561,16 @@ Two approaches can be used to obtain controllers with reasonable order:
 </div>
 </div>
 
-<div id="outline-container-orgb5da593" class="outline-2">
-<h2 id="orgb5da593"><span class="section-number-2">6</span> Mixed-Sensitivity \(\mathcal{H}_\infty\) Control - Example</h2>
+<div id="outline-container-orgdfad810" class="outline-2">
+<h2 id="orgdfad810"><span class="section-number-2">6</span> Mixed-Sensitivity \(\mathcal{H}_\infty\) Control - Example</h2>
 <div class="outline-text-2" id="text-6">
 <p>
-<a id="org099ff7f"></a>
+<a id="org8a12722"></a>
 </p>
 </div>
 
-<div id="outline-container-org366bd83" class="outline-3">
-<h3 id="org366bd83"><span class="section-number-3">6.1</span> Control Problem</h3>
+<div id="outline-container-org22bd096" class="outline-3">
+<h3 id="org22bd096"><span class="section-number-3">6.1</span> Control Problem</h3>
 <div class="outline-text-3" id="text-6-1">
 <ul class="org-ul">
 <li class="off"><code>[&#xa0;]</code> Control Diagram</li>
@@ -2622,8 +2623,8 @@ d(t<span class="org-type">&gt;</span>0.5) = 5e<span class="org-type">-</span>4;
 </div>
 </div>
 
-<div id="outline-container-org7004a2e" class="outline-3">
-<h3 id="org7004a2e"><span class="section-number-3">6.2</span> Control Design Procedure</h3>
+<div id="outline-container-org02f4803" class="outline-3">
+<h3 id="org02f4803"><span class="section-number-3">6.2</span> Control Design Procedure</h3>
 <div class="outline-text-3" id="text-6-2">
 <table border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
 
@@ -2670,7 +2671,7 @@ d(t<span class="org-type">&gt;</span>0.5) = 5e<span class="org-type">-</span>4;
 </table>
 
 
-<div id="org60c3cac" class="figure">
+<div id="org1467f47" class="figure">
 <p><img src="figs/mixed_sensitivity_control_schematic.png" alt="mixed_sensitivity_control_schematic.png" />
 </p>
 <p><span class="figure-number">Figure 36: </span>Generalized Plant used for the Mixed Sensitivity Synthesis</p>
@@ -2692,11 +2693,11 @@ d(t<span class="org-type">&gt;</span>0.5) = 5e<span class="org-type">-</span>4;
 </div>
 </div>
 
-<div id="outline-container-org23a1031" class="outline-3">
-<h3 id="org23a1031"><span class="section-number-3">6.3</span> Step 1 - Shaping of \(S\)</h3>
+<div id="outline-container-org741a631" class="outline-3">
+<h3 id="org741a631"><span class="section-number-3">6.3</span> Step 1 - Shaping of \(S\)</h3>
 <div class="outline-text-3" id="text-6-3">
 <p>
-<a id="org4e8fc87"></a>
+<a id="orga096629"></a>
 </p>
 
 <p>
@@ -2728,7 +2729,7 @@ W3 = tf(0.1);
 </pre>
 </div>
 
-<pre class="example" id="orge75e58a">
+<pre class="example" id="orgcca6715">
 K1 = hinfsyn(Pw, 1, 1, 'Display', 'on');
 
   Test bounds:  0.5 &lt;=  gamma  &lt;=  0.552
@@ -2789,11 +2790,11 @@ xlabel(<span class="org-string">'Time [s]'</span>); ylabel(<span class="org-stri
 </div>
 </div>
 
-<div id="outline-container-org759fd41" class="outline-3">
-<h3 id="org759fd41"><span class="section-number-3">6.4</span> Step 2 - Shaping of \(KS\)</h3>
+<div id="outline-container-orga7db01d" class="outline-3">
+<h3 id="orga7db01d"><span class="section-number-3">6.4</span> Step 2 - Shaping of \(KS\)</h3>
 <div class="outline-text-3" id="text-6-4">
 <p>
-<a id="org4234f5f"></a>
+<a id="org7fa931c"></a>
 </p>
 
 <div class="org-src-container">
@@ -2814,7 +2815,7 @@ xlabel(<span class="org-string">'Time [s]'</span>); ylabel(<span class="org-stri
 </pre>
 </div>
 
-<pre class="example" id="orgaf1200c">
+<pre class="example" id="org7d3c166">
 K1 = hinfsyn(Pw, 1, 1, 'Display', 'on');
 
   Test bounds:  0.51 &lt;=  gamma  &lt;=  1.2
@@ -2873,11 +2874,11 @@ xlabel(<span class="org-string">'Time [s]'</span>); ylabel(<span class="org-stri
 </div>
 </div>
 
-<div id="outline-container-org5f57abd" class="outline-3">
-<h3 id="org5f57abd"><span class="section-number-3">6.5</span> Step 3 - Shaping of \(T\)</h3>
+<div id="outline-container-orge4c24c4" class="outline-3">
+<h3 id="orge4c24c4"><span class="section-number-3">6.5</span> Step 3 - Shaping of \(T\)</h3>
 <div class="outline-text-3" id="text-6-5">
 <p>
-<a id="org9db980e"></a>
+<a id="org8351347"></a>
 </p>
 
 <div class="org-src-container">
@@ -2898,7 +2899,7 @@ xlabel(<span class="org-string">'Time [s]'</span>); ylabel(<span class="org-stri
 </pre>
 </div>
 
-<pre class="example" id="org8bda36c">
+<pre class="example" id="org985e051">
 K3 = hinfsyn(Pw, 1, 1, 'Display', 'on');
 
   Test bounds:  0.578 &lt;=  gamma  &lt;=  1.66
@@ -2960,18 +2961,18 @@ xlabel(<span class="org-string">'Time [s]'</span>); ylabel(<span class="org-stri
 </div>
 </div>
 
-<div id="outline-container-orgb1a415e" class="outline-2">
-<h2 id="orgb1a415e"><span class="section-number-2">7</span> Conclusion</h2>
+<div id="outline-container-org46694c3" class="outline-2">
+<h2 id="org46694c3"><span class="section-number-2">7</span> Conclusion</h2>
 <div class="outline-text-2" id="text-7">
 <p>
-<a id="org1607071"></a>
+<a id="org6a1aa57"></a>
 </p>
 </div>
 </div>
 
-<div id="outline-container-org1e48135" class="outline-2">
-<h2 id="org1e48135">Resources</h2>
-<div class="outline-text-2" id="text-org1e48135">
+<div id="outline-container-org86a09c0" class="outline-2">
+<h2 id="org86a09c0">Resources</h2>
+<div class="outline-text-2" id="text-org86a09c0">
 <p>
 <div class="yt"><iframe width="100%" height="100%" src="https://www.youtube.com/embed/?listType=playlist&list=PLn8PRpmsu08qFLMfgTEzR8DxOPE7fBiin" frameborder="0" allowfullscreen></iframe></div>
 </p>
@@ -2991,7 +2992,7 @@ xlabel(<span class="org-string">'Time [s]'</span>); ylabel(<span class="org-stri
 </div>
 <div id="postamble" class="status">
 <p class="author">Author: Dehaeze Thomas</p>
-<p class="date">Created: 2020-12-02 mer. 19:53</p>
+<p class="date">Created: 2020-12-02 mer. 20:03</p>
 </div>
 </body>
 </html>
diff --git a/index.org b/index.org
index cd56204..6c81822 100644
--- a/index.org
+++ b/index.org
@@ -11,6 +11,7 @@
 
 #+HTML_HEAD: <link rel="stylesheet" type="text/css" href="https://research.tdehaeze.xyz/css/style.css"/>
 #+HTML_HEAD: <script type="text/javascript" src="https://research.tdehaeze.xyz/js/script.js"></script>
+#+HTML_HEAD: <meta name="viewport" content="width=device-width,initial-scale=1.0">
 
 #+HTML_MATHJAX: align: center tagside: right font: TeX
 
